We establish an improved Sobolev trace inequality of order two in the Euclidean unit ball under the vanishing of higher order moments of the boundary volume element, and construct precise test functions to show that such inequalities are almost optim
al. Our arguments can be adapted to the fourth order Sobolev trace inequalities in higher dimensional unit ball.
Multi-view clustering (MVC) has been extensively studied to collect multiple source information in recent years. One typical type of MVC methods is based on matrix factorization to effectively perform dimension reduction and clustering. However, the
existing approaches can be further improved with following considerations: i) The current one-layer matrix factorization framework cannot fully exploit the useful data representations. ii) Most algorithms only focus on the shared information while ignore the view-specific structure leading to suboptimal solutions. iii) The partition level information has not been utilized in existing work. To solve the above issues, we propose a novel multi-view clustering algorithm via deep matrix decomposition and partition alignment. To be specific, the partition representations of each view are obtained through deep matrix decomposition, and then are jointly utilized with the optimal partition representation for fusing multi-view information. Finally, an alternating optimization algorithm is developed to solve the optimization problem with proven convergence. The comprehensive experimental results conducted on six benchmark multi-view datasets clearly demonstrates the effectiveness of the proposed algorithm against the SOTA methods.
We present a novel Relightable Neural Renderer (RNR) for simultaneous view synthesis and relighting using multi-view image inputs. Existing neural rendering (NR) does not explicitly model the physical rendering process and hence has limited capabilit
ies on relighting. RNR instead models image formation in terms of environment lighting, object intrinsic attributes, and light transport function (LTF), each corresponding to a learnable component. In particular, the incorporation of a physically based rendering process not only enables relighting but also improves the quality of view synthesis. Comprehensive experiments on synthetic and real data show that RNR provides a practical and effective solution for conducting free-viewpoint relighting.
Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n$ with smooth boundary $partial M$, admitting a scalar-flat conformal metric. We prove that the supremum of the isoperimetric ratio over the scalar-flat conformal class is strictly la
rger than the best constant of the isoperimetric inequality in the Euclidean space, and consequently is achieved, if either (i) $9le nle 11$ and $partial M$ has a nonumbilic point; or (ii) $7le nle 9$, $partial M$ is umbilic and the Weyl tensor does not vanish identically on the boundary. This is a continuation of the work cite{Jin-Xiong} by the second named author and Xiong.
We first present a warped product manifold with boundary to show the non-uniqueness of the positive constant scalar curvature and positive constant boundary mean curvature equation. Next, we construct a smooth counterexample to show that the compactn
ess of the set of lower energy solutions to the above equation fails when the dimension of the manifold is not less than $62$.
We present a study of the transport properties of thermally generated spin currents in an insulating ferrimagnetic-antiferromagnetic-ferrimagnetic trilayer over a wide range of temperature. Spin currents generated by the spin Seebeck effect (SSE) in
a yttrium iron garnet (YIG) YIG/NiO/YIG trilayer on a gadolinium gallium garnet (GGG) substrate were detected using the inverse spin Hall effect in Pt. By studying samples with different NiO thicknesses, the NiO spin diffusion length was determined to be 4.2 nm at room temperature. Interestingly, below 30 K, the inverse spin Hall signals are associated with the GGG substrate. The field dependence of the signal follows a Brillouin function for a S=7/2 spin ($mathrm{Gd^{3+}}$) at low temperature. Sharp changes in the SSE signal at low fields are due to switching of the YIG magnetization. A broad peak in the SSE response was observed around 100 K, which we associate with an increase in the spin-diffusion length in YIG. These observations are important in understanding the generation and transport properties of spin currents through magnetic insulators and the role of a paramagnetic substrate in spin current generation.
We study a fractional conformal curvature flow on the standard unit sphere and prove a perturbation result of the fractional Nirenberg problem with fractional exponent $sigma in (1/2,1)$. This extends the result of Chen-Xu (Invent. Math. 187, no. 2,
395-506, 2012) for the scalar curvature flow on the standard unit sphere.
The Han-Li conjecture states that: Let $(M,g_0)$ be an $n$-dimensional $(ngeq 3)$ smooth compact Riemannian manifold with boundary having positive (generalized) Yamabe constant and $c$ be any real number, then there exists a conformal metric of $g_0$
with scalar curvature $1$ and boundary mean curvature $c$. Combining with Z. C. Han and Y. Y. Lis results, we answer this conjecture affirmatively except for the case that $ngeq 8$, the boundary is umbilic, the Weyl tensor of $M$ vanishes on the boundary and has a non-zero interior point.
We study the problem of deforming a Riemannian metric to a conformal one with nonzero constant scalar curvature and nonzero constant boundary mean curvature on a compact manifold of dimension $ngeq 3$. We prove the existence of such conformal metrics
in the cases of $n=6,7$ or the manifold is spin and some other remaining ones left by Escobar. Furthermore, in the positive Yamabe constant case, by normalizing the scalar curvature to be $1$, there exists a sequence of conformal metrics such that their constant boundary mean curvatures go to $+infty$.
We study analysis aspects of the sixth order GJMS operator $P_g^6$. Under conformal normal coordinates around a point, the expansions of Greens function of $P_g^6$ with pole at this point are presented. As a starting point of the study of $P_g^6$, we
manage to give some existence results of prescribed $Q$-curvature problem on Einstein manifolds. One among them is that for $n geq 10$, let $(M^n,g)$ be a closed Einstein manifold of positive scalar curvature and $f$ a smooth positive function in $M$. If the Weyl tensor is nonzero at a maximum point of $f$ and $f$ satisfies a vanishing order condition at this maximum point, then there exists a conformal metric $tilde g$ of $g$ such that its $Q$-curvature $Q_{tilde g}^6$ equals $f$.
