an inverse problem for circle packing and conformal mapping figure 1 1 illustrates the fact that if a region is almost packed with circles of radius e in the hexagonal pattern and if the unit disk is packed in an isomorphic pattern with circles of varying radii then after suitable normalization the correspondence of circles converges to the riemann mapping function as e 0 see 15 in the present paper an inverse of this result is obtained as illustrated by figure 1 2 namely if the unit disk is almost packed with e circles there is an isomorphic circle packing almost filling the region such that after suitable normalization the circle correspondence converges to the conformal map of the disk onto the region as e 0

the split levinson algorithm the classical levinson algorithm for computing the predictor polynomial relative to a real positive definite toeplitz matrix is shown to be redundant in complexity it can be broken down into two simpler algorithms either of which needs only to be processed this result can be interpreted in the framework of the theory of orthogonal polynomials on the real line as follows the symmetric and antisymmetric parts of the predictors relative to the sequence of toeplitz matrices constitute two families of polynomials orthogonal on the interval 1 1 with respect to some even weight functions it turns out that the recurrence relations for these orthogonal polynomials can be used efficiently to compute the desired predictor the resulting split levinson algorithm requires roughly one half the number of multiplications and the same number of additions as the classical levinson algorithm a simple derivation of cybenko s method for computing the pisarenko frequencies is obtained from the recurrence relations underlying the split levinson algorithm

combined effect of pores concavity and aspect ratio on the elastic properties of a porous material the present paper focuses on materials containing superellipsoidal pores described by equation x 1 2 p x 2 2 p x 3 2 p 1 this shape is concave when 0 p p 0 5 they are oblate when 1 combined effect of two shape factors parameter p describing the concavity and aspect ratio describing extend of oblateness prolateness on compliance contribution tensor is analyzed numerically and approximated analytically for most important case of p 1 it is shown that in this case the combined effect is equal to the product of two effects considered separately the obtained approximate expressions are used to calculate the effective elastic properties of a heterogeneous material containing pores of superellipsoidal shape

variational principles for branched circle patterns abstract we are concerned with variational principles for branched circle patterns we use critical point theory to establish existence and uniqueness furthermore we apply the ricci flow method to give an approximation scheme which converges exponentially to the branched circle pattern metric

constructing conformal maps of triangulated surfaces abstract we describe a means of computing the uniformizing conformal map from a triangulated surface whose triangles are realized as euclidean triangles in r 3 onto a fundamental domain in the unit disc d plane c or sphere s 2 mapping such triangulated surfaces arises in a number of applications such as conformal brain flattening we use the circle packing technique of bowers hurdal stephenson et al to first create a quasiconformal approximation to the conformal map then we apply a discrete form of conformal welding to reduce the distortion and converge to the conformal map in the limit

is the concept of average shape legitimate for a mixture of inclusions of diverse shapes abstract in heterogeneous materials inhomogeneities pores cracks particles usually represent a mixture of diverse shapes in modeling the effective elastic and conductive properties they are often replaced by identical spheroidal shapes for which exact solutions are available that intend to represent certain average shape for cracks for example the circular shape is routinely used we find that in anisotropic cases of non random orientations such a replacement cannot generally be done with the exception of flat cracks in other words the concept of average spheroid is not legitimate and its use may lead to large errors in predicting the effective properties information on shape distribution is needed in cases of overall isotropy the replacement is possible in certain cases even in these cases however the aspect ratio of the average shape may not be intuitively obvious

a branched andreev thurston theorem for circle packings of the sphere andreev thurston theorem let r be a triangulation of s which is not simplicially equivalent to a tetrahedron and let r 0 n be given where t denotes the set of edges of x suppose that a if eu e2 e3 e r 1 form a closed loop on s and if 2 i e n then eif e2 e3 are the three edges of a two simplex of x b if ex e2 e3 ea e r 1 form a closed loop on s and if 2 i e 2k then ex e2 e2 e4 form the boundary of the union of two adjacent twosimplexes of x

a symmetric algorithm for toeplitz systems abstract we derive an algorithm for real symmetric toeplitz systems with an arbitrary right hand side it differs from the levinson algorithm in that the solution is built up from its middle component s outwards rather than from top to bottom we then exploit the symmetry of this method by solving separately for the even and odd parts of the right hand side of the system on a sequential machine the complexity of our algorithm for a system of order n is 7 2 n 2 o n flops compared to 4n 2 o n flops for levinson s algorithm the algorithm can be extended to nonsymmetric systems just like levinson s algorithm

exact degenerate scales in plane elasticity using complex variable methods a recent work has shown that using conformal mapping can lead to exact values of the degenerate scales in plane elasticity we elaborate on this work by introducing some algebraic tools when this conformal mapping is a rational fraction transforming the outside of the unit circle into the outside of the considered domain using these tools new cases are solved including shortened hypotrochoid arc of circle new approximates of equilateral triangle and square or symmetric joukowski profiles another method makes it possible to obtain the degenerate scales for plane elasticity from the degenerate scale for laplace s equation for some multiply connected sets the cases of segments on a line or of arcs of circle with a n fold symmetry in these last cases the exact values of the degenerate scales are obtained when the degenerate scale for the laplace problem is known

effective elastic properties of solids with defects of irregular shapes pores and defects in real materials often have very irregular shapes thus micromechanical modeling based on the analytical solutions of elasticity becomes inapplicable the objective of this paper is to present a computational procedure to calculate the contribution of the irregularly shaped defects into the effective moduli of two dimensional elastic solids in this procedure the cavity compliance tensor is constructed numerically for an individual defect and then used in the elastic potential based approach to predict the effective moduli of porous solids two computational methods are used in this paper to calculate the components of a cavity compliance tensor finite element analysis and numerical conformal mapping application of this procedure to the regular hole shapes produces results that are in good correspondence with analytical predictions

shear compliance of two dimensional pores possessing n fold axis of rotational symmetry we use the complex variable method and conformal mapping to derive a closed form expression for the shear compliance parameters of some two dimensional pores in an elastic material the pores have

comparison of full field and single pore approaches to homogenization of linearly elastic materials with pores of regular and irregular shapes abstract two approaches to predict the overall elastic properties of solids with regularly and irregularly shaped pores are compared the first approach involves direct finite element simulations of periodic representative volume elements containing arrangements of pores a simplified algorithm of collective rearrangement type is developed for generating microstructures with the desired density of randomly distributed pores of regular and irregular shapes homogeneity and isotropy where appropriate of the microstructures are confirmed by generating two point statistics functions the second approach utilizes mori tanaka and maxwell micromechanical models implemented via the cavity compliance contribution tensor h tensor formalism the effects of pore shape and matrix poisson s ratio on compliance contribution parameters of different shapes are discussed h tensors of cubical octahedral and tetrahedral pores for several values of matrix poisson s ratio are published in explicit form for the first time good correspondence between the direct finite element simulations and micromechanical homogenization is observed for randomly oriented and parallel pores of the same shape as well as mixtures of pores of various shapes up to 0 25 pore volume fractions

conformal mapping for the helmholtz equation acoustic wave scattering by a two dimensional inclusion with irregular shape in an ideal fluid the complex variables method with mapping function was extended to solve the linear acoustic wave scattering by an inclusion with sharp smooth corners in an infinite ideal fluid domain the improved solutions of helmholtz equation shown as bessel function with mapping function as the argument and fractional order bessel function were analytically obtained based on the mapping function the initial geometry as well as the original physical vector can be transformed into the corresponding expressions inside the mapping plane as all the physical vectors are calculated in the mapping plane this method can lead to potential vast savings of computational resources and memory in this work the results are validated against several published works in the literature the different geometries of the inclusion with sharp corners based on the proposed mapping functions for irregular polygons are studied and discussed the findings show that the variation of angles and frequencies of the incident waves ha

a two step even odd split levinson algorithm for toeplitz systems abstract we use a two step durbin method rather than the single step version in the even odd split levinson algorithm for strongly nonsingular real symmetric toeplitz systems with arbitrary right hand side thereby slightly reducing the complexity of this algorithm to 19 8 n 2 o n flops on a sequential machine we also present extensive numerical results comparing several levinson type methods

effective elastic properties of solids with two dimensional inclusions of irregular shapes abstract a computational procedure to calculate the contribution of irregularly shaped inclusions into the effective moduli of two dimensional elastic solids is proposed this procedure is based on the analysis of a representative volume element subjected to a prescribed macrostress for each type of inclusion the compliance contribution tensor is constructed using a combination of kolosov muskhelishvili complex variable technique and numerical conformal mapping application of this procedure to regularly shaped inhomogeneities produces results that are in good correspondence with analytical predictions the effective properties are first derived in the approximation of non interacting inclusions then several first order micromechanical models for interacting inclusions are considered

compressibility of two dimensional pores having n fold axes of symmetry the complex variable method and conformal mapping are used to derive a closed form expression for the compressibility of an isolated pore in an infinite two dimensional isotropic elastic body the pore is assumed to have an n fold axis of symmetry and be represented by at most four terms in the mapping function that conformally maps the exterior of the pore into the interior of the unit circle the results are validated against some special cases available in the literature and against boundary element calculations by extrapolation of the results for pores obtained from three and four terms of the schwarz christoffel mapping function for regular polygons the compressibilities of a triangle square pentagon and hexagon are found to at least three digits specific results for some other pore shapes more general than the quasi polygons obtained from the schwarz christoffel mapping are also presented an approximate scaling law for the compressibility in terms of the ratio of perimeter squared to area is also tested this expression gives a reasonable approximation to the pore compressibility but may overestimate it by as much as 20

analysis of stress concentrations in plates with rectangular openings by a combined conformal mapping finite element approach plates with rectangular openings develop stress concentrations under bending while these stresses can be determined using finite elements in many problems this would be difficult because a high density mesh would be needed in the neighborhood of every opening corner in this paper it is shown how a complex variable conformal mapping approach can be numerically coupled with the finite element method to analyze these corner stresses this approach can be used even with relatively coarse meshes where the finite element results by themselves do not resolve the stress concentrations in essence the method relies on finite element analysis to obtain information on the stress field in a region surrounding the plate opening this information is subsequently used to set up the parameters of the conformal mapping approach to obtain the near field stresses at the opening corners

branched circle packings and discrete blaschke products in this paper we introduce the notion of discrete blaschke products via circle packing we first establish necessary and sufficient conditions for the existence of finite branched circle packings next discrete blaschke products are defined as circle packing maps from univalent circle packings that properly fill d z izi 1 to the corresponding branched circle packings that properly cover d it is verified that such maps have all geometric properties of their classical counterparts finally we show that any classical finite blaschke product can be approximated uniformly on compacta of d by discrete ones

schwarz s lemma for circle packings burt rodin department of mathematics university of california la jolla ca 92093 usa 1 introduction conformal mappings can be approximated by circle packing isomorphisms this is proved in 10 roughly speaking see sect 5 for a more detailed description a bounded region f2 is almost filled by e circles 7 from the regular hexagonal e circle packing of the plane denote this approximate circle pack ing of q by q by results of andreev and thurston there is a combinatorially isomorphic circle packing d of the unit disk d denote this isomorphism suitably normalized by 7 7 f2 d let f be the piecewise linear quasicon formal mapping determined by the associated triangulations as e 0 f con verges uniformly on compacta to the riemann mapping function f f2 d let

numerical experience with a superfast real toeplitz solver abstract we briefly describe the generalized schur algorithm for the superfast solution of positive definite toeplitz systems of equations and its relationship with schur s algorithm and the szego recursions we then present some experimental results obtained with our fortran implementation of this superfast toeplitz solver the algorithm displays favorable behavior in that the growth rates of the resulting residuals and errors are comparable with those of the szego recursions

stress fields around two pores in an elastic body exact quadrature domain solutions analytical solutions are given for the stress fields in both compression and far field shear in a two dimensional elastic body containing two interacting non circular pores the two complex potentials governing the solutions are found by using a conformal mapping from a pre image annulus with those potentials expressed in terms of the schottky klein prime function for the annulus solutions for a three parameter family of elastic bodies with two equal symmetric pores are presented and the compressibility of a special family of pore pairs is studied in detail the methodology extends to two unequal pores the importance for boundary value problems of plane elasticity of a special class of planar domains known as quadrature domains is also elucidated this observation provides the route to generalization of the mathematical approach here to finding analytical solutions for the stress fields in bodies containing any finite number of pores