arXiv research

Characterizes nodal volumes of Gaussian fields on manifolds, extending previous work.

Study on predicting sequences with Gaussian constraints, linking to intrinsic volumes and metric complexity.

New chaos formula simplifies variance calculation for Gaussian nodal volumes.

Mirzakhani volumes of moduli spaces are polylogarithmic.

It is shown that $3$ disjoint sets with fixed Gaussian volumes that partition $\mathbb{R}^{n}$ with nearly minimum total Gaussian surface area must be close to adjacent $120$ degree sectors, when $n\geq2$. These same results hold for any number $m\leq n+1$ of sets partitioning $\mathbb{R}^{n}$, conditional on the solut…

It is shown that $m$ disjoint sets with fixed Gaussian volumes that partition $\mathbb{R}^{n}$ with minimum Gaussian surface area must be $(m-1)$-dimensional. This follows from a second variation argument using infinitesimal translations. The special case $m=3$ proves the Double Bubble problem for the Gaussian measure,…

We introduce Gaussian-type measures on the manifold of all metrics with a fixed volume form on a compact Riemannian manifold of dimension $\geq 3$. For this random model we compute the characteristic function for the $L^2$ (Ebin) distance to the reference metric. In the Appendix, we study Lipschitz-type distance betwee…

Consider a $d\times d$ matrix $M$ whose rows are independent centered non-degenerate Gaussian vectors $ξ_1,...,ξ_d$ with covariance matrices $Σ_1,...,Σ_d$. Denote by $\mathcal{E}_i$ the location-dispersion ellipsoid of $ξ_i:\mathcal{E}_i={\mathbf{x}\in\mathbb{R}^d : \mathbf{x}^\topΣ_i^{-1} \mathbf{x}\leqslant1}$. We sh…

Stein variational gradient descent improves inference in Gaussian process models.

We obtain sharp volume bound for a conic 2-sphere in terms of its Gaussian curvature bound. We also give the geometric models realizing the extremal volume. In particular, when the curvature is bounded in absolute value by $1$, we compute the minimal volume of a conic sphere in the sense of Gromov. In order to apply th…

We use the explicit relation between genus filtrated $s$-loop means of the Gaussian matrix model and terms of the genus expansion of the Kontsevich--Penner matrix model (KPMM), which is the generating function for volumes of discretized (open) moduli spaces $M_{g,s}^{disc}$ (discrete volumes), to express Gaussian means…

Improved sample efficiency for private learning of Gaussian mixtures.

Proves existence and uniqueness of solutions to the Lp Gaussian Minkowski problem.

Market-based asset price probability depends on trade volumes and values, improving forecasts and reliability.

The article proves Randers Poincaré disc satisfies isoperimetric equality.

The abstract investigates how volume ratios relate to curvature in geometric surfaces.

In this pre-print we explore the multi-fractal properties of 1 minute traded volume of the equities which compose the Dow Jones 30. We also evaluate the weights of linear and non-linear dependences in the multi-fractal structure of the observable. Our results show that the multi-fractal nature of traded volume comes es…

The paper sets limits on the accuracy of macroeconomic forecasts based on statistical moments and trade volumes.

In this article, we study properly immersed complete noncompact submanifolds in a complete shrinking gradient Ricci soliton with weighted mean curvature vector bounded in norm. We prove that such a submanifold must have polynomial volume growth under some mild assumption on the potential function. On the other hand, if…

The daily volume of transaction on the New York Stock Exchange and its day-to-day fluctuations are analysed with respect to power-law tails as well long-term trends. We also model the transition to a Gaussian distribution for longer time intervals, like months instead of days.

The paper proves that Gaussian field critical points have finite moments.

Geometrically proves majorizing measure theorem on Hadamard manifolds.

We prove that the ring $\Aff{\R}{M}$ of all polynomials defined on a real algebraic variety $M\subset\R^n$ is dense in the Hilbert space $L^2(M,e^{-|x|^2}\deμ)$, where $\deμ$ denotes the volume form of $M$ and $\deν=e^{-|x|^2}\deμ$ the Gaussian measure on $M$.

The normal map given by Birkhoff orthogonality yields extensions of principal, Gaussian and mean curvatures to surfaces immersed in three-dimensional spaces whose geometry is given by an arbitrary norm and which are also called Minkowski spaces. We obtain characterizations of the Minkowski Gaussian curvature in terms o…

Solves a long-standing convex geometry problem about mixed volumes.

Model estimates lung well-aerated volume from CT images, independent of patient and imaging parameters.

We study stable smooth solutions to the isoperimetric type problem for a Gaussian weight on Euclidean Space. That is, we study hypersurfaces $Σ^n \subset \mathbb R^{n+1}$ that are second order stable critical points of compact variations that minimize Gaussian weighted area and preserve Gaussian weighted volume. We sho…

In this paper, we show that any ancient solution to the Ricci flow with the reduced volume whose asymptotic limit is sufficiently close to that of the Gaussian soliton is isometric to the Euclidean space for all time. This is a generalization of Anderson's result for Ricci-flat manifolds. As a corollary, a gap theorem …

We prove that an approximated version of the Brunn--Minkowski inequality with volume distortion coefficient implies a Gaussian concentration-of-measure phenomenon. Our main theorem is applicable to discrete spaces.

We determine the expected curvature polynomial of random real projective varieties given as the zero set of independent random polynomials with Gaussian distribution, whose distribution is invariant under the action of the orthogonal group. In particular, the expected Euler characteristic of such random real projective…

The study challenges the reliability of VaR due to market randomness.

Synthetic scalar curvature defined via Gaussian integrals, applied to manifolds and flows.

Gradient descent on LSE objectives implicitly performs EM, leading to collapse without volume control.

Using high-frequency time series of stock prices and share volumes sizes from January 2002-May 2009, this paper investigates whether the effects of the onset of high-frequency trading, most prominent since 2005, are apparent in the dynamics of the dollar traded volume. Indeed it is found in almost all of 14 heavily tra…

The hypercube's perimeter is significantly larger than expected near half volume.

A new distillation framework predicts stock trading volumes more accurately with less model size.

Three-candidate plurality voting is stable for small correlations.

Paper introduces PHI to identify structurally distinct payment patterns in UK municipal procurement.

The paper proves lower bounds for Gaussian-weighted curvature integrals of self-shrinkers.

For any closed Riemannian manifold $X$ we prove that large isoperimetric regions in $X\times{\mathbb R}^n$ are of the form $X\times$(Euclidean ball). We prove that if $X$ has non-negative Ricci curvature then the only soap bubbles enclosing a large volume are the products $X\times$(Euclidean sphere). We give an example…

Proposes a new model for traffic flow on directed graphs.

Study finds Calabi-Yau models' operator spectra match random matrix theory.

This is the fourth article of our series. Here, we study weighted norm inequalities for the Riesz transform of the Laplace-Beltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Gaussian upper bounds.

A novel probabilistic approach forecasts imbalance prices in Belgium.

The paper explores how market trade values and volumes affect price and return statistics.

This is first of series papers on new two-side Gaussian bounds for the heat kernel $H(x,y,t)$ on a complete manifold $(M,g)$. In this paper, on a complete manifold $M$ with $Ric(M)\geq 0$, we obtain new two-side Gaussian bounds for the heat kernel $H(x,y,t)$, which improve the well-known Li-Yau's two-side bounds. As ap…

The paper proves that symmetric sets with minimal Gaussian surface area are nearly convex cylinders.

New weighted surface area measures for convex bodies with applications.