fourier transform of self affine measures abstract suppose f is a self affine set on r d d 2 which is not a singleton associated to affine contractions f j a j b j a j gl d r b j r d j a for some finite a we prove that if the group generated by the matrices a j j a forms a proximal and totally irreducible subgroup of gl d r then any self affine measure p j f j p j 1 0 p j 1 j a on f is a rajchman measure the fourier transform 0 as as an application this shows that self affine sets with proximal and totally irreducible linear parts are sets of rectangular multiplicity for multiple trigonometric series moreover if the zariski closure of is connected real split lie group in the zariski topology then has a power decay at infinity hence is l p improving for all 1 p and f has positive fourier dimension in dimension d 2 3 the irreducibility of and non compactness of the image of in pgl d r is enough for power decay of the proof is based on quantitative renewal theorems for random walks on the sphere s d 1

on the singular set for lipschitzian critical points of polyconvex functionals abstract partial regularity is proved for lipschitzian critical points of polyconvex functionals provided d u l is small enough in particular the singular set for a lipschitzian critical point has hausdorff dimension strictly less than n when d u l is small enough model problems treated include u 2 det u 2 where u r 2 r 2 and u 2 u s ad u s det u s where u r 3 r 3 with s 2 moreover it is shown that the singular set of a lipschitzian global minimizer has hausdorff dimension strictly less than n

a condition for the holder regularity of local minimizers of a nonlinear elastic energy in two dimensions we prove the local holder continuity of strong local minimizers of the stored energy functional e u int omega lambda nabla u 2 h rm det nabla u rm d x subject to a condition of positive twist the latter turns out to be equivalent to requiring that u maps circles to suitably star shaped sets the convex function h s grows logarithmically as s to 0 linearly as s to infty and satisfies h s infty if s leqq 0 these properties encode a constitutive condition which ensures that material does not interpenetrate during a deformation and is one of the principal obstacles to proving the regularity of local or global minimizers the main innovation is to prove that if a local minimizer has positive twist a e on a ball then an euler lagrange type inequality holds and a caccioppoli inequality can be derived from it the claimed holder continuity then follows by adapting some well known elliptic regularity theory we also demonstrate the regularizing effect that the term int omega h rm det nabla u rm d x can have by analysing the regularity of local minimizers in the class of shear maps in this setting a more easily verifiable condition than that of positive twist is imposed with the result that local minimizers are holder continuous

on the fourier transform of coin tossing type measures abstract we give explicit estimates of the fourier decay rate of some coin tossing type measures which generalizes a classical result of hartman and kershner using an elementary method originated from cassels and schmidt we also prove that almost every point is absolutely normal with respect to such measures as an application we present some new examples of measures whose fourier decay rate could be as slow as possible yet almost all points are absolutely normal which complements a result of r lyons on the set of non normal numbers

the singular set of lipschitzian minima of multiple integrals the singular set of any lipschitzian minimizer of a general quasiconvex functional is uniformly porous and hence its hausdorff dimension is strictly smaller than the space dimension

fourier asymptotics of fractal measures abstract a measure on r n will be called locally uniformly dimensional if br x cr for all r 1 and all x where br x denotes the ball of radius r about x for l 2 d the measure d is in i so d is well defined we show it is locally in l2 and sup r 1 r n b r y f d 2 d c f 2 under additional hypotheses we show that lim r r n b r y f d 2 2 d is comparable in size to 2 2 a number of other related results are established the special case when is an integer and is the surface measure on a c1 manifold was treated by s agmon and l hormander j analyse math 30 1976 1 38

partial regularity of strong local minimizers in the multi dimensional calculus of variations let n be a bounded domain and f a given strongly quasiconvex integrand of class c2 satisfying the growth condition f xi le c 1 xi p for some c 0 and 2 p consider the multiple integral i u int omega f nabla u where u w1 p n the main result of the paper is the proof that any strong local minimizer of i is of class c1 loc for any 0 1 on an open set of full n dimensional measure in the case of weak local minimizers we establish the same result under the extra assumption that the oscillations in the gradient of the minimizer are not too large without such an assumption weak local minimizers need not be partially regular as we show by a class of examples we also briefly discuss the question of existence of strong local minimizers for i and connections of our results to extensions of weierstrass sufficiency theorem to the multi dimensional setting

semicontinuity problems in the calculus of variations soit f r 3 r une fonction deux fois continument differentiable et pour u w 1 1 a b f u a b a b f x u x u x dx si f r n r m r nm r satisfait 1 f est mesurable en x continue en cs et 2 0 f x s a x c s p p ou p 1 a est une fonction localement sommable non negative et c une constante non negative alors la fonctionnelle f u a b est sequentiellement faiblement semi continue inferieurement sur w 1 p r m si et seulement si f x s est quasi convexe pour chaque s et presque chaque x

phase transitions for countable markov shifts we study the analyticity of the topological pressure for some one parameter families of potentials on countable markov shifts we relate the non analyticity of the pressure to changes in the recurrence properties of the system we give sufficient conditions for such changes to exist and not to exist we apply these results to the manneville pomeau map and use them to construct examples with different critical behavior

large deviations for countable to one markov systems in this paper we study large deviation properties for countable to one markov systems associated to weak gibbs measures for non holder potentials furthermore we establish multifractal large deviation laws for countable to one piecewise conformal markov systems which are derived systems constructed over hyperbolic regions for certain nonhyperbolic systems exhibiting intermittency we apply our results to higher dimensional number theoretical transformations

refinable functions with non integer dilations refinable functions and distributions with integer dilations have been studied extensively since the pioneer work of daubechies on wavelets however very little is known about refinable functions and distributions with non integer dilations particularly concerning its regularity in this paper we study the decay of the fourier transform of refinable functions and distributions we prove that uniform decay can be achieved for any dilation this leads to the existence of refinable functions that can be made arbitrarily smooth for any given dilation factor we exploit the connection between algebraic properties of dilation factors and the regularity of refinable functions and distributions our work can be viewed as a continuation of the work of erdos p erdos on the smoothness properties of a family of bernoulli convolutions amer j math 62 1940 180 186 kahane j p kahane sur la distribution de certaines series aleatoires in colloque de theorie des nombres univ bordeaux bordeaux 1969 mem soc math france 25 1971 119 122 in french and solomyak b solomyak on the random series n an erdos problem ann of math 2 142 1995 611 625 on bernoulli convolutions we also construct explicitly a class of refinable functions whose dilation factors are certain algebraic numbers and whose fourier transforms have uniform decay this extends a classical result of garsia a m garsia arithmetic properties of bernoulli convolutions trans amer math soc 102 1962 409 432

coarse grid correction scheme for implicit multiblock euler calculations a coarse grid correction scheme is used to bring global influence to implicit multiblock calculations compared to using only explicit coupling between the blocks a considerable reduction in the number of time steps needed to reach steady state has been observed for transonic and subsonic flows the additional cpu time per time step associated with the method is small resulting in a highly efficient parallel implicit scheme

on a problem in simultaneous diophantine approximation littlewood s conjecture a consequence of hurwitz s theorem is that the right hand side of the above inequality cannot be improved by an arbitrary positive constant s more precisely for s l x 5 there exist real numbers a e i for which the inequality ilqall cq 1 has at most a finite number of solutions these a are the badly approximable numbers and we will denote by b a d the set of all such numbers that is

the measure of non normal sets it is well known that almost every number in 0 1 is normal in base 2 in the sense of lebesgue measure kahane and salem asked whether the same is true with respect to any borel measure whose fourier stieltjes coefficients vanish at infinity in other words whether the set of non normal numbers is a set of uniqueness in the wide sense we show that this is not the case in fact we give best possible conditions on the rate of decay of hat mu n in order that almost every number be normal the techniques include on the one hand probability measures with respect to which the binary digits in 0 1 are independent only by blocks rather than individually and on the other hand the strong law of large numbers for weakly correlated random variables

implicit multiblock euler and navier stokes calculations implicit multiblock computations have been carried out for a large number of blocks using explicit coupling between the blocks the convergence rate of this method is very sensitive to the block partitioning for the navier stokes equations the height of the blocks should be greater than the boundary layer thickness also excessively thin blocks will cause breakdown of the algorithm for transonic calculations the best convergence rate was obtained using a red black gauss seidel approach it is concluded that the method is well suited for massively parallel computers

fourier transforms of gibbs measures for the gauss map we investigate under which conditions a given invariant measure mu for the dynamical system defined by the gauss map x mapsto 1 x mathrm mod 1 is a rajchman measure with polynomially decaying fourier transform begin aligned widehat mu xi o xi eta quad text as quad xi rightarrow infty end aligned we show that this property holds for any gibbs measure mu of hausdorff dimension greater than 1 2 with a natural large deviation assumption on the gibbs potential in particular we obtain the result for the hausdorff measure and all gibbs measures of dimension greater than 1 2 on badly approximable numbers which extends the constructions of kaufman and queffelec ramare our main result implies that the fourier stieltjes coefficients of the minkowski s question mark function decay to 0 polynomially answering a question of salem from 1943 as an application of the davenport erd s leveque criterion we obtain an equidistribution theorem for gibbs measures which extends in part a recent result by hochman shmerkin our proofs are based on exploiting the nonlinear and number theoretic nature of the gauss map and large deviation theory for hausdorff dimension and lyapunov exponents

the singular set of minima of integral functionals in this paper we provide upper bounds for the hausdorff dimension of the singular set of minima of general variational integrals open image in new window where f is suitably convex with respect to dv and holder continuous with respect to x v in particular we prove that the hausdorff dimension of the singular set is always strictly less than n where open image in new window

decrease of fourier coefficients of stationary measures let mu be a borel probability measure on mathrm sl 2 mathbb r with a finite exponential moment and assume that the subgroup vargamma mu generated by the support of mu is zariski dense let nu be the unique mu stationary measure on mathbb p 1 we prove that the fourier coefficients hat nu k of nu converge to 0 as k tends to infinity our proof relies on a generalized renewal theorem for the cartan projection

self similar measures and their fourier transforms ii a self similar measure on r n was defined by hutchinson to be a probability measure satisfying formule here s j x j r j x b j is a contractive similarity 0 j 1 r j orthogonal and the weights a j satisfy 0 a j 1 j 1 m a j 1 by analogy we define a self similar distribution by the same identity but allowing the weights a j to be arbitrary complex numbers we give necessary and sufficient conditions for the existence of a solution to among distributions of compact support and show that the space of such solutions is always finite dimensional if f denotes the fourier transformation of a self similar distribution of compact support let formule where is defined by the equation j 1 m j a j 2 1

the regularity of critical points of polyconvex functionals in this paper we are concerned with the question of regularity of critical points for functionals of the type eq1 we construct a smooth strongly polyconvex eq2 and lipschitzian weak solutions eq3 to the corresponding euler lagrange system which are nowhere c1 moreover we show that f can be chosen in such a way that these irregular weak solutions are weak local minimisers

on irregular weak solutions of the energy momentum equations irregular mappings that are weak solutions of the energy momentum equations are presented one example is discontinuous at a countable number of points while the other is c 1 but not c 2 these mappings are not solutions of the usual euler lagrange equations

parallel implicit time accurate navier stokes computations using coarse grid correction a parallel implicit navier stokes solver is constructed using a block by block inversion of the resulting linear system of equations followed by a global coarse grid correction to account for the block coupling this solver is used to compute two different cases of vortex shedding e ows the efe ciency of our method is compared to the efe ciency of the method based on pseudotime stepping and multigrid acceleration this comparison is based on direct comparison of the two methods applied to identical test cases as well as results reported in the literature emphasis is put on scalability as well as mesh sensitivity it is concluded that the solver is not sensitive to grid stretching and scales well with both problem size and number of processors and it is shown that measured cpu times compare well with the multigrid approach

coin tossing measures and their fourier transforms abstract a coin tossing measure on 0 1 is a probability measure satisfying n 1 p n 0 1 p n 1 2 n where p n 0 1 x denotes the probability atom at x and the convergence is in the weak sense we study the asymptotic behavior of averages of the fourier transform of x for p 2 and 0 we prove that x r x p d x o r 1 p r where p 1 lim sup n 1 n n 1 n log 2 1 a n p a n 2 p n 1 this extends some results due to r strichartz for measures which are not self similar we also study the sobolev exponent of x p and its scaling exponent as well as the asymptotic behavior of sums of the walsh fourier coefficients of

m sets and measures the measures pt to be investigated are defined on the torus t and satisfy lim suplii n i 0 lyons 8 recently proved the striking theorem that every measure t of this type must have positive mass in some closed set of type uo later lyons asked whether uo could be improved to u and provided 9 ingenious counter examples for the classical examples of u sets definitions and elementary results in harmonic analysis may be found in works cited a few lines below we prove the following

quasiconvexity and partial regularity in the calculus of variations we prove partial regularity of minimizers of certain functionals in the calculus of variations under the principal assumption that the integrands be uniformly strictly quasiconvex this is of interest since quasiconvexity is known in many circumstances to be necessary and sufficient for the weak sequential lower semicontinuity of these functionals on appropriate sobolev spaces examples covered by the regularity theory include functionals with integrands which are convex in the determinants of various submatrices of the gradient matrix

on the modulus of continuity for spectral measures in substitution dynamics abstract the paper gives first quantitative estimates on the modulus of continuity of the spectral measure for weak mixing suspension flows over substitution automorphisms which yield information about the fractal structure of these measures the main results are first a holder estimate for the spectral measure of almost all suspension flows with a piecewise constant roof function second a log holder estimate for self similar suspension flows and third a holder asymptotic expansion of the spectral measure at zero for such flows our second result implies log holder estimates for the spectral measures of translation flows along stable foliations of pseudo anosov automorphisms a key technical tool in the proof of the second result is an arithmetic diophantine proposition which has other applications in appendix a this proposition is used to derive new decay estimates for the fourier transforms of bernoulli convolutions

boundary regularity in variational problems we prove that if u omega subset mathbb r n to mathbb r n is a solution to the dirichlet variational problem mathop rm min limits w int omega f x w dw rm d x quad rm subject to quad w equiv u 0 rm on partial omega involving a regular boundary datum u 0 and a regular integrand f x w dw strongly convex in dw and satisfying suitable growth conditions then mathcal h n 1 almost every boundary point is regular for u in the sense that du is holder continuous in a relative neighborhood of the point the existence of even one such regular boundary point was previously not known except for some very special cases treated by jost meier math ann 262 549 561 1983 our results are consequences of new up to the boundary higher differentiability results that we establish for minima of the functionals in question the methods also allow us to improve the known boundary regularity results for solutions to non linear elliptic systems and in some cases to improve the known interior singular sets estimates for minimizers moreover our approach allows for a treatment of systems and functionals with rough coefficients belonging to suitable sobolev spaces of fractional order