Large language models (LLMs), endowed with exceptional reasoning capabilities, are adept at discerning profound user interests from historical behaviors, thereby presenting a promising avenue for the advancement of recommendation systems. However, notable discrepancy persists between sparse collaborative semantics typically found in systems and dense token representations within LLMs. In our study, we propose novel framework that harmoniously merges traditional prowess We initiate this...

Given a calibrated Riemannian manifold \overline{M} with parallel calibration \Omega of rank m and M an orientable m-submanifold mean curvature H , we prove that if \cos\theta is bounded away from zero, where \theta the -angle has zero Cheeger constant, then minimal. In particular case complete Ricci^M\geq 0 may replace boundedness condition on by \cos\theta\geq Cr^{-\beta} when r\rightarrow+\infty < \beta 1 C > are constants r distance function to point in . Our proof surprisingly...

This paper concerns closed hypersurfaces of dimension $n(\geq 2)$ in the hyperbolic space ${\mathbb{H}}_{\kappa}^{n+1}$ constant sectional curvature $\kappa$ evolving direction its normal vector, where speed is given by a power $\beta (\geq 1/m)$ $m$th mean plus volume preserving term, including case powers and $\mbox{Gaus}$ curvature. The main result that if initial hypersurface satisfies ratio biggest smallest principal close enough to 1 everywhere, depending only on $n$, $m$, $\beta$...

This paper concerns the evolution of a closed hypersurface hyperbolic space, convex by horospheres, in direction its inner unit normal vector, where speed equals positive power <TEX>${\beta}$</TEX> mean curvature. It is shown that flow exists on finite maximal interval, convexity horospheres preserved and hypersurfaces shrink down to single point as final time approached.

By studying the monotonicity of first nonzero eigenvalues Laplace and p-Laplace operators on a closed convex hypersurface $M^n$ which evolves under inverse mean curvature flow in $\mathbb{R}^{n+1}$, isoperimetric lower bounds for both were founded.

This paper concerns closed hypersurfaces of dimension <TEX>$n{\geq}2$</TEX> in the hyperbolic space <TEX>${\mathbb{H}}_{\kappa}^{n+1}$</TEX> constant sectional curvature evolving direction its normal vector, where speed equals a power <TEX>${\beta}{\geq}1$</TEX> mean curvature. The main result is that if initial closed, weakly h-convex hypersurface satisfies ratio biggest and smallest principal at everywhere close enough to 1, depending only on n <TEX>${\beta}$</TEX>, then under flow this...

In this paper, we study the eigenvalue problem of a system sub-elliptic equations on abounded domain in Heisenberg group and obtain some universal inequalities. Moreover, for lower order eigenvalues problem, also give

Recently, B. Y. Chen introduced a new invariant δ(n1,n2,…,nk) of Riemannian manifold and proved basic inequality between the extrinsic if, where H is mean curvature an immersion Mn in real space form Rm(ε) constant ε. He pointed out that such also holds for totally complex form. The called ideal (by Chen) if it satisfies equality case identically. In this paper we classify semi-parallel immersions Euclidean their normal bundle flat, prove every Lagrangian geodesic, moreover result...

Abstract An initial entire graph with bounded second fundamental form in

关键词 旋转对称空间 星形超曲面 Minkowski

We show the generalization of Hsiung-Minkowski integral formula for anisotropic capillary hypersurfaces in half-space, which includes weighted and classical Minkowski identity closed as special cases. As applications, we prove some Alexandrov-type theorems rigidity results hypersurfaces. Specially, uniqueness solution to Orlicz-Christoffel-Minkowski problem is obtained.

Abstract In this paper, we consider the closed spacelike solution to a class of Hessian quotient equations in de Sitter space. Under mild assumptions, obtain an existence result using standard degree theory based on priori estimates.