majorization for a optimal designs in a regression setting the greatest lower bound for the trace of the covariance matrix of the simple least squares estimator is given based on the idea of majorization the problem raised by n n chan schur convexity for a optimal designs j math anal appl 122 1987 1 6 thus possesses a concise solution

an inhomogeneous wave equation and non linear diophantine approximation abstract a non linear diophantine condition involving perfect squares and arising from an inhomogeneous wave equation on the torus guarantees the existence of a smooth solution the exceptional set associated with the failure of the diophantine condition and hence of the existence of a smooth solution is studied both the lebesgue and hausdorff measures of this set are obtained

equivalence relations on separated nets arising from linear toral flows in 1998 burago kleiner and mcmullen independently proved the existence of separated nets in r which are not bi lipschitz equivalent bl to a lattice a finer equivalence relation than bl is bounded displacement bd separated nets arise naturally as return times to a section for minimal r actions we analyze the separated nets which arise via these constructions focusing particularly on nets arising from linear r actions on tori we show that generically these nets are bl to a lattice and for some choices of dimensions and sections they are generically bd to a lattice we also show the existence of such nets which are not bd to a lattice

self affine delone sets and deviation phenomena we study the growth of norms of ergodic integrals for the translation action on spaces coming from expansive self affine delone sets the linear map giving the self affinity induces a renormalization map on the pattern space and we show that the rate of growth of ergodic integrals is controlled by the induced action of the renormalizing map on the cohomology of the pattern space up to boundary errors we explore the consequences for the diffraction of such delone sets and explore in detail what the picture is for substitution tilings as well as for cut and project sets which are self affine we also explicitly compute some examples

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

continuity of eigenfunctions of uniquely ergodic dynamical systems and intensity of bragg peaks we study uniquely ergodic dynamical systems over locally compact sigma compact abelian groups we characterize uniform convergence in wiener wintner type ergodic theorems in terms of continuity of the limit our results generalize and unify earlier results of robinson and assani respectively

entire majorants via euler maclaurin summation it is the aim of this article to give extremal majorants of type 2 for the class of functions f n x sgn x x n where n n as applications we obtain positive definite extensions to r of it m defined on r 1 1 where m n optimal bounds in hilbert type inequalities for the class of functions it m and majorants of type 2 for functions whose graphs are trapezoids

continuously many bounded displacement non equivalences in substitution tiling spaces abstract we consider substitution tilings in r d that give rise to point sets that are not bounded displacement bd equivalent to a lattice and study the cardinality of bd x the set of distinct bd class representatives in the corresponding tiling space x we prove a sufficient condition under which the tiling space contains continuously many distinct bd classes and present such an example in the plane in particular we show here for the first time that this cardinality can be greater than one

extremal functions in de branges and euclidean spaces ii this paper presents the gaussian subordination framework to generate optimal one sided approximations to multidimensional real valued functions by functions of prescribed exponential type such extremal problems date back to the works of beurling and selberg and provide a variety of applications in analysis and analytic number theory here we majorize and minorize on bbb r n the gaussian bf x mapsto e pi lambda bf x 2 where lambda0 is a free parameter by functions with distributional fourier transforms supported on euclidean balls optimizing weighted l 1 errors by integrating the parameter lambda against suitable measures we solve the analogous problem for a wide class of radial functions applications to inequalities and periodic analogues are discussed the constructions presented here rely on the theory of de branges spaces of entire functions and on new interpolation tools derived from the theory of laplace transforms of laguerre p o lya functions

interpolation formulas with derivatives in de branges spaces ii abstract we investigate necessary and sufficient conditions under which entire functions in de branges spaces can be recovered from function values and values of derivatives our main focus is on spaces with a structure function whose logarithmic derivative is bounded in the upper half plane

analysis of uncertain structural systems using interval analysis the imprecision or uncertainty present in many engineering analysis design problems can be modeled using probabilistic fuzzy or interval methods this work considers the modeling of uncertain structural systems using interval analysis by representing each uncertain input parameter as an interval number a static structural analysis problem can be expressed in the form of a system of linear interval equations in addition to the direct and gaussian elimination based solution approaches a combinatorial approach based on an exhaustive combination of the extreme values of the interval numbers and an inequality hased method are presented for finding the solution of interval equations the range or interval of the solution vector response parameters is found to increase with increasing size of the problem in all of the methods an interval truncation approach is proposed to limit the growth of intervals of response parameters so that realistic and accurate solutions can be obtained in the presence of large amounts of uncertainty numerical examples are presented to illustrate the computational aspects of the methods and also to indicate the importance of the truncation approach in practical problems the utility of interval methods in predicting the extreme values of the response parameters of structures is discussed

a optimality for regression designs abstract use is made of a result of horn amer j math 76 1954 620 630 on the existence of a symmetric matrix with prescribed diagonal elements and eigenvalues a necessary and sufficient condition is then given for the existence of an a optimal design for a regression experiment in the dorogovcev selected transl math statist probab 10 1971 35 41 setting

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

anti diagonals of symmetric and skew symmetric matrices with prescribed eigenvalues abstract the complete relationship between the kth anti diagonals and the eigenvalues of an n n k n real symmetric matrix is obtained when k is even the relationship is weak majorization the hermitian case is also studied similar results are obtained for real and complex skew symmetric matrices

systems of linear interval equations this paper presents theory and methods for computing the exact bounds on the solution of a system of n linear equations in n variables whose coefficients and right hand sides vary in some real intervals finite and iterative methods are given based on results from linear complementarity theory also regularity conditions regularity testing and computing the exact inverse of an interval matrix are dealt with

finite step algorithms for constructing optimal cdma signature sequences a description of optimal sequences for direct spread code division multiple access ds cdma is a byproduct of recent characterizations of the sum capacity this paper restates the sequence design problem as an inverse singular value problem and shows that the problem can be solved with finite step algorithms from matrix theory it proposes a new one sided algorithm that is numerically stable and faster than previous methods

cohomology of quasiperiodic patterns and matching rules quasiperiodic patterns described by polyhedral atomic surfaces are considered it is shown that under certain rationality conditions which coincide with the necessary conditions for the existence of matching rules the cohomology ring of the continuous hull of such patterns is isomorphic to that of the complement of a torus tn to an arrangement a of thickened affine tori of codimension 2 explicit computation of betti numbers for several two dimensional tilings and for the icosahedral ammann kramer tiling confirms in most cases the results obtained previously by different methods the cohomology groups of tn a have a natural structure of a right module over the group ring of the space symmetry group of the pattern and can be decomposed in a direct sum of its irreducible representations an example of such decomposition is shown for the ammann kramer tiling

exceptional sets in kolmogorov arnol d moser theory the exceptional sets in the autonomous and periodic kam theorems consist of sets of near resonant frequencies w w w for which the method of proof of the theorems breaks down they are residual and of measure 0 in this paper their hausdorff dimension is deduced from that of a related set of well approximable linear forms the hausdortl dimension of these linear forms is a special case of a more general number theoretic result on systems of linear forms established recently but a self contained proof is given the proof relies on the resonant hyperplanes being reasonably well distributed a fairly general statistical second moment argument is used but some geometrical ideas are introduced to simplify and improve the original proof when the number of degrees of freedom is large the hausdorff dimension is nearly maximal so that although of measure 0 the exceptional sets are roughly speaking close to sets of positive lebesgue measure the implications for the stability of hamiltonian systems with many degrees of freedom and for the onset of certain kinds of instability are discussed briefly

the static displacement and the stress analysis of structures with bounded uncertainties using the vertex solution theorem abstract in practical engineering on account of physical imperfections model inaccuracies and system complexities almost all structures have physical and geometrical uncertainties in some degrees in this paper we focus on the static response problem and the stress distribution of structures with bounded uncertainties in terms of the vertex notation in interval analysis a new mathematical proof of the vertex solution theorem which is used to determine the supremum and the infimum of the static responses of structures with bounded uncertainties is given then we extend this theorem for calculating the interval stress and the deformation distributions which was only used to determine the supremum and the infimum of the static responses of structures with bounded uncertainties the basic idea of the vertex solution theorem is to convert the interval linear equations into systems of equivalent deterministic linear equations the resulting deterministic linear equations are then solved by using the familiar techniques like the gauss elimination method or the gauss seidel iteration method another advantage of this method is that the increase in number of the uncertain parameters in a structure element will not result in the increase in number of the uncertain elements in the stiffness matrix for practical engineering problem with bounded uncertainties the chief thing is to avoid the disadvantages of large calculation and much runtime of the vertex solution in this paper we present the parallel arithmetic of the vertex solution theorem which can be used in large scale computations in practical engineering to avoid much runtime on the other hand we can use the symmetrical characteristic of the stiffness matrix and the substructure method to reduce the computational effort in practical engineering the numerical results of the typical example presented by deif calculated by the proposed vertex solution theorem show that the width between the supremum and the infimum on the static response yielded by the vertex solution theorem is tighter than those produced by the kcn s method presented by koyluoglu cakmak and nielson the numerical examples of the two dimensional shell with an edge fixed and the two dimensional shell with surrounding fixed are used to illustrate the computational feasibility of the vertex solution theorem

bounding s t and s_1 t on the riemann hypothesis let pi s t denote the argument of the riemann zeta function zeta s at the point s frac 1 2 it assuming the riemann hypothesis we present two proofs of the bound begin aligned s t le left frac 1 4 o 1 right frac log t log log t end aligned for large t this improves a result of goldston and gonek by a factor of 2 the first method consists of bounding the auxiliary function s 1 t int 0 t s u text d u using extremal functions constructed by carneiro littmann and vaaler we then relate the size of s t to the size of the functions s 1 t pm h s 1 t when h asymp 1 log log t the alternative approach bounds s t directly relying on the solution of the beurling selberg extremal problem for the odd function f x arctan left frac 1 x right frac x 1 x 2 this draws upon recent work by carneiro and littmann

hilbert spaces and the pair correlation of zeros of the riemann zeta function montgomery s pair correlation conjecture predicts the asymptotic behavior of the function n t defined to be the number of pairs and of ordinates of nontrivial zeros of the riemann zetafunction satisfying 0 0 using montgomery s formula and some extremal functions of exponential type these functions are optimal in the sense that they majorize and minorize the characteristic function of the interval in a way to minimize the l1 r 1 sin x x 2 dx error we give a complete solution for this extremal problem using the framework of reproducing kernel hilbert spaces of entire functions this extends previous work of p x gallagher 18 in 1985 where the case 1 2 n was considered using non extremal majorants and minorants

invariant measures and the set of exceptions to littlewood s conjecture we classify the measures on sl k r sl k z which are invariant and ergodic under the action of the group a of positive diagonal matrices with positive entropy we apply this to prove that the set of exceptions to littlewood s conjecture has hausdorff dimension zero

badly approximable points on manifolds this paper is motivated by two problems in the theory of diophantine approximation namely davenport s problem regarding badly approximable points on submanifolds of a euclidean space and schmidt s problem regarding the intersections of the sets of weighted badly approximable points the problems have been recently settled in dimension two but remain open in higher dimensions in this paper we develop new techniques that allow us to tackle them in full generality the techniques rest on lattice points counting and a powerful quantitative result of bernik kleinbock and margulis the main theorem of this paper implies that any finite intersection of the sets of weighted badly approximable points on any analytic nondegenerate submanifold of mathbb r n has full dimension one of the consequences of this result is the existence of transcendental real numbers badly approximable by algebraic numbers of any bounded degree

pair correlation estimates for the zeros of the zeta function via semidefinite programming abstract in this paper we study the distribution of the non trivial zeros of the riemann zeta function s and other l functions using montgomery s pair correlation approach we use semidefinite programming to improve upon numerous asymptotic bounds in the theory of s including the proportion of distinct zeros counts of small gaps between zeros and sums involving multiplicities of zeros

unified polynomial expansion for interval and random response analysis of uncertain structure acoustic system with arbitrary probability distribution abstract for structure acousticsystem with uncertainties the interval model the random model and the hybrid uncertain model have been introduced in the interval model and the random model the uncertain parameters are described as either the random variable with well defined probability density function pdf or the interval variable without any probability information whereas in the hybrid uncertain model both interval variable and random variable exist simultaneously for response analysis of these three uncertain models of structure acoustic problem involving arbitrary pdfs a unified polynomial expansion method named as the interval and random arbitrary polynomial chaos method irapcm is proposed in irapcm the response of the structure acoustic system is approximated by apc expansion in a unified form particularly only the weight function of polynomial basis is required to be changed to construct the apc expansion for the response of different uncertain models through the unified apc expansion the uncertain properties of the response of three uncertain models can be efficiently obtained as the apc expansion can provide a free choice of the polynomial basis the optimal polynomial basis for the random variable with arbitrary pdfs can be obtained by using the proposed irapcm the irapcm has been employed to solve a mathematical problem and a structure acoustic problem and the effectiveness of the unified irapcm for response analysis of three uncertain models is demonstrated by fully comparing it with the hybrid first order perturbation method and several existing polynomial chaos methods

extremal functions in de branges and euclidean spaces abstract in this work we obtain optimal majorants and minorants of exponential type for a wide class of radial functions on r n these extremal functions minimize the l 1 r n x 2 2 n d x distance to the original function where 1 is a free parameter to achieve this result we develop new interpolation tools to solve an associated extremal problem for the exponential function f x e x where 0 in the general framework of de branges spaces of entire functions we then specialize the construction to a particular family of homogeneous de branges spaces to approach the multidimensional euclidean case finally we extend the result from the exponential function to a class of subordinated radial functions via integration on the parameter 0 against suitable measures applications of the results presented here include multidimensional versions of hilbert type inequalities extremal one sided approximations by trigonometric polynomials for a class of even periodic functions and extremal one sided approximations by polynomials for a class of functions on the sphere s n 1 with an axis of symmetry

on a problem in simultaneous diophantine approximation schmidt s conjecture for any i j 0 with i j 1 let bad i j denote the set of points x y 2 r 2 for which maxfkqxk 1 i kqyk 1 j g c q for all q 2 n here c c x y is a positive constant our main result implies that any nite intersection of such sets has full dimension this settles a conjecture of wolfgang m schmidt in the theory of simultaneous diophantine approximation

diagonal elements and eigenvalues of a real symmetric matrix abstract a simple and constructive proof is given for the existence of a real symmetric matrix with prescribed diagonal elements and eigenvalues numerically implementable algorithms for constructing such a matrix are discussed

gaussian subordination for the beurling selberg extremal problem we determine extremal entire functions for the problem of ma jorizing minorizing and approximating the gaussian function e x 2 by en tire functions of exponential type the combination of the gaussian and a general distribution approach provides the solution of the extremal problem for a wide class of even functions that includes most of the previously known examples for instance 3 4 10 and 17 plus a variety of new interesting functions such asjxj for 1 log x 2 2 x 2 2 for 0 log x2 2 andx2n logx2 forn2 n further applications to number theory include optimal approximations of theta functions by trigonometric polynomi als and optimal bounds for certain hilbert type inequalities related to the discrete hardy littlewood sobolev inequality in dimension one

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

a class of extremal functions for the fourier transform we determine a class of real valued integrable functions f x and corresponding functions m x such that f x 1 and the value of mfo is miimi several applications of these functions to number theory and analysis are given

extremal problems in de branges spaces the case of truncated and odd functions in this paper we find extremal one sided approximations of exponential type for a class of truncated and odd functions with a certain exponential subordination these approximations optimize the l 1 mathbb r e x 2 mathrm d x error where e is an arbitrary hermite biehler entire function of bounded type in the upper half plane this extends the work of holt and vaaler duke math j 83 203 247 1996 for the signum function we also provide periodic analogues of these results finding optimal one sided approximations by trigonometric polynomials of a given degree to a class of periodic functions with exponential subordination these extremal trigonometric polynomials optimize the l 1 mathbb r mathbb z mathrm d vartheta error where vartheta is an arbitrary nontrivial measure on mathbb r mathbb z the periodic results extend the work of li and vaaler indiana univ math j 48 1 183 236 1999 who considered this problem for the sawtooth function with respect to jacobi measures our techniques are based on the theory of reproducing kernel hilbert spaces of entire functions and of polynomials and on the construction of suitable interpolations with nodes at the zeros of laguerre polya functions

reliability analysis for multidisciplinary systems with random and interval variables tremendous efforts have been devoted to developing efficient approaches to reliability analysis for multidisciplinary systems most of the approaches are only capable of dealing with random variables modeled by probability distributions both random and interval variables however may exist in multidisciplinary systems their propagation through coupled subsystems make s reliability analysis computationally expensive in this work a unified reliability analysis framework is proposed to deal with both random and interval variables in multidisciplinary systems the framework is an extension of an existing unified uncertainty analysis framework for single disciplinary problems the new framework involves probabilistic analysis and interval analysis both probabilistic analysis and interval analysis are decoupled from each other and are performed sequentially the first order reliability method is used for probabilistic analysis three supporting algorithms are developed the effectiveness of the algorithms is demonstrated with a mathematical example and an engineering application

reliability analysis of stiffened aircraft panels using adjusting mean value method with regard to real world engineering structures an adjusting mean value called the self adjusting mean value smv is proposed for improving the efficiency and robustness of a maximum performance

perfectly ordered quasicrystals and the littlewood conjecture linearly repetitive cut and project sets are mathematical models for perfectly ordered quasicrystals in a previous paper we presented a characterization of linearly repetitive cut and project sets in this paper we extend the classical definition of linear repetitivity to try to discover whether or not there is a natural class of cut and project sets which are models for quasicrystals which are better than perfectly ordered in the positive direction we demonstrate an uncountable collection of such sets in fact a collection with large hausdorff dimension for every choice of dimension of the physical space on the other hand we show that for many natural versions of the problems under consideration the existence of these sets turns out to be equivalent to the negation of a well known open problem in diophantine approximation the littlewood conjecture

the ruelle sullivan map for actions of ℝ n the ruelle sullivan map for an r n action on a compact metric space with invariant probability measure is a graded homomorphism between the integer cech cohomology of the space and the exterior algebra of the dual of r n we investigate flows on tori to illuminate that it detects geometrical structure of the system for actions arising from delone sets of finite local complexity the existence of canonical transversals and a formulation in terms of pattern equi variant functions lead to the result that the ruelle sullivan map is even a ring homomorphism provided the measure is ergodic

a note on the zeros of zeta and l functions let pi s t denote the argument of the riemann zeta function at the point s tfrac 1 2 it assuming the riemann hypothesis we give a new and simple proof of the sharpest known bound for s t we discuss a generalization of this bound for a large class of l functions including those which arise from cuspidal automorphic representations of gl m over a number field we also prove a number of related results including bounding the order of vanishing of an l function at the central point and bounding the height of the lowest zero of an l function

universal grey system theory for analysis of uncertain structural systems the uncertainty present in many structural systems is commonly modeled using probabilistic fuzzy or interval approaches depending on the nature of uncertainty present the probabilistic approach

bounded interpolations between lattices it is shown that given two arbitrary lattices of equal density in the euclidean space rn a bounded quasi periodic and piecewise affine vector field nu on rn a so called modulation field can be built so that the second lattice is the image of the first one under the map x to x v x the proof relies on a factorization lemma for matrices with determinant equal to one each factor represents a shear like transformation of rn which in turn is closely approximated by a periodic set of slips in the lattice

fuzzy finite element approach for the analysis of imprecisely defined systems many engineering systems are too complex to be defined in precise mathematical terms they often contain information and features that are vague imprecise qualitative linguistic or incomplete the traditional deterministic and probabilistic techniques are not adequate to analyze such systems this paper aims at developing a fuzzy finite element approach for the analysis of imprecisely defined systems the development of the methodology starts from the basic concepts of fuzzy numbers of fuzzy arithmetic and implements suitably defined fuzzy calculus concepts such as differentiation and integration for the derivation manipulation and solution of the finite element equations simple stress analysis problems involving vaguely defined geometry material properties external loads and boundary conditions are solved to establish and to illustrate the new procedure the approach developed is applicable to systems that are described in linguistic terms as well as those that are described by incomplete information if complete data are known the method handles the information similar to that of a probabilistic approach the present approach represents a unique methodology that enables us to handle certain types of imprecisely known data more realistically compared with the existing procedures

some extremal functions in fourier analysis ii we obtain extremal majorants and minorants of exponential type for a class of even functions on which includes log x and x where 1 1 we also give periodic versions of these results in which the majorants and minorants are trigonometric polynomials of bounded degree as applications we obtain optimal estimates for certain hermitian forms which include discrete analogues of the one dimensional hardy littlewood sobolev inequalities a further application provides an erdos turan type inequality that estimates the sup norm of algebraic polynomials on the unit disc in terms of power sums in the roots of the polynomials

a simple condition for bounded displacement abstract we study separated nets y that arise from primitive substitution tilings of r d we show that the question whether y is a bounded displacement of z d or not can be reduced in many cases to a simple question on the eigenvalues and eigenspaces of the substitution matrix

uncertainty based structural optimization using universal grey number theory several types of uncertainties are encountered in the design of machines and structures the geometric parameters or dimensions are specified in terms of nominal values and tolerances the experime

melnikov s method and arnold diffusion for perturbations of integrable hamiltonian systems we start with an unperturbed system containing a homoclinic orbit and at least two families of periodic orbits associated with action angle coordinates we use kolmogorov arnold moser kam theory to show that some of the resulting tori persist under small perturbations and use a vector of melnikov integrals to show that under suitable hypotheses their stable and unstable manifolds intersect transversely this transverse intersection is ultimately responsible for arnold diffusion on each energy surface the method is applied to a pendulum oscillator system

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

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