topological tensor current of p branes in the φ mapping theory we present a new topological tensor current of p branes by making use of the mapping theory it is shown that the current is identically conserved and behaves as and every isolated zero of the vector field x corresponds to a magnetic p brane using this topological current the generalized nambu action for multi p branes is given and the field strength f corresponding to this topological tensor current is obtained it is also shown that the magnetic charges carried by p branes are topologically quantized and labeled by hopf index and brouwer degree the winding number of the mapping

a crossover in the scaling law of the lyapunov exponent the authors consider products of random matrices appearing in the study of schrodinger operators or dynamical systems they show how a crossover in the scaling law of the lyapunov exponent takes place when the standard deviation of the process increases

the billiard algorithm and ks entropy there is known to be a close relation between the kolmogorov sinai entropy sum of the positive lyapunov exponents of an ergodic dynamical system and the algorithmic complexity of encoding trajectories of the system with respect to some partition the authors explicitly gives an encoding which demonstrates this relation for the square sinai billiard the encoding depends on the fact that the collision criterion for the billiard is an example of rational approximants the method may be used to achieve very fast simulation times for the system

on the measure of maximal entropy for finite horizon sinai billiard maps the sinai billiard map t on the two torus i e the periodic lorentz gas is a discontinuous map assuming finite horizon we propose a definition h for the topological entropy of t we prove that h is not smaller than the value given by the variational principle and that it is compatible with the definitions of bowen using spanning or separating sets if h ge log 2 our actual condition is weaker then we get more first using a transfer operator acting on a space of anisotropic distributions we construct an invariant probability measure mu of maximal entropy for t i e h mu t h we show that mu has full support and is bernoulli and we prove that mu is different from the smooth invariant measure except if all non grazing periodic orbits have multiplier equal to h second h is compatible with the bowen pesin pitskel topological entropy of the restriction of t to a non compact domain of continuity last applying results of lima and matheus the map t has at least c e pnh periodic points of period pn for all n and some p ge 1

attractors for the 3d baroclinic quasi geostrophic equations of large scale atmosphere abstract the existence of absorbing sets and a compact connected global attractor to the 3d baroclinic quasi geostrophic equations of large scale atmosphere are obtained using cft theory an upper bound of the hausdorff and fractal dimensions of the global attractor in terms of the physically relevant parameters re the reynolds number ro the rossby number and f is derived moreover similar results are also established for the 2d vorticity equations of globe atmosphere

classical representation of the one dimensional anderson model a new approach is applied to the one dimensional anderson model by using a two dimensional hamiltonian map for a weak disorder this approach allows for a simple derivation of correct expressions for the localization length both at the centre and at the edge of the energy band where standard perturbation theory fails approximate analytical expressions for strong disorder are also obtained

real representations of finite clifford algebras i classification a classification of real matrix irreducible representations of finite dimensional real clifford algebras has been made in contrast to the case of complex representation three distinct types of representations can be obtained which we call normal almost complex and quaternionic the dimension of the latter two cases is twice as large as that of the normal representation a criteria for a given clifford algebra to possess a particular type of the representations is also given with some applications

topological structure of gauss bonnet chern density and its topological current the decomposition of spin connection on the sphere bundle of a compact and oriented even dimensional riemannian manifold is detailed it is shown that the gauss bonnet chern form can be expressed in terms of a smooth vector field and taken the form of the function the topological current of the gauss bonnet chern theorem and its topological structure are discussed

a simple global model for the general circulation of the atmosphere in this article we present a new method of rossby asymptotics for the equations of the atmosphere similar to the geostrophic asymptotics we depart from the classical geostrophics see j g charney 5 and our previous article 29 by considering an asymptotics valid for the whole atmosphere not only in midlatitude regions and by taking into account the spherical form of the earth we obtain in this way a very simple global circulation model of the atmosphere for which the equations of motion for wind and temperature are linear evolution equations similar to the linear stokes equations furthermore the solutions are independent of longitude and winds travel exactly to the east or to the west in this mathematically oriented article we do not discuss the physical significance of the model that we derive except for observing that this picture coincides in general terms with the classically averaged data obtained by experimental measurements we also note that different global geostrophic asymptotics called planetary geostrophic asymptotics are considered elsewhere in the literature for example in j pedlosky 33 in a less mathematically rigorous way h lamb 21 observed that the only globally valid geostrophic flow that maintains a slow time scale is zonally symmetric see the comments in n phillips 35 and an explicit derivation in h jeffreys 19 c 1997 john wiley sons inc

soft billiard systems abstract we study a variation of a sinai type billiard system where a circular scatterer of radius r is centered on a 2 d torus and given a finite potential height u measured in units of kinetic energy for 1 u 0 the kolmogorov sinai entropy of a point particle travelling on the torus is numerically calculated as a function of u and r and empirically seen to be a universal function of u r 2 for u r sufficiently small for 0 u uc r we prove the non ergodicity of the system and discuss the resulting complicated structure of the phase space uc r is also explicitly given the u 0 and u uc transitions are also discussed

inner topological structure of hopf invariant in light of mapping topological current theory the inner topological structure of hopf invariant is investigated it is revealed that hopf invariant is just the winding number of gauss mapping according to the inner structure of topological current a precise expression for hopf invariant is also presented it is the total sum of all the self linking and all the linking numbers of the knot family

the topological structure of nieh yan form and the chiral anomaly in spaces with torsion the topological structure of the nieh yan form in a four dimensional manifold is given by making use of the decomposition of spin connection the case of the generalized nieh yan form on a 2d dimensional manifold is discussed with an example of an eight dimensional case which is studied in detail the chiral anomaly with nonvanishing torsion is also studied the further contributions from the torsional part to the chiral anomaly are found to estimate from the zeros of some fields under the pure gauge condition

a remark on the three dimensional baroclinic quasi geostrophic dynamics1this research was supported by the national science foundation grant dms 9704345 and a clemson university research grant this research was performed while the author was visiting the isaac newton institute for mathematical sciences at cambridge university uk 1 we consider time periodic patterns of the dissipative three dimensional baroclinic quasi geostrophic model in spherical coordinates under time dependent forcing we show that when the forcing is time periodic and the spatial square integral of the forcing is bounded in time the model has time periodic solutions

hopf bifurcation in quasi geostrophic channel flow in this article we conduct a rigorous stability and bifurcation analysis for a highly idealized model of planetary scale atmospheric and oceanic flows the model is governed by the two dimensional quasi geostrophic equation for the conservation of vorticity in an east west oriented periodic channel the main result is the existence of hopf bifurcation of the flow as the reynolds number crosses a critical value the key idea in proving this result is translating the eigenvalue problem into a difference equation and treating the latter by continued fraction methods numerical results are obtained by using a finite difference scheme with high spatial resolution and these results agree closely with the theoretical predictions the spatio temporal structure of the limit cycle corresponds to a wave that propagates slowly westward and is symmetric about the midaxis of the channel for plausible paramater values that correspond to midlatitude atmospheric flows the period of this wave is 20 25 days

existence of the universal attractor for the 3 d viscous primitive equations of large scale moist atmosphere abstract in this paper we consider the initial boundary value problem for the three dimensional viscous primitive equations of large scale moist atmosphere which are used to describe the turbulent behavior of long term weather prediction and climate changes by obtaining the existence and uniqueness of global strong solutions for the problem and studying the long time behavior of strong solutions we prove the existence of the universal attractor for the dynamical system generated by the primitive equations of large scale moist atmosphere

scaling law for the maximum lyapunov characteristic exponent of infinite product of random matrices the authors present a simple derivation for the scaling behaviour of the maximum lyapunov characteristic exponent lambda of infinite product of symplectic random matrices the considered random matrices depend on a parameter epsilon and lambda 0 for epsilon 0 they obtain lambda varies as epsilon beta with either beta 1 2 or beta 2 3 depending on the probability distribution of the matrix elements the results are in agreement with a previous numerical simulation

existence of weak solutions and trajectory attractors for the moist atmospheric equations in geophysics in this paper we consider the initial boundary value problem for the primitive equations of moist atmospheric dynamics that are used to describe the turbulent behavior of long term weather prediction and climate changes by the faedo galerkin method we obtain the existence of global weak solutions to the problem in a large scale atmosphere by studying the long time behavior of solutions we obtain trajectory and global attractors for the primitive equations of moist atmosphere

supersymmetric yang mills theory on a four manifold by exploiting standard facts about n 1 and n 2 supersymmetric yang mills theory the donaldson invariants of four manifolds that admit a kahler metric can be computed the results are in agreement with available mathematical computations and provide a powerful check on the standard claims about supersymmetric yang mills theory

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

billiards correlation functions we discuss various experiments on the time decay of velocity autocorrelation functions in billiards we perform new experiments and find results which are compatible with an exponential mixing hypothesis first put forward by friedman and martin fm they do not seem compatible with the stretched exponentials believed in spite of fm and more recently of chernov to describe the mixing the analysis leads to several byproducts we obtain information about the normal diffusive nature of the motion and we consider the probability distribution of the number of collisions in timetm astm finding a strong dependence on some geometric characteristics of the locus of the billiard obstacles

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

viscosity for a periodic two disk fluid an existence proof we express the momentum current stress tensor for a periodic fluid with two hard disks per unit cell in terms of a single particle billiard we establish a central limit theorem for the time integrated stress tensor and thereby prove the existence of a strictly positive shear and bulk viscosity

the general decomposition theory of su 2 gauge potential topological structure and bifurcation of su 2 chern density by means of the geometric algebra the general decomposition of su 2 gauge potential on the sphere bundle of a compact and oriented four dimensional manifold is given using this decomposition theory the su 2 chern density has been studied in detail it shows that the su 2 chern density can be expressed in terms of the function one can also find that the zero points of the vector fields are essential to the topological properties of a manifold it is shown that there exists the crucial case of branch process at the zero points based on the implicit function theorem and the taylor expansion the bifurcation of the chern density is detailed in the neighborhoods of the bifurcation points of it is pointed out that since the chern density is a topological invariant the sum topological chargers of the branches will remain constant during the bifurcation process

inertial forms of navier stokes equations on the sphere abstract the navier stokes equations for incompressible flows past a two dimensional sphere are considered in this article the existence of an inertial form of the equations is established furthermore for the first time for fluid equations we derive an upper bound on the dimension of the differential system inertial manifold which fully reproduces the infinite dimensional dynamics this bound is expressed in terms of grashof numbers

physical insight into superdiffusive dynamics of sinai billiard through collision statistics abstract we report on distinct steady motion dynamic regimes in chaotic sinai billiard sb a numerical study on elastic reflections from the sb boundary square wall of length l and circle obstacle of radius r is carried out for different r l the research is based on the exploration of the generalized diffusion equation and on the analysis of wall collision and the circle collision distributions observed at late times the asymptotes for the diffusion coefficient d r and the corresponding diffusion exponent z r are established for all geometries the universal r independent diffusion with d 1 t 1 3 and z 1 1 5 replaces the ballistic motion regime z 0 1 attributed to square billiard r 0 geometrically this superdiffusive regime is bounded by small radii 0 r r 1 r 1 l 2 4 when both diagonal and non diagonal bleher s corridors are open in the correspondent square lattice of lorentz gas lg model the relaxation dynamics observed is ensured by the universal diffusive propagation of the regular type and the bouncing ball type orbits within the random walk scheme this superdiffusive regime is due to the levy flights between the long distant scatterers with the increase in circle radius r 1 r r 2 r 2 l 2 the diagonal corridor closes but the arc touching effects continue to bring the long living bouncing ball orbits back to the non diagonal infinite corridors this transient non universal dynamics with 1 5 z r 2 also associated with the trapping of regular orbits seems to be characteristic of non fully hyperbolic billiard systems in sb with finite horizon r r 2 all the principal corridors are closed and the interplay between square and circle boundaries generates the known chaotic dynamics attributed to the fully hyperbolic systems this is also observed through the normal brownian diffusion z 2 2 and the gaussian statistics proved for both kinds of collisions

slow relaxation in weakly open rational polygons the interplay between the regular piecewise linear and irregular vertex angle boundary effects in nonintegrable rational polygonal billiards of m equal sides is discussed decay dynamics in polygons of perimeter p m and small opening delta is analyzed through the late time survival probability s m approximately equal t delta two distinct slow relaxation channels are established the primary universal channel exhibits relaxation of regular sliding orbits with delta 1 the secondary channel is given by delta 1 and becomes open when m p m delta it originates from vertex order disorder dual effects and is due to relaxation of chaoticlike excitations

non abelian chern simons action is topological invariant on 3 simple knot under su 2 gauge transformation the non abelian chern simons action is invariant on a class of three dimensional manifold 3 simple knot

kolmogorov sinai entropy lyapunov exponents and mean free time in billiard systems we perform new experiments on the kolmogorov sinai entropy lyapunov exponents and the mean free time in billiards we study their dependence on the geometry of the scatterers made up of two interpenetrating square lattices each one with circular scatterers with different radius we find in particular that the above quantities are continuous functions of the ratio of the scatterer radius however it seems that their derivative is discontinuous around the radius ratio which separates the diffusive and nondiffusive types of geometries

intrinsic stickiness and chaos in open integrable billiards tiny border effects rounding border effects at the escape point of open integrable billiards are analyzed via the escape time statistics and emission angles the model is the rectangular billiard and the shape of the escape point is assumed to have a semicircular form stickiness chaos and self similar structures for the escape times and emission angles are generated inside backgammon like stripes of initial conditions these stripes are born at the boundary between two different emission angles but with the same escape times and when rounding effects increase they start to overlap generating a very rich dynamics tiny rounded borders around 0 1 from the whole billiard size are shown to be sufficient to generate the sticky motion with power law decay esc 1 27 while borders larger than 10 are enough to produce escape times related to the chaotic motion escape exponents in the interval 1 esc 2 are generated due to marginal unstable periodic orbits trapping alternately in time regular and chaotic trajectories

generalized lyapunov exponents in high dimensional chaotic dynamics and products of large random matrices we study the behavior of the generalized lyapunov exponents for chaotic symplectic dynamical systems and products of random matrices in the limit of large dimensionsd for products of random matrices without any particular structure the generalized lyapunov exponents become equal in this limit and the value of one of the generalized lyapunov exponents is obtained by simple arguments on the contrary for random symplectic matrices with peculiar structures and for chaotic symplectic maps the generalized lyapunov exponents remains different ford indicating that high dimensionality cannot always destroy intermittency

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

conformal transformations and the effective action in the presence of boundaries the conformal properties of the heat kernel expansion are used to determine the local form of the coefficients in a manifold with boundary the conformal transformation of the effective action is obtained a novel derivation of the boundary term in the gauss bonnet chern theorem is detailed

distribution of characteristic exponents in the thermodynamic limit the existence of the thermodynamic limit for the spectrum of the lyapunov characteristic exponents is numerically investigated for the fermi pasta ulam p model we show that the shape of the spectrum for energy density well above the equipartition threshold e allows the kolmogorov sinai entropy to be expressed simply in terms of the maximum exponent i the presence of a power law behaviour e is investigated the analogies with similar results obtained from the dynamics of symplectic random matrices seem to indicate the possibility of interpreting chaotic dynamics in terms of some universal properties

potentials on the two torus for which the hamiltonian flow is ergodic we consider the motion of a test particle in a compound central potential field on a two dimensional torus we discuss three different classes of potentials attracting repelling and mixed that lead to hamiltonian systems which have positive lyapunov exponent almost everywhere and are ergodic included among the mixed potentials are smooth potentials without singularities

lie groups as spin groups it is shown that every lie algebra can be represented as a bivector algebra hence every lie group can be represented as a spin group thus the computational power of geometric algebra is available to simplify the analysis and applications of lie groups and lie algebras the spin version of the general linear group is thoroughly analyzed and an invariant method for constructing real spin representations of other classical groups is developed moreover it is demonstrated that every linear transformation can be represented as a monomial of vectors in geometric algebra

on the dimension of the attractors in two dimensional turbulence abstract using a new version of the sobolev lieb thirring inequality we derive an upper bound for the dimension of the universal attractor for two dimensional space periodic navier stokes equations this estimate is optimal up to a logarithmic correction the relevance of this estimate to turbulence and related results are also briefly discussed

a random attractor for the stochastic quasi geostrophic dynamical system on unbounded domains abstract in this paper we prove the existence of a random attractor for the stochastic two dimensional quasi geostrophic equation on an unbounded domain which models a class of large scale geophysical flows it is sufficient for a closed absorbing set and the asymptotic compactness of the stochastic system to guarantee the existence of a random attractor for a continuous random dynamical system hence the uniform estimates of solutions including the estimates on the tails of solutions are derived first second the d pullback asymptotic compactness of the stochastic system is proved by uniform estimates on solutions for large space and time variables

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