relaxation of three solenoidal wells and characterization of extremal three phase h measures we fully characterize quasiconvex hulls for three arbitrary solenoidal divergence free wells in dimension three with this aim we establish weak lower semicontinuity of certain functionals with integrands restricted to generic two dimensional planes and convex in up to three rank 2 directions within the planes within the framework of the theory of compensated compactness the latter represents an example when the differential constraints fail the constant rank condition but nevertheless the so called convexity still implies lower semicontinuity and fancyscript a quasiconvexity which essentially means that rank 2 convexity implies s quasiconvexity that is quasiconvexity in the sense of the divergence free differential constraints on the planes the proof employs a version of muller s estimates of haar wavelet projections in terms of the riesz transform the above semicontinuity result is then applied to the three solenoidal wells problem via analogs of sverak s nontrivial quasiconvex functions and connectedness properties of the rank 2 envelopes as another application of the semicontinuity result we obtain a geometric result of a more general nature characterization of certain extremal three point h measures for three phase mixtures of three characteristic functions in dimension three we also discuss the applicability of the results to problems with other differential constrains in particular to three linear elastic wells and further generalizations

improved bounds for the overall properties of a nonlinear composite dielectric the construction of generalized hashin shtrikman bounds for nonlinear composite problems relies on the introduction of a comparison material in recent work a nonlinear comparison medium has been used however this requires detailed knowledge of the properties of the trial fields that are employed the fields used have the property of bounded mean oscillation and this enables the size of the penalty exacted by using a nonlinear comparison material to be controlled some recent results concerning riesz transforms and the beurling operator are used here to reduce the effect of the penalty and hence to generate improved bounds in addition an exact solution is established for a particular class of composites

numerical range and functional calculus in hilbert space we prove an inequality related to polynomial functions of a square matrix involving the numerical range of the matrix we also show extensions valid for bounded and also unbounded operators in hilbert spaces which allow the development of a functional calculus

multiplication and mixed differentiation in generalized fock spaces the classical distribution theory founded by laurent schwartz 26 has an intrinsic multiplication problem he indicates this in his paper 27 this problem is clearly demonstrated by selecting two particular distributions and attempting to define a multiplication that enjoys commutative associative and unit identity properties this process leads to the contradiction 1 ep i lo a new generalized function theory developed by j colombeau 4 91 provides a solution to the multiplication problem and also develops a general integration theory t todorov reexamines the new generalized function theory in the spirit of non denumerable sequences termed ultrapowers 29 the l schwartz distribution theory was reexamined in the spirit of fundamental sequences 16 333 however there are new considerations in the theory of new generalized function theory requiring an algebraic notion of a particular ideal contained in a specified test function space a brief review of these constructions can be found in section 7 and a comprehensive review can be found in ref 25 a principle application for our development will be modeling a physical system consisting of an infinite number of particles a system of n particles termed bosoms or fermions has previously been modeled by n dimensional symmetric or antisymmetric functions when this system is enlarged to an infinite number of particles the model for the state space vector has the form

rearrangeability of 2 protectn 1 stage shuffle exchange networks rearrangeable networks can realize each and every permutation in one pass through the network shuffle exchange networks provide an efficient interconnection scheme for implementing various types of parallel processes whether 2n stage shuffle exchange networks with n 2n inputs outputs are rearrangeable has remained an open question for approximately three decades this question has been answered affirmatively in this paper an important corollary of the main result is the proof that two passes through an omega network are sufficient and necessary to implement any permutation in obtaining the main results of this paper frames that look like grids with horizontal links of different lengths are shown to be remarkable tools for identifying and characterizing the binary matrix representations of permutations

h measures a new approach for studying homogenisation oscillations and concentration effects in partial differential equations new mathematical objects called h measures are introduced for studying oscillations and concentration effects in partial differential equations applications to transport properties and to homogenisation are given as an example of the new results which can be obtained by this approach

algebraic and geometric aspects of rational γ inner functions abstract the set g def z w z w z 1 w 1 c 2 has intriguing complex geometric properties it has a 3 parameter group of automorphisms its distinguished boundary is a ruled surface homeomorphic to the mobius band and it has a special subvariety which is the only complex geodesic of g that is invariant under all automorphisms we exploit the geometry of g to develop an explicit and detailed structure theory for the rational maps from the unit disc to the closure of g that map the boundary of the disc to the distinguished boundary of

a complete parameterization of all positive rational extensions of a covariance sequence in this paper we formalize the observation that filtering and interpolation induce complementary or dual decompositions of the space of positive real rational functions of degree less than or equal to n from this basic result about the geometry of the space of positive real functions we are able to deduce two complementary sets of conclusions about positive rational extensions of a given partial covariance sequence on the one hand by viewing a certain fast filtering algorithm as a nonlinear dynamical system defined on this space we are able to develop estimates on the asymptotic behavior of the schur parameters 1918 of positive rational extensions on the other hand we are also able to provide a characterization of all positive rational extensions of a given partial covariance sequence indeed motivated by its application to signal processing speech processing and stochastic realization theory this characterization is in terms of a complete parameterization using familiar objects from systems theory and proves a conjecture made by georgiou 1983 1987 our basic result however also enables us to analyze the robustness of this parameterization with respect to variations in the problem data the methodology employed is a combination of complex analysis geometry linear systems and nonlinear dynamics

upper and lower bounds on norms of functions of matrices abstract given an n by n matrix a we look for a set s in the complex plane and positive scalars m and m such that for all functions p bounded and analytic on s and throughout a neighborhood of each eigenvalue of a the inequalities m inf f l s f a p a p a m inf f l s f a p a hold we show that for 2 by 2 matrices if s is the field of values then one can take m 1 and m 2 we show that for a perturbed jordan block a matrix a that is an n by n jordan block with eigenvalue 0 except that its n 1 entry is with 0 1 if s is the unit disk then m m 1 we argue however that in general due to the behavior of minimal norm interpolating functions it may be very difficult or impossible to find such a set s for which the ratio m m is of moderate size

new approach to bounding effective properties of random heterogeneous materials a new approach to bounding effective properties of random heterogeneous materials is developed in this paper the expectation material properties are computed for a stochastic unit cell to represent the random heterogeneous materials this stochastic unit cell is then used to compute the upper bound and lower bound of effective properties using the variational asymptotic method for unit cell homogenization a recently developed general purpose micromechanics code in comparison with other approaches the present approach has two main advantages 1 the numerical implementation is not restricted to statically homogeneous and isotropic microstructures 2 both the lower bound and upper bound converge to the effective properties of a deterministic unit cell with decreasing randomness of the stochastic unit cell to illustrate the application of this new approach various examples are analyzed and compared with existing theories

stochastic functional expansion for random media with perfectly disordered constitution the random field created by a system of random points which are centers of perfectly disordered equi sized spheres of a finite radius is introduced and named the pds field the notion of perfect disorder is defined statistically correctly taking the cumulants instead of moments to be delta functions the volterra wiener functional expansion with the pds field as a basis function is considered and a system of orthogonal wiener functionals is explicitly constructed the expansion is employed to the problem of specifying the overall conductivity for a random medium containing an array of perfectly disordered spherical inclusions an infinite hierarchy of coupled equations for the kernels of the wiener functionals is derived it is shown that the wiener expansion with respect to the pds field is also a virial one i e the nth order term contributes quantities of order c n where c is the volume fraction of the inclusions this allows the full stochastic solution to the problem of heat conduction throu

functional models of γn contractions and characterization of γn isometries abstract we show that a weighted bergman space on the polydisc splits up into the orthogonal direct sum of subspaces each corresponding to an irreducible representation of the symmetric group the multiplication by the n tuple of elementary symmetric functions is a n contraction on each of these subspaces a necessary condition for a commuting n tuple to be a n contraction is obtained in terms of pencil of operators we provide a model theory for n isometries a beurling lax halmos type representation for the invariant subspaces of n isometries is given

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

multicomponent composites electrical networks and new types of continued fraction i the development of bounds on the complex effective conductivity tensor that relates the average current to the average electric field in a multicomponent composite has been hindered by lack of a suitable continued fraction representation for here a new field equation recursion method is developed which gives an expression for as a continued fraction of a novel form incorporating as coefficients the component conductivities and a set of fundamental geometric parameters reflecting the composite geometry a hierarchy of field equations is set up such that the solutions of the j 1 th order equation generate the solutions of thejth order equation consequently the effective tensor j associated with thejth order field equation is expressible as a fractional linear matrix transformation of j 1 these transformations combine to form the continued fraction expansion for 0 which is exploited in the following paper part ii to obtain bounds crude bounds on j forj 1 give narrow bounds on the continued fraction is a generalization to multivariate functions of the continued fraction expansion of single variable stieltjes functions that proved important in the development of the theory of pade approximants asymptotic analysis and the theory of orthogonal polynomials in the last century the results extend to other transport problems including conduction in polycrystalline media the viscoelasticity of composites and the response of multicomponent multiterminal linear electrical networks

the nevanlinna pick problem on the closed unit disk minimal norm rational solutions of low degree for the nevanlinna pick interpolation problem with n interpolation conditions interior and boundary we construct a family of rational solutions of degree at most n 1 we also establish necessary and sufficient conditions for the existence and the uniqueness of a solution with the minimally possible h norm and construct a family of minimal norm rational solutions of degree at most n 1 in the indeterminate case finally we supplement a result of ruscheweyh and jones showing that in case the interpolation nodes and the target values are all unimodular any rational solution of degree at most n 1 is necessarily a finite blaschke product

bounds and correlation approximation for the effective conductivity of heterogeneous plates the size effect obtained when studying the effective properties of plates is investigated by producing a third order correlation approximation and hashin shtrikman bounds for the effective in plane conductivity of heterogeneous plates the boundary condition of the plates is taken into account by obtaining the exact green operator for the boundary problem results are obtained for a two dimensional 2d random distribution of disks and a 3d distribution of spheres all results recover those obtained for an infinite medium when the heterogeneity size becomes small compared to plate thickness they show that the size effect is more significant in the case of a 2d distribution of heterogeneities than for a 3d distribution

linear preservers of polynomial numerical hulls of matrices abstract let m n be the algebra of all n n complex matrices 1 k n 1 be an integer and m n m n be a linear operator in this paper it is shown that preserves the polynomial numerical hull of order k if and only if there exists a unitary matrix u m n such that either a u a u for all a m n or a u a t u for all a m n

optimal boundary interpolation and one block optimal control problem with invariant zeros on the imaginary axis and infinity we formulate a matrix interpolation problem with existing interpolation points on the imaginary axis and infinity and existing equal left and right interpolation points using the concept of parametrisation of stabilising controllers then we solve the problem of obtaining all its solutions if interpolation points at infinity are absent we show that the introduced problem is equivalent to the existing one we apply this result to solve the problem of optimal interpolation with existent interpolation points on the imaginary axis and infinity we show by an example that the solution of optimal interpolation is directly applicable to the one block optimal control with existent invariant zeros on the imaginary axis and infinity it is seen from the example that not only the transfer matrix of the closed loop system is constrained on the extended imaginary axis but also its derivatives

crouzeix s conjecture and perturbed jordan blocks abstract crouzeix conjectured in m crouzeix bounds for analytical functions of matrices integr equ oper theory 48 2004 461 477 that for any square matrix a and any polynomial p p a 2 max p z z w a where w a is the field of values of a and denotes the spectral norm in this paper we show that the conjecture holds for matrices of the form 1 1 where and are complex numbers the technique for 1 and for 1 is to show that if g is a bijective conformal mapping from the field of values of such a matrix a to the unit disk mapping to 0 then g a is similar to a contraction via a similarity transformation with condition number at most 2

stochastic functional expansion in elasticity of heterogeneous solids abstract the volterra wiener functional expansion is employed to the analysis of statistic properties for random heterogeneous solids for simplicity the technique is displayed on an elastic suspension of spheres the basis function in the expansion is chosen as that corresponding to the so called perfect disorder of spheres pds recently introduced by the authors an infinite hierarchy of equations for the kernels in the expansion is derived whose truncating after the nth equation is shown to yield results for the averaged statistical characteristics which are valid to order c n f where c f is the volume fraction of the spheres the kernels for the first and the second approximations n 1 2 are found and related to the displacement fields in an infinite elastic body containing respectively one and two spherical inhomogeneities within the frame of the so called singular approximation the overall tensor of elastic moduli for a suspension of perfectly disordered spheres is shown to coincide to the order c 2 f with a formula earlier obtained by means of the method of the effective field

strong fatou 1 points of blaschke products this paper shows that to every countable set m on the unit circle there corresponds a blaschke product whose set of strong fatou 1 points contains m it also shows that some blaschke products have an uncountable set of strong fatou 1 points

radial limits of interpolating blaschke products it is shown that if n is a sequence of distinct points on the unit circle then for a sequence an of points in the closed unit disk there exists an interpolating blaschke product b with b n an for all n if and only if an is bounded away from zero this complements results of cargo carroll colwell belna and piranian on prescribing radial limits for blaschke products

rearrangeability of bit permutation networks in this paper we introduce the concept of routing grid as a tool for analyzing realizability of permutations on bit permutation networks bpns we extend a result by linial and tarsi which characterizes permutations realizable on shuffle exchange networks to any bpns a necessary condition for a bpn to he rearrangeable is given and the rearrangeability of two families of bpns are explored finally we present a method which may help to tackle one kind of balanced matrix problems whose solution implies an answer to benes conjecture hopefully our treatment brings some new insight into the problem of permutation routing

finite blaschke products and the construction of rational γ inner functions abstract let def z w z w z 1 w 1 c 2 a inner function is a holomorphic map h from the unit disc d to whose boundary values at almost all points of the unit circle t belong to the distinguished boundary b of a rational inner function h induces a continuous map h t from t to b the latter set is topologically a mobius band and so has fundamental group z the degree of h is defined to be the topological degree of h t in a previous paper the authors showed that if h s p is a rational inner function of degree n then s 2 4 p has exactly n zeros in the closed unit disc d counted with an appropriate notion of multiplicity in this paper with the aid of a solution of an interpolation problem for finite blaschke products we explicitly construct the rational inner functions of degree n with the n zeros of s 2 4 p prescribed

upper and lower bounds for the overall properties of a nonlinear composite dielectric ii periodic microgeometry the study of the effective properties of a nonlinear composite dielectric begun in the companion to this paper is continued in the context of a simple cubic array of inclusions embedded in a matrix use of a nonlinear comparison material in the nonlinear hashin shtrikman variational principles permits the generation of both upper and lower bounds for any composite it is straightforward to apply the new approach to the simple composite considered here as the trial electric field which is substituted into the hashin shtrikman variational principle is finite everywhere the results obtained indicate the likely improvement over the simple classical bounds that can be obtained using the hashin shtrikman principles

the polynomial numerical hulls of jordan blocks and related matrices abstract the polynomial numerical hull of degree k for a square matrix a is a set designed to give useful information about the norms of polynomial functions of the matrix it is defined as z c p a p z for all p of degree k or less whilethese sets have been computed numerically for a number of matrices the computations have not been verified analytically in most cases in this paper we show analytically that the 2 norm polynomial numerical hulls of degrees 1 through n 1 for an n by n jordan block are disks about the eigenvalue with radii approaching 1 as n and we prove a theorem characterizing these radii rk n in the special case where k n 1 this theorem leads to a known result in complex approximation theory for n even rn 1 n is the positive root of 2rn r 1 0 and for n odd it satisfies a similar formula for large n this means that rn 1 n 1 log 2n n log log 2n n these results are used to obtain bounds on the polynomial numerical hulls of certain degrees for banded triangular toeplitz matrices and for block diagonal matrices with triangular toeplitz blocks

generalizations of the field of values useful in the study of polynomial functions of a matrix for a given square matrix a and positive integer k we consider sets in the complex plane satisfying p a maxz p z for all polynomials p of degree k or less the largest such set referred to as the polynomial numerical hull of degree k was introduced by o nevanlinna convergence of iterations for linear equations birkhauser basel 1993 and a number of properties of this set were derived for both matrices and linear operators we give several equivalent characterizations of the polynomial numerical hull of degree k and we actually compute these sets for several matrices for k 1 this set is just the field of values of a and for k m where m is the degree of the minimal polynomial of a it is the spectrum of a for 1 k m these sets are intermediate between the field of values and the spectrum and sometimes resemble pseudospectra

rigidity and lack of rigidity for solenoidal matrix fields the problem of finding a curl free matrix valued field e with values in an assigned set of matrices has received considerable attention given a bounded connected open set and a compact set of m n matrices in this paper we establish existence or non existence results for the following problem find b l m n such that div b 0 in in the sense of distributions under the constraint that b x almost everywhere in we consider the case of a b rank a b n and we establish non existence both for the case of exact solutions described above and for the case of approximate solutions described in 1 we also prove existence of approximate solutions for a suitably chosen triple a 1 a 2 a 3 of matrices with rank a i a j n i j and i j 1 2 3 we give examples when the differential constraints are of a different type and present some applications to composites

application of volterra wiener series for bounding the overall conductivity of hetergeneous media i general procedure a general method of placing optimal bounds on the overall conductivity of random heterogeneous media is proposed it utilizes truncated volterra wiener functional series generated by the random conductivity field k x as classes of trial functions for the classical variational principles the method simplifies unifies and or generalizes the earlier proposed variational techniques of beran 3 dederichs and zeller 10 hori 15 kr6ner 17 and prager 25 the general procedure is displayed in detail in the simplest case of interest the construction of the optimal third order bounds that requires knowledge of the two and three point correlation functions for k x the evaluation of these bounds is reduced to the solution of an integrodifferential equation whose coefficients and kernels are expressed through the said correlation functions a perturbation solution to this equation is given for miller s cell materials the equation is explicitly solved and the obtained optimal bounds are shown to coincide with those of beran miller

a new routing algorithm for a class of rearrangeable networks this paper presents a routing algorithm for a class of multistage interconnection networks specifically the concatenation of two omega networks which has 2 log sub 2 n stages is treated it is shown that this kind of asymmetric omega omega network can be converted into a symmetric omega sup 1spl times omega network or a symmetric omega spl times omega sup 1 network however they have butterfly connections between the two center stages a general algorithm is developed which routes a class of symmetric networks the algorithm routes the network from center stages to outer stages at both the input and the output sides simultaneously the algorithm presented is simpler and more flexible than the well known looping algorithm in that it can be applied adaptively according to the structure of the network it can be applied to routing the omega based networks regardless of the center stage connection patterns i e straight skewed straight simple butterfly or skewed butterfly as long as the networks are symmetric the sufficient conditions for proper routing are shown and proved in addition an example is shown to demonstrate the algorithm

a convex optimization approach to the rational covariance extension problem in this paper we present a convex optimization problem for solving the rational covariance extension problem given a partial covariance sequence and the desired zeros of the modeling filter the poles are uniquely determined from the unique minimum of the corresponding optimization problem in this way we obtain an algorithm for solving the covariance extension problem as well as a constructive proof of georgiou s seminal existence result and his conjecture a stronger version of which we have resolved in byrnes et al ieee trans automat control ac 40 1995 pp 1841 1857

a commutant lifting theorem for a domain in c2and spectral interpolation abstract we characterise the pairs of commuting operators on hilbert space for which the domain 1 2 1 2 1 1 2 1 is a complete spectral set we give an application to the spectral nevanlinna pick problem we obtain a necessary condition for the existence of an analytic 2 2 matrix function satisfying interpolation conditions and bounds on eigenvalues

spectral sets and distinguished varieties in the symmetrized bidisc abstract we show that for every pair of matrices s p having the closed symmetrized bidisc as a spectral set there is a one dimensional complex algebraic variety in such that for every matrix valued polynomial f z 1 z 2 f s p max z 1 z 2 f z 1 z 2 the variety is shown to have the determinantal representation s p det f p f s i 0 where f is the unique matrix of numerical radius not greater than 1 that satisfies s s p i p p 1 2 f i p p 1 2 when s p is a strict contraction then is a distinguished variety in the symmetrized bidisc i e a one dimensional algebraic variety that exits the symmetrized bidisc through its distinguished boundary we characterize all distinguished varieties of the symmetrized bidisc by a determinantal representation as above

on the factorial functional series and their application to random media functional series with a pointwise random input the density field of a random set of points bf x j are considered the series are rearranged so as the so called factorial fields of the set bf x j appear the obtained series are called factorial the basic result of the paper states that the factorial series possess virial property this means that if a random field u bf x is expanded as a factorial series the truncation u p bf x of the latter after the p tuple term coincides in the statistical sense with u bf x to the order n p where n is the number density of the set bf x j p 1 2 cdots the performance of the factorial series in random media problems is illustrated on the example of steady state diffusion in a random dispersion of spheres whose sink strength differs from that of the matrix the full statistical solution of this problem correct to the order c 2 is obtained in particular the effective sink strength of the dispersion i

finite blaschke product interpolation on the closed unit disc abstract we show how to construct all finite blaschke product solutions and the minimal scaled blaschke product solution to the nevanlinna pick interpolation problem in the open unit disc by solving eigenvalue problems of the interpolation data based on a result of jones and ruscheweyh we note that there always exists a finite blaschke product of degree at most n 1 that maps n distinct points in the closed unit disc of which at least one is on the unit circle into n arbitrary points in the closed unit disc provided that the points inside the unit circle form a positive semi definite pick matrix of full rank finally we discuss a numerical limiting procedure

multiple nevanlinna pick interpolation with both interior and boundary data and its connection with the power moment problem abstract the so called hankel vector approach is used to handle three boundary versions of the multiple nevanlinna pick interpolation problem in the nevanlinna class n involving both interior and boundary data it turns out that each of these boundary interpolation problems can be reduced to what amounts to a certain truncated standard or nonstandard hamburger moment problem associated with the hankel vector of the former with some possible constraint on distribution functions that assign no mass to any of the real nodes in particular this leads to solvability criteria for each of these interpolation problems and the description of solutions by using results from theory of moments

application of volterra wiener series for bounding the overall conductivity of heterogeneous media ii suspensions of eui sized spheres the variational procedure of part i of this paper this journal 47 1987 pp 831 849 is employed for bounding the overall conductivity k of a suspension of equi sized spheres the procedure uses classes of trial functions in the form of truncated volterra wiener series generated here by the random density function of the system of sphere centers integrodifferential equations are derived whose solutions yield bounds upper and lower on k for given two and three point distribution functions f2 f3 of the system of spheres the bounds are explicitly calculated to the order c2 where c is the volume fraction of the spheres they include a statistical parameter m2 that can be simply evaluated by means of the two point distribution function f2 and that reflects the tendency of the spheres to form clusters at higher c the predictions of various approximate models of composite mechanics are checked against the c2 bounds the check yields a number of curious results concerning the random constitution of suspensions of impenetrable spheres that can be described by means of these models in this paper we develop the technique of the volterra wiener series for bounding the overall conductivity in random suspensions of equi sized spheres in 6 we describe the statistics of such suspensions by means of the multipoint distribution functions f for the system of sphere centers in 7 the two and three point moments correlation functions m and m3 of the random conductivity field k x of the suspension are explicitly expressed by f2 and f3 in 8 the variational procedure of i 2 is slightly modified and an integrodifferential equation is derived whose solution leads to an upper bound k 3 on the overall conductivity that depends on f2 and f3 only in 9 a virial solution to this equation is given and the bound k 3 is explicitly calculated to the order c2 for a wide class of suspensions denoted here by a the bound includes a positive scalar parameter m2 that depends on the two point function f2 in 10 the lower bound to the order c2 is given for the same class of suspensions s it includes again the parameter m2 the so obtained c2 bounds are not optimal for f2 given because additional statistical information is tacitly employed when restricting the analysis to the class a the bounds however coincide to the order k 3 in 11 a simple formula for the statistical parameter m2 is derived and it is calculated for certain kinds of sphere distribution an interpretation of m2 is also proposed in 12 certain known results and approximations that have been proposed to evaluate k are compared with the bounds in order to establish the range of applicability of the former in this way it appears possible to extract some nontrivial statistical information about the kind of random suspensions for which the reviewed models of mechanics of composite materials can be employed when predicting the overall conductivity

new bounds on the effective thermal conductivity of n phase materials based on the perturbation solution to the effective thermal conductivity problem for an n component material we derive new third and fourth order bounds for the effective thermal conductivity of the composite the new third order bounds are accurate to third order in epsilon a epsilon b where epsilon a is the thermal conductivity of phase a and require a total of frac 1 2 n 1 2 third order correlation parameters in their evaluation for two phase composites these bounds require only one geometrical parameter and are identical to beran s bounds for n geqslant 3 the third order bounds use the same information as beran s bounds but always more restrictive than beran s bounds the new fourth order bounds are accurate to fourth order in epsilon a epsilon b and require an additional frac 1 4 n 2 n 1 2 fourth order correlation parameters for n 2 only two parameters are needed and the fourth order bounds are shown to be more restrictive than the third order beran s bounds and the second order hashin shtrikman s bounds the application of the third order bounds to a spherical cell material of miller is illustrated

on the correctness of inside out routing algorithm recently a new routing algorithm called inside out routing algorithm was proposed for routing an arbitrary permutation in the omega based 2log sub 2 n stage networks this paper discusses the problems of the inside out routing algorithm and shows that the suggested condition for proper routing in the omega omega network is insufficient an extended necessary and sufficient condition for proper routing in the omega omega network is also suggested however it is unknown if any permutation can be successfully routed by a heuristic algorithm which follows the condition thus the rearrangeability of the omega omega network still remains an open problem

bounds which incorporate morphological information for a nonlinear composite dielectric the nonlinear hashin shtrikman variational structure relies on the introduction of a comparison material in recent work a nonlinear comparison composite has been employed this has enabled both upper and lower bounds to be found that depend on the three point statistics of the microstructure through a parameter introduced by g w milton at least one of the bounds requires an estimate of the size of the trial field to be known and even if the best estimate available is used and the milton parameter is known for a particular microstructure the bounds can still be widely separated o p bruno has obtained bounds for a linear composite consisting of a matrix containing spherical inclusions where the inclusions are coated with the matrix phase the bounds remain non trivial even when the properties of the inclusions tend to either zero or infinity these bounds are used to improve the bounds for a nonlinear composite comprising either a nonlinear matrix containing linear inclusions or a linear matrix containing nonlinear inclusions for the microgeometry considered here it is possible to construct a bounded trial field and a simple example involving this approach is also described

dielectric breakdown optimal bounds we study dielectric breakdown for composites made of two isotropic phases we show that sachs s bound is optimal this simple example is used to illustrate a variational principle which departs from the traditional one we also derive the usual variational principle by elementary means without appealing to the technology of convex duality

a case of mu synthesis as a quadratic semidefinite program we analyze a special case of the robust stabilization problem under structured uncertainty we obtain a new criterion for the solvability of the spectral nevanlinna pick problem which is a special case of the mu synthesis problem of h infty control in which mu is the spectral radius given n distinct points lambda 1 dots lambda n in the unit disc and 2 times 2 nonscalar complex matrices w 1 ldots w n the problem is to determine whether there is an analytic 2 times 2 matrix function f on the disc such that f lambda j w j for each j and the supremum of the spectral radius of f lambda is less than 1 for lambda in the disc the condition is that the minimum of a quadratic function of pairs of positive 3n square matrices subject to certain linear matrix inequalities in the data be attained and be zero

the scalar nevanlinna pick interpolation problem with boundary conditions we show that if the nevanlinna pick interpolation problem is solvable by a function mapping into a compact subset of the unit disc then the problem remains solvable with the addition of any number of boundary interpolation conditions provided the boundary interpolation values have modulus less than unity we give new inductive proofs of the nevanlinna pick interpolation problem with any finite number of interpolation points in the interior and on the boundary of the domain of interpolation the right half plane or unit disc with function values and any finite number of derivatives specified our solutions are analytic on the closure of the domain of interpolation our proofs only require a minimum of matrix theory and operator theory we also give new straightforward algorithms for obtaining minimal h norm solutions finally some numerical examples are given

some simple explicit bounds for the overall behaviour of nonlinear composites abstract variational expressions developed over the last few years provide bounds for the overall energy functions of a range of nonlinear composite materials the evaluation of a bound requires the solution of a system of nonlinear algebraic equations and this generally involves a computation there are however certain simple composites comprising a nonlinear matrix containing either rigid inclusions or cavities for which very simple explicit formulae can be given these formulae are displayed here at a level of generality greater than in any previous presentation the energy density function of the matrix is arbitrary and the microgeometry of the composite appears through an expression which bounds the energy of a linear composite with the same geometry to the extent that such linear bounds can be developed making allowance for any amount of statistical information on the composite the new nonlinear bounds collect this new results at the level of employing bounds of hashin shtrikman type for the linear problem are given for an incompressible matrix reinforced by aligned rigid platelets or weakened by aligned cracks in the course of the work a recently derived formula more general than any available previously is presented and developed explicitly for any two phase composite

interpolation with constraint on frequency region and systems control application abstract this paper elaborates on a bitangential interpolation problem with a norm constraint of the interpolant on a given frequency region and on its generalization and proves that it admits a solution a motivation from systems control for the problem is given and the solution is applied for construction of a stabilizing control for one block plants such that the closed loop transfer matrix is norm constrained on the frequency region illustrative numerical examples are given

essentially normal composition operators on h2 abstract we prove a simple criterion for essential normality of composition operators on the hardy space induced by maps in a reasonably large class s of analytic self maps of the unit disk by combining this criterion with boundary caratheodory fejer interpolation theory we exhibit a parametrization for all rational self maps of the unit disk which induce essentially normal composition operators

the coherent potential approximation is a realizable effective medium scheme the effective electrical conductivity of an aggregate composed of grains of various conductivities is frequently estimated by the coherent potential approximation which embodies a local effective medium concept it is proved rigorously that this approximation is exact for a wide class of hierarchical model composites made of spherical grains the starting material 0 in the hierarchy is chosen arbitrarily otherwise materialj 1 2 consists of equisized spheres sayj spheres of arbitrary conductivities embedded in materialj 1 the spatial distribution of thej spheres must satisfy a mild homogeneity condition and their radiusr j must asymptotically increase faster than exponentially withj furthermore the minimum spacing 2s j between thej spheres is such that the ratios j r j diverges on the basis of these and some further ancillary conditions it is established that the coherent potential approximation becomes asymptotically exact for the effective conductivity of materialj the results extend to other effective parameters of the composites including the thermal conductivity dielectric constant and magnetic permeability in addition the model composites and the proof of realizability may be generalized to allow non spherical grains

homogenization and optimal bounds in linear elasticity in their celebrated paper of 1963 hashin shtrikman 1 addressed the problem of determining optimal bounds for the bulk and shear moduli of a statistically isotropic elastic composite material with arbitrary isotropic phase geometry they derived a set of bounds for these moduli in the physically meaningful case of three dimensional elasticity in the case of a two phase composite let k 1 1 and k 2 2 respectively denote the bulk and shear moduli for the first and the second phase let k denote their analogues for the composite and let 0 stand for the volume fraction of the first phase in the composite under the ordering restriction that both

the interpolation problem with a degree constraint the author previously 1983 1987 showed that there is a correspondence between nonnegative hermitian trigonometric polynomials of degree spl les n and solutions to the standard nevanlinna pick caratheodory interpolation problem with n 1 constraints which are rational and also of degree spl les n it was conjectured that the correspondence under suitable normalization is bijective and thereby that it results in a complete parametrization of rational solutions of degree spl les n the conjecture was proven by byrnes et al 1995 along with a detailed study of this parametrization however byrnes et al used a slightly restrictive assumption that the trigonometric polynomials are positive and accordingly the corresponding solutions have positive real part the purpose of the present note is to extend the result to the case of nonnegative trigonometric polynomials as well we present the arguments in the context of the general nevanlinna pick caratheodory fejer interpolation

nevanlinna pick meromorphic interpolation the degenerate case and minimal norm solutions abstract the nevanlinna pick interpolation problem is studied in the class s of meromorphic functions f with poles inside the unit disk d and with f l t 1 in the indeterminate case the parametrization of all solutions is given in terms of a family of linear fractional transformations with disjoint ranges a necessary and sufficient condition for the problem being determinate is given in terms of the pick matrix of the problem the result is then applied to obtain necessary and sufficient conditions for the existence of a meromorphic function with a given pole multiplicity which satisfies nevanlinna pick interpolation conditions and has the minimal possible l norm on the unit circle t

frames a simple characterization of permutations realized by frequently used networks rearrangeable multistage networks such as the benes network realize any permutation yet their routing algorithms are not cost effective on the other hand there exist inexpensive routing algorithms for nonrearrangeable networks but no simple technique exists to characterize all the permutations realized on these networks this paper introduces the concept of frame and shows how it can be used to characterize all the permutations realized on various multistage interconnection networks they include subnetworks of baseline benes and cascaded baseline and shuffle exchange networks

wrong side interpolation by positive real rational functions abstract using polynomial interpolation along with structural properties of the family of rational positive real functions we here show that a set of m nodes in the open left half of the complex plane can always be mapped to anywhere in the complex plane by rational positive real functions whose degree is at most m moreover we introduce an easy to find parametrization in r 2 m 3 of a large subset of these interpolating functions

topologies of combined 2 log n 1 stage interconnection networks a combined 2 log n 1 stage interconnection network denoted by spl delta spl oplus spl delta is constructed by concatenating two omega equivalent networks spl delta and spl delta with the rightmost stage of spl delta and the leftmost stage of spl delta overlapped benes network and the 2 log n 1 stage shuffle exchange network are two examples of such networks although these two networks have received intensive studies the research on the topology of entire class of spl delta spl oplus spl delta networks is very limited so far in this paper we study the topological structure of spl delta spl oplus spl delta networks and propose an algorithm for determining topological equivalence between two given spl delta spl oplus spl delta networks we also present a simplified spl omega equivalence checking algorithm as a supporting result

a variational approach to the overall sink strength of a nonlinear lossy composite medium an extension of the hashin shtrikman variational structure to nonlinear problems initiated by j r willis j appl mech 50 1202 1209 1983 and developed generally by d r s talbot j r willis ima j appl math 35 39 54 1985 is applied to the problem of homogenization of the semilinear steady state diffusion equation nabla 2c partial w partial c c x k x 0 where the function tilde w is convex in c but varies randomly with position x a strict definition for an overall function w which defines the homogenized medium is given and its properties are discussed bounds for tilde w are then developed from variational principles classical energy principles yield bounds analogous to the voigt and reuss bounds of elasticity theory while the new hashin shtrikman structure yields tighter bounds which are sensitive to the two point statistics of the medium explicit results are presented for a particular two phase composite medium for which one phase is linear so that w is quadratic whereas the second has a quadratic nonlinearity so that w is cubic

linear maps preserving the polynomial numerical radius of matrices abstract let n 2 be a fixed integer and denote by m n the algebra of all n n complex matrices fix also an integer k such that 1 k n we prove that if m n m n is a linear map which preserves the polynomial numerical radius of order k there exist then a unitary n n complex matrix u and a complex number of modulus one such that either t u t u for all t m n or t u t t u for all t m n

dilations of γ contractions by solving operator equations for a contraction p and a bounded commutant s of p we seek a solution x of the operator equation s s p 1 p p 1 2 x 1 p p 1 2 where x is a bounded operator on ran over bar 1 p p 1 2 with numerical radius of x being not greater than 1 a pair of bounded operators s p which has the domain gamma z 1 z 2 z 2 vertical bar z 1 vertical bar 1 vertical bar z 2 vertical bar 1 subset of c 2 as a spectral set is called a p contraction in the literature we show the existence and uniqueness of solution to the operator equation above for a gamma contraction s p this allows us to construct an explicit gamma isometric dilation of a gamma contraction s p we prove the other way too i e for a commuting pair s p with parallel to p parallel to 1 and the spectral radius of s being not greater than 2 the existence of a solution to the above equation implies that s p is a gamma contraction we show that for a pure f contraction s p there is a bounded operator c with numerical radius not greater than 1 such that s c c p any gamma isometry can be written in this form where p now is an isometry commuting with c and c any gamma unitary is of this form as well with p and c being commuting unitaries examples of gamma contractions on reproducing kernel hilbert spaces and their gamma isometric dilations are discussed c 2012 elsevier inc all rights reserved

a reformulation of augmented basic interpolation problem and an application to h control abstract in this paper we present a new formulation of the augmented basic interpolation problem abip with rational matrices in terms of the stability of four rational matrices so that the abip transforms into a purely linear algebraic problem actually the existing interpolation condition given by an integral is replaced by stability of a rational matrix the new condition is applied to the h optimal control of one block plants having imaginary axis invariant zeros a new parameter in the parametrization of h optimal controllers is revealed which is given in terms of the derivative of the closed loop system at the imaginary axis invariant zeros of the plant the h optimal control algorithm is illustrated by an example and compared to the existing algorithms

bounded universal functions in one and several complex variables we show how to obtain functions that are universal for the ball of h infty omega where omega subset mathbb c n the existence of our functions will follow from universality criteria but we also show how to construct them then we study the connection between certain interpolating sequences runaway automorphisms and the existence of universal functions on domains in mathbb c n

extremal holomorphic maps and the symmetrized bidisc we analyze the 3 extremal holomorphic maps from the unit disc mathbb d to the symmetrized bidisc mathcal g mathop limits mathrm def z w zw z w in mathbb d with a view to the complex geometry and function theory of mathcal g these are the maps whose restriction to any triple of distinct points in mathbb d yields interpolation data that are only just solvable we find a large class of such maps they are rational of degree at most 4 it is shown that there are two qualitatively different classes of rational mathcal g inner functions of degree at most 4 to be called aligned and caddywhompus functions the distinction relates to the cyclic ordering of certain associated points on the unit circle the aligned ones are 3 extremal we describe a method for the construction of aligned rational mathcal g inner functions with the aid of this method we reduce the solution of a 3 point interpolation problem for aligned holomorphic maps from mathbb d to mathcal g to a collection of classical nevanlinna pick problems with mixed interior and boundary interpolation nodes proofs depend on a form of duality for mathcal g

a new approximation for the three point probability function statistical continuum theory based approaches are commonly used for the computation of the effective properties of heterogeneous materials statistical distribution and morphology of the microstructure are represented by n point probability function one point probability function statistical representation of the microstructure leads to volume fraction dependent homogenization however second and higher order probability functions include the information of phase distribution and morphology most statistical based homogenization methods are limited to two point probability function due to the lack of simple approximation of higher order probability functions that can be easily exploited in this paper a new approximation of the three point probability function is proposed and discussed the new approximation results are compared to existing approximations from the literature and to the real probability functions calculated from a computer generated two phase micrographs

bounds for effective parameters of heterogeneous media by analytic continuation we give a mathematical formulation of a method for obtaining bounds on effective parameters developed by d bergman and g w milton this method in contrast to others used before does not rely on a variational principle but exploits the properties of the effective parameter as an analytic function of the component parameters the method is at present restricted to two component media