arXiv research

New lattices are linked to higher hypergeometric functions.

New proof of rearrangement lemma for noncommutative tori using hypergeometric functions.

Study spherical Fourier transform on hypergeometric type harmonic manifolds.

Unified method for computing modular curvature on toric noncommutative manifolds using hypergeometric functions.

We extend projectively equivariant quantization and symbol calculus to symbols of pseudo-differential operators. An explicit expression in terms of hypergeometric functions with noncommutative arguments is given. Some examples are worked out, one of them yielding a quantum length element on $S^3$.

New harmonic Hadamard manifolds defined via hypergeometric equations.

Study of line congruences for Appell's rank-4 hypergeometric functions.

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

Study functional relations on noncommutative tori using transformations.

We present a global representation for surfaces in 3-dimensional hyperbolic space with constant mean curvature 1 (CMC-1 surfaces) in terms of holomorphic spinors. This is a modification of Bryant's representation. It is used to derive explicit formulas in hypergeometric functions for CMC-1 surfaces of genus 0 with thre…

Constructs explicit nontrivial cycles in Habiro cohomology of smooth varieties.

The aim of this text is to establish some relations between Markov chains in Dirichlet Environments on directed graphs and certain hypergeometric integrals associated with a particular arrangement of hyperplanes. We deduce from these relations and the computation of the connexion obtained by moving one hyperplane of th…

Estimates unknown population sizes using the hypergeometric distribution.

Harmonic manifolds of hypergeometric type have entropy bounds related to real hyperbolic spaces.

We derive a factorization of the Alexander polynomial of the 4-strand Turk's head knot using hypergeometric representations.

Proposes a differentiable hypergeometric distribution for learning group importance.

New method identifies graph structure from data using generalized hypergeometric distributions.

Explicit computation of Kontsevich weights for symplectic Poisson structures.

In the genus one case, we make explicit some constructions of Veech on flat surfaces and generalize some geometric results of Thurston about moduli spaces of flat spheres as well as some equivalent ones but of an analytico-cohomological nature of Deligne-Mostow, which concern the monodromy of Appell-Lauricella hypergeo…

New quasi-Einstein metrics found on a sphere.

New formulas for colored Jones polynomials of double twist knots generalize series and duality.

This work presents an exact solution to the generalized Heston model, where the model parameters are assumed to have linear time dependence The solution for the model in expressed in terms of confluent hypergeometric functions.

The SABR model is shortly presented and the volatility swap explained. The fair value for a volatility swap is then computed using the usual theory in financial mathematics. An analytical solution using confluent hypergeometric functions is found. The solution is then verified using Rama Cont's functional calculus.

This paper establishes certain existence and classification results for solutions to $SU(n)$ Toda systems with three singular sources at 0, 1, and $\infty$. First, we determine the necessary conditions for such an $SU(n)$ Toda system to be related to an $n$th order hypergeometric equation. Then, we construct solutions …

We investigate the pricing of financial options under the 2-hypergeometric stochastic volatility model. This is an analytically tractable model that reproduces the volatility smile and skew effects observed in empirical market data. Using a regular perturbation method from asymptotic analysis of partial differential eq…

Formulates integrals for hypersphere arrangements using cohomology and Cayley-Menger determinants.

Develops tropical geometry for weighted Hurwitz numbers, generalizing previous results.

Expanded Local Variance Gamma model adds drift and simplifies calibration.

We give a list of Heun equations which are Picard-Fuchs associated to families of algebraic varieties. Our list is based on the classification of families of elliptic curves with four singular fibers done by Herfurtner. We also show that pullbacks of hypergeometric functions by rational Belyi functions with restricted …

Classical solvable stochastic volatility models (SVM) use a CEV process for instantaneous variance where the CEV parameter $γ$ takes just few values: 0 - the Ornstein-Uhlenbeck process, 1/2 - the Heston (or square root) process, 1- GARCH, and 3/2 - the 3/2 model. Some other models were discovered in \cite{Labordere2009…

I present the technique which can analyse some interest rate models: Constantinides-Ingersoll, CIR-model, geometric CIR and Geometric Brownian Motion. All these models have the unified structure of Whittaker function. The main focus of this text is closed-form solutions of the zero-coupon bond value in these models. In…

Formula derived for special q-hypergeometric series at roots of unity.

A new distribution family extends the $\alpha$-stable distribution with a degree of freedom parameter.

Researchers create a fundamental domain for all Deligne-Mostow lattices in PU(2,1).

New identities for simplex volume in Euclidean space.

New groups discovered with unique properties in a specific space.

We prove some value of the harmonic volume for the Klein quartic $C$ is nonzero modulo ${1/2}\{mathbb Z}$, using special values of the generalized hypergeometric function ${}_3F_2$. This result tells us the algebraic cycle $C-C^-$ is not algebraically equivalent to zero in the Jacobian variety $J(C)$.

Researchers found new functions for spherical clothoids using special functions.

After introducing the sub-Riemannian geometry of the Heisenberg group Hn, n \geq 1, we recall some basics about hypersurfaces endowed with the H-perimeter measure and horizontal Green's formulas. Then, we describe a class of compact closed hypersurfaces of constant horizontal mean curvature called "Isoperimetric Profil…

In this paper we develop a methodology to analyze and compare multiple global networks. We focus our analysis on the relation between human migration and trade. First, we identify the subset of products for which the presence of a community of migrants significantly increases trade intensity. To assure comparability ac…

The study of Farey polynomials connects geometry, topology, and combinatorics.

This paper studies an optimal investment problem under M-CEV with power utility function. Using Laplace transform we obtain explicit expression for optimal strategy in terms of confluent hypergeometric functions. For obtained representations we derive asymptotic and approximation formulas contains only elementary funct…

Proves formula for 3D index change with Dehn filling.

Differential Galois theory connects connections with parameters to isomonodromic deformations.

Although this article can be read independently, it is a continuation of the introduction to integrable systems aspects of quantum cohomology given in part 1 (math.DG/0104274). In the same elementary style, i.e. assuming basic properties of quantum cohomology and concentrating on the simplest nontrivial examples, the q…

This is a glossary of notions and methods related with the topological theory of collections of affine planes, including braid groups, configuration spaces, order complexes, stratified Morse theory, simplicial resolutions, complexes of graphs, Orlik--Solomon rings, Salvetti complex, matroids, Spanier--Whitehead duality…

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

A function of several variables is called holonomic if, roughly speaking, it is determined from finitely many of its values via finitely many linear recursion relations with polynomial coefficients. Zeilberger was the first to notice that the abstract notion of holonomicity can be applied to verify, in a systematic and…