Intersection Matrix Research Articles

Path intersection matrices and applications to networks

For a network G, we introduce a non-singular symmetric matrix, called a path intersection matrix, that will provide a new method for computing the ratio k(G)/k(G/ab) where k(G) is the tree-number of G and G/ab is obtained from G∪ab by contracting the new edge ab between two distinct nodes a and b. The quantities k(G)/k(G/ab) appear as invariants for various networks such as effective conductance for an electrical network and an ingredient for information centrality for a social network. We will review several examples of networks where path intersection matrices can be applied.

Enriching the semantics of the directed polyline–polygon topological relationships: the $$\mathcal {DLP}$$ DLP -intersection matrix

In this paper, we define an intersection matrix for enriching the semantics of the topological relationships between a directed polyline and a polygon. In particular, we propose the \(\mathcal {DLP}\)-intersection matrix which enables us to model the origin and destination points, as well as the right- and left-hand sides of the directed polyline. This matrix overcomes the limitation of the well-known DE-9IM, because it allows the representation of the different dimensions of the intersection results at the same time. Accordingly, the geo-operators have been revised and extended in order to address the notions of right- and left-hand sides of a directed polyline, as well as additional notions related to the orientation of the polyline. The \(\mathcal {DLP}\)-intersection matrix has been implemented by extending the Java Topology Suite methods in order to address the new geo-operators based on the notion of orientation.

Ribbon graphs and bialgebra of Lagrangian subspaces

To each ribbon graph we assign a so-called [Formula: see text]-space, which is a Lagrangian subspace in an even-dimensional vector space with the standard symplectic form. This invariant generalizes the notion of the intersection matrix of a chord diagram. Moreover, the actions of Morse perestroikas (or taking a partial dual) and Vassiliev moves on ribbon graphs are reinterpreted nicely in the language of [Formula: see text]-spaces, becoming changes of bases in this vector space. Finally, we define a bialgebra structure on the span of [Formula: see text]-spaces, which is analogous to the 4-bialgebra structure on chord diagrams.

ASSET: Analysis of Sequences of Synchronous Events in Massively Parallel Spike Trains.

With the ability to observe the activity from large numbers of neurons simultaneously using modern recording technologies, the chance to identify sub-networks involved in coordinated processing increases. Sequences of synchronous spike events (SSEs) constitute one type of such coordinated spiking that propagates activity in a temporally precise manner. The synfire chain was proposed as one potential model for such network processing. Previous work introduced a method for visualization of SSEs in massively parallel spike trains, based on an intersection matrix that contains in each entry the degree of overlap of active neurons in two corresponding time bins. Repeated SSEs are reflected in the matrix as diagonal structures of high overlap values. The method as such, however, leaves the task of identifying these diagonal structures to visual inspection rather than to a quantitative analysis. Here we present ASSET (Analysis of Sequences of Synchronous EvenTs), an improved, fully automated method which determines diagonal structures in the intersection matrix by a robust mathematical procedure. The method consists of a sequence of steps that i) assess which entries in the matrix potentially belong to a diagonal structure, ii) cluster these entries into individual diagonal structures and iii) determine the neurons composing the associated SSEs. We employ parallel point processes generated by stochastic simulations as test data to demonstrate the performance of the method under a wide range of realistic scenarios, including different types of non-stationarity of the spiking activity and different correlation structures. Finally, the ability of the method to discover SSEs is demonstrated on complex data from large network simulations with embedded synfire chains. Thus, ASSET represents an effective and efficient tool to analyze massively parallel spike data for temporal sequences of synchronous activity.

Finite-dimensional Representations for a Class of Generalized Intersection Matrix Lie Algebras

In this paper, the authors study a class of generalized intersection matrix Lie algebras gim(Mn), and prove that its every finite-dimensional semisimple quotient is of type M(n, a, c, d). Particularly, any finite dimensional irreducible gim(Mn) module must be an irreducible module of Lie algebra of type M(n, a, c, d) and any finite dimensional irreducible module of Lie algebra of type M(n, a, c, d) must be an irreducible module of gim(Mn).

On a class of generalized Kac–Moody algebras arising from 2-fold affinizations

We study a class of generalized Kac–Moody algebras which are fixed point subalgebras of the covering Kac–Moody algebras of some generalized intersection matrix (GIM) algebras of Slodowy.

Open intersection numbers, matrix models and MKP hierarchy

In this paper we conjecture that the generating function of the intersection numbers on the moduli spaces of Riemann surfaces with boundary, constructed recently by R. Pandharipande, J. Solomon and R. Tessler and extended by A. Buryak, is a tau-function of the KP integrable hierarchy. Moreover, it is given by a simple modification of the Kontsevich matrix integral so that the generating functions of open and closed intersection numbers are described by the MKP integrable hierarchy. Virasoro constraints for the open intersection numbers naturally follow from the matrix integral representation.

Some gim algebras and the associated indefinite Kac–Moody algebras

Let A[2] be the 2-fold affinization of a Cartan matrix A of ADE type. Associated to the generalized intersection matrix algebra gim(A[2]) there are two indefinite Kac–Moody algebras g(A[2]˜) and g(A[2]˜τ), where A[2]˜ is the covering matrix of A[2] and A[2]˜τ is obtained from a diagram automorphism τ of g(A[2]˜). We show that there is a bijection between non-isotropic positive imaginary roots of g(A[2]˜τ) and those of g(A[2]˜), which is applied to describe the 0-root space and the center of gim(A[2]). Derivations of gim(A[2]) which can be extended to derivations of g(A[2]˜) are determined.

Knots in lattice homology

Assume that \Gamma _{v_0} is a tree with vertex set \mathrm{Vert}(\Gamma _{v_0})=\{ v_0, v_1, \ldots , v_n \} , and with an integral framing (weight) attached to each vertex except v_0 . Assume furthermore that the intersection matrix of G=\Gamma _{v_0}-\{v_0\} is negative definite. We define a filtration on the chain complex computing the lattice homology of G and show how to use this information in computing lattice homology groups of a negative definite graph we get by attaching some framing to v_0 . As a simple application we produce new families of graphs which have arbitrarily many bad vertices for which the lattice homology groups are isomorphic to the corresponding Heegaard Floer homology groups.

A class of Lie algebras arising from intersection matrices

In present work, we find a class of Lie algebras, which are defined from the symmetrizable generalized intersection matrices. However, such algebras are different from generalized intersection matrix algebras and intersection matrix algebras. Moreover, such Lie algebras generated by semi-positive definite matrices can be classified by the modified Dynkin diagrams.

Realizations ofBCr-graded intersection matrix algebras with grading subalgebras of typeBr,r≥ 3

Realizations ofBCr-graded intersection matrix algebras with grading subalgebras of typeBr,r≥ 3

On Generalized Intersection Matrix Algebras Arising from 2-Fold Affinization of Cartan Matrices

Let 𝔤𝔦𝔪(A [2]) be the generalized intersection matrix algebra associated to the 2-fold affinization A [2] of a simply-laced Cartan matrix . The root system, nondegenerate invariant bilinear form, and Weyl group of 𝔤𝔦𝔪(A [2]) are discussed.

Wild quotient singularities of surfaces

Let \((B,\mathcal{M }_B)\) be a noetherian regular local ring of dimension \(2\) with residue field \(B/\mathcal{M }_B\) of characteristic \(p>0\). Assume that \(B\) is endowed with an action of a finite cyclic group \(H\) whose order is divisible by \(p\). Associated with a resolution of singularities of \(\mathrm{Spec}B^H\) is a resolution graph \(G\) and an intersection matrix \(N\). We prove in this article three structural properties of wild quotient singularities, which suggest that in general, one should expect when \(H= \mathbb{Z }/p\mathbb{Z }\) that the graph \(G\) is a tree, that the Smith group \(\mathbb{Z }^n/\mathrm{Im}(N)\) is killed by \(p\), and that the fundamental cycle \(Z\) has self-intersection \(|Z^2|\le p\). We undertake a combinatorial study of intersection matrices \(N\) with a view towards the explicit determination of the invariants \(\mathbb{Z }^n/\mathrm{Im}(N)\) and \(Z\). We also exhibit explicitly the resolution graphs of an infinite set of wild \(\mathbb{Z }/2\mathbb{Z }\)-singularities, using some results on elliptic curves with potentially good ordinary reduction which could be of independent interest.

Constraint relaxation of the polygon-polyline topological relation for geographic pictorial query languages

This work addresses the problem of relaxing spatial constraints for pictorial queries having null answers in geographical databases. It focuses on the polygon-polyline topological relationship and proposes a computational model which is based on the notions of Operator Conceptual Neighborhood (OCN) graph and the relative 16-intersection matrix. The former is addressed to represent the conceptual topological neighborhood between pairs of Symbolic Graphical Objects and indicates how spatial constraints can be relaxed. The nodes of the OCN graph are labeled with geo-operators that have been formalized and their semantics has been enriched in order to capture user query details. The latter is a 16- intersection matrix which provides additional information about the query with respect to the well-known 9-intersection matrix proposed in the literature. It has been conceived to identify, among the approximate answers, the one closer to the user needs. In particular, it allows us to quantify the difference among the user query and the configurations corresponding to the proposed approximate answers on the basis of the OCN graph. The main characteristics of our approach are highlighted through some query examples.

On the $F$-rationality and cohomological properties of matrix Schubert varieties

We characterize complete intersection matrix Schubert varieties, generalizing the classical result on one-sided ladder determinantal varieties. We also give a new proof of the $F$-rationality of matrix Schubert varieties. Although it is known that such varieties are $F$-regular (hence $F$-rational) by the global $F$-regularity of Schubert varieties, our proof is of independent interest since it does not require the Bott–Samelson resolution of Schubert varieties. As a consequence, this provides an alternative proof of the classical fact that Schubert varieties in flag varieties are normal and have rational singularities.

ON QUANTIZED GENERALIZED INTERSECTION MATRIX ALGEBRAS ASSOCIATED TO 2-FOLD AFFINIZATION OF CARTAN MATRICES

Let A[2] be the generalized intersection matrix (gim for short) given by the 2-fold affinization of a Cartan matrix of ADE type and [Formula: see text] its covering matrix. We show that the quantized gim algebra given by A[2] is isomorphic to a right coideal subalgebra of the quantized enveloping algebra associated to [Formula: see text], which is a quantum analogue of S. Berman's theorem on involutory subalgebras.

Adapted hyperbolic polygons and symplectic representations for group actions on Riemann surfaces

We prove that given a finite group G together with a set of fixed geometric generators, there is a family of special hyperbolic polygons that uniformize the Riemann surfaces admitting the action of G with the given geometric generators. From these special polygons, we obtain geometric information for the action: a basis for the homology group of surfaces, its intersection matrix, and the action of the given generators of G on this basis. We then use the Frobenius algorithm to obtain a symplectic representation G of G corresponding to this action. The fixed point set of G in the Siegel upper half-space corresponds to a component of the singular locus of the moduli space of principally polarized abelian varieties.We also describe an implementation of the algorithm using the open source computer algebra system SAGE.

Symmetrizable Intersection Matrix Lie Algebras

Associated to every symmetrizable generalized intersection matrix A, we define a Lie algebra, called an SIM-Lie algebra. We prove that SIM-Lie algebras keep unchange under braid-equivalences. Two special cases are considered. In the case when A is a symmetrizable generalized Cartan matrix, we show that the corresponding SIM-Lie algebra is just the Kac–Moody Lie algebra. In another case when A is an intersection matrix, we prove that the corresponding SIM-Lie algebra is just the intersection matrix Lie algebra in the sense of Slodowy.

Microstructure Modeling and Further Effective Property Prediction for Plain Weave C/SiC Composites Considering Manufacturing Process

The aim of this paper is to propose a systemic methodology to predict effective properties of plain weave C/SiC composites based on finite element computational micromechanics (FECM) method. Utilizing photomicrographs taken by scanning electron microscope (SEM), we established an accurate macro representative volume element (macro-RVE) model for plain weave C/SiC composites and a sub-RVE model for their carbon fiber bundles. Four classes of manufacturing porosity were taken into consideration, namely intersection voids; matrix cracks around fiber bundles; inter-fiber micro-voids; and interfacial micro-cracks. The presence of each of these classes of porosity has been confirmed by SEM observations. The undulation equations of carbon fiber bundles have been obtained from measurements of SEM photomicrographs of sectioned C/SiC composites to describe the actual fiber bundle architectures. Mix mode of equilateral triangle and square arrays was adopted to characterize the distributions of carbon fibers in fiber bundles. For the case of void-like porosity, periodicity and volume fraction have been used; and for crack-like porosity, periodicity of the cracks and their lengths have been measured and quantified. Intersection voids and matrix cracks around fiber bundles were implemented into the macro-RVE model and the rest two classes into sub-RVE model. The procedures to determine the elastic moduli using FECM method on the basis of homogenization theory were presented, in which the in situ properties of carbon fiber after manufacturing were used. Comparisons with experimental data show the validity of the proposed methodology.

On Matrix Unit Test Plan for Testing of Product Properties

This article applies the method of formulating those test projects affecting product system properties into a matrix with tabular units. According to the complexity of product system, the complexity in test operation, economical cost for a test, etc., the paper proposes the concept of whole matrix unit, intersection matrix unit and particular matrix unit test plan to test the product properties. The result indicates that there are various complicated factors influencing the properties of the product, aggravating the unstability of products system in operation. By utilizing the unit test matrix method, we can obtain experimental data chain for advancing the product reliability design, optimizing product systematical structure. With the existing scientific and technical results, together with the experiment data analysis, mean value and strategy of reliability design, we can promote product reliability and robustness.