arXiv research

Real-time fall detection system using DVS-TN on low-computing hardware.

Decision tree predicts DV recidivism with interpretable models.

Paper proposes a framework to analyze DV on social media.

Adaptive sampling improves graph diffusion models by maintaining uniform information speed.

By improving the analysis developed in the study of $\s_k$-Yamabe problem, we prove in this paper that the De Lellis-Topping inequality is true on 3-dimensional Riemannian manifolds of nonnegative scalar curvature. More precisely, if $(M^3, g)$ is a 3-dimensional closed Riemannian manifold with non-negative scalar curv…

In this note, we compute the second variational formula for the functional $\int_M v^{(6)}(g)dv_g$, which was introduced by Graham-Juhl and the first variational formula was obtained by Chang-Fang. We also prove that Einstein manifolds (with dimension $\ge 7$) with positive scalar curvature is a strict local maximum wi…

We consider the Riemannian functional defined on the space of Riemannian metrics with unit volume on a closed smooth manifold $M$ given by $\mathcal{R}_{\frac{n}{2}}(g):= \int_M |R(g)|^{\frac{n}{2}}dv_g$ where $R(g)$, $dv_g$ denote the Riemannian curvature and volume form corresponding to $g$. We show that there are lo…

Given a compact Riemannian Manifold (M,g) of dimension n > 2, a point x_0 in M and s in (0,2). We let 2*(s) = 2(n-s)/(n-2) be the critical Hardy-Sobolev exponent. The Hardy-Sobolev embedding yields the existence of A,B > 0 such that (\int_M|u|^{2*(s)}dv_g)^{2/2*(s)} \leq A\int_M |\nabla u|_g^2 dv_g +B\int_M u^2 dv_g fo…

Sharp constants in curl-Sobolev inequalities on spheres determined.

Machine learning improves design verification, achieving better coverage than random methods.

The paper proves new rigidity results for critical metrics of quadratic curvature functionals.

We show that a complete $m$-dimensional immersed submanifold $M$ of $\mathbb{R}^{n}$ with $a(M)<1$ is properly immersed and have finite topology, where $a(M)\in [0,\infty]$ is an scaling invariant number that gives the rate that the norm of the second fundamental form decays to zero at infinity. The class of submanifol…

We consider polyharmonic maps $φ:(M,g)\rightarrow$\mathbb{E}^n$of order k from a complete Riemannian manifold into the Euclidean space and let$p$be a real constant satisfying$1<p<\infty$. (i) If,$\int_M|W^{k-1}|^p dv_g<\infty,$and$\int_M|\bar \nabla W^{k-2}|^2dv_g<\infty.$Then$φ$ is a polyharmonic map of orde…

We consider a complete noncompact smooth metric measure space $(M^n,g,e^{-f} dv)$ and the associated drifting Laplacian. We find sufficient conditions on the geometry of the space so that every nonnegative $f$-subharmonic function with bounded weighted $L^1$ norm is constant.

SLAYER improves SNN training by backpropagating spike errors.

Constructs a moment map for maps to balanced manifolds.

Efficient SNN on Loihi achieves high gesture recognition accuracy.

The study proves a compactness theorem for manifolds with specific curvature conditions.

We consider a complete biharmonic hypersurface with nowhere zero mean curvature vector field $φ:(M^m,g)\rightarrow (S^{m+1},h)$ in a sphere. If the squared norm of the second fundamental form $B$ is bounded from above by m, and $\int_M H^{- p }dv_g<\infty$, for some $0<p<\infty$, then the mean curvature is constant.

The present note deals with the properties of metric connections $\nabla$ with vectorial torsion $V$ on semi-Riemannian manifolds $(M^n,g)$. We show that the $\nabla$-curvature is symmetric if and only if $V^{\flat}$ is closed, and that $V^\perp$ then defines an $(n-1)$-dimensional integrable distribution on $M^n$. If …

I In this paper, first we study a complete smooth metric measure space $(M^n,g, e^{-f}dv)$ with the ($\infty$)-Bakry-Émery Ricci curvature $\textrm{Ric}_f\ge \frac a2g$ for some positive constant $a$. It is known that the spectrum of the drifted Laplacian $Δ_f$ for $M$ is discrete and the first nonzero eigenvalue of $Δ…

Liouville property proven for certain harmonic functions on metric measure spaces.

New insights link RLHF and contrastive learning for better model alignment.

This paper relates the boundary term in the Chern-Gauss-Bonnet formula on 4-manifolds M with the renormalized volume V, as defined in the AdS/CFT correspondence, for asymptotically hyperbolic Einstein metrics on M. In addition, we compute and discuss the differential or variation dV of V, or equivalently the variation …

Computes a new metric quantity Y(M) for Riemannian 2d-manifolds.

New rigidity results for critical metrics of a quadratic curvature functional.

In this short paper we study $L_f^p$-Liouville property with $0<p<1$ for nonnegative $f$-subharmonic functions on a complete noncompact smooth metric measure space $(M,g,e^{-f}dv)$ with $\mathrm{Ric}_f^m$ bounded below for $0<m\leq\infty$. We prove a sharp $L_f^p$-Liouville theorem when $0<m<\infty$. We also prove an $…

Ricci flow preserves standard sphere's curvature for certain conditions.

We consider the Riemannian functional defined on the space of Riemannian metrics with unit volume on a closed smooth manifold M given by $R_p(g) :=\int_M|R(g)|^pdvg$ where $R(g)$, $dv_g$ denote the corresponding Riemannian curvature, volume form and p is a real number greater than or equal to 2. We prove that $R_p$ res…

The Bakry-Emery Ricci tensor of a metric-measure space (M,g,e^{-f}dv_{g}) plays an important role in both geometric measure theory and the study of Hamilton's Ricci flow. Under a uniform positivity condition on this tensor and with bounded Ricci curvature we show the underlying space has finite f-volume. As a consequen…

We consider a complete biharmonic submanifold $φ:(M,g)\rightarrow (N,h)$ in a Riemannian manifold with sectional curvature bounded from above by a non-negative constant $c$. Assume that the mean curvature is bounded from below by $\sqrt c$. If (i) $\int_M (|{\bf H}|^2-c)^{p}dv_g<\infty$, for some $0<p<\infty$, or (ii) …

An isometric immersion $x:M^n\rightarrow S^{n+p}$ is called Willmore if it is an extremal submanifold of the Willmore functional: $W(x)=\int_{M^n} (S-nH^2)^{\frac{n}{2}}dv$, where $S$ is the norm square of the second fundamental form and $H$ is the mean curvature. Examples of Willmore submanifolds in the unit sphere ar…

We study some function-theoretic properties on a complete smooth metric measure space $(M,g,e^{-f}dv)$ with Bakry-Émery Ricci curvature bounded from below. We derive a Moser's parabolic Harnack inequality for the $f$-heat equation, which leads to upper and lower Gaussian bounds on the $f$-heat kernel. We also prove $L^…

The paper classifies hypersurfaces in Spin$^c$ manifolds that satisfy a specific inequality.

Study solves Yamabe problems on metric measure spaces with or without boundary.

The study defines norming sets for holomorphic sections on complex manifolds.

We derive a Harnack inequality for positive solutions of the $f$-heat equation and Gaussian upper and lower bounds for the $f$-heat kernel on complete smooth metric measure spaces $(M, g, e^{-f}dv)$ with Bakry-Émery Ricci curvature bounded below. The lower bound is sharp. The main argument is the De Giorgi-Nash-Moser t…

We derive a local Gaussian upper bound for the $f$-heat kernel on complete smooth metric measure space $(M,g,e^{-f}dv)$ with nonnegative Bakry-Émery Ricci curvature, which generalizes the classic Li-Yau estimate. As applications, we obtain a sharp $L_f^1$-Liouville theorem for $f$-subharmonic functions and an $L_f^1$-u…

In this note we prove a new ε-regularity theorem for the Ricci flow. Let (M^n,g(t)) with t\in [-T,0] be a Ricci flow and H_{x} the conjugate heat kernel centered at a point (x,0) in the final time slice. Substituting H_{x} into Perelman's W-functional produces a monotone function W_{x}(s) of s \in [-T,0], the pointed e…

New moving average adapts weight dynamically based on polynomial and wavefunction.

We study both function theoretic and spectral properties on complete noncompact smooth metric measure space $(M,g,e^{-f}dv)$ with nonnegative Bakry-Émery Ricci curvature. Among other things, we derive a gradient estimate for positive $f$-harmonic functions and obtain as a consequence the strong Liouville property under…

The paper provides gradient estimates for specific evolution equations on metric measure spaces.

The paper estimates gradients for a weighted parabolic equation under geometric flow.

Let $(M,g)$ be a simple Riemannian manifold with boundary and consider the geodesic ray transform of symmetric 2-tensor fields. Let the integral of $f$ along maximal geodesics vanish on an appropriate open subset of the space of geodesics in $M$. Under the assumption that the metric $g$ is real-analytic, it is shown th…

We show that a diffeological bundle gives rise to an exact sequence of internal tangent spaces. We then introduce two new classes of diffeological spaces, which we call weakly filtered and filtered diffeological spaces, whose tangent spaces are easier to understand. These are the diffeological spaces whose categories o…

The paper derives gradient estimates for porous medium and fast diffusion equations on metric measure spaces.

The paper derives estimates and proves theorems for a specific type of nonlinear parabolic equation.

LAVA values data without needing a specific learning algorithm.