arXiv research

Study counts and equidistributes rational points in quaternionic Heisenberg groups.

We strongly develop the relationship between complex hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on complex hyperbolic spaces, especially in dimension $2$. We prove a Mertens' formula for the integer points over a quadratic imaginary num…

We provide a general Doob-Meyer decomposition for $g$-supermartingale systems, which does not require any right-continuity on the system. In particular, it generalizes the Doob-Meyer decomposition of Mertens (1972) for classical supermartingales, as well as Peng's (1999) version for right-continuous $g$-supermartingale…

In the first part of the paper, we study reflected backward stochastic differential equations (RBSDEs) with lower obstacle which is assumed to be right upper-semicontinuous but not necessarily right-continuous. We prove existence and uniqueness of the solutions to such RBSDEs in appropriate Banach spaces. The result is…

We first introduce the concept of $\mathscr{Y}^{g,ξ}$-submartingale systems, where the nonlinear operator $\mathscr{Y}^{g,ξ}$ corresponds to the first component of the solution of a reflected BSDE with generator $g$ and lower obstacle $ξ$. We first show that, in the case of a left-limited right-continuous obstacle, any…

Sharpe ratio is widely used in asset management to compare and benchmark funds and asset managers. It computes the ratio of the excess return over the strategy standard deviation. However, the elements to compute the Sharpe ratio, namely, the expected returns and the volatilities are unknown numbers and need to be esti…

The counting function on the natural numbers defines a discrete Morse-Smale complex with a cohomology for which topological quantities like Morse indices, Betti numbers or counting functions for critical points of Morse index are explicitly given in number theoretical terms. The Euler characteristic of the Morse filtra…

Paper develops a new probabilistic method for American options using entropy regularization.

In approachability with full monitoring there are two types of conditions that are known to be equivalent for convex sets: a primal and a dual condition. The primal one is of the form: a set C is approachable if and only all containing half-spaces are approachable in the one-shot game; while the dual one is of the form…

Proves a general connected sum formula for families Seiberg-Witten invariants.

Derives an integral formula for G2-structures.

Derives integral formulae on weighted manifolds.

Paper derives trace formula for magnetic Laplacian at zero energy.

The Gauss formula is extended to various Laplacians on submanifolds.

The paper is devoted to the problem of finding explicit combinatorial formulae for the Pontryagin classes. We discuss two formulae, the classical Gabrielov-Gelfand-Losik formula based on investigation of configuration spaces and the local combinatorial formula obtained by the author in 2004. The latter formula is based…

We prove two tropical gluing formulae for Gromov-Witten invariants of exploded manifolds, useful for calculating Gromov-Witten invariants of a symplectic manifold using a normal-crossing degeneration. The first formula generalizes the symplectic-sum formula for Gromov-Witten invariants. The second formula is stronger, …

Note on new cancellation formulas for manifolds.

The main result of the present paper is a coincidence formula for foliated manifolds. To prove this we establish Kuenneth formula, Poincare duality and intersection product in the context of tangential de Rham cohomology and homology of tangential currents. We apply the formula to get a dynamical Lefschetz formula for …

Formula calculates volume of two-bridge knots.

Introduces a universal Bochner formula for scalar curvature.

Formula connects surgeries to Seiberg-Witten invariants.

It has been shown that the Alvarez-Gaum$\mathrm{\acute{e}}$-Witten miraculous anomaly cancellation formula in type IIB superstring theory and its various generalizations can be derived from modularity of certain characteristic forms. In this paper, we show that the Green-Schwarz formula and the Schwarz-Witten formula i…

Proves a formula for a special invariant of 4-manifolds.

Proves a special case of the Gaussian kinematic formula using large sphere limits.

New Crofton formulae derived from existing ones.

Formulae for non-symmetric connections derived from covariant derivatives.

Formula connects curvature to volume in special geometric spaces.

Kenmotsu's formula describes surfaces in Euclidean 3-space by their mean curvature functions and Gauss maps. In Lorentzian 3-space, Akutagawa-Nishikawa's formula and Magid's formula are Kenmotsu-type formulas for spacelike surfaces and for timelike surfaces, respectively. We apply them to a few problems concerning rota…

The paper proves T-duality and Hori formulae for winding loop spaces.

New formulas for measuring geometric properties of definable sets.

Guillemin trace formula adapted for group actions.

Unified entropy formula for real, complex, and quaternionic DLNs.

Proves an Euler-type formula for Möbius strip partitions.

We prove a quasi-Poisson bracket formula for the space of representations of the fundamental groupoid of a surface with boundary, which generalizes Goldman's Poisson bracket formula. We also deduce a similar formula for quasi-Poisson cross-sections.

Paper derives matrix formulae and proves skein relations for non-orientable surfaces in quasi-cluster algebras.

New formulas for coassociative submanifolds' volume variation.

New methods derive a generalized Frenkel trace formula for Lie groups.

Alternative closed-form formula for spread call option prices under log-normal models.

The paper derives new Gauss-Bonnet formulas for frontal bundles over surfaces with boundary.

In this paper, we obtain asymptotic formulas with error estimates for the implied volatility associated with a European call pricing function. We show that these formulas imply Lee's moment formulas for the implied volatility and the tail-wing formulas due to Benaim and Friz. In addition, we analyze Pareto-type tails o…

Proves a formula in Heegaard Floer homology using combinatorial methods.

We give a formula of the colored Alexander invariant in terms of the homological representation of the braid groups which we call truncated Lawrence's representation. This formula generalizes the famous Burau representation formula of the Alexander polynomial.

We derive formulas for F measures' standard error and confidence intervals.

Derives Selberg trace formula on Riemann surfaces and generalizes to other spaces.

This paper derives an explicit formula for Branson's Q-curvature in even-dimensional conformal geometry. The ingredients in the formula come from the Poincare metric in one higher dimension; hence the formula is called holographic. When specialized to the conformally flat case, the holographic formula expresses Q-curva…

Wave trace singularity formula for fibre bundles generalizes Poisson summation.

We give an immersion formula, the Sym-Bobenko formula, for minimal surfaces in the 3-dimensional Heisenberg space. Such a formula can be used to give a generalized Weierstrass type representation and construct explicit examples of minimal surfaces.

Proves Juhl formulas for curved Ovsienko--Redou operators, confirming conjectures.