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11.
Let $M$ be a compact immersed submanifold of the Euclidean space, the hyperbolic space or the standard sphere. For any continuous
potential q on M, we give a sharp upper bound for the second eigenvalue of the operator −Δ+q in terms of the total mean curvature of M and the mean value of q. Moreover, we analyze the case where this bound is achieved. As a consequence of this result we obtain an alternative proof
for the Alikakos–Fusco conjecture concerning the stability of the interface in the Allen–Cahn reaction diffusion model.
Received: 18 June 1999 / Accepted: 6 July 1999 相似文献
12.
We study a variant of the sparse PCA (principal component analysis) problem in the “hard” regime, where the inference task is possible yet no polynomial-time algorithm is known to exist. Prior work, based on the low-degree likelihood ratio, has conjectured a precise expression for the best possible (subexponential) runtime throughout the hard regime. Following instead a statistical physics-inspired point of view, we show bounds on the depth of free energy wells for various Gibbs measures naturally associated to the problem. These free energy wells imply hitting time lower bounds that corroborate the low-degree conjecture: we show that a class of natural MCMC (Markov chain Monte Carlo) methods (with worst-case initialization) cannot solve sparse PCA with less than the conjectured runtime. These lower bounds apply to a wide range of values for two tuning parameters: temperature and sparsity misparametrization. Finally, we prove that the overlap gap property (OGP), a structural property that implies failure of certain local search algorithms, holds in a significant part of the hard regime. © 2022 Wiley Periodicals, Inc. 相似文献
13.
Let M be a compact submanifold with boundary of a Euclidean space or a Sphere. In this paper, we derive an upper bound for the first non-zero eigenvalue p1 of Steklov problem on M in terms of the r-th mean curvatures of its boundary ∂M. The upper bound obtained is sharp. 相似文献
14.
In this paper, we obtain a new abstract formula relating eigenvalues of a self-adjoint operator to two families of symmetric and skew-symmetric operators and their commutators. This formula generalizes earlier ones obtained by Harrell, Stubbe, Hook, Ashbaugh, Hermi, Levitin and Parnovski. We also show how one can use this abstract formulation both for giving different and simpler proofs for all the known results obtained for the eigenvalues of a power of the Laplace operator (i.e. the Dirichlet Laplacian, the clamped plate problem for the bilaplacian and more generally for the polyharmonic problem on a bounded Euclidean domain) and to obtain new ones. In a last paragraph, we derive new bounds for eigenvalues of any power of the Kohn Laplacian on the Heisenberg group. 相似文献
15.
In this paper, we investigate critical points of the eigenvalues of the Laplace operator considered as functionals on the space of Riemannian metrics or a conformal class of metrics on a compact manifold. We introduce a natural notion of the critical metric of such a functional and obtain necessary and sufficient conditions for a metric to be critical. We derive specific consequences concerning possible locally maximizing metrics. We also characterize critical metrics of the ratio of two consecutive eigenvalues. 相似文献
16.
Compression of periodic complementary sequences and applications 总被引:1,自引:0,他引:1
17.
Noncovalent Interactions of π Systems with Sulfur: The Atomic Chameleon of Molecular Recognition
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Prof. William B. Motherwell Dr. Rafael B. Moreno Dr. Ilias Pavlakos Josephine R. T. Arendorf Dr. Tanzeel Arif Dr. Graham J. Tizzard Prof. Simon J. Coles Dr. Abil E. Aliev 《Angewandte Chemie (International ed. in English)》2018,57(5):1193-1198
The relative strength of noncovalent interactions between a thioether sulfur atom and various π systems in designed top pan molecular balances was determined by NMR spectroscopy. Compared to its oxygen counterpart, the sulfur atom displays a remarkable ability to interact with almost equal facility over the entire range of π systems studied, with the simple alkene emerging as the most powerful partner. With the exception of the O???heteroarene interaction, all noncovalent interactions of sulfur with π systems are favoured over oxygen. 相似文献
18.
Ilias M. Mahmud 《Tetrahedron》2008,64(50):11420-11432
A series of Fréchet-type dendron functionalized [60]fullerene derivatives that bear a 1,2,3-triazole linkage group, referred to as triazole-linked dendro[60]fullerenes, were prepared via a modular synthetic protocol based on a Cu-catalyzed [3+2] cycloaddition (‘click’) reaction. Electronic properties of these dendro[60]fullerenes were investigated by UV-vis spectroscopy and cyclic voltammetry. Interfacial supramolecular self-assembly behavior of these dendro[60]fullerenes was studied using atomic force microscopy (AFM). The resulting self-assemblies showed different nanoscale packing geometries and morphologies on the surface, which are controllable by parameters such as the generation of dendron, the nature of peripheral functionalities, and the experimental conditions (e.g., acidity) applied. Correlations between molecular structure and self-assembling outcome were surveyed and discussed. The results of this study suggest a new avenue to gain better ‘bottom-up’ control over the self-assembly of dendrimer-fullerene hybrid materials in terms of shape and size. 相似文献
19.
The vibronic and spin-orbit-induced interactions among the (3)Sigma(-), (1)Delta, and (1)Sigma(+) electronic states arising from a half-filled pi orbital of a linear triatomic molecule are considered, employing the microscopic (Breit-Pauli) spin-orbit coupling operator. The 6 x 6 Hamiltonian matrix is derived in a diabatic spin-orbital electronic basis set, including terms up to fourth order in the expansion of the molecular Hamiltonian in the bending normal coordinate about the linear geometry. The symmetry properties of the Hamiltonian are analyzed. Aside from the nonrelativistic fourth-order Renner-Teller vibronic coupling within the (1)Delta state and the second-order nonrelativistic vibronic coupling between the (1)Sigma(+) and (1)Delta states, there exist zeroth-order, first-order, as well as third-order vibronic coupling terms of spin-orbit origin. The latter are absent when the phenomenological expression for the spin-orbit coupling operator is used instead of the microscopic form. The effects of the nonrelativistic and spin-orbit-induced vibronic coupling mechanisms on the (3)Sigma(-), (1)Delta, and (1)Sigma(+) adiabatic potential energy surfaces as well as on the spin-vibronic energy levels are discussed for selected parameter values. 相似文献
20.
Weighing Matrices and String Sorting 总被引:1,自引:0,他引:1
Ilias S. Kotsireas Christos Koukouvinos Jennifer Seberry 《Annals of Combinatorics》2009,13(3):305-313
In this paper we establish a fundamental link between the search for weighing matrices constructed from two circulants and
the operation of sorting strings, an operation that has been studied extensively in computer science. In particular, we demonstrate
that the search for weighing matrices constructed from two circulants using the power spectral density criterion and exploiting
structural patterns for the locations of the zeros in candidate solutions, can be viewed as a string sorting problem together
with a linear time algorithm to locate common strings in two sorted arrays. This allows us to bring into bear efficient algorithms
from the string sorting literature. We also state and prove some new enhancements to the power spectral density criterion,
that allow us to treat successfully the rounding error effect and speed up the algorithm. Finally, we use these ideas to find
new weighing matrices of order 2n and weights 2n – 13, 2n – 17 constructed from two circulants. 相似文献