best approximation in c algebras on developpe des methodes permettant d obtenir les meilleures approximations des ideaux de sous algebres et de sous espaces des c algebres

comments on stability analysis of a family of matrices by y k foo and y c soh the robust stability problem of a convex hull of a finite number of matrices was treated in the above titled paper by y k foo and y c soh ibid vol 35 no 11 p 1257 9 nov 1990 and a sufficient condition was presented the commenters give another sufficient condition and show using an example that their condition can be less conservative than the result they also give a necessary and sufficient condition for the robust stability problems treated by foo and soh

a broad class of shellable lattices abstract we introduce a new class of lattices the modernistic lattices and their duals the comodernistic lattices we show that every modernistic or comodernistic lattice has shellable order complex we go on to exhibit a large number of examples of co modernistic lattices we show comodernism for two main families of lattices that were not previously known to be shellable the order congruence lattices of finite posets and a weighted generalization of the k equal partition lattices we also exhibit many examples of co modernistic lattices that were already known to be shellable to begin with the definition of modernistic is a common weakening of the definitions of semimodular and supersolvable we thus obtain a unified proof that lattices in these classes are shellable subgroup lattices of solvable groups form another family of comodernistic lattices that were already proven to be shellable we show not only that subgroup lattices of solvable groups are comodernistic but that solvability of a group is equivalent to the comodernistic property on its subgroup lattice indeed the definition of comodernistic exactly requires on every interval a lattice theoretic analogue of the composition series in a solvable group thus the relation between comodernistic lattices and solvable groups resembles in several respects that between supersolvable lattices and supersolvable groups

an euler relation for valuations on polytopes abstract a locally finite point set such as the set z n of integral points gives rise to a lattice of polytopes in euclidean space taking vertices from the given point set we develop the combinatorial structure of this polytope lattice and derive euler type relations for valuations on polytopes using the language of mobius inversion in this context a new family of inversion relations is obtained thereby generalizing classical relations of euler dehn sommerville and macdonald

squarefree p modules and the cd index abstract in this paper we introduce a new algebraic concept which we call squarefree p modules this concept is inspired from karu s proof of the non negativity of the cd indices of gorenstein posets and supplies a way to study cd indices from the viewpoint of commutative algebra indeed by using the theory of squarefree p modules we give several new algebraic and combinatorial results on cw posets first we define an analogue of the cd index for any cw poset and prove its non negativity when a cw poset is cohen macaulay this result proves that the h vector of the barycentric subdivision of a cohen macaulay regular cw complex is unimodal second we prove that the stanley reisner ring of the barycentric subdivision of an odd dimensional cohen macaulay polyhedral complex has the weak lefschetz property third we obtain sharp upper bounds of the cd indices of gorenstein posets for a fixed rank generating function

metric geometry in homogeneous spaces of the unitary group of a c algebra part i minimal curves abstract we study the metric geometry of homogeneous spaces p of the unitary group of a c algebra a modulo the unitary group of a c subalgebra b where the invariant finsler metric is induced by the quotient norm of a b the main results include the following in the von neumann algebra context for each tangent vector x at a point p there is a geodesic t 0 x which is obtained by the action on of a 1 parameter group in the unitary group of a this geodesic is minimizing up to length 2

lifting inequalities for polytopes we present a method of lifting linear inequalities for the flag f vector of polytopes to higher dimensions known inequalities that can be lifted using this technique are the non negativity of the toric g vector and that the simplex minimizes the cd index we obtain new inequalities for six dimensional polytopes in the last section we present the currently best known inequalities for dimensions 5 through 8

orthogonality to matrix subspaces and a distance formula abstract we obtain a necessary and sufficient condition for a matrix a to be birkhoff james orthogonal to any subspace w of m n c using this we obtain an expression for the distance of a from any unital c subalgebra of m n c

group actions on stanley reisner rings and invariants of permutation groups abstract let h be a group of permutations of x 1 x n and let q h x 1 x 2 x n denote the ring of h invariant polynomials in x 1 x 2 x n with rational coefficients combinatorial methods for the explicit construction of free bases for q h x 1 x 2 x n as a module over the symmetric polynomials are developed the methods are developed by studying the action of the symmetric group on the stanley reisner ring of the subset lattice some general results are also obtained by studying the action of a coxeter group on the stanley reisner ring of the corresponding coxeter complex in the case of a weyl group a purely combinatorial construction of certain invariants first considered by r steinberg topology 14 1975 173 177 is obtained some applications to representation theory are also included

on the cd index and γ vector of s shellable cw spheres we show that the vector of the order complex of any polytope is the f vector of a balanced simplicial complex this is done by proving this statement for a subclass of stanley s s shellable cw spheres which includes all polytopes the proof shows that certain parts of the cd index when specializing c 1 and considering the resulted polynomial in d are the f polynomials of simplicial complexes that can be colored with few colors we conjecture that the cd index of a regular cw sphere is itself the flag f vector of a colored simplicial complex in a certain sense

a characterization of solvability for finite groups in terms of their frame abstract the frame of a group is the poset of conjugacy classes of all its proper subgroups in this paper we will prove that a finite group is solvable if and only if every collection of maximal elements of its frame has a well defined meet and the poset consisting of all such meets including the meet of the empty set is a modular lattice

peak quasisymmetric functions and eulerian enumeration abstract via duality of hopf algebras there is a direct association between peak quasisymmetric functions and enumeration of chains in eulerian posets we study this association explicitly showing that the notion of cd index long studied in the context of convex polytopes and eulerian posets arises as the dual basis to a natural basis of peak quasisymmetric functions introduced by stembridge thus eulerian posets having a nonnegative cd index for example face lattices of convex polytopes correspond to peak quasisymmetric functions having a nonnegative representation in terms of this basis we diagonalize the operator that associates the basis of descent sets for all quasisymmetric functions to that of peak sets for the algebra of peak functions and study the g polynomial for eulerian posets as an algebra homomorphism

discrete morse theory for totally non negative flag varieties abstract in a seminal 1994 paper lusztig 1994 26 lusztig extended the theory of total positivity by introducing the totally non negative part g p 0 of an arbitrary generalized partial flag variety g p he referred to this space as a remarkable polyhedral subspace and conjectured a decomposition into cells which was subsequently proven by the first author rietsch 1998 33 in williams 2007 40 the second author made the concrete conjecture that this cell decomposed space is the next best thing to a polyhedron by conjecturing it to be a regular cw complex that is homeomorphic to a closed ball in this article we use discrete morse theory to prove this conjecture up to homotopy equivalence explicitly we prove that the boundaries of the cells are homotopic to spheres and the closures of cells are contractible the latter part generalizes a result of lusztig s 1998 28 that g p 0 the closure of the top dimensional cell is contractible concerning our result on the boundaries of cells even the special case that the boundary of the top dimensional cell g p 0 is homotopic to a sphere is new for all g p other than projective space

andre permutations lexicographic shellability and the cd index of a convex polytope the cd index of a polytope was introduced by fine it is an integer valued noncommutative polynomial obtained from the flag vector a result of bayer and fine states that for any integer flag vector the existence of the cd index is equivalent to the holding of the generalized dehn sommerville equations of bayer and billera for the flag vector the coefficients of the cd index are conjectured to be nonnegative we show a connection between the cd index of a polytope p and any cl shelling of the lattice of faces of p this enables us to prove that each andre polynomial of foata and schutzenberger is the cd index of a simplex

a characterization of minimal hermitian matrices abstract we describe properties of a hermitian matrix m m n c having minimal quotient norm in the following sense m m d for all real diagonal matrices d m n c here denotes the operator norm we show a constructive method to obtain all the minimal matrices of any size

coxeter cones and their h vectors coxeter cones are formed by intersecting the nonnegative sides of a collection of root hyperplanes in some root system they are shellable subcomplexes of the coxeter complex and their h vectors record the distribution of descents among their chambers we identify a natural class of graded coxeter cones with the property that their h vectors are symmetric and unimodal thereby generalizing recent theorems of reiner welker and branden about the eulerian polynomials of graded partially ordered sets

the colored symmetric and exterior algebras abstract we study colored generalizations of the symmetric algebra and its koszul dual the exterior algebra the symmetric group s n acts on the multilinear components of these algebras while s n acts trivially on the multilinear components of the colored symmetric algebra we use poset topology techniques to understand the representation on its koszul dual we introduce an s n poset of weighted subsets that we call the weighted boolean algebra and we prove that the multilinear components of the colored exterior algebra are s n isomorphic to the top cohomology modules of its maximal intervals we use a technique of sundaram to compute group representations on cohen macaulay posets to give a generating formula for the frobenius series of the colored exterior algebra we exploit that formula to find an explicit expression for the expansion of the corresponding representations in terms of irreducible s n representations we show that the two colored koszul dual algebras are koszul in the sense of priddy

counter example to a recent result on the stability of interval matrices by s bialas abstract in bialas 1983 a condition is given which is supposedly necessary and sufficient for the stability of interval matrices we demonstrate via a counter example that the condition given is in fact not sufficient

from the weak bruhat order to crystal posets we investigate the ways in which fundamental properties of the weak bruhat order on a weyl group can be lifted or not to a corresponding highest weight crystal graph viewed as a partially ordered set the latter projects to the weak order via the key map first a crystal theoretic analogue of the statement that any two reduced expressions for the same coxeter group element are related by coxeter moves is proven for all lower intervals hat 0 v in a simply or doubly laced crystal on the other hand it is shown that no finite set of moves exists even in type a for arbitrary crystal graph intervals in fact it is shown that there are relations of arbitrarily high degree amongst crystal operators that are not implied by lower degree relations second for crystals associated to kac moody algebras it is shown for lower intervals that the mobius function is always 0 or 1 and in finite type this is also proven for upper intervals with a precise formula given in each case moreover the order complex for each of these intervals is proven to be homotopy equivalent to a ball or to a sphere of some dimension despite often not being shellable for general intervals examples are constructed with arbitrarily large mobius function again even in type a any interval having mobius function other than 0 or 1 is shown to contain within it a relation amongst crystal operators that is not implied by the relations giving rise to the local structure of the crystal making precise a tight relationship between the mobius function and these somewhat unexpected relations appearing in crystals new properties of the key map are also derived the key is shown to be determined entirely by the edge colored poset theoretic structure of the crystal and a recursive algorithm is given for calculating it in finite types the fiber of the longest element of any parabolic subgroup of the weyl group is also proven to have a unique minimal and a unique maximal element this property fails for more general elements of the weyl group

on borel fixed ideals generated in one degree abstract we construct a shellable polytopal cell complex that supports a minimal free resolution of a borel fixed ideal which is borel generated by just one monomial in s k x 1 x 2 x n this includes the case of powers of the homogeneous maximal ideal x 1 x 2 x n as a special case in our most general result we prove that for any borel fixed ideal i generated in one degree there exists a polytopal cell complex that supports a minimal free resolution of i

generalized quotients in coxeter groups for w s a coxeter group we study sets of the form w v w e w i l wv 1 w i v for all v e v where v c w such sets w v here called generalized quotients are shown to have much of the rich combinatorial structure under bruhat order that has previously been known only for the case when v c s i e for minimal coset representatives modulo a parabolic subgroup we show that bruhat intervals in w v for general v c w are lexicographically shellable the mobius function on w v under bruhat order takes values in 1 0 1 for finite groups w generalized quotients are the same thing as lower intervals in the weak order this is however in general not true connections with the weak order are explored and it is shown that w v is always a complete meet semilattice and a convex order ideal as a subset of w under weak order descent classes di w e w il1 ws 1 w xt s e i for all s e s i c s are also analyzed using generalized quotients it is shown that each descent class as a poset under bruhat order or weak order is isomorphic to a generalized quotient under the corresponding ordering the latter half of the paper is devoted to the symmetric group and to the study of some specific examples of generalized quotients which arise in combinatorics for instance the set of standard young tableaux of a fixed shape or the set of linear extensions of a rooted forest suitably interpreted form generalized quotients we prove a factorization result for the quotients that come from rooted forests which shows that algebraically these quotients behave as a system of minimal coset representatives of a subset which is in general not a subgroup we also study the rank generating function for certain quotients in the symmetric group

proof of the k pi 1 conjecture for affine artin groups we prove the k pi 1 conjecture for affine artin groups the complexified complement of an affine reflection arrangement is a classifying space this is a long standing problem due to arnol d pham and thom our proof is based on recent advancements in the theory of dual coxeter and artin groups as well as on several new results and constructions in particular we show that all affine noncrossing partition posets are el shellable we use these posets to construct finite classifying spaces for dual affine artin groups we introduce new cw models for the orbit configuration spaces associated with arbitrary coxeter groups we construct finite classifying spaces for the braided crystallographic groups introduced by mccammond and sulway

best approximation by diagonal compact operators abstract we study the existence and characterization properties of compact hermitian operators c on a hilbert space h such that c c d for all d d k h h or equivalently c min d d k h h c d dist c d k h h where d k h h denotes the space of compact self adjoint diagonal operators in a fixed base of h and is the operator norm we also exhibit a positive trace class operator that fails to attain the minimum in a compact diagonal

discrete morse functions from lexicographic orders this paper shows how to construct a discrete morse function with a relatively small number of critical cells for the order complex of any finite poset with o and i from any lexicographic order on its maximal chains specifically if we attach facets according to the lexicographic order on maximal chains then each facet contributes at most one new face which is critical and at most one betti number changes facets which do not change the homotopy type also do not contribute any critical faces dimensions of critical faces as well as a description of which facet attachments change the homotopy type are provided in terms of interval systems associated to the facets as one application the mobius function may be computed as the alternating sum of morse numbers the above construction enables us to prove that the poset n s of partitions of a set 1 1 k k with repetition is homotopy equivalent to a wedge of spheres of top dimension when a is a hook shaped partition it is likely that the proof may be extended to a larger class of a and perhaps to all a despite a result of ziegler 1986 which shows that n s is not always cohen macaulay

eigenspace arrangements of reflection groups the lattice of intersections of reflecting hyperplanes of a complex reflection group w may be considered as the poset of 1 eigenspaces of the elements of w in this paper we replace 1 with an arbitrary eigenvalue and study the topology and homology representation of the resulting poset after posing the main question of whether this poset is shellable we show that all its upper intervals are geometric lattices and then answer the question in the affirmative for the infinite family g m p n of complex reflection groups and the first 31 of the 34 exceptional groups by constructing cl shellings in addition we completely determine when these eigenspaces of w form a k 1 resp free arrangement for the symmetric group we also extend the combinatorial model available for its intersection lattice to all other eigenvalues by introducing balanced partition posets presented as particular upper order ideals of dowling lattices study the representation afforded by the top co homology group and give a simple map to the posets of pointed d divisible partitions 1 main question and results let v denote an n dimensional c vector space a reflection in v is any nonidentity element r in gl v of finite order that fixes some hyperplane hr and a finite subgroup w of gl v is called a reflection group if it is generated by reflections example the action of the symmetric group sn on n 1 2 n gives rise to a faithful action on c via ei e i where e1 e2 en denote the standard basis vectors since the transpositions act as reflections and generate the group this representation realizes sn as a reflection group in gl c we shall refer to it as the defining representation for an element g w and root of unity let v g denote the eigenspace of g in v define e w to be the w poset partially ordered set of all such eigenspaces v g g w ordered by reverse inclusion with w action given by h v g v hgh the eigenvalue 1 gives the lattice lw of intersections of reflecting hyperplanes for w see 29 lemma 4 4 the minimal elements of e w i e the inclusion maximal eigenspaces forw are the focus of springer s theory of regular elements 35 and each has dimension equal to the number a d of degrees d1 d2 dn of w that are divisible by d the order of see proposition 3 7 below when w is crystallographic the poset e w itself appears in broue malle and michel s sylow theory 8 2010 mathematics subject classification primary 20f55 05e10 05e18 55p20 14n20

shellable nonpure complexes and posets ii this is a direct continuation of shellable nonpure complexes and posets i which appeared in transactions of the american mathematical society 348 1996 1299 1327 8 interval generated lattices and dominance order in this section and the following one we will continue exemplifying the applicability of lexicographic shellability to nonpure posets let f i1 i2 in be a family of intervals of integers by which is meant sets of the form a b a a 1 b a b we assume that there are no containments among these intervals and that they are ordered so that their left and right endpoints are increasing let l f be the lattice of all sets that are unions of subfamilies of f ordered by inclusion such interval generated lattices l f were introduced and studied by greene g define an edge labeling of l f as follows if a b is a covering and a max b a then a b a if a 1 a and a is the left endpoint of some i f

on the combinatorics of b b orbits on group compactifications abstract it is shown that there is an order isomorphism from the poset v of b b orbits on the wonderful compactification of a semi simple adjoint group g with weyl group w to an interval in reverse chevalley bruhat order on a non canonically associated coxeter group w in general neither finite nor affine moreover preserves the corresponding kazhdan lusztig polynomials springer s partly conjectural construction of kazhdan lusztig polynomials for the analogues of v for general coxeter groups w is completed by reducing it by a similar order isomorphism to known results involving a twisted chevalley bruhat order on w

natural variational problems in the grassmann manifold of a c algebra with trace we consider a family of natural variational problems in the grassmannian of a c algebra with trace which can be considered as slightly degenerate finsler metrics we show that all these problems have as solutions the standard geodesics and that the short standard geodesics are absolute minima of the functionals restricted to a special class of curves

random unistochastic matrices an ensemble of random unistochastic orthostochastic matrices is defined by taking squared moduli of elements of random unitary orthogonal matrices distributed according to the haar measure on u n or o n an ensemble of symmetric unistochastic matrices is obtained with use of unitary symmetric matrices pertaining to the circular orthogonal ensemble we study the distribution of complex eigenvalues of bistochastic unistochastic and orthostochastic matrices in the complex plane we compute averages entropy traces over the ensembles of unistochastic matrices and present inequalities concerning the entropies of products of bistochastic matrices

some characterizations of minimal compact normal operators abstract in this note we consider properties of a compact normal operator a such that a a d for all d d k h or equivalently a dist a d k h where denotes the usual operator norm and d k h is the subalgebra of diagonal compact operators in a fixed orthonormal basis of the hilbert space h

simultaneous eigenvalue lower bounds for the riccati matrix equation the author derives n simultaneous lower bounds on the sums of the eigenvalues of the solution of the algebraic riccati equation and of the lyapunov matrix equation these bounds extend the single inequalities shown in the literature for the trace and for the maximum eigenvalue two inequality versions are derived and evaluated for application purposes and a comparison to relevant bounds in the literature is given

minimal matrices and the corresponding minimal curves on flag manifolds in low dimension abstract in general c algebras elements with minimal norm in some equivalence class are introduced and characterized we study the set of minimal hermitian matrices in the case where the c algebra consists of 3 3 complex matrices and the quotient is taken by the subalgebra of diagonal matrices we thoroughly study the set of minimal matrices particularly because of its relation to the geometric problem of finding minimal curves in flag manifolds for the flag manifold of four mutually orthogonal complex lines in c 4 it is shown that there are infinitely many minimal curves joining arbitrarily close points in the case of the flag manifold of three mutually orthogonal complex lines in c 3 we show that the phenomenon of multiple minimal curves joining arbitrarily close points does not occur

on the co homology of the poset of weighted partitions we consider the poset of weighted partitions w in troduced by dotsenko and khoroshkin in their study of a certain pair of dual operads the maximal intervals of w provide a gen eralization of the lattice n of partitions which we show possesses many of the well known properties of n in particular we prove these intervals are el shellable we show that the mobius invari ant of each maximal interval is given up to sign by the number of rooted trees on node setf1 2 ng having a xed number of de scents we nd combinatorial bases for homology and cohomology and we give an explicit sign twisted sn module isomorphism from cohomology to the multilinear component of the free lie algebra with two compatible brackets we also show that the characteris tic polynomial of w has a nice factorization analogous to that of n

supports for minimal hermitian matrices abstract we study certain pairs of subspaces v and w of c n we call supports that consist of eigenspaces of the eigenvalues m of a minimal hermitian matrix m m m d for all real diagonals d for any pair of orthogonal subspaces we define a non negative invariant called the adequacy to measure how close they are to form a support and to detect one this function is the minimum of another map f defined in a product of spheres of hermitian matrices we study the gradient hessian and critical points of f in order to approximate these results allow us to prove that the set of supports has interior points in the space of flag manifolds

spectral analysis of random to random markov chains abstract we compute the eigenvalues and eigenspaces of random to random markov chains we use a family of maps which reveal a remarkable recursive structure of the eigenspaces yielding an explicit and effective construction of all eigenbases starting from bases of the kernels

lattice paths and kazhdan lusztig polynomials in their fundamental paper 18 kazhdan and lusztig defined for every coxeter group w a family of polynomials indexed by pairs of elements of w which have become known as the kazhdan lusztig polynomials of w see e g 17 chap 7 these polynomials are intimately related to the bruhat order of w and to the geometry of schubert varieties and have proven to be of fundamental importance in representation theory the purpose of this paper is to present a new non recursive combinatorial formula for these polynomials more precisely we show that each directed path in the bruhat graph of w has a naturally associated set of lattice paths with the property that the kazhdan lusztig polynomial of u v is the sum over all the lattice paths associated to all the paths going from u to v of l r o d r q i v i u r 1 r 2 where f o d f and f l f are three natural statistics on the lattice path we believe that this formula is the most explicit non recursive formula known for the kazhdan lusztig polynomials which holds in complete generality the organization of the paper is as follows in section 3 we define and study the r polynomial of a chain this polynomial reduces to the usual r polynomial for a chain of length one and is fundamental in all that follows in section 4 we show that the antisymmetrization of the kazhdan lusztig polynomial of two elements u v of w equals the alternating sum over all the chains going from u to v of the corresponding r polynomials in section 5 we derive some consequences of this result which although not needed in what follows are of independent interest more precisely we derive explicit formulas for each coefficient of any kazhdan lusztig polynomial and comparing these to known formulas we obtain some new identities for the r polynomials in section 6 we introduce and study a family of polynomials indexed by sequences of positive integers which is needed in the proof of our main result these polynomials are independent of w are easily computable by simple recursions and can be interpreted as counting certain lattice paths in section 7 we prove our main theorem which is obtained by combining the results in sections 4 and 6 this expresses the kazhdan lusztig polynomial of two elements u v in w as

constructible complexes and recursive division of posets shellability has been extensively studied by many researchers since mcmullen solved the upper bound theorem for convex polytopes there are also some important notions weaker than shellability and in this paper we treat constructibility among these and define a notion of recursively dividable posets which corresponds to the notion of constructible complexes when seeing their face posets also we define a notion of strongly constructible complexes and correspondingly strongly dividable posets by strengthening the conditions and prove that strongly dividable posets are signable this result means that strongly constructible simplicial complexes are partitionable

some characterizations of the trace norm triangle equality abstract in this paper we investigate some equalities for the trace class operators on a hilbert space for two trace class operators a and b we get some equivalent conditions for a 1 b 1 a b 1 where a 1 is the trace norm of the trace class operator a particularly we show that a 1 b 1 a b 1 if and only if a b a b for two trace class operators a and b this condition is related to a result of ando and hayashi moreover some characterizations of the equality a 1 a 1 a a 1 and other relevant results are given

minimal curves in u n and gl n with respect to the spectral and the trace norms abstract consider the lie group of n n complex unitary matrices u n endowed with the bi invariant finsler metric given by the spectral norm x u u x x for any x tangent to a unitary operator u given two points in u n in general there exist infinitely many curves of minimal length in this paper we provide a complete description of such curves and as a consequence we give an equivalent condition for uniqueness similar studies are done for the grassmann manifolds on the other hand consider the cone of n n positive invertible matrices g l n endowed with the bi invariant finsler metric given by the trace norm x 1 a a 1 2 x a 1 2 1 for any x tangent to a g l n in this context also exist infinitely many curves of minimal length in this paper we provide a complete description of such curves proving first a characterization of the minimal curves joining two hermitian matrices x y h n the last description is also used to construct minimal paths in the group of unitary matrices u n endowed with the bi invariant finsler metric x 1 u u x 1 x 1 for any x tangent to u u n we also study the set of intermediate points in all the previous contexts

euler flag enumeration of whitney stratified spaces abstract the flag vector contains all the face incidence data of a polytope and in the poset setting the chain enumerative data it is a classical result due to bayer and klapper that for face lattices of polytopes and more generally eulerian graded posets the flag vector can be written as a c d index a non commutative polynomial which removes all the linear redundancies among the flag vector entries this result holds for regular cw complexes we relax the regularity condition to show the c d index exists for whitney stratified manifolds by extending the notion of a graded poset to that of a quasi graded poset this is a poset endowed with an order preserving rank function and a weighted zeta function this allows us to generalize the classical notion of eulerianness and obtain a c d index in the quasi graded poset arena we also extend the semi suspension operation to that of embedding a complex in the boundary of a higher dimensional ball and study the simplicial shelling components

a geometric formulation of the law of aboav weaire in two and three dimensions the law of aboav weaire is a simple mathematical expression deriving from empirical observations that the number of sides of a grain is related to the average number of sides of the neighboring grains and is usually restricted to natural two dimensional microstructures numerous attempts have been made to justify this relationship theoretically or to derive an analogous relation in three dimensions this paper provides several exact geometric results with expressions similar to that of the usual law of aboav weaire though with additional terms that may be used to establish when the law of abaov weaire is a suitable approximation specifically we derive several local relations that apply to individual grain clusters and a corresponding global relation that is identical in two and three dimensions except for a single parameter the derivation requires the definition and investigation of the average excess curvature a previously unconsidered physical quantity an approximation to our exact result is compared to the results of extensive simulations in two and three dimensions and we provide a compact expression that strikes a balance between complexity and accuracy

comments on a necessary and sufficient condition for the stability of interval matrices in a recent paper by bialas a new proof of a known result concerning interval polynomials was given this result is summarized here and a much stronger result is mentioned bialas main theorem on the stability of interval matrices is shown to be false

stability analysis of a family of matrices the authors extend the use of the lyapunov equation method to check the stability of a family of matrices the family of interval matrices is defined as the convex hull of a finite number of matrices and is a special case where each element has independent perturbation while the stability of interval matrices cannot be inferred from the stability of its vertex matrices the authors show that the family of matrices will be stable if the symmetric matrices associated with the vertex matrices have matrix measures less than two

peak algebras paths in the bruhat graph and kazhdan lusztig polynomials we obtain a nonrecursive combinatorial formula for the kazhdan lusztig polynomials which holds in complete generality and which is simpler and more explicit than any existing one and which cannot be linearly simplified our proof uses a new basis of the peak subalgebra of the algebra of quasisymmetric functions

defect of a unitary matrix abstract we analyze properties of a map f sending a unitary matrix u of size n into a doubly stochastic matrix b f u defined by bi j ui j 2 for any u we define its defect determined by the dimension of the image df t u u of the space t u u tangent to the manifold of unitary matrices u at u under the tangent map df corresponding to f the defect of u equal to zero for a generic unitary matrix gives an upper bound for the dimension of a smooth orbit a manifold stemming from u of inequivalent unitary matrices mapped into the same doubly stochastic matrix b f u we demonstrate several properties of the defect and prove an explicit formula for the defect of the fourier matrix fn of size n in this way we obtain an upper bound for the dimension of a smooth orbit of inequivalent unitary complex hadamard matrices stemming from fn it is equal to zero iff n is prime and coincides with the dimension of the known orbits if n is a power of a prime two constructions of these orbits are presented at the end of this work

homotopy type of the boolean complex of a coxeter system in any coxeter group the set of elements whose principal order ideals are boolean forms a simplicial poset under the bruhat order this simplicial poset defines a cell complex called the boolean complex in this paper it is shown that for any coxeter system of rank n the boolean complex is homotopy equivalent to a wedge of n 1 dimensional spheres the number of such spheres can be computed recursively from the unlabeled coxeter graph and defines a new graph invariant called the boolean number specific calculations of the boolean number are given for all finite and affine irreducible coxeter systems as well as for systems with graphs that are disconnected complete or stars one implication of these results is that the boolean complex is contractible if and only if a generator of the coxeter system is in the center of the group

the pure symmetric automorphisms of a free group form a duality group abstract the pure symmetric automorphism group of a finitely generated free group consists of those automorphisms which send each standard generator to a conjugate of itself we prove that these groups are duality groups

monotonicity of the cd index for polytopes we prove that the cd index of a convex polytope satisfies a strong monotonicity property with respect to the cd indices of any face and its link as a consequence we prove for d dimensional polytopes a conjecture of stanley that the cd index is minimized on the d dimensional simplex moreover we prove the upper bound theorem for the cd index namely that the cd index of any d dimensional polytope with n vertices is at most that of c n d the d dimensional cyclic polytope with n vertices

flag numbers and floating bodies abstract we investigate weighted floating bodies of polytopes we show that the weighted volume depends on the complete flags of the polytope this connection is obtained by introducing flag simplices which translate between the metric and combinatorial structure our results are applied in spherical and hyperbolic space this leads to new asymptotic results for polytopes in these spaces we also provide explicit examples of spherical and hyperbolic convex bodies whose floating bodies behave completely different from any convex body in euclidean space

f vectors of barycentric subdivisions for a simplicial complex or more generally boolean cell complex we study the behavior of the f and h vector under barycentric subdivision we show that if has a non negative h vector then the h polynomial of its barycentric subdivision has only simple and real zeros as a consequence this implies a strong version of the charney davis conjecture for spheres that are the subdivision of a boolean cell complex or the subdivision of the boundary complex of a simple polytope for a general d 1 dimensional simplicial complex the h polynomial of its n th iterated subdivision shows convergent behavior more precisely we show that among the zeros of this h polynomial there is one converging to infinity and the other d 1 converge to a set of d 1 real numbers which only depends on d