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

geometry of free loci and factorization of noncommutative polynomials abstract the free singularity locus of a noncommutative polynomial f is defined to be the sequence of hypersurfaces z n f x m n k g det f x 0 the main theorem of this article shows that f is irreducible if and only if z n f is eventually irreducible a key step in the proof is an irreducibility result for linear pencils arising from this is a free singularity locus nullstellensatz for noncommutative polynomials apart from consequences to factorization in a free algebra the paper also discusses its applications to invariant subspaces in perturbation theory and linear matrix inequalities in real algebraic geometry

quantum field theory and hopf algebra cohomology we exhibit a hopf superalgebra structure of the algebra of field operators of quantum field theory qft with the normal product based on this we construct the operator product and the time ordered product as a twist deformation in the sense of drinfeld our approach yields formulae for perturbative products and expectation values that allow for a significant enhancement in computational efficiency as compared to traditional methods employing hopf algebra cohomology sheds new light on the structure of qft and allows the extension to interacting not necessarily perturbative qft we give a reconstruction theorem for time ordered products in the spirit of streater and wightman and recover the distinction between free and interacting theory from a property of the underlying cocycle we also demonstrate how non trivial vacua are described in our approach solving a problem in quantum chemistry

inverse numerical range and determinantal representation abstract the numerical range w a of an n n matrix a is the collection of quadratic forms z a over unit sphere 1 the inverse numerical range problem aims to find a unit vector which corresponds to a given point z of the numerical range w a given a matrix a the helton vinnikov theorem produces a symmetric matrix s so that a and s have the same numerical range w a w s in this paper we investigate the inverse numerical range problem for the boundary points and points on the boundary generating curve of the numerical range in place of the construction of unit vectors satisfying z a we express the kernel vector function of the linear pencil x s y s z i n as a function on the abel jacobi variety of the associated elliptic curve of a the kernel function plays a key role for the inverse numerical problem we perform this process when s is a generic 3 3 symmetric matrix

scaling analysis for the adsorption transition in a watermelon network of n directed non intersecting walks the partition function for the problem of n directed non intersecting walks interacting via contact potentials with a wall parallel to the direction of the walks has previously been calculated as an n n determinant here we describe how to analyse the scaling behaviour of this problem using alternative representations of the solution in doing so we derive the asymptotics of the partition function of a watermelon network of n such walks for all temperatures and so calculate the associated network exponents in the three regimes desorbed adsorbed and at the adsorption transition furthermore we derive the full scaling function around the adsorption transition for all n at the adsorption transition we also derive a simple product form for the partition function these results have in part been derived using recurrence relations satisfied by the original determinantal solution

convex matrix inequalities versus linear matrix inequalities most linear control problems lead directly to matrix inequalities mis many of these are badly behaved but a classical core of problems are expressible as linear matrix inequalities lmis in many engineering systems problems convexity has all of the advantages of a lmi since lmis have a structure which is seemingly much more rigid than convex mis there is the hope that a convexity based theory will be less restrictive than lmis how much more restrictive are lmis than convex mis there are two fundamentally different classes of linear systems problems dimension free and dimension dependent a dimension free mi is a mi where the unknowns are g tuples of matrices and appear in the formulas in a manner which respects matrix multiplication most of the classic mis of control theory are dimension free dimension dependent mis have unknowns which are tuples of numbers the two classes behave very differently and this survey describes what is known in each case about the relation between convex mis and lmis the proof techniques involve and necessitate new developments in the field of semialgebraic geometry

on an easy transition from operator dynamics to generating functionals by clifford algebras clifford geometric algebras of multivectors are treated in detail these algebras are built over a graded space and exhibit a grading or multivector structure the careful study of the endomorphisms of this space makes it clear that opposite clifford algebras have to be used for obtaining all endomorphisms based on these mathematics we give a fully clifford algebraic account on generating functionals which is thereby geometric the field operators are shown to be clifford and opposite clifford maps this picture relying on geometry does not need positivity in principle furthermore we propose a transition from operator dynamics to generating functionals which is based on the algebraic techniques as a calculational benefit this transition is considerably short compared to standard ones the transition is not injective unique and depends additionally on the choice of an ordering we obtain a direct and constructive connection between orderings and the explicit form of the functional hamiltonian these orderings depend on the propagator of the theory and thus on the ground state this is invisible in path integral formulations the method is demonstrated within two examples a nonlinear spinor field theory and spinor qed antisymmetrized and normal ordered functional equations are derived in both cases

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

linear matrix inequality representation of sets this article concerns the question which subsets of m can be represented with linear matrix inequalities lmis this gives some perspective on the scope and limitations of one of the most powerful techniques commonly used in control theory also before having much hope of representing engineering problems as lmis by automatic methods one needs a good idea of which problems can and cannot be represented by lmis little is currently known about such problems in this article we give a necessary condition that we call rigid convexity which must hold for a set m in order for to have an lmi representation rigid convexity is proved to be necessary and sufficient when m 2 this settles a question formally stated by pablo parrilo and berndt sturmfels in 15 as shown by lewis parillo and ramana 11 our main result also establishes in the case of three variables a 1958 conjecture by peter lax on hyperbolic polynomials 2006 wiley periodicals inc

the bethe ansatz as a matrix product ansatz the bethe ansatz in its several formulations is a common tool for the exact solution of one dimensional quantum hamiltonians this ansatz asserts that several eigenfunctions of the hamiltonians are given in terms of a sum of permutations of plane waves we present results that induce us to expect that alternatively the eigenfunctions of all the exact integrable quantum chains can also be expressed by a matrix product ansatz in this ansatz several components of the eigenfunctions are obtained through the algebraic properties of properly defined matrices this ansatz allows an unified formulation of several exact integrable hamiltonians we show how to formulate this ansatz for a large family of quantum chains such as the anisotropic heisenberg model fateev zamolodchikov model izergin korepin model sutherland model t j model hubbard model etc

real zero polynomials and a horn s problem abstract a horn s problem concerns finding two self adjoint matrices so that a b and a b have prescribed spectrum in this paper we show how it connects to an interpolation problem for two variable real zero polynomials and a tracial moment problem in addition we outline an algorithm to construct a pair a b

obstructions to determinantal representability there has recently been ample interest in the question of which sets can be represented by linear matrix inequalities lmis a necessary condition is that the set is rigidly convex and it has been conjectured that rigid convexity is also sufficient to this end helton and vinnikov conjectured that any real zero polynomial admits a determinantal representation with symmetric matrices we disprove this conjecture by relating the question of finding lmi representations to the problem of determining whether a polymatroid is representable over the complex numbers we find a real zero polynomial such that no power of it admits a determinantal representation the proof uses recent results of wagner and wei on matroids with the half plane property and the polymatroids associated to hyperbolic polynomials introduced by gurvits

non representable hyperbolic matroids the generalized lax conjecture asserts that each hyperbolicity cone is a linear slice of the cone of positive semidefinite matrices hyperbolic polynomials give rise to a class of hyperbolic matroids which properly contains the class of matroids representable over the complex numbers this connection was used by the first author to construct counterexamples to algebraic stronger versions of the generalized lax conjecture by considering a non representable hyperbolic matroid the va mos matroid and a generalization of it are to this day the only known instances of non representable hyperbolic matroids we prove that the non pappus and non desargues matroids are non representable hyperbolic matroids by exploiting a connection due to jordan between euclidean jordan algebras and projective geometries we further identify a large class of hyperbolic matroids that are parametrized by uniform hypergraphs and prove that many of them are non representable finally we explore consequences to algebraic versions of the generalized lax conjecture

hybrid nlie for the mirror ads5 s5 we revisit the derivation of hybrid nonlinear integral equations of the xxx model starting from the linearization of the t system related to spinon variables we obtain two sets of equations corresponding to two linearly independent solutions of the a1 tq relation recalling that the tq relations in the horizontal strips of the hook is of a1 type we replace the corresponding y functions by a finite number of auxiliary variables

determinantal representations of singular hypersurfaces in pn abstract a global determinantal representation of projective hypersurface x p n is a matrix whose entries are linear forms in homogeneous coordinates and whose determinant defines the hypersurface we study the properties of such representations for singular possibly reducible or non reduced hypersurfaces in particular we obtain the decomposability criteria for determinantal representations of globally reducible hypersurfaces further we classify the determinantal representations in terms of the corresponding kernel sheaves on x finally we extend the results to the case of symmetric self adjoint representations with implications to hyperbolic polynomials and the generalized lax conjecture

classification of three state hamiltonians solvable by the coordinate bethe ansatz we classify all hamiltonians with rank 1 symmetry and nearest neighbour interactions acting on a periodic three state spin chain and solvable through generalization of the coordinate bethe ansatz cba in this way we obtain four multi parametric extensions of the known 19 vertex hamiltonians such as zamolodchikov fateev izergin korepin and bariev hamiltonians apart from the 19 vertex hamiltonians there exist 17 vertex and 14 vertex hamiltonians that cannot be viewed as subcases of the 19 vertex ones in the case of 17 vertex hamiltonians we get a generalization of the genus 5 special branch found by martins plus three new ones we also get two 14 vertex hamiltonians we solve all these hamiltonians using cba and provide their spectrum eigenfunctions and bethe equations special attention is given to provide the specifications of our multi parametric hamiltonians that give back known hamiltonians

hyperbolic polynomials interlacers and sums of squares hyperbolic polynomials are real polynomials whose real hypersurfaces are maximally nested ovaloids the innermost of which is convex these polynomials appear in many areas of mathematics including optimization combinatorics and differential equations here we investigate the special connection between a hyperbolic polynomial and the set of polynomials that interlace it this set of interlacers is a convex cone which we write as a linear slice of the cone of nonnegative polynomials in particular this allows us to realize any hyperbolicity cone as a slice of the cone of nonnegative polynomials using a sums of squares relaxation we then approximate a hyperbolicity cone by the projection of a spectrahedron a multiaffine example coming from the vamos matroid shows that this relaxation is not always exact using this theory we characterize the real stable multiaffine polynomials that have a definite determinantal representation and construct one when it exists

complete description of determinantal representations of smooth irreducible curves abstract determinantal representations of algebraic curves are interesting in themselves and their classification is equivalent to the simultaneous classification of triples of matrices we present a complete description of determinantal representations of smooth irreducible curves over any algebraically closed field we used the notion of the class of divisors of the vector bundle corresponding to a determinantal representation we prove that two determinantal representations of a smooth curve f are equivalent if and only if the classes of divisors of the corresponding vector bundles coincide we give a precise characterization of those classes of divisors that arise from vector bundles corresponding to determinantal representations of f then we obtain a parametrization of determinantal representations of f up to equivalence by the points of the jacobian variety of f not on some exceptional subvariety in particular it follows that any smooth curve of order 3 or greater possesses an infinite number of nonequivalent determinantal representations we also specialize our results to symmetrical and self adjoint representations

the algebraic degree of semidefinite programming given a generic semidefinite program specified by matrices with rational entries each coordinate of its optimal solution is an algebraic number we study the degree of the minimal polynomials of these algebraic numbers geometrically this degree counts the critical points attained by a linear functional on a fixed rank locus in a linear space of symmetric matrices we determine this degree using methods from complex algebraic geometry such as projective duality determinantal varieties and their chern classes

determinantal representations and bezoutians a major open question in convex algebraic geometry is whether all hyperbolicity cones are spectrahedral i e the solution sets of linear matrix inequalities we will use sum of squares decompositions of certain bilinear forms called bezoutians to approach this problem more precisely we show that for every smooth hyperbolic polynomial h there is another hyperbolic polynomial q such that q cdot h has a definite determinantal representation besides commutative algebra the proof relies on results from real algebraic geometry

quantum convex support abstract convex support the mean values of a set of random variables is central in information theory and statistics equally central in quantum information theory are mean values of a set of observables in a finite dimensional c algebra a which we call quantum convex support the convex support can be viewed as a projection of the state space of a and it is a projection of a spectrahedron spectrahedra are increasingly investigated at least since the 1990s boom in semi definite programming we recall the geometry of the positive semi definite cone and of the state space we write a convex duality for general self dual convex cones this restricts to projections of state spaces and connects them to results on spectrahedra our main result is an analysis of the face lattice of convex support by mapping this lattice to a lattice of orthogonal projections using natural isomorphisms the result encodes the face lattice of the convex support into a set of projections in a and enables the integration of convex geometry with matrix calculus or algebraic techniques

singular points of the algebraic curves of symmetric hyperbolic forms abstract we investigate singular points of the algebraic curve defined by a hyperbolic form satisfying some symmetric properties we show that all singular points of such a curve are real nodes

vertex normal ordering as a consequence of nonsymmetric bilinear forms in clifford algebras we consider clifford algebras with nonsymmetric bilinear forms they parametrize the chosen ideal in the isomorphism class of the standard symmetric ones since the content of physical theories depends on the injection n nv cl v q one has to transform to the standard construction the injection is described by the antisymmetric part of the bilinear form this process results in the appropriate vertex normal ordering terms which are now obtained from the theory itself and not added ad hoc via a regularization argument

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

containment problems for polytopes and spectrahedra we study the computational question of whether a given polytope or spectrahedron s a as given by the positive semidefiniteness region of a linear matrix pencil a x is contained in another s b first we classify the computational complexity extending results on the polytope polytope case by gritzmann and klee to the polytope spectrahedron case for various restricted containment problems np hardness is shown we then study in detail semidefinite conditions to certify containment building upon work by ben tal and nemirovski and helton klep and mccullough in particular we discuss variations of a sufficient semidefinite condition to certify containment of a spectrahedron in a spectrahedron it is shown that these sufficient conditions even provide exact semidefinite characterizations for containment in several important cases including containment of a spectrahedron in a polyhedron moreover in the case of bounded s a the criteria will always succeed in certifying containment of some scale

livsic type determinantal representations and hyperbolicity abstract hyperbolic homogeneous polynomials with real coefficients i e hyperbolic real projective hypersurfaces and their determinantal representations play a key role in the emerging field of convex algebraic geometry in this paper we consider a natural notion of hyperbolicity for a real subvariety x p d of an arbitrary codimension l with respect to a real l 1 dimensional linear subspace v p d and study its basic properties we also consider a class of determinantal representations that we call livsic type and a nice subclass of these that we call very reasonable much like in the case of hypersurfaces l 1 the existence of a definite hermitian very reasonable livsic type determinantal representation implies hyperbolicity we show that every curve admits a very reasonable livsic type determinantal representation our basic tools are cauchy kernels for line bundles and the notion of the bezoutian for two meromorphic functions on a compact riemann surface that we introduce we then proceed to show that every real curve in p d hyperbolic with respect to some real d 2 dimensional linear subspace admits a definite hermitian or even definite real symmetric very reasonable livsic type determinantal representation

the lee yang and polya schur programs ii theory of stable polynomials and applications in the first part of this series we characterized all linear operators on spaces of multivariate polynomials preserving the property of being nonvanishing in products of open circular domains for such sets this completes the multivariate generalization of the classification program initiated by polya and schur for univariate real polynomials we build on these classification theorems to develop here a theory of multivariate stable polynomials applications and examples show that this theory provides a natural framework for dealing in a uniform way with lee yang type problems in statistical mechanics combinatorics and geometric function theory in one or several variables in particular we answer a question of hinkkanen on multivariate apolarity

deciding polyhedrality of spectrahedra spectrahedra are linear sections of the cone of positive semidefinite matrices which as convex bodies generalize the class of polyhedra in this paper we investigate the problem of recognizing when a spectrahedron is polyhedral we generalize and strengthen results of m v ramana polyhedra spectrahedra and semidefinite programming in topics in semidefinite and interior point methods fields inst commun 18 ams providence ri 1998 pp 27 38 regarding the structure of spectrahedra and we devise a normal form of representations of spectrahedra this normal form is effectively computable and leads to an algorithm for deciding polyhedrality

remarks on the lax conjecture for hyperbolic polynomials abstract a conjecture of lax p lax differential equations difference equations and matrix theory commun pure appl math 11 1958 175 194 says that every hyperbolic polynomial in two space variables is the determinant of a symmetric hyperbolic matrix the conjecture has recently been proved by lewis parillo ramana based on previous work of dubrovin and helton vinnikov in this note we prove related results for polynomials in several space variables which have rotational symmetries

construction of determinantal representation of trigonometric polynomials abstract for a pair of n n hermitian matrices h and k a real ternary homogeneous polynomial defined by f t x y det ti n xh yk is hyperbolic with respect to 1 0 0 the fiedler conjecture or lax conjecture is recently affirmed namely for any real ternary hyperbolic polynomial f t x y there exist real symmetric matrices s 1 and s 2 such that f t x y det ti n xs 1 ys 2 in this paper we give a constructive proof of the existence of symmetric matrices for the ternary forms associated with trigonometric polynomials

exposed faces of semidefinitely representable sets a linear matrix inequality lmi is a condition stating that a symmetric matrix whose entries are affine linear combinations of variables is positive semidefinite motivated by the fact that diagonal lmis define polyhedra the solution set of an lmi is called a spectrahedron linear images of spectrahedra are called semidefinitely representable sets part of the interest in spectrahedra and semidefinitely representable sets arises from the fact that one can efficiently optimize linear functions on them by semidefinite programming such as one can do on polyhedra by linear programming it is known that every face of a spectrahedron is exposed this is also true in the general context of rigidly convex sets we study the same question for semidefinitely representable sets lasserre proposed a moment matrix method to construct semidefinite representations for certain sets our main result is that this method can work only if all faces of the considered set are exposed this necessary condition complements sufficient conditions recently proved by lasserre helton and nie

estimating mutual information we present two classes of improved estimators for mutual information m x y from samples of random points distributed according to some joint probability density mu x y in contrast to conventional estimators based on binnings they are based on entropy estimates from k nearest neighbor distances this means that they are data efficient with k 1 we resolve structures down to the smallest possible scales adaptive the resolution is higher where data are more numerous and have minimal bias indeed the bias of the underlying entropy estimates is mainly due to nonuniformity of the density at the smallest resolved scale giving typically systematic errors which scale as functions of k n for n points numerically we find that both families become exact for independent distributions i e the estimator m x y vanishes up to statistical fluctuations if mu x y mu x mu y this holds for all tested marginal distributions and for all dimensions of x and y in addition we give estimators for redundancies between more than two random variables we compare our algorithms in detail with existing algorithms finally we demonstrate the usefulness of our estimators for assessing the actual independence of components obtained from independent component analysis ica for improving ica and for estimating the reliability of blind source separation

shared information in classical mean field models universal scaling of entanglement estimators of critical quantum systems has drawn a lot of attention in the past recent studies indicate that similar universal properties can be found for bipartite information estimators of classical systems near phase transition opening a new direction in the study of critical phenomena we explore this subject by studying the information estimators of classical spin chains with general mean field interactions in our explicit analysis of two different bipartite information estimators in the canonical ensemble we find that away from criticality both the estimators remain finite in the thermodynamic limit on the other hand along the critical line there is a logarithmic divergence with increasing system size the coefficient of the logarithm is fully determined by the mean field interaction and it is the same for the class of models we consider the scaling function however depends on the details of each model in addition we study the information estimators in the micro canonical ensemble where they are shown to exhibit a different universal behavior we verify our results using numerical calculation for two specific cases of the general hamiltonian

applications of stable polynomials to mixed determinants johnson s conjectures unimodality and symmetrized fischer products for n x n matrices a and b define eta a b sigma s det a s det b s where the summation is over all subsets of 1 n s is the complement of s and a s is the principal submatrix

positive polynomials and projections of spectrahedra this article is concerned with different aspects of spectrahedra and their projections sets that are important in semidefinite optimization we prove results on the limitations of so called lasserre and theta body relaxation methods for semialgebraic sets and varieties as a special case we obtain the main result of netzer plaumann and schweighofer siam j optim 20 2010 pp 1944 1955 on nonexposed faces we also solve the open problems from that work we further give a unified account of several results on convex hulls of curves and images of polynomial maps we finally prove a positivstellensatz for projections of spectrahedra which exceeds the known results that only work for basic closed semialgebraic sets

noncommutative convexity arises from linear matrix inequalities this paper concerns polynomials in g noncommutative variables x x1 xg inverses of such polynomials and more generally noncommutative rational expressions with real coefficients which are formally symmetric and analytic near 0 the focus is on rational expressions r r x which are matrix convex near 0 i e those rational expressions r for which there is an 0 such that if x x1 xg is a g tuple of n n symmetric matrices satisfying in x12 xg2 is positive definite and y is also then the symmetric matrix tr x 1 t r y r tx 1 t y is positive semidefinite for all numbers t 0 t 1 this article gives a complete classification of matrix convex rational expressions see theorem 3 3 by representing such r in terms of a symmetric linear pencil l x id jajxj 0d 100 1 r 0 in the noncommuting variables xj where aj are symmetric d d matrices namely for a real number r is a schur complement of the linear pencil l moreover given a matrix convex r the set consisting of g tuples x of n n symmetric matrices 0 1 x r x iis negative definite has component containing 0 which is the same as the negativity set 0 2 x l x is negative definite for l conditions like l x is negative definite are known as linear matrix inequalities lmis in the engineering literature and arguably the main advance in linear systems theory in the 1990s was the introduction of lmi techniques in this language what we have shown in 0 1 vs 0 2 is that the set of solutions to a convex matrix inequality with noncommutative unknowns is the same as the set of solutions to some lmi in many engineering systems problems convexity would have all of the advantages of lmis indeed convexity guarantees that solutions are global and convexity bodes well for reliability of the numerics since lmis have a structure which is seemingly much more rigid than convexity there is the continual hope that a convexity based theory will be more far reaching than lmis but will it there are two natural situations one where the unknowns are scalars and one where the unknowns are matrices appearing in formulas which respect matrix multiplication these latter problems mathematically yield expressions with noncommutative unknowns and they arise in engineering systems problems which are dimensionless in the sense that they scale automatically with dimension as do most of the classics of control theory that is the case we study here and the result stated above suggests the surprising conclusion that for dimensionless systems problems convexity offers no greater generality than lmis indeed the result proves this for a class of model problems furthermore we show that existing algorithms together with algorithms described here construct the lmis above which are equivalent to the matrix inequalities based on the given matrix convex rational function r in a very different direction we prove that a symmetric polynomial p in g noncommutative symmetric variables has a symmetric determinantal representation namely there are symmetric matrices a0 ag in srd d with a0 invertible such that 0 3 detp x det a0 la x for each x a g tuple of symmetric n n matrices of course taking n 1 implies immediately that a commuting variables polynomial p on rg has a symmetric determinantal representation for g 2 much stronger commutative results can be obtained using tools of algebraic geometry but these do not seem to generalize to the higher dimensional case on the other hand a nonsymmetric commutative determinantal representation for any g is due to valiant universality of determinant in algebraic complexity theory our determinantal representation theorem is a bi product of the theory of systems realizations of noncommutative rational functions and can be read independently of much of the rest of the paper while the notion of noncommutative rational functions is standard the equivalence relation we use on rational expressions in our construction based on evaluating rational expressions on matrices is new and gives a new approach to noncommutative rational functions

minimal determinantal representations of bivariate polynomials abstract for a square free bivariate polynomial p of degree n we introduce a simple and fast numerical algorithm for the construction of n n matrices a b and c such that det a x b y c p x y this is the minimal size needed to represent a bivariate polynomial of degree n combined with a square free factorization one can now compute n n matrices for any bivariate polynomial of degree n the existence of such symmetric matrices was established by dixon in 1902 but up to now no simple numerical construction has been found even if the matrices can be nonsymmetric such representations may be used to efficiently numerically solve a system of two bivariate polynomials of small degree via the eigenvalues of a two parameter eigenvalue problem the new representation speeds up the computation considerably

low temperature asymptotics of integrable systems in an external field an asymptotic low temperature expansion is performed for an integrable bosonic lattice model and for the critical spin 1 2 heisenberg chain in a magnetic field the results apply to the integrable bose gas as well we also comment on a high temperature expansion of the bosonic lattice model

singularities of rational functions and minimal factorizations the noncommutative and the commutative setting we show that the singularities of a matrix valued noncommutative rational function which is regular at zero coincide with the singularities of the resolvent in its minimal state space realization the proof uses a new notion of noncommutative backward shifts as an application we establish the commutative counterpart of the singularities theorem the singularities of a matrix valued commutative rational function which is regular at zero coincide with the singularities of the resolvent in any of its fornasini marchesini realizations with the minimal possible state space dimension the singularities results imply the absence of zero pole cancellations in a minimal factorization both in the noncommutative and in the commutative setting

semidefinite descriptions of low dimensional separable matrix cones abstract let k e k e be convex cones residing in finite dimensional real vector spaces an element y in the tensor product e e is k k separable if it can be represented as finite sum y l x l x l where x l k and x l k for all l let s n h n q n be the spaces of n n real symmetric complex hermitian and quaternionic hermitian matrices respectively let further s n h n q n be the cones of positive semidefinite matrices in these spaces if a matrix a h mn h m h n is h m h n separable then it fulfills also the so called ppt condition i e it is positive semidefinite and has a positive semidefinite partial transpose the same implication holds for matrices in the spaces s m s n h m s n and for m 2 in the space q m s n we provide a complete enumeration of all pairs n m when the inverse implication is also true for each of the above spaces i e the ppt condition is sufficient for separability we also show that a matrix in q n s 2 is q n s 2 separable if and only if it is positive semidefinite

noncommutative polynomials nonnegative on a variety intersect a convex set abstract this paper gives a precise algebraic certificate for a noncommutative polynomial to be nonnegative on a free convex semialgebraic set intersect the variety of a free left ideal

strict convexity of the joint c numerical range abstract let h h 1 h 2 h m be an m tuple of n n hermitian matrices and c c 1 c 2 c n r n the joint c numerical range w c h of h is defined for x x 1 x 2 x m r m the form associated with h is defined as f h t x det ti n x 1 h 1 x 2 h 2 x m h m we prove that the convex hull of w c h is strictly convex if the roots of f h t x 0 have constant multiplicities for all x on the unit sphere in r m the result is illustrated with several examples

combinatorics of n point functions via hopf algebra in quantum field theory we use a coproduct on the time ordered algebra of field operators to derive simple relations between complete connected and 1 particle irreducible n point functions compared to traditional functional methods our approach is much more intrinsic and leads to efficient algorithms suitable for concrete computations it may also be used to efficiently perform tree level computations

slowly evolving connectivity in recurrent neural networks i the extreme dilution regime we study extremely diluted spin models of neural networks in which the connectivity evolves in time although adiabatically slowly compared to the neurons according to stochastic equations which on average aim to reduce frustration the fast neurons and slow connectivity variables equilibrate separately but at different temperatures our model is exactly solvable in equilibrium we obtain phase diagrams upon making the condensed ansatz i e recall of one pattern these show that as the connectivity temperature is lowered the volume of the retrieval phase diverges and the fraction of mis aligned spins is reduced still one always retains a region in the retrieval phase where recall states other than the one corresponding to the condensed pattern are locally stable so the associative memory character of our model is preserved

renormalization a number theoretical model we analyse the dirichlet convolution ring of arithmetic number theoretic functions it turns out to fail to be a hopf algebra on the diagonal due to the lack of complete multiplicativity of the product and coproduct a related hopf algebra can be established which however overcounts the diagonal we argue that the mechanism of renormalization in quantum field theory is modelled after the same principle singularities hence arise as a now continuously indexed overcounting on the diagonals renormalization is given by the map from the auxiliary hopf algebra to the weaker multiplicative structure called hopf gebra rescaling the diagonals

exact solutions of exactly integrable quantum chains by a matrix product ansatz most of the exact solutions of quantum one dimensional hamiltonians are obtained thanks to the success of the bethe ansatz on its several formulations according to this ansatz the amplitudes of the eigenfunctions of the hamiltonian are given by a sum of permutations of appropriate plane waves in this paper alternatively we present a matrix product ansatz that asserts that those amplitudes are given in terms of a matrix product the eigenvalue equation for the hamiltonian defines the algebraic properties of the matrices defining the amplitudes the consistency of the commutativity relations among the elements of the algebra implies the exact integrability of the model the matrix product ansatz we propose allows an unified and simple formulation of several exact integrable hamiltonians in order to introduce and illustrate this ansatz we present the exact solutions of several quantum chains with one and two global conservation laws and periodic boundaries such as the xxz chain spin 1 fateev zamolodchikov model izergin korepin model sutherland model t j model hubbard model etc formulation of the matrix product ansatz for quantum chains with open ends is also possible as an illustration we present the exact solution of an extended xxz chain with z magnetic fields at the surface and arbitrary hard core exclusion among the spins

interlacing families ii mixed characteristic polynomials and the kadison singer problem we use the method of interlacing polynomials introduced in our previous article to prove two theorems known to imply a positive solution to the kadison singer problem the rst is weaver s conjecture ks 2 which is known to imply kadison singer via a projection paving conjecture of akemann and anderson the second is a formulation due to casazza et al of anderson s original paving conjecture s for which we are able to compute explicit paving bounds the proof involves an analysis of the largest roots of a family of polynomials that we call the mixed characteristic polynomials of a collection of matrices

roots of bivariate polynomial systems via determinantal representations we give two determinantal representations for a bivariate polynomial they may be used to compute the zeros of a system of two of these polynomials via the eigenvalues of a two parameter eigenvalue problem the first determinantal representation is suitable for polynomials with scalar or matrix coefficients and consists of matrices with asymptotic order n 2 4 where n is the degree of the polynomial the second representation is useful for scalar polynomials and has asymptotic order n 2 6 the resulting method to compute the roots of a system of two bivariate polynomials is very competitive with some existing methods for polynomials up to degree 10 as well as for polynomials with a small number of terms

symmetric determinantal representations in characteristic 2 abstract this paper studies symmetric determinantal representations sdr in characteristic 2 that is the representation of a multivariate polynomial p by a symmetric matrix m such that p det m and where each entry of m is either a constant or a variable we first give some sufficient conditions for a polynomial to have an sdr we then give a non trivial necessary condition which implies that some polynomials have no sdr answering a question of grenet et al a large part of the paper is then devoted to the case of multilinear polynomials we prove that the existence of an sdr for a multilinear polynomial is equivalent to the existence of a factorization of the polynomial in certain quotient rings we develop some algorithms to test the factorizability in these rings and use them to find sdrs when they exist altogether this gives us polynomial time algorithms to factorize the polynomials in the quotient rings and to build sdrs we conclude by describing the case of alternating determinantal representations in any characteristic

nonequilibrium steady states of matrix product form a solver s guide we consider the general problem of determining the steady state of stochastic nonequilibrium systems such as those that have been used to model among other things biological transport and traffic flow we begin with a broad overview of this class of driven diffusive systems which includes exclusion processes focusing on interesting physical properties such as shocks and phase transitions we then turn our attention specifically to those models for which the exact distribution of microstates in the steady state can be expressed in a matrix product form in addition to a gentle introduction to this matrix product approach how it works and how it relates to similar constructions that arise in other physical contexts we present a unified pedagogical account of the various means by which the statistical mechanical calculations of macroscopic physical quantities are actually performed we also review a number of more advanced topics including nonequilibrium free energy functionals the classification of exclusion processes involving multiple particle species existence proofs of a matrix product state for a given model and more complicated variants of the matrix product state that allow various types of parallel dynamics to be handled we conclude with a brief discussion of open problems for future research

sturm and sylvester algorithms revisited via tridiagonal determinantal representations first we show that sturm algorithm and sylvester algorithm which compute the number of real roots of a given univariate polynomial lead to two dual tridiagonal determinantal representations of the polynomial next we show that the number of real roots of a polynomial given by a tridiagonal determinantal representation is greater than the signature of this representation

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

detecting rigid convexity of bivariate polynomials abstract given a polynomial x r n p x in n 2 variables a symbolic numerical algorithm is first described for detecting whether the connected component of the plane sublevel set p x p x 0 containing the origin is rigidly convex or equivalently whether it has a linear matrix inequality lmi representation or equivalently if polynomial p x is hyperbolic with respect to the origin the problem boils down to checking whether a univariate polynomial matrix is positive semidefinite an optimization problem that can be solved with eigenvalue decomposition when the variety c x p x 0 is an algebraic curve of genus zero a second algorithm based on bezoutians is proposed to detect whether p has an lmi representation and to build such a representation from a rational parametrization of c finally some extensions to positive genus curves and to the case n 2 are mentioned

zero pole interpolation for matrix meromorphic functions on a compact riemann surface and a matrix fay trisecant identity this paper presents a new approach to constructing a meromorphic bundle map between flat vector bundles over a compact riemann surface having a prescribed weil divisor i e having prescribed zeros and poles with directional as well as multiplicity information included in the vector case this new formalism unifies the earlier approach of ball clancey in the setting of trivial bundles over an abstract riemann surface with an earlier approach of the authors where the riemann surface was assumed to be the normalizing riemann surface for an algebraic curve embedded in c 2 with determinantal representation and the vector bundles were assumed to be presented as the kernels of linear matrix pencils the main tool is a version of the cauchy kernel appropriate for flat vector bundles over the riemann surface our formula for the interpolating bundle map in the special case of a single zero and a single pole can be viewed as a generalization of the fay trisecant identity from the usual line bundle case to the vector bundle case in terms of cauchy kernels in particular we obtain a new proof of the fay trisecant identity

smooth hyperbolicity cones are spectrahedral shadows hyperbolicity cones are convex algebraic cones arising from hyperbolic polynomials a well understood subclass of hyperbolicity cones is that of spectrahedral cones and it is conjectured that every hyperbolicity cone is spectrahedral in this paper we prove a weaker version of this conjecture by showing that every smooth hyperbolicity cone is the linear projection of a spectrahedral cone that is a spectrahedral shadow

the yang baxter equation for mathcal pt invariant 19 vertex models we study the solutions of the yang baxter equation associated with 19 vertex models invariant by the parity time symmetry from the perspective of algebraic geometry we determine the form of the algebraic curves constraining the respective boltzmann weights and find that they possess a universal structure this allows us to classify the integrable manifolds into four different families reproducing three known models besides uncovering a novel 19 vertex model in a unified way the introduction of the spectral parameter on the weights is made via the parameterization of the fundamental algebraic curve which is a conic the diagonalization of the transfer matrix of the new vertex model and its thermodynamic limit properties are discussed we point out a connection between the form of the main curve and the nature of the excitations of the corresponding spin 1 chains

a randomized approximation algorithm for the minimal norm static output feedback problem a new randomized algorithm is suggested for extracting static output stabilizing feedbacks with approximately minimal norm for lti systems the algorithm has two similar stages where in the first one the feasibility problem is solved and in the second one the optimization problem is solved the formulation is unified for the feasibility and for the optimization problems as well as for continuous time or discrete time systems the method is demonstrated by applying it to the hard conjectured to be np hard problem of the minimal gain static output stabilizing feedback and to the hard conjectured to be np hard problem of regional pole placement via static output feedback in non convex or unconnected regions a proof of convergence in probability that captures the two rounds of the algorithm is given and complexity analysis is provided under some mild assumptions

from the quantum jacobi trudi and giambelli formula to a nonlinear integral equation for thermodynamics of the higher spin heisenberg model we propose a nonlinear integral equation nlie with only one unknown function which gives the free energy of the integrable one dimensional heisenberg model with arbitrary spin in deriving the nlie the quantum jacobi trudi and giambelli formula bazhanov reshetikhin formula which gives the solution of the t system plays an important role in addition we also calculate the high temperature expansion of the specific heat and the magnetic susceptibility

polynomial matrix inequality and semidefinite representation consider a convex set s x in cal d g x succeq 0 where g x is a symmetric matrix whose every entry is a polynomial or rational function d rn is a domain on which g x is defined and g x succeq 0 means g x is positive semidefinite the set s is called semidefinite representable if it equals the projection of a higher dimensional set that is defined by a linear matrix inequality lmi this paper studies sufficient conditions guaranteeing semidefinite representability of s we prove that s is semidefinite representable in the following cases i d rn g x is a matrix polynomial and matrix sos concave ii d is compact convex g x is a matrix polynomial and strictly matrix concave on d iii g x is a matrix rational function and q module matrix concave on d explicit constructions of semidefinite representations are given some examples are illustrated

determinantal representations of hyperbolic forms via weighted shift matrices we characterize hyperbolic ternary forms of degrees 3 4 and 5 that admit determinantal representations via cyclic weighted shift matrices

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

polynomials with and without determinantal representations abstract the problem of writing real zero polynomials as determinants of linear matrix polynomials has recently attracted a lot of attention helton and vinnikov 9 have proved that any real zero polynomial in two variables has a determinantal representation branden 2 has shown that the result does not extend to arbitrary numbers of variables disproving the generalized lax conjecture we prove that in fact almost no real zero polynomial admits a determinantal representation there are dimensional differences between the two sets the result follows from a general upper bound on the size of linear matrix polynomials we then provide a large class of surprisingly simple explicit real zero polynomials that do not have a determinantal representation we finally characterize polynomials of which some power has a determinantal representation in terms of an algebra with involution having a finite dimensional representation we use the characterization to prove that any quadratic real zero polynomial has a determinantal representation after taking a high enough power taking powers is thereby really necessary in general the representations emerge explicitly and we characterize them up to unitary equivalence

bianalytic maps between free spectrahedra linear matrix inequalities lmis i d sum j 1 g a jx j sum j 1 g a j x j succeq 0 play a role in many areas of applications the set of solutions of an lmi is a spectrahedron lmis in dimension free matrix variables model most problems in linear systems engineering and their solution sets are called free spectrahedra free spectrahedra are exactly the free semialgebraic convex sets this paper studies free analytic maps between free spectrahedra and under certain generically valid irreducibility assumptions classifies all those that are bianalytic the foundation of such maps turns out to be a very small class of birational maps we call convexotonic the convexotonic maps in g variables sit in correspondence with g dimensional algebras if two bounded free spectrahedra mathcal d a and mathcal d b meeting our irreducibility assumptions are free bianalytic with map denoted p then p must after possibly an affine linear transform extend to a convexotonic map corresponding to a g dimensional algebra spanned by u i a 1 ldots u i a g for some unitary u furthermore b and ua are unitarily equivalent the article also establishes a positivstellensatz for free analytic functions whose real part is positive semidefinite on a free spectrahedron and proves a representation for a free analytic map from mathcal d a to mathcal d b not necessarily bianalytic another result shows that a function analytic on any radial expansion of a free spectrahedron is approximable by polynomials uniformly on the spectrahedron these theorems are needed for classifying free bianalytic maps

shared information in stationary states of stochastic processes we present four estimators of the shared information or interdepency in ground states given that the coefficients appearing in the wave function are all real non negative numbers and therefore can be interpreted as probabilities of configurations such ground states of hermitian and non hermitian hamiltonians can be given for example by superpositions of valence bond states which can describe equilibrium but also stationary states of stochastic models we consider in detail the last case the system being a classical not a quantum one using analytical and numerical methods we compare the values of the estimators in the directed polymer and the raise and peel models which have massive conformal invariant and nonconformal invariant massless phases we show that like in the case of the quantum problem the estimators verify the area law with logarithmic corrections when phase transitions take place

grobner bases for finite temperature quantum computing and their complexity following the recent approach of using order domains to construct grobner bases from general projective varieties we examine the parity and time reversal arguments relating to the wightman axioms of quantum field theory and propose that the definition of associativity in these axioms should be introduced a posteriori to the cluster property in order to generalize the anyon conjecture for quantum computing to indefinite metrics we then show that this modification which we define via ideal quotients does not admit a faithful representation of the braid group because the generalized twisted inner automorphisms that we use to reintroduce associativity are only parity invariant for the prime spectra of the exterior algebra we then use a coordinate prescription for the quantum deformations of toric varieties to show how a faithful representation of the braid group can be reconstructed and argue that for a degree reverse lexicographic monomial ordered grobner basis the complexity class of this problem is bounded quantum polynomial

a complete characterization of determinantal quadratic polynomials abstract the problem of expressing a multivariate polynomial as the determinant of a monic definite symmetric or hermitian linear matrix polynomial lmp has drawn a huge amount of attention due to its connection with optimization problems in this paper we provide a necessary and sufficient condition for the existence of monic hermitian determinantal representation as well as monic symmetric determinantal representation of size 2 for a given quadratic polynomial further we propose a method to construct such a monic determinantal representation mdr of size 2 if it exists it is known that a quadratic polynomial f x x t a x b t x 1 has a symmetric mdr of size n 1 if a is negative semidefinite we prove that if a quadratic polynomial f x with a which is not negative semidefinite has an mdr of size greater than 2 then it has an mdr of size 2 too finally we characterize all quadratic polynomials that exhibit mdrs of any size

unitary similarity of the determinantal representation of unitary bordering matrices abstract let f x y z be a hyperbolic ternary form of degree n the helton vinnikov theorem asserts that there exists an n n complex symmetric matrix s such that f s x y z f x y z where the determinantal hyperbolic ternary form of an n n matrix b is defined by f b x y z det x b y b z i n let a be an n n unitary bordering matrix it is known that a is unitarily similar to a symmetric matrix in this paper we investigate the unitary similarity between the unitary bordering matrix a and the helton vinnikov symmetric matrix s admitted the determinantal representation of the ternary form f a x y z satisfying f s x y z f a x y z for n 3 4

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

bianalytic free maps between spectrahedra and spectraballs abstract linear matrix inequalities lmis are ubiquitous in real algebraic geometry semidefinite programming control theory and signal processing lmis with dimension free matrix unknowns are central to the theories of completely positive maps and operator algebras operator systems and spaces and serve as the paradigm for matrix convex sets the matricial feasibility set of an lmi is called a free spectrahedron in this article the bianalytic maps between a very general class of ball like free spectrahedra examples of which include row or column contractions and tuples of contractions and arbitrary free spectrahedra are characterized and seen to have an elegant algebraic form they are all highly structured rational maps in the case that both the domain and codomain are ball like these bianalytic maps are explicitly determined and the article gives necessary and sufficient conditions for the existence of such a map with a specified value and derivative at a point in particular this result leads to a classification of automorphism groups of ball like free spectrahedra the proofs depend on a novel free nullstellensatz established only after new tools in free analysis are developed and applied to obtain fine detail geometric in nature locally and algebraic in nature globally about the boundary of ball like free spectrahedra

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

symmetric determinantal representation of polynomials abstract we give an elementary proof only using linear algebra of a result due to helton mccullough and vinnikov which says that any polynomial over the reals can be written as the determinant of a symmetric affine linear pencil we give explicit determinantal representation formulas and extend our results to polynomials with coefficients in a ring of characteristic different from 2

the raise and peel model of a fluctuating interface we propose a one dimensional nonlocal stochastic model of adsorption and desorption depending on one parameter the adsorption rate at a special value of this parameter the model has some interesting features for example the spectrum is given by conformal field theory and the stationary non equilibrium probability distribution is given by the two dimensional equilibrium distribution of the ice model with domain wall type boundary conditions this connection is used to find exact analytic expressions for several quantities of the stochastic model vice versa some understanding of the ice model with domain wall type boundary conditions can be obtained by the study of the stochastic model at the special point we study several properties of the model such as the height fluctuations as well as cluster and avalanche distributions the latter has a long tail which shows that the model exhibits self organized criticality we also find in the stationary state a special surface phase transition without enhancement and with a crossover exponent 2 3 furthermore we study the phase diagram of the model as a function of the adsorption rate and find two massive phases and a scale invariant phase where conformal invariance is broken