This paper introduces TDA and TSI for better business analytics.
problem Nonlinear, multi-scale business datasets under-represented by traditional tools.
method Topological Data Analysis (TDA) and Topological Stability Index (TSI).
result TSI reveals structural variability in business data.
Introduces TSI, a variance-based measure for persistence barcodes.
problem Capturing structural variability in persistence barcodes.
method Variance-based scalar measure, TSI, and complementary TSigI.
result TSI captures structural variability complementary to entropy.
We obtain a quantitative estimate on the generalised index of translators for the mean curvature flow with bounded norm of the second fundamental form. The estimate involves the dimension of the space of weighted square integrable f-harmonic 1-forms. By the adaptation to the weighted setting of Li-Tam theory developed …
The study classifies translating and self-expanding solitons in 3D space.
problem Characterizing the topology and index of solitons in mean curvature flow.
method Analyzing the spectrum and index of expanding and translating solitons in R3. result Translating and self-expanding solitons have finite topology under certain conditions.
We prove that for every nonnegative integer g, there exists a bound on the number of ends of a complete, embedded minimal surface M in R3 of genus g and finite topology. This bound on the finite number of ends when M has at least two ends implies that M has finite stability index which is bounded …
The paper studies capillary surfaces, proving stability and curvature estimates.
problem Stability and curvature estimates for capillary surfaces.
method Analyzes the index and geometry/topology of capillary surfaces in 3-manifolds.
result Stable capillary surfaces in a half-space are half-planes.
Minimal 7D hypersurfaces degenerate under stability or bounded index constraints.
problem Degeneration of minimal hypersurfaces under stability or bounded index constraints.
method Analysis of sequences of minimal hypersurfaces, parameterization with controlled maps, and topological finiteness results.
result Minimal hypersurfaces can degenerate to singular ones with controlled geometry, topology, and singular set.
New framework models high-Hopf-index hopfions using generalized fold maps.
problem Modeling and understanding high-Hopf-index hopfions.
method Proposes a new mathematical framework based on generalized Hopf maps and fold maps.
result Establishes a robust bridge between singular maps and experimentally observed hopfion topologies.
In this paper, we study stability and instability problem for type-II partitioning problem. First, we make a complete classification of stable type-II stationary hypersurfaces in a ball in a space form as totally geodesic n-balls. Second, for general ambient spaces and convex domains, we give some topological restric…
Finite p-group actions on manifolds have limited stabilizer subgroups.
problem Understanding the structure of stabilizer subgroups in group actions on manifolds.
method Bounding the index of a subgroup H in a finite p-group G acting on a compact manifold M, ensuring a controlled number of stabilizers.
result The existence of a subgroup H with a controlled index and limited stabilizers.
A smooth five-dimensional s-cobordism becomes a smooth product if stabilized by a finite number n of S2xS2x[0,1]'s. We show that for amenable fundamental groups, the minimal n is subextensive in covers, i.e., n(cover)/index(cover) has limit 0. We focus on the notion of sweepout width, which is a bridge between 4-di…
We show that after stabilizations of opposite parity and braid isotopy, any two braids in the same topological link type cobound embedded annuli. We use this to prove the generalized Jones conjecture relating the braid index and algebraic length of closed braids within a link type, following a reformulation of the prob…
The paper proves rigidity and stability properties of self-shrinking surfaces in 3D space.
problem Rigidity and stability of self-shrinking surfaces in R3. method Analyzing the mean curvature flow and L-index of self-shrinkers. result No stable two-dimensional self-shrinker in R3 exists without properness. 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. Critical surfaces are defined by Bachman as topological index 2 surfaces, generalizing incompressible surfaces and strongly irreducible surfaces. In this paper we give a condition to obtain critical Heegaard surfaces by amalgamation. As a special case, we obtain critical Heegaard surfaces by boundary stabilization. It …
It is known that the totally umbilical hypersurfaces in the (n+1)-dimensional spheres are characterized as the only hypersurfaces with weak stability index 0. That is, a compact hypersurface with constant mean curvature, cmc, in S^{n+1}, different from an Euclidean sphere, must have stability index greater than or equa…
Stability of Yang-Mills connections' Morse indices and nullity in 4D.
problem Stability of Yang-Mills connections' Morse indices and nullity in 4D under weak convergence.
method Proves stability results of the Morse index plus nullity of Yang-Mills connections in dimension 4 under weak convergence.
result Stability of the sum of Morse indices and nullity of a sequence of Yang-Mills connections.
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…
We review the concepts of the index of a Fredholm operator, the spectral flow of a curve of self-adjoint Fredholm operators, the Maslov index of a curve of Lagrangian subspaces in symplectic Hilbert space, and the eta invariant of operators of Dirac type on closed manifolds and manifolds with boundary. We emphasize var…
The topological index of a surface was previously introduced by the first author as the topological analogue of the index of an unstable minimal surface. Here we show that surfaces of arbitrarily high topological index exist.
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.
The paper studies stability and index of biharmonic hypersurfaces in Riemannian manifolds.
problem Analyzing stability and index of biharmonic hypersurfaces.
method Using a second variation formula for biharmonic hypersurfaces, computing stability index, and proving non-existence of unstable hypersurfaces.
result Proves non-existence of unstable proper biharmonic hypersurfaces in Euclidean space or hyperbolic space.
Study stability of surfaces in spacetimes, proving new estimates and theorems.
problem Stability of surfaces in spacetime and their applications.
method Variational techniques, Christodoulou-Yau estimate, Cohn-Vossen inequality, global theorem, capillary stability, area inequality, diameter estimate.
result Established new estimates and theorems for stable surfaces in spacetime.
The paper finds new eigenfunctions for minimal immersions and their stability index.
problem Finding new eigenfunctions for minimal immersions and their stability index.
method Explicitly showed new eigenfunctions for the stability operator.
result The stability index of minimal immersions is at least kℓ+3k+3ℓ+8. Study links K-stability of certain surfaces to binary forms, proving stability and non-stability conditions.
problem Investigating K-stability of specific del Pezzo surfaces.
method Relating K-stability to GIT stability of binary forms, proving stability and non-stability conditions.
result K-polystability and non-K-stability of quasi-smooth hypersurfaces.
Continuous analysis techniques for deforming domains in manifolds.
problem Analyzing continuity of analysis objects in deforming domains.
method Introducing quasi-Lipschitz domains and studying monotone C0-deformations. result Global Morse index theorem holds for arbitrary Lipschitz domains.
Global Morse index theorem applied to Jacobi fields on CMC surfaces.
problem Existence and structural theorem of Jacobi fields on CMC surfaces.
method Global Morse index theorem proof via set-continuity of domain shapes and eigenvalue continuity.
result Global Morse index theorem provides structural existence of Jacobi fields.
Stability of Morse index for Yang-Mills connections in 4D.
problem Stability of critical points in Yang-Mills energy relaxation.
method Establishing lower semi-continuity of Morse index and upper continuity of Morse index plus nullity.
result Yang-Mills fields are more stable than harmonic maps in 4D.
Stability of mapping spaces is shown to be related to the D-topology.
problem Understanding the relationship between stability and the D-topology of mapping spaces.
method Reformulation and proof of stability theorems in diffeological étale manifolds.
result Stable classes of mapping spaces are D-open.
The paper refines the stability index for a specific minimal hypersurface and verifies Yau's conjecture.
problem Stability of minimal hypersurfaces in spheres and eigenvalue multiplicity.
method Analytical and numerical methods to study eigenvalues and stability indices.
result The multiplicity of the eigenvalue for the Carlotto-Schulz minimal embedding is at least 2n+1+n^2.
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.
Stability of Morse index for harmonic maps on degenerating surfaces analyzed.
problem Analyzing stability of Morse index for harmonic maps on degenerating Riemann surfaces.
method Analysis of second variation of energy, identification of conditions for upper semicontinuity, explicit contribution of geodesics.
result Sharper control of spectrum of Jacobi operator, explicit contribution of geodesic segments to Morse index.
Study index theory on Lie group homogeneous spaces using topological and analytic methods.
problem Index theory on homogeneous spaces of Lie groups.
method Topological and analytic approaches: Riemann-Roch formula and heat kernel methods.
result Local index formula representing higher indices of equivariant elliptic operators.
The paper computes torsion invariants for groups acting on complexes.
problem Computing torsion invariants for groups acting on complexes.
method Analyzes residually finite groups acting cocompactly on contractible complexes with specific stabilizers.
result Torsion limits to the torsion of the boundary subcomplex, independent of the chain of subgroups.
The disk complex of a surface in a 3-manifold is used to define its {\it topological index}. Surfaces with well-defined topological index are shown to generalize well-known classes, such as incompressible, strongly irreducible, and critical surfaces. The main result is that one may always isotope a surface H with top…
We prove that every smooth Fano complete intersection of index 1 and codimension r in Pn+r is birationally superrigid and K-stable if n≥10r. We also propose a generalization of Tian's criterion of K-stability and, as an application, prove the K-stability of the complete intersection of a quadric …
Study topological quantum mechanics on orbifolds with geometric interpretation.
problem Quantum mechanical models on symplectic orbifolds.
method Explicit orbifold version of quantum HKR map and exact semi-classical approximation.
result Geometric and quantum field theoretic interpretation of orbifold algebraic index.
Graph braid groups' complexity stabilizes for most graphs.
problem Stabilization of topological complexity in graph braid groups.
method Geometric lower bounds on configuration spaces.
result Topological complexity stabilizes for most graphs.
Atiyah-Singer theorem links math fields, predicts topological insights.
problem Understanding the interplay between analysis, geometry, and topology.
method Analyzes and generalizes topological invariants in differential geometry.
result Predicts the index of elliptic operators based on topology.
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.
In this paper we prove the existence of a natural mapping from the surgery exact sequence for topological manifolds to the analytic surgery exact sequence of N. Higson and J. Roe. This generalizes the fundamental result of Higson and Roe, but in the treatment given by Piazza and Schick, from smooth manifolds to topolog…
Study finds minimal hypersurfaces grow linearly in index, contrary to 3D.
problem Understanding index growth of minimal hypersurfaces.
method Partitioning methods for compact Lie groups, applied to hypersurfaces.
result Linear index growth for hypersurfaces of fixed topological type.
We show that except for n=2 if a bridge surface for a knot is an index n topologically minimal surface, then after a perturbation it is still topologically minimal with index at most n+1.
New Bailey pairs derived for tetrahedron index, linking knot invariants.
problem Deriving knot invariants using Bailey pairs for the tetrahedron index.
method Developed new Bailey pairs in terms of q-series.
result New Bailey pairs express the pentagon identity of the tetrahedron index.
A new method tracks index using topological data analysis for sparse portfolios.
problem Sparse index tracking with robust risk management.
method Topological learning via Vietoris-Rips filtration for sparse regularization.
result The method outperforms state-of-the-art techniques in various market conditions.
The paper proves stability of minimal embeddings in spheres and relates it to Yau's conjecture.
problem Stability of minimal embeddings in spheres and Yau's conjecture.
method Analyzes stability index and solves differential equation to relate to Yau's conjecture.
result Stability index of minimal hypersurfaces is at least n^2+4n+3 and Yau's conjecture holds under specific conditions.
New invariant for classifying 4-manifolds up to cobordism.
problem Classifying closed, oriented topological 4-manifolds up to s-cobordism. method Introducing a stable range invariant after stabilization by a fixed number of S2imesS2. result A new invariant for classifying 4-manifolds up to s-cobordism. Study uses VIX for zero-coupon Treasury rates, proving long-term stability and returns.
problem Modeling zero-coupon Treasury rates with VIX for volatility.
method Multivariate autoregressive stochastic volatility model, proving stability and Law of Large Numbers.
result VIX accurately models zero-coupon Treasury rates and returns.