plethystic hopf algebras abstract the notion of a plethystic algebra associated with a hopf algebra endowed with a suitable bilinear form is defined a special case is the hopf algebra of symmetric functions

transformation groups for soliton equations iv a new hierarchy of soliton equations of kp type abstract a new approach to soliton equations based on functions or hirota s dependent variables vertex operators and the clifford algebra of free fermions is applied to study a new hierarchy of kadomtsev petviashvili type equations the bkp hierarchy the infinite dimensional orthogonal group acts on the space of bkp functions the sawada kotera equation is obtained as a reduction of bkp its infinitesimal transformations constitute the euclidean lie algebra a2 2

exact operator solution of the calogero sutherland model the wave functions of the calogero sutherland model are known to be expressible in terms of jack polynomials a formula which allows to obtain the wave functions of the excited states by acting with a string of creation operators on the wave function of the ground state is presented and derived the creation operators that enter in this formula of rodrigues type for the jack polynomials involve dunkl operators

loewner matrices and operator convexity let f be a function from mathbb r into itself a classic theorem of k lowner says that f is operator monotone if and only if all matrices of the form left frac f p i f p j p i p j right vphantom x x 1 are positive semidefinite we show that f is operator convex if and only if all such matrices are conditionally negative definite and that f t t g t for some operator convex function g if and only if these matrices are conditionally positive definite elementary proofs are given for the most interesting special cases f t t r and f t t log t several consequences are derived

analogies between finite dimensional irreducible representations of so 2n and infinite dimensional irreducible representations of sp 2n r i characters and products the analogy between the finite dimensional spin representation of so 2n and the infinite dimensional representation of sp 2n r is made precise it is then shown that this analogy can be extended so as to provide a precise link between each finite dimensional unitary irreducible representation of so 2n and a corresponding infinite dimensional unitary irreducible representation of sp 2n r the analogy shows itself at the level of the corresponding characters and difference characters and involves the use of schur function methods to express both characters and difference characters of so 2n and sp 2n r in terms of characters of irreducible representations of their common subgroup u n the analogy is extended still further to cover the explicit decomposition of not only tensor products of and with other unitary irreducible representations of so 2n and sp 2n r respectively but also arbitrary tensor powers of and

a weak majorization involving the matrices a b and ab abstract let a and b be n n positive definite matrices and let the eigenvalues of a b and ab be arranged in decreasing order then for all r 0 i k n i r ab i k n i r a b for k 1 n furthermore it is shown that i k n i r a b i k n i r ab for k 1 n is a limiting case of the weak majorization above which settles a conjecture of bapat and johnson

plethystic algebras and vector symmetric functions an isomorphism is established between the plethystic hopf algebra pleth super l and the algebra of vector symmetric functions the hall inner product of symmetric function theory is extended to the hopf algebra pleth super l

some results on kwong functions and related inequalities abstract we investigate some relations between kwong functions and operator monotone functions as an application we present an arithmetic geometric mean type inequality by showing that for two continuous functions f g on 0 such that h t f t g t is a kwong function and f t g t t any positive matrices a b and any matrix x it holds that f a x g b g a x f b a x x b for each unitarily invariant norm

sequences of symmetric polynomials and combinatorial properties of tableaux abstract in 1977 g p thomas showed that the sequence of schur polynomials associated to a partition can be comfortably generated from the sequence of variablesx x1 x2 x3 by the application of mixed baxter multiplication operators which in turn can be easily computed from the set syt of standard young tableaux of shape we generalize this construction thereby making possible the explicit and effective computation of the hall littlewood jack and macdonald polynomials used in representation theory combinatorics multivariate statistics and quantum algebra these generalized formulas have a pleasing recursive structure with respect to the young lattice and they can easily be specialized to yield skew forms in all cases and super forms in the schur case we introduce and investigate 1 the descent polynomial of a partition which arises naturally in the enumeration of semistandard young tableaux of shape 2 the boolean latticeg associated to any syt which is fundamental for the weighted generalization of thomas approach to schur polynomials and 3 an action of the symmetric groups on semistandard young tableaux which is connected with knuth s combinatorial proof of the symmetry of schur functions moreover we argue that a generalization of thomas approach is a natural starting point in search of universal weighted symmetric functions

a singular value inequality for heinz means we prove a matrix inequality for matrix monotone functions and apply it to prove a singular value inequality for heinz means recently conjectured by zhan

information based objective functions for active data selection learning can be made more efficient if we can actively select particularly salient data points within a bayesian learning framework objective functions are discussed that measure the expected informativeness of candidate measurements three alternative specifications of what we want to gain information about lead to three different criteria for data selection all these criteria depend on the assumption that the hypothesis space is correct which may prove to be their main weakness

more results on singular value inequalities of matrices abstract the arithmetic geometric mean inequality for singular values due to bhatia and kittaneh says that 2 s j ab s j a a b b j 1 2 for any matrices a b we give a new equivalent form and some relevant generalizations of this inequality in particular we show that s j a 1 4 b 3 4 a 3 4 b 1 4 s j a b j 1 n for any n n positive semidefinite matrices a b which proves a special case of zhan s conjecture posed in 2000

positive semidefinite solutions of the operator equation j 1nan jxaj 1 b abstract let a be a positive definite operator and b be a self adjoint operator we discuss the existence of positive semidefinite solutions of the operator equation j 1 n a n j xa j 1 b via an order preserving operator inequality by using the solutions we also give concrete examples of positive semidefinite matrices

some combinatorial properties of jack symmetric functions if 1 l n then write 2 n or 1 n if p is another partition then write p c j if pi 6 for all i i e if the diagram of 1 contains the diagram of p if ipl ii 1 then write p 2 1 reverse lexicographic order if either j i or the first nonvanishing difference e pi is positive for instance writing 2 1 i for ai i it 5 5 41 32 5 311 221 2111 11111 finally write p i if ipl lal and p1 pl2 pli a1 a2 li for all i macdonald m p 63 call the partial ordering the natural ordering but we will call it the dominance ordering let x x x2 be an infinite set of indeterminates as in mi we

an operator inequality abstract for hilbert space operators with s invertible hermitian it is proved that sts 1 s 1 ts 2 t

holomorphic discrete series and harmonic series unitary irreducible representations of non compact lie groups sp 2n r u p q and so 2n generalised characters of the infinite dimensional holomorphic discrete series unitary irreducible representations of the non compact groups u p q sp 2n r and so 2n are explicitly expressed in terms of characters of finite dimensional unitary group representations these formulae are remarkably succinct despite involving certain infinite series of schur functions similar formulae are derived for harmonic series unitary representation of both u p q and sp 2n r consideration of the branching rules from u p q to u q p and from sp 2n r to u n enables holomorphic representations to be identified as a subset of the harmonic representations the branching rules are established in full generality and are then used in the evaluation of tensor products of both holomorphic and harmonic representations in the case of the former a known result is recast in terms of closed formulae involving schur functions and for the latter various generalisations of these formulae are given a conjecture is also made regarding what might be the simplest possible formulae covering all holomorphic and harmonic representations of sp 2n r and u p q illustrative examples are presented

interpolating the arithmetic geometric mean inequality and its operator version two families of means called heinz means and heron means that interpolate between the geometric and the arithmetic mean are considered comparison inequalities between them are established operator versions of these inequalities are obtained failure of such extensions in some cases is illustrated by a simple example

determinantal forms for composite schur and q functions via the boson fermion correspondence the boson fermion correspondence is used to represent a composite schur function as a fermionic expectation value wick s theorem then enables it to be written as a determinant generalizing the giambelli formula for ordinary schur functions as single hook determinants the schur q function analogue recently introduced in the context of the bkp hierarchy is also given

vertex operators arising from jacobi trudi identities we give an interpretation of the boson fermion correspondence as a direct consequence of the jacobi trudi identity this viewpoint enables us to construct from a generalized version of the jacobi trudi identity the action of a clifford algebra on the polynomial algebras that arrive as analogues of the algebra of symmetric functions a generalized giambelli identity is also proved to follow from that identity as applications we obtain explicit formulas for vertex operators corresponding to characters of the classical lie algebras shifted schur functions and generalized schur symmetric functions associated to linear recurrence relations

a problem of cayley from 1857 and how he could have solved it abstract in a paper of 1857 cayley suggests the problem of determining all the symmetric functions of the common solutions of a linear cubic homogeneous system of ternary equations as functions of the coefficients of the equations we show how cayley could have solved the problem himself if he had taken another look at it all the facts necessary for the solution were known to cayley including a crucial formula known as cayley s identity we study the problem by mixing techniques from vector symmetric function theory with techniques from the study of determinants and permanents we develop some new formulas for determinants of matrices which arise by the hadamard or pointwise product from other matrices

symmetric function products and plethysms and the boson fermion correspondence we use the boson fermion correspondence for s and q functions to establish some interesting properties concerning outer products and plethysms of s functions or q functions by power sum symmetric functions the techniques which are developed are also applied to computing the inverse kostka foulkes matrix which is the transition matrix between hall littlewood symmetric functions and s functions in some simple cases

notes on matrix arithmetic geometric mean inequalities abstract for positive semi definite n n matrices the inequality 4 ab a b 2 isshown to hold for every unitarily invariant norm the connection of this with some other matrix arithmetic geometric mean inequalities and trace inequalities is discussed

on the hopf algebraic origin of wick normal ordering a combinatorial formula of rota and stein is taken to perform wick reordering in quantum field theory wick s theorem becomes a hopf algebraic identity called cliffordization the combinatorial method relying on hopf algebras is highly efficient in computations and yields closed algebraic expressions

improved young and heinz inequalities for matrices we give refinements of the classical young inequality for positive real numbers and we use these refinements to establish improved young and heinz inequalities for matrices

plethysms replicated schur functions and series with applications to vertex operators specializations of schur functions are exploited to define and evaluate the schur functions s x and plethysms s s x for any integer real or complex plethysms are then used to define pairs of mutually inverse infinite series of schur functions m and l specified by arbitrary partitions these are used in turn to define and provide generating functions for formal characters s of certain groups h thereby extending known results for orthogonal and symplectic group characters each of these formal characters is then given a vertex operator realization first in terms of the seriesm m 0 and various l dual to l and then more explicitly in the exponential form finally the replicated form of such vertex operators are written down the characters of the orthogonal and symplectic groups have been found by schur 34 and weyl 35 respectively the method used is transcendental and depends on integration over the group manifold these characters however may be obtained by purely algebraic methods this algebraic method would seem to offer a better prospect of successful application to other restricted groups than the method of group integration littlewood d e 1944 phil trans r soc london ser a 239 809 392 pacs numbers 02 10 v 02 10 de 02 20 a 02 20 hj mathematics subject classification 05e05 17b69 11e57 16w30 20e22 33d52 43a40 1751 8113

convexity of the inverse and moore penrose inverse abstract convexity properties of the inverse of positive definite matrices and the moore penrose inverse of nonnegative definite matrices with respect to the partial ordering induced by nonnegative definiteness are studied for the positive definite case null space characterizations are derived and lead naturally to a concept of strong convexity of a matrix function extending the conventional concept of strict convexity the positive definite results are shown to allow for a unified analysis of problems in reproducing kernel hilbert space theory and inequalities involving matrix means the main results comprise a detailed study of the convexity properties of the moore penrose inverse providing extensions and generalizations of all the earlier work in this area

non commutative hopf algebra of formal diffeomorphisms abstract this paper deals with two hopf algebras which are the non commutative analogues of two different groups of formal power series the first group is the set of invertible series with the group law being multiplication of series while the second is the set of formal diffeomorphisms with the group law being a composition of series the motivation to introduce these hopf algebras comes from the study of formal series with non commutative coefficients invertible series with non commutative coefficients still form a group and we interpret the corresponding new non commutative hopf algebra as an alternative to the natural hopf algebra given by the co ordinate ring of the group which has the advantage of being functorial in the algebra of coefficients for the formal diffeomorphisms with non commutative coefficients this interpretation fails because in this case the composition is not associative anymore however we show that for the dual non commutative algebra there exists a natural co associative co product defining a non commutative hopf algebra moreover we give an explicit formula for the antipode which represents a non commutative version of the lagrange inversion formula and we show that its coefficients are related to planar binary trees then we extend these results to the semi direct co product of the previous hopf algebras and to series in several variables finally we show how the non commutative hopf algebras of formal series are related to some renormalization hopf algebras which are combinatorial hopf algebras motivated by the renormalization procedure in quantum field theory and to the renormalization functor given by the double tensor algebra on a bi algebra

adaptive relevance matrices in learning vector quantization we propose a new matrix learning scheme to extend relevance learning vector quantization rlvq an efficient prototype based classification algorithm toward a general adaptive metric by introducing a full matrix of relevance factors in the distance measure correlations between different features and their importance for the classification scheme can be taken into account and automated and general metric adaptation takes place during training in comparison to the weighted euclidean metric used in rlvq and its variations a full matrix is more powerful to represent the internal structure of the data appropriately large margin generalization bounds can be transferred to this case leading to bounds that are independent of the input dimensionality this also holds for local metrics attached to each prototype which corresponds to piecewise quadratic decision boundaries the algorithm is tested in comparison to alternative learning vector quantization schemes using an artificial data set a benchmark multiclass problem from the vci repository and a problem from bioinformatics the recognition of splice sites for c elegans

on some inequalities for unitarily invariant norms and singular values abstract in this note we improve and generalize some existing inequalities for unitarily invariant norms and singular values and present some other inequalities for unitarily invariant norms

vertex operators and composite supersymmetric s functions the s function basis for the fock space of vertex operator constructions is considered where states are associated with partitions mod lambda the authors show that the matrix element lambda mod valpha z mod mu is essentially given by the composite s function slambda mu z where the argument z identical to z 1 is replicated mod alpha mod times matrix elements of products of vertex operators lambda mod valpha z v beta w mod mu are given up to functions of z w independent of lambda and mu by the composite supersymmetric s function slambda mu z w where the z w are replicated mod alpha mod mod beta mod times the supersymmetric arguments being those associated with negative roots

on infinite divisibility of positive definite functions arising from operator means abstract for a point wisely non negative positive definite function a certain criterion for its infinite divisibility i e all its fractional powers are also positive definite is obtained this criterion enables us to show infinite divisibility for many positive definite functions appearing naturally in study of operator means in particular we determine when the function cosh x s cosh x s 0 1 s s 1 1 is infinitely divisible

a quantitative version of the observation that the hadamard product is a principal submatrix of the kronecker product abstract the hadamard and kronecker products of two n m matrices a b are related by a b p t 1 a b p 2 where p 1 p 2 are partial permutation matrices after establishing several properties of the p matrices this relationship is employed to demonstrate how a simplified theory of the hadamard product can be developed during this process the well known result a b a b aa bb is extended to a b a b 1 2 aa bb ab ba aa bb showing an inherent link between the hadamard product and conventional product of two matrices this leads to a sharper bound on the spectral norm of a b a b 1 2 a 2 b 2 ab 2 1 2 a b and an improvement on the weak majorization of a b 2 a b w 1 2 2 a 2 b 2 ab w 2 a 2 b for a real non singular matrix x and invertible diagonal matrices d e the spectral condition number is shown to be if scaled bounded below as follows dxe 2 x x t 2 x x t 2 1 2 x x t 1 for a 0 we have i a 2 1 2 i a 2 a a i a 2 and a 1 2 a 1 2 2 1 2 i a a 1 a a 1 when a 0 the latter inequality is compared to styan s inequality a a 1 1 2 i a a 1 when a is a correlation matrix and is shown to possess stronger properties of ordering finally the relationship a b p t 1 a b p 2 is applied to determine conditions of singularity of certain orderings of the hadamard products of matrices

parametric extensions of shannon inequality and its reverse one in hilbert space operators we shall state the following parametric extensions of shannon inequality and its reverse one in hilbert space operators let p 0 1 and also let a1 a2 an and b1 b2 bn be two sequences of strictly positive operators on a hilbert space h such that j 1naj pbj i then j 1nsp 1 aj bj j 1n aj p 1bj i j 1naj pbj log j 1n aj p 1bj i j 1naj pbj log j 1n aj p 1bj i j 1naj pbj j 1nsp aj bj log j 1n aj p 1bj i j 1naj pbj j 1n aj p 1bj i j 1naj pbj log j 1n aj p 1bj i j 1naj pbj j 1nsp 1 aj bj where sq a b a12a 12ba 12qloga 12ba 12a12 for a 0 b 0 and any real number q and a qb a12a 12ba 12qa12 for a 0 b 0 and any real number q in particular if j 1naj j 1nbj i then j 1ns2 aj bj j 1nbjaj 1bjlog j 1nbjaj 1bj log j 1nbjaj 1bj j 1ns1 aj bj 0 j 1ns aj bj log j 1najbj 1aj j 1najbj 1ajlog j 1najbj 1aj j 1ns 1 aj bj where s a b s0 a b a12loga 12ba 12a12 which is the relative operator entropy of a 0 and b 0 our results can be considered as parametric extensions of the following celebrated shannon inequality ann math statistics 22 1951 79 bull syst tech j 27 1948 379 pitman monographs and surveys in pure and applied mathematics 97 addison wesley longman 1998 p 233 which is very useful and so famous in information theory let a1 a2 an and b1 b2 bn be two probability vectors then 0 j 1najlogbj j 1najlogaj see inequalities 2 4 of corollary 2 4

some results on matrix monotone functions abstract let a and b be hermitian matrices we say that a b if a b is nonnegative definite a function 0 r is said to be matrix monotone m m if a b 0 implies that a b matrix convex and concave functions can be similarly defined in this paper several known results some in sharper forms are given alternative proofs these include the following a a positive function t is m m iff t t is m m b if t is m m then both t t and t t are matrix convex and are almost always strictly convex on the other hand t is always concave and almost always strictly concave the concavity of m m functions and an inequality of hansen are consequences of a more general inequality on m m functions some complementary inequalities are also established we then prove an extension of lyapunov s theorem let a and p be positive definite matrices and either or 1 be a m m function then the unique matrix solution x of the matrix equation ax xa a p p a is positive definite a similar result holds for a related equation most of the results are obtained by exploiting the integral representation of m m functions

vertex operator realization of symplectic and orthogonal s functions it is shown that symplectic and orthogonal schur functions can be realized by the action of the modes of certain vertex operators on the function the properties of these vertex operators allow one to reproduce the basic propreties of these types of symmetric functions the vertex operator modes are shown to obey free fermionic relations which provide applications such as calculating products and plethysms as well as generating hirota type partial differential equations pdes which have symplectic s functions as functions

critical behaviour in aperiodic systems the author considers layered systems with aperiodically modulated couplings and study the effect of inhomogeneity on the critical behaviour using scaling arguments a relevance irrelevance type criterion is formulated and for relevant inhomogeneities the scaling form of singular quantities is determined the author has performed exact calculations for the surface magnetization of aperiodic quantum ising chains and for a 1 1 dimensional directed polymer in a longitudinal aperiodic environment the results for both systems are in accord with scaling considerations for a marginal aperiodicity the critical exponents are found to be non universal

on the definiteness of the solutions of certain matrix equations abstract it is shown that for each dimension n 2 there is a real number t n such that for all t 2 t n the unique n n solution x of the matrix equation a 2 x xa 2 taxa b must be positive semidefinite for any positive definite a and positive semidefinite b this generalizes the classical lyapunov result it is established that t n 8 4 for n 2 3 4 respectively and that t n 2 for all n upper bounds are given for n up to 6 related equations are also discussed

operator inequalities related to the corach porta recht inequality abstract we prove some refinements of an inequality due to x zhan in an arbitrary complex hilbert space by using some results on the heinz inequality we present several related inequalities as well as new variants of the corach porta recht inequality we also characterize the class of operators satisfying sxs 1 s 1 xs kx k 2 x under certain conditions

soc monotone and soc convex functions vs matrix monotone and matrix convex functions abstract the soc monotone function respectively soc convex function is a scalar valued function that induces a map to preserve the monotone order respectively the convex order when imposed on the spectral factorization of vectors associated with second order cones socs in general hilbert spaces in this paper we provide the sufficient and necessary characterizations for the two classes of functions and particularly establish that the set of continuous soc monotone respectively soc convex functions coincides with that of continuous matrix monotone respectively matrix convex functions of order 2

a characterization of the class of partial isometries abstract let b h be the set of all bounded linear operators on a hilbert space h in this paper we show that if s is a closed range operator with r s r s then s s s s ast 2 if and only if s is a non zero real scalar of a normal partial isometry also we find some other characterizations of this space using some inequalities related to corach porta recht inequality

wronskian determinants the kp hierarchy and supersymmetric polynomials by using the wronskian representation of the solutions of the bilinear kp hierarchy a connection between hirota derivatives and supersymmetric polynomials is brought to light this correspondence is used in order to give an alternative construction of the hierarchy

novel matrix singular value inequalities and their applications to uncertain mimo channels novel matrix singular value inequalities are established for a sum product of three matrices their application to the uncertain compound multiple input multiple output mimo channel subject to normed additive uncertainty establishes the saddle point property for a wide range of performance metrics monotonic in the channel singular values including among others the mutual information mmse error exponent and pairwise error probability this in turn implies that the transmission on the eigenmodes of the nominal or worst case channel is also optimal for the whole set of channels under a general power constraint and hence achieves the compound channel capacity the worst case channel turns out to be antiparallel of the nominal one for all these performance metrics an application of these results to beamforming over compound mimo channels is discussed an optimal robust precoder for the uncertain mimo channel is obtained in a closed form under the sum mse criterion and the total power constraint the saddle point property is shown to hold and the optimal strategy is to diagonalize the nominal or worst case channel

plethystic vertex operators and boson fermion correspondences we study the algebraic properties of plethystic vertex operators introduced in 2010 j phys a math theor 43 405202 underlying the structure of symmetric functions associated with certain generalized universal character rings of subgroups of the general linear group defined to stabilize tensors of young symmetry type characterized by a partition of arbitrary shape here we establish an extension of the well known boson fermion correspondence involving schur functions and their associated bernstein vertex operators for each the modes generated by the plethystic vertex operators and their suitably constructed duals satisfy the anticommutation relations of a complex clifford algebra the combinatorial manipulations underlying the results involve exchange identities exploiting the hopf algebraic structure of certain symmetric function series and their plethysms

tensor fields of mixed young symmetry type and n complexes we construct n complexes of non completely antisymmetric irreducible tensor fields on d which generalize the usual complex n 2 of differential forms although for n 3 the generalized cohomology of these n complexes is nontrivial we prove a generalization of the poincare lemma to that end we use a technique reminiscent of the green ansatz for parastatistics several results which appeared in various contexts are shown to be particular cases of this generalized poincare lemma we furthermore identify the nontrivial part of the generalized cohomology many of the results presented here were announced in 10

quantum schubert polynomials where in is the ideal generated by symmetric polynomials in x1 xn without constant term another geometric description of the cohomology ring of the flag manifold is based on the decomposition of fln into schubert cells these are even dimensional cells indexed by the elements w of the symmetric group sn the corresponding cohomology classes oa called schubert classes form an additive basis in h fln 2 to relate the two descriptions one would like to determine which elements of 2 xl xn in correspond to the schubert classes under the isomorphism 1 1 this was first done in 2 see also 8 for a general case of an arbitrary complex semisimple lie group later lascoux and schiitzenberger 22 came up with a combinatorial version of this theory for the type a by introducing remarkable polynomial representatives of the schubert classes oa called schubert polynomials and denoted gw recently motivated by ideas that came from the string theory 31 30 mathematicians defined for any kahler algebraic manifold x the small quantum cohomology ring qh x 2 which is a certain deformation of the classical cohomology ring see e g 28 19 14 and references therein the additive structure of qh x 2 is essentially the same as that of ordinary cohomology in particular qh fln z is canonically isomorphic as an abelian group to the tensor product h fln 2 0 z ql qn 1 where the qi are formal variables deformation parameters the multiplicative structure of the quantum cohomology is however

mean matrices and infinite divisibility abstract we consider matrices m with entries mij m i j where 1 n are positive numbers and m is a binary mean dominated by the geometric mean and matrices w with entries wij 1 m i j where m is a binary mean that dominates the geometric mean we show that these matrices are infinitely divisible for several much studied classes of means

vertex operators and hall littlewood symmetric functions abstract we consider vertex operators on space v with a parameter t their components form an associative algebra which is a generalization of the clifford algebra a distinguished orthogonal basis of v is proved to be the hall littlewood symmetric functions we show that kostka foulkes polynomials or certain kazhdan lusztig polynomials for the affine weyl group of type a are matrix coefficients on the space v we also obtain certain generating functions for the product of hall littlewood functions and the kostka foulkes polynomials

the epstein glaser approach to perturbative quantum field theory graphs and hopf algebras the paper aims at investigating perturbative quantum field theory in the approach of epstein and glaser eg and in particular its formulation in the language of graphs and hopf algebras has various has are encountered each one associated with a special combination of physical concepts such as normalization localization pseudounitarity causal regularization and renormalization the algebraic structures representing the perturbative expansion of the s matrix are imposed on operator valued distributions equipped with appropriate graph indices translation invariance ensures the algebras to be analytically well defined and graded total symmetry allows to formulate bialgebras the algebraic results are given embedded in the corresponding physical framework covering the two eg versions by fredenhagen and scharf that differ with respect to the concrete recursive implementation of causality besides the ultraviolet divergences occurring in feynman s representation are mathematically reasoned as a final result the change of the renormalization scheme in the context of eg is modeled via a ha and interpreted as the eg analog of kreimer s ha the paper aims at investigating perturbative quantum field theory in the approach of epstein and glaser eg and in particular its formulation in the language of graphs and hopf algebras has various has are encountered each one associated with a special combination of physical concepts such as normalization localization pseudounitarity causal regularization and renormalization the algebraic structures representing the perturbative expansion of the s matrix are imposed on operator valued distributions equipped with appropriate graph indices translation invariance ensures the algebras to be analytically well defined and graded total symmetry allows to formulate bialgebras the algebraic results are given embedded in the corresponding physical framework covering the two eg versions by fredenhagen and scharf that differ with respect to the concrete recursive implementation of causality besides the ultraviolet divergences occurring in feynman s representation are mathematically reasoned as a

hall littlewood symmetric functions and the bkp equation the connection between solutions of the bkp equation and hall littlewood symmetric functions is utilised in a unified approach to soliton and polynomial solutions this is analogous to the wronskian formulation of the solution of the kp equation as a by product two novel expressions for certain hall littlewood functions in terms of pfaffians are derived

new s function series and non compact lie groups a number of new infinite series of s functions are described in terms of their generating functions and s function content applications to the character theory of non compact lie groups are noted

symmetric function means of positive operators abstract many authors have studied means involving ratios of the elementary symmetric functions this work involves generalizing these means to an operator theoretic setting and extending many of the classical scalar inequalities to operators it is shown that these operator means are related to electrical networks

zhan s inequality on a g mean inequalities abstract in this paper we show some generalizations of zhan s inequality by virtue of hadamard product as follows let a b x be n n matrices such that a b are positive definite and f g be two continuous functions on 0 k max f g a b where a is the spectrum of a if f x g x is kwong i e the matrix f g i f g j i j is positive semidefinite for any positive numbers 1 n then for any 0 i a 1 2 f a x g b g a x f b b 1 2 k 2 1 2 2 r 0 a x b 4 1 r 0 t 2 a 2 x t a x b x b 2 holds for 2 t 2 2 where r 0 min 1 2 1 r 1 1 r i i a 1 2 f a x g b g a x f b b 1 2 2 k t 2 a 2 x t a x b x b 2 holds for any 2 t 2 where is any unitarily invariant norm

regularization in matrix relevance learning in this paper we present a regularization technique to extend recently proposed matrix learning schemes in learning vector quantization lvq these learning algorithms extend the concept of adaptive distance measures in lvq to the use of relevance matrices in general metric learning can display a tendency towards oversimplification in the course of training an overly pronounced elimination of dimensions in feature space can have negative effects on the performance and may lead to instabilities in the training we focus on matrix learning in generalized lvq glvq extending the cost function by an appropriate regularization term prevents the unfavorable behavior and can help to improve the generalization ability the approach is first tested and illustrated in terms of artificial model data furthermore we apply the scheme to benchmark classification data sets from the uci repository of machine learning we demonstrate the usefulness of regularization also in the case of rank limited relevance matrices i e matrix learning with an implicit low dimensional representation of the data

on the expansion of schur and schubert polynomials into standard elementary monomials abstract motivated by the recent discovery of a simple quantization procedure for schubert polynomials we study the expansion of schur and schubert polynomials into standard elementary monomials sem the sem expansion of schur polynomials can be described algebraically by a simple variant of the jacobi trudi formula and combinatorially by a rule based on posets of staircase box diagrams these posets are seen to be rank symmetric and order isomorphic to certain principal order ideals in the bruhat order of symmetric groups ranging between the full symmetric group and the respective maximal boolean sublattice we prove and conjecture extensions of these results for general schubert polynomials the featured conjectures are 1 an interpretation of sem expansions as alternating approximations and 2 surprising properties of different numbers naturally associated to sem expansions this hints at as yet undiscovered deeper symmetry properties of the sem expansion of schubert polynomials

some convex and monotone matrix functions we prove some matrix monotonicity and matrix convexity properties for functions derived from operator connections as a consequence we prove that a certain product of matrix monotone functions is a matrix monotone function similar results are proved for matrix convex functions these results include some known results for matrix monotone and matrix convex functions

inequalities theory of majorization and its applications by albert w marshall and ingram olkin abstract in the years since the appearance of marshall and olkin s masterful survey the influence of this work has been so great that some comments however incomplete and idiosyncratic on subsequent developments will be of value figure 1 an appendix lists minor errata to the first printing of the book

concrete examples of operator monotone functions obtained by an elementary method without appealing to lowner integral representation we study concrete examples of operator monotone functions obtained by an elementary method without appealing to lowner integral representation

plethysms of schur functions and the shell model we present a method for evaluating plethysms of schur functions that is conceptually simpler than existing methods moreover the algorithm can be easily implemented with an algebraic computer language plethysms of sums differences and products of s functions are dealt with in exactly the same manner as plethysms of simple s functions sums and differences of s functions are of importance for the description of multi shell configurations in the shell model the number of variables in which the s functions are expressed can be specified in advance significantly simplifying the calculations in typical applications to many body problems the method relies on an algorithm that we have developed for the product of monomial symmetric functions we present a new way of calculating the kostka numbers using gel fand patterns and give as well a new formula for the littlewood richardson coefficients

symmetric functions and the kp and bkp hierarchies we study the kp hierarchy through its relationship with s functions using results from the classical theory of symmetric functions the plucker equations for the hierarchy are derived from the tau function bilinear identity and are given in terms of composite s functions their connection to the hirota bilinear form of the hierarchy is clarified a novel combinatorial proof is given of the fact that schur polynomials solve the kp hierarchy we show how the analysis can be carried through for the bkp hierarchy in a completely parallel fashion with the s functions replaced by schur q functions

regularized margin based conditional log likelihood loss for prototype learning the classification performance of nearest prototype classifiers largely relies on the prototype learning algorithm the minimum classification error mce method and the soft nearest prototype classifier snpc method are two important algorithms using misclassification loss this paper proposes a new prototype learning algorithm based on the conditional log likelihood loss cll which is based on the discriminative model called log likelihood of margin logm a regularization term is added to avoid over fitting in training as well as to maximize the hypothesis margin the cll in the logm algorithm is a convex function of margin and so shows better convergence than the mce in addition we show the effects of distance metric learning with both prototype dependent weighting and prototype independent weighting our empirical study on the benchmark datasets demonstrates that the logm algorithm yields higher classification accuracies than the mce generalized learning vector quantization glvq soft nearest prototype classifier snpc and the robust soft learning vector quantization rslvq and moreover the logm with prototype dependent weighting achieves comparable accuracies to the support vector machine svm classifier

hopf algebras distributive laplace pairings and hash products a unified approach to tensor product decompositions of group characters we show for bicommutative graded connected hopf algebras that a certain distributive laplace subgroup of the convolution monoid of 2 cochains parameterizes certain well behaved hopf algebra deformations using the laplace group or its frobenius subgroup we define higher derived hash products and develop a general theory to study their main properties applying our results to the universal bicommutative graded connected hopf algebra of symmetric functions we show that classical tensor product and character decompositions such as those for the general linear group mixed co and contravariant or rational characters orthogonal and symplectic group characters thibon and reduced symmetric group characters are special cases of higher derived hash products in the appendix we discuss a relation to formal group laws

new branching rules induced by plethysm we derive group branching laws for formal characters of subgroups of leaving invariant an arbitrary tensor t of young symmetry type where is an integer partition the branchings and fixing a vector vi a symmetric tensor gij gji and an antisymmetric tensor fij fji respectively are obtained as special cases all new branchings are governed by schur function series obtained from plethysms of the schur function s by the basic m series of complete symmetric functions and the l m 1 series of elementary symmetric functions our main technical tool is that of hopf algebras and our main result is the derivation of a coproduct for any schur function series obtained by plethysm from another such series therefrom one easily obtains generalized newell littlewood formulae and the algebra of the formal group characters of these subgroups is established concrete examples and extensive tabulations are displayed for and showing their involved and nontrivial representation theory the nature of the subgroups is shown to be in general affine and in some instances non reductive we discuss the complexity of the coproduct formula and give a graphical notation to cope with it we also discuss the way in which the group branching laws can be reinterpreted as twisted structures deformed by highly nontrivial 2 cocycles the algebra of subgroup characters is identified as a cliffordization of the algebra of symmetric functions for formal characters modification rules are beyond the scope of the present paper but are briefly discussed

enhancing prototype reduction schemes with lvq3 type algorithms abstract various prototype reduction schemes have been reported in the literature foremost among these are the prototypes for nearest neighbor pnn the vector quantization vq and the support vector machines svm methods in this paper we shall show that these schemes can be enhanced by the introduction of a post processing phase that is related but not identical to the lvq3 process although the post processing with lvq3 has been reported for the som and the basic vq methods in this paper we shall show that an analogous philosophy can be used in conjunction with the svm and pnn rules our essential modification to lvq3 first entails a partitioning of the respective training sets into two sets called the placement set and the optimizing set which are instrumental in determining the lvq3 parameters such a partitioning is novel to the literature our experimental results demonstrate that the proposed enhancement yields the best reported prototype condensation scheme to date for both artificial data sets and for samples involving real life data sets

on the injective norm and characterization of some subclasses of normal operators by inequalities or equalities abstract let b h be the c algebra of all bounded linear operators acting on a complex hilbert space h in this note we shall show that if s is an invertible normal operator in b h the following estimation holds s s 1 s 1 s s s 1 1 s s 1 where is the injective norm on the tensor product b h b h this last inequality becomes an equality when s is invertible self adjoint on the other hand we shall characterize the set of all invertible normal operators s in b h satisfying the relation s s 1 s 1 s s s 1 1 s s 1 and also we shall give some characterizations of some subclasses of normal operators in b h by inequalities or equalities

cauchy schwarz norm inequalities for weak integrals of operator valued functions for a finite measures on and weakly measurable families at t and bt t of hilbert space operators we have the non commutative cauchy schwarz inequalities in schatten p ideals axbd p a aa d q 1ad 2qx b b bd r 1b d 2rp for all x p h and for all p q r 1 such that 1q 1r 2p if both at t and bt t consists of commuting normal operators then axbd a ad x b bd for all unitarily invariant norms and all x h if additionally a ad i and b bd i then i a ad xi b bd h and i a ad xi b bd x axbd applications include young s and arithmetic geometric logarithmic means inequalities for operators and the mean value theorem for operator monotone functions

a recursion and a combinatorial formula for jack polynomials the jack polynomials j x form a remarkable class of symmetric polynomials they are indexed by a partition and depend on a parameter one of their properties is that several classical families of symmetric functions can be obtained by specializing e g the monomial symmetric functions m the elementary functions e 0 the schur functions s 1 and nally the two classes of zonal polynomials 2 1 2 the jack polynomials can be de ned in various ways e g a as an orthogonal family of functions which is compatible with the canonical ltration of the ring of symmetric functions or b as simultaneous eigenfunctions of certain di erential operators the sekiguchi debiard operators recently opdam o constructed a similar family f x of nonsymmetric polynomials the index runs now through all compositions nn they are de ned in a completely similar fashion e g the sekiguchi debiard operators are being replaced by the cherednik di erential re ection operators see sect 3 it is becoming more and more clear that these polynomials are as important as their symmetric counterparts the purpose of this paper is to add to the existing characterizations of jack polynomials two further ones c a recursion formula among the f together with two formulas to obtain j from them d combinatorial formulas of both j and f in terms of certain generalized tableaux

an operator monotone function tlogt t 1log2t and strictly chaotic order we consider the function f t defined by f t tlogt t 1log2t firstly we shall show direct and simplified proofs of operator monotonicity of f t and its dual function f t tlog2ttlogt t 1 without appealing of lowner theorem then we shall show the following loga logb thereexists 0 1 suchthatf a f b holds forall 0 by using the result we summarize the following interesting contrast between a b 0 and a b 0 a b 0 thereexists 0 1 suchthatf a f b holds i forall 0 ii a b 0 f a f b forall 0 1 finally we cite a counterexample related to as follows thereexiststritlypositiveoperatorsaandbsuchthatloga logb c butf a f b doesnotholdforany 0

on singular value inequalities for matrix means abstract for positive semidefinite n n matrices a and b the singular value inequality 2 t s j a r b 2 r a 2 r b r 2 s j a 2 t a b b 2 is shown to hold for r 1 2 1 3 2 and all 2 t 2

the hopf algebra structure of the character rings of classical groups the character ring of covariant irreducible tensor representations of the general linear group admits a hopf algebra structure isomorphic to the hopf algebra of symmetric functions here we study the character rings and of the orthogonal and symplectic subgroups of the general linear group within the same framework of symmetric functions we show that and also admit natural hopf algebra structures that are isomorphic to that of and hence to the isomorphisms are determined explicitly along with the specification of standard bases for and analogous to those used for a major structural change arising from the adoption of these bases is the introduction of new orthogonal and symplectic schur hall scalar products significantly the adjoint with respect to multiplication no longer coincides as it does in the case with a foulkes derivative or skew operation the adjoint and foulkes derivative now require separate definitions and their properties are explored here in the orthogonal and symplectic cases moreover the hopf algebras and are not self dual the dual hopf algebras and are identified finally the hopf algebra of the universal rational character ring of mixed irreducible tensor representations of the general linear group is introduced and its structure maps identified

a hopf laboratory for symmetric functions an analysis of symmetric function theory is given from the perspective of the underlying hopf and bi algebraic structures these are presented explicitly in terms of standard symmetric function operations particular attention is focused on laplace pairing sweedler cohomology for 1 and 2 cochains and twisted products rota cliffordizations induced by branching operators in the symmetric function context the latter are shown to include the algebras of symmetric functions of orthogonal and symplectic type a commentary on related issues in the combinatorial approach to quantum field theory is given

soft nearest prototype classification we propose a new method for the construction of nearest prototype classifiers which is based on a gaussian mixture ansatz and which can be interpreted as an annealed version of learning vector quantization lvq the algorithm performs a gradient descent on a cost function minimizing the classification error on the training set we investigate the properties of the algorithm and assess its performance for several toy data sets and for an optical letter classification task results show 1 that annealing in the dispersion parameter of the gaussian kernels improves classification accuracy 2 that classification results are better than those obtained with standard learning vector quantization lvq 2 1 lvq 3 for equal numbers of prototypes and 3 that annealing of the width parameter improved the classification capability additionally the principled approach provides an explanation of a number of features of the heuristic lvq methods

hook young diagrams with applications to combinatorics and to representations of lie superalgebras definition et etude des tableaux k l semi standards applications a la combinatoire et aux representations des superalgebres de lie des diagrammes de young a crochets