scholarly journals A Survey on Extremal Problems of Eigenvalues

2012 ◽  
Vol 2012 ◽  
pp. 1-26 ◽  
Author(s):  
Ping Yan ◽  
Meirong Zhang
Keyword(s):  
Variational Method ◽  
Lebesgue Space ◽  
Upper Bounds ◽  
Extremal Problems ◽  
Lower And Upper Bounds ◽  
Open Problems ◽  
One Dimensional ◽  
Sturm Liouville ◽  
The One ◽  
Neumann Eigenvalues

Given an integrable potentialq∈L1([0,1],ℝ), the Dirichlet and the Neumann eigenvaluesλnD(q)andλnN(q)of the Sturm-Liouville operator with the potentialqare defined in an implicit way. In recent years, the authors and their collaborators have solved some basic extremal problems concerning these eigenvalues when theL1metric forqis given;∥q∥L1=r. Note that theL1spheres andL1balls are nonsmooth, noncompact domains of the Lebesgue space(L1([0,1],ℝ),∥·∥L1). To solve these extremal problems, we will reveal some deep results on the dependence of eigenvalues on potentials. Moreover, the variational method for the approximating extremal problems on the balls of the spacesLα([0,1],ℝ),1<α<∞will be used. Then theL1problems will be solved by passingα↓1. Corresponding extremal problems for eigenvalues of the one-dimensionalp-Laplacian with integrable potentials have also been solved. The results can yield optimal lower and upper bounds for these eigenvalues. This paper will review the most important ideas and techniques in solving these difficult and interesting extremal problems. Some open problems will also be imposed.

scholarly journals Variational approach to the Schrödinger equation with a delta-function potential

2021 ◽  
Author(s):  
Francisco Marcelo Fernandez
Keyword(s):  
Variational Method ◽  
Delta Function ◽  
Basis Set ◽  
One Dimensional ◽  
Dirac Delta ◽  
Introductory Course ◽  
Relevant Role ◽  
Secular Equations ◽  
The One ◽  
Computer Algebra Software

Abstract We obtain accurate eigenvalues of the one-dimensional Schr\"{o}dinger equation with a Hamiltonian of the form $H_{g}=H+g\delta (x)$, where $\delta (x)$ is the Dirac delta function. We show that the well known Rayleigh-Ritz variational method is a suitable approach provided that the basis set takes into account the effect of the Dirac delta on the wavefunction. Present analysis may be suitable for an introductory course on quantum mechanics to illustrate the application of the Rayleigh-Ritz variational method to a problem where the boundary conditions play a relevant role and have to be introduced carefully into the trial function. Besides, the examples are suitable for motivating the students to resort to any computer-algebra software in order to calculate the required integrals and solve the secular equations.

scholarly journals Tiling the Line with Triples

2001 ◽  
Vol DMTCS Proceedings vol. AA,... (Proceedings) ◽  
Author(s):  
Aaron Meyerowitz
Keyword(s):  
Greedy Algorithm ◽  
Direct Proof ◽  
Open Problems ◽  
One Dimensional ◽  
Proof By Contradiction ◽  
International Audience ◽  
The Subject ◽  
The One ◽  
Finite Digraph ◽  
Elegant Proof

International audience It is known the one dimensional prototile $0,a,a+b$ and its reflection $0,b,a+b$ always tile some interval. The subject has not received a great deal of further attention, although many interesting questions exist. All the information about tilings can be encoded in a finite digraph $D_{ab}$. We present several results about cycles and other structures in this graph. A number of conjectures and open problems are given.In [Go] an elegant proof by contradiction shows that a greedy algorithm will produce an interval tiling. We show that the process of converting to a direct proof leads to much stronger results.

scholarly journals A Numerical Variational Method for Calculating Accurate Vibrational Energy Separations of Small Molecules and Their Ions

1986 ◽  
Vol 39 (5) ◽  
pp. 749 ◽  
Author(s):  
G Doherty ◽  
MJ Hamilton ◽  
PG Burton ◽  
EI von Nagy-Felsobuki
Keyword(s):  
Finite Element ◽  
Variational Method ◽  
Small Molecules ◽  
Vibrational Energy ◽  
Basis Functions ◽  
Variational Procedure ◽  
One Dimensional ◽  
Trial Wavefunction ◽  
Vibrational Energies ◽  
The One

A combination of known methods have been spliced together in order to calculate accurate vibrational energies and wavefunctions. The algorithm is based on the Rayleigh-Ritz variational procedure in which the trial wavefunction is a linear combination of configuration products of one-dimensional basis functions. The Hamiltonian is that due to Carney and Porter (1976). The kernel of the algorithm consists o( the one-dimensional basis functions, which are the finite element solutions of the associated one-dimensional problems.

scholarly journals Generalization of the Hardy-Littlewood theorem on Fourier series

2021 ◽  
Vol 104 (4) ◽  
pp. 49-55
Author(s):  
S. Bitimkhan ◽  
Keyword(s):  
Fourier Series ◽  
Trigonometric Series ◽  
Lebesgue Space ◽  
Fourier Coefficients ◽  
Double Fourier Series ◽  
Function Spaces ◽  
Dimensional Case ◽  
One Dimensional ◽  
Littlewood Theorem ◽  
The One

In the theory of one-dimensional trigonometric series, the Hardy-Littlewood theorem on Fourier series with monotone Fourier coefficients is of great importance. Multidimensional versions of this theorem have been extensively studied for the Lebesgue space. Significant differences of the multidimensional variants in comparison with the one-dimensional case are revealed and the strengthening of this theorem is obtained. The Hardy-Littlewood theorem is also generalized for various function spaces and various types of monotonicity of the series coefficients. Some of these generalizations can be seen in works of M.F. Timan, M.I. Dyachenko, E.D. Nursultanov, S. Tikhonov. In this paper, a generalization of the Hardy-Littlewood theorem for double Fourier series of a function in the space L_qϕ(L_q)(0,2π]^2 is obtained.

scholarly journals Rayleigh-Ritz variational method with suitable asymptotic behaviour

Open Physics ◽  
2014 ◽  
Vol 12 (8) ◽  
Author(s):  
Francisco Fernández ◽  
Javier Garcia
Keyword(s):  
Variational Method ◽  
Asymptotic Behaviour ◽  
Orthogonal Polynomial ◽  
Rate Of Convergence ◽  
Orthogonal Basis ◽  
Basis Sets ◽  
One Dimensional ◽  
Variational Calculations ◽  
Dimensional Models ◽  
The One

AbstractThis paper considers the Rayleigh-Ritz variational calculations with non-orthogonal basis sets that exhibit the correct asymptotic behaviour. This approach is illustrated by constructing suitable basis sets for one-dimensional models such as the two double-well oscillators recently considered by other authors. The rate of convergence of the variational method proves to be considerably greater than the one exhibited by the recently developed orthogonal polynomial projection quantization.

scholarly journals Alternating and variable controls for the wave equation

2020 ◽  
Vol 26 ◽  
pp. 38 ◽  
Author(s):  
Antonio Agresti ◽  
Daniele Andreucci ◽  
Paola Loreti
Keyword(s):  
Wave Equation ◽  
Sufficient Conditions ◽  
Dimensional Case ◽  
Equivalent Condition ◽  
Open Problems ◽  
One Dimensional ◽  
Constant Rate ◽  
Exact Observability ◽  
Single Exchange ◽  
The One

The present article discusses the exact observability of the wave equation when the observation subset of the boundary is variable in time. In the one-dimensional case, we prove an equivalent condition for the exact observability, which takes into account only the location in time of the observation. To this end we use Fourier series. Then we investigate the two specific cases of single exchange of the control position, and of exchange at a constant rate. In the multi-dimensional case, we analyse sufficient conditions for the exact observability relying on the multiplier method. In the last section, the multi-dimensional results are applied to specific settings and some connections between the one and multi-dimensional case are discussed; furthermore some open problems are presented.

scholarly journals New upper bounds for noncentral chi-square cdf

2020 ◽  
pp. 70-74
Author(s):  
Valeriy A. Voloshko ◽  
Egor V. Vecherko
Keyword(s):  
Density Function ◽  
Upper Bounds ◽  
Cumulative Density Function ◽  
Coordinate Systems ◽  
Chi Square ◽  
One Dimensional ◽  
Chi Squared ◽  
Cartesian Coordinate ◽  
The One ◽  
Inverse Trigonometric Functions

Some new upper bounds for noncentral chi-square cumulative density function are derived from the basic symmetries of the multidimensional standard Gaussian distribution: unitary invariance, components independence in both polar and Cartesian coordinate systems. The proposed new bounds have analytically simple form compared to analogues available in the literature: they are based on combination of exponents, direct and inverse trigonometric functions, including hyperbolic ones, and the cdf of the one dimensional standard Gaussian law. These new bounds may be useful both in theory and in applications: for proving inequalities related to noncentral chi-square cumulative density function, and for bounding powers of Pearson’s chi-squared tests.

Are You Sure That This Happened? Assessing the Factuality Degree of Events in Text

2012 ◽  
Vol 38 (2) ◽  
pp. 261-299 ◽  
Author(s):  
Roser Saurí ◽  
James Pustejovsky
Keyword(s):  
Computational Model ◽  
Upper Bounds ◽  
Epistemic Modality ◽  
The Other ◽  
Proof Of Concept ◽  
Lower And Upper Bounds ◽  
Information Level ◽  
Two Measures ◽  
The One ◽  
Event Factuality

Identifying the veracity, or factuality, of event mentions in text is fundamental for reasoning about eventualities in discourse. Inferences derived from events judged as not having happened, or as being only possible, are different from those derived from events evaluated as factual. Event factuality involves two separate levels of information. On the one hand, it deals with polarity, which distinguishes between positive and negative instantiations of events. On the other, it has to do with degrees of certainty (e.g., possible, probable), an information level generally subsumed under the category of epistemic modality. This article aims at contributing to a better understanding of how event factuality is articulated in natural language. For that purpose, we put forward a linguistic-oriented computational model which has at its core an algorithm articulating the effect of factuality relations across levels of syntactic embedding. As a proof of concept, this model has been implemented in De Facto, a factuality profiler for eventualities mentioned in text, and tested against a corpus built specifically for the task, yielding an F1 of 0.70 (macro-averaging) and 0.80 (micro-averaging). These two measures mutually compensate for an over-emphasis present in the other (either on the lesser or greater populated categories), and can therefore be interpreted as the lower and upper bounds of the De Facto's performance.

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