8,893 research outputs found
Eigenvalues of the Laplacian on Riemannian manifolds
For a bounded domain with a piecewise smooth boundary in a complete
Riemannian manifold , we study eigenvalues of the Dirichlet eigenvalue
problem of the Laplacian. By making use of a fact that eigenfunctions form an
orthonormal basis of in place of the Rayleigh-Ritz formula, we
obtain inequalities for eigenvalues of the Laplacian. In particular, for lower
order eigenvalues, our results extend the results of Chen and Cheng \cite{CC}.Comment: 17 page
Relationships Between Generalized Bernoulli Numbers and Polynomials and Generalized Euler Numbers and Polynomials
In this paper, concepts of the generalized Bernoulli and Euler numbers and polynomials are introduced, and some relationships between them are established
Pulse Width Modulation for Speeding Up Quantum Optimal Control Design
This paper focuses on accelerating quantum optimal control design for complex
quantum systems. Based on our previous work [{arXiv:1607.04054}], we combine
Pulse Width Modulation (PWM) and gradient descent algorithm into solving
quantum optimal control problems, which shows distinct improvement of
computational efficiency in various cases. To further apply this algorithm to
potential experiments, we also propose the smooth realization of the optimized
control solution, e.g. using Gaussian pulse train to replace rectangular
pulses. Based on the experimental data of the D-Norleucine molecule, we
numerically find optimal control functions in -qubit and -qubit systems,
and demonstrate its efficiency advantage compared with basic GRAPE algorithm
Bounds for eigenvalue ratios of the Laplacian
For a bounded domain with a piecewise smooth boundary in an
-dimensional Euclidean space , we study eigenvalues of the
Dirichlet eigenvalue problem of the Laplacian. First we give a general
inequality for eigenvalues of the Laplacian. As an application, we study lower
order eigenvalues of the Laplacian and derive the ratios of lower order
eigenvalues of the Laplacian.Comment: 14 page
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