the minimum rank of symmetric matrices described by a graph a survey abstract the minimum rank of a simple graph g is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry for i j is nonzero whenever i j is an edge in g and is zero otherwise this paper surveys the current state of knowledge on the problem of determining the minimum rank of a graph and related issues

periodically forced double homoclinic loops to a dissipative saddle abstract in this paper we present a comprehensive theory on the dynamics of strange attractors in periodically perturbed second order differential equations assuming that the unperturbed equations have two homoclinic loops to a dissipative saddle fixed point we prove the existence of many complicated dynamical objects for a large class of non autonomous second order equations ranging from attractive quasi periodic torus to newhouse sinks and henon like attractors and to rank one attractors with srb measures and full stochastic behavior this theory enables us to apply rigorously many profound dynamics theories on non uniformly hyperbolic maps developed in the last forty years including the newhouse theory the theory of srb measures the theory of henon like attractors and the theory of rank one attractors to the analysis of the strange attractors in a periodically perturbed duffing equation

abundance of strange attractors in 1976 he h non performed a numerical study of the family of diffeomorphisms of the plane ha b x y 1 ax2 y bx and detected for parameter values a l 4 b 0 3 what seemed to be a non trivial attractor with a highly intricate geometric structure this family has since then been the subject of intense research both numerical and theoretical but its dynamics is still far from being completely understood in particular one could not exclude the possibility that the attractor observed by h6non were just a periodic orbit with a very high period recently in a remarkable paper bc2 benedicks and carleson were able to show that this is not the case at least for a positive lebesgue measure set of parameter values near a 2 b o more precisely they showed that if b 0 is small enough then for a positive measure set of a values near a 2 the corresponding diffeomorphism ha b exhibits a strange attractor their argument is a very creative extension of the techniques they had previously developed in bull for the study of the quadratic family on the real line and no doubt it will be important for the understanding of several other situations of complicated nonhyperbolic dynamics when acquainted in 1985 with the work by benedicks and carleson then in progress palls suggested that one should in this context think of the h6non family as a particular although important model for the creation of a horseshoe and that the emphasis should be put on the occurrence of unfoldings of homoclinic tangencies he proposed that the correct setting for benedicks carleson s results is within this more general framework of homoclinic bifurcations and stated the following

toward a theory of rank one attractors introduction 1 statement of results part i preparation 2 relevant results from one dimension 3 tools for analyzing rank one maps part ii phase space dynamics 4 critical structure and orbits 5 properties of orbits controlled by critical set 6 identification of hyperbolic behavior formal inductive procedure 7 global geometry via monotone branches 8 completion of induction 9 construction of srb measures part iii parameter issues 10 dependence of dynamical structures on parameter 11 dynamics of curves of critical points 12 derivative growth via statistics 13 positive measure sets of good parameters appendices

melnikov transforms bernoulli bundles and almost periodic perturbations in this paper we study nonlinear time varying perturbations of an autonomous vector field in the plane r2 we assume that the unperturbed equation i e the given vector field has a homoclinic orbit and we present a gen eralization of the melnikov method which allows us to show that the perturbed equation has a transversal homoclinic trajectory the key to our generalization is the concept of the melnikov transform which is a linear transformation on the space of perturbation functions the appropriate dynamical setting for studying these perturbation is the concept of a skew product flow the concept of transversality we require is best understood in this context under conditions whereby the perturbed equation admits a transversal homoclinic trajectory we also study the dynamics of the perturbed vector field in the vicinity of this trajectory in the skew product flow we show the dynamics near this trajectory can have the exotic behavior of the bernoulli shift the exact description of this dynamical phenomenon is in terms of a flow on a fiber bundle which we call the bernoulli bundle we allow all perturbations which are bounded and uniformly continuous in time thus our theory includes the classical periodic perturbations studied by melnikov quasi periodic and almost periodic perturbations as well as toroidal perturbations which are close to quasi periodic perturbations

a correction orthogonal representations and connectivity of graphs abstract this note corrects an error in the proof of the main result of the authors paper orthogonal representations and connectivity of graphs which appeared in linear algebra and its applications 114 115 1989 439 454

strange attractors in periodically kicked limit cycles and hopf bifurcations we prove the emergence of chaotic behavior in the form of horseshoes and strange attractors with srb measures when certain simple dynamical systems are kicked at periodic time intervals the settings considered include limit cycles and stationary points undergoing hopf bifurcations

strange attractors with one direction of instability we give simple conditions that guarantee for strongly dissipative maps the existence of strange attractors with a single direction of instability and certain controlled behaviors only the d 2 case is treated in this paper although our approach is by no means limited to two phase dimensions we develop a dynamical picture for the attractors in this class proving they have many of the statistical properties associated with chaos positive lyapunov exponents existence of srb measures and exponential decay of correlations other results include the geometry of fractal critical sets nonuniform hyperbolic behavior symbolic coding of orbits and formulas for topological entropy

functional connectivity in resting state fmri is linear correlation sufficient functional connectivity fc analysis is a prominent approach to analyzing fmri data especially acquired under the resting state condition the commonly used linear correlation fc measure bears an implicit assumption of gaussianity of the dependence structure if only the marginals but not all the bivariate distributions are gaussian linear correlation consistently underestimates the strength of the dependence to assess the suitability of linear correlation and the general potential of nonlinear fc measures we present a framework for testing and estimating the deviation from gaussianity by means of comparing mutual information in the data and its gaussianized counterpart we apply this method to 24 sessions of human resting state fmri for each session matrix of connectivities between 90 anatomical parcel time series is computed using mutual information and compared to results from its multivariate gaussian surrogate that conserves the correlations but cancels any nonlinearity while the group level tests confirmed non gaussianity in the fc the quantitative assessment revealed that the portion of mutual information neglected by linear correlation is relatively minor on average only about 5 of the mutual information already captured by the linear correlation the marginality of the non gaussianity was confirmed in comparisons using clustering of the parcels the disagreement between clustering obtained from mutual information and linear correlation was attributable to random error we conclude that for this type of data practical relevance of nonlinear methods trying to improve over linear correlation might be limited by the fact that the data are indeed almost gaussian

dynamics of homoclinic tangles in periodically perturbed second order equations we obtain a comprehensive description on the overall geometrical and dynamical structures of homoclinic tangles in periodically perturbed second order ordinary differential equations with dissipation let be the size of perturbation and be the entire homoclinic tangle we prove in particular that i for infinitely many disjoint open sets of contains nothing else but a horseshoe of infinitely many branches ii for infinitely many disjoint open sets of contains nothing else but one sink and one horseshoe of infinitely many branches and iii there are positive measure sets of so that admits strange attractors with sinai ruelle bowen measure we also use the equation d2qdt2 q2 dqdt q q2 q2sin t to illustrate how to apply our theory to the analysis and to the numerical simulations of a given equation

the dynamics of the henon map the purpose of this paper is to develop machinery which is suitable for a study of the long time behavior of systems such as the henon map with expansion combined with strong contraction our study is modeled on the treatment of the one dimensional system x 1 ax 2 and we study the perturbation of a from the value a 2 and b small a surprising number of complications arise when b 0 the computer simulations give some indication of the complicated structure of the attractor but it can safely be stated that the real difficulties do not show up the central aim of the paper is to introduce what we consider the relevant concepts and techniques needed to study mappings such as the henon one

pairwise network information and nonlinear correlations reconstructing the structural connectivity between interacting units from observed activity is a challenge across many different disciplines the fundamental first step is to establish whether or to what extent the interactions between the units can be considered pairwise and thus can be modeled as an interaction network with simple links corresponding to pairwise interactions in principle this can be determined by comparing the maximum entropy given the bivariate probability distributions to the true joint entropy in many practical cases this is not an option since the bivariate distributions needed may not be reliably estimated or the optimization is too computationally expensive here we present an approach that allows one to use mutual informations as a proxy for the bivariate probability distributions this has the advantage of being less computationally expensive and easier to estimate we achieve this by introducing a novel entropy maximization scheme that is based on conditioning on entropies and mutual informations this renders our approach typically superior to other methods based on linear approximations the advantages of the proposed method are documented using oscillator networks and a resting state human brain network as generic relevant examples

maximum generic nullity of a graph abstract for a graph g of order n the maximum nullity of g is defined to be the largest possible nullity over all real symmetric n n matrices a whose i j th entry for i j is nonzero whenever i j is an edge in g and is zero otherwise maximum nullity and the related parameter minimum rank of the same set of matrices have been studied extensively a new parameter maximum generic nullity is introduced maximum generic nullity provides insight into the structure of the null space of a matrix realizing maximum nullity of a graph it is shown that maximum generic nullity is bounded above by edge connectivity and below by vertex connectivity results on random graphs are used to show that as n goes to infinity almost all graphs have equal maximum generic nullity vertex connectivity edge connectivity and minimum degree

the minrank of random graphs the minrank of a directed graph g is the minimum rank of a matrix m that can be obtained from the adjacency matrix of g by switching some ones to zeros i e deleting edges and then setting all diagonal entries to one this quantity is closely related to the fundamental information theoretic problems of linear index coding bar yossef et al network coding effros et al and distributed storage mazumdar isit 2014 we prove tight bounds on the minrank of directed erd s renyi random graphs g n p for all regimes of p in 0 1 in particular for any constant p we show that mathsf minrk g theta n log n with high probability where g is chosen from g n p this bound gives a near quadratic improvement over the previous best lower bound of omega sqrt n haviv and langberg and partially settles an open problem raised by lubetzky and stav our lower bound matches the well known upper bound obtained by the clique covering solution and settles the linear index coding problem for random knowledge graphs

the bifurcation to homoclinic tori in the quasiperiodically forced duffing oscillator abstract using recently developed techniques of wiggins 1988 we study the bifurcation to homoclinic tori in the quasiperiodically forced duffing oscillator we show how homoclinic tori give rise to chaotic dynamics for single degree of freedom quasiperiodically forced oscillators in much the same way as smale horseshoes in the periodically forced case we give a complete description of the bifurcation set in the five dimensional parameter space for the two frequency forced duffing oscillator and then show how the two frequency results can be used to understand the effects of an arbitrary but finite number of frequencies

sinai bowen ruelle measures for certain henon maps we study maps t a b 2 2 defined by t a b left x y right left 1 a x 2 y bx right 0 0

from invariant curves to strange attractors we prove that simple mechanical systems when subjected to external periodic forcing can exhibit a surprisingly rich array of dynamical behaviors as parameters are varied in particular the existence of global strange attractors with fully stochastic properties is proved for a class of second order odes

an upper bound for the minimum rank of a graph abstract for a graph g of order n the minimum rank of g is defined to be the smallest possible rank over all real symmetric n n matrices a whose i j th entry for i j is nonzero whenever i j is an edge in g and is zero otherwise we prove an upper bound for minimum rank in terms of minimum degree of a vertex is valid for many graphs including all bipartite graphs and conjecture this bound is true over for all graphs and prove a related bound for all zero nonzero patterns of not necessarily symmetric matrices most of the results are valid for matrices over any infinite field but need not be true for matrices over finite fields

dissipative homoclinic loops of two dimensional maps and strange attractors with one direction of instability we prove that when subjected to periodic forcing of the form p t d h x y c sin t certain two dimensional vector fields with dissipative homoclinic loops generate strange attractors with sinai ruelle bowen measures for a set of forcing parameters of positive lebesgue measure the proof extends ideas of afraimovich and shilnikov and applies the recent theory of rank 1 maps developed by wang and young we prove a general theorem and then apply this theorem to an explicit model a forced duffing equation of the form d 2 q

three connected graphs whose maximum nullity is at most three for a graph g v e with vertex set v 1 2 n let s g be the set of all n n real valued symmetric matrices a which represent g the maximum nullity of a graph g denoted by m g is the largest possible nullity of any matrix a s g fiedler showed that a graph g has m g 1 if and only if g is a path johnson et al gave a characterization of all graphs g with m g 2 independently hogben and van der holst gave a characterization of all 2 connected graphs with m g 2 in this paper we show that k connected graphs g have m g k that k connected partial k graphs g have m g k and that for 3 connected graphs g m g 3 if and only if g is a partial 3 path

statistical mechanics of letters in words we consider words as a network of interacting letters and approximate the probability distribution of states taken on by this network despite the intuition that the rules of english spelling are highly combinatorial and arbitrary we find that maximum entropy models consistent with pairwise correlations among letters provide a surprisingly good approximation to the full statistics of words capturing ensuremath sim 92 mathrm of the multi information in four letter words and even discovering words that were not represented in the data these maximum entropy models incorporate letter interactions through a set of pairwise potentials and thus define an energy landscape on the space of possible words guided by the large letter redundancy we seek a lower dimensional encoding of the letter distribution and show that distinctions between local minima in the landscape account for ensuremath sim 68 mathrm of the four letter entropy we suggest that these states provide an effective vocabulary which is matched to the frequency of word use and much smaller than the full lexicon

note on von neumann and renyi entropies of a graph abstract we conjecture that all connected graphs of order n have von neumann entropy at least as great as the star k 1 n 1 and prove this for almost all graphs of order n we show that connected graphs of order n have renyi 2 entropy at least as great as k 1 n 1 and for 1 k n maximizes renyi entropy over graphs of order n we show that adding an edge to a graph can lower its von neumann entropy

orthogonal representations and connectivity of graphs abstract it is proved that a graph on n nodes is k connected if and only if its nodes can be represented by real vectors in dimension n k such that a nonadjacent nodes are represented by orthogonal vectors and b any n k of them are linearly independent we show that the closure of the set of all representations with properties a and b is irreducible as an algebraic variety and study the question of irreducibility of the variety of all representations with property a

on the number of zero patterns of a sequence of polynomials let f f1 fm be a sequence of polynomials of degree d in n variables m n over a field f the zero pattern of f at u f n is the set of those i 1 i m for which fi u 0 let zf f denote the number of zero patterns of f as u ranges over f we prove that zf f pn j 0 m j for d 1 and 1 zf f md n for d 2 for m nd these bounds are optimal within a factor of 7 25 the bound 1 improves the bound 1 md proved by j heintz 1983 using the dimension theory of affine varieties over the field of real numbers bounds stronger than heintz s but slightly weaker than 1 follow from results of j milnor 1964 h e warren 1968 and others their proofs use techniques from real algebraic geometry in contrast our half page proof is a simple application of the elementary linear algebra bound heintz applied his bound to estimate the complexity of his quanti fier elimination algorithm for algebraically closed fields we give sev eral additional applications the first two establish the existence of certain combinatorial objects our first application motivated by the branching program model in the theory of computing asserts that over any field f most graphs with v vertices have projective dimension p v log v the implied constant is absolute this result was previ ously known over the reals pudlak rodl the second application con cerns a lower bound in the span program model for computing boolean functions the third application motivated by a paper by n alon gives nearly tight ramsey bounds for matrices whose entries are defined by zero patterns of a sequence of polynomials we conclude the paper with a number of open problems computer and automation research institute hungarian academy of sci ences h 1111 budapest lagymanyosi u 11 hungary e mail address lajos nyest ilab sztaki hu department of computer science university of chicago chicago il 60637 e mail address laci cs uchicago edu department of computer science university of chicago chicago il 60637 e mail address gmkrishn cs uchicago edu

expected values of parameters associated with the minimum rank of a graph abstract we investigate the expected value of various graph parameters associated with the minimum rank of a graph including minimum rank maximum nullity and related colin de verdiere type parameters let g v p denote the usual erd s renyi random graph on v vertices with edge probability p we obtain bounds for the expected value of the random variables mr g v p m g v p g v p and g v p which yield bounds on the average values of these parameters over all labeled graphs of order v

on the minimum rank of a graph over finite fields abstract in this paper we deal with two aspects of the minimum rank of a simple undirected graph g on n vertices over a finite field f q with q elements which is denoted by mr f q g in the first part of this paper we show that the average minimum rank of simple undirected labeled graphs on n vertices over f 2 is 1 e n n were lim n e n 0 in the second part of this paper we assume that g contains a clique k k on k vertices we show that if q is not a prime then mr f q g n k 1 for 4 k n 1 and n 5 it is known that mr f q g 3 for k n 2 n 4 and q 4 we show that for k n 2 and each n 10 there exists a graph g such that mr f 3 g 3 for k n 3 n 5 and q 4 we show that mr f q g 4