arXiv research

Study rationality of meromorphic functions between real algebraic sets in the plane.

Extends Thurston's combinatorial characterization to all branched coverings of the 2-sphere.

Classifies real rational knots and curves in a specific quadric space.

We develop Fourier methods to expand translation-invariant kernels.

We consider CR submersive mappings between generic submanifolds in complex space. We show that, under suitable conditions on the manifolds, there is an integer k such that any jet of the CR mapping at a given point is a rational function of its k-jet at that point. As a consequence, it is shown that the stability group…

We give a characterization of closed, simply connected, rationally elliptic 6-manifolds in terms of their rational cohomology rings and a partial classification of their real cohomology rings. We classify rational, real and complex homotopy types of closed, simply connected, rationally elliptic 7-manifolds. We give par…

Rational bubbles form in nonstationary models of real assets.

Paper introduces rational Gaussian wavelets for efficient signal approximation.

Generalized Thurston's characterization for branched coverings of the 2-sphere.

In this paper we classify, up to rigid isotopy, non-singular real rational curves of degrees less than or equal to 6 in a quadric homeomorphic to the 3-sphere. We also study their connections with rigid isotopy classes of real rational knots in $\mathbb{RP}^3$.

Graph manifolds' Thurston norms are sums of linear functionals, and every such norm can be realized.

In a finite-dimensional real vector space furnished with a rational structure with respect to a subfield of the field of real numbers, every (simplicial) rational semifan is contained in a complete (simplicial) rational semifan. In this paper this result is proved constructively on use of techniques from polyhedral geo…

We discuss the question of geometric formality for rationally elliptic manifolds of dimension $6$ and $7$. We prove that a geometrically formal six-dimensional biquotient with $b_{2}=3$ has the real cohomology of a symmetric space. We also show that a rationally hyperbolic six-dimensional manifold with $b_2\leq 2$ and …

Characterizes mappings preserving Pythagorean-hodograph curves.

Rational neural networks approximate functions more efficiently with less depth.

We show that the problem of constructing a real rational knot of a reasonably low degree can be reduced to an algebraic problem involving the pure braid group: expressing an associated element of the pure braid group in terms of the standard generators of the pure braid group. We also predict the existence of a real ra…

Study optimal investment with herd behavior using rational decision decomposition.

We develop a multi-curve term structure setup in which the modelling ingredients are expressed by rational functionals of Markov processes. We calibrate to LIBOR swaptions data and show that a rational two-factor lognormal multi-curve model is sufficient to match market data with accuracy. We elucidate the relationship…

For node level graph encoding, a recent important state-of-art method is the graph convolutional networks (GCN), which nicely integrate local vertex features and graph topology in the spectral domain. However, current studies suffer from several drawbacks: (1) graph CNNs relies on Chebyshev polynomial approximation whi…

Neural networks and rational functions efficiently approximate each other. In more detail, it is shown here that for any ReLU network, there exists a rational function of degree $O(\text{polylog}(1/ε))$ which is $ε$-close, and similarly for any rational function there exists a ReLU network of size $O(\text{polylog}(1/ε…

Homological stability fails for Cremona groups, rational varieties, and function fields.

In this paper we study rational real algebraic knots in $\R P^3$. We show that two real algebraic knots of degree $\leq5$ are rigidly isotopic if and only if their degrees and encomplexed writhes are equal. We also show that any irreducible smooth knot which admits a plane projection with less than or equal to four cro…

Compactifies stability space for $A_2$ category, introducing $q$-deformed rational numbers.

Satake has constructed compactifications of symmetric spaces D=G/K which (under a condition called geometric rationality by Casselman) yield compactifications of the corresponding locally symmetric spaces. The different compactifications depend on the choice of a representation of G. One example is the Baily-Borel-Sata…

Study real Mordell-Weil group and real lines on rational elliptic surfaces and del Pezzo surfaces.

The abstract discusses connections between K-stability, heights, and rational points on Fano varieties.

In this paper we show that the space of nodal rational curves, which is so called a Severi variety (of rational curves), on any non-singular projective surface is always equipped with a natural Einstein-Weyl structure, if the space is 3-dimensional. This is a generalization of the Einstein-Weyl structure on the space o…

We show that the loop spaces of real projective spaces are topologically approximated by the spaces of rational maps from RP(1) to RP(n). As a byproduct of our constructions we obtain an interpretation of the Kronecker characteristic (degree) of an ornament via particle spaces.

Study on mapping classes of real rational surface automorphisms, focusing on reducible maps and pseudo-Anosov maps.

In 1973, J. Cheeger and J. Simons raised the following question that still remains open and is known as the Rational Simplex Problem: Given a geodesic simplex in the spherical 3-space so that all of its interior dihedral angles are rational multiples of $π$, is it true that its volume is a rational multiple of the volu…

Let $F$ be a countable family of rational functions of two variables with real coefficients. Each rational function $f\in F$ can be thought as a continuous function $f:dom(f)\to\bar R$ taking values in the projective line $\bar R=R\cup\{\infty\}$ and defined on a cofinite subset $dom(f)$ of the torus $\bar R^2$. Then t…

We show that if the fundamental group of the complement of a rationally homologically fibered knot in a rational homology 3-sphere is bi-orderable, then its Alexander polynomial has at least one positive real root. Our argument can be applied for a finitely generated group which is an HNN extension with certain propert…

Let W -> X be a real smooth projective 3-fold fibred by rational curves. J. Kollár proved that, if W(R) is orientable, then a connected component N of W(R) is essentially either a Seifert fibred manifold or a connected sum of lens spaces. Our Main Theorem, answering in the affirmative three questions of Kollár, gives s…

We characterise simply-connected biquotients which potentially admit metrics of holonomy G_2. We prove that there are at most three real homotopy types of rationally elliptic such manifolds---all of them being formal. In the course of this examination we classify rationally elliptic homotopy types and characterise 7-di…

We investigate random complex dynamics of rational or polynomial maps on the Riemann sphere. We show that regarding random complex dynamics of polynomials, generically, the chaos of the averaged system disappears at any point in the Riemann sphere due to the automatic coopeartion of many kinds of maps in the system, ev…

Study wave functions in complex Chern-Simons theory, finding integrality and rational points.

We describe rational knots with any of the possible combinations of the properties (a)chirality, (non-)positivity, (non-)fiberedness, and unknotting number one (or higher), and determine exactly their number for a given number of crossings in terms of their generating functions. We show in particular how Fibonacci numb…

We explain the main concepts of Prospect Theory and Cumulative Prospect Theory within the framework of rational dynamic asset pricing theory. We derive option pricing formulas when asset returns are altered with a generalized Prospect Theory value function or a modified Prelec weighting probability function and introdu…

Moduli space of genus zero stable maps to the projective three-space naturally carries a real structure such that the fixed locus is a moduli space for real rational spatial curves with real marked points. The latter is a normal projective real variety. The singular locus being in codimension at least two, a first Stie…

The paper reconfirms a lower bound on rational genus using Heegaard Floer homology.

Study counts rational curves on hyperKähler ALE 4-manifolds.

In this paper we prove the following result: if two 2-dimensional 2-homogeneous rational vector fields commute, then either both vector fields can be explicitly integrated to produce rational flows with orbits being lines through the origin, or both flows can be explicitly integrated in terms of algebraic functions. In…

Three methods solve spatial rational curves with rational arc length.

A new approach to rationalization identifies true rationales by considering causal relationships.

The paper explores rational functions with 3 branching points on the Riemann sphere.

Ricci-positive manifolds span the kernel of the $\hat A$-genus in rational Spin bordism.

We give some characterizations for the critical values at infinity of a rational function in two complex variables in terms of the Euler characteristic, the Malgrange condition and the M-tameness

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 …