arXiv research

A new method for SSMF improves upon existing algorithms.

Derives a new maximum principle for Riemannian manifolds with volume growth constraints.

Proves a conjecture about the maximum tet-volume of triangulations of a 2-sphere.

The maximum hyperbolic polyhedron volume is found to be the rectification of its skeleton.

Econophysics and econometrics agree that there is a correlation between volume and volatility in a time series. Using empirical data and their distributions, we further investigate this correlation and discover new ways that volatility and volume interact, particularly when the levels of both are high. We find that the…

In this paper, we show that Gromov-Thurston's principle works for hyperbolic 3-manifolds of infinite volume and with finitely generated fundamental group. As an application, we have a new proof of Ending Lamination Theorem. Our proof essentially relays only on Maximum Volume Law for hyperbolic 3-simplices.

Weaving knots are alternating knots with the same projection as torus knots, and were conjectured by X.-S. Lin to be among the maximum volume knots for fixed crossing number. We provide the first asymptotically correct volume bounds for weaving knots, and we prove that the infinite weave is their geometric limit.

Study on isoperimetric problem in Randers planes achieving maximum area.

In this paper we present a proof of a Neumann type maximum principle for the Laplace operator on compact Riemannian manifolds. A key p oint is the simple geometric nature of the constant in the a priori estimate of this maximum principle. In particular, this maximum principle can be applied to manifolds with Ricci curv…

We propose an approach to find constant curvature metrics on triangulated closed 3-manifolds using a finite dimensional variational method whose energy function is the volume. The concept of an angle structure on a tetrahedron and on a triangulated closed 3-manifold is introduced following the work of Casson, Murakami …

New method speeds up deep neural networks inference.

Consider a structured matrix factorization model where one factor is restricted to have its columns lying in the unit simplex. This simplex-structured matrix factorization (SSMF) model and the associated factorization techniques have spurred much interest in research topics over different areas, such as hyperspectral u…

The paper introduces a new price model based on entropy that better fits high-frequency market data.

MaxVol NMF maximizes the volume of $H$ in NMF for better sparse and interpretable solutions.

Study shows spheres in high dimensions have maximum volume if they are smooth and have a specific reach.

How large can be the width of Riemannian three-spheres of the same volume in the same conformal class? If a maximum value is attained, how does a maximising metric look like? What happens as the conformal class changes? In this paper, we investigate these and other related questions, focusing on the context of Simon-Sm…

Given a compact Alexadrov $n$-space $Z$ with curvature curv $\ge κ$, and let $f: Z\to X$ be a distance non-increasing onto map to another Alexandrov $n$-space with curv $\ge κ$. The relative volume rigidity conjecture says that if $X$ achieves the relative maximal volume i.e. $vol(Z)=vol(X)$, then $X$ is isometric to $…

Method identifies regions of maximum dissimilarity in stochastic processes.

We study the asymptotic behavior of smooth, origin-symmetric, strictly convex bodies under the centro-affine normal flows. By means of a stability version of the Blaschke-Santaló inequality, we obtain regularity of the solutions provided that initial convex bodies have almost maximum Mahler volume. We prove that suitab…

We investigate the formation of singularities for surfaces evolving by volume preserving mean curvature flow. For axially symmetric flows - surfaces of revolution - in $\mathbb{R}^3$ with Neumann boundary conditions, we prove that the first developing singularity is of Type I. The result is obtained without any additio…

In this thesis we give a review on Ricci flow, an overview on Poincare conjecture, maximum principle, Li-Yau-Perelman estimate, Two functional F and W of Perelman, Reduced volume and reduced length and k-non collapsing estimate

Researchers calculate the volume of Seifert representations for graph manifolds and their covers.

The Schwarz--Pick lemma is a fundamental result in complex analysis. It is well-known that Yau generalized it to the higher dimensional manifolds by applying his maximum principle for complete Riemannian manifolds. Jeffres obtained Schwarz lemma for volume forms of conical Kähler metrics, based on a barrier function an…

We consider the volume-normalized Ricci flow close to compact shrinking Ricci solitons. We show that if a compact Ricci soliton $(M,g)$ is a local maximum of Perelman's shrinker entropy, any normalized Ricci flow starting close to it exists for all time and converges towards a Ricci soliton. If $g$ is not a local maxim…

We consider a convex Euclidean hypersurface that evolves by a volume or area preserving flow with speed given by a general nonhomogeneous function of the mean curvature. For a broad class of possible speed functions, we show that any closed convex hypersurface converges to a round sphere. The proof is based on the mono…

We obtain a maximum principle, and "a priori" upper estimates for solutions of a class of non linear singular elliptic differential inequalities on Riemannian manifolds under the sole geometrical assumption of volume growth conditions. Various applications of the results obtained are presented.

It is our purpose to study complete self-shrinkers in Euclidean space. First of all, we show some examples of complete self-shrinkers without polynomial volume growth. By making use of the generalized maximum principle for $\mathcal{L}$-operator, we give a complete classification for 2-dimensional complete self-shrinke…

We prove that the first positive eigenvalue, normalized by the volume, of the sub-Laplacian associated with a strictly pseudoconvex pseudo-Hermitian structure $\θ$ on the CR sphere S 2n+1 $\subset$ C n+1 , achieves its maximum when $\θ$ is the standard contact form.

Graphs with nonnegative Bakry-Émery curvature have volume doubling and Poincaré inequalities.

Proves necessity of at least log2(n) layers to compute maximum of n numbers.

For a non-vanishing gradient-like vector field on a compact manifold $X^{n+1}$ with boundary, a discrete set of trajectories may be tangent to the boundary with reduced multiplicity $n$, which is the maximum possible. (Among them are trajectories that are tangent to $\partial X$ exactly $n$ times.) We prove a lower bou…

Some optimization problems coming from the Differential Geometry, as for example, the minimal submanifolds problem and the harmonic maps problem are solved here via interior solutions of appropriate multitime optimal control problems. Section 1 underlines some science domains where appear multitime optimal control prob…

We prove the existence of extremal domains for the first eigenvalue of the Laplace-Beltrami operator in some compact Riemannian manifolds of dimension $n \geq 2$, with volume close to the volume of the manifold. If the first (positive) eigenfunction $φ_0$ of the Laplace-Beltrami operator over the manifold is a nonconst…

It is our purpose to study complete self-shrinkers in Euclidean space. By introducing a generalized maximum principle for $\mathcal{L}$-operator, we give estimates on supremum and infimum of the squared norm of the second fundamental form of self-shrinkers without assumption on \emph{polynomial volume growth}, which is…

Given the fundamental group $Γ$ of a finite-volume complete hyperbolic $3$-manifold $M$, it is possible to associate to any representation $ρ:Γ\rightarrow \text{Isom}(\mathbb{H}^3)$ a numerical invariant called volume. This invariant is bounded by the hyperbolic volume of $M$ and satisfies a rigidity condition: if the …

We study some basic problems of translating solitons: the volume growth, generalized maximum principle, Gauss maps and certain functions related to the Gauss maps, finally we carry out point-wise estimates and integral estimates for the squared norm of the second fundamental form. Those estimates give rigidity theorems…

The paper modifies asset pricing models using Taylor series expansions and market-based averages.

Using an idea of Voronoi in the geometric theory of positive definite quadratic forms, we give a transparent proof of John's characterization of the unique ellipsoid of maximum volume contained in a convex body. The same idea applies to the 'hard part' of a generalization of John's theorem and shows the difficulties of…

Consider vector valued harmonic maps of at most linear growth, defined on a complete non-compact Riemannian manifold with non-negative Ricci curvature. For the norm square of the pull-back of the target volume form by such maps, we report a strong maximum principle, and equalities among its supremum, its asymptotic ave…

The paper investigates topic models, ensuring their statistical identifiability and accuracy.

Time and Sales of corn futures traded electronically on the CME Group Globex are studied. Theories of continuous prices turn upside down reality of intra-day trading. Prices and their increments are discrete and obey lattice probability distributions. A function for systematic evolution of futures trading volume is pro…

Geodesic balls in a simply connected space forms $\mathbb{S}^n$, $\mathbb{R}^{n}$ or $\mathbb{H}^{n}$ are distinguished manifolds for comparison in bounded Riemannian geometry. In this paper we show that they have the maximum possible boundary volume among Miao-Tam critical metrics with connected boundary provided that…

This paper introduces new invariants for time series analysis.

We introduce a natural extension of the concept of gradient Ricci soliton: the Ricci almost soliton. We provide existence and rigidity results, we deduce a-priori curvature estimates and isolation phenomena, and we investigate some topological properties. A number of differential identities involving the relevant geome…

We study a volume/area preserving curvature flow of hypersurfaces that are convex by horospheres in the hyperbolic space, with velocity given by a generic positive, increasing function of the mean curvature, not necessarly homogeneous. For this class of speeds we prove the exponential convergence to a geodesic sphere. …

Study on ratio of intrinsic to extrinsic metrics and its relation to surface area.

In this paper we extend to non-compact Riemannian manifolds with boundary the use of two important tools in the geometric analysis of compact spaces, namely, the weak maximum principle for subharmonic functions and the integration by parts. The first one is a new form of the classical Ahlfors maximum principle whereas …

Since $n$-dimensional $λ$-hypersurfaces in the Euclidean space $\mathbb {R}^{n+1}$ are critical points of the weighted area functional for the weighted volume-preserving variations, in this paper, we study the rigidity properties of complete $λ$-hypersurfaces. We give a gap theorem of complete $λ$-hypersurfaces with po…