arXiv research

Model interpretability is an increasingly important component of practical machine learning. Some of the most common forms of interpretability systems are example-based, local, and global explanations. One of the main challenges in interpretability is designing explanation systems that can capture aspects of each of th…

We describe an algorithm for determining whether a finite quandle is isomorphic to an Alexander quandle by finding all possible Alexander presentations of the quandle. We give an implementation of this algorithm in Maple.

We use the methods of geometric control theory to study extremal trajectories of vertical rolling disk. We focus on the role of symmetries of the underlying geometric structures. We demonstrate the computations in the CAS Maple package DifferentialGeometry.

New method uses quandle rings to distinguish knots and their mirrors.

Recently examples of Riemannian homogeneous spaces with linear Jacobi relations were found. We calculate the Singer invariants of these spaces with the computer algebra program Maple and discuss the results by means of the Jet Isomorphism Theorem of pseudo-Riemannian geometry.

The 2-bridge knots are a family of knots with bridge number 2. In this paper, we compute the Kauffman polynomials of 2-bridge knots using the Kauffman skein theory and linear algebra techniques. Our calculation can be easily carried out using Mathematica, Maple, Mathcad, etc.

We present a MAPLE program for the explicit computation of the curvature of calibrated ordinary webs in codimension one in any dimension (recall that all planar webs are calibrated and ordinary). The vanishing of this curvature means the maximality of the rank. We give some first examples.

Investigates locally symmetric polynomial metrics in Riemannian and Finslerian surfaces.

It is still a long-standing open problem in Finsler geometry, is there any regular Landsberg metric which is not Berwaldian. However, there are non-regular Landsberg metrics which are not Berwladian. The known examples are established by G. S. Asanov and Z. Shen. In this paper, we use the Maple program to study some ex…

A method of computing a basis for the second Yang-Baxter cohomology of a finite biquandle with coefficients in Q and Z_p from a matrix presentation of the finite biquandle is described. We also describe a method for computing the Yang-Baxter cocycle invariants of an oriented knot or link represented as a signed Gauss c…

In this article we give necessary and sufficient conditions for an irreducible Kähler C-space with $b_2=1$ to have nonnegative or positive quadratic bisectional curvature, assuming the space is not Hermitian symmetric. In the cases of the five exceptional Lie groups $E_6, E_7, E_8, F_4, G_2$, the computer package MAPLE…

Using Maple, we compute some analytical solutions of a modified Black-Scholes equation, recently proposed, in the case of the European put option. We show that the modified Black-Scholes equation with the European put option is exactly solvable in terms of associated Laguerre polynomials. We make some numerical experim…

We present algorithms to compute the topology of 2D and 3D hyperelliptic curves. The algorithms are based on the fact that 2D and 3D hyperelliptic curves can be seen as the image of a planar curve (the Weierstrass form of the curve), whose topology is easy to compute, under a birational mapping of the plane or the spac…

Using Maple, we compute a new exact series solution of a modified Black-Scholes equation, recently proposed, for the case of the Aunt Michaela option with a maturity condition of gamma type. We show that the modified Black-Scholes equation with the Aunt Michaela option is exactly solvable in terms of associated Laguerr…

The curvature of web curves is studied in 3D manifolds.

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…

Algorithms are described and Maple implementations are provided for finding all quandles of order $n$, as well as computing all homomorphisms between two finite quandles or from a finitely presented quandle (e.g., a knot quandle) to a finite quandle, computing the automorphism group of a finite quandle, etc. Several of…

A systematic algorithm for building integrating factors of the form mu(x,y') or mu(y,y') for non-linear second order ODEs is presented. When such an integrating factor exists, the algorithm determines it without solving any differential equations. Examples of ODEs not having point symmetries are shown to be solvable us…

Paper finds maximum curvature of Bézier-spline curves.

In the previous work, the notion of the Finsleroid--Finsler space have been formulated and the necessary and sufficient conditions for the space to be of the Landsberg type have been found. In the present paper, starting with particular spray coefficients, we demonstrate how the Landsberg condition can explicitly appea…

Complete classification of Deligne-Mostow lattice representations into PGL(3,C).

This note (originally from 2015) provides a proof of a 1985 conjecture of Montiel and Ros concerning the conformal volume of tori. This updated version adds a proof of the claim made in Remark 5 about the value of the conformal volume of tori in the cases not covered by the conjecture of Montiel and Ros. Originally, I …

We describe a way of representing finite biquandles with n elements as 2n x 2n block matrices. Any finite biquandle defines an invariant of virtual knots through counting homomorphisms. The counting invariants of non-quandle biquandles can reveal information not present in the knot quandle, such as the non-triviality o…

In this paper, we define, from a finite set E of functions, a family of holomorphic webs ${\cal W}(n;E)$ of codimension one in any dimension $n$. We prove that it is sufficient to check a finite number of conditions for these webs to be all ordinary and to have all maximal rank. Moreover, these conditions may be prac…

We prove that the automorphism group of the dihedral quandle with n elements is isomorphic to the affine group of the integers mod n, and also obtain the inner automorphism group of this quandle. In [9], automorphism groups of quandles (up to isomorphisms) of order less than or equal to 5 were given. With the help of t…

We present and implement two algorithms for analytic asymptotic evaluation of the marginal likelihood of data given a Bayesian network with hidden nodes. As shown by previous work, this evaluation is particularly hard for latent Bayesian network models, namely networks that include hidden variables, where asymptotic ap…

The aim of this paper is mainly, after some theoretical explanations, to provide a program on Maple for computing, whatever be d, the curvature of the planar d-web implicitely defined by a differential equation F(x,y,y')=0, F being polynomial of degree d with respect to y'. Moreover, we prove in the appendix a "concent…

We perform detailed computations of Lie algebras of infinitesimal CR-automorphisms associated to three specific model real analytic CR-generic submanifolds in C^9 by employing differential algebra computer tools -- mostly within the Maple package DifferentialAlgebra -- in order to automate the handling of the arising h…

The paper analyzes Lie symmetries in a specific geometric context.

The local geometry of a Riemannian symmetric space is described completely by the Riemannian metric and the Riemannian curvature tensor of the space. In the present article I describe how to compute these tensors for any Riemannian symmetric space from the Satake diagram, in a way that is suited for the use with comput…

The book "Handbook of Finsler geometry" has been included with a CD containing an elegant Maple package, FINSLER, for calculations in Finsler geometry. Using this package, an example concerning a Finsler generalization of Einstein's vacuum field equations was treated. In this example, the calculation of the components …

New method detects projective equivalences and symmetries in rational 3D curves.

We consider the significant class of holomorphically nondegenerate CR manifolds of finite type that are represented by some weighted homogeneous polynomials and we derive some useful features which enable us to set up a fast effective algorithm to compute their Lie algebras of infinitesimal CR-automorphisms. This algor…

Machine learning improves Buchberger's algorithm for polynomial systems.

The paper defines and characterizes 2-Ruled hypersurfaces in Minkowski 4-space using octonions.

Geometric framework for dynamic feedback linearization of control systems with symmetry.

We present a set of 26 finite quandles that distinguish (up to reversal and mirror image) by number of colorings, all of the 2977 prime oriented knots with up to 12 crossings. We also show that 1058 of these knots can be distinguished from their mirror images by the number of colorings by quandles from a certain set of…

We present a novel certified and complete algorithm to compute arrangements of real planar algebraic curves. It provides a geometric-topological analysis of the decomposition of the plane induced by a finite number of algebraic curves in terms of a cylindrical algebraic decomposition. From a high-level perspective, the…

Bayesian framework estimates label shift for improved classifier performance.

We are concerned with bifurcation analysis and control of nonlinear Eulerian flows with non-resonant n-tuple Hopf singularity. The analysis is involved with CW complex bifurcations of flow-invariant Clifford hypertori, where we refer to these toral manifolds by toral CW complexes. We observe from primary to tertiary fl…

Study minimal and CMC tori in spheres and their duals.