arXiv research

The paper explores properties of Lie algebra g2 and related geometric structures.

The purpose of this note is to provide yet another example of the link between certain conformal geometries and ordinary differential equations, along the lines of the examples discussed by Nurowski in math.DG/0406400. In this particular case, I consider the equivalence problem for 3-plane fields D on 6-manifolds M tha…

We obtain defining equations of the smooth equivariant compactification of the Grassmannian of the complex associative $3$-planes in $\C^7$, which is the parametrizing variety of all quaternionic subalgebras of the algebra of complex octonions $\OO\cong \C^8$. By studying the torus fixed points, we compute the Poincaré…

In this article, we derive a topological obstruction to the removal of a isolated degenerate complex tangent to an embedding of a 3-manifold into $\mathbb{C}^3$ (without affecting the structure of the remaining complex tangents). We demonstrate how the vanishing of this obstruction is both a necessary and sufficient co…

By a theorem of Mclean, the deformation space of an associative submanifold Y of an integrable G_2 manifold (M,φ) can be identified with the kernel of a Dirac operator D:Ω^{0}(ν) -->Ω^{0}(ν) on the normal bundle νof Y. Here, we generalize this to the non-integrable case, and also show that the deformation space becomes…

In this paper, we prove various results on the topology of the Grassmannian of oriented 3-planes in Euclidean 6-space and compute its cohomology ring. We give self-contained proofs. These spaces come up when studying submanifolds of manifolds with calibrated geometries. We collect these results here for the sake of com…

We introduce a class of links strictly containing quasi-alternating links for which mod 2 reduced Khovanov homology is always thin. We compute the framed instanton homology for double branched covers of such links. Aligning certain dotted markings on a link with bundle data over the branched cover, we also provide many…

Proves stability of Minkowski space for specific initial data.

The main purpose of this paper is to give a mathematical definition of ``mirror symmetry'' for Calabi-Yau and G_2 manifolds. More specifically, we explain how to assign a G_2 manifold (M,φ,Λ), with the calibration 3-form φand an oriented 2-plane field Λ, a pair of parametrized tangent bundle valued 2 and 3-forms of M. …

Let S be an infinite-dimensional manifold of all symplectic, or hyperkahler, structures on a compact manifold M, and $Diff_0$ the connected component of its diffeomorphism group. The quotient $S/\Diff_0$ is called the Teichmuller space of symplectic (or hyperkahler) structures on M. MBM classes on a hyperkahler manifol…

Researchers solve Nielsen Realization problems for K3 surfaces in various categories.

Study explores associated groups of symmetric quandles and their properties.

When response variables are nominal and populations are cross-classified with respect to multiple polytomies, questions often arise about the degree of association of the responses with explanatory variables. When populations are known, we introduce a nominal association vector and matrix to evaluate the dependence of …

Discovering associations is of central importance in scientific practices. Currently, most researches consider only linear association measured by correlation coefficient, which has its theoretical limitations. In this paper, we propose a new method for discovering association with copula entropy -- a universal applica…

We study deformations of associative submanifolds $Y^3\subset M^7$ of a $G_2$ manifold $M^7$. We show that the deformation space can be perturbed to be smooth, and it can be made compact and zero dimensional by constraining it with an additional equation. This allows us to associate local invariants to associative subm…

This study addresses transitions in conically singular associative submanifolds and their desingularizations.

Study of associative submanifolds in Berger space SO(5)/SO(3).

Model for associative submanifolds in K3 fibrations.

This paper studies the associativity of gluing of trajectories in Morse theory. We show that the associativity of gluing follows from of the existence of compatible manifold with face structures on the compactified moduli spaces. Using our previous work, we obtain the associativity of gluing in certain cases. In partic…

In this paper, we propose a novel structure for a cross-modal data association, which is inspired by the recent research on the associative learning structure of the brain. We formulate the cross-modal association in Bayesian inference framework realized by a deep neural network with multiple variational auto-encoders …

It is well-known that in any codimension a simply connected Euclidean minimal surface has an associated one-parameter family of minimal isometric deformations. In this paper, we show that this is just a special case of the associated family to any simply connected elliptic surface for which all curvature ellipses of a …

Study constructs associative submanifolds in $G_2$-manifolds from orbifolds.

DEDACT breaks down feature importance into direct and associative components.

A pseudo $H$-type Lie algebra naturally gives rise to a conformal pseudo-subriemannian fundamental graded Lie algebras. In this paper we investigate the prolongations of the associated fundamental graded Lie algebra and the associated conformal pseudo-subriemannian fundamental graded Lie algebra. In particular, we show…

Method constructs rigid associative submanifolds in twisted G2-manifolds.

Proofs and descriptions of totally geodesic submanifolds in symmetric spaces.

The associator of a non-associative algebra is the curvature of the Hochschild quasi-complex. The relationship ``curvature-associator'' is investigated. Based on this generic example, we extend the geometric language of vector fields to a purely algebraic setting, similar to the context of Gerstenhaber algebras. We int…

A local classification of the Hermitian manifolds with flat associated connection is given. Hermitian manifolds admitting locally a conformal metric with flat associated connection are characterized by a curvature identity. Locally conformal Kaehler manifolds as well as Hermitian surfaces with vanishing associated conf…

Framework uncovers symmetric and asymmetric species associations from data.

OMBA learns product and user representations for better online market basket analysis.

Computes quandle associated groups using group homology.

The Long Short-Term Memory (LSTM) neural network based data association algorithm named as DeepDA for multi-target tracking in clutters is proposed to deal with the NP-hard combinatorial optimization problem in this paper. Different from the classical data association methods involving complex models and accurate prior…

Develops G-MLKM for better data-target association in constrained spaces.

New algebraic structure for vector bundles with special properties.

New mechanics on non-associative octonions discovered.

Drinfeld associator is a key tool in computing the Kontsevich integral of knots. A Drinfeld associator is a series in two non-commuting variables, satisfying highly complicated algebraic equations - hexagon and pentagon. The logarithm of a Drinfeld associator lives in the Lie algbera L generated by the symbols a,b,c mo…

Associative submanifolds of the 7-sphere S^7 are 3-dimensional minimal submanifolds which are the links of calibrated 4-dimensional cones in R^8 called Cayley cones. Examples of associative 3-folds are thus given by the links of complex and special Lagrangian cones in C^4, as well as Lagrangian submanifolds of the near…

Graph network predicts circRNA-disease associations using multi-source similarity features.

Poisson algebra is usually defined to be a commutative algebra together with a Lie bracket, and these operations are required to satisfy the Leibniz rule. We describe Poisson structures in terms of a single bilinear operation. This enables us to explore Poisson algebras in the realm of non-associative algebras. We stud…

We consider minimal immersions in MxR. We study existence and uniqueness of associate and conjugate isometric immersions to a given minimal surface. We use the theory of univalent harmonic map between surfaces. Then we study the geometry of associate minimal vertical graphs. We prove that an associate surface of a vert…

The aim of this paper is to present the stochastic Poisson equations associated to Lie algebroids. The stochastic Poisson equations associated to a refinement of a concrete principal bundle are determined.

Survey of recent measures of association, including a new coefficient.

This paper gives two methods for constructing associative 3-folds in R^7, based around the fundamental idea of evolution equations, and uses these methods to construct examples of these geometric objects. The paper is a generalisation of the work by Joyce in math.DG/0008021, math.DG/0008155, math.DG/0010036 and math.DG…

In this paper we investigate two variants of association rules for preference data, Label Ranking Association Rules and Pairwise Association Rules. Label Ranking Association Rules (LRAR) are the equivalent of Class Association Rules (CAR) for the Label Ranking task. In CAR, the consequent is a single class, to which th…

We solve the regularized Knizhnik-Zamolodchikov equation and find an explicit expression for the Drinfeld associator. We restrict to the case of the fundamental representation of $gl(N)$. Several tests of the results are presented. It can be explicitly seen that components of this solution for the associator coincide w…

Building on ideas from [DT98; DS11; Wal17; Hay17], we outline a proposal for constructing Floer homology groups associated with a G2-manifold. These groups are generated by associative submanifolds and solutions of the ADHM Seiberg-Witten equations. The construction is motivated by the analysis of various transitions w…

The paper studies scaling laws for associative memory mechanisms.

The article develops deformation theory for ACyl associative submanifolds in ACyl G2-manifolds.