arXiv research

The paper examines sequences of metric spaces converging to compact limits with specific properties.

Let \gh = \gh_{-k}\oplus \cdots \oplus \gh_{l} (k >0, l \geq 0) be a finite dimensional real graded Lie algebra, with a Euclidian metric \langle \cdot , \cdot \rangle adapted to the gradation. The metric \langle\cdot , \cdot \rangle is called admissible if the codifferentials \partial^{*} : C^{k+1}(\gh_{-}, \gh ) \ra C…

The study examines sequences of Riemannian manifolds with uniform Sobolev bounds and their convergence properties.

Defines new metrics for Lorentzian spaces and their convergence.

Bounds and constructions for Gromov-Hausdorff distance between spheres.

The paper constructs manifolds with infinite holes from a given manifold.

The abstract discusses a new type of space and its properties.

Survey of nonnegative scalar curvature sequences and their limits.

Proves $L^2$ curvature bound for manifolds with bounded Ricci curvature.

We consider (flat) Cauchy-complete GH spacetimes, i.e., globally hyperbolic flat lorentzian manifolds admitting some Cauchy hypersurface on which the ambient lorentzian metric restricts as a complete riemannian metric. We define a family of such spacetimes - model spacetimes - including four subfamilies: translation sp…

GC Stein manifolds characterized with embeddings and functions.

Study shows no new Euclidean factors can appear in the limit of CAT(0) spaces.

GH-PID uses guided harmonic paths for efficient SOT with interpretable diagnostics.

This paper extends FP's method to complex hyperbolic branched covers to find Einstein metrics.

The paper improves estimates on singular sets in manifolds with integral curvature bounds.

Consider a limit space $(M_α,g_α,p_α)\stackrel{GH}{\rightarrow} (Y,d_Y,p)$, where the $M_α^n$ have a lower Ricci curvature bound and are volume noncollapsed. The tangent cones of $Y$ at a point $p\in Y$ are known to be metric cones $C(X)$, however they need not be unique. Let $\barΩ_{Y,p}\subseteq\cM_{GH}$ be the close…

Study on null hypersurfaces in complex contact manifolds.

A Riemannian or Finsler metric on a compact manifold M gives rise to a length function on the free loop space ΛM, whose critical points are the closed geodesics in the given metric. If X is a homology class on ΛM, the minimax critical level cr(X) is a critical value. Let M be a sphere of dimension >2, and fix a metric …

We present a combinatorial proof for the existence of the sign refined grid homology in lens spaces, and a self contained proof that $\partial_{\mathbb{Z}}^2 = 0$. We also present a Sage program that computes $\widehat{\mathrm{GH}} (L(p,q),K;\mathbb{Z})$, and provide empirical evidence supporting the absence of torsion…

Study sequences of static spacetimes using null distance convergence.

Quantizes moduli space of 3D gravity metrics.

Generalizes global hyperbolicity to higher signatures and proves compactness.

In this paper, we are concerned with the regularity of noncollapsed Riemannian manifolds $(M^n,g)$ with bounded Ricci curvature, as well as their Gromov-Hausdorff limit spaces $(M^n_j,d_j)\stackrel{d_{GH}}{\longrightarrow} (X,d)$, where $d_j$ denotes the Riemannian distance. Our main result is a solution to the codimen…

Holomorphic curves found in compact quotients of SL(2,C).

Consider a hyperbolic group G and a quasiconvex subgroup H of infinite index. We construct a set-theoretic section s of the quotient map (of sets) from G to G/H such that s(G/H) is a net in G; that is, any element of G is a bounded distance from s(G/H). This section arises naturally as a set of points minimizing word-l…

4-manifolds with Ricci curvature bounds can be approximated by non-manifold spaces.

Paper introduces MSP Network for harmonizing images from different scanners.

We study here limit spaces $(M_α,g_α,p_α)\stackrel{GH}{\rightarrow} (Y,d_Y,p)$, where the $M_α$ have a lower Ricci curvature bound and are volume noncollapsed. Such limits $Y$ may be quite singular, however it is known that there is a subset of full measure $\cR(Y)\subseteq Y$, called {\it regular} points, along with c…

Enhanced probabilistic sampling on manifolds using Double Diffusion Maps and Geometric Harmonics.

Let $M$ be an $n$-dimensional complete Riemannian manifold with Ricci curvature $\ge n-1$. In \cite{colding1, colding2}, Tobias Colding, by developing some new techniques, proved that the following three condtions: 1) $d_{GH}(M, S^n)\to 0$; 2) the volume of $M$ ${\text{Vol}}(M)\to{\text{Vol}}(S^n)$; 3) the radius of $M…

Study of induced Hausdorff metrics on Lie groups and their convergence.

Rectifies singular sets in noncollapsed spaces with Ricci curvature bounds.

In this paper, we quantitatively investigate the statistical properties of a statistical ensemble of stock prices. We selected 1200 stocks traded on the Tokyo Stock Exchange, and formed a statistical ensemble of daily stock prices for each trading day in the 3-year period from January 4, 1999 to December 28, 2001, corr…

We study the local differential geometry of varieties $X^n\subset \Bbb C\Bbb P^{n+a}$ with degenerate secant and tangential varieties. We show that the second fundamental form of a smooth variety with degenerate tangential variety is subject to certain rank restrictions. The rank restrictions imply a slightly refined v…

Polynomial algorithm estimates mGH distance between unweighted graphs.

The article introduces a new convergence concept for Lorentzian spaces and applies it to generalized cones.

Study proposes a new portfolio selection method using non-Gaussian models and Esscher transform.

The NIG model outperforms others in pricing S&P 500 index options.

A para-Kähler manifold can be defined as a pseudo-Riemannian manifold $(M,g)$ with a parallel skew-symmetric para-complex structures $K$, i.e. a parallel field of skew-symmetric endomorphisms with $K^2 = \mathrm{Id}$ or, equivalently, as a symplectic manifold $(M,ω)$ with a bi-Lagrangian structure $L^\pm$, i.e. two c…

The paper establishes a new link between isometries and scaling nonvanishing property in manifolds with Ricci curvature.

This paper studies the relationship between fundamental groups of manifolds and their effective regular sets.

Adaptive wave model for financial option pricing is proposed, as a high-complexity alternative to the standard Black--Scholes model. The new option-pricing model, representing a controlled Brownian motion, includes two wave-type approaches: nonlinear and quantum, both based on (adaptive form of) the Schrödinger equatio…

In this paper, we quantitatively investigate the properties of a statistical ensemble of stock prices. We focus attention on the relative price defined as $X(t) = S(t)/S(0)$, where $S(0)$ is the initial price. We selected approximately 3200 stocks traded on the Japanese Stock Exchange and formed a statistical ensem…

Paper proposes new Bayesian neural network models for efficient learning.

This work defines a new metric on quotient spaces and studies its geometry.

Proves cosets of certain subgroups in hyperbolic 3-manifold groups are conjugacy distinguished.

We consider maximum solution $g(t)$, $t\in [0, +\infty)$, to the normalized Ricci flow. Among other things, we prove that, if $(M, ω)$ is a smooth compact symplectic 4-manifold such that $b_2^+(M)>1$ and let $g(t),t\in[0,\infty)$, be a solution to (1.3) on $M$ whose Ricci curvature satisfies that $|\text{Ric}(g(t))|\l…

The paper proves rigidity theorems for Poincaré-Einstein manifolds with specific conformal properties.