arXiv research

3-manifold triangulation can be reconstructed from its intersection matrix.

Algorithm counts intersections of normal curves efficiently.

We extend the notion of intersection graphs for knots in the theory of finite type invariants to string links. We use our definition to develop weight systems for string links via the adjacency matrix of the intersection graph, and show that these weight systems are related to the weight systems induced by the Conway a…

New bounds on curves on torus with few intersections.

A new method clusters intersecting lines using hypergraphs.

The paper proves embedding conditions for complexes in manifolds using matrix rank criteria.

Non-negative matrix factorization (NMF) is a natural model of admixture and is widely used in science and engineering. A plethora of algorithms have been developed to tackle NMF, but due to the non-convex nature of the problem, there is little guarantee on how well these methods work. Recently a surge of research have …

In a previous paper, we defined an operation $μ$ that generalizes Turaev's cobracket for loops on a surface. We showed that, in contrast to the cobracket, this operation gives a formula for the minimum number of self-intersections of a loop in a given free homotopy class. In this paper we consider the corresponding que…

The paper connects Riemannian Gaussian distributions to random matrix theory and diffusion kernels.

We investigate some geometric properties of the real algebraic variety $Δ$ of symmetric matrices with repeated eigenvalues. We explicitly compute the volume of its intersection with the sphere and prove a Eckart-Young-Mirsky-type theorem for the distance function from a generic matrix to points in $Δ$. We exhibit conne…

Study of Hamiltonian flows on character varieties for self-intersecting curves.

In this paper, we show that the Lie superalgebra $\mathfrak{spo}(2l+2|n)$ is into the intersection of Lie superalgebra of contact vector fields $\mathcal{K}(2l+1|n)$ and the Lie superalgebra of projective vector fields $\mathfrak{pgl}(2l+2|n)$. We use mainly the embedding used by P. Mathonet and F. Radoux in "\textit{ …

Basket links are shown to be isotopic to $T(2,n+1)$.

Investigates linearity of group amalgams and new examples of non-linear groups.

We introduce a differential refinement of Cohomotopy cohomology theory, defined on Penrose diagram spacetimes, whose cocycle spaces are unordered configuration spaces of points. First we prove that brane charge quantization in this differential 4-Cohomotopy theory implies intersecting p/(p+2)-brane moduli given by orde…

New method reduces computational cost for nonnegative low rank matrix approximation.

BEGIN network models binary data without parametric assumptions.

The increasing size and complexity of scientific data could dramatically enhance discovery and prediction for basic scientific applications. Realizing this potential, however, requires novel statistical analysis methods that are both interpretable and predictive. We introduce Union of Intersections (UoI), a flexible, m…

This paper concerns cluster algebras with principal coefficients A(S,M) associated to bordered surfaces (S,M), and is a companion to a concurrent work of the authors with Schiffler [MSW2]. Given any (generalized) arc or loop in the surface -- with or without self-intersections -- we associate an element of (the fractio…

Knots and 4-manifolds linked via matrix kinking.

Traffic signal control has long been considered as a critical topic in intelligent transportation systems. Most existing learning methods mainly focus on isolated intersections and suffer from inefficient training. This paper aims at the cooperative control for large scale multi-intersection traffic signal, in which a …

Given a matrix the seriation problem consists in permuting its rows in such way that all its columns have the same shape, for example, they are monotone increasing. We propose a statistical approach to this problem where the matrix of interest is observed with noise and study the corresponding minimax rate of estimatio…

It can be conjectured that the colored Jones function of a knot can be computed in terms of counting paths on the graph of a planar projection of a knot. On the combinatorial level, the colored Jones function can be replaced by its weight system. We give two curious formulas for the weight system of a colored Jones fun…

New method for triclustering with reduced arbitrariness.

Tensor networks improve anomaly detection at LHC for new physics.

We develop a family of reformulations of an arbitrary consistent linear system into a stochastic problem. The reformulations are governed by two user-defined parameters: a positive definite matrix defining a norm, and an arbitrary discrete or continuous distribution over random matrices. Our reformulation has several e…

To each ribbon graph we assign a so-called L-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 Vassil…

The paper examines how closed curves on surfaces intersect and how this intersection determines the curves.

We introduce a method that uses the Cauchy-Crofton formula and a new curvature formula from integral geometry to reweight the sampling probabilities of Metropolis-within-Gibbs algorithms in order to increase their convergence speed. We consider algorithms that sample from a probability density conditioned on a manifold…

The paper finds diffeomorphic complex intersections with distinct Hodge numbers.

New polynomials defined for virtual knots, calculated up to crossing 4.

In this paper we present the algorithms for calculating the differential geometric properties {t,n,b1,b2,b3,k1,k2,k3,k4} along-with geodesic curvature and geodesic torsion of the transversal intersection curve of four hypersurfaces (given by parametric representation) in Euclidean space R^5. In transversal intersection…

We generalize the PL intersection product for chains on PL manifolds and for intersection chains on PL stratified pseudomanifolds to products of locally finite chains on non-compact spaces that are natural with respect to restriction to open sets. This is necessary to sheafify the intersection product, an essential ste…

Conditions for curves on a torus with specific pairwise intersections.

Study self-intersections of arcs on a pair of pants, proving natural number spectrum.

Virtual knots with same writhe polynomial have equivalent intersection graphs.

Study properties of self-similar continua with finite intersection property.

Estimates intersection pairing in hyperbolic 4-manifolds.

By considering a (not necessarily locally-flat) PL knot as the singular locus of a PL stratified pseudomanifold, we can use intersection homology theory to define intersection Alexander polynomials, a generalization of the classical Alexander polynomial invariants for smooth or PL locally-flat knots. We show that the i…

Classifies arcs on a 4-punctured sphere that intersect at most once.

James McClure recently showed that the domain for the intersection pairing of PL chains on a PL manifold $M$ is a subcomplex of $C_*(M)\otimes C_*(M)$ that is quasi-isomorphic to $C_*(M)\otimes C_*(M)$ and, more generally, that the intersection pairing endows $C_*(M)$ with the structure of a partially-defined commutati…

Novel approach for large genus intersection number asymptotics.

The paper calculates self-intersections on a pair of pants using Bowen and Series' coding.

Generic potential primes have no self-intersections or intersections.

Paper proves curves can be smoothed to reduce self-intersection by exactly 1.

The paper extends intersection theory for b-divisors, proving monotonicity and volume inequalities.

Study intersection polynomials of long virtual knots with supporting genera.

New invariant $c_{sm}$ simplifies computing geometric invariants of recursive group orbits.