Real Vertex Research Articles

Perfect matchings, rank of connection tensors and graph homomorphisms

AbstractWe develop a theory of graph algebras over general fields. This is modelled after the theory developed by Freedman et al. (2007, J. Amer. Math. Soc.20 37–51) for connection matrices, in the study of graph homomorphism functions over real edge weight and positive vertex weight. We introduce connection tensors for graph properties. This notion naturally generalizes the concept of connection matrices. It is shown that counting perfect matchings, and a host of other graph properties naturally defined as Holant problems (edge models), cannot be expressed by graph homomorphism functions with both complex vertex and edge weights (or even from more general fields). Our necessary and sufficient condition in terms of connection tensors is a simple exponential rank bound. It shows that positive semidefiniteness is not needed in the more general setting.

The effect of high-frequency rTMS of the left dorsolateral prefrontal cortex on the resolution of response, semantic and task conflict in the colour-word Stroop task

Previous work investigating the effect of rTMS of left Dorso-Lateral Prefrontal Cortex (DLPFC) on Stroop task performance reports no changes to the Stroop effect but reduced reaction times on both congruent and incongruent trials relative to sham stimulation; an effect attributed to an enhanced attentional (or task) set for colour classification. The present study tested this account by investigating whether, relative to vertex stimulation, rTMS of the left DLPFC modifies task conflict, a form of conflict that arises when task sets for colour classification and word reading compete, given that this particular type of conflict would be reduced by an enhanced task set for colour classification. Furthermore, the present study included measures of other forms of conflict present in the Stroop task (response and semantic conflict), the potential effects on which would have been hidden in previous studies employing only incongruent and congruent stimuli. Our data showed that left DLPFC stimulation had no effect on the magnitude of task conflict, nor did it affect response, semantic or overall conflict (where the null is supported by sensitive Bayes Factors in most cases). However, consistent with previous research left DLPFC stimulation had the general effect of reducing reaction times. We, therefore, show for the first time that relative to real vertex stimulation left DLPFC stimulation does not modify Stroop interference. Alternative accounts of the role of the left DLPFC in Stroop task performance in which it either modifies response thresholds or facilitates responding by keeping the correct response keys active in working memory are discussed.

On partition functions for 3-graphs

A cyclic graph is a graph with at each vertex a cyclic order of the edges incident with it specified. We characterize which real-valued functions on the collection of cubic cyclic graphs are partition functions of a real vertex model (P. de la Harpe, V.F.R. Jones (1993) [4]). They are characterized by ‘weak reflection positivity’, which amounts to the positive semidefiniteness of matrices based on the ‘k-join’ of cubic cyclic graphs (for all k∈Z+).Basic tools are the representation theory of the symmetric group and geometric invariant theory, in particular the Hanlon–Wales theorem on the decomposition of Brauer algebras and the Procesi–Schwarz theorem on inequalities defining orbit spaces.

CINOEDV: a co-information based method for detecting and visualizing n-order epistatic interactions.

BackgroundDetecting and visualizing nonlinear interaction effects of single nucleotide polymorphisms (SNPs) or epistatic interactions are important topics in bioinformatics since they play an important role in unraveling the mystery of “missing heritability”. However, related studies are almost limited to pairwise epistatic interactions due to their methodological and computational challenges.ResultsWe develop CINOEDV (Co-Information based N-Order Epistasis Detector and Visualizer) for the detection and visualization of epistatic interactions of their orders from 1 to n (n ≥ 2). CINOEDV is composed of two stages, namely, detecting stage and visualizing stage. In detecting stage, co-information based measures are employed to quantify association effects of n-order SNP combinations to the phenotype, and two types of search strategies are introduced to identify n-order epistatic interactions: an exhaustive search and a particle swarm optimization based search. In visualizing stage, all detected n-order epistatic interactions are used to construct a hypergraph, where a real vertex represents the main effect of a SNP and a virtual vertex denotes the interaction effect of an n-order epistatic interaction. By deeply analyzing the constructed hypergraph, some hidden clues for better understanding the underlying genetic architecture of complex diseases could be revealed.ConclusionsExperiments of CINOEDV and its comparison with existing state-of-the-art methods are performed on both simulation data sets and a real data set of age-related macular degeneration. Results demonstrate that CINOEDV is promising in detecting and visualizing n-order epistatic interactions. CINOEDV is implemented in R and is freely available from R CRAN: http://cran.r-project.org and https://sourceforge.net/projects/cinoedv/files/.Electronic supplementary materialThe online version of this article (doi:10.1186/s12859-016-1076-8) contains supplementary material, which is available to authorized users.

INTERMEDIATE SUMS ON POLYHEDRA II: BIDEGREE AND POISSON FORMULA

We continue our study of intermediate sums over polyhedra, interpolating between integrals and discrete sums, which were introduced by A. Barvinok [Computing the Ehrhart quasi-polynomial of a rational simplex, Math. Comp. 75 (2006), 1449-1466]. By well-known decompositions, it is sufficient to consider the case of affine cones s+c, where s is an arbitrary real vertex and c is a rational polyhedral cone. For a given rational subspace L, we integrate a given polynomial function h over all lattice slices of the affine cone s + c parallel to the subspace L and sum up the integrals. We study these intermediate sums by means of the intermediate generating functions $S^L(s+c)(\xi)$, and expose the bidegree structure in parameters s and $\xi$, which was implicitly used in the algorithms in our papers [Computation of the highest coefficients of weighted Ehrhart quasi-polynomials of rational polyhedra, Found. Comput. Math. 12 (2012), 435-469] and [Intermediate sums on polyhedra: Computation and real Ehrhart theory, Mathematika 59 (2013), 1-22]. The bidegree structure is key to a new proof for the Baldoni--Berline--Vergne approximation theorem for discrete generating functions [Local Euler--Maclaurin expansion of Barvinok valuations and Ehrhart coefficients of rational polytopes, Contemp. Math. 452 (2008), 15-33], using the Fourier analysis with respect to the parameter s and a continuity argument. Our study also enables a forthcoming paper, in which we study intermediate sums over multi-parameter families of polytopes.

Optimized Real Time Vertex Based Deformation Using Octree and Two Neighborhood Method

Soft tissue simulation is very important in medical simulation and learning procedures. But such simulations require intensive computation. With the force feedback devices, the computation required should be much faster as touch sensation is approximately 20 times faster than that of visual. Efficient collision detection techniques are required to quickly locate the touched node of the model and a few triangles of the model need to be rendered in the haptic loop to get further optimization while achieving the same haptic sensation. In this study an octree space partitioning method is used for collision detection to find the touched node quickly and two circular rings of neighbors of the touched node are rendered in the haptic loop for further optimization. This technique is implemented in our previously developed real time vertex based deformation. The results are compared with the previous method which shows better performance.

The rank of edge connection matrices and the dimension of algebras of invariant tensors

We characterize the rank of edge connection matrices of partition functions of real vertex models, as the dimension of the homogeneous components of the algebra of G-invariant tensors. Here G is the subgroup of the real orthogonal group that stabilizes the vertex model. This answers a question of Balázs Szegedy from 2007.

The real topological vertex at work

We develop the real vertex formalism for the computation of the topological string partition function with D-branes and O-planes at the fixed point locus of an anti-holomorphic involution acting non-trivially on the toric diagram of any local toric Calabi–Yau manifold. Our results cover in particular the real vertex with non-trivial fixed leg. We give a careful derivation of the relevant ingredients using duality with Chern–Simons theory on orbifolds. We show that the real vertex can also be interpreted in terms of a statistical model of symmetric crystal melting. Using this latter connection, we also assess the constant map contribution in Calabi–Yau orientifold models. We find that there are no perturbative contributions beyond one-loop, but a non-trivial sum over non-perturbative sectors, which we compare with the non-perturbative contribution to the closed string expansion.

ESTUDIO MORFOLOGICO DEL ACETABULO HUMANO: BIOMETRIA

The anatomic study of the acetabulum is made to help the resolution of the mechanical loss problem of the cotyle in the total replacement of the hip. Heigh, ant post diameter, depth, the distance between the horns of the articulated half -moon and the distance between the real vertex of the acetabulum and the vertex of the issosceles triangle, which base is the distance between the two horns, were measured in 150 acetabulum. It has been proved that the acetabulum is not hemispheric, but ogival. There exists a predominance of heigh upon the ant post diameter of 5,86 mm, ring level average, that is related to charges transmission. It is shown that the distance between the real vertex of the acetabulum and the vertex of the issosceles triangle, which joins the two horns of the articulated half-moon, is 24,3 mm average. This constitutes the elasticity zone in the moment of the charges transmission

Characterizations of strongly chordal graphs

In this paper we present several characterizations of the class of strongly chordal graphs. These include a forbidden induced subgraph characterization and two characterizations in terms of totally balanced matrices. Another characterization yields a polynomial recognition algorithm. Interest in these graphs arises in several ways. First, the problems of locating minimum weight dominating sets and minimum weight independent dominating sets in strongly chordal graphs with real vertex weights can be solved in polynomial time, whereas each of these problems is NP-hard for chordal graphs. Moreover, every well-described class of graphs for which polynomial algorithms to solve either of these problems have been presented is a subclass of the class of strongly chordal graphs. Second, these graphs have surprisingly nice structural properties and are intimately related to the class of totally balanced matrices.

Reggeon phases and the dual bootstrap program

Analyticity requirements lead to complex Regge vertices in multiparticle production amplitudes. Their effect is discussed within a dual, unitary and exchange-degenerate multi-Regge model. A resulting complex Regge vertex is incorporated in fixed- t model calculations of the reggeon bootstrap and the topological pomeron. The bootstrap condition for particle-particle scattering is unaffected but new j-plane singularities appear in Regge-Regge scattering amplitudes. The pomeron and reggeon intercepts determine the two parameters of our Regge vertex and admit two possible solutions: one is close to the naive phase rule (real vertex) and the other is complex and leads to a high lying non-leading pole in the pomeron channel.