arXiv research

This paper calculates stick numbers for rail arcs and knot classes.

Survey of Weber's class number problem and related topics.

Extends Euler class result to symplectic group.

The study finds infinite commensurability classes of hyperbolic manifolds with Salem number lengths.

This paper studies symplectic structures on elliptic surfaces with positive Euler number.

The group of bordism classes of unoriented surfaces in 4-space is determined. The bordism classes are characterized by normal Euler numbers,double linking numbers, and triple linking numbers.

The paper studies $p$-adic limits of class numbers in $\mathbb{Z}_p$-extensions and covers.

Study shows mapping class groups are one-ended for surfaces with at least one end.

The paper establishes inequalities for Chern classes and numbers on polarized manifolds and nef vector bundles.

The objective of the paper is to study accuracy of multi-class classification in high-dimensional setting, where the number of classes is also large ("large $L$, large $p$, small $n$" model). While this problem arises in many practical applications and many techniques have been recently developed for its solution, to t…

The paper bounds the handle number of sutured manifolds using Morse-Novikov numbers and tunnel numbers.

Training of one-vs.-rest SVMs can be parallelized over the number of classes in a straight forward way. Given enough computational resources, one-vs.-rest SVMs can thus be trained on data involving a large number of classes. The same cannot be stated, however, for the so-called all-in-one SVMs, which require solving a …

We compute the automorphism group of OT manifolds of simple type. We show that the graded pieces under a natural filtration are related to a certain ray class group of the underlying number field. This does not solve the open question whether the geometry of the OT manifold sees the class number directly, but brings us…

The paper proposes a test to determine the number of latent classes in ordinal categorical data.

In this paper, we discuss the region unknotting number of different classes of 2-bridge knots. In particular, we provide region unknotting number for the classes of $2$-bridge knots whose Conway notation is $C(m,\ n), C(m,\ 2,\ m),$ $C(m,\ 2,\ m\pm1)$ and $C(2,\ m,\ 2,\ n)$. By generalizing, we also provide a sharp up…

We propose a method for learning embeddings for few-shot learning that is suitable for use with any number of ways and any number of shots (shot-free). Rather than fixing the class prototypes to be the Euclidean average of sample embeddings, we allow them to live in a higher-dimensional space (embedded class models) an…

In this paper, we address the stability of a broad class of discrete-time hypercomplex-valued Hopfield-type neural networks. To ensure the neural networks belonging to this class always settle down at a stationary state, we introduce novel hypercomplex number systems referred to as real-part associative hypercomplex nu…

The problem on the minimal number (with respect to deformation) of intersection points of two closed curves on a surface is solved. Following the Nielsen approach, we define classes of intersection points and essential classes of intersection points, which "are preserved under deformation" and whose total number is cal…

New geometric interpretation of a group class using circle action and rotation numbers.

Italy's daily COVID-19 cases stratified by age groups.

We consider training probabilistic classifiers in the case of a large number of classes. The number of classes is assumed too large to perform exact normalisation over all classes. To account for this we consider a simple approach that directly approximates the likelihood. We show that this simple approach works well o…

Novel link classification connects quadratic forms and knot theory.

In recent years, deep neural network is widely used in machine learning. The multi-class classification problem is a class of important problem in machine learning. However, in order to solve those types of multi-class classification problems effectively, the required network size should have hyper-linear growth with r…

In an orientable surface with boundary, free homotopy classes of curves on surfaces are in one to one correspondence with cyclic reduced words in a set of standard generators of the fundamental group. The combinatorial length of a class is the number of letters of the corresponding word. The self-intersection of a free…

Proves a conjecture for annular links using homology classes.

Given a flexible $n$-gon with generic side lengths, the moduli space of its configurations in $\mathbb{R}^2$ as well as in $\mathbb{R}^3$ is a smooth manifold. It is equipped with $n$ \textit{tautological} line bundles whose definition is motivated by M. Kontsevich's tautological bundles over $\mathcal{M}_{0,n}$. We st…

Oriented closed curves on an orientable surface with boundary are described up to continuous deformation by reduced cyclic words in the generators of the fundamental group and their inverses. By self-intersection number one means the minimum number of transversal self-intersection points of representatives of the class…

The number of possible methods of generalizing binary classification to multi-class classification increases exponentially with the number of class labels. Often, the best method of doing so will be highly problem dependent. Here we present classification software in which the partitioning of multi-class classification…

We show that the number of twisted conjugacy classes is infinite for any automorphism of non-elementary, Gromov hyperbolic group . An analog of Selberg theory for twisted conjugacy classes is proposed.

Study non-vanishing $\ell^2$-Betti numbers for specific groups.

For every orientable surface of finite negative Euler characteristic, we find a right-angled Artin group of cohomological dimension two which does not embed into the associated mapping class group. For a right-angled Artin group on a graph $\gam$ to embed into the mapping class group of a surface $S$, we show that the …

The minimum number of self-intersection points for members of a free homotopy class of curves on the punctured torus is bounded above in terms of the number L of letters required for a minimal description of the class in terms of the generators of the fundamental group and their inverses: it is less than or equal to (L…

We study mapping class group orbits of homotopy and isotopy classes of curves with self-intersections. We exhibit the asymptotics of the number of such orbits of curves with a bounded number of self-intersections, as the complexity of the surface tends to infinity. We also consider the minimal genus of a subsurface tha…

Mapping class group dynamics tracked through Teichmüller space.

We show that the integral of the first Pontrjagin class is given by an integer and it is identified with instanton number of the U(n) gauge theory on noncommutative ${\bf R^4}$. Here the dimension of the vector space $V$ that appear in the ADHM construction is called Instanton number. The calculation is done in operato…

New growth rate for pseudo-Anosov conjugacy classes in Teichmüller space.

Predicts optimal training dataset sizes per class for machine learning models.

A number of classification problems need to deal with data imbalance between classes. Often it is desired to have a high recall on the minority class while maintaining a high precision on the majority class. In this paper, we review a number of resampling techniques proposed in literature to handle unbalanced datasets …

Novel approach for large genus intersection number asymptotics.

This work studies the impact of intra-/inter-class diversity on pre-training datasets and finds a balance for optimal performance.

The difficulty of multi-class classification generally increases with the number of classes. Using data from a subset of the classes, can we predict how well a classifier will scale with an increased number of classes? Under the assumptions that the classes are sampled identically and independently from a population, a…

Estimates the number of closed curves on surfaces with power-saving error terms.

New approach calibrates predictions for better decision-making.

One can define the complexity of a smooth 4-manifold as the minimal sum of the number of disks, strands and crossings in a Kirby diagram. Martelli proved that the number of homeomorphism classes of complexity less than n grows as $n^2$. In this paper we prove that the number of diffeomorphism classes grows at least as …

We calculate explicitly the Betti numbers of a class of barely G2 manifolds - that is, G2 manifolds that are realised as a product of a Calabi-Yau manifold and a circle, modulo an involution. The particular class which we consider are those spaces where the Calabi-Yau manifolds are complete intersections of hypersurfac…

Let M be an n-dimensional Kähler manifold with numerically effective Ricci class. In this note we prove that, if the first Betti number b_1(M)=2n, then M is biholomorphic to the complex torus T^n_C.

The study of special Lagrangian classes and semistable Mukai vectors on K3 surfaces.

The paper provides a uniform lower bound for intersection numbers of psi-classes on moduli spaces.