pv cohomology of the pinwheel tilings their integer group of coinvariants and gap labeling in this paper we first remind how we can see the hull of the pinwheel tiling as an inverse limit of simplicial complexes anderson and putnam in ergod th dynam sys 18 509 537 1998 and we then adapt the pv cohomology introduced in savinien and bellissard ergod th dynam sys 29 997 1031 2009 to define it for pinwheel tilings we then prove that this cohomology is isomorphic to the integer cech cohomology of the quotient of the hull by s1 which let us prove that the top integer cech cohomology of the hull is in fact the integer group of coinvariants of the canonical transversal of the hull the gap labeling for pinwheel tilings is then proved and we end this article by an explicit computation of this gap labeling showing that mu t left c xi mathbb z right frac 1 264 mathbb z left frac 1 5 right

graphs and gromov hyperbolicity of non constant negatively curved surfaces in this paper we obtain the equivalence of the gromov hyperbolicity between an extensive class of complete riemannian surfaces with pinched negative curvature and certain kind of simple graphs whose edges have length 1 constructed following an easy triangular design of geodesics in the surface

bifurcations of relative periodic orbits in nls gp with a triple well potential abstract the nonlinear schrodinger gross pitaevskii nls gp equation is considered in the presence of three equally spaced potentials the problem is reduced to a finite dimensional hamiltonian system by a galerkin truncation families of oscillatory orbits are sought in the neighborhoods of the system s nine branches of standing wave solutions normal forms are computed in the neighborhood of these branches various hamiltonian hopf and saddle node bifurcations showing how the oscillatory orbits change as a parameter is increased numerical experiments show agreement between normal form theory and numerical solutions to the reduced system and nls gp near the hamiltonian hopf bifurcations and some subtle disagreements near the saddle node bifurcations due to exponentially small terms in the asymptotics

laplace transformations of hydrodynamic type systems in riemann invariants periodic sequences the conserved densities of hydrodynamic type systems in riemann invariants satisfy a system of linear second order partial differential equations for linear systems of this type darboux introduced laplace transformations generalizing the classical transformations in the scalar case it is demonstrated that laplace transformations can be pulled back to the transformations of the corresponding hydrodynamic type systems we discuss periodic laplace sequences of hydrodynamic type systems with emphasis on the simplest nontrivial case of period 2 for systems in riemann invariants a complete discription of closed quadruples is proposed they turn out to be related to a special quadratic reduction of the 2 1 dimensional 3 wave system which can be reduced to a triple of pairwise commuting monge ampere equations in terms of the lame and rotation coefficients laplace transformations have a natural interpretation as the symmetries of the dirac operator associated with the 2 1 dimensional n wave system the 2 component laplace transformations can also be interpreted as the symmetries of the 2 1 dimensional integrable equations of davey stewartson type laplace transformations of hydrodynamic type systems originate from a canonical geometric correspondence between systems of conservation laws and line congruences in projective space

spaces with almost euclidean dehn function we prove that any proper geodesic metric space whose dehn function grows asymptotically like the euclidean one has asymptotic cones which are non positively curved in the sense of alexandrov thus are mathrm cat 0 this is new already in the setting of riemannian manifolds and establishes in particular the borderline case of a result about the sharp isoperimetric constant which implies gromov hyperbolicity our result moreover provides a large scale analog of a recent result of lytchak and the author which characterizes proper mathrm cat 0 in terms of the growth of the dehn function at all scales we finally obtain a generalization of this result of lytchak and the author namely we show that if the dehn function of a proper geodesic metric space is sufficiently close to the euclidean dehn function up to some scale then the space is not far in a suitable sense from being mathrm cat 0 up to that scale

laplace maps and constraints for a class of third order partial differential operators we explore the existence of a class of generalised laplace maps for third order partial differential operators of the form 1 2 3 a1 2 3 a2 1 3 a3 1 2 a12 3 a23 1 a13 2 a123 and related first order 3 systems and show that they require the satisfaction of constraints on the invariants for such operators

periodic orbits near a bifurcating slow manifold this paper studies a class of one degree of freedom hamiltonian systems with a slowly varying phase the slowly varying phase is assumed to unfold a hamiltonian pitchfork bifurcation the main result of the paper is that there exists an order of ln 2 1 many periodic orbits that all stay within an o 1 3 distance from the union of the normally elliptic slow manifolds that occur as a result of the bifurcation as they pass through the bifurcation the time scales are comparable here 1 measures the time scale separation these periodic orbits are typically moderately unstable this is in contrast with the periodic orbits that remain an o 1 distance from the slow manifold the effect of approaching the normally elliptic slow manifold is therefore to reduce the stability region the smallest stable orbits that are persistently obtained remain further away from the slow manifold being distant by an order o 1 3 ln 1 2 ln 1 the proofs of these statements are based on averaging of two blow up systems allowing one to estimate the effect of the singularity combined with results on asymptotics of the second painleve equation

exponential estimates of symplectic slow manifolds in this paper we prove the existence of an almost invariant symplectic slow manifold for analytic hamiltonian slow fast systems with finitely many slow degrees of freedom for which the error field is exponentially small we allow for infinitely many fast degrees of freedom the method we use is motivated by a paper of mackay from 2004 the method does not notice resonances and therefore we do not pose any restrictions on the motion normal to the slow manifold other than it being fast and analytic we also present a stability result and obtain a generalization of a result of gelfreich and lerman on an invariant slow manifold to finitely many fast degrees of freedom

on asymptotic volume of finsler tori minimal surfaces in normed spaces and symplectic filling volume the main unconditional result of this paper theorem 3 states that every two dimensional affine disc in a normed space that is a disc contained in a two dimensional affine subspace is an area minimizing surface among all immersed discs with the same boundary with respect to the symplectic holmes thompson surface area to emphasize that this is not at all obvious it may be worth mentioning that a similar statement with rational chains in place of immersed discs is incorrect theorem 2 and that it is not known for surfaces that may not be topological discs the result still may not sound too exciting to the reader who never looked at the problem before even though it goes back to busemann s works in the 50 s see bes th and references there and the proof heavily relies on asymptotic geometry of tori we believe that it is more important that we embed this problem into a whole area of mostly open problems as well as give some partial results and suggest certain directions of how to attack them we begin with a trivial statement

limit theorems for self similar tilings we study deviation of ergodic averages for dynamical systems given by self similar tilings on the plane and in higher dimensions the main object of our paper is a special family of finitely additive measures for our systems an asymptotic formula is given for ergodic integrals in terms of these finitely additive measures and as a corollary limit theorems are obtained for dynamical systems given by self similar tilings

estimates on invariant tori near an elliptic equilibrium point of a hamiltonian system abstract we give a precise statement for the kam theorem in a neighbourhood of an elliptic equilibrium point of a hamiltonian system if the frequencies of the elliptic point are nonresonant up to a certain order k 4 and a nondegeneracy condition is fulfilled we get an estimate for the measure of the complement of the kam tori in a neighbourhood of given radius moreover if the frequencies satisfy a diophantine condition with exponent we show that in a neighbourhood of radius r the measure of the complement is exponentially small in 1 r 1 1 we also give a related result for quasi diophantine frequencies which is more useful for practical purposes the results are obtained by putting the system in birkhoff normal form up to an appropiate order and the key point relies on giving accurate bounds for its terms

oscillations of the zero dispersion limit of the korteweg de vries equation we attack the multiphase averaged systems for the zero dispersion limit of the kdv equation attention is paid to the most important case the single phase oscillations a scheme is developed to solve the whitham averaged system single phase averaged system this system under our scheme is transformed to a linear over determined system of euler poisson darboux type whose solution can be written down explicitly we show that for any smooth initial data which has only one hump or is a nontrivial monotone function the weak limit has single phase oscillations within a cusp in the x t plane for a short time after the breaking time for the corresponding burgers equation outside the cusp the limit satisfies the burgers equation more surprisingly we also show that the weak limit has global single phase oscillations within a cusp for any smooth nontrivial monotone initial data with only one inflection point 1993 john wiley sons inc

canonical substitutions tilings of ammann beenker type we present a large class of parallelogram tilings that have two important features in common with the ammann beenker tiling they can be constructed by canonical projection from z4 r4 to a plane and they admit substitution rules

the isoperimetric problem for pinwheel tilings in aperiodic pinwheel tilings of the plane there exist unions of tiles with ratio area perimeter 2 arbitrarily close to that of a circle such approximate circles can be constructed with arbitrary center and any sufficiently large radius the existence of such circles follows from the metric on pinwheel space being almost euclidean at large distances ifp andq are points separated by large euclidean distancer then the shortest path along tile edges fromp toq has lengthr o r

systems of conservation laws within the framework of the projective theory of congruences the levy transformations of semi hamiltonian systems we continue the investigation of the recently proposed geometric correspondence between systems of conservation laws and congruences of lines in projective space the relationship between the additional conservation laws and hypersurfaces conjugate to a congruence is established thus providing a basis for the levy transformations of semi hamiltonian systems similarly the correspondence between commuting flows and certain families of planes containing the lines of the congruence gives rise to the adjoint levy transformations

convergence of random walks on the circle generated by an irrational rotation fix 0 1 consider the random walk on the circle s1 which proceeds by repeatedly rotating points forward or backward with probability 1 2 by an angle 2 this paper analyzes the rate of convergence of this walk to the uniform distribution under discrepancy distance the rate depends on the continued fraction properties of the number 2 we obtain bounds for rates when is any irrational and a sharp rate when is a quadratic irrational in that case the discrepancy falls as k 1 2 up to constant factors where k is the number of steps in the walk this is the first example of a sharp rate for a discrete walk on a continuous state space it is obtained by establishing an interesting recurrence relation for the distribution of multiples of which allows for tighter bounds on terms which appear in the erd s turan inequality

area minimizing discs in metric spaces we solve the classical problem of plateau in the setting of proper metric spaces precisely we prove that among all disc type surfaces with prescribed jordan boundary in a proper metric space there exists an area minimizing disc which moreover has a quasi conformal parametrization if the space supports a local quadratic isoperimetric inequality for curves we prove that such a solution is locally holder continuous in the interior and continuous up to the boundary our results generalize corresponding results of douglas rado and morrey from the setting of euclidean space and riemannian manifolds to that of proper metric spaces

invariants of hyperbolic partial differential operators we present a construction of a large class of laplace invariants for linear hyperbolic partial differential operators of fairly general form and arbitrary order

cohomology of canonical projection tilings we define the cohomology of a tiling as the cocycle cohomology of its associated groupoid and consider this cohomology for the class of tilings which are obtained from a higher dimensional lattice by the canonical projection method in schlottmann s formulation we prove the cohomology to be equivalent to a certain cohomology of the lattice we discuss one of its qualitative features namely that it provides a topological obstruction for a generic tiling to be substitutional we develop and demonstrate techniques for the computation of cohomology for tilings of codimension smaller than or equal to 2 presenting explicit formulae these in turn give computations for the k theory of certain associated non commutative c algebras

spaces of tilings finite telescopic approximations and gap labeling the continuous hull of a repetitive tiling t in d with the finite pattern condition fpc inherits a minimal d lamination structure with flat leaves and a transversal open image in new window which is a cantor set this class of tiling includes the penrose the amman benkker ones in 2d as well as the icosahedral tilings in 3d we show that the continuous hull with its canonical d action can be seen as the projective limit of a suitable sequence of branched oriented and flat compact d manifolds as a consequence the longitudinal cohomology and the k theory of the corresponding c algebra open image in new window are obtained as direct limits of cohomology and k theory of ordinary manifolds moreover the space of invariant finite positive measures can be identified with a cone in the d th homology group canonically associated with the orientation of d at last the gap labeling theorem holds given an invariant ergodic probability measure on the hull the corresponding integrated density of states ids of any selfadjoint operators affiliated to open image in new window takes on values on spectral gaps in the module generated by the occurrence probabilities of finite patches in the tiling

exactly solvable disordered sphere packing model in arbitrary dimensional euclidean spaces we introduce a generalization of the well known random sequential addition rsa process for hard spheres in d dimensional euclidean space mathbb r d we show that all of the n particle correlation functions of this nonequilibrium model in a certain limit called the ghost rsa packing can be obtained analytically for all allowable densities and in any dimension this represents the first exactly solvable disordered sphere packing model in an arbitrary dimension the fact that the maximal density ensuremath phi ensuremath infty 1 2 d of the ghost rsa packing implies that there may be disordered sphere packings in sufficiently high d whose density exceeds minkowski s lower bound for bravais lattices the dominant asymptotic term of which is 1 2 d indeed we report on a conjectural lower bound on the density whose asymptotic behavior is controlled by 2 ensuremath 0 778 65 dots d thus providing the putative exponential improvement on minkowski s 100 year old bound our results suggest that the densest packings in sufficiently high dimensions may be disordered rather than periodic implying the existence of disordered classical ground states for some continuous potentials

are there chaotic tilings we develop a class of examples in the form of tiling dynamical systems for use as toy models in statistical mechanics to analyze the possible existence of disordered crystals we give the first such models which are disordered in the sense of having no discrete spectrum

quaquaversal tilings and rotations we construct a hierarchical tiling of 3 dimensional euclidean space based on a triangular prism using repeated rotations about orthogonal axes by angles 2 m and 2 n to analyze the structure of the tiling we are led to determine the group g m n generated by such a pair of rotations for m n 3 and for m 3 n 4

normally elliptic singular perturbations and persistence of homoclinic orbits abstract we consider a dynamical system possibly infinite dimensional or non autonomous with fast and slow time scales which is oscillatory with high frequencies in the fast directions we first derive and justify the limit system of the slow variables assuming a steady state persists we construct the stable unstable center stable center unstable and center manifolds of the steady state of a size of order o 1 and give their leading order approximations finally using these tools we study the persistence of homoclinic solutions in this type of normally elliptic singular perturbation problems

gromov hyperbolic spaces and the sharp isoperimetric constant in this article we exhibit the largest constant in a quadratic isoperimetric inequality which ensures that a geodesic metric space is gromov hyperbolic as a particular consequence we obtain that euclidean space is a borderline case for gromov hyperbolicity in terms of the isoperimetric function we prove similar results for the linear filling radius inequality our results strengthen and generalize theorems of gromov papasoglu and others

a multidimensional critical factorization theorem the critical factorization theorem is one of the principal results in combinatorics on words it relates local periodicities of a word to its global periodicity in this paper we give a multidimensional extension of it more precisely we give a new proof of the critical factorization theorem but in a weak form where the weakness is due to the fact that we loose the tightness of the local repetition order in exchange we gain the possibility of extending our proof to the multidimensional case indeed this new proof makes use of the theorem of fine and wilf that has several classical generalizations to the multidimensional case

a hierarchical strongly aperiodic set of tiles in the hyperbolic plane we give a new construction of strongly aperiodic set of tiles in h 2 exhibiting a kind of hierarchical structure simplifying the central framework of margenstern s proof that the domino problem is undecidable in the hyperbolic plane margenstern 2008 16

subgroups ofso 3 associated with tilings abstract we give a thorough analysis of those subgroups ofso 3 generated by rotations about perpendicular axes by 2 pand 2 q a corollary is that such a group is the free product of the cyclic groups of rotations about the separate axes if and only ifp q 3 and are both odd these groups are naturally associated with a family of hierarchical tilings of euclidean 3 space

long periodic orbits and invariant tori in a singularly perturbed hamiltonian system in this paper we study a singularly perturbed two degree of freedom hamiltonian system with a normally elliptic slow manifold we prove that the slow manifold persists but can have a large number e 1 of exponentially small e c e gaps we demonstrate the existence of kam tori in a neighborhood of the slow manifold in addition we investigate a bifurcation which describes the creation of a gap in the slow manifold and derive its normal form 2002 elsevier science b v all rights reserved

large time behavior of disturbed planar fronts in the allen cahn equation abstract we consider the allen cahn equation in r n with n 2 and study how a planar front behaves when arbitrarily large but bounded perturbation is given near the front region we first show that the behavior of the disturbed front can be approximated by that of the mean curvature flow with a drift term for all large time up to t using this observation we then show that the planar front is asymptotically stable in l r n under spatially ergodic perturbations which include quasi periodic and almost periodic ones as special cases as a by product of our analysis we present a result of a rather general nature which states that for a large class of evolution equations the unique ergodicity of the initial data is inherited by the solution at any later time

the pinwheel tilings of the plane prototiles in euclidean n space en for n 2 the prototiles are usually required to be rather nice topologically at least homeomorphs of the closed unit ball one then makes arbitrarily many congruent copies called tiles of these prototiles and considers all ways called tilings that such tiles may provide a simultaneous covering and packing of en a tiling is thus an unordered collection of tiles for which the union is all of en but such that the interiors of each pair of tiles do not intersect wang s original problem was to determine if it was possible to have a finite set of prototiles with associated tilings of space for which all the tilings were nonperiodic a tiling is nonperiodic if it is not invariant under any single simultaneous translation of its tiles this was settled in the affirmative in the thesis of wang s student robert berger 1

drift of slow variables in slow fast hamiltonian systems abstract we study the drift of slow variables in a slow fast hamiltonian system with several fast and slow degrees of freedom keeping the slow variables frozen for any periodic trajectory of the fast subsystem we define an action for a family of periodic orbits the action is a scalar function of the slow variables and can be considered as a hamiltonian function which generates some slow dynamics these dynamics depend on the family of periodic orbits assuming that for the frozen slow variables the fast system has a pair of hyperbolic periodic orbits connected by two transversal heteroclinic trajectories we prove that for any path composed of a finite sequence of slow trajectories generated by action hamiltonians there is a trajectory of the full system whose slow component shadows the path

symmetries of quasicrystals we consider tiling models of round quasicrystals which would have diffraction patterns which are fully rotation invariant rings instead of bragg peaks they can be distinguished from glasses by self similarity of the pattern of radii of the rings

averaging slaving and balance dynamics in a simple atmospheric model we report numerical results from a study of balance dynamics using a simple model of atmospheric motion that is designed to help address the question of why balance dynamics is so stable the non autonomous hamiltonian model has a chaotic slow degree of freedom representing vortical modes coupled to one or two linear fast oscillators representing inertia gravity waves the system is said to be balanced when the fast and slow degrees of freedom are separated we find adiabatic invariants that drift slowly in time this drift is consistent with a random walk behaviour at a speed which qualitatively scales even for modest time scale separations as the upper bound given by neishtadt s and nekhoroshev s theorems moreover a similar type of scaling is observed for solutions obtained using a singular perturbation slaving technique in resonant cases where nekhoroshev s theorem does not apply we present evidence that the smaller lyapunov exponents of the system scale exponentially as well the results suggest that the observed stability of nearly slow motion is a consequence of the approximate adiabatic invariance of the fast motion

tiling spaces are inverse limits let m be an arbitrary riemannian homogeneous space and let be a space of tilings of m with finite local complexity relative to some symmetry group and closed in the natural topology then is the inverse limit of a sequence of compact finite dimensional branched manifolds the branched manifolds are finite unions of cells constructed from the tiles themselves and the group this result extends previous results of anderson and putnam of ormes radin and sadun of bellissard benedetti and gambaudo and of gahler in particular the construction in this paper is a natural generalization of gahler s

existence of perturbed solitary wave solutions to a model equation for water waves abstract we prove the existence of travelling wave solutions to a fifth order partial differential equation which is a formal asymptotic approximation for water waves with surface tension these travelling waves are arbitrarily small perturbations of solitary waves but are not solitary waves themselves because they approach small amplitude oscillations for large values of the independent variable this result suggests that for bond numbers less than one third there are branches of travelling wave solutions to the water wave equations which are perturbations of supercritical elevation solitary waves and which bifurcate from froude number one and bond number one third

meyer s concept of quasicrystal and quasiregular sets this paper relates two mathematical concepts of long range order of a set of atoms each of which is based on restrictions on the set of interatomic distances a set in n is ameyer set if is a delone set and there is a finite setf such that lambda lambda subseteq lambda f text y meyer proposed that such sets include the possible structures of quasicrystals he obtained a structure theory for such sets which reformulates results that he obtained in harmonic analysis around 1970 and which relates these sets to cut and project sets in geometric crystallography v i galiulin introduced the concept ofquasiregular set which is a set such that both and are delone sets this paper shows that these two concepts are identical

an algorithm for two dimensional mesh generation based on the pinwheel tiling we propose a new two dimensional meshing algorithm called pinw able to generate meshes that accurately approximate the distance between any two domain points by paths composed only of cell edges this technique is based on an extension of pinwheel tilings proposed by radin and conway see c radin ann of math 2 139 1994 pp 661 702 we prove that the algorithm produces triangles of bounded aspect ratio this kind of mesh would be useful in cohesive interface finite element modeling when the crack propagation path is an outcome of a simulation process

spatial convergence of crack nucleation using a cohesive finite element model on a pinwheel based mesh we consider the use of initially rigid cohesive interface models in a two dimensional dynamic finite element solution of a fracture process our focus is on convergence of finite element solutions to a solution of the undiscretized medium as the mesh spacing x and therefore time step t tends to zero we propose the use of pinwheel meshes which possess the isoperimetric property that for any curve c in the computational domain there is an approximation to c using mesh edges that tends to c including a correct representation of its length as the grid size tends to zero we suggest that the isoperimetric property is a necessary condition for any possible spatial convergence proof in the general case that the crack path is not known in advance conversely we establish that if the pinwheel mesh is used the discrete interface first activated in the finite element model will converge to the initial crack in the undiscretized medium finally we carry out a mesh refinement experiment to check convergence of both nucleation and propagation our results indicate that the crack path computed in the pinwheel mesh is more stable as the mesh is refined compared to other types of meshes copyright 2006 john wiley sons ltd

conjugacies for tiling dynamical systems we consider tiling dynamical systems and topological conjugacies between them we prove that the criterion of being of finite type is invariant under topological conjugacy for substitution tiling systems under rather general conditions including the penrose and pinwheel systems we show that substitutions are invertible and that conjugacies are generalized sliding block codes

cheeger constants of surfaces and isoperimetric inequalities we show that if the isoperimetric profile of a bounded genus non compact surface grows faster than vt then it grows at least as fast as a linear function this generalizes a result of gromov for simply connected surfaces we study the isoperimetric problem in dimension 3 we show that if the filling volume function in dimension 2 is euclidean while in dimension 3 it is sub euclidean and there is a g such that minimizers in dimension 3 have genus at most g then the filling function in dimension 3 is almost linear

rectifiability of flat chains in banach spaces with coefficients in zp in this paper we study the behaviour of a class of currents with coefficients in zp in metric spaces obtained by taking a suitable quotient of currents with coefficients in z we prove under fairly general assumption on the ambient space the rectifiability of the so called slice mass and under more restrictive assumptions the rectifiability of the mass measure

effective beam forming networks for large arbitrary array of antennas a deterministic and effective strategy to reduce as much as possible the number of amplitude controls points in large arrays is herein proposed and applied to cases of practical interest the proposed strategy exploits two important concepts the first one coming from a special class of aperiodic tiling and the second one based on nufft non uniform fast fourier transform techniques the first one allows designing innovative front end architectures made of a reduced number of control points while nufft algorithms lead to computationally effective synthesis procedures the overall results and design guidelines allow to achieve simplified beamforming networks derived from aperiodic tilings and to know the performances that can be achieved using this approach particularly in case of low sidelobe requirements to confirm the usefulness of the proposed tools numerical examples are discussed conclusion follows

rapid convergence to frequency for substitution tilings of the plane this paper concerns self similar tilings of the euclidean plane we consider the number of occurrences of a given tile in any domain bounded by a jordan curve for a large class of self similar tilings including many well known examples we give estimates of the oscillation of this number of occurrences around its average frequency times the total number of tiles in the domain which depend only on the jordan curve

on a conjecture on bidimensional words we prove that given a double sequence w over the alphabet a i e a mapping from z2 to a if there exists a pair n0 m0 z2 such that pw n0 m0 1 100n0m0 then w has a periodicity vector where pw is the complexity function in rectangles of w