a new reiterated structure with optimal macroscopic behavior we consider homogenization of sequences of integral functionals of the type mathcal f h u int omega f h x du x dx omega subset mathbf r n the functions fh are defined in the first variable via a new reiterated structure with m microlevels in particular our results imply that the corresponding homogenized functional becomes optimal as m turns to infinity both the scalar case the conductivity problem and the vector valued case the elasticity problem are considered

singular continuous spectrum for palindromic schrodinger operators we give new examples of discrete schrodinger operators with potentials taking finitely many values that have purely singular continuous spectrum if the hullx of the potential is strictly ergodic then the existence of just one potentialx inx for which the operator has no eigenvalues implies that there is a generic set inx for which the operator has purely singular continuous spectrum a sufficient condition for the existence of such anx is that there is az x that contains arbitrarily long palindromes thus we can define a large class of primitive substitutions for which the operators are purely singularly continuous for a generic subset inx the class includes well known substitutions like fibonacci thue morse period doubling binary non pisot and ternary non pisot we also show that the operator has no absolutely continuous spectrum for allx x ifx derives from a primitive substitution for potentials defined by circle maps x n 1 j 0 n we show that the operator has purely singular continuous spectrum for a generic subset inx for all irrational and every half open intervalj

on the numerical evaluation of electrostatic fields in dense random dispersions of cylinders we consider the interface boundary value problem which arises in the evaluation of electrostatic fields in composite materials consisting of dense random dispersions of cylinders in a uniform background this is a well studied problem from the viewpoint of homogenization and effective medium theory but one for which accurate numerical simulations have been difficult to obtain size effects in particular have been neglected due to the expense of solving the field equation in the presence of large numbers of close to touching inclusions such features require very fine discretizations even with the use of adaptive gridding and cause the linear systems which arise to be highly ill conditioned in this paper we present a new integral equation method for the solution of the interface problem which uses a recently developed method of images to resolve the close to touching interactions and the fast multipole method to compute far field interactions only minutes of workstation time are needed to solve the field equation with thousands of inclusions allowing us to carry out large scale statistical studies of the effective conductivity of random two phase materials at a variety of volume fractions and contrast ratios

bounds for effective parameters of multicomponent media by analytic continuation recently d bergman introduced a method for obtaining bounds for the effective dielectric constant or conductivity of a two component medium this method does not rely on a variational principle but instead exploits the properties of the effective parameter as an analytic function of the ratio of the component parameters we extend the method to multicomponent media using techniques of several complex variables

invariant properties of the stress in plane elasticity and equivalence classes of composites attention is drawn to the invariance of the stress field in a two dimensional body loaded at the boundary by fixed forces when the compliance tensor scr g chi is shifted uniformly by ell text i lambda lambda where lambda is an arbitrary constant and scr g text i kappa mu is the compliance tensor of a isotropic material with two dimensional bulk and shear moduli kappa and mu this invariance is explained from two simple observations first that in two dimensions the tensor scr g text i frac 1 2 frac 1 2 acts to locally rotate the stress by 90 degrees and the second that this rotated field is the symmetrized gradient of a vector field and therefore can be treated as a strain for composite materials the invariance of the stress field implies that the effective compliance tensor ell ast also gets shifted by scr g text i lambda lambda when the constituent moduli are each shifted by ell text i lambda lambda this imposes constraints on the functional dependence of ell ast on the material moduli of the components applied to an isotropic composite of two isotropic components it implies that when the inverse bulk modulus is shifted by the constant 1 lambda and the inverse shear modulus is shifted by 1 lambda then the inverse effective bulk and shear moduli undergo precisely the same shifts in particular it explains why the effective young s modulus of a two dimensional media with holes does not depend on the poisson s ratio of the matrix material

eigenfunctions transfer matrices and absolutely continuous spectrum of one dimensional schrodinger operators on l r and its half line problem h on l2 0 with u 0 0 boundary conditions we will focus on a new approach to the absolutely continuous spectrum ac h and more generally ac h the essential support of the a c part of the spectral measures what is new in our approach is that it relies on estimates on the transfer matrix that is the 2 2 matrix te n m which takes u m 1 u m to u n 1 u n for solutions u of hu eu in the continuum case use u x u x instead of u x 1 u x we let te n te n 0 for example we will prove the following

note a note on the hardy hilbert inequality abstract in this note it is shown that the hardy hilbert inequality for double series can be improved by introducing a proper weight function of the form sin p o r n n 1 1 r with o r n 0 into either of the two single summations when r 2 the classical hilbert inequality is improved

dynamical localization for the random dimer schrodinger operator we study the one dimensional random dimer model with hamiltonian h v where for all x mathbb z v 2x v 2x 1 and where the v 2x are i i d bernoulli random variables taking the values v v 0 we show that for all values of vand with probability one in the spectrum of his pure point if v 1 and v 1 sqrt 2 the lyapunov exponent vanishes only at the two critical energies given by e v for the particular value v 1 sqrt 2 respectively v sqrt 2 we show the existence of new additional critical energies at e 3 sqrt 2 respectively e 0 on any compact interval inot containing the critical energies the eigenfunctions are then shown to be semi uniformly exponentially localized and this implies dynamical localization for all q 0 and for all ell 2 mathbb z with sufficiently rapid decrease mathop sup limits t r psi i left q right kern 1pt left t right mathop sup limits t left langle p i left h omega right psi t left x right q p i left h omega right psi t right rangle infty here psi t e ih omega t psi and pi h is the spectral projector of h onto the interval i in particular if v 1 and v sqrt 2 these results hold on the entire spectrum so that one can take i h

spectra and transport in almost periodic dimers we study spectral properties of discrete schrodinger operators with potentials obtained via dimerization of a class of aperiodic sequences it is shown that both the nature of the autocorrelation measure of a regular sequence and the presence of generic full probability singular continuous spectrum in the hull of primitive and palindromic four block substitution potentials are robust under dimerization generic results also hold for circle potentials we illustrate these results with numerical studies of the quantum mean square displacement as a function of time the numerical techniques provide a very fast algorithm for the time evolution of wave packets

some remarks on discrete aperiodic schrodinger operators we consider schrodinger operators onl2 with deterministic aperiodic potential and schrodinger operators on the l2 space of the set of vertices of penrose tilings and other aperiodic self similar tilings the operators onl2 fit into the formalism of ergodic random schrodinger operators hence their lyapunov exponent integrated density of states and spectrum are almost surely constant we show that they are actually constant the lyapunov exponent for one dimensional schrodinger operators with potential defined by a primitive substitution the integrated density of states and the spectrum in arbitrary dimension if the system is strictly ergodic we give examples of strictly ergodic schrodinger operators that include several kinds of almost periodic operators that have been studied in the literature for schrodinger operators on penrose tilings we prove that the integrated density of states exists and is independent of boundary conditions and the particular penrose tiling under consideration

integral equation methods for elliptic problems with boundary conditions of mixed type laplace s equation with mixed boundary conditions that is dirichlet conditions on parts of the boundary and neumann conditions on the remaining contiguous parts is solved on an interior planar domain using an integral equation method rapid execution and high accuracy is obtained by combining equations which are of fredholm s second kind with compact operators on almost the entire boundary with a recursive compressed inverse preconditioning technique then an elastic problem with mixed boundary conditions is formulated and solved in an analogous manner and with similar results this opens up for the rapid and accurate solution of several elliptic problems of mixed type

proof of a conjecture on the conductivity of checkerboards in 1985 mortola and steffe conjectured a formula for the effective conductivity tensor of a checkerboard structure where the unit cell of periodicity is square and subdivided into four equal squares each having a different conductivity in this article their conjecture is proven the key idea is to superimpose suitably reflected potentials to obtain the solution to the dual problem this is then related back to the original problem using a well known theorem of keller thereby proving the conjecture the analysis also yields formulas relating the potentials in the four squares independently craster and obnosov have obtained a completely different proof of the conjecture

fast finite element calculation of effective conductivity of random continuum microstructures the recursive poincare steklov operator method abstract the effective conductivity of a block of composite can be extracted from the dirichlet to neumann poincare steklov operator pso for that block in this paper a domain decomposition method for computing the pso over a finite element mesh is discussed a new numerical strategy is introduced to accelerate the computation of this operator using the schur complement to calculate the pso for the smallest subdomains then recursively merging subdomain psos up to the full domain at each step of the algorithm information extraneous to the pso is discarded the effective conductivity values computed by this method are identical to those obtained from a basic finite element method an order of magnitude faster and with much less computer memory consumed as proof of concept effective conductivity measurements are presented for a percolating random fractal like microstructure across a range of phase fractions

quantum return probability for substitution potentials we propose an effective exponent ruling the algebraic decay of the average quantum return probability for discrete schrodinger operators we compute it for some non periodic substitution potentials with different degrees of randomness and do not find a complete qualitative agreement with the spectral type of the substitution sequences themselves i e the more random the sequence the smaller such an exponent

necessary and sufficient conditions for the two point phase probability function of two phase random media constructing realizations of random media with a specified two point phase probability function s2 has attracted considerable attention in the recent literature however little is known about cond

on effective resistivity and related parameters for periodic checkerboard composites an approach for finding the effective properties of two dimensional periodic rectangular checkerboard media is presented it is demonstrated that one can take a relatively simple model problem an appropriate system of joined quarter planes and that after appropriate conformal transformation one can use this to solve for either periodic or doubly periodic structures illustrative calculations involving three phase media are presented that in limiting cases recover earlier analyses useful asymptotic relations are also given

relation between effective conductivity and susceptibility of two component rhombic checkerboard the heterogeneity of composite leads to extra charge concentration at the boundaries of different phases which results in essentially nonzero effective electric susceptibility the relation between tensors of effective electric susceptibility ef and effective conductivity ef of the infinite two dimensional two component regular composite with rhombic cell structure has been established the degrees of electric field singularity at corner points of cells are found by constructing the integral equation for the effective conductivity problem the limits of weak and strong contrast of partial conductivities 1 2 are considered the results are valid for thin films and cylindrical samples

a generalization of van der corput s inequality in this article van der corput s inequality is generalized by using the well known euler maclaurin sum formula and other analytic techniques

on hardy s inequality abstract in this paper some new inequalities on the constant e are deduced as an application we build a strengthened hardy s inequality

electrostatics and heat conduction in high contrast composite materials we present a robust integral equation method for the calculation of the electrostatic and thermal properties of systems made of piecewise homogeneous high contrast materials by high contrast we mean that the electrical or thermal conductivity ratios are high our approach involves a modification of the standard integral representation using in electrostatic terms the polarization charge as an unknown it is related to the perturbation approach proposed by tausch and white j tausch j white capacitance extraction of 3 d conductor systems in dielectric media with high permittivity ratios ieee trans microwave theory tech 47 1999 18 26 but based on a simple form of projection rather than on the solution of auxiliary integral equations numerical implementation is straightforward in the multiple dielectric case

the effective conductivity of random checkerboards an algorithm is presented for the fast and accurate solution of the electrostatic equation on multi component random checkerboards it relies on a particular choice of integral equation extended as to separate ill conditioning due to singular fields in corners from ill conditioning due to interaction of clusters of well conducting squares at large distances two separate preconditioners take care of the two separate phenomena in a series of numerical examples effective conductivities are computed for random checkerboards containing up to 104 squares with conductivity ratios of up to 106 the achievable relative precision in these examples is on the order of 10 11

periodic heterogeneous structures new explicit solutions and effective characteristics of refraction of an imposed field complex variable methods are applied to study three double periodic two phase planar heterogeneous structures explicit expressions of functionals of energy dissipation and of effective resistivities conductivities are found on the basis of strict analytical solutions of a boundary value problem for an arbitrary direction of an imposed external field comparisons with the results of bakhvalov and panasenko homogenization of processes in periodic media nauka moscow 1984 dykhne zh eksp teor fiz 59 1970 pp 110 113 soviet phys jetp 32 1971 pp 63 65 emets zh eksp teor fiz 96 1989 pp 701 711 soviet phys jetp 96 1989 keller j appl phys 34 1963 pp 991 993 and mendelson j appl phys 46 1975 pp 4740 4741 are made in particular the dissipation in the general case is shown essentially to depend on the direction of the external field for a rectangular checkerboard dissipation within the two phases does not coincide it is rigorously pr

note on the carleman s inequality and hardy s inequality in this article using the properties of power mean and induction new strengthened carleman s inequality and hardy s inequality are obtained we also give an answer to the conjectures proposed by x yang in the literature yang 2001 5

checkerboard composites with separated phases recently the authors have developed a method siam j appl math 61 1839 1556 2001 capable of solving in closed form boundary value problems for four phase doubly periodic checkerboard composites with continuity between the different phases the method is based upon a novel conformal mapping that preserves the doubly periodic nature of the physical problem the aim of the current article is to explore generalizations of that approach where we now replace continuity between some phases by nonconducting or perfectly conducting strips thereby modeling debonding or electrodes in electrochemical devices the specific objective is to determine effective resistivities and related parameters for these four phase objects in a concise and explicit form

perturbation theory for the random dimer model the authors study a one dimensional tight binding model with a random potential taking two values or nu with the restriction that on pairs of two neighbouring sites the potentials take the same value this random dimer model proposed by dunlap phillips and wu 1990 has a vanishing lyapunov exponent for the energy e or nu they compute the lyapunov exponent and the density of states perturbatively in the vicinity of this energy using the invariant measure formalism

critical energies in random palindrome models we investigate the occurrence of critical energies where the lyapunov exponent vanishes in random schrodinger operators when the potentials have some local order which we call random palindrome models we give necessary and sufficient conditions for the presence of such critical energies the commutativity of finite word elliptic transfer matrices finally we perform some numerical calculations of the lyapunov exponents showing their behavior near the critical energies and the respective time evolution of an initially localized wave packet obtaining the exponent ruling the algebraic growth of the second momentum we also consider special random palindrome models with one letter bounded gap property the transport effects of such long range order are showed numerically

regular hexagonal three phase checkerboard abstract a two dimensional doubly periodic three phase hexagonal structure is considered the flow in the structure is generated by three sets of vortexes sinks sources which are the same in each phase and are located in the centers of the hexagons complex analysis methods are utilized to reduce the doubly periodic r linear conjugation problem to the simpler one riemann hilbert rh problem on a three sheeted riemann surface in turn the latter problem is reduced to a rh problem involving three joined sectors on the plane which was previously investigated in 3 the limiting cases with one non conducting phase and two phases of the same conductivities are investigated all solutions derived are verified both numerically and analytically examples of relevant flow networks streamlines and equipotentials are plotted in the whole structure and separately in each phase

corner singularities for elliptic problems integral equations graded meshes quadrature and compressed inverse preconditioning we take a fairly comprehensive approach to the problem of solving elliptic partial differential equations numerically using integral equation methods on domains where the boundary has a large number of corners and branching points use of non standard integral equations graded meshes interpolatory quadrature and compressed inverse preconditioning are techniques that are explored developed mixed and tested on some familiar problems in materials science the recursive compressed inverse preconditioning the major novelty of the paper turns out to be particularly powerful and when it applies eliminates the need for mesh grading completely in an electrostatic example for a multiphase granular material with about two thousand corners and triple junctions and a conductivity ratio between phases up to a million we compute a common functional of the solution with an estimated relative error of 10 1 2 in another example five times as large but with a conductivity ratio of only a hundred we achieve an estimated relative error of 10 1 4

on the evaluation of layer potentials close to their sources when solving elliptic boundary value problems using integral equation methods one may need to evaluate potentials represented by a convolution of discretized layer density sources against a kernel standard quadrature accelerated with a fast hierarchical method for potential field evaluation gives accurate results far away from the sources close to the sources this is not so cancellation and nearly singular kernels may cause serious degradation this paper presents a new scheme based on a mix of composite polynomial quadrature layer density interpolation kernel approximation rational quadrature high polynomial order corrected interpolation and differentiation temporary panel mergers and splits and a particular implementation of the gmres solver criteria for which mix is fastest and most accurate in various situations are also supplied the paper focuses on the solution of the dirichlet problem for laplace s equation in the plane in a series of examples we demonstrate the efficiency of the new scheme for interior domains and domains exterior to up to 2000 close to touching contours densities are computed and potentials are evaluated rapidly and accurate to almost machine precision at points that lie arbitrarily close to the boundaries

singular continuous spectrum on a cantor set of zero lebesgue measure for the fibonacci hamiltonian it is rigorously proven that the spectrum of the tight binding fibonacci hamiltonian h mn m n 1 m n 1 m n n 1 n where 5 1 2 and means integer part is a cantor set of zero lebesgue measure for all real nonzero and the spectral measures are purely singular continuous this follows from a recent result by kotani coupled with the vanishing of the lyapunov exponent in the spectrum

anderson localization for bernoulli and other singular potentials we prove exponential localization in the anderson model under very weak assumptions on the potential distribution in one dimension we allow any measure which is not concentrated on a single point and possesses some finite moment in particular this solves the longstanding problem of localization for bernoulli potentials i e potentials that take only two values in dimensions greater than one we prove localization at high disorder for potentials with holder continuous distributions and for bounded potentials whose distribution is a convex combination of a holder continuous distribution with high disorder and an arbitrary distribution these include potentials with singular distributions we also show that for certain bernoulli potentials in one dimension the integrated density of states has a nontrivial singular component

rapid solution of integral equations of classical potential theory an algorithm is described for rapid solution of classical boundary value problems dirichlet an neumann for the laplace equation based on iteratively solving integral equations of potential theory cpu time requirements for previously published algorithms of this type are proportional to n2 where n is the number of nodes in the discretization of the boundary of the region the cpu time requirements for the algorithm of the present paper are proportional to n making it considerably more practical for large scale problems

hybrid quasicrystals transport and localization in products of minimal sets we consider convex combinations of finite valued almost periodic sequences mainly substitution sequences and put them as potentials of one dimensional tight binding models we prove that these sequences are almost periodic we call such combinations hybrid quasicrystals and these studies are related to the minimality under the shift on both coordinates of the product space of the respective minimal hulls we observe a rich variety of behaviors on the quantum dynamical transport ranging from localization to transport

on deriving the inverse of a sum of matrices available expressions are reviewed and new ones derived for the inverse of the sum of two matrices one of them being nonsingular particular attention is given to text bf a text bf ubv 1 where textbf a is nonsingular and textbf u b and textbf v may be rectangular generalized inverses of textbf a textbf ubv are also considered several statistical applications are discussed

four phase checkerboard composites two dimensional periodic rectangular checkerboard media are considered in the situation where mean fluxes are prescribed across the structure the closed form solution is obtained in the general case where the checkerboard is constructed using four rectangular cells each having a different constant resistivity this four cell structure then repeats doubly periodically to cover the whole plane this general solution is then used to calculate the effective properties thus this four phase checkerboard encapsulates many limiting and special cases as a starting point we develop a concise closed form solution to a basic problem involving four joined quarter planes each of a different resistivity subsequent manipulations yield solutions to problems posed in increasingly convoluted domains while retaining the essentially simple structure found for joined quarter planes

a high order accurate algorithm for electrostatics of overlapping disks a fast and accurate algorithm for the computation of effective electric and mathematically equivalent properties of composites with nonsmooth interfaces is reported the algorithm is based on an integral equation reformulation of the electrostatic partial differential equation and a fast hierarchical technique for potential field evaluation in a numerical example 200 large and strongly inhomogeneous aggregates of randomly overlapping disks are solved with a relative error of 0 0005

exact solution of a boundary value problem for a rectangular checkerboard field a class of two phase composite materials with a biperiodic structure is investigated by the methods of complex analysis two interface conditions continuity of normal component of a desired vector w and tangential component of w at the contact boundary as well as the double periodicity condition are involved in rigorous form the exact analytic solution of the corresponding generalized riemann boundary value problem is obtained the explicit values of the effective parameters namely effective resistivity and dissipation of energy of an elementary cell and resistivities along the symmetry axes are calculated in closed analytic form the coincidence of our formulae with the well known effective resistivity conductivity formula of keller 1964 dykhne 1970 and mendelson 1975 and the dissipation formula of dykhne 1970 is shown in the case of square checkerboard field the keller 1963 identity is generalized for the heterogeneous structure studied

a three phase tessellation solution and effective properties two dimensional doubly periodic three phase structures are considered in the situation where mean fluxes are applied across the structure the approach is to utilize complex variables and to use a mapping that reduces the doubly periodic problem to a m uch simpler one involving joined sectors this is a model composite structure in electrostatics and mathematically analogous areas such as porous media anti plane elasticit y heat conduction and we find various eectiv e parameters and investigate limiting cases the structure is also amenable to asymptotic methods in the case of highly varying composition and we provide these solutions partly as a check upon our analysis and partly as they are useful in their own right

conduction in a two phase plane with diamond shaped tiling the effective conductivity of a two phase two dimensional composite with diamond shaped tiling is considered this analysis based on a projection of the boundary conditions on linear combinations of solutions to the electrostatic equation that are orthonormal on the boundary generalizes results by keller j math phys 28 2516 1987 and others numerical results are given for several conductivity ratios of the two phases and for varying obtuse angles of the tiles special emphasis is given to very large and very small conductivity differences and very elongated tiles

the effective conductivity of arrays of squares large random unit cells and extreme contrast ratios an integral equation based scheme is presented for the fast and accurate computation of effective conductivities of two component checkerboard like composites with complicated unit cells at very high contrast ratios the scheme extends recent work on multi component checkerboards at medium contrast ratios general improvement include the simplification of a long range preconditioner the use of a banded solver and a more efficient placement of quadrature points this together with a reduction in the number of unknowns allows for a substantial increase in achievable accuracy as well as in tractable system size results accurate to at least nine digits are obtained for random checkerboards with over a million squares in the unit cell at contrast ratio 106 furthermore the scheme is flexible enough to handle complex valued conductivities and using a homotopy method purely negative contrast ratios examples of the accurate computation of resonant spectra are given

operators with singular continuous spectrum i general operators the baire category theorem implies that the family f of dense sets g in fixed metric space x is a candidate for generic sets since it is closed under countable intersections and if x is perfect has no isolated point then a f has uncountable intersections with any open ball in x

efficient discretization of laplace boundary integral equations on polygonal domains we describe a numerical procedure for the construction of quadrature formulae suitable for the efficient discretization of boundary integral equations over very general curve segments while the procedure has applications to the solution of boundary value problems on a wide class of complicated domains we concentrate in this paper on a particularly simple case the rapid solution of boundary value problems for laplace s equation on two dimensional polygonal domains we view this work as the first step toward the efficient solution of boundary value problems on very general singular domains in both two and three dimensions the performance of the method is illustrated with several numerical examples

expansion formulae for the homogenized determinant of anisotropic checkerboards in this paper some effective properties of anisotropic four phase periodic checkerboards are studied in two dimensional electrostatics an explicit low contrast second order expansion for the determinant of the effective conductivity is given in the case of a twophase checkerboard with commuting conductivities the expansion reduces to an explicit formula for the effective determinant valid for any contrast as soon as the second order term vanishes such an explicit formula cannot be extended to four phase checkerboards a counter example with high contrast conductivities is provided the construction of the counter example is based on a factorization principle due to astala nesi which allows us to pass from an anisotropic four phase square checkerboard to an isotropic one with the same effective determinant

noether lefschetz space curves and mathematical instantons there are three areas of study of geometric entities living in the three dimensional projective space over the complex numbers which have proved to be strictly connected the classification of space curves surfaces and vector bundles despite all the production in these areas several key unsolved problems still challenge us after the work of many classical and modern authors see h1 h2 for a bibliography the problem of classifying space curves has been divided into two main parts 1 the so called halphen problem find all the triples of integers n g s such that there exists a smooth irreducible curve c ip 3 of degree n and genus g lying on a surface of degree s but not on any surface of lower degree 2 study the properties of the hilbert scheme hn g of smooth irreducible curves in ip namely its irreducible components dimension singular points let g n s be the maximum genus of a curve of degree n not contained in a surface of degree less than s then part 1 above falls in turn into three ranges for n and s

a fast direct solver for the integral equations of scattering theory on planar curves with corners we describe an approach to the numerical solution of the integral equations of scattering theory on planar curves with corners it is rather comprehensive in that it applies to a wide variety of boundary value problems here we treat the neumann and dirichlet problems as well as the boundary value problem arising from acoustic scattering at the interface of two fluids it achieves high accuracy is applicable to large scale problems and perhaps most importantly does not require asymptotic estimates for solutions instead the singularities of solutions are resolved numerically the approach is efficient however only in the low and mid frequency regimes once the scatterer becomes more than several hundred wavelengths in size the performance of the algorithm of this paper deteriorates significantly we illustrate our method with several numerical experiments including the solution of a neumann problem for the helmholtz equation given on a domain with nearly 10000 corner points

absence of localization in a class of schrodinger operators with quasiperiodic potential we prove that a class of discrete schrodinger operators with a quasiperiodic potential taking only a finite number of values exhibits purely continuous spectrum in particular they cannot have localized eigenvectors

spectral properties of one dimensional schrodinger operators with potentials generated by substitutions we investigate one dimensional discrete schrodinger operators whose potentials are invariant under a substitution rule the spectral properties of these operators can be obtained from the analysis of a dynamical system called the trace map we give a careful derivation of these maps in the general case and exhibit some specific properties under an additional easily verifiable ypothesis concerning the structure of the trace map we present an analysis of their dynamical properties that allows us to prove that the spectrum of the underlying schrodinger operator is singular and supported on a set of zero lebesgue measure a condition allowing to exclude point spectrum is also given the application of our theorems is explained on a series of examples

approximation formulas for the constant e and an improvement to a carleman type inequality abstract we give an explicit formula for the determination of the coefficients c j appearing in the expansion x 1 j 1 q c j x j x 1 2 1 x e o 1 x q 1 for x and q n 1 2 we also derive a pair of recurrence relations for the determination of the constants l and l in the expansion 1 1 x x e 1 l 1 l x l 2 l 1 as x based on this expansion we establish an inequality for 1 1 x x as an application we give an improvement to a carleman type inequality

irreducibility of a subscheme of the hilbert scheme of complex space curves abstract denoting by j d g 3 the subscheme of the hubert scheme whose general point corresponds to smooth irreducible and nondegenerate curve of degree d and genus g in p 3 it is proved that j d g 3 is irreducible in the following cases i d g 3 ii d g 2 and g 5 iii d g 1 and g 11

convergence of iterative methods based on neumann series for composite materials theory and practice iterative fast fourier transform methods are useful for calculating the fields in composite materials and their macroscopic response by iterating back and forth until convergence the differential constraints are satisfied in fourier space and the constitutive law in real space the methods correspond to series expansions of appropriate operators and to series expansions for the effective tensor as a function of the component moduli it is shown that the singularity structure of this function can shed much light on the convergence properties of the iterative fast fourier transform methods we look at a model example of a square array of conducting square inclusions for which there is an exact formula for the effective conductivity obnosov theoretically some of the methods converge when the inclusions have zero or even negative conductivity however the numerics do not always confirm this extended range of convergence and show that accuracy is lost after relatively few iterations there is little point in iterating beyond this accuracy improves when the grid size is reduced showing that the discrepancy is linked to the discretization finally it is shown that none of the three iterative schemes investigated over performs the others for all possible microstructures and all contrasts

approximation for constant e and its applications abstract a new approximation for the constant e is obtained and its application to carleman s inequality and hardy s inequality are obtained the results of this paper generalize some early results on this topic

effective medium approximation and exact formulae for electrokinetic phenomena in porous media electrokinetic phenomena in porous media are studied by application of the effective medium theory and the theory of duality transformation we deduce new exact relations and analytical formulae for the effective constants of the macroscopic tensor we also prove that the effective tensors obtained by these approaches coincide for 2d problems the obtained results for the electrokinetic processes are closely related to similar results derived for piezoelectric composites because of a common mathematical background