dimer statistics of honeycomb lattices on klein bottle mobius strip and cylinder abstract dimer statistics is a central problem in statistical physics in this paper the enumerations of close packed dimers of honeycomb lattices on klein bottle mobius strip and cylinder are considered by establishing a pfaffian orientation or a crossing orientation and then computing the determinants of the skew symmetric matrices of the resulting orientation graphs we obtain explicit expressions of the number of close packed dimers of the klein bottle polyhex the mobius polyhex and the cylindrical polyhex

the decomposition of lie powers let g be a group f a field of prime characteristic p and v a finite dimensional fg module let l v denote the free lie algebra on v regarded as an fg submodule of the free associative algebra or tensor algebra t v for each positive integer r let l r v and t r v be the r th homogeneous components of l v and t v respectively here l r v is called the r th lie power of v our main result is that there are submodules b 1 b 2 of l v such that for all r b r is a direct summand of t r v and whenever m geqslant 0 and k is not divisible by p the module l p mk v is the direct sum of l p m b k l p m 1 b pk l 1 b p mk thus every lie power is a direct sum of lie powers of p power degree the approach builds on an analysis of t r v as a bimodule for g and the solomon descent algebra

an exactly solvable random satisfiability problem we introduce a new model for the generation of random satisfiability problems it is an extension of the hyper sat model of ricci tersenghi weigt and zecchina which is a variant of the famous k sat model it is extended to q state variables and relates to a different choice of the statistical ensemble the model has an exactly solvable statistic the critical exponents and scaling functions of the sat unsat transition are calculable at zero temperature with no need of replicas also with exact finite size corrections we also introduce an exact duality of the model and show an analogy of thermodynamic properties with the random energy model of disordered spin system theory relations with error correcting codes are also discussed

spectral properties of a matrix of redheffer abstract define n n matrices d n d ij and c n c ij by d ij 1 if i j 0 otherwise and c n 0 1 1 1 t 1 0 0 0 let a n d n c n the matrix is of number theoretic significance because det a n m n is mertens function we give a simplified derivation of the characteristic polynomial of a n give precise asymptotic estimates for its two large eigenvalues and use rouche s theorem to significantly improve a previous bound on its small eigenvalues finally we present some attractive conjectures about these small eigenvalues based on numerical evidence

chernoff bound of trellis coded modulation over correlated fading channels we derive a chernoff upper bound for the pairwise error probability in the presence of an additive white gaussian noise and a rayleigh or rice correlated fading the bound is useful for situations where perfect interleaving cannot be achieved we use it to determine some indications in the design of optimum trellis coded modulation for correlated fading channels

a generalization of permanents and determinants a method is developed for expanding arbitrary powers of the characteristic polynomial of a matrix the coefficients are expressed in terms of matrix functions generalizing those of the permanent and determinant the expansions have applications in probability theory

polynomials with the half plane property and matroid theory a polynomial f is said to have the half plane property if there is an open half plane h subset of c whose boundary contains the origin such that f is non zero whenever all the variables are in h

a unified snr dependent analysis of space time codes in fading channels the pairwise error probability pep for multiple input multiple output mimo radio interfaces is investigated by means of a novel formulation based on compound matrices the proposed approach is suitable for any mimo system having average upper bounded pep written as det i gamma a zeta where a is a hermitian matrix zeta an integer number and gamma the signal to noise ratio snr that bound frequently results in mimo single and multicarrier transmissions it is shown that the minimization of the bounded pep should consider the whole set of nonzero compound matrices of a in particular the snr of interest marks the compound matrix that mainly drives the system performance both diversity advantage and coding gain are given as continuous functions of the variable gamma hence their asymptotic behaviors are taken as important case of studies the interaction effects between channel code and propagation environment are also discussed it is shown how the eigenvectors and eigenvalues of the autocorrelation channel matrix may be considered for code design it is also proved the maximization of the code rank is not always a necessary requirement for performance improvement being its optimal value fixed by the channel structure and snr of interest finally the analysis is applied to space time trellis coded transmissions over spatially correlated slow rayleigh fading channels

the complex wishart distribution and the symmetric group let v be the space of r r hermitian matrices and let omega be the cone of the positive definite ones we say that the random variable s taking its values in overline omega has the complex wishart distribution gamma p sigma if mathbb e exp tr theta s det i r sigma theta p where sigma and sigma 1 theta are in omega and where p 1 2 r 1 or p r 1 in this paper we compute all moments of s and s 1 the techniques involve in particular the use of the irreducible characters of the symmetric group

exact pairwise error probability of space time codes we describe a simple technique for the numerical calculation within any desired degree of accuracy of the pairwise error probability pep of space time codes over fading channels this method applies also to the calculation of e q spl radic spl xi for any nonnegative random variable spl xi whose moment generating function spl phi sub spl xi s e exp s spl xi is known its application to the multiple antenna independent rayleigh fading channel and to the rayleigh block fading channel is discussed and illustrated by two simple examples

use of geometric algebra compound matrices and the determinant of the sum of two matrices in this paper we demonstrate the capabilities of geometric algebra by the derivation of a formula for the determinant of the sum of two matrices in which both matrices are separated in the sense that the resulting expression consists of a sum of traces of products of their compound matrices for the derivation we introduce a vector of grassmann elements associated with an arbitrary square matrix we recall the concept of compound matrices and summarize some of their properties this paper introduces a new derivation and interpretation of the relationship between p forms and the pth compound matrix and it demonstrates the use of geometric algebra which has the potential to be applied to a wide range of problems

extensions and generalizations of a theorem of widder and of the theory of symmetric local semigroups the theory of symmetric local semigroups due to a klein and l landau j funct anal 44 1981 121 136 is generalized to semigroups indexed by subsets of rn for n 1 the result implies a similar result of a e nussbaum j funct anal 48 1982 213 223 it is further generalized to semigroups that are symmetric local in some directions and unitary in others the results are used to give a simple proof of a devinatz s duke math j 22 1955 185 192 and n i akhiezer s the classical moment problem and some related questions hafner new york 1965 generalization of a theorem of widder concerning the representation of functions as laplace integrals this result is extended to the representation as a laplace integral of a function taking values in b r the set of bounded linear operators on a hilbert space r also a theorem is proved encompassing both the result of devinatz and akhiezer and bochner s theorem on the representation of positive definite functions as fourier integrals

generalizing completeness results for loop checks in logic programming abstract loop checking is a mechanism for pruning infinite sld derivations in bol apt and klop 1991 simple loop checks were introduced and their soundness completeness and relative strength was studied since no sound and complete simple loop check exists even in the absence of function symbols subclasses of programs were determined for which the sound loop checks introduced by bol are complete in this paper the generalization theorem is proved this theorem presents a method to extend under certain conditions a class of programs for which a given loop check is complete to a larger class for which the loop check is still complete then this theorem is applied to the results of bol giving rise to stronger completeness theorems it appears that unnecessary complications in the proof of the theorem can be avoided by introducing a normal form for sld derivations allowing only certain most general unifiers this normal form might have other applications than those in the area of loop checking

narain gupta s three normal subgroup problem and group homology abstract this paper is about application of various homological methods to classical problems in the theory of group rings it is shown that the third homology of groups plays a key role in narain gupta s three normal subgroup problem for a free group f and its normal subgroups r s t and the corresponding ideals in the integral group ring z f r r 1 z f s s 1 z f t t 1 z f a complete description of the normal subgroup f 1 rst is given provided r t and the third fourth and fifth homology groups of r r s are torsion groups

the hypertree poset and the l 2 betti numbers of the motion group of the trivial link we give explicit formulae for the euler characteristic and l2 cohomology of the group of motions of the trivial link or isomorphically the group of free group automorphisms that send each standard generator to a conjugate of itself the method is primarily combinatorial and ultimately relies on a computation of the mobius function for the poset of labelled hypertrees

three observations on the determinantal range abstract let a c m n the algebra of n n complex matrices the set of complex numbers c a det a ucu u u i n is the c determinantal range of a in this note it is proved that c a is an elliptical disc for a c m 2 a necessary and sufficient condition for c a to be a line segment is given when a and c are normal matrices with pairwise distinct eigenvalues the linear operators l that satisfy the linear preserver property c a c l a for all a c m n are characterized

ruga formation instabilities of a graded stiffness boundary layer in a neo hookean solid we present an analysis of ruga formation instabilities arising in a graded stiffness boundary layer of a neo hookean half space caused by lateral plane strain compression in this study we repres

loop checks for logic programs with functions two complete loop checking mechanisms have been presented in the literature for logic programs with functions os check and eva check os check is computationally efficient but quite unreliable in that it often misidentifies infinite loops whereas eva check is reliable for a majority of cases but quite expensive in this paper we develop a series of new complete loop checking mechanisms called vaf checks the key technique we introduce is the notion of expanded variants which captures a key structural characteristic of in finite loops we show that our approach is superior to both os check and eva check in that it is as efficient as os check and as reliable as eva check copyright 2001 elsevier science b v

on the eigenstructure of sparse matrices related to the prime number theorem abstract the sparse 0 1 matrix that was introduced by redheffer has the property that its nth leading principal minor equals i n i where is the mobius function recently a substantially sparser 0 1 matrix that possesses the same property has been described let r n denote the nth leading principal submatrix of this sparser matrix our main result establishes asymptotically tight estimates for the two dominant eigenvalues of r n we also state explicit formulae for the eigenvectors of r n that are associated with the eigenvalues that are different from 1 finally we state several conjectures about the eigenstructure of a related class of matrix

milnor numbers spanning trees and the alexander conway polynomial abstract we study relations between the alexander conway polynomial l and milnor higher linking numbers of links from the point of view of finite type vassiliev invariants we give a formula for the first non vanishing coefficient of l of an m component link l all of whose milnor numbers i 1 i p vanish for p n we express this coefficient as a polynomial in milnor numbers of l depending on whether the parity of n is odd or even the terms in this polynomial correspond either to spanning trees in certain graphs or to decompositions of certain 3 graphs into pairs of spanning trees our results complement determinantal formulas of traldi and levine obtained by geometric methods

the design of trellis coded mpsk for fading channels performance criteria it has been well established that the appropriate criterion for optimum trellis coded modulation design on the additive white gaussian noise channel is maximization of the free euclidean distance it is shown that when the trellis coded modulation is used on a rician fading channel with interleaving deinterleaving the design of the code of optimum performance is guided by other factors in particular the length of the shortest error event path and the product of branch distances possibly normalized by the euclidean distance of the path along the path although maximum free distance d sub free is still an important consideration it plays a less significant role the more severe the fading is on the channel these considerations lead to the definition of a new distance measure of optimization of trellis codes transmitted over rician fading channels if no interleaving deinterleaving is used then once again the design of the trellis code is guided by maximizing d sub free

free centre by abelian by exponent 2 groups abstract we study free centre by abelian by exponent 2 groups our main result is a complete description of the centre it is isomorphic to a direct sum of a free abelian group and a torsion subgroup the latter is a direct sum of cyclic groups of order two and cyclic groups of order four we exhibit a generating set consisting of elements of infinite order order 2 and order 4 such that the centre is the direct sum of cyclic subgroups generated by those generators our approach makes essential use of homological methods

on the spectral radius of a 0 1 matrix related to merten s function abstract define n n matrices dn dij and cn cij by dij 1 if i j 0 otherwise and cn 0 1 1 1 t 1 0 0 0 let an dn cn we use the directed graph of an in to obtain the characteristic polynomial of an then we show that all but log2n 1 of the eigenvalues of an are equal to 1 and that an is asymptotically equal to n as n

symmetric matrices related to the mertens function abstract in this paper we explore a family of congruences over n from which one builds a sequence of symmetric matrices related to the mertens function from the results of numerical experiments we formulate a conjecture about the growth of the quadratic norm of these matrices which implies the riemann hypothesis this suggests that matrix analysis methods may come to play a more important role in this classical and difficult problem

loop checking in sld derivations by well quasi ordering of goals abstract loop checking mechanisms are used to detect and prune infinite sld derivations through run time checks which are introduced in logic program interpreters simple loop checks i e checks which do not depend on the specific logic program have been widely studied in literature since no sound and complete loop check exists even in the case of function free programs several subclasses have been characterized for which sound and complete loop checks can be determined in this paper a theoretical framework for analysing properties of loop check mechanisms for logic programs is proposed which exploits general mathematical results about well quasi ordered wqo sets in a way the method can be viewed as a counterpart of well known techniques based on well founded partial ordering used in termination proofs for rewriting systems and for logic programs the main results are obtained on the basis of a combinatorial analysis of properties of wqo sets of goals as shown in the paper subclasses of programs for which sound and complete simple loop checks exist can be easily framed in the wqo approach reasons for the different behaviours of subsumption loop checks based on list and multiset goal comparisons are also plainly highlighted

performance and design of space time coding in fading channels the pairwise error probability upper bounds of space time codes stcs in independent rician fading channels are derived based on the performance analysis novel code design criteria for slow and fast rayleigh fading channels are developed it is found that in fading channels the stc design criteria depend on the value of the possible diversity gain of the system in slow fading channels when the diversity gain is smaller than four the code error performance is dominated by the minimum rank and the minimum determinant of the codeword distance matrix however when the diversity gain is larger than or equal to four the performance is dominated by the minimum squared euclidean distance based on the proposed design criteria new codes are designed and evaluated by simulation

some remarks on a conjecture of de oliveira abstract let u n be the group of the unitary n n matrices let a diag 1 n b diag 1 n where 1 n 1 n are complex numbers a class of normal matrices satisfying the inclusion det a ubu u n co n i 1 i i s n where co is the convex hull of and s n is the symmetric group of degree n is obtained

a limit approach to group homology abstract in this paper we consider for any free presentation g f r of a group g the coinvariance h 0 g r a b n of the nth tensor power of the relation module r a b and show that the homology group h 2 n g z may be identified with the limit of the groups h 0 g r a b n where the limit is taken over the category of these presentations of g we also consider the free lie ring generated by the relation module r a b in order to relate the limit of the groups n r n r f to the n torsion subgroup of h 2 n g z

a sparser matrix representation of the mertens function abstract redheffer introduced a sparse 0 1 matrix whose determinant provides an unorthodox representation of the mertens function the growth of the mertens function is intimately tied to the locations of the nontrivial zeros of the riemann zeta function we introduce a 0 1 matrix whose determinant also provides a nontrivial representation of the mertens function but it is sparser than the matrix of redheffer it is sparser in two ways the matrix has fewer nonzero entries o n compared to o n log n and it has substantially fewer eigenvalues different from 1 this construction is a corollary of a more general result that holds for multiplicative sequences

a space time code design for cpm diversity order and coding gain sufficient conditions are derived under which m ary partial and full response continuous phase modulation cpm space time st codes will attain both full spatial diversity and optimal coding gain general code construction rules are desirable due to the nonlinearity and inherent memory of the cpm signals which makes manual design or computer search difficult using a linear decomposition of cpm signals with tilted phase we identify a rank criterion for m ary partial and full response cpm that specifies the set of allowable modulation indices we also propose a coding gain design criterion optimization of the coding gain for cpm st codes is shown to depend on the cpm frequency phase shaping pulse modulation index and codewords the modulation indices and phase shaping functions that improve the coding gain are specified finally optimization of coding gain for st cpm and orthogonal st cpm codewords is discussed

unimodal beam elements abstract a uniform prismatic elastic element in small displacements theory can be modeled as consisting of extension torsion and flexural elements acting in parallel the extension and torsion elements are unimodal whereas the flexural elements are bimodal due to the combined action of bending moments and shear forces this dissimilarity between the elements is the source of inconsistency between the analysis of a truss and a framed structure especially in a force approach it is shown that by an appropriate change of basis one can uncouple the flexural element into a unimodal moment element which carries the average moment of the element and a unimodal shear element which carries the shear force and related moment as a result any framed structure can be viewed as a generalized truss and analyzed accordingly in a standard form it is further shown that it is possible to physically construct a moment element and a shear element both exhibiting unimodal deformation patterns consequently one can replace a classical beam element by a moment element and a shear element assembled in parallel the approach is tested numerically in the case of beams of constant height subjected to several loading conditions preliminary results indicate that in theory substantial weight reductions can be obtained when designing structures composed of parallel unimodal elements

on the solution of a minimum weight elastoplastic problem involving displacement and complementarity constraints this paper deals with a special class of minimum weight design problem involving discretized structures holonomic reversible plasticity and constraints on displacements a key feature of the optimization problem is the presence of complementarity conditions involving the orthogonality of two sign constrained vectors this problem falls within the important class of so called mathematical programs with equilibrium constraints mpecs two simple algorithms are proposed to solve our synthesis problem and application is illustrated by some examples concerning truss like structures

a method to derive approximate explicit solutions for structural mechanics problems the availability of explicit solutions i e analytical relationships between the structural response and the design variables allows a more direct and plain treatment of several structural problems this paper is devoted to derive approximate explicit solutions in the framework of linear static analysis of finite element modeled structures with a given layout fixed node positions the proposed procedure is based on a factorization of the element stiffness matrix following the unimodal components concept which allows a non conventional assembly of the global stiffness matrix the exact inversion of that matrix is a trivial task for the case of statically determinate structures structures with few redundancies or few design variables an approximate inverse of the stiffness matrix is herein derived for more general structural problems by resorting to the sherman morrison woodbury formula 2006 elsevier ltd all rights reserved

on the eversion of compressible elastic cylinders we consider the eversion problem for compressible hyperelastic isotropic thick walled cylinders we give a new strain energy function for a highly compressible material that admits an exact solution to the problem with point wise end conditions previously only averaged loads could be accounted for secondly we investigate the bifurcation problem for a wide range of material models we find qualitively similar results to those obtained for incompressible materials while thin walled cylinders can be everted into other right circular cylinders if the wall thickness exceeds some critical value bifurcation into non cylindrical shapes is possible

a dominant negative eigenvalue of a matrix of redheffer abstract define the matrix an as the sum of dn dij and cn where dij 1 when i j and zero when i j and where cn 0 1 1 1 t 1 0 0 0 we use the gersgorin disc theorem and previous results regarding the spectral radiusand the characteristic polynomial of an to prove the existence of a real eigenvalue which is asymptotically equal to n as n we also show that all eigenvalues except the spectral radius and eigenvalue of order n are for large n contained in the circle z c z 1 n log n where an is the spectral radius

the group with grassmann structure uosp 1 2 the finite dimensional representations of the lie superalgebraosp 1 2 and the group with grassmann structureosp 1 2 have been studied the explicit expression of the projection operator of the superalgebraosp 1 2 has been found the operator permits an arbitrary finite dimensional representation to be expanded in the components multiple to the irreducible ones the clebsch gordan coefficients for the tensor product of two arbitrary irreducible representations have been obtained the matrix elements of the irreducible representations of the groupuosp 1 2 the analoque of the compact form of the grouposp 1 2 are studied the explicit form of these matrix elements the differential equations satisfied by them and the integral of their product have been found

on the existence of nonterminating queries for a restricted class of prolog clauses abstract this work studies the existence of nonterminating queries for a restricted class of recursive predicates the class is restricted to predicates defined by a set of clauses containing only one direct recursive clause moreover this clause must be of the form p p for this restricted class of predicates we prove a necessary and sufficient condition for the existence of a query which is nonterminating in the absence of the occur check the condition is based upon concepts from rational trees

mertens equimodular matrices of redheffer type abstract asymptotic behavior of the mertens function m n k 0 n k which is equal to the determinant of the n n redheffer matrix is known to be closely related to the riemann hypothesis an infinite matrix whose nth leading principal minor is equal to m n for all n 1 is called a mertens equimodular matrix we use riordan matrices to find a large class of mertens equimodular matrices each element of which is called by us a riordan redheffer matrix briefly an r r matrix we also give the generating function for the characteristic polynomials of r r matrices as a result we introduce several examples of r r matrices that reveal interesting spectral properties further we pose two conjectures on the eigenvalues of those r r matrices finally we find a sufficient condition for the riemann hypothesis using the smallest singular value of a r r matrix

the lukacs olkin rubin characterization of wishart distributions on symmetric cones we characterize the wishart distributions on a symmetric cone c if c 0 this has been done by lukacs in 1955 if c is the cone of positive definite symmetric matrices this has been done by olkin and rubin in 1962 we both shorten and extend the olkin rubin proof sometimes obscure by using three modern ideas i try to avoid artificial coordinates in differential geometry ii the variance function of a natural exponential family f characterizes f iii symmetric matrices are a particular example of a euclidean simple jordan algebra

controlling recursive inference abstract loosely speaking recursive inference occurs when an inference procedure generates an infinite sequence of similar subgoals in general the control of recursive inference involves demonstrating that recursive portions of a search space will not contribute any new answers to the problem beyond a certain level we first review a well known syntactic method for controlling repeating inference inference where the conjuncts processed are instances of their ancestors provide a proof that it is correct and discuss the conditions under which the strategy is optimal we also derive more powerful pruning theorems for cases involving transitivity axioms and cases involving subsumed subgoals the treatment of repeating inference is followed by consideration of the more difficult problem of recursive inference that does not repeat here we show how knowledge of the properties of the relations involved and knowledge about the contents of the system s database can be used to prove that portions of a search space will not contribute any new answers

bicolouring random hypergraphs we study the problem of bicolouring random hypergraphs both numerically and analytically we apply the zero temperature cavity method to find analytical results for the phase transitions dynamic and static in the one step replica symmetry breaking 1rsb approximation these points appear to be in agreement with the results of the numerical algorithm in the second part we implement and test the survey propagation algorithm for specific bicolouring instances in the so called hard sat phase

redundancy elimination and loop checks for logic programs a simple analysis of the arguments developed by bol et al theoret comput sci 86 35 79 1991 shows that an actual reason for the nonexistence of a complete sound simple check for all function free programs is the presence in the resolvents of potentially unlimited sequences of atoms chained by common variables this hints that a limitation of the number of variables generating this kind of chain could guarantee the applicability of complete simple loop checks this line is followed in the paper and quite general classes of logic programs are characterized without any direct imposition on the structures of the rules this objective is accomplished by exploiting a variant of sld resolution which is able to perform a systematic elimination of redundant atoms from resolvents as a notable result it turns out that the equality loop check is complete for our class of logic programs this seems to suggest that the necessity of using subsumption loop checks instead of equality checks is essentially due to the presence of redundant atoms in resolvents

matrices satisfying a conjecture of g n de oliveira on determinants abstract it is shown that the determinant of the sum of a positive definite hermitian matrix and a skew hermitian matrix with eigenvalues j and i j j 1 n respectively is contained in the convex hull of the points j 1 n j i j s n as usual s n denotes the symmetric group of degree n this result proves in a particular case a conjecture of g n de oliveira on determinants

complete monotonicity for inverse powers of some combinatorially defined polynomials we prove the complete monotonicity on 0 infty n for suitable inverse powers of the spanning tree polynomials of graphs and more generally of the basis generating polynomials of certain classes of matroids this generalizes a result of szeg and answers among other things a long standing question of lewy and askey concerning the positivity of taylor coefficients for certain rational functions our proofs are based on two ab initio methods for proving that p beta is completely monotone on a convex cone c the determinantal method and the quadratic form method these methods are closely connected with harmonic analysis on euclidean jordan algebras or equivalently on symmetric cones we furthermore have a variety of constructions that given such polynomials can create other ones with the same property among these are algebraic analogues of the matroid operations of deletion contraction direct sum parallel connection series connection and 2 sum the complete monotonicity of p beta for some beta 0 can be viewed as a strong quantitative version of the half plane property hurwitz stability for p and is also related to the rayleigh property for matroids

chernoff bounds on pairwise error probabilities of space time codes we derive chernoff bounds on pairwise error probabilities of coherent and noncoherent space time signaling schemes first general chernoff bound expressions are derived for a correlated ricean fading channel and correlated additive gaussian noise then we specialize the obtained results to the cases of space time separable noise white noise and uncorrelated fading we derive approximate chernoff bounds for high and low signal to noise ratios snrs and propose optimal signaling schemes we also compute the optimal number of transmitter antennas for noncoherent signaling with unitary mutually orthogonal space time codes

potts models and random cluster processes with many body interactions known differential inequalities for certain ferromagnetic potts models with pair interactions may be extended to potts models with many body interactions as a major application of such differential inequalities we obtain necessary and sufficient conditions on the set of interactions of such a potts model in order that its critical point be astrictly monotonic function of the strengths of interactions the method yields some ancillary information concerning the equality of certain critical exponents for potts models this amounts to a small amount of rigorous universality these results are achieved in the context of a fortuin kasteleyn representation of potts models with many body interactions for such a potts model the corresponding random cluster process is a random hypergraph

omc for scalar multiples of unitaries abstract we establish the following case of the determinantal conjecture of marcus m marcus derivations plucker relations and the numerical range indiana univ math j 22 1973 1137 1149 and de oliveira g n de oliveira research problem normal matrices linear and multilinear algebra 12 1982 153 154 let a and b be unitary n n matrices with prescribed eigenvalues a1 an and b1 bn respectively then for any scalars t and s det ta sb co j 1 n ta j sb j s n where s n denotes the group of all permutations of 1 n and co the convex hull taken in the complex plane

on a conjecture of marcus and de oliveira abstract let x y be two normal n n matrices over c with respective spectra x 1 x n and y 1 y n marcus and de oliveira conjectured that the determinant of x y lies in the convex hull generated by the n complex points n i 1 x i y i where ranges over the symmetric group s n we prove the conjecture for the case that n 2 of the y i are equal first the problem is reduced to the question of nonnegative solvability of a certain system of linear equations then the required nonnegative solvability is demonstrated by involving a measure theoretic generalization of the marriage lemma our main result forms a positive counterpart to a recent example given by drury

asymptotics of multivariate sequences part iii quadratic points abstract we consider a number of combinatorial problems in which rational generating functions may be obtained whose denominators have factors with certain singularities specifically there exist points near which one of the factors is asymptotic to a nondegenerate quadratic we compute the asymptotics of the coefficients of such a generating function the computation requires some topological deformations as well as fourier laplace transforms of generalized functions we apply the results of the theory to specific combinatorial problems such as aztec diamond tilings cube groves and multi set permutations

the explicit inverse of the stiffness matrix abstract in two recent papers the author has given the exact analytic expression of the internal forces in linear elastic structures composed of uniform prismatic elements it was shown that the member forces are the ratios of two multilinear homogeneous polynomials in the unimodal stiffnesses of the elements of the structure the order of the polynomials is equal to the number of nodal degrees of freedom of the structure the number of terms of each polynomial is equal to the number of statically determinate stable substructures which can be derived from the original structure the coefficients of the polynomials can be computed by employing the equilibrium equations and by enforcing global compatibility of deformations it was found empirically that the coefficients of the polynomial in the denominator were numerically equal to the square of the determinants of the statics matrices of the respective statically determinate substructures as a consequence the denominator became the sum of the stiffness matrices of the statically determinate substructures this is in fact the binet cauchy form of the determinant of the stiffness matrix of the structure bearing in mind that the inverse of the stiffness matrix can be expressed as the ratio of the adjoint matrix of the stiffness matrix divided by the determinant of the stiffness matrix it became clear that the expressions of the stress resultants stem from an explicit expression of the adjoint the explicit expression of the adjoint of the stiffness matrix lies at the heart of this paper it is shown that the adjoint is a congruent transformation of the n 1 compound of the stiffness matrix where n is the number of degrees of freedom of the structure this cleared the way to use the binct cauchy theorem on the product of compound matrices to obtain an explicit expression for the adjoint and ipso facto for the inverse of the stiffness matrix having now the displacements of the structure the expression of the stress resultants which was obtained independently emerges in a very elegant manner the member forces in a structure can be expressed as the weighted sum of the member forces in all its determinate substructures when subjected to the applied loads the weighting factors are the ratios of the determinants of he stiffness matrices of the substructures to the determinant of the stiffness matrix of the original structure both the explicit inverse of the stiffness matrix and the expression of the internal forces in the structure are at present of a theoretical nature the number of terms involved in the polynomials is simply excessive for common engineering structures however ongoing reseach may yield more applicable expressions to be used for instance in he field of automated design of structures the theory is illustrated with the explicit analysis of a stayed mast

bordered hermitian matrices and sums of the mobius function abstract matrices in r n n with determinant equal to i n i where represents the mobius function have been studied for decades the motivation to study such matrices is the close connection that they have to the prime number theorem dirichlet convolution and related concepts we introduce two parameterized families of bordered hermitian matrices that possess similar properties each family is comprised of matrices m s c n 1 n 1 that satisfy det m 0 i n i det m 1 i n i 2 and we show det m s is a quadratic polynomial in s we apply the cauchy interlacing theorem to show that for each matrix in one of the families the product of all of the subdominant eigenvalues is bounded above by 6 2 o n 1 2

twenty combinatorial examples of asymptotics derived from multivariate generating functions let a bf r bf r in mathbb n d be a d dimensional array of numbers for which the generating function f bf z sum bf r a bf r bf z bf r is meromorphic in a neighborhood of the origin for example f may be a rational multivariate generating function we discuss recent results that allow the effective computation of asymptotic expansions for the coefficients of f our purpose is to illustrate the use of these techniques on a variety of problems of combinatorial interest the survey begins by summarizing previous work on the asymptotics of univariate and multivariate generating functions next we describe the morse theoretic underpinnings of some new asymptotic techniques we then quote and summarize these results in such a way that only elementary analyses are needed to check hypotheses and carry out computations the remainder of the survey focuses on combinatorial applications such as enumeration of words with forbidden substrings edges and cycles in graphs polyominoes and descents in permutations after the individual examples we discuss three broad classes of examples namely functions derived via the transfer matrix method those derived via the kernel method and those derived via the method of lagrange inversion these methods have the property that generating functions derived from them are amenable to our asymptotic analyses and we describe further machinery that facilitates computations for these classes of examples

spanning forests and the q state potts model in the limit q 0 we study the q state potts model with nearest neighbor coupling v e j 1 in the limit q v 0 with the ratio w v q held fixed combinatorially this limit gives rise to the generating polynomial of spanning forests physically it provides information about the potts model phase diagram in the neighborhood of q v 0 0 we have studied this model on the square and triangular lattices using a transfer matrix approach at both real and complex values of w for both lattices we have computed the symbolic transfer matrices for cylindrical strips of widths 2 l 10 as well as the limiting curves b of partition function zeros in the complex w plane for real w we find two distinct phases separated by a transition point w w0 where w0 1 4 resp w0 0 1753 0 0002 for the square resp triangular lattice for w w0 we find a non critical disordered phase that is compatible with the predicted asymptotic freedom as w for w w0 our results are compatible with a massless berker kadanoff phase with central charge c 2 and leading thermal scaling dimension xt 1 2 marginally irrelevant operator at w w0 we find a first order critical point the first derivative of the free energy is discontinuous at w0 while the correlation length diverges as w w0 and is infinite at w w0 the critical ehavior at w w0 seems to be the same for both lattices and it differs from that of the berker kadanoff phase our results suggest that the central charge is c 1 the leading thermal scaling dimension is xt 1 0 and the critical exponents are 1 d 1 2 and 1

bochner and bernstein theorems via the nuclear integral representation theorem abstract we show how using the nuclear integral representation theorem the bernstein choquet theorem and bochner schwartz theorem may be derived in the case of the bernstein choquet theorem we give an example determining the representing measure explicitly

the validity of the marcus de oliveira conjecture for essentially hermitian matrices abstract let x y be matrices of spectra x 1 x n and y 1 y n respectively for a wide class of pairs of essentially hermitian matrices x y we prove that the determinant of x y belongs to the convex hull of the n points n i 1 x i y i s n

some notes on de oliveira s determinantal conjecture abstract let a b u n x n a diag a j b diag b j u unitary d det a u b u h z j 1 n a j b j s n de oliveira s conjecture d is a convex combination of the z s is shown to be true if all the z s are collinear the question of representing u m 2 u m is the m th compound of u as a convex combination of the p m s is also studied here and 2 are understood elementwise and p denotes the permutation matrix with rows corresponding to

interval analysis of structures with uncertain but bounded axial stiffness abstract a method to evaluate the static response of structures with interval axial stiffnesses is proposed the procedure overcomes the drawbacks arising in traditional interval analysis due to dependency phenomenon the inversion of the interval stiffness matrix is pursued by means of a novel approximate relationship based on an interval valued sherman morrison woodbury formula numerical results on truss and frame structures evidence the accuracy and the effectiveness of the proposed approach to evaluate bounds of displacements and internal forces

the jordan 1 structure of a matrix of redheffer abstract let 1 1 0 0 and 1 1 1 be complex vectors of length n and let d n d ij with d ij 1 if i j and 0 otherwise be an n n complex matrix then the sizes of the jordan blocks of the matrix t 1 d n corresponding to the eigenvalue 1 are log 2 n 3 1 log 2 n 5 1 log 2 n n 1 where n is the greatest odd integer less than or equal to n

negative dependence and the geometry of polynomials we introduce the class of strongly rayleigh probability measures by means of geometric properties of their generating polynomials that amount to the stability of the latter this class covers important models such as determinantal measures e g product measures and uniform random spanning tree measures and distributions for symmetric exclusion processes we show that strongly rayleigh measures enjoy all virtues of negative dependence and we also prove a series of conjectures due to liggett pemantle and wagner respectively moreover we extend lyons recent results on determinantal measures and we construct counterexamples to several conjectures of pemantle and wagner on negative dependence and ultra log concave rank sequences

on the determinantal conjecture of marcus and de oliveira abstract we provide a theorem and a counterexample relating to the determinantal conjecture of marcus and de oliveira

joint transmitter receiver optimization for multi input multi output systems with decision feedback the joint transmit receive optimization problem for multiuser communication systems with decision feedback is investigated it is shown that minimization of the geometric mean squared error defined as the determinant of the error covariance matrix leads to a tractable transmitter optimization problem for general multi input multi output decision feedback systems several computational results are included that highlight system performance for a variety of useful transmission scenarios

solutions to two problems on permanents in this note we settle two open problems in the theory of permanents by using recent results from other areas of mathematics both problems were recently discussed in bapat s survey 2 bapat conj

function spaces and reproducing kernels on bounded symmetric domains there are several natural hilbert spaces of holomorphic functions on a bounded symmetric domain such are the bergman type spaces on which the holomorphic discrete series operates and the hardy type spaces which are related to the analytic continuation of the holomorphic discrete series also closely related is the bergmann space of entire functions on the ambient which arises as the closure of the polynomials with respect to a natural inner product the space of holomorphic polynomials decomposes into irreducible subspaces under the action of the isotropy group k of the domain the main facts about this decomposition were proved by schmid 21 for another proof see 22 each irreducible subspace contains a unique normalized l invariant spherical polynomial where l is the isotropy group of the shilov boundary in k our first main result is the explicit computation of the norms of the spherical polynomials with respect to each of the hilbert spaces considered for the domains of classical type they were considered by hua 9 and for some of the hilbert spaces this was done before by upmeier 24 using different methods for certain others there are partial results in 22 we are able to do this in a fairly simple unified way by making strong use of gindikin s generalized gamma function s next we obtain a description of the reproducing kernels of the k irreducible subspaces in each of our hilbert spaces and an expansion in terms of these for

multivariate stable polynomials theory and applications univariate polynomials with only real roots while special do occur often enough that their properties can lead to interesting conclusions in diverse areas due mainly to the recent work of two young mathematicians julius borcea and petter branden a very successful multivariate generalization of this method has been developed the first part of this paper surveys some of the main results of this theory of multivariate stable polynomials the most central of these results is the characterization of linear transformations preserving stability of polynomials the second part presents various applications of this theory in complex analysis matrix theory probability and statistical mechanics and combinatorics

strict diagonal dominance and a gersgorin type theorem in euclidean jordan algebras abstract for complex square matrices the levy desplanques theorem asserts that a strictly diagonally dominant matrix is invertible the well known gersgorin theorem on the location of eigenvalues is equivalent to this in this article we extend the levy desplanques theorem to an object in a euclidean jordan algebra when its peirce decomposition with respect to a jordan frame is given as a consequence we prove a gersgorin type theorem for the spectral eigenvalues of an object in a euclidean jordan algebra

simplest random k satisfiability problem we study a simple and exactly solvable model for the generation of random satisfiability problems these consist of gamman random boolean constraints which are to be satisfied simultaneously by n logical variables in statistical mechanics language the considered model can be seen as a diluted p spin model at zero temperature while such problems become extraordinarily hard to solve by local search methods in a large region of the parameter space still at least one solution may be superimposed by construction the statistical properties of the model can be studied exactly by the replica method and each single instance can be analyzed in polynomial time by a simple global solution method the geometrical and topological structures responsible for dynamic and static phase transitions as well as for the onset of computational complexity in the local search method are thoroughly analyzed numerical analysis on very large samples allows for a precise characterization of the critical scaling behavior

an exact decomposition theorem and a unified view of some related distributions for a class of exponential transformation models on symmetric cones a class of exponential transformation models is defined on symmetric cones q with the group of automorphisms on q as the acting group we show that these models are reproductive and the exponent of their joint distribution for a given sample of size q can be split into q independent components introducing one sample point at a time the automorphism group can be factorized into the group of positive dilation and another group accordingly the symmetric cone q can be seen as the direct product of r and a unit orbit and every x in q can be identified by its orbital decomposition we derive the distributions of the independent components of the exponent of the length of x the direction of x the conditional distribution of the direction given the length and other distributions for a given sample the wishart distribution and the hyperboloid distribution are two special cases in the class we define we also give a unified view of several distributions which are usually treated separately

on the c determinantal range for special classes of matrices let a and c be square complex matrices of size n the c determinantal range of a is the subset of the complex plane det a u c u u u i n if a c are both hermitian matrices then by a result of fiedler 1971 11 this set is a real line segment in our paper we study this set for the case when c is a hermitian matrix our purpose is to revisit and improve two well known results on this topic the first result is due to li concerning the c numerical range of a hermitian matrix see condition 5 1 a in li 1994 20 the second one is due to c k li y t poon and n s sze about necessary and sufficient conditions for the c determinantal range of a to be a subset of the line see li et al 2008 21 theorem 3 3

geometrical relations between space time block code designs and complexity reduction in this work the geometric relation between space time block code design for the coherent channel and its noncoherent counterpart is exploited to get an analog of the information theoretic inequality i x s lesi x h s in terms of diversity it provides a lower bound on the performance of noncoherent codes when used in coherent scenarios this leads in turn to a code design decomposition result splitting coherent code design into two complexity reduced subtasks moreover a geometrical criterion for high performance space time code design is derived

grassmann integral representation for spanning hyperforests given a hypergraph g we introduce a grassmann algebra over the vertex set and show that a class of grassmann integrals permits an expansion in terms of spanning hyperforests special cases provide the generating functions for rooted and unrooted spanning hyper forests and spanning hyper trees all these results are generalizations of kirchhoff s matrix tree theorem furthermore we show that the class of integrals describing unrooted spanning hyper forests is induced by a theory with an underlying osp 1 2 supersymmetry

theorems of perron frobenius type for matrices without sign restrictions abstract the paper attempts to solve a problem which was called a challenge for the future in linear algebra appl we define and investigate a new quantity for real matrices the sign real spectral radius and derive various characterizations bounds and properties of it in certain aspects our quantity shows similar behavior to the perron root of a nonnegative matrix it is shown that our quantity also has intimate connections to the componentwise distance to the nearest singular matrix relations to the perron root of the entrywise absolute value of the matrix and to the number are given as well

norms on terms and their use in proving universal termination of a logic program abstract in this paper semi linear norms a class of functions to weight the terms occurring in a program are defined and studied all the functions in this class have the nice property of allowing a syntactical characterization of rigid terms i e terms whose weight does not change under substitution based on these norms a general proof method for universal termination of pure prolog programs can be adapted to deal with a large class of programs in a simple way the proof method requires pre post specifications well behaved with respect to substitutions to be associated with each predicate symbol in the program and ordering functions not increasing with respect to substitutions to be associated with cycles in the program the specification collects information on term properties which are useful to prove that the ordering functions actually decrease at each traversal of each cycle some examples of termination proof are also given

analytic representation of member forces in linear elastic redundant trusses abstract the process of designing a truss of given geometry and material properties where the design variables are the cross sectional areas of the bars is hampered by the need to reanalyse the structure many times until an acceptable design is obtained currently approximate explicit analysis models based on truncated linear taylor series expansions are used to evaluate the structural response at the various candidate design points due to the approximate nature of the analysis model the structure is designed iteratively until convergence of both the analysis equations and the design process this paper presents for the first time the exact analytic expressions of the internal loads in a truss which is subjected to static loads the stress resultants are the ratio of two multilinear polynomials in the element stillness the number of terms of the polynomials is equal to the number of combinations of statically determinate stable structures which can be derived from the original structures the coefficients of the polynomial expansions can be obtained from equilibrium considerations and from enforcing global compatibility of deformations the expressions are explicit in both the external loads and the element stiffness the applicability of the analytic equations hinges on the number of combinations of statically determinate stable substructures in the case of small size structures the present explicit equations circumvent the need for approximate reanalysis in common engineering structures the number of stable subsets is prohibitively large which renders the analytic expressions intractable the exact analytic expressions may however constitute a starting point for constructing approximate explicit analysis equations of improved quality