the dirichlet problem for nonlinear second order elliptic equations iii functions of the eigenvalues of the hessian on etudie le probleme de dirichlet dans un domaine borne de r n a frontiere lisse f d 2 u dans u sur

the mathematics of eigenvalue optimization optimization problems involving the eigenvalues of symmetric and nonsymmetric matrices present a fascinating mathematical challenge such problems arise often in theory and practice particularly in engineering design and are amenable to a rich blend of classical mathematical techniques and contemporary optimization theory this essay presents a personal choice of some central mathematical ideas outlined for the broad optimization community i discuss the convex analysis of spectral functions and invariant matrix norms touching briefly on semidefinite representability and then outlining two broader algebraic viewpoints based on hyperbolic polynomials and lie algebra analogous nonconvex notions lead into eigenvalue perturbation theory the last third of the article concerns stability for polynomials matrices and associated dynamical systems ending with a section on robustness the powerful and elegant language of nonsmooth analysis appears throughout as a unifying narrative thread

new approaches to plactic monoid via grobner shirshov bases abstract we give two explicit quadratic presentations of the plactic monoid in row and column generators correspondingly then we give direct independent proofs that these presentations are grobner shirshov bases of the plactic algebra in deg lex orderings of generators from composition diamond lemma for associative algebras it follows that the set of young tableaux is the knuth normal form for plactic monoid 30 see also ch 5 in 32

liouville type results of curvature operators of fuchsian convex surfaces abstract in this paper we deduce some liouville type results of curvature operators of fuchsian convex surfaces in minkowski space

parabolic monge ampere equations on riemannian manifolds on a compact riemannian manifold m g we consider the parabolic monge ampere equation t x t logdet g x hess x t detg x x t f x x 0 phiv 0 x here is a real parameter andf 0 m r are smooth functions we show existence of for all timestindependent of if 0 then t t converges exponentially towards a solution of the stationary problem ast in the special case 0 f 0 one has 0 and we determine the convergence rate t in thel2 norm more precisely

accelerated first order methods for hyperbolic programming we develop a framework for applying accelerated methods to general hyperbolic programming including linear second order cone and semidefinite programming as special cases the approach replaces a hyperbolic program with a convex optimization problem whose smooth objective function is explicit and for which the only constraints are linear equations one more linear equation than for the original problem virtually any first order method can be applied an iteration bound for a representative accelerated method is derived

generalized monte carlo significance tests summary simple monte carlo significance testing has many applications particularly in the preliminary analysis of spatial data the method requires the value of the test statistic to be ranked among a random sample of values generated according to the null hypothesis however there are situations in which a sample of values can only be conveniently generated using a markov chain initiated by the observed data so that independence is violated this paper describes two methods that overcome the problem of dependence and allow exact tests to be carried out the methods are applied to the rasch model to the finite lattice ising model and to the testing of association between spatial processes power is discussed in a simple case

synchronized rational relations of finite and infinite words abstract the purpose of this paper is a comprehensive study of a family of rational relations both of finite and infinite words namely those that are computable by automata where the reading heads move simultaneously on the n input tapes and that we thus propose to call synchronized rational relations

hyperbolic polynomials and interior point methods for convex programming hyperbolic polynomials have their origins in partial differential equations we show in this paper that they have applications in interior point methods for convex programming each homogeneous hyperbolic polynomial p has an associated open and convex cone called its hyperbolicity cone we give an explicit representation of this cone in terms of polynomial inequalities the function fx log px is a logarithmically homogeneous self concordant barrier function for the hyperbolicity cone with barrier parameter equal to the degree of p the function fx possesses striking additional properties that are useful in designing long step interior point methods for example we show that the long step primal potential reduction methods of nesterov and todd and the surface following methods of nesterov and nemirovskii extend to hyperbolic barrier functions we also show that there exists a hyperbolic barrier function on every homogeneous cone

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

the dirichlet problem for nonlinear second order elliptic equations i monge ampegre equation on considere le probleme de dirichlet pour des equations elliptiques non lineaires pour une fonction reelle u definie dans la fermeture d un domaine borne dans r n avec une frontiere c

local superlinear convergence of polynomial time interior point methods for hyperbolicity cone optimization problems in this paper we establish the local superlinear convergence property of some polynomial time interior point methods for an important family of conic optimization problems the main structural property used in our analysis is the logarithmic homogeneity of self concordant barrier function which must have negative curvature we propose a new path following predictor corrector scheme which works only in the dual space it is based on an easily computable gradient proximity measure which ensures an automatic transformation of the global linear rate of convergence to the local superlinear one under some mild assumptions our step size procedure for the predictor step is related to the maximum step size maintaining feasibility as the optimal solution set is approached our algorithm automatically tightens the neighborhood of the central path proportionally to the current duality gap

detection of topological patterns in complex networks correlation profile of the internet abstract a general scheme for detecting and analyzing topological patterns in large complex networks is presented in this scheme the network in question is compared with its properly randomized version that preserves some of its low level topological properties statistically significant deviation of any topological property of a network from this null model likely reflects its design principles and or evolutionary history we illustrate this basic scheme using the example of the correlation profile of the internet quantifying correlations between degrees of its neighboring nodes this profile distinguishes the internet from previously studied molecular networks with a similar scale free degree distribution we finally demonstrate that the clustering in a network is very sensitive to both the degree distribution and its correlation profile and compare the clustering in the internet to the appropriate null model

on finite complete rewriting systems finite derivation type and automaticity for homogeneous monoids this paper investigates the class of finitely presented monoids defined by homogeneous length preserving relations from a computational perspective the properties of admitting a finite complete rewriting system having finite derivation type being automatic and being biautomatic are investigated for this class of monoids the first main result shows that for any consistent combination of these properties and their negations there is a homogeneous monoid with exactly this combination of properties we then introduce the new concept of abstract rees commensurability an analogue of the notion of abstract commensurability for groups in order to extend this result to show that the same statement holds even if one restricts attention to the class of n ary homogeneous monoids where every side of every relation has fixed length n we then introduce a new encoding technique that allows us to extend the result partially to the class of n ary multihomogenous monoids

a finiteness condition for rewriting systems abstract we present a purely combinatorial approach to the question of whether or not a finitely presented monoid has a finite canonical presentation our approach is based on the notion of finite derivation type which is a combinatorial condition satisfied by certain rewriting systems our main result states that if a monoid m has a finite canonical presentation then all finite presentations of m have finite derivation type by proving that a certain monoid s1 does not have finite derivation type we show in addition that the homological finiteness condition fp is not sufficient to guarantee that a finitely presented monoid with a decidable word problem has a finite canonical presentation

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

on the self concordance of the universal barrier function let k be a regular convex cone in hbox bbb r n and let f x be its universal barrier function let d kf x h ldots h be the k th order directional derivative at the point x in k 0 and direction h in hbox bbb r n we show that for every m ge3 there exists a constant c m 0 depending only on m such that d mf x h ldots h le c m d 2f x h h m 2 for m 3 this is the self concordance inequality of nesterov and nemirovskii our proof uses a powerful recent result of bourgain

mixing patterns in networks we study assortative mixing in networks the tendency for vertices in networks to be connected to other vertices that are like or unlike them in some way we consider mixing according to discrete characteristics such as language or race in social networks and scalar characteristics such as age as a special example of the latter we consider mixing according to vertex degree i e according to the number of connections vertices have to other vertices do gregarious people tend to associate with other gregarious people we propose a number of measures of assortative mixing appropriate to the various mixing types and apply them to a variety of real world networks showing that assortative mixing is a pervasive phenomenon found in many networks we also propose several models of assortatively mixed networks both analytic ones based on generating function methods and numerical ones based on monte carlo graph generation techniques we use these models to probe the properties of networks as their level of assortativity is varied in the particular case of mixing by degree we find strong variation with assortativity in the connectivity of the network and in the resilience of the network to the removal of vertices

constrained markovian dynamics of random graphs we introduce a statistical mechanics formalism for the study of constrained graph evolution as a markovian stochastic process in analogy with that available for spin systems deriving its basic properties and highlighting the role of the mobility the number of allowed moves for any given graph as an application of the general theory we analyze the properties of degree preserving markov chains based on elementary edge switchings we give an exact yet simple formula for the mobility in terms of the graph s adjacency matrix and its spectrum this formula allows us to define acceptance probabilities for edge switchings such that the markov chains become controlled glauber type detailed balance processes designed to evolve to any required invariant measure representing the asymptotic frequencies with which the allowed graphs are visited during the process as a corollary we also derive a condition in terms of simple degree statistics sufficient to guarantee that in the limit where the number of nodes diverges even for state independent acceptance probabilities of proposed moves the invariant measure of the process will be uniform we test our theory on synthetic graphs and on realistic larger graphs as studied in cellular biology showing explicitly that for instances where the simple edge swap dynamics fails to converge to the uniform measure a suitably modified markov chain instead generates the correct phase space sampling

characterization of the barrier parameter of homogeneous convex cones we characterize the smallest best barrier parameter of self concordant barriers for homogeneous convex cones in particular we prove that this parameter is the same as the rank of the cone which is the number of steps in a recursive construction of the cone siegel domain construction we also provide lower bounds on the barrier parameter in terms of the caratheodory number of the cone the bounds are tight for homogeneous self dual cones 1998 the mathematical programming society inc published by elsevier science b v

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

self concordant barriers for hyperbolic means the geometric mean and the function det 1 m on the m by m positive definite matrices are examples of hyperbolic means functions of the form p 1 m where p is a hyperbolic polynomial of degree m a homogeneous polynomial p is hyperbolic with respect to a vector d if the polynomial t p x td has only real roots for every vector x any hyperbolic mean is positively homogeneous and concave on a suitable domain we present a self concordant barrier for its hypograph with barrier parameter o m2 our approach is direct and shows for example that the function mlog det 1 is an m2 self concordant barrier on a natural domain such barriers suggest novel interior point approaches to convex programs involving hyperbolic means

multi parameter surfaces of analytic centers and long step surface following interior point methods we develop a long step polynomial time version of the method of analytic centers for non linear convex problems the method traces a multi parameter surface of analytic centers rather than the usual path which allows to handle cases with noncentered and possibly infeasible starting point

improved inference for respondent driven sampling data with application to hiv prevalence estimation respondent driven sampling is a form of link tracing network sampling which is widely used to study hard to reach populations often to estimate population proportions previous treatments of this process have used a with replacement approximation which we show induces bias in estimates for large sample fractions and differential network connectedness by characteristic of interest we present a treatment of respondent driven sampling as a successive sampling process unlike existing representations our approach respects the essential without replacement feature of the process while converging to an existing with replacement representation for small sample fractions and to the sample mean for a full population sample we present a successive sampling based estimator for population means based on respondent driven sampling data and demonstrate its superior performance when the size of the hidden population is known we present sensitivity analyses for unknown population sizes in addition we note tha

generation of uncorrelated random scale free networks uncorrelated random scale free networks are useful null models to check the accuracy and the analytical solutions of dynamical processes defined on complex networks we propose and analyze a model capable of generating random uncorrelated scale free networks with no multiple and self connections the model is based on the classical configuration model with an additional restriction on the maximum possible degree of the vertices we check numerically that the proposed model indeed generates scale free networks with no two and three vertex correlations as measured by the average degree of the nearest neighbors and the clustering coefficient of the vertices of degree k respectively complex networks constitute a general framework for the topological characterization of many natural and technological systems whose complexity prevents a more detailed microscopic description f1 3g within this framework these systems are represented in terms of networks or graphs f4g in which vertices stand for the units composing the system while edges among vertices represent the interactions or relations between pairs of units the focus is thus shift to the topological characterization of the representative network a task which is largely more feasible and yields nevertheless a noticeable amount of information on the structure and properties of the original system the empirical analysis of many real complex networks has unveiled the presence of several typical properties widely found in systems belonging to a large variety of realms one of the most relevant is given by the scale free nature of the degree distribution pskd f1 3 5g defined as the probability that a randomly chosen vertex has degree k si e it is connected to other k verticesd in mathematical terms the scale free property translates into a power law function of the form psk d k g

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

cone free primal dual path following and potential reduction polynomial time interior point methods we present a framework for designing and analyzing primal dual interior point methods for convex optimization we assume that a self concordant barrier for the convex domain of interest and the legendre transformation of the barrier are both available to us we directly apply the theory and techniques of interior point methods to the given good formulation of the problem as is without a conic reformulation using the very usual primal central path concept and a less usual version of a dual path concept we show that many of the advantages of the primal dual interior point techniques are available to us in this framework and therefore they are not intrinsically tied to the conic reformulation and the logarithmic homogeneity of the underlying barrier function

geometry of homogeneous convex cones duality mapping and optimal self concordant barriers we study homogeneous convex cones we first characterize the extreme rays of such cones in the context of their primal construction due to vinberg and also in the context of their dual construction due to rothaus then using these results we prove that every homogeneous cone is facially exposed we provide an alternative proof of a result of guler and tunc el that the siegel rank of a symmetric cone is equal to its caratheodory number our proof does not use the jordan von neumann wigner characterization of the symmetric cones but it easily follows from the primal construction of the homogeneous cones and our results on the geometry of homogeneous cones in primal and dual forms we study optimal self concordant barriers in this context we briefly discuss the duality mapping in the context of automorphisms of convex cones and prove using numerical integration that the duality mapping is not an involution on certain self dual cones

graph animals subgraph sampling and motif search in large networks we generalize a sampling algorithm for lattice animals connected clusters on a regular lattice to a monte carlo algorithm for graph animals i e connected subgraphs in arbitrary networks as with the algorithm in n kashtan et al bioinformatics 20 1746 2004 it provides a weighted sample but the computation of the weights is much faster linear in the size of subgraphs instead of superexponential this allows subgraphs with up to ten or more nodes to be sampled with very high statistics from arbitrarily large networks using this together with a heuristic algorithm for rapidly classifying isomorphic graphs we present results for two protein interaction networks obtained using the tandem affinity purification tap method one of escherichia coli with 230 nodes and 695 links and one for yeast saccharomyces cerevisiae with roughly ten times more nodes and links we find in both cases that most connected subgraphs are strong motifs z scores 10 or antimotifs z scores 10 when the null model is the ensemble of networks with fixed degree sequence strong differences appear between the two networks with dominant motifs in e coli being nearly bipartite graphs and having many pairs of nodes that connect to the same neighbors while dominant motifs in yeast tend towards completeness or contain large cliques we also explore a number of methods that do not rely on measurements of z scores or comparisons with null models for instance we discuss the influence of specific complexes like the 26s proteasome in yeast where a small number of complexes dominate the k cores with large k and have a decisive effect on the strongest motifs with 6 8 nodes we also present zipf plots of counts versus rank they show broad distributions that are not power laws in contrast to the case when disconnected subgraphs are included

tailored graph ensembles as proxies or null models for real networks i tools for quantifying structure we study the tailoring of structured random graph ensembles to real networks with the objective of generating precise and practical mathematical tools for quantifying and comparing network topologies macroscopically beyond the level of degree statistics our family of ensembles can produce graphs with any prescribed degree distribution and any degree degree correlation function its control parameters can be calculated fully analytically and as a result we can calculate asymptotically formulae for entropies and complexities and for information theoretic distances between networks expressed directly and explicitly in terms of their measured degree distribution and degree correlations

origin of degree correlations in the internet and other networks it has been argued that the observed anticorrelation between the degrees of adjacent vertices in the network representation of the internet has its origin in the restriction that no two vertices have more than one edge connecting them here we propose a formalism for modeling ensembles of graphs with single edges only and derive values for the exponents and correlation coefficients characterizing them our results confirm that the conjectured mechanism does indeed give rise to correlations of the kind seen in the internet although only a part of the measured correlation can be accounted for in this way

finite grobner shirshov bases for plactic algebras and biautomatic structures for plactic monoids this paper shows that every plactic algebra of finite rank admits a finite grobner shirshov basis the result is proved by using the combinatorial properties of young tableaux to construct a finite complete rewriting system for the corresponding plactic monoid which also yields the corollaries that plactic monoids of finite rank have finite derivation type and satisfy the homological finiteness properties left and right fp fp also answering a question of zelmanov we apply this rewriting system and other techniques to show that plactic monoids of finite rank are biautomatic

bifurcation and admissible solutions for the hessian equation abstract we study the following eigenvalue problem of k hessian equation s k d 2 u k f u in b u 0 on b global bifurcation result is established for this problem as applications of the bifurcation result we determine the intervals of for the existence nonexistence uniqueness and multiplicity of radially symmetric k admissible solutions for this problem

nonsmooth analysis of lorentz invariant functions a real valued function g x t on mathbb r n times mathbb r is called a lorentz invariant if g x t g ux t for all n times n orthogonal matrices u and all x t in the domain of g in other words g is invariant under the linear orthogonal transformations preserving the lorentz cone x t in mathbb r n times mathbb r t ge x it is easy to see that every lorentz invariant function can be decomposed as g f circ beta where f mathbb r 2 rightarrow mathbb r is a symmetric function and beta is the root map of the hyperbolic polynomial p x t t 2 x 1 2 cdots x n 2 we investigate a variety of important variational and nonsmooth properties of g and characterize them in terms of the symmetric function f

sequential monte carlo methods for statistical analysis of tables we describe a sequential importance sampling sis procedure for analyzing two way zero one or contingency tables with fixed marginal sums an essential feature of the new method is that it samples the columns of the table progressively according to certain special distributions our method produces monte carlo samples that are remarkably close to the uniform distribution enabling one to approximate closely the null distributions of various test statistics about these tables our method compares favorably with other existing monte carlo based algorithms and sometimes is a few orders of magnitude more efficient in particular compared with markov chain monte carlo mcmc based approaches our importance sampling method not only is more efficient in terms of absolute running time and frees one from pondering over the mixing issue but also provides an easy and accurate estimate of the total number of tables with fixed marginal sums which is far more difficult for an mcmc method to achieve

barrier functions in interior point methods we show that the universal barrier function of a convex cone introduced by nesterov and nemirovskii is the logarithm of the characteristic function of the cone this interpretation demonstrates the invariance of the universal barrier under the automorphism group of the underlying cone this provides a simple method for calculating the universal barrier for homogeneous cones we identify some known barriers as the universal barrier scaled by an appropriate constant we also calculate some new universal barrier functions our results connect the field of interior point methods to several branches of mathematics such as lie groups jordan algebras siegel domains differential geometry complex analysis of several variables etc

on cone of nonsymmetric positive semidefinite matrices abstract in this paper we analyze and characterize the cone of nonsymmetric positive semidefinite matrices ns psd firstly we study basic properties of the geometry of the ns psd cone and show that it is a hyperbolic but not homogeneous cone secondly we prove that the ns psd cone is a maximal convex subcone of p 0 matrix cone which is not convex but the interior of the ns psd cone is not a maximal convex subcone of p matrix cone as the byproducts some new sufficient and necessary conditions for a nonsymmetric matrix to be positive semidefinite are given finally we present some properties of metric projection onto the ns psd cone

finite transducers for divisibility monoids divisibility monoids are a natural lattice theoretical generalization of mazurkiewicz trace monoids namely monoids in which the distributivity of the involved divisibility lattices is kept as an hypothesis but the relations between the generators are not supposed to necessarily be commutations here we show that every divisibility monoid admits an explicit finite transducer which allows to compute normal forms in quadratic time in addition we prove that every divisibility monoid is biautomatic

communities clustering phase transitions and hysteresis pitfalls in constructing network ensembles ensembles of networks are used as null models in many applications however simple null models often show much less clustering than their real world counterparts in this paper we study a biased rewiring model where clustering is enhanced by means of a fugacity as in the strauss or triangle model but where the number of links attached to each node is strictly preserved similar models have been proposed previously in milo et al science 298 824 2002 our model exhibits phase transitions as the fugacity is changed for regular graphs identical degrees for all nodes with degree kg2 we find a single first order transition for all nonregular networks that we studied including erd os r enyi scale free and several real world networks multiple jumps resembling first order transitions appear the jumps coincide with the sudden emergence of cluster cores groups of highly interconnected nodes with higher than average degrees where each edge participates in many triangles hence clustering is not smoothly distributed throughout the network once formed the cluster cores are difficult to remove leading to strong hysteresis to study the cluster cores visually we introduce q clique adjacency plots cluster cores constitute robust communities that emerge spontaneously from the triangle generating process rather than being put explicitly into the definition of the model all the quantities we measured including the modularity assortativity clustering and number of four and five cliques exhibit simultaneous jumps and are equivalent order parameters finally we point out that cluster cores produce pitfalls when using the present and similar models as null models for strongly clustered networks due to strong hysteresis which leads to broken ergodicity on realistic sampling time scales

artin groups of large type are shortlex automatic with regular geodesics we prove that any artin group of large type is shortlex automatic with respect to its standard generating set and that the set of all geodesic words over the same generating set satisfies the falsification by fellow traveller property fftp and hence is regular

garding s theory of hyperbolic polynomials this paper presents a simple self contained account of garding s theory of hyperbolic polynomials together with a recent convexity result of bauschke guler lewis sendov and an inequality of gurvits this account begins by establishing some new results the first concerns the existence of a pointwise arrangement of the eigenvalues so that they become global real analytic functions the second asserts that the associated branches are independent of the choice of hyperbolic direction 2013 wiley periodicals inc

link and subgraph likelihoods in random undirected networks with fixed and partially fixed degree sequences the simplest null models for networks used to distinguish significant features of a particular network from a priori expected features are random ensembles with the degree sequence fixed by the specific network of interest these fixed degree sequence fds ensembles are however famously resistant to analytic attack in this paper we introduce ensembles with partially fixed degree sequences pfds and compare analytic results obtained for them with monte carlo results for the fds ensemble these results include link likelihoods subgraph likelihoods and degree correlations we find that local structural features in the fds ensemble can be reasonably well estimated by simultaneously fixing only the degrees of a few nodes in addition to the total number of nodes and links as test cases we use two protein interaction networks escherichia coli saccharomyces cerevisiae the internet on the autonomous system as level and the world wide web fixing just the degrees of two nodes gives the mean neighbor degree as a function of node degree k k in agreement with results explicitly obtained from rewiring for power law degree distributions we derive the disassortativity analytically in the pfds ensemble the partition function can be expanded diagrammatically we obtain an explicit expression for the link likelihood to lowest order which reduces in the limit of large sparse undirected networks with l links and with k max l to the simple formula p k k kk 2l kk in a similar limit the probability for three nodes to be linked into a triangle reduces to the factorized expression p delta k 1 k 2 k 3 p k 1 k 2 p k 1 k 3 p k 2 k 3

polynomial sized semidefinite representations of derivative relaxations of spectrahedral cones we give explicit polynomial sized in n n and k k semidefinite representations of the hyperbolicity cones associated with the elementary symmetric polynomials of degree k k in n n variables these convex cones form a family of non polyhedral outer approximations of the non negative orthant that preserve low dimensional faces while successively discarding high dimensional faces more generally we construct explicit semidefinite representations polynomial sized in k m k m and n n of the hyperbolicity cones associated with k kth directional derivatives of polynomials of the form p x det sum i 1 n a i x i p x det i 1naixi where the a i ai are m times m m m symmetric matrices these convex cones form an analogous family of outer approximations to any spectrahedral cone our representations allow us to use semidefinite programming to solve the linear cone programs associated with these convex cones as well as their less well understood dual cones

determinantal representations of invariant hyperbolic plane curves abstract we study hyperbolic polynomials with nice symmetry and express them as the determinant of a hermitian matrix with special structure the goal of this paper is to answer a question posed by chien and nakazato in 2015 3 by properly modifying a determinantal representation construction of dixon 1902 5 we show for every hyperbolic polynomial of degree n invariant under the cyclic group of order n there exists a determinantal representation admitted via some cyclic weighted shift matrix moreover if the polynomial is invariant under the action of the dihedral group of order n the associated cyclic weighted shift matrix is unitarily equivalent to one with real entries

hermite s method of separation of solutions of systems of algebraic equations and its applications abstract the paper may be considered as a supplement to the book by m krein and m naimark problem find the number of real solutions of an algebraic system f1 x y 0 f2 x y 0 satisfying an algebraic condition g x y 0 we give a review of hermite s method which permits one to solve the problem using a finite number of algebraic operations on the coefficients of f1 f2 g the method reduces the problem to an investigation of the quadratic form in x0 xn 1 j 1 n j j x 0 x 1 j x n 1 j n 1 2 where j j is a solution of the system and n deg f1 deg f2 methods involved elimination theory the theory of symmetric functions and the theory of hankel quadratic forms applications analogue of sturm series in r 2 calculation of the kronecker poincare index of an algebraic field with respect to an algebraic curve in r 2 conditions for the sign definiteness of a homogeneous higher order polynomial of three variables we discuss also the possibility of extension of the method to r 2

tropical plactic algebra the cloaktic monoid and semigroup representations abstract a new tropical plactic algebra is introduced in which the knuth relations are inferred from the underlying semiring arithmetic encapsulating the ubiquitous plactic monoid p n this algebra manifests a natural framework for accommodating representations of p n and equivalently of young tableaux and its moderate coarsening the cloaktic monoid k n and the co cloaktic monoid the faithful linear representations of k n and by tropical matrices which constitute a tropical plactic algebra are shown to provide linear representations of the plactic monoid to this end the paper develops a special type of configuration tableaux corresponding bijectively to semi standard young tableaux these special tableaux allow a systematic encoding of combinatorial properties in numerical algebraic ways including algorithmic benefits the interplay between these algebraic combinatorial structures establishes a profound machinery for exploring semigroup attributes in particular satisfying of semigroup identities this machinery is utilized here to prove that k n and admit all the semigroup identities satisfied by n n triangular tropical matrices which holds also for p 3

the dirichlet problem for nonlinear second order elliptic equations ii complex monge ampere and uniformaly elliptic equations on considere le probleme de dirichlet pour des equations elliptiques non lineaires d ordre 2 pour une fonction reelle dans un domaine borne de r n a frontiere lisse

plactic monoids a braided approach abstract young tableaux carry an associative product described by the schensted algorithm they thus form a monoid pl called plactic it is central in numerous combinatorial and algebraic applications in this paper the tableaux product is shown to be completely determined by an idempotent braiding on the much simpler set of columns col here a braiding is a set theoretic solution to the yang baxter equation as an application we identify the hochschild cohomology of pl which resists classical approaches with the more accessible braided cohomology of col the cohomological dimension of pl is obtained as a corollary also the braiding is proved to commute with the classical crystal reflection operators s i

the shifted plactic monoid we introduce a shifted analog of the plactic monoid of lascoux and schutzenberger the shifted plactic monoid it can be defined in two different ways via the shifted knuth relations or using haiman s mixed insertion applications include a new combinatorial derivation and a new version of the shifted littlewood richardson rule similar results for the coefficients in the schur expansion of a schur p function a shifted counterpart of the lascoux schutzenberger theory of noncommutative schur functions in plactic variables a characterization of shifted tableau words and more

continuity estimates for solutions to the prescribed curvature dirichlet problem on etudie des solutions au probleme de dirichlet pour l equation de courbure moyenne prescrite ainsi que pour des equations de weingarten plus generales on etablit des modules d estimation de continuite dependant seulement des valeurs limites

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

structured matrices continued fractions and root localization of polynomials we give a detailed account of various connections between several classes of objects hankel hurwitz toeplitz vandermonde and other structured matrices stietjes and jacobi type continued fractions cauchy indices moment problems total positivity and root localization of univariate polynomials along with a survey of many classical facts we provide a number of new results

the constraint equations of lovelock gravity theories a new σk yamabe problem this paper is devoted to the study of the constraint equations of the lovelock gravity theories in the case of a conformally flat time symmetric and space like manifold we show that the hamiltonian constraint equation becomes a generalisation of the k yamabe problem that is to say the prescription of a linear combination of the k curvatures of the manifold we search solutions in a conformal class for a compact manifold using the existing results on the k yamabe problem we describe some cases in which they can be extended to this new problem this requires to study the concavity of some polynomial we do it in two ways regarding the concavity of a root of this polynomial which is connected to algebraic properties of the polynomial and seeking analytically a concavifying function this gives several cases in which a conformal solution exists finally we show an implicit function theorem in the case of a manifold with negative scalar curvature and find a conformal solution when the lovelock theories are close to general relativity this paper is devoted to the study of the constraint equations of the lovelock gravity theories in the case of a conformally flat time symmetric and space like manifold we show that the hamiltonian constraint equation becomes a generalisation of the k yamabe problem that is to say the prescription of a linear combination of the k curvatures of the manifold we search solutions in a conformal class for a compact manifold using the existing results on the k yamabe problem we describe some cases in which they can be extended to this new problem this requires to study the concavity of some polynomial we do it in two ways regarding the concavity of a root of this polynomial which is connected to algebraic properties of the polynomial and seeking analytically a concavifying function this gives several cases in which a conformal solution exists finally we show an implicit function theorem in the case of a manifold with negative scalar curvature and find a conformal solution when the lovelock t

irreducible representations of the plactic algebra of rank four abstract irreducible representations of the plactic monoid m of rank four are studied certain concrete families of simple modules over the plactic algebra k m over a field k are constructed it is shown that the jacobson radical j k m of k m is nilpotent moreover the congruence on m determined by j k m coincides with the intersection of the congruences determined by the primitive ideals of k m corresponding to the constructed simple modules in particular m is a subdirect product of the images of m in the corresponding endomorphism algebras

nonsmooth solutions for a class of fully nonlinear pde s on lie groups abstract in this paper we prove the existence of nonsmooth viscosity solutions for dirichlet problems involving a class a fully non linear operators on lie groups in particular we consider the elementary symmetric functions of the eigenvalues of the hessian built with left invariant vector fields

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

homogeneous complex networks abstract we discuss various ensembles of homogeneous complex networks and a monte carlo method of generating graphs from these ensembles the method is quite general and can be applied to simulate micro canonical canonical or grand canonical ensembles for systems with various statistical weights it can be used to construct homogeneous networks with desired properties or to construct a non trivial scoring function for problems of advanced motif searching

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

correlated random networks we develop a statistical theory of networks a network is a set of vertices and links given by its adjacency matrix c and the relevant statistical ensembles are defined in terms of a partition function z summation operator exp betah c the simplest cases are uncorrelated random networks such as the well known erdos renyi graphs here we study more general interactions h c which lead to correlations for example between the connectivities of adjacent vertices in particular such correlations occur in optimized networks described by partition functions in the limit beta infinity they are argued to be a crucial signature of evolutionary design in biological networks

testing for nonlinearity in time series the method of surrogate data we describe a statistical approach for identifying nonlinearity in time series the method first specifies some linear process as a null hypothesis then generates surrogate data sets which are consistent with this null hypothesis and finally computes a discriminating statistic for the original and for each of the surrogate data sets if the value computed for the original data is significantly different than the ensemble of values computed for the surrogate data then the null hypothesis is rejected and nonlinearity is detected we discuss various null hypotheses and discriminating statistics the method is demonstrated for numerical data generated by known chaotic systems and applied to a number of experimental time series which arise in the measurement of superfluids brain waves and sunspots we evaluate the statistical significance of the evidence for nonlinear structure in each case and illustrate aspects of the data which this approach identifies

surrogate time series before we apply nonlinear techniques e g those inspired by chaos theory to dynamical phenomena occurring in nature it is necessary to first ask if the use of such advanced techniques is justified by the data while many processes in nature seem very unlikely a priori to be linear the possible nonlinear nature might not be evident in specific aspects of their dynamics the method of surrogate data has become a very popular tool to address such a question however while it was meant to provide a statistically rigorous foolproof framework some limitations and caveats have shown up in its practical use in this paper recent efforts to understand the caveats avoid the pitfalls and to overcome some of the limitations are reviewed and augmented by new material in particular we will discuss specific as well as more general approaches to constrained randomisation providing a full range of examples new algorithms will be introduced for unevenly sampled and multivariate data and for surrogate spike trains the main limitation which lies in the interpretability of the test results will be illustrated through instructive case studies we will also discuss some implementational aspects of the realisation of these methods in the tisean software package

on the homology of associative algebras we present a new free resolution for k as an g module where g is an associative augmented algebra over a field k the resolution reflects the combinatorial properties of g introduction let k be a field and let g be an associative augmented k algebra for many purposes one wishes to have a projective resolution of k as a g module the bar resolution is always easy to define but it is often too large to use in practice at the other extreme minimal resolutions may exist but they are often hard to write down in a way that is amenable to calculations the main theorem of this paper presents a compromise resolution though rarely minimal it is small enough to offer some bounds but explicit enough to facilitate calculations as it relies heavily upon combinatorial constructions it is best suited for analyzing otherwise tricky algebras given via generators and relations since several results we get as consequences of the main theorem have been obtained before through other means this paper may be viewed as generalizing and unifying several seemingly unrelated ideas in particular we are generalizing priddy s results on koszul algebras 12 extending homology computations by govorov 9 and backelin 3 and complementing bergman s methods regarding the diamond lemma 6 three results may be of interest the homology of the mod p steenrod algebra is given in terms of the homology of a new chain complex smaller than the a algebra in theorem 3 5 formula 16 offers an efficient algorithm for the determination of hilbert series and theorem 4 2 asserts the existence of new bounds on the torsion groups of commutative graded rings 1 definitions and the main theorem throughout this paper k denotes any field and g is an associative k algebra with unity the field k embeds in g via 77 k g and we suppose that g has an augmentation i e a k algebra map s g k for which 77 is a right inverse s denotes a set of generators for g as a k algebra and k s is the free associative k algebra with unity on s there is a canonical surjection f k s g once s is chosen and the augmentation e is determined once we know s x for each x e s in particular this means that k s may be augmented such that f becomes a map of augmented algebras to s we associate a function e s z called a grading in the absence of a more compelling choice we often take e to be grading by length i e e x 1 for received by the editors may 23 1983 and in revised form february 22 1984 this paper was the subject of an invited one hour address in boulder colorado during the week long ams summer program on combinatorics and algebra june 1983 1980 mathematics subject classification 1985 revision primary 16a62 secondary 13d03 55s10 1986 american mathematical society 0002 9947 86 1 00 25 per page

subgraph ensembles and motif discovery using an alternative heuristic for graph isomorphism a heuristic based on vertex invariants is developed to rapidly distinguish nonisomorphic graphs to a desired level of accuracy the method is applied to sample subgraphs from an escherichia coli protein interaction network and as a probe for discovery of extended motifs the network s structure is described using statistical properties of its n node subgraphs for n ensuremath leqslant 14 the zipf plots for subgraph occurrences are robust power laws that do not change when rewiring the network while fixing the degree sequence although many specific subgraphs exchange rank the exponent for the zipf law depends on n studying larger subgraphs highlights some striking patterns for various n motifs or connected pieces that are overabundant in the ensemble of subgraphs have more edges for a given number of nodes than antimotifs and generally display a bipartite structure or tend toward a complete graph in contrast antimotifs which are underabundant connected pieces are mostly trees or contain at most a single small loop

invariance and efficiency of convex representations we consider two notions for the representations of convex cones g representation and lifted g representation the former represents a convex cone as a slice of another the latter allows in addition the usage of auxiliary variables in the representation we first study the basic properties of these representations we show that some basic properties of convex cones are invariant under one notion of representation but not the other in particular we prove that lifted g representation is closed under duality when the representing cone is self dual we also prove that strict complementarity of a convex optimization problem in conic form is preserved under g representations then we move to study efficiency measures for representations we evaluate the representations of homogeneous convex cones based on the smoothness of the transformations mapping the central path of the representation to the central path of the represented optimization problem

an introduction to a class of matrix cone programming in this paper we define a class of linear conic programming which we call matrix cone programming or mcp involving the epigraphs of five commonly used matrix norms and the well studied symmetric cone mcp has recently been found to have many important applications for example in nuclear norm relaxations of affine rank minimization problems in order to make the defined mcp tractable and meaningful we must first understand the structure of these epigraphs so far only the epigraph of the frobenius matrix norm which can be regarded as a second order cone has been well studied here we take an initial step to study several important properties including its closed form solution calm bouligand differentiability and strong semismoothness of the metric projection operator over the epigraph of the l 1 l infty spectral or operator and nuclear matrix norm respectively these properties make it possible to apply augmented lagrangian methods which have recently received a great deal of interests due to their high efficiency in solving large scale semidefinite programming to this class of mcp problems the work done in this paper is far from comprehensive rather it is intended as a starting point to call for more insightful research on mcp so that it can serve as a basic tool to solve more challenging convex matrix optimization problems in years to come

on the complexity of a practical interior point method the theory of self concordance in convex optimization has been used to analyze the complexity of interior point methods based on newton s method for large problems it may be impractical to use newton s method here we analyze a truncated newton method in which an approximation to the newton search direction is used in addition practical interior point methods often include enhancements such as extrapolation that are absent from the theoretical algorithms analyzed previously we derive theoretical results that apply to such an algorithm one similar to a sophisticated computer implementation of a barrier method the results for a single barrier subproblem are a satisfying extension of the results for newton s method when extrapolation is used in the overall barrier method however our results are more limited we indicate by both theoretical arguments and examples why more elaborate results may be difficult to obtain

on local behavior of analytic functions we prove local inequalities for analytic functions defined on a convex body in r n which generalize well known classical inequalities for polynomials

subgraphs in random networks understanding the subgraph distribution in random networks is important for modeling complex systems in classic erdos networks which exhibit a poissonian degree distribution the number of appearances of a subgraph g with n nodes and g edges scales with network size as approximately n n g however many natural networks have a non poissonian degree distribution here we present approximate equations for the average number of subgraphs in an ensemble of random sparse directed networks characterized by an arbitrary degree sequence we find scaling rules for the commonly occurring case of directed scale free networks in which the outgoing degree distribution scales as p k approximately k gamma considering the power exponent of the degree distribution gamma as a control parameter we show that random networks exhibit transitions between three regimes in each regime the subgraph number of appearances follows a different scaling law approximately nalpha where alpha n g s 1 for gamma gamma c where s is the maximal outdegree in the subgraph and gamma c s 1 we find that certain subgraphs appear much more frequently than in erdos networks these results are in very good agreement with numerical simulations this has implications for detecting network motifs subgraphs that occur in natural networks significantly more than in their randomized counterparts