dimension of slices through the sierpinski carpet for lebesgue typical the intersection of the sierpinski carpet with a line has if non empty dimension where fix the slope then we shall show on the one hand that this dimension is strictly less than for lebesgue almost every on the other hand for almost every according to the angle projection of the natural measure on this dimension is at least for any we find a connection between the box dimension of this intersection and the local dimension of at

a finite element spectral method for approximating the time harmonic maxwell system in r 3 we describe and analyze a method for computing an approximation to the time harmonic electromagnetic field scattered by a bounded inhomogeneity the method is to couple a finite element scheme on a bounded domain with a series solution outside the bounded domain the main result of the paper is to show that the proposed numerical scheme possesses a unique solution with quasi optimal approximation properties we do this by verifying the conditions of the babuska brezzi theory for saddle point problems in analyzing the continuous problem we also provide a new variational proof of the existence of solutions of the continuous maxwell problem

shrinkage priors for bayesian prediction we investigate shrinkage priors for constructing bayesian predictive distributions it is shown that there exist shrinkage predictive distributions asymptotically dominating bayesian predictive distributions based on the jeffreys prior or other vague priors if the model manifold satisfies some differential geometric conditions kullback leibler divergence from the true distribution to a predictive distribution is adopted as a loss function conformal transformations of model manifolds corresponding to vague priors are introduced we show several examples where shrinkage predictive distributions dominate bayesian predictive distributions based on vague priors

local entropy averages and projections of fractal measures we show that for families of measures on euclidean space which satisfy an ergodic theoretic form of self similarity under the operation of re scaling the dimension of linear images of the measure behaves in a semi continuous way we apply this to prove the following conjecture of furstenberg let m n be integers which are not powers of the same integer and let x y be closed subsets of the unit interval which are invariant respectively under times m mod 1 and times n mod 1 then for any non zero t dim x ty min 1 dim x dim y a similar result holds for invariant measures and gives a simple proof of the rudolph johnson theorem our methods also apply to many other classes of conformal fractals and measures as another application we extend and unify results of peres shmerkin and nazarov and of moreira concerning projections of products self similar measures and gibbs measures on regular cantor sets we show that under natural irreducibility assumptions on the maps in the ifs the image measure has the maximal possible dimension under any linear projection other than the coordinate projections we also present applications to bernoulli convolutions and to the images of fractal measures under differentiable maps

singular value shrinkage priors for bayesian prediction we develop singular value shrinkage priors for the mean matrix parameters in the matrix variate normal model with known covariance matrices introduced priors are superharmonic and put more weight on matrices with smaller singular values they are a natural generalization of the stein prior bayes estimators and bayesian predictive densities based on introduced priors are minimax and dominate those based on the uniform prior in finite samples the risk reduction is large when the true value of the parameter has small singular values in particular introduced priors perform stably well in the low rank case we apply this result to multivariate linear regression problems by considering priors depending on the future samples introduced priors are compatible with reduced rank regression

estimation of the mean of a multivariate normal distribution estimation of the means of independent normal random variables is considered using sum of squared errors as loss an unbiased estimate of risk is obtained for an arbitrary estimate and certain special classes of estimates are then discussed the results are applied to smoothing by use of moving averages and to trimmed analogs of the james stein estimate a suggestion is made for calculating approximate confidence sets for the mean vector centered at an arbitrary estimate

a dimension formula for bernoulli convolutions we present a dynamical approach to the study of the singularity of infinitely convolved bernoulli measuresv for the golden section we introducev as the transverse measure of the maximum entropy measure on the repelling set invariant for contracting maps of the square the fat baker s transformation our approach strongly relies on the markov structure of the underlying dynamical system indeed if golden mean the fat baker s transformation has a very simple markov coding the ambiguity of order two of this coding which appears when projecting on the line due to passages for the central overlapping zone can be expressed by means of products of matrices of order two this product has a markov distribution inherited by the markov structure of the map the dimension of the projected measure is therefore associated to the growth of this product our dimension formula appears in a natural way as a version of the furstenberg guivarch formula our technique provides an explicit dimension formula and most important provides a formalism well suited for the multifractal analysis of this measure as we will show in a forthcoming paper

a self similar measure with dense rotations singular projections and discrete slices abstract we construct a planar homogeneous self similar measure with strong separation dense rotations and dimension greater than 1 such that there exist lines for which dimension conservation does not hold and the projection of the measure is singular in fact the set of such directions is residual and the typical slices of the measure perpendicular to these directions are discrete

random countable iterated function systems with overlaps and applications abstract we study invariant measures for random countable finite or infinite conformal iterated function systems ifs with arbitrary overlaps we do not assume any type of separation condition we prove under a mild assumption of finite entropy the dimensional exactness of the projections of invariant measures from the shift space and we give a formula for their dimension in the context of random infinite conformal iterated function systems with overlaps there exist many differences between our case and the finite deterministic case studied in 7 and we introduce new methods specific to the infinite and random case we apply our results towards a problem related to a conjecture of lyons about random continued fractions 11 and show that for all parameters 0 the invariant measure is exact dimensional and in addition we give estimates for the pointwise and hausdorff dimension of for in a certain interval the finite ifs determining these continued fractions is not hyperbolic but we can associate to it a random infinite ifs of contractions which have overlaps we study then also other large classes of random countable iterated function systems with overlaps namely a several types of random iterated function systems related to kahane salem sets and b randomized infinite ifs in the plane which have uniformly bounded number of disc overlaps for all the above classes we prove dimensional exactness and we find lower and upper estimates for the pointwise and hausdorff packing dimensions of the projection measures

convergence of the infinite element methods for the helmholtz equation in separable domains to the best knowledge of the authors this work presents the first convergence analysis for the infinite element method iem for the helmholtz equation in exterior domains the approximation applies to separable geometries only combining an arbitrary finite element fe discretization on the boundary of the domain with a spectral like approximation in the radial direction with shape functions resulting from the separation of variables the principal idea of the presented analysis is based on the spectral decomposition of the problem

equilibrium states for factor maps between subshifts abstract let x y be a factor map where x x and y y are subshifts over finite alphabets assume that x satisfies weak specification let a a 1 a 2 r 2 with a 1 0 and a 2 0 let f be a continuous function on x with sufficient regularity holder continuity for instance we show that there is a unique shift invariant measure on x that maximizes f d a 1 h x a 2 h 1 y in particular taking f 0 we see that there is a unique invariant measure on x that maximizes the weighted entropy a 1 h x a 2 h 1 y which answers an open question raised by gatzouras and peres 1996 in 15 an extension is given to high dimensional cases as an application we show that for each compact invariant set k on the k torus under a diagonal endomorphism if the symbolic coding of k satisfies weak specification then there is a unique invariant measure supported on k so that dim h dim h k

hausdorff dimension of the level sets of self affine functions abstract given a partition of 0 1 by closed intervals i 1 i k 0 1 with k 2 given 0 1 and closed intervals j i i 1 k with i j j j given 1 k 0 1 let 0 1 0 1 be the compact set satisfying that i 1 k i i 1 j i i where for an interval i a b 0 1 and 1 1 i 0 1 i is the linear map such that i 1 0 a i 1 1 b and i 1 0 b i 1 1 a such is a graph of a borel function f almost surely and is called a self affine set of function type we obtain the hausdorff dimension of the level set y x x y in the case that f 1 has a bounded density with respect to where is the lebesgue measure on 0 1

dimension and product structure of hyperbolic measures we prove that every hyperbolic measure invariant under a c 1 fi difieomorphism of a smooth riemannian manifold possesses asymptotically almost local product structure i e its density can be approximated by the product of the densities on stable and unstable manifolds up to small exponentials this has not been known even for measures supported on locally maximal hyperbolic sets using this property of hyperbolic measures we prove the long standing eckmann ruelle conjecture in dimension theory of smooth dynamical systems the pointwise dimension of every hyperbolic measure invariant under a c 1 fi difieomorphism exists almost everywhere this implies the crucial fact that virtually all the characteristics of dimension type of the measure including the hausdorfi dimension box dimension and information dimension coincide this provides the rigorous mathematical justiflcation of the concept of fractal dimension for hyperbolic measures

local dimensions of measures on infinitely generated self affine sets abstract we show the existence of the local dimension of an invariant probability measure on an infinitely generated self affine set for almost all translations this implies that an ergodic probability measure is exactly dimensional furthermore the local dimension equals the minimum of the local lyapunov dimension and the dimension of the space

a variable order infinite acoustic wave envelope element abstract a new infinite wave envelope element for analyzing acoustic radiation and scattering problems in an unbounded domain is presented the element is based on a finite to infinite geometry mapping and a wave representation within the shape function it allows for the specification of an arbitrary number of acoustic degrees of freedom in the radial infinite direction yielding a 1 r expansion for the proper modelling of the amplitude decay both two dimensional and axisymmetric problems are presented to show the use and accuracy of this n th order infinite wave envelope element numerical results are compared with analytical and boundary element solutions

inadmissibility of the best equivariant predictive density in the unknown variance case where is an estimator of and the expectation is taken with respect to random quantities usedin the construction of this phenomenon the stein e ect was rst brought to lightin the context of parameter estimation for such a problem many classes of estimatorsdominating the average have been proposed in parallel a similar phenomenon has been observed for predictive density estimation for instance let us observe nindependent d dimensional vectors x

modeling of electromagnetic absorption scattering problems using hp adaptive finite elements a model problem for the steady state form of maxwell s equations is considered the problem is formulated in a weak form using a lagrange multiplier laying down a foundation for a general class of novel hp adaptive fe approximations a convergence proof for affine elements is presented the proposed method is illustrated and verified with a series of 2d experiments including elements with curved boundaries and nonhomogeneous media

simultaneous prediction of independent poisson observables simultaneous predictive distributions for independent poisson observables are investigated a class of improper prior distributions for poisson means is introduced the bayesian predictive distributions based on priors from the introduced class are shown to be admissible under the kullback leibler loss a bayesian predictive distribution based on a prior in this class dominates the bayesian predictive distribution based on the jeffreys prior

absolute continuity of non homogeneous self similar measures abstract we prove that self similar measures on the real line are absolutely continuous for almost all parameters in the super critical region in particular confirming a conjecture of s m ngai and y wang while recently there has been much progress in understanding absolute continuity for homogeneous self similar measures this is the first improvement over the classical transversality method in the general non homogeneous case in the course of the proof we establish new results on the dimension and fourier decay of a class of random self similar measures

ledrappier young formula and exact dimensionality of self affine measures in this paper we solve the long standing open problem on exact dimensionality of self affine measures on the plane we show that every self affine measure on the plane is exact dimensional regardless of the choice of the defining iterated function system in higher dimensions under certain assumptions we prove that self affine and quasi self affine measures are exact dimensional in both cases the measures satisfy the ledrappier young formula

asymptotically minimax bayes predictive densities given a random sample from a distribution with density function that depends on an unknown parameter 0 we are interested in accurately estimating the true parametric density function at a future observation from the same distribution the asymptotic risk of bayes predictive density estimates with kullback leibler loss function d f f f f log f f is used to examine various ways of choosing prior distributions the principal type of choice studied is minimax we seek asymptotically least favorable predictive densities for which the corresponding asymptotic risk is minimax a result resembling stein s paradox for estimating normal means by maximum likelihood holds for the uniform prior in the multivariate location family case when the dimensionality of the model is at least three the jeffreys prior is minimax though inadmissible the jeffreys prior is both admissible and minimax for one and two dimensional location problems

admissible predictive density estimation let xj np v xi and y j np v yi be independent p dimensional multivariate normal vectors with common unknown mean based on observing x x we consider the problem of estimating the true predictive density p yj of y under expected kullback leibler loss our focus here is the characterization of admissible procedures for this problem we show that the class of all generalized bayes rules is a complete class and that the easily interpretable conditions of brown and hwang 1982 are sucient for a formal bayes rule to be admissible 1 introduction let xj np v xi and yj np v yi be independent p dimensional multivariate normal vectors with a common unknown mean 2 r p we assume that vx 0 and vy 0 are known we let p xj and p yj denote the conditional densities of x and y suppressing the dependence onvx and vy throughout based on observing only x x we consider the problem of estimating the density p yj of y the natural action spacea0 consists of all proper densities on r p i e

on the dimension of furstenberg measure for sl _ 2 mathbb r random matrix products let mu be a measure on sl 2 mathbb r generating a non compact and totally irreducible subgroup and let nu be the associated stationary furstenberg measure for the action on the projective line we prove that if mu is supported on finitely many matrices with algebraic entries then begin aligned dim nu min big 1 frac h mathrm mathrm rw mu 2 chi mu big end aligned where h mathrm mathrm rw mu is the random walk entropy of mu chi mu is the lyapunov exponent for the random matrix product associated with mu and dim denotes pointwise dimension in particular for every delta 0 there is a neighborhood u of the identity in sl 2 mathbb r such that if a measure mu in mathcal p u is supported on algebraic matrices with all atoms of size at least delta and generates a group which is non compact and totally irreducible then its stationary measure nu satisfies dim nu 1

solution of 3d laplace and helmholtz equations in exterior domains using hp infinite elements abstract this work is devoted to a convergence study for infinite element discretizations for laplace and helmholtz equations in exterior domains the proposed approximation applies to separable geometries only combining an hp fe discretization on the boundary of the domain with a spectral like representation resulting from the separation of variables in the radial direction the presentation includes a convergence proof for the laplace equation and a stability analysis for the variational formulation of the helmholtz equation in weighted sobolev spaces the theoretical investigations are verified and illustrated with numerical examples for the exterior spherical domain

the metric entropy of diffeomorphisms part i characterization of measures satisfying pesin s entropy formula soit m une variete de riemann compacte soit f m m un diffeomorphisme et soit m une mesure de probabilite de borel f invariante sur m on identifie les mesures pour lesquelles l inegalite de margulis et ruelle qui relie l entropie aux exposants de lyapunov atteint l egalite on demontre une formule valable pour toutes les mesures invariantes

the hausdorff and dynamical dimensions of self affine sponges a dimension gap result we construct a self affine sponge in mathbb r 3 whose dynamical dimension i e the supremum of the hausdorff dimensions of its invariant measures is strictly less than its hausdorff dimension this resolves a long standing open problem in the dimension theory of dynamical systems namely whether every expanding repeller has an ergodic invariant measure of full hausdorff dimension more generally we compute the hausdorff and dynamical dimensions of a large class of self affine sponges a problem that previous techniques could only solve in two dimensions the hausdorff and dynamical dimensions depend continuously on the iterated function system defining the sponge implying that sponges with a dimension gap represent a nonempty open subset of the parameter space

on multifractal formalism for self similar measures with overlaps let mu be a self similar measure generated by an ifs varphi phi i i 1 ell of similarities on mathbb r d d ge 1 when varphi is dimensional regular see definition 1 1 we give an explicit formula for the l q spectrum tau mu q of mu over 0 1 and show that tau mu is differentiable over 0 1 and the multifractal formalism holds for mu at any alpha in tau mu 1 tau mu 0 we also verify the validity of the multifractal formalism of mu over tau mu infty tau mu 0 for two new classes of overlapping algebraic ifss by showing that the asymptotically weak separation condition holds for one of them the proof appeals to the recent result of shmerkin ann math 2 189 2 319 391 2019 on the l q spectrum of self similar measures

a finite element method for solving helmholtz type equations in waveguides and other unbounded domains a finite element method is described for solving helmholtz type boundary value problems in unbounded regions including those with infinite boundaries typical examples include the propagation of acoustic or electromagnetic waves in waveguides the radiation condition at infinity is based on separation of variables and differs from the classical sommerfeld radiation condition it is shown that the problem may be replaced by a boundary value problem on a fixed bounded domain the behavior of the solution near infinity is incorporated in a nonlocal boundary condition this problem is given a weak or variational formulation and the finite element method is then applied it is proved that optimal error estimates hold

multifractal formalism for almost all self affine measures we conduct the multifractal analysis of self affine measures for almost all family of affine maps besides partially extending falconer s formula of lq spectrum outside the range 1 q 2 the multifractal formalism is also partially verified

local time of self affine sets of brownian motion type revisited abstract in 5 we studied self affine sets of brownian motion type we obtained the image measure induced by the lebesgue measure on the domain space if it is absolutely continuous the density function is called the local time the local times are obtained by solving so called jigsaw puzzle in this paper we are specially interested in the self affine set with 3 pieces with the orientations up down up see the picture below

the conjugated vs the unconjugated infinite element method for the helmholtz equation in exterior domains abstract this work is devoted to a convergence study of infinite element ie discretizations for the helmholtz equation in exterior domains the different behavior of the conjugated and the unconjugated ie formulation is analyzed in context of 1 1 a formulation following a mathematical existence theory by leis 2 2 a formulation not based on an available existence theory following burnett four variational formulations are presented and the differences in implementing them are discussed the effect of using or not using the complex conjugate in the weak formulation is carefully studied the numerical and theoretical results clearly indicate which of the four presented formulations is the most efficient that can still give reliable results

an infinite element for maxwell s equations a new infinite element ie approximation for steady state maxwell s equations is proposed the ie is compatible with a new class of hp finite element fe discretizations presented in 13 32 following 12 a convergence analysis for the proposed coupled fe ie approximations is presented

local time of self affine sets of brownian motion type and the jigsaw puzzle problem abstract let 0 1 0 1 be the solution of the set equation i 1 k i i 1 j i i where for an interval i a b 0 1 and 1 1 i 0 1 i is the linear map such that i 1 0 a i 1 1 b i 1 0 b i 1 1 a and i i i 1 k is a partition of 0 1 with j i i i 1 2 thus is a graph of a borel function f almost surely and it is called a self affine set of brownian motion type let be the lebesgue measure on 0 1 and let f 1 the density d d if it exists is called the local time of and it has been studied it is known that dim h 3 2 if exists in the present study is obtained by solving the so called jigsaw puzzle on j i i i 1 k i e the problem of decomposing into a sum of its self similar images with the support j i and the orientation i for i 1 k

multifractal measures and a weak separation condition abstract we define a new separation property on the family of contractive similitudes that allows certain overlappings this property is weaker than the open set condition of hutchinson it includes the well known class of infinite bernoulli convolutions associated with the p v numbers and the solutions of the two scale dilation equations our main purpose in this paper is to prove the multifractal formalism under such condition

hausdorff dimension for randomly perturbed self affine attractors in this paper we shall consider a self affine iterated function system in mathbb r d d 2 where we allow a small random translation at each application of the contractions we compute the dimension of a typical attractor of the resulting random iterated function system complementing a famous deterministic result of falconer which necessarily requires restrictions on the norms of the contraction however our result has the advantage that we do not need to impose any additional assumptions on the norms this is of benefit in practical applications where such perturbations would correspond to the effect of random noise we also give analogous results for the dimension of ergodic measures in terms of their lyapunov dimension finally we apply our method to a problem originating in the theory of fractal image compression

on the complexity of the set of codings for self similar sets and a variation on the construction of champernowne abstract let f p 0 p n be a collection of points in r d the set f naturally gives rise to a family of iterated function systems consisting of contractions of the form s i x x 1 p i i 0 n where 0 1 and x r d given f and it is well known that there exists a unique non empty compact set x satisfying x i 0 n s i x for each x x there exists a sequence a j j 1 0 n n satisfying x lim j s a 1 s a j 0 we call such a sequence a coding of x in this paper we prove that for any f and k n there exists k f 0 such that if 1 k f 1 then every point in the interior of x has a coding which is k simply normal similarly we prove that there exists u n i f 0 such that if 1 u n i f 1 then every point in the interior of x has a coding containing all finite words for some specific choices of f we obtain lower bounds for k f and u n i f we also prove some weaker statements that hold in the more general setting when the similarities in our iterated function systems exhibit different rates of contraction our proofs rely on a variation of a well known construction of a normal number due to champernowne and an approach introduced by erd s and komornik

on self similar sets with overlaps and inverse theorems for entropy we study the dimension of self similar sets and measures on the line we show that if the dimension is less than the generic bound of minf1 sg where s is the similarity dimension then there are superexponentially close cylinders at all small enough scales this is a step towards the conjecture that such a dimension drop implies exact overlaps and conrms it when the generating similarities have algebraic coecients as applications we prove furstenberg s conjecture on projections of the one dimensional sierpinski gasket and achieve some progress on the bernoulli convolutions problem and more generally on problems about parametric families of self similar measures the key tool is an inverse theorem on the structure of pairs of probability measures whose mean entropy at scale 2 n has only a small amount of growth under convolution

mapped wave envelope elements for acoustical radiation and scattering wave envelope elements provide an alternative to boundary element methods for the solution of exterior wave problems applications to date have been to axisymmetric geometries and have involved the use of large but finite wave envelope meshes compatibly matched to denser near field meshes of conventional elements in the present paper the wave envelope idea is extended by the inclusion of an element mapping taken directly from infinite element methodology and by modifications to the residual formulation taken in conjunction these two measures permit the outer boundary of the wave envelope region to be moved to infinity while preserving the boundedness of the acoustic mass and stiffness integrals the present formulation is developed initially with reference to a simple one dimensional model and is then applied in two and three dimensions comparisons between computed results and those obtained from analytic solutions where applicable and from boundary element codes demonstrate the effectiveness of the method

on asymptotic properties of predictive distributions summary second order asymptotic theory of predictive distributions is investigated it is shown that estimative distributions with asymptotically efficient estimators can be improved to predictive distributions that do not belong to the model the model is assumed to be a multidimensional curved exponential family predictive distributions can be constructed by shifting estimative distributions in a direction orthogonal to the model the average kullback leibler divergence from the true distribution to a predictive distribution is represented as the sum of two components one of them depends on the choice of the estimative distribution and the other depends on the shift orthogonal to the model the optimal shift orthogonal to the model is obtained a quantity that can be regarded as the mixture mean curvature of the model in the space of all probability distributions is introduced the difference in average kullback leibler divergence between an estimative distribution and the optimal predictive distribution obtained by shifting it can be regarded as the mixture mean curvature of the model an asymptotic expression for bayesian predictive distributions is also obtained

the limited rademacher functions and bernoulli convolutions associated with pisot numbers abstract in this paper we give a systematical study of the local structures and fractal indices of the limited rademacher functions and bernoulli convolutions associated with pisot numbers for a given pisot number in the interval 1 2 we construct a finite family of non negative matrices maybe non square such that the corresponding fractal indices can be re expressed as some limits in terms of products of these non negative matrices we are especially interested in the case that the associated pisot number is a simple pisot number i e the unique positive root of the polynomial x k x k 1 x 1 k 2 3 in this case the corresponding products of matrices can be decomposed into the products of scalars based on which the precise formulas of fractal indices as well as the multifractal formalism are obtained

hausdorff dimension of the limit sets of some planar geometric constructions abstract we determine the hausdorff and box dimension of the limit sets for some class of planar non moran like geometric constructions generalizing the bedford mcmullen general sierpinski carpets the class includes affine constructions generated by an arbitrary partition of the unit square by a finite number of horizontal and vertical lines as well as some non affine examples e g the flexed sierpinski gasket

exact minimax estimation of the predictive density in sparse gaussian models we consider estimating the predictive density under kullback leibler loss in an l0 sparse gaussian sequence model explicit expressions of the first order minimax risk along with its exact constant asymptotically least favorable priors and optimal predictive density estimates are derived compared to the sparse recovery results involving point estimation of the normal mean new decision theoretic phenomena are seen suboptimal performance of the class of plug in density estimates reflects the predictive nature of the problem and optimal strategies need diversification of the future risk we find that minimax optimal strategies lie outside the gaussian family but can be constructed with threshold predictive density estimates novel minimax techniques involving simultaneous calibration of the sparsity adjustment and the risk diversification mechanisms are used to design optimal predictive density estimates

improved minimax predictive densities under kullback leibler loss let x n p u x i and y n p v y i be independentp dimensional multivariate normal vectors with common unknown mean based on only observing x x we consider the problem of obtaining a predictive density p y x for y that is close to p y as measured by expected kullback leibler loss a natural procedure for this problem is the formal bayes predictive density p u y x under the uniform prior u 1 which is best invariant and minimax we show that any bayes predictive density will be minimax if it is obtained by a prior yielding a marginal that is superharmonic or whose square root is superharmonic this yields wide classes of minimax procedures that dominate p u y x including bayes predictive densities under superharmonic priors fundamental similarities and differences with the parallel theory of estimating a multivariate normal mean under quadratic loss are described

numerical solution for exterior problems we solve the helmholtz equation in an exterior domain in the plane a perfect absorption condition is introduced on a circle which contains the obstacle this boundary condition is given explicitly by bessel functions we use a finite element method in the bounded domain an explicit formula is used to compute the solution out of the circle we give an error estimate and we present relevant numerical results

dimensions of invariant sets of expanding maps we consider a compact invariant set of an expanding map of a manifoldm and give upper and lowerbounds for the hausdorff dimension dimh and box dimensionsdimb and dimb these bounds are given in terms of the topological entropy topological pressure and uniform lyapunov exponents of the map

a shrinkage predictive distribution for multivariate normal observables summary we investigate shrinkage methods for constructing predictive distributions we consider the multivariate normal model with a known covariance matrix and show that there exists a shrinkage predictive distribution dominating the bayesian predictive distribution based on the vague prior when the dimension is not less than three kullback leibler divergence from the true distribution to a predictive distribution is adopted as a loss function somiie key w ords invariance james stein estimator kullback leibler divergence stein s prior vague prior

dimension theory of iterated function systems let si l i 1 be an iterated function system ifs on d with attractor k let denote the one sided full shift over the alphabet 1 l we define the projection entropy function h on the space of invariant measures on e associated with the coding map k and develop some basic ergodic properties about it this concept turns out to be crucial in the study of dimensional properties of invariant measures on k we show that for any conformal ifs respectively the direct product of finitely many conformal ifss without any separation condition the projection of an ergodic measure under is always exactly dimensional and its hausdorff dimension can be represented as the ratio of its projection entropy to its lyapunov exponent respectively the linear combination of projection entropies associated with several coding maps furthermore for any conformal ifs and certain affine ifss we prove a variational principle between the hausdorff dimension of the attractors and that of projections of ergodic measures 2008 wiley periodicals inc

harmonic bayesian prediction under alpha divergence we investigate bayesian shrinkage methods for constructing predictive distributions we consider the multivariate normal model with a known covariance matrix and show that the bayesian predictive density with respect to stein s harmonic prior dominates the best invariant bayesian predictive density when the dimension is greater than or equal to 3 alpha divergence from the true distribution to a predictive distribution is adopted as a loss function