arXiv research

Paper connects AJ conjecture and colored Jones polynomial potential function.

Study on unimodality of plucking polynomial with delay function.

Study on polynomial growth functions and forms on gradient Ricci solitons.

We study a class of 2-variable polynomials called exact polynomials which contains $A$-polynomials of knot complements. The Mahler measure of these polynomials can be computed in terms of a volume function defined on the vanishing set of the polynomial. We prove that the local extrema of the volume function are on the …

The paper generalizes polynomial functions on Lie groups and their properties.

Classifies polynomial growth solutions to drift-harmonic equations on asymptotically paraboloidal manifolds.

Study polynomial growth harmonic functions on infinite penny graphs.

Ancient solutions to biharmonic heat equation bounded by polynomial dimensions.

Analytic functions on specific domains are characterized by their smoothness and composites with polynomial curves.

Paper proves BGW tau-function can be represented as Q-polynomials.

In this work we describe a new invariant of virtual knots. We show that this transcendental function invariant generalizes several polynomial invariants of virtual knots, such as the writhe polynomial, the affine index polynomial and the zero polynomial.

Ancient caloric functions on manifolds with polynomial growth are studied under volume doubling barrier.

Proves a formula for Kontsevich-Witten tau-function using Schur Q-polynomials.

New polynomial invariants for virtual links are stronger than F-polynomials.

This work improves polynomial approximations for functions with asymmetric behavior.

Study slice-regular polynomial functions via twistor space group actions.

Shallow neural networks can represent polynomials efficiently.

In this paper, we study harmonic and caloric functions of polynomial growth on a complete non-compact gradient shrinking Ricci soliton. On one hand, when the scalar curvature satisfies at least quadratic decay, we prove that the space of harmonic functions with fixed polynomial growth degree is finite dimensional. We a…

In this article we have studied some properties of subharmonic functions in a strongly symmetric Riemannian manifold with a pole. As a generalization of polynomial growth of a function we have introduced the notion of polynomial growth of some degree of a function with respect to a real function and proved that any non…

Characterizes values at infinity for real polynomial maps with 2D fibers.

For any manifold with polynomial volume growth, we show: The dimension of the space of ancient caloric functions with polynomial growth is bounded by the degree of growth times the dimension of harmonic functions with the same growth. As a consequence, we get a sharp bound for the dimension of ancient caloric functions…

Holomorphic functions grow polynomially on Kähler-Ricci shrinkers, proving ring finitely generated.

Formula for arborescent link tails using theta functions.

It is shown that, in the 1-jet space of the circle, the swapping and the flyping procedures, which produce topologically equivalent links, can produce nonequivalent legendrian links. Each component of the links considered is legendrian isotopic to the 1-jet of the 0-function, and thus cannot be distinguished by the cla…

Develops a Gaussian model to compute the Alexander polynomial of knots.

Deep neural networks with piecewise-polynomial activations can approximate smooth functions and their derivatives.

All link types arise from semiholomorphic polynomials.

The paper shows neural networks can approximate functions over non-compact domains with non-polynomial activation.

We formulate a conjecture (already proven by A. Kricker) about the structure of Kontsevich integral of a knot. We describe its value in terms of the generating functions for the numbers of external edges attached to closed 3-valent diagrams. We conjecture that these functions are rational functions of the exponentials …

The Jones polynomial of a knot in 3-space is a Laurent polynomial in $q$, with integer coefficients. Many people have pondered why is this so, and what is a proper generalization of the Jones polynomial for knots in other closed 3-manifolds. Our paper centers around this question. After reviewing several existing defin…

Non-negative $L_1$-approximating polynomials for Gaussian distributions are proven for certain classes of sets.

We show that the standard stochastic gradient decent (SGD) algorithm is guaranteed to learn, in polynomial time, a function that is competitive with the best function in the conjugate kernel space of the network, as defined in Daniely, Frostig and Singer. The result holds for log-depth networks from a rich family of ar…

Unified ADO and colored Jones polynomials for knots.

Explicit polynomial bound found for subgroup Dehn function.

We study deep neural networks with polynomial activations, particularly their expressive power. For a fixed architecture and activation degree, a polynomial neural network defines an algebraic map from weights to polynomials. The image of this map is the functional space associated to the network, and it is an irreduci…

In the present paper we introduce the class of slice-polynomial functions: slice regular functions {defined over the quaternions, outside the real axis,} whose restriction to any complex half-plane is a polynomial. These functions naturally emerge in the twistor interpretation of slice regularity introduced in \cite{ge…

Study harmonic functions on submanifolds and their cones.

The pioneering work of Jones and Kauffman unveiled a fruitful relationship between statistical mechanics and knot theory. Recently, Jones introduced two subgroups $\vec{F}$ and $\vec{T}$ of the Thompson groups $F$ and $T$, respectively, together with a procedure that associates an oriented link diagram to any element o…

Polynomial error equidistribution for SL2 groups.

3D gauge theories link knot polynomials to vortex partition functions.

In an earlier paper the first author defined a non-commutative A-polynomial for knots in 3-space, using the colored Jones function. The idea is that the colored Jones function of a knot satisfies a non-trivial linear q-difference equation. Said differently, the colored Jones function of a knot is annihilated by a non-z…

Study introduces a new method for multiple parameter regularization in polynomial functional regression.

Study on colored Jones polynomial and link complements.

Research on mixed polynomials, extending non-degeneracy concepts to complex variables.

New $L$-functions for 3-manifolds connect to Witten invariants and relate to generalized Bernoulli polynomials.

This paper studies a specific blow-up algorithm for sop polynomials and their RLCT.

We show that for a special alternating link diagram, the following three polynomials are essentially the same: a) the part of the HOMFLY polynomial that corresponds to the leading term in the Alexander polynomial; b) the $h$-vector for a triangulation of the root polytope of the Seifert graph and c) the enumerator of p…

In many applications (in particular information systems, such as pattern recognition, machine learning, cheminformatics, bioinformatics to name but a few) the assessment of uncertainty is essential - i.e., the estimation of the underlying probability distribution function. More often than not, the form of this function…