arXiv research

Computes presentations for specific arithmetic hyperbolic lattices.

New computations show various properties of bounded cohomology in finitely presented groups.

In this note, we present a new method for computing fundamental groups of curve complements using a variation of the Zariski-Van Kampen method on general ruled surfaces. As an application we give an alternative (computation-free) proof for the fundamental group of generic $(p,q)$-torus curves.

New findings on stable commutator lengths in recursively presented groups.

The computation of the fundamental group of the complement of an algebraic plane curve has been theoretically solved since Zariski-van Kampen, but actual computations are usually cumbersome. In this work, we describe the notion of Wirtinger presentation of such a group relying on the real picture of the curve and with …

The paper computes presentations of cluster modular groups and verifies their generation by Dehn twists.

Algorithms compute the topology of hyperelliptic curves in 2D and 3D.

State-sum invariants for knotted curves and surfaces using quandle cohomology were introduced by Laurel Langford and the authors in math.GT/9903135 In this paper we present methods to compute the invariants and sample computations. Computer calculations of cohomological dimensions for some quandles are presented. For c…

Here we present the results of the NSF-funded Workshop on Computational Topology, which met on June 11 and 12 in Miami Beach, Florida. This report identifies important problems involving both computation and topology.

Paper computes knot symmetric quandle for surface-links and finds infinitely many distinct surface-knots.

We consider computational complexity of problems related to the fundamental group and the first homology group of (embeddable) $2$-complexes. We show, as an extension of an earlier work, that computing first homology of $2$-complexes is equivalent in computational complexity to matrix diagonalization. That is, the usua…

We present a combinatorial proof for the existence of the sign refined grid homology in lens spaces, and a self contained proof that $\partial_{\mathbb{Z}}^2 = 0$. We also present a Sage program that computes $\widehat{\mathrm{GH}} (L(p,q),K;\mathbb{Z})$, and provide empirical evidence supporting the absence of torsion…

Study a specific line arrangement and compute its fundamental group via braid monodromy.

Paper presents a new method to train deep neural networks with reduced memory access.

Method for computing Khovanov homology of tangles.

Efficient algorithm approximates discrete random variables with minimal Kolmogorov distance.

Improved PQM for pattern classification on quantum computers.

Learning codes for non-linear computations improves resilience in machine learning.

Szabó recently introduced a combinatorially-defined spectral sequence in Khovanov homology. After reviewing its construction and explaining our methodology for computing it, we present results of computations of the spectral sequence. Based on these computations, we make a number of conjectures concerning the structure…

Efficient method computes ${ m SL}(3,\mathbb{C})$-character varieties for two-generator groups.

Computes group of ring motions for a specific link structure.

Formulas previously presented for the Casson-Walker invariant are generalized to Lescop's extension. These formulas in terms of linking numbers and surgery coefficients compute the change in Lescop's invariant under crossing changes in a framed link presenting a 3-manifold. This leads us to revisit an old formula for a…

Computes cohomology of Steenrod algebra for k ≤ 5.

The paper studies convergence of discrete harmonic maps to smooth ones.

Paper presents privacy-preserving techniques for HD computing.

We design an algorithm writing down presentations of graph braid groups. Generators are represented in terms of actual motions of robots moving without collisions on a given graph. A key ingredient is a new motion planning algorithm whose complexity is linear in the number of edges and quadratic in the number of robots…

We describe a collection of computer scripts written in PARI/GP to compute, for reflection groups determined by finite-volume polyhedra in $\mathbb{H}^3$, the commensurability invariants known as the invariant trace field and invariant quaternion algebra. Our scripts also allow one to determine arithmeticity of such gr…

Defines computable learning for binary classification over metric spaces.

We present two paradigms relating algebraic, topological and quantum computational statistics for the topological model for quantum computation. In particular we suggest correspondences between the computational power of topological quantum computers, computational complexity of link invariants and images of braid grou…

This study compares Matlab and OpenCV for machine learning algorithms.

We present in this paper a new premium computation principle based on the use of prior information from multiple sources for computing the premium charged to a policyholder. Under this framework, based on the use of Ordered Weighted Averaging (OWA) operators, we propose alternative collective and Bayes premiums and des…

The aim of this study was to develop methods for evaluating the American-style option prices when the volatility of the underlying asset is described by a stochastic process. As part of this problem were developed techniques for modeling the early exercise surface of the American option. These methods of present work a…

Researchers prove quantum invariants remain hard even when restricted.

Non-associtive algebras is a research direction gaining much attention these days. New developments show that associative algebras and some not-associative structures can be unified at the level of Yang-Baxter structures. In this paper, we present a unification for associative algebras, Jordan algebras and Lie algebras…

Quantum computing speeds up linear regression training.

In this paper the possibility of computing equilibrium in pure exchange and production economies by a homotopy method is investigated. The performance of the algorithm is tested on examples with known equilibria taken from the literature on general equilibrium models and numerical results are presented. In computing eq…

Paper presents a neural network method for efficient xVA computation and risk management.

Paper presents a faster method for computing cost of equity and performing comparable company analysis.

Two methods using Chebyshev tensors improve accuracy and speed in computing Dynamic Initial Margin.

The paper computes various invariants of Legendrian knots in open book presentations.

We present a generic compact computational framework relying on structured random matrices that can be applied to speed up several machine learning algorithms with almost no loss of accuracy. The applications include new fast LSH-based algorithms, efficient kernel computations via random feature maps, convex optimizati…

We discuss the relative merits of optimistic and randomized approaches to exploration in reinforcement learning. Optimistic approaches presented in the literature apply an optimistic boost to the value estimate at each state-action pair and select actions that are greedy with respect to the resulting optimistic value f…

A new method simplifies feature explanation for complex models.

The computation of Greeks for exponential Lévy models are usually approached by Malliavin Calculus and other methods, as the Likelihood Ratio and the finite difference method. In this paper we obtain exact formulas for Greeks of European options based on the Lewis formula for the option value. Therefore, it is possible…

Study compares methods for computing hypergradients in machine learning problems.

RO-TD learns sparse value functions efficiently.

Paper presents RSVD for better recommender system performance.

EdgeAI aims to deploy deep learning on IoT devices.