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We study the asymptotic behaviour of$N$(α)—the number of negative eigenvalues of the operator (-τ)^{$l$}-α$V$in$L$_{2}($R$^{$d$}) for an even$d$and$2l≥d$. This is the only case where the previously known results were far from being complete. In order to describe our results we introduce an auxiliary ordinary differential operator (system) on the semiaxis. Depending on the spectral properties of this operator we can distinguish between three cases where$N$(α) is of the Weyl-type,$N$(α) is of the Weyl-order but not the Weyl-type coefficient and finally where$N$(α)=$O$(α^{q}) with$q>d/2l$.

We study derived categories arising from quivers with potential associated to a decorated marked surface S_Delta, in the sense taken in a paper by Qiu. We prove two conjectures from Qiu’s paper in which, under a bijection between certain objects in these categories and certain arcs in S, the dimensions of morphisms between these objects equal the intersection numbers between the corresponding arcs.

Motivated by the sampling problems and heterogeneity issues common in high-dimensional big datasets, we consider a class of discordant additive index models. We propose method of moments based procedures for estimating the indices of such discordant additive index models in both low and high-dimensional settings. Our estimators are based on factorizing certain moment tensors and are also applicable in the overcomplete setting, where the number of indices is more than the dimensionality of the datasets. Furthermore, we provide rates of convergence of our estimator in both high and low-dimensional setting. Establishing such results requires deriving tensor operator norm concentration inequalities that might be of independent interest. Finally, we provide simulation results supporting our theory. Our contributions extend the applicability of tensor methods for novel models in addition to making progress on understanding theoretical properties of such tensor methods.

1. Statement of results Let Mn be a compact manifold of dimension n endowed with a Riemannian metric. The spectrum of the Laplacian,, acting on functions form a discrete set of the form {0< 1 2 k}. In 1970, Joseph Hersch [5] gave a sharp upper bound for the first non-zero eigenvalue 1 for any Riemannian metric on the 2-sphere in terms of its volume alone. Similar estimates for 1 on any compact oriented surfaces were derived by Yang-Yau [7]. The second and the third authors [6] studied the nonorientable surfaces and pointed out the relationship of 1 and the conformal class of the surface. In fact, their estimates were applied to study the Willmore problem. Another application of these types of upper bounds was found by Choi-Schoen [3] in relation to the set of all minimal surfaces in a compact 3-manifold of positive Ricci curvature. The purpose of this paper is to prove a higher dimensional generalization of the above results. It was pointed out by Marcel Berger [1] that Herschs theorem fails in higher dimensional spheres. In view of the relationship between 1 and the conformal