The paper verifies a conjecture about the index of symmetric spaces.
problem Calculating the minimal codimension of totally geodesic submanifolds in symmetric spaces of higher rank.
method Developed several approaches to tackle the problem, including a conjecture.
result Verified the conjecture for the index of symmetric spaces.
This paper identifies and analyzes biases in risk-adjusted index weighting methods, affecting social welfare and market fairness.
problem Biases in risk-adjusted index weighting methods lead to tracking errors and fraud in indices and ETFs.
method Characterizes and analyzes the biases and adverse effects of risk-adjusted index weighting methods.
result These biases reduce social welfare and can enable harmful arbitrage activities.
Researchers calculate Morse index and nullity for two specific minimal hypersurfaces.
problem Calculating Morse index and nullity for homogeneous minimal hypersurfaces with g=4 or 6. method Analyzing two specific homogeneous minimal hypersurfaces in Sn with g=4. result Obtained irrational eigenvalues in Laplace spectra for the two 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.
problem Calculating indices and nullities of focal manifolds in spheres.
method Analyzes isoparametric hypersurfaces in spheres with three distinct principal curvatures.
result Index equals the ambient space dimension, nullity determined by Killing vector fields.
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.
problem Analyzing the stability of sphere immersions.
method Calculation of Morse indices and stability indices for specific sphere immersions.
result Bounds on stability index of associative cone in R7. 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.
problem Stability and discriminatory power of binary choice models.
method Derives the real Gini index and incorporates PSI and KS statistics into the model.
result The real Gini index should be less than the calculated Gini index when the population distribution changes.
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.
problem Computing superbridge index of knots.
method Polygonal realizations with odd edges and linear programming.
result Exact superbridge index of many new knots, including 9- and 12-crossing knots.
Researchers prove an equivariant index theorem on Euclidean space.
problem Calculating the equivariant index of the Bott-Dirac operator on R2n. method Continuous field of C∗-algebras and equivariant index theorem. result Explicit calculation of the equivariant index of the Bott-Dirac operator on R2n. Develops methods to calculate global index of real polynomials.
problem Calculating the global index of real polynomials.
method Two methods: via atypical fibres and Milnor arcs clusters.
result Derives upper bounds for the global index, refining Durfee's degree-based bound.
Short note on braid index and quasipositivity of certain pretzel knots.
problem Calculating braid index and identifying quasipositive status for specific pretzel knots.
method Used Morton-Franks-Williams inequalities and Khovanov-Rozansky concordance homomorphisms.
result Determined braid index and identified quasipositivity for knots with even crossings in one strand.
The study calculates braid indices for two-bridge knots and proves inequalities.
problem Calculating the braid index of two-bridge knots.
method Proved inequalities and provided average braid index for knots of a given crossing number.
result Average braid index for two-bridge knots with a given crossing number.
A new clustering evaluation index based on density estimation.
problem Improving internal clustering evaluation indices.
method The index is a mixture of Ambiguous and Similarity sub-indices, calculated using density estimation.
result The new index significantly outperforms other internal clustering evaluation indices.
Study on symmetric braid index of ribbon knots, deriving bounds and characterizations.
problem Understanding the symmetric braid index of ribbon knots.
method Defining symmetric braid index, using Khovanov homology, and calculating bounds.
result Existence of knots with symmetric braid index greater than braid index.
Paper calculates Morse index of Y-singular minimal surfaces.
problem Computing Morse index of Y-singular minimal surfaces.
method Utilized two simpler problems: fixed boundary problem and Dirichlet-to-Neumann map.
result Index of Y-catenoid is one.
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→M in the total space of a semi-Riemannian submersion π:M→B by determining an explicit relation with the corresponding objects along the projected geodesic π∘γ:I→B in the base space. We use this result to calculate the focal Maslov in…
Paper calculates indices for group actions using cocycles.
problem Calculating equivariant indices for group actions.
method Construct cyclic cocycles on Harish-Chandra's Schwartz algebra, compute pairings with equivariant indices.
result Index formula completely determines equivariant indices via topological expressions.
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.
problem Computing the index for Cauchy-Riemann operators on surfaces with boundary punctures.
method Large antilinear deformations method, generalized to punctured surfaces.
result Involves a non-standard weighted count of boundary zeros in the Euler characteristic term.
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.
problem Calculating the index of Heisenberg elliptic operators on contact manifolds.
method Introduced a new trace on Heisenberg pseudodifferential operators and constructed a cocycle in periodic cyclic cohomology.
result Simplified the construction of the trace on Heisenberg pseudodifferential operators.
This paper uses machine learning to improve VIX index calculation and detect market manipulation.
problem Inaccuracies and potential market manipulation in VIX index calculation.
method Replicates VIX index using a subset of SP options and neural networks.
result A small number of SP options can accurately replicate the VIX index.
The study connects knot complements to 3d theories via half-index calculations.
problem Understanding the relationship between knot complements and 3d theories.
method Using half-index calculations and inverted Habiro series, the study realizes knot complements as homological blocks.
result The colored Jones polynomial is derived from choosing specific poles in the half-index integral expression.
The paper proves a disk's energy minimizer is holomorphic and calculates its Morse index.
problem Analyzing the Morse index of a non-holomorphic disk in pseudoconvex domains.
method Proof of holomorphic minimizers and Morse index calculation.
result Non-holomorphic critical disks have a Morse index of at least n-1.
The paper calculates critical points of systole function on Teichmüller space.
problem Critical points with pathological feature in surfaces of large genus.
method Integer linear programming with symmetry breaking technique.
result Found minimal filling sets of systoles in genus 5 with 8 geodesics.
An expression is found for the L2-index of a Dirac operator coupled to a connection on a Un vector bundle over S1×R3. 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 L2-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.
problem Distinguishing 3-manifolds and gauge theories phases.
method Dehn surgery presentation, ideal triangulation, and enhanced flavor symmetries.
result Invariance of refined index under various transformations.
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.
problem Computing the index and nullity of Fraser-Sargent surfaces.
method Analytical and numerical methods to compute index and nullity; provides bounds.
result Lower and upper bounds on the index of Fraser-Sargent surfaces inside the ball.
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.
problem Lag in macroeconomic index availability and autoregressive models' limitations in complex scenarios.
method Use financial transactions data to estimate macroeconomic indexes using neural networks and smart sampling.
result Neural network approach outperforms baseline methods on hand-crafted features based on transactions.
New RL method improves financial index tracking accuracy.
problem Optimizing financial index tracking with dynamic market information.
method Discrete-time dynamic model, Banach fixed point iteration, deep reinforcement learning.
result Proposed RL method outperforms benchmarks in tracking accuracy.
Paper derives second variational formula for statistical manifold mappings.
problem Variational formulas for mappings between statistical manifolds.
method Develops second variational formula for harmonic mappings, defines stability, index, and nullity.
result Shows weakly stability for harmonic mappings into statistical manifolds of non-positive curvature.
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.
problem Calculating indices for families of Fredholm operators and their extensions.
method Passing from a Fredholm operator to its graph, deforming the horizontal subspace.
result Index formulas for families of Fredholm realizations and self-adjoint extensions.
This study compares Markowitz and Single-Index models for Malaysian stocks.
problem Optimizing portfolio selection for Malaysian stocks using different models.
method Applied Markowitz and Single-Index models to 10-year historical data of 10 stocks and a risk-free asset.
result Comparison of minimum variance and maximum Sharpe portfolios for both models under various constraints.
The paper calculates ranks and bounds for Stiefel manifolds over different fields.
problem Computing ranks and bounds for Stiefel manifolds over various fields.
method Computation of upper characteristic ranks and cup lengths, providing bounds and necessary conditions for maps.
result Bounds and necessary conditions for S3-maps between quaternionic Stiefel manifolds. The paper calculates Morse indices and nullities for embedded networks on spheres.
problem Computing Morse indices and nullities for embedded networks on spheres.
method Using the Dirichlet-to-Neumann map and properties of eigenvalues and eigenfunctions.
result For all stationary triple junction networks in S2, there is only one eigenvalue -1. OGRe simplifies tensor calculations in general relativity.
problem Complex tensor calculations in general relativity.
method Object-oriented design for tensor calculus, automatic transformations, and optimized algorithms.
result Eliminates user errors and simplifies tensor calculations.