time domain procedures for testing that a stationary time series is gaussian a class of time domain procedures for testing that a stationary time series is gaussian is presented and analyzed these tests are based on the deviations of the sample value of finite memory nonlinear transformations of the process from their ensemble averaged counterparts asymptotic distributions of these tests are derived under the null hypothesis of gaussianity and under a class of local and fixed alternatives specific tests are then developed based respectively on higher order moments and on the characteristic functions practical construction of the test statistics is discussed with a special emphasis on the estimation of the covariance of the sample statistics which appears to play a key role in the performance of the tests when dealing with small samples

an improvement to the bit stuffing algorithm the bit stuffing algorithm is a technique for coding constrained sequences by the insertion of bits into an arbitrary data sequence this approach was previously introduced and applied to d k constrained codes results show that the maximum average rate of the bit stuffing code achieves capacity when k d 1 or k spl infin while it is suboptimal for all other d k pairs furthermore this technique was generalized to produce codes with an average rate that achieves capacity for all d k pairs however this extension results in a more complicated scheme this correspondence proposes a modification to the bit stuffing algorithm that maintains its simplicity we show analytically that the proposed algorithm achieves improved average rates over bit stuffing for most d k constraints we further determine all constraints for which this scheme produces codes with an average rate equal to the shannon capacity

a characterization of dual quermassintegrals and the roots of dual steiner polynomials abstract let m 1 r 0 0 r 1 r m be a tuple of distinct real numbers and n 2 we provide a characterization of those tuples 0 1 m of real numbers such that there exist n dimensional star bodies k l with w r j k l j j 0 m where w r k l denotes the r th dual relative quermassintegral of k and l this may be regarded as an analogue within the dual brunn minkowski theory of shephard s classification of quermassintegrals of two convex bodies it turns out that the characterization of dual quermassintegrals is related to the moment problem and based on this relation we also derive new determinantal inequalities among the dual quermassintegrals moreover this characterization will be the key tool in order to investigate structural properties of the set of roots of dual steiner polynomials of star bodies

the lp electrostatic q capacitary minkowski problem for general measures abstract existence of solutions to the l p minkowski problem of electrostatic q capacity for discrete measures was very recently given by g xiong j xiong and l xu jfa 2019 when 0 p 1 and 1 q 2 in this paper an existence result of solutions to this problem for general measures is proved when 0 p 1 and 1 q n which is a supplement to the previous result

an efficient coding system for long source sequences the elias source coding scheme is modified to permit a source sequence of practically unlimited length to be coded as a single codeword using arithmetic of only limited precision the result is shown to be a nonblock arithmetic code of the first in first out fifo type source symbols are decoded in the same order as they were encoded codeword lengths which are near optimum for the specified statistical properties of the source can be achieved explicit encoding and decoding algorithms are provided which effectively implement the coding scheme applications to data compression and cryptography are suggested

the affine polya szego principle equality cases and stability a brothers ziemer type theorem for the affine polya szego principle and a quantitative affine polya szego principle are established

liouville type results of curvature operators of fuchsian convex surfaces abstract in this paper we deduce some liouville type results of curvature operators of fuchsian convex surfaces in minkowski space

brunn minkowski and zhang inequalities for convolution bodies abstract a quantitative version of minkowski sum extending the definition of convolution of convex bodies is studied to obtain extensions of the brunn minkowski and zhang inequalities as well as other interesting properties on convex geometry involving convolution bodies or polar projection bodies the extension of this new version to more than two sets is also given

a mass transportation approach to quantitative isoperimetric inequalities a sharp quantitative version of the anisotropic isoperimetric inequality is established corresponding to a stability estimate for the wulff shape of a given surface tension energy this is achieved by exploiting mass transportation theory especially gromov s proof of the isoperimetric inequality and the brenier mccann theorem a sharp quantitative version of the brunn minkowski inequality for convex sets is proved as a corollary

new affine measures of symmetry for convex bodies abstract grunbaum introduced measures of symmetry for convex bodies that measure how far a given convex body is from a centrally symmetric one here we introduce new measures of symmetry that measure how far a given convex body is from one with enough symmetries to define these new measures of symmetry we use affine covariant points we give examples of convex bodies whose affine covariant points are far apart in particular we give an example of a convex body whose centroid and santalo point are far apart

new orlicz affine isoperimetric inequalities abstract the orlicz brunn minkowski theory received considerable attention recently and many results in the l p brunn minkowski theory have been extended to their orlicz counterparts the aim of this paper is to develop orlicz l affine and geominimal surface areas for a single convex body as well as for multiple convex bodies which generalize the l p mixed affine and geominimal surface areas fundamental concepts in the l p brunn minkowski theory our extensions are different from the general affine surface areas by ludwig in 21 moreover our definitions for orlicz l affine and geominimal surface areas reveal that these affine invariants are essentially the infimum supremum of v k l the orlicz mixed volume of k and the polar body of l where l runs over all star bodies and all convex bodies respectively with volume of l equal to the volume of the unit euclidean ball b 2 n properties for the orlicz l affine and geominimal surface areas such as affine invariance and monotonicity are proved related orlicz affine isoperimetric inequalities are also established

a classification of sl n invariant valuations a classification of upper semicontinuous and sl n invariant valuations on the space of n dimensional convex bodies is established as a consequence complete characterizations of cen t ro affine and l p affine surface areas are obtained the proofs make use of a new sl n shaping process for convex bodies

the sharp quantitative isoperimetric inequality a quantitative sharp form of the classical isoperimetric inequality is proved thus giving a positive answer to a conjecture by hall

adaptive entropy coded subband coding of images the authors describe a design approach called 2 d entropy constrained subband coding ecsbc based upon recently developed 2 d entropy constrained vector quantization ecvq schemes the output indexes of the embedded quantizers are further compressed by use of noiseless entropy coding schemes such as huffman or arithmetic codes resulting in variable rate outputs depending upon the specific configurations of the ecvq and the ecpvq over the subbands many different types of sbc schemes can be derived within the generic 2 d ecsbc framework among these the authors concentrate on three representative types of 2 d ecsbc schemes and provide relative performance evaluations they also describe an adaptive buffer instrumented version of 2 d ecsbc called 2 d ecsbc aec for use with fixed rate channels which completely eliminates buffer overflow underflow problems this adaptive scheme achieves performance quite close to the corresponding ideal 2 d ecsbc system

blowup analysis for a nonlinear equation with negative exponent abstract in this paper we study the blowup behavior of the minimizing sequence to certain nonlinear elliptic equation with negative critical exponents we analyze the location for blowup point and the asymptotic behavior of solutions near the blowup point

a remark on rotationally symmetric solutions to the centroaffine minkowski problem abstract in this paper we study the solvability of the rotationally symmetric centroaffine minkowski problem by delicate blow up analyses we remove a technical condition in the existence result obtained by lu and wang 30

arithmetic coding for finite state noiseless channels the authors analyze the expected delay for infinite precision arithmetic codes and suggest a practical implementation that closely approximates the idealized infinite precision model

the first variation of the total mass of log concave functions and related inequalities abstract on the class of log concave functions on r n endowed with a suitable algebraic structure we study the first variation of the total mass functional which corresponds to the volume of convex bodies when restricted to the subclass of characteristic functions we prove some integral representation formulae for such a first variation which suggest to define in a natural way the notion of area measure for a log concave function in the same framework we obtain a functional counterpart of minkowski s first inequality for convex bodies as corollaries we derive a functional form of the isoperimetric inequality and a family of logarithmic type sobolev inequalities with respect to log concave probability measures finally we propose a suitable functional version of the classical minkowski s problem for convex bodies and prove some partial results towards its solution

improved lp mixed volume inequality for convex bodies abstract a sharp quantitative version of the l p mixed volume inequality is established this is achieved by exploiting an improved jensen inequality this inequality is a generalization of pinsker csiszar kullback inequality for the tsallis entropy finally a sharp quantitative version of the l p brunn minkowski inequality is also proved as a corollary

dual orlicz brunn minkowski theory dual orlicz lϕ affine and geominimal surface areas abstract this paper aims to develop the basic theory for the dual orlicz l affine and geominimal surface areas of star bodies basic properties of these new affine invariants are established moreover related orlicz affine isoperimetric cyclic santalo type and alexandrov fenchel type inequalities are obtained

valuations and busemann petty type problems projection and intersection bodies define continuous and gl n contravariant valuations they played a critical role in the solution of the shephard problem for projections of convex bodies and its dual version for sections the busemann petty problem we consider the question whether fkfl implies v k v l where f is a homogeneous continuous operator on convex or star bodies which is an so n equivariant valuation important previous results for projection and intersection bodies are extended to a large class of valuations

fisher information and matrix valued valuations abstract all continuous and affinely contravariant matrix valued valuations on the sobolev space w 1 2 r n are completely classified it is shown that there is a unique such valuation this valuation turns out to be the fisher information matrix

lp mixed geominimal surface area abstract this paper deals with l p geominimal surface area and its extension to l p mixed geominimal surface area we give an integral formula of l p geominimal surface area by the p petty body and introduce the concept of l p mixed geominimal surface area which is a natural extension of l p geominimal surface area some inequalities such as analogues of alexandrov fenchel inequalities blaschke santalo inequalities and affine isoperimetric inequalities for l p mixed geominimal surface areas are obtained

an asymptotically optimal detector for gaussianity testing the aim of this paper is to provide a novel approach to the classical problem of testing the gaussianity of a stationary process where the data can be assumed to be independent and identically distributed some results of large deviations theory are used to select a set of asymptotically optimal test statistics so that their combination results in a suitable detection procedure the johnson system of distributions has been used as the basis for the design of the detector the fact that the good asymptotical properties can be extended to finite samples with small sizes provides the motivation for the rest of the work in this respect the performance of the proposed procedure is also compared with that of some known gaussianity tests the possibility to directly calculate a reliable measure of support such as the bayesian posterior probabilities as opposed to p values is an additional advantage furthermore it will be shown how the proposed procedure can provide some extra information concerning the type of deviation from gaussianity which is present in the data

capacity achieving code constructions for two classes of d k constraints this correspondence presents two variable rate encoding algorithms that achieve capacity for the d k constraint when k 2d 1 or when k d 1 is not prime the first algorithm symbol sliding is a generalized version of the bit flipping algorithm introduced by aviran in addition to achieving capacity for d 2d 1 constraints it comes close to capacity in other cases the second algorithm is based on interleaving and is a generalized version of the bit stuffing algorithm introduced by bender and wolf this method uses fewer than k d biased bit streams to achieve capacity for d k constraints with k d 1 not prime in particular the encoder for d d 2 sup m 1 constraints 2 spl les m spl infin requires only m biased bit streams

adaptive entropy coded subband coding of image sequences a new approach to image sequence coding based on variable rate entropy constrained subband coding ecsbc is described the corresponding practical implementation of the ecsbc scheme for fixed rate channels is developed by extending recent adaptive entropy coded aec quantization techniques although the entropy constrained design of subband coding systems provides improve coding efficiency compared to level constrained design approaches the resulting coding system generates variable rate outputs which must be buffered before fixed rate transmissions in this case the finite buffer however large that interfaces the encoder and the channel will eventually overflow or underflow resulting in a catastrophic loss of encoder decoder synchronism with an associated large amount of distortion a buffer adaptive arithmetic coded implementation of the ecsbc scheme called adaptive entropy coded subband coding ecsbc aec is described to completely eliminate the associated encoder buffer overflow underflow problems even with a very small encoder buffer

on the uniqueness of lp minkowski problems the constant p curvature case in r3 abstract we study the c 4 smooth convex bodies k r n 1 satisfying k x u x 1 p where x s n k is the gauss curvature of k u is the support function of k and p is a constant in the case of n 2 either when p 1 0 or when p 0 1 in addition to a pinching condition we show that k must be the unit ball this partially answers a conjecture of lutwak yang and zhang about the uniqueness of the l p minkowski problem in r 3 moreover we give an explicit pinching constant depending only on p when p 0 1

rotationally symmetric solutions to the l p minkowski problem abstract in this paper we study the l p minkowski problem for p n 1 which corresponds to the critical exponent in the blaschke santalo inequality we first obtain volume estimates for general solutions then establish a priori estimates for rotationally symmetric solutions by using a kazdan warner type obstruction finally we give sufficient conditions for the existence of rotationally symmetric solutions by a blow up analysis we also include an existence result for the l p minkowski problem which corresponds to the super critical case of the blaschke santalo inequality

runlength codes from source codes a class of binary runlength codes also known as d k codes is analyzed these codes are developed by constructing a lossless source code that maps runlengths into unconstrained binary sequences the source code is constructed for the maxentropic distribution on runlengths the inverse of the source code which outputs runlengths guided toward the ideal maxentropic distribution is the d k code four types of source codes are investigated for this purpose huffman enumerative variable length to block and elias or arithmetic the rates of the codes are each proven to converge to the capacity with increasing complexity the codes are not state dependent and are variable rate except for the fixed rate enumerative code a combined source d k code is presented that is based on the arithmetic code

brunn minkowski and prekopa leindler s inequalities under projection assumptions abstract brunn minkowski s theorem says that vol 1 k l 1 n for k l convex bodies is a concave function in and assuming a common hyperplane projection of k and l it is known that the volume itself is concave the a priori natural hypothesis of a common n k plane projection of the sets turned out in the end not to imply the 1 k th concavity of the volume function in this paper we show which is the somehow best projection type assumption that is needed in order to get concavity for vol 1 k l 1 k characterizing also the equality case in the corresponding inequality moreover we consider the same problem for its functional analogue the prekopa leindler inequality

boundary holder estimates for nonlinear singular elliptic equations abstract in this paper we study the dirichlet problem of a class of nonlinear elliptic equations which includes monge ampere equation k hessian equation and the usual linear elliptic equation this problem becomes singular near the boundary of the domain by carefully constructing sub solutions we obtain the boundary holder estimate for the convex solution to the problem when the domain satisfies exterior sphere condition

the lp brunn minkowski inequality for p 1 abstract in this paper we confirm the l p brunn minkowski inequality conjecture for p close to 1 the logarithmic brunn minkowski inequality is also verified for convex bodies close to the unit ball in hausdorff distance

toward a relative q entropy abstract we address the question and related controversy of the formulation of the q entropy and its relative entropy counterpart for models described by continuous non discrete sets of variables we notice that an l p normalized functional proposed by lutwak yang zhang lyz which is essentially a variation of a properly normalized relative renyi entropy up to a logarithm has extremal properties that make it an attractive candidate which can be used to construct such a relative q entropy we comment on the extremizing probability distributions of this lyz functional its relation to the escort distributions a generalized fisher information and the corresponding cramer rao inequality we point out potential physical implications of the lyz entropic functional and of its extremal distributions

a filter based bit allocation scheme for subband compression of hdtv the authors compare the subband compression capabilities of eight filter sets consisting of linear phase quadrature mirror filters qmfs perfect reconstruction filters and nonlinear phase wavelets at different bit rates using a filter based bit allocation procedure using dpcm and pcm in hdtv subband coding it is found that qmfs have an edge over the rest

time domain tests for gaussianity and time reversibility statistical signal processing algorithms often rely upon gaussianity and time reversibility two important notions related to the probability structure of stationary random signals and their symmetry parametric models obtained via second order statistics sos are appropriate when the available data is gaussian and time reversible on the other hand evidence of nonlinearity non gaussianity or time irreversibility favors the use of higher order statistics hos in order to validate gaussianity and time reversibility and quantify the tradeoffs between sos and hos consistent time domain chi squared statistical tests are developed exact asymptotic distributions are derived to estimate the power of the tests including a covariance expression for fourth order sample cumulants a modification of existing linearity tests in the presence of additive gaussian noise is discussed briefly the novel gaussianity statistic is computationally attractive leads to a constant false alarm rate test and is well suited for parametric modeling because it employs the minimal hos lags which uniquely characterize arma processes simulations include comparisons with an existing frequency domain approach and an application to real seismic data time reversibility tests are also derived and their performance is analyzed both theoretically and experimentally

variational characterization for the planar dual minkowski problem abstract in this paper we give a variational analysis to the planar dual minkowski problem in the sobolev space with the new variational characterization we can deal with existence results for prescribed not necessarily positive data meanwhile functional inequalities and multiple solutions are also obtained

subband finite state scalar quantization a new image coding algorithm that exploits the relationship between various subbands of an image is presented in this correspondence using local variance for state selection and uniform threshold quantizers on subbands we have achieved a reduction in quantization noise compared to other published techniques for lena at various bit rates

the use of contextual information in the reversible compression of medical images the authors investigate the use of conditioning events or contexts in improving the performances of known compression methods by building a source model with multiple contexts to code the decorrelated pixels three methods for reversible compression namely dpcm differential pulse code modulation wht walsh hadamard transform and hint hierarchical interpolation employing respectively predictive decorrelation transform decorrelation and multiresolution decorrelation are considered it is shown that the performance of these methods can be enhanced significantly sometimes even up to 40 by using contexts the enhanced dpcm method is found to perform the best for mr and ut ultrasound medical images the enhanced wht method is found to be the best for x ray images the source models used in the enhanced models employ several hundred contexts

on the 2 dimensional dual minkowski problem abstract we study the 2 dimensional dual minkowski problem which is the following nonlinear problem on unit circle u u g u 1 u 2 u 2 2 k 2 s for any given positive continuous function g with 2 m periodic we prove that it is solvable for all k 1 and m 3 4 5

optimal boundary regularity for a singular monge ampere equation abstract in this paper we study the optimal global regularity for a singular monge ampere type equation which arises from a few geometric problems we find that the global regularity does not depend on the smoothness of domain but it does depend on the convexity of the domain we introduce a type to describe the convexity as a result we show that the more convex is the domain the better is the regularity of the solution in particular the regularity is the best near angular points

renyi divergence and lp affine surface area for convex bodies abstract we show that the fundamental objects of the l p brunn minkowski theory namely the l p affine surface areas for a convex body are closely related to information theory they are exponentials of renyi divergences of the cone measures of a convex body and its polar we give geometric interpretations for all renyi divergences d not just for the previously treated special case of relative entropy which is the case 1 now no symmetry assumptions are needed and if at all only very weak regularity assumptions are required previously the relative entropies appeared only after performing second order expansions of certain expressions now already first order expansions make them appear thus in the new approach we detect faster details about the boundary of a convex body

testing that a stationary time series is gaussian on propose une classe de procedures pour tester les hypotheses composites selon lesquelles un processus stochastique stationnaire est gaussien

the lp minkowski problem for torsion abstract in the present paper we introduce the so called l p torsion measure l p mixed torsion a hadamard variational formula for torsion under the p sum is established as an application the existence of solutions to a l p minkowski problem for torsion is proved for any fixed p 1 and p n 2 which extends a result of colesanti and fimiani 24 a l p minkowski inequality for torsion is also obtained which implies the uniqueness of the solutions for the l p minkowski problem mentioned above

universal modeling and coding the problems arising in the modeling and coding of strings for compression purposes are discussed the notion of an information source that simplifies and sharpens the traditional one is axiomatized and adaptive and nonadaptive models are defined with a measure of complexity assigned to the models a fundamental theorem is proved which states that models that use any kind of alphabet extension are inferior to the best models using no alphabet extensions at all a general class of so called first in first out fifo arithmetic codes is described which require no alphabet extension devices and which therefore can be used in conjunction with the best models because the coding parameters are the probabilities that define the model their design is easy and the application of the code is straightforward even with adaptively changing source models

arithmetic coding with dual symbol sets and its performance analysis we propose a novel adaptive arithmetic coding method that uses dual symbol sets a primary symbol set that contains all the symbols that are likely to occur in the near future and a secondary symbol set that contains all other symbols the simplest implementation of our method assumes that symbols that have appeared in the previously are highly likely to appear in the near future it therefore fills the primary set with symbols that have occurred in the previously symbols move dynamically between the two symbol sets to adapt to the local statistics of the symbol source the proposed method works well for sources such as images that are characterized by large alphabets and alphabet distributions that are skewed and highly nonstationary we analyze the performance of the proposed method and compare it to other arithmetic coding methods both theoretically and experimentally we show experimentally that in certain contexts e g with a wavelet based image coding scheme that has appeared in the literature the compression performance of the proposed method is better than that of the conventional arithmetic coding method and the zero frequency escape arithmetic coding method

affine vs euclidean isoperimetric inequalities abstract it is shown that every even zonal measure on the euclidean unit sphere gives rise to an isoperimetric inequality for sets of finite perimeter which directly implies the classical euclidean isoperimetric inequality the strongest member of this large family of inequalities is shown to be the only affine invariant one among them the petty projection inequality as an application a family of sharp sobolev inequalities for functions of bounded variation is obtained each of which is stronger than the classical sobolev inequality moreover corresponding families of l p isoperimetric and sobolev type inequalities are also established

the dual orlicz brunn minkowski theory this paper introduces the dual orlicz brunn minkowski theory for star sets a radial orlicz addition of two or more star sets is proposed and a corresponding dual orlicz brunn minkowski inequality is established based on a radial orlicz linear combination of two star sets a formula for the dual orlicz mixed volume is derived and a corresponding dual orlicz minkowski inequality proved the inequalities proved yield as special cases the precise duals of the conjectured log brunn minkowski and log minkowski inequalities of boroczky lutwak yang and zhang a new addition of star sets called radial m addition is also introduced and shown to relate to the radial orlicz addition

lutwak petty projection inequalities for minkowski valuations and their duals abstract lutwak s volume inequalities for polar projection bodies of all orders are generalized to polarizations of minkowski valuations generated by even zonal measures on the euclidean unit sphere this is based on analogues of mixed projection bodies for such minkowski valuations and a generalization of the notion of centroid bodies a new integral representation is used to single out lutwak s inequalities as the strongest among these families of inequalities which in turn are related to a conjecture on affine quermassintegrals in the dual setting a generalization of volume inequalities for intersection bodies of all orders by leng and lu is proved these results are related to grinberg s inequalities for dual affine quermassintegrals

lp minkowski valuations on polytopes abstract for 1 p ludwig haberl and parapatits classified l p minkowski valuations intertwining the special linear group with additional conditions such as homogeneity and continuity in this paper a complete classification of l p minkowski valuations intertwining the special linear group on polytopes without any additional conditions is established for p 1 including p for n 3 and p 1 there exist valuations not mentioned before

linear time adaptive arithmetic coding the issue of how arithmetic coding should be implemented is addressed a data structure is described and shown to support adaptive arithmetic coding on an arbitrary sized alphabet in time linear in the size of the inputs and outputs experimental results that show the method to be useful even on relatively small alphabets are given

efficient multiplication free arithmetic codes arithmetic coding is a powerful lossless data compression technique that has attracted much attention it provides more flexibility and better efficiency than huffman coding does however the multiplications needed in its encoding and decoding algorithms are very undesirable rissanen and mohiuddin 1989 have proposed a simple scheme to avoid the multiplications the present authors found that the performance of their proposed scheme might degrade significantly in some cases they propose a multiplication free multialphabet arithmetic code which can be shown to have minor performance degradation in all cases in the proposed scheme each multiplication is replaced by a single shift and add the authors prove by both theoretical analysis and simulation results that the degradation of the proposed multiplication free scheme is always several times 2 7 times in the present experiments smaller than that of the rissanen mohiuddin s scheme

a comparison of enumerative and adaptive codes the problem of noiselessly encoding a message when prior statistics are known is considered the close relationship of arithmetic and enumerative coding for this problem is shown by computing explicit arithmetic coding probabilities for various enumerative coding examples this enables a comparison to be made of the coding efficiency of markov models and enumerative codes as well as a new coding scheme intermediate between the two these codes are then extended to messages whose statistics are not known em a priori two adaptive codes are described for this problem whose coding efficiency is upper bounded by the extended enumerative codes on some practical examples the adaptive codes perform significantly better than the nonadaptive ones

cone volume measure and stability abstract we prove stability results for two central inequalities involving the cone volume measure of a centered convex body the subspace concentration conditions and the u functional volume inequality

sharp stability theorems for the anisotropic sobolev and log sobolev inequalities on functions of bounded variation abstract combining rearrangement techniques with gromov s proof via optimal mass transportation of the 1 sobolev inequality we prove a sharp quantitative version of the anisotropic sobolev inequality on b v r n we also deduce as a corollary of this result a sharp stability estimate for the anisotropic 1 log sobolev inequality

motion vector quantization for video coding a new algorithm is developed for the vector quantization of motion vectors this algorithm called motion vector quantization mvq simultaneously estimates and vector quantizes the motion vectors by reinterpreting the block matching algorithm as a type of vector quantization an iterative design algorithm based on this concept is developed in addition to reducing rate for fixed length encoding the algorithm also reduces the computation considerably we include coding simulation results on the flower garden sequence

on the planar dual minkowski problem abstract in this paper we resolve the planar dual minkowski problem proposed by huang et al 2016 31 for all positive indices without any symmetry assumption more precisely given any q 0 and function f on s 1 bounded by two positive constants we show that there exists a convex body in the plane containing the origin in its interior whose dual curvature measure c q has density f in particular if f is smooth then is also smooth

trispectral analysis of stationary random time series this paper investigates the statistical properties of the trispectrum the trispectrum is the fourier transform of the fourth order joint lagged cumulant of a zero mean stationary random time series a complete representation of the principal domain of the trispectrum derived from its inherent symmetries is detailed the large sample variance for a consistent estimator of the trispectrum is presented and used in the development of the fourth order generalizations of the bispectrum based hinich tests of gaussianity and linearity a test for stationarity is developed using an amended version of the trispectrum based test of gaussianity an arithmetic frame averaging procedure is used to compute consistent trispectral estimates for a zero mean bandlimited real valued stationary random process the trispectral based tests are applied to time series of ship noise recorded by a sonobouy in the eastern atlantic ocean

bootstrapping bispectra an application to testing for departure from gaussianity of stationary signals we propose a bootstrap version of a bispectrum based test for departure from gaussianity that achieves high power while maintaining the level of significance even for small sample sizes the proposed procedure can be also used to set confidence bands for a measure of the bicoherence of stationary random signals

an extension of minkowski s theorem and its applications to questions about projections for measures abstract minkowski s theorem asserts that every centered measure on the sphere which is not concentrated on a great subsphere is the surface area measure of some convex body and the surface area measure determines a convex body uniquely up to a shift in this manuscript we prove an extension of minkowski s theorem consider a measure on r n with positive degree of concavity and positive degree of homogeneity we show that a surface area measure of a convex set k weighted with respect to determines a convex body uniquely up to measure zero we also establish an existence result under natural conditions including symmetry we apply this result to extend the solution to classical shephard s problem to do this we introduce a new notion which relates projections of convex bodies to a given measure and is a generalization of the lebesgue volume of a projection