arXiv research

In 1980, Onishchik introduced the index of a Riemannian symmetric space as the minimal codimension of a (proper) totally geodesic submanifold. He calculated the index for symmetric spaces of rank less than or equal to 2, but for higher rank it was unclear how to tackle the problem. In earlier papers we developed severa…

This paper identifies and analyzes biases in risk-adjusted index weighting methods, affecting social welfare and market fairness.

Researchers calculate Morse index and nullity for two specific minimal hypersurfaces.

We discuss the relation between arc index, maximal Thurston--Bennequin number, and Khovanov homology for knots. As a consequence, we calculate the arc index and maximal Thurston--Bennequin number for all knots with at most 11 crossings. For some of these knots, the calculation requires a consideration of cables which a…

We calculate the equivariant index formula for an infinite dimensional Clifford module canonically associated to any Riemannian manifold. It encompasses the fractional index formula of the projective Dirac operator by Mathai--Melrose--Singer.

We investigate the problem of calculating the Fredholm index of a geometric Dirac operator subject to local (e.g. Dirichlet and Neumann) and non-local (APS) boundary conditions posed on the strata of a manifold with corners. The boundary strata of the manifold with corners can intersect in higher codimension. To calcul…

Study calculates indices and nullities of focal manifolds in spheres.

Main theorem of this paper states that Floer cohomology groups in a Hilbert space are isomorphic to the cohomological Conley Index. It is also shown that calculating cohomological Conley Index does not require finite dimensional approximations of the vector field. Further directions are discussed.

Study on sphere immersions and their stability indices.

In this paper, we study the inequality indices for some models of wealth exchange. We calculated Gini index and newly introduced k-index and compare the results with reported empirical data available for different countries. We have found lower and upper bounds for the indices and discuss the efficiencies of the models…

The paper examines the stability of binary choice models using Gini index and scoring indicators.

We calculate the alternating number of torus knots with braid index 4 and less. For the lower bound, we use the upsilon-invariant recently introduced by Ozsváth, Stipsicz, and Szabó. For the upper bound, we use a known bound for braid index $3$ and a new bound for braid index $4$. Both bounds coincide, so that we obtai…

New superbridge index calculations for knots with odd edges.

Researchers prove an equivariant index theorem on Euclidean space.

Develops methods to calculate global index of real polynomials.

Short note on braid index and quasipositivity of certain pretzel knots.

The study calculates braid indices for two-bridge knots and proves inequalities.

A new clustering evaluation index based on density estimation.

Study on symmetric braid index of ribbon knots, deriving bounds and characterizations.

Paper calculates Morse index of Y-singular minimal surfaces.

Let $M$ be a compact manifold. and $D$ a Dirac type differential operator on $M$. Let $A$ be a $C^*$-algebra. Given a bundle $W$ of $A$-modules over $M$ (with connection), the operator $D$ can be twisted with this bundle. One can then use a trace on $A$ to define numerical indices of this twisted operator. We prove an …

We study focal points and Maslov index of a horizontal geodesic $γ:I\to M$ in the total space of a semi-Riemannian submersion $π:M\to B$ by determining an explicit relation with the corresponding objects along the projected geodesic $π\circγ:I\to B$ in the base space. We use this result to calculate the focal Maslov in…

Paper calculates indices for group actions using cocycles.

The index of a Riemannian symmetric space is the minimal codimension of a proper totally geodesic submanifold (Onishchik, 1980). There is a conjecture by the first two authors for how to calculate the index. In this paper we give an affirmative answer to this conjecture for the exceptional Riemannian symmetric spaces a…

Formula calculates index for CR operators on surfaces with boundary punctures.

We discuss modelling of SPX and DAX index option prices using the Shifted Log-Normal (SLN) model, (also known as Displaced Diffusion), and the SABR model. We found out that for SPX options, an example of strongly skewed option prices, SLN can produce a quite accurate fit. Moreover, for both types of index options, the …

Clarifies a trace for Heisenberg operators on contact manifolds.

This paper uses machine learning to improve VIX index calculation and detect market manipulation.

The study connects knot complements to 3d theories via half-index calculations.

The paper proves a disk's energy minimizer is holomorphic and calculates its Morse index.

The paper calculates critical points of systole function on Teichmüller space.

An expression is found for the $L^2$-index of a Dirac operator coupled to a connection on a $U_n$ vector bundle over $S^1\times{\mathbb R}^3$. Boundary conditions for the connection are given which ensure the coupled Dirac operator is Fredholm. Callias' index theorem is used to calculate the index when the connection i…

We prove the Fredholmness of Dirac operators of monopoles with Dirac-type singularities on oriented complete Riemannian $3$-folds, and we also calculate the $L^2$-indices of them.

Prime knots of genus one admitting diagram with at most five classical crossings were classified by Akimova and Matveev in 2014. In 2018 Kaur, Prabhakar and Vesnin introduced families of L-polynomials and F-polynomials for virtual knots which are generalizations of affine index polynomial. Here we introduce a notion of…

Refined 3D index uses surgery and gradings to distinguish 3-manifolds.

When calculating the index of a minimal surface, the set of smooth functions on a domain with compact support is the standard setting to describe admissible variations. We show that the set of admissible variations can be widened in a geometrically meaningful manner by considering the difference of area functional, lea…

The paper calculates the index and nullity of Fraser-Sargent surfaces and provides bounds.

We study the stability of coassociative 4-folds with conical singularities under perturbations of the ambient G_2 structure by defining an integer invariant of a coassociative cone which we call the stability index. The stability index of a coassociative cone is determined by the spectrum of the curl operator acting on…

Bank transactions help predict macroeconomic indexes faster and more accurately.

New RL method improves financial index tracking accuracy.

Paper derives second variational formula for statistical manifold mappings.

We study the de Rham complex on a smooth manifold with a periodic end modeled on an infinite cyclic cover X' \to X. The completion of this complex in exponentially weighted L^2-norms is Fredholm for all but finitely many exceptional weights determined by the eigenvalues of the covering translation map H_*(X') \to H_*(X…

In this work, a version of Fermat's principle for causal curves with the same energy in time orientable Finsler spacetimes is proved. We calculate the secondvariation of the {\it time arrival functional} along a geodesic in terms of the index form associated with the Finsler spacetime Lagrangian. Then the character of …

The paper calculates indices for families of Fredholm operators and their extensions.

This study compares Markowitz and Single-Index models for Malaysian stocks.

The paper calculates ranks and bounds for Stiefel manifolds over different fields.

The paper calculates Morse indices and nullities for embedded networks on spheres.

OGRe simplifies tensor calculations in general relativity.