283,437 research outputs found

    On the graph complement conjecture for minimum rank

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    AbstractThe minimum rank of a graph has been an interesting and well studied parameter investigated by many researchers over the past decade or so. One of the many unresolved questions on this topic is the so-called graph complement conjecture, which grew out of a workshop in 2006. This conjecture asks for an upper bound on the sum of the minimum rank of a graph and the minimum rank of its complement, and may be classified as a Nordhaus–Gaddum type problem involving the graph parameter minimum rank. The conjectured bound is the order of the graph plus two. Other variants of the graph complement conjecture are introduced here for the minimum semidefinite rank and the Colin de Verdière type parameter ν. We show that if the ν-graph complement conjecture is true for two graphs then it is true for the join of these graphs. Related results for the graph complement conjecture (and the positive semidefinite version) for joins of graphs are also established. We also report on the use of recent results on partial k-trees to establish the graph complement conjecture for graphs of low minimum rank

    Cut and join operator ring in Aristotelian tensor model

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    Recent advancement of rainbow tensor models based on their superintegrability (manifesting itself as the existence of an explicit expression for a generic Gaussian correlator) has allowed us to bypass the long-standing problem seen as the lack of eigenvalue/determinant representation needed to establish the KP/Toda integrability. As the mandatory next step, we discuss in this paper how to provide an adequate designation to each of the connected gauge-invariant operators that form a double coset, which is required to cleverly formulate a tree-algebra generalization of the Virasoro constraints. This problem goes beyond the enumeration problem per se tied to the permutation group, forcing us to introduce a few gauge fixing procedures to the coset. We point out that the permutation-based labeling, which has proven to be relevant for the Gaussian averages is, via interesting complexity, related to the one based on the keystone trees, whose algebra will provide the tensor counterpart of the Virasoro algebra for matrix models. Moreover, our simple analysis reveals the existence of nontrivial kernels and co-kernels for the cut operation and for the join operation respectively that prevent a straightforward construction of the non-perturbative RG-complete partition function and the identification of truly independent time variables. We demonstrate these problems by the simplest non-trivial Aristotelian RGB model with one complex rank-3 tensor, studying its ring of gauge-invariant operators, generated by the keystone triple with the help of four operations: addition, multiplication, cut and join.Comment: 55 page

    Equivariant Nica-Pimsner quotients associated with strong compactly aligned product systems

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    We parametrise the gauge-invariant ideals of the Toeplitz-Nica-Pimsner algebra of a strong compactly aligned product system over Z+d\mathbb{Z}_+^d by using 2d2^d-tuples of ideals of the coefficient algebra that are invariant, partially ordered, and maximal. We give an algebraic characterisation of maximality that allows the iteration of a 2d2^d-tuple to the maximal one inducing the same gauge-invariant ideal. The parametrisation respects inclusions and intersections, while we characterise the join operation on the 2d2^d-tuples that renders the parametrisation a lattice isomorphism. The problem of the parametrisation of the gauge-invariant ideals is equivalent to the study of relative Cuntz-Nica-Pimsner algebras, for which we provide a generalised Gauge-Invariant Uniqueness Theorem. We focus further on equivariant quotients of the Cuntz-Nica-Pimsner algebra and provide applications to regular product systems, C*-dynamical systems, strong finitely aligned higher-rank graphs, and product systems on finite frames. In particular, we provide a description of the parametrisation for (possibly non-automorphic) C*-dynamical systems and row-finite higher-rank graphs, which squares with known results when restricting to crossed products and to locally convex row-finite higher-rank graphs.Comment: 104 page

    The condition number of join decompositions

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    The join set of a finite collection of smooth embedded submanifolds of a mutual vector space is defined as their Minkowski sum. Join decompositions generalize some ubiquitous decompositions in multilinear algebra, namely tensor rank, Waring, partially symmetric rank and block term decompositions. This paper examines the numerical sensitivity of join decompositions to perturbations; specifically, we consider the condition number for general join decompositions. It is characterized as a distance to a set of ill-posed points in a supplementary product of Grassmannians. We prove that this condition number can be computed efficiently as the smallest singular value of an auxiliary matrix. For some special join sets, we characterized the behavior of sequences in the join set converging to the latter's boundary points. Finally, we specialize our discussion to the tensor rank and Waring decompositions and provide several numerical experiments confirming the key results

    Tensor decomposition and homotopy continuation

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    A computationally challenging classical elimination theory problem is to compute polynomials which vanish on the set of tensors of a given rank. By moving away from computing polynomials via elimination theory to computing pseudowitness sets via numerical elimination theory, we develop computational methods for computing ranks and border ranks of tensors along with decompositions. More generally, we present our approach using joins of any collection of irreducible and nondegenerate projective varieties X1,…,Xk⊂PNX_1,\ldots,X_k\subset\mathbb{P}^N defined over C\mathbb{C}. After computing ranks over C\mathbb{C}, we also explore computing real ranks. Various examples are included to demonstrate this numerical algebraic geometric approach.Comment: We have added two examples: A Coppersmith-Winograd tensor, Matrix multiplication with zeros. (26 pages, 1 figure

    Collective responses to antipredator recruitment calls in the jackdaw (Corvus monedula)

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    MbyRes thesis on collective responses to antipredator recruitment call in the jackdaw with particular focus on numerical assessment and the influence of dominance rank.Collective behaviour, whereby multiple individuals act together in a coherent, coordinated manner, occurs throughout nature. Self-organisation theory suggests that the maintenance of the collective behaviours shown by bird flocks and fish shoals, emerge as the result of simple rules of attraction among neighbours, with no need for leadership. Individuals in the models of self-organisation are assumed to be identical but in reality animals differ in aspects such as personality or motivation and in their social bonds with other conspecifics. Individual variation among group members means that certain individuals may exert disproportionate effects on the behaviour of groups and so in some cases, leadership is important for the initiation of collective behaviours. Moreover, cognitive processing of information about other individuals may play an important, but hitherto neglected role in the coordination of collective behaviours. This thesis examines the role of cognitive processes and of leadership in the formation of collective antipredator mobbing events in wild jackdaws (Corvus monedula). Mobbing presents a collective action problem as it entails substantial risks for individuals but, by driving away potential predators, provides collective benefits. Individuals may therefore benefit from processing information about the likely costs and benefits when deciding whether to join a mob. Recruitment to a mob is initiated through distinctive scolding calls, and information contained in these calls may be very valuable for individuals when deciding whether to join mobbing events. Chapter Two tested whether the number of callers influenced the number of individuals recruited to a mob. Individuals are expected to join a larger group because for an individual there is a reduced risk of predation. As predicted, I found that a greater number of callers does recruit a greater number of individuals. This work shows evidence that jackdaws discriminate between the calls of different individuals and is the first to show numerical assessment in an antipredator collective behaviour. Chapter Three tested whether the dominance rank of caller influenced the number of individuals recruited to a mob. A dominant individual may be expected to recruit more individuals as it is likely to be stronger and more able to drive away a predator. A dominant may also punish others for not responding. However, I found no influence of dominance rank of caller on the number of recruits. Punishment is unlikely to be important in jackdaw societies and any effect of rank may be obscured by other factors such as direct benefits from joining the mob, not contingent on dominance rank. Social relationships between the recruits and the caller have not been considered in this study and individuals may be more likely to respond to a close affiliate than pay attention to an individual’s dominance rank. Together, these results highlight the importance of investigating the cognitive processes involved in collective behaviours and also the characteristics of individuals who initiate collective behaviours

    Join-Reachability Problems in Directed Graphs

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    For a given collection G of directed graphs we define the join-reachability graph of G, denoted by J(G), as the directed graph that, for any pair of vertices a and b, contains a path from a to b if and only if such a path exists in all graphs of G. Our goal is to compute an efficient representation of J(G). In particular, we consider two versions of this problem. In the explicit version we wish to construct the smallest join-reachability graph for G. In the implicit version we wish to build an efficient data structure (in terms of space and query time) such that we can report fast the set of vertices that reach a query vertex in all graphs of G. This problem is related to the well-studied reachability problem and is motivated by emerging applications of graph-structured databases and graph algorithms. We consider the construction of join-reachability structures for two graphs and develop techniques that can be applied to both the explicit and the implicit problem. First we present optimal and near-optimal structures for paths and trees. Then, based on these results, we provide efficient structures for planar graphs and general directed graphs
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