multiscale resolution in the computation of crystalline microstructure this paper addresses the numerical approximation of microstructures in crystalline phase transitions without surface energy it is shown that branching of different variants near interfaces of twinned martensite and austenite phases leads to reduced energies in finite element approximations such behavior of minimizing deformations is understood for an extended model that involves surface energies moreover the closely related question of the role of different growth conditions of the employed bulk energy is discussed by explicit construction of discrete deformations in lowest order finite element spaces we prove upper bounds for the energy and thereby clarify the question of the dependence of the convergence rate upon growth conditions and lamination orders for first order laminates the estimates are optimal

polynomial matrices with given determinant abstract let a be a q n polynomial matrix 1 q a if suitable polynomials 1 n exist such that one can write 1 a i 1 n a i n where ai 1 ai n is one row in a and 1 n is a unimodular row then a can be row bordered to an n n polynomial matrix m with det m some examples are discussed

approximation of a laminated microstructure for a rotationally invariant double well energy density we give error estimates for the approximation of a laminated microstructure which minimizes the energy int omega phi nabla v x dx for a rotationally invariant double well energy density phi a we present error estimates for the convergence of the deformation in l 2 the convergence of directional derivatives of the deformation in the twin planes the weak convergence of the deformation gradient the convergence of the microstructure or young measure of the deformation gradients and the convergence of nonlinear integrals of the deformation gradient

feedback stabilization over commutative rings the matrix case this paper provides a solution of the feedback stabilization problem over commutative rings for matrix transfer functions stabilizability of a transfer matrix is realised as local stabilizability over the entire spectrum of the ring for stabilizable plants certain modules generated by its fractions and that of the stabilizing controller are shown to be projective compliments of each other in the case of rings with irreducible spectrum this geometric relationship shows that the plant is stabilizable if and only if the above modules of the plant are projective of ranks equal to the number of inputs and the outputs if the maxspectrum of the ring is noetherian and of zero krull dimension then this result shows that the stabilizable plants have doubly coprime fractions over unique factorization domains the above stabilizability condition is interpreted in terms of matrices formed by the fractions of the plant certain invariants of these matrices known as elementary factors are defined and it is shown that the plant is stabilizable if and only if these elementary factors generate the whole ring this condition thus provides a generalization of the coprime factorizability as a condition for stabilizabilty a formula for the class of all stabilizing controllers is then developed that generalizes the previous well known formula in factorization theory for multidimensional transfer functions these results provide concrete conditions for stabilizabilty finally it is shown that the class of polynomial rings over principal ideal domains is an additional class of rings over which stabilizable plants always have doubly coprime fractions

the simply laminated microstructure in martensitic crystals that undergo a cubic to orthorhombic phase transformation we study simply laminated microstructures of a martensitic crystal capable of undergoing a cubic to orthorhombic transformation of type mathcal p 432 to mathcal p 222 the free energy density modeling such a crystal is minimized on six energy wells that are pairwise rank one connected we consider the energy minimization problem with dirichlet boundary data compatible with an arbitrary but fixed simple laminate we first show that for all but a few isolated values of transformation strains this problem has a unique young measure solution solely characterized by the boundary data that represents the simply laminated microstructure we then present a theory of stability for such a microstructure and apply it to the conforming finite element approximation to obtain the corresponding error estimates for the finite element energy minimizers

numerical approximation of the solution of variational problem with a double well potential variational problems with a double well potential are not lower semicontinuous and can fail to attain a minimum value rather the gradients of minimizing sequences do not converge pointwise and can have oscillations however the gradients do converge in the weak topology i e their local spatial averages converge such functionals arise in the description of equilibria of crystals or other ordered materials stable configurations for solid crystals which have symmetry related martensitic energy wells have a fine scale microstructure which can be related to the oscillations that energy minimizing sequences for the bulk energy exhibit an analysis is given of approximation methods for variational problems with a double well potential to give a rigorous justification for the use of such numerical methods to model the behavior of this class of solid crystals

optimal order error estimates for the finite element approximation of the solution of a nonconvex variational problem nonconvex variational problems arise in models for the equilibria of crystals and other ordered materials the solution of these variational prob lems must be described in terms of a microstructure rather than in terms of a deformation moreover the numerical approximation of the deformation gra dient often does not converge strongly as the mesh is refined nevertheless the probability distribution of the deformation gradients near each material point does converge recently we introduced a metric to analyze this convergence in this paper we give an optimal order error estimate for the convergence of the deformation gradient in a norm which is stronger than the metric used earlier

common canonical variates canonical correlation analysis measures the linear relationship between two random vectors x 1 and x 2 as the maximum correlation between linear combinations of x 1 and linear combinations of x 2 several generalisations of canonical correlation analysis to k 2 random vectors x 1 x k have been proposed in the literature kettenring 1971 1985 based on the principle of maximising some generalised measure of correlation in this paper we propose an alternative generalisation called common canonical variates based on the assumption that the canonical variates have the same coefficients in all k sets of variables this generalisation is applicable in situations where all x i have the same dimension we present normal theory maximum likelihood estimation of common canonical variates and illustrate their use on a morphometric data set

estimates for numerical approximations of rank one convex envelopes summary we present a convergence analysis of an algorithm for the numerical computation of the rank one convex envelope of a function f mathrm m m times n rightarrow mathbb r a rate of convergence for the scheme is established and numerical experiments are presented to illustrate the analytical results and applications of the algorithm

on the computation of the rank one convex hull of a function we report on several numerical experiments where the rank one convexification of an energy density is computed the explicit examples cover a whole spectrum of typical situations one may encounter one of those is especially relevant for the computation of microstructures in crystalline solids

an algorithmic proof of suslin s stability theorem for polynomial rings abstract let k be a field then gaussian elimination over k and the euclidean division algorithm for the univariate polynomial ring k x allow us to write any matrix in sln k or sln k x n 2 as a product of elementary matrices suslin s stability theorem states that the same is true for sln k xl xm with n 3 and m 1 in this paper we present an algorithmic proof of suslin s stability theorem thus providing a method for finding an explicit factorization of a given polynomial matrix into elementary matrices grobner basis techniques may be used in the implementation of the algorithm

solving linear systems via pfaffians abstract an explicit solution is given for the system of linear equations e e t where are alternating matrices of pfaffian 1 which at most differ in their first row and column and e is of the form i 2 2 r e 1 i x i if v v w r r 1 with v w 1 v w then a sequence of cohn transforms with respect to the fixed w which takes v w to v w is prescribed

constitutive theory for some constrained elastic crystals abstract some of the observations of a 15 superconductors near cubic tetragonal phase transformations suggest treating them as thermoelastic bodies subject to certain material constraints here we begin to develop a theory of this kind

on the numerical analysis of non convex variational problems we discuss a numerical method for finding young measure valued minimizers of non convex variational problems to have any hope of a convergence theorem one must work in a setting where the minimizer is unique and minimizing sequences converge strongly this paper has two main goals i we specify a method for producing strongly convergent minimizing sequences despite the failure of strict convexity and ii we show how uniqueness of the young measure can be parlayed into a numerical convergence theorem the treatment of ii is done in the setting of two model problems one involving scalar valued functions and a multiwell energy the other from micromagnetics

numerical analysis of oscillations in multiple well problems variational problems which fail to be convex occur often in the study of ordered materials such as crystals in these problems the energy density for the material has multiple potential wells in this paper we study multiple well problems by first determining the analytic properties of energy minimizing sequences and then by estimating the continuous problem by an approximation using piecewise linear finite elements we show that even when there is no minimizer of the energy the approximations still take on a predictable structure

algorithms for the quillen suslin theorem abstract we give an algorithm for computing a free basis of a projective c x1 xn module which is presented as the image kernel or cokernel of a polynomial matrix our method can be implemented using grobner bases and it provides an elementary constructive proof of the quillen suslin theorem

on algorithms for nonconvex optimization in the calculus of variations a simple algorithm for the computation of local minima of nonconvex problems in one dimension is proposed this algorithm avoids certain well known local minima that are distant from the global minimum and can be used to improve the minima obtained using classical descent methods the calculated minima have mesh scale oscillations typical of weakly converging sequences it is shown that these local minima converge weakly to the correct limit for certain simple problems however numerical evidence suggests that the gradients may not converge to the correct weak limits and this is verified using asymptotic expansions extensions to multiple dimensions and some numerical examples in two dimensions are considered

numerical approximations in variational problems with potential wells in this paper some numerical aspects of variational problems which fail to be convex are studied it is well known that for such a problem in general the infimum of the energy the functional that has to be minimized fails to be attained instead minimizing sequences develop oscillations which allow them to decrease the energy it is shown that there exists a minimizes for an approximation of the problem and the oscillations in the minimizing sequence are analyzed it is also shown that these minimizing sequences choose their gradients in the vicinity of the wells with a probability which tends to be constant an estimate of the approximate deformation as it approximates a measure and some numerical results are also given

strong convergence of numerical solutions to degenerate variational problems numerical approximations of strongly degenerate variational problems of the form j u 0 1 f u u f 2 are considered where f is assumed convex but may have intervals where f 0 it is shown that in spite of the degeneracy natural numerical approximations still converge in w 1 p rates in weaker norms and the connection with nonconvex variational problems are also considered

approximation of martensitic microstructure with general homogeneous boundary data we consider the approximation of martensitic microstructure for a class of martensitic transformations we model such microstructures by multi well energy minimization problems with general homogeneous boundary data under our assumptions on such boundary data the underlying microstructure can be nonunique we first show that any energy minimizing sequence converges strongly to a unique macroscopic deformation that is precisely the homogeneous deformation in the boundary condition we then prove a series of estimates for the approximation of admissible deformations to the unique macroscopic deformation of the microstructure and for the closeness of the gradients of admissible deformations to the energy wells

a generalized serre problem abstract a possible generalization of the serre problem quillen suslin theorem on the freeness of projective modules over polynomial rings which seems to be of interest in systems theory applications is shown to follow from the solution to the original serre problem

an algorithm for unimodular completion over laurent polynomial rings abstract we present a new and simple algorithm for completion of unimodular vectors with entries in a multivariate laurent polynomial ring r k x 1 x k over an infinite field k more precisely given n 3 and a unimodular vector v t v 1 v n r n that is such that v 1 v n r the algorithm computes a matrix m in m n r whose determinant is a monomial such that m v t 1 0 0 and thus m 1 is a completion of v to an invertible matrix

approximated convex envelope of a function the goal of this paper is to introduce the approximated convex envelope of a function and to estimate how it differs from its convex envelope such a problem arises in various physical situations where the function considered is some energy that has to be minimized this study is a first step toward understanding how to approximate the quasi convex envelope of a function the importance of this issue is due to the various applications that are encountered in particular in the field of material science 1994 copyright society for industrial and applied mathematics

relaxation of some multi well problems mathematical models of phase transitions in solids lead to the variational problem minimize w du dx where w has a multi well structure i e w 0 on a multi well set k and w 0 otherwise we study this problem in two dimensions in the case of equal determinant i e for k so 2 u 1 so 2 u k or k o 2 u 1 o 2 uk for u 1 u k m 2 2 with det u i in three dimensions when the matrices u i are essentially two dimensional and also for k so 3 u 1 so 3 u k for u 1 u k m 3 3 with adj u i tu i 33 2 which arises in the study of thin films here u i denotes the 3 2 matrix formed with the first two columns of u i we characterize generalized convex hulls including the quasiconvex hull of these sets prove existence of minimizers and identify conditions for the uniqueness of the minimizing young measure finally we use the characterization of the quasiconvex hull to propose approximate relaxed energies quasiconvex functions which vanish on the quasiconvex hull of k and grow quadratically away from it

numerical analysis of compatible phase transitions in elastic solids the variational model of phase transitions for elastic materials based on linearized elasticity leads to a nonconvex minimization problem p in which a minimum need not be attained in the design of advanced materials the main interest is in reliable numerical predictions of certain macroscopic quantities such as the global deformation and the stress field determined in a relaxed problem qp an explicit formula of the quasi convexified energy density in qp due to r v kohn provides us with a well posed numerical problem first a mathematical a priori and a posteriori error analysis is established for the finite element approximation of the stress variable then the residual based error indicator is implemented within an adaptive mesh refinement algorithm numerical examples illustrate that the macroscopic properties of the materials are computed efficiently with appropriate error control

numerical solution of the scalar double well problem allowing microstructure the direct numerical solution of a non convex variational problem p typically faces the difficulty of the finite element approximation of rapid oscillations although the oscillatory discrete minimisers are properly related to corresponding young measures and describe real physical phenomena they are costly and difficult to compute in this work we treat the scalar double well problem by numerical solution of the relaxed problem rp leading to a degenerate convex minimisation problem the problem rp has a minimiser u and a related stress field dw u which is known to coincide with the stress field obtained by solving p in a generalised sense involving young measures if u h is a finite element solution h dw u h is the related discrete stress field we prove a priori and a posteriori estimates for h in l 4 3 and weaker weighted estimates for u u h the a posteriori estimate indicates an adaptive scheme for automatic mesh refinements as illustrated in numerical experiments

a generalization of serre s conjecture and some related issues abstract several topics concerned with multivariate polynomial matrices like unimodular matrix completion matrix determinantal or primitive factorization matrix greatest common factor existence and subsequent extraction along with relevant primeness and coprimeness issues are related to a conjecture which may be viewed as a type of generalization of the original serre problem conjecture solved nonconstructively in 1976 and constructively more recently this generalized serre conjecture is proved to be equivalent to several other unsettled conjectures and therfore all these conjectures constitute a complete set in the sense that solution to any one also solves all the remaining

the appearance of microstructures in problems with incompatible wells and their numerical approach the goal of the paper is to analyze the creation of microstructure in problems of calculus of variations with wells more precisely we consider a case with strong incompatibility between the wells this forces the minimizing sequences to use other gradients than the wells in a puzzling way using a p 1 method we are then able to single out discrete minimizing sequences and to give energy estimates in terms of the mesh size

on syzygy modules for polynomial matrices abstract in this paper we apply the theory of multivariate polynomial matrices to the study of syzygy modules for a system of homogeneous linear equations with multivariate polynomial coefficients several interesting structural properties of syzygy modules are presented and illustrated with examples

on lin bose problem abstract this paper study generalized serre problem proposed by lin and bose in multidimensional system theory context multidimens systems and signal process 10 1999 379 linear algebra appl 338 2001 125 this problem is stated as follows let f a l m be a full row rank matrix and d be the greatest common divisor of all the l l minors of f assume that the reduced minors of f generate the unit ideal where a k x 1 x n is the polynomial ring in n variables x 1 x n over any coefficient field k then there exist matrices g a l l and f 1 a l m such that f gf 1 with det g d and f 1 is a zlp matrix we provide an elementary proof to this problem and treat non full rank case

finite element analysis of a class of stress free martensitic microstructures this work is concerned with the finite element approximation of a class of stress free martensitic microstructures modeled by multi well energy minimization finite element energy minimizing sequences are first constructed to obtain bounds on the minimum energy over all admissible finite element deformations a series of error estimates are then derived for finite element energy minimizers

nonconforming finite element approximation of crystalline microstructure we consider a class of nonconforming nite element approxima tions of a simply laminated microstructure which minimizes the nonconvex variational problem for the deformation of martensitic crystals which can un dergo either an orthorhombic to monoclinic double well or a cubic to tetrag onal triple well transformation we rst establish a series of error bounds in terms of elastic energies for the l 2 approximation of derivatives of the de formation in the direction tangential to parallel layers of the laminate for the l 2 approximation of the deformation for the weak approximation of the de formation gradient for the approximation of volume fractions of deformation gradients and for the approximation of nonlinear integrals of the deformation gradient we then use these bounds to give corresponding convergence rates for quasi optimal nite element approximations

numerical approach to double well problems this paper deals with two sided numerical energy estimates of a solution to two dimensional double well problems these estimates are obtained by computing the energy appropriate for the polyconvex envelope and an upper estimate of the energy appropriate for the rank one convex envelope of the energy density examples of computed laminated microstructures are shown

numerical analysis of a nonconvex variational problem related to solid solid phase transitions the description of equilibria of shape memory alloys or other ordered materials gives rise to nonconvex variational problems in this paper a two dimensional model of such materials is studied due to the fact that the corresponding functional has two symmetry related martensitic energy wells the numerical approximation of the deformation gradient does not converge but tends to oscillate between the two wells as the size of the mesh is refined these oscillations may be interpreted in terms of microstructures using a nonconforming mathcal p 1 finite element an estimate is given for the rate of convergence of the probability for the approximated deformation to have its gradient near one of the two martensitic wells

further remarks on multivariate polynomial matrix factorizations abstract multivariate n d polynomial matrix factorizations are basic research problems in multidimensional systems and signal processing in this paper youla s mlp lemma 19 is extended to the general case based on this extension generalizations of some results in 16 are proved which might be useful for further investigations on problems of factorizations of multivariate polynomial matrices

null lagrangians weak continuity and variational problems of arbitrary order abstract we consider the problem of minimizing integral functionals of the form i u f x k u x dx where r p u r and k u denotes the set of all partial derivatives of u with orders k the method is based on a characterization of null lagrangians l ku depending only on derivatives of order k applications to elasticity and other theories of mechanics are given

analysis of the finite element approximation of microstructure in micromagnetics the solution to nonconvex variational problems is often characterized by microstructure the variational problem for the magnetization field in micromagnetics has a nonconvex constraint and energy minimizing sequences of magnetization fields can have oscillations whose scale converges to zero but whose amplitude remains finite it is shown that the finite element approximation of the magnetization field for this variational problem does not converge pointwise as the mesh is refined but that nonlinear functions of the approximate magnetization fields converge weakly or equivalently all local spatial averages of nonlinear functions of the approximate magnetization field converge as the mesh is refined which implies the convergence as the mesh is refined of the probability distribution of the approximate magnetization fields in local spatial domains a norm is given to measure the convergence of this microstructure and we prove a rate of convergence in this norm

restrictions on microstructure we consider the following question given a set of matrices with no rank one connections does it support a nontrivial young measure limit of gradients our main results are these a a young measure can be supported on four incompatible matrices b in two space dimensions a young measure cannot be supported on finitely many incompatible elastic wells c in three or more space dimensions a young measure can be supported on three incompatible elastic wells and d if supports a nontrivial young measure with mean value 0 then the linear span of must contain a matrix of rank one

on the convergence of finite element approximations of a relaxed variational problem the accuracy of finite element approximations to a convex but not strictly convex variational problem is considered convergence is proved for a finite element approximation of a particular vector field related to the solution in a special one dimensional case 0 h convergence is shown for a piecewise linear approximation of the derivative h denotes the size of each element domain numerical results are also presented for this one dimensional case

common canonical variates in k independent groups abstract suppose that data are collected from k independent but closely related populations and that a canonical correlation analysis is to be performed within each group in such a situation it may turn out that the transformation to canonical variates is similar across all k groups the common canonical variates model studied in this article assumes that the transformation to canonical variates is actually identical across groups while the canonical correlations may vary we develop some algebra of the common canonical variates model discuss normal theory maximum likelihood estimation and likelihood ratio testing give illustrative examples

an informational measure of association and dimension reduction for multiple sets and groups with applications in morphometric analysis in this article we propose a new general index that measures relationships between multiple sets of random vectors this index is based on kullback leibler kl information which measures linear or nonlinear dependence between multiple sets using joint and marginal densities of affine transformations of the random vectors estimates of the matrixes are obtained by maximizing the kl information and are shown to be consistent the motivation for introducing such an index comes from morphological integration studies a topic in biological science as a special case of this index we define an overall measure of association and two other measures for dimension reduction the use of these measures is illustrated through real data analysis in morphometric studies and extensive simulations and their performance is compared with that of approaches based on canonical correlation analysis extensions of the aforementioned measures to multiple groups are also discussed in contrast to canonical correlation analysi

new results on multivariate polynomial matrix factorizations abstract multivariate n d polynomial matrix factorizations are basic research problems in multidimensional systems and signal processing in this paper several extensions of the existing results on multivariate matrix factorizations are proved moreover by means of the general theory of multivariate matrix factorizations recently developed a new proof is given to the bivariate polynomial matrix factorizations this new approach produces also a recursive algorithm for the actual computation of the general bivariate matrix factorizations which relies on the algorithm of the grobner bases for modules and does not involve calculations over the function fields

finite element analysis of microstructure for the cubic to tetragonal transformation martensitic crystals which can undergo a cubic to tetragonal phase transformation have a nonconvex energy density with three symmetry related rotationally invariant energy wells we give estimates for the numerical approximation of a first order laminate for such martensitic crystals we give bounds for the l2 convergence of directional derivatives in the twin plane for the l2 convergence of the deformation for the weak convergence of the deformation gradient for the convergence of the microstructure and for the convergence of nonlinear integrals of the deformation gradient

raw data maximum likelihood estimation for common principal component models a state space approach the specifications of state space model for some principal component related models are described including the independent group common principal component cpc model the dependent group cpc model and principal component based multivariate analysis of variance some derivations are provided to show the equivalence of the state space approach and the existing wishart likelihood approach for each model a numeric example is used to illustrate the state space approach in addition a simulation study is conducted to evaluate the standard error estimates under the normality and nonnormality conditions in order to cope with the nonnormality conditions the robust standard errors are also computed finally other possible applications of the state space approach are discussed at the end

multivariate association and dimension reduction a generalization of canonical correlation analysis summary in this article we propose a new generalized index to recover relationships between two sets of random vectors by finding the vector projections that minimize an l 2 distance between each projected vector and an unknown function of the other the unknown functions are estimated using the nadaraya watson smoother extensions to multiple sets and groups of multiple sets are also discussed and a bootstrap procedure is developed to detect the number of significant relationships all the proposed methods are assessed through extensive simulations and real data analyses in particular for environmental data from los angeles county we apply our multiple set methodology to study relationships between mortality weather and pollutants vectors here we detect existence of both linear and nonlinear relationships between the dimension reduced vectors which are then used to build nonlinear time series regression models for the dimension reduced mortality vector these findings also illustrate potential use of our method in many other applications a comprehensive assessment of our methodologies along with their theoretical properties are given in a web appendix

numerical analysis of oscillations in nonconvex problems we study numerically the pattern of the minimizing sequences of nonconvex problems which do not admit a minimizer

primeness of multivariate polynomial matrices abstract different characterizations of primeness of polynomial matrices that are mutually equivalent in the one dimensional case of univiriate polynomials lead to various primeness concepts in the multivariate situation applied to systems theory this fact gives rise to a more refined view to observability and controllability of multidimensional systems algorithms for testing primeness properties are given based on computer algebraic techniques