Meaning of Kato's Formulas for Upper and Lower Bounds to Eigenvalues of Hermitian Operators

1971 ◽  
Vol 49 (2) ◽  
pp. 218-223 ◽  
Author(s):  
Dallas T. Hayes
Keyword(s):  
Eigenvalue Problem ◽  
Recent Work ◽  
Lower Bounds ◽  
Trial Function ◽  
Hermitian Operator ◽  
Upper And Lower Bounds ◽  
The One ◽  
Independent Derivation

Using an independent derivation by Kohn, the full meaning of Kato's formulas for upper and lower bounds to eigenvalues of a Hermitian operator is shown. These bounds are the best possible when the only information available on a particular eigenvalue problem is a suitable trial function and an estimate of the neighboring eigenvalues to the one in question. This was asserted by Kato but not proved. A comparison is made of Kato's bounds with those derived in papers by Stevenson and Crawford and by Cohen and Feldmann. Under the conditions which result in Kato's bounds it is shown that the Stevenson–Crawford and Cohen–Feldmann bounds reduce to those of Kato. When more information is available these bounds are an improvement upon Kato's. This makes more precise the recent work of Walmsley and Cohen–Feldmann, whose results appear to prove in general the greater accuracy of the Stevenson–Crawford and Cohen–Feldmann bounds over those of Kato. A general discussion of all three sets of bounds is given in terms of the parameter λ appearing in the Stevenson–Crawford formulation.

NOTES ON THE DISTRIBUTION OF PHASE SHIFTS

2008 ◽  
Vol 22 (23) ◽  
pp. 2163-2175 ◽  
Author(s):  
MIKLÓS HORVÁTH
Keyword(s):  
Inverse Scattering ◽  
Lower Bounds ◽  
Three Dimensional ◽  
Variational Formula ◽  
Phase Shifts ◽  
Upper And Lower Bounds ◽  
One Dimensional ◽  
Fixed Energy ◽  
Symmetrical Potential ◽  
The One

We consider three-dimensional inverse scattering with fixed energy for which the spherically symmetrical potential is nonvanishing only in a ball. We give exact upper and lower bounds for the phase shifts. We provide a variational formula for the Weyl–Titchmarsh m-function of the one-dimensional Schrödinger operator defined on the half-line.

The accuracy of Rayleigh’s method of calculating the natural frequencies of vibrating systems

1952 ◽  
Vol 211 (1105) ◽  
pp. 204-224 ◽  
Keyword(s):  
Lower Bounds ◽  
Natural Frequencies ◽  
Hermitian Operator ◽  
Normal Modes ◽  
Comparison Theorems ◽  
Upper And Lower Bounds ◽  
Subsequent Paper ◽  
Numerical Examples ◽  
Modes Of Vibration ◽  
Vibrating Systems

In this paper a theorem of Kato (1949) which provides upper and lower bounds for the eigenvalues of a Hermitian operator is modified and generalized so as to give upper and lower bounds for the normal frequencies of oscillation of a conservative dynamical system. The method given here is directly applicable to a system specified by generalized co-ordinates with both elastic and inertial couplings. It can be applied to any one of the normal modes of vibration of the system. The bounds obtained are much closer than those given by Rayleigh’s comparison theorems in which the inertia or elasticity of the system is changed, and they are in fact the ‘best possible’ bounds. The principles of the computation of upper and lower bounds is explained in this paper and will be illustrated by some numerical examples in a subsequent paper.

scholarly journals Dimension estimates for sets of uniformly badly approximable systems of linear forms

2015 ◽  
Vol 11 (07) ◽  
pp. 2037-2054 ◽  
Author(s):  
Ryan Broderick ◽  
Dmitry Kleinbock
Keyword(s):  
Hausdorff Dimension ◽  
Lower Bounds ◽  
Higher Dimensions ◽  
Upper And Lower Bounds ◽  
Homogeneous Dynamics ◽  
Linear Forms ◽  
One Dimensional ◽  
Approximation Constant ◽  
The One ◽  
Badly Approximable

The set of badly approximable m × n matrices is known to have Hausdorff dimension mn. Each such matrix comes with its own approximation constant c, and one can ask for the dimension of the set of badly approximable matrices with approximation constant greater than or equal to some fixed c. In the one-dimensional case, a very precise answer to this question is known. In this note, we obtain upper and lower bounds in higher dimensions. The lower bounds are established via the technique of Schmidt games, while for the upper bound we use homogeneous dynamics methods, namely exponential mixing of flows on the space of lattices.

Upper and Lower Bounds for the Number of Conjugated Patterns in Carbocyclic and Heterocyclic Compounds

1994 ◽  
Vol 49 (10) ◽  
pp. 973-976
Author(s):  
Tetsuo Morikawa
Keyword(s):  
Lower Bounds ◽  
Special Class ◽  
Heterocyclic Compounds ◽  
The Other ◽  
Upper And Lower Bounds ◽  
The One

Abstract It is possible to regard two polygonal skeletons as the same in a special class of carbocyclic and heterocyclic compounds, if the one is reducible to the other by means of the contraction of cyclic subskeletons, and if the numbers of conjugated patterns in them are equal to each other. In such polygonal skeletons, three forms of cyclic subskeletons are defined; the one is called “alternate”, and the others, involving the one called “inclusive”, have a path (b, b), where (b) is a conjugated vertex connecting with three vertices. Successive eliminations of the cyclic subskeletons enable to estimate the upper and lower bounds for the number of conjugated patterns in a given polygonal skeleton.

Harmonic Waves in Layered Composites: New Bounds on Eigenfrequencies

1978 ◽  
Vol 45 (4) ◽  
pp. 829-833 ◽  
Author(s):  
C. O. Horgan ◽  
K.-W. Lang ◽  
S. Nemat-Nasser
Keyword(s):  
Lower Bounds ◽  
Eigenvalue Problems ◽  
Frequency Estimation ◽  
Discontinuous Coefficients ◽  
Upper And Lower Bounds ◽  
Harmonic Waves ◽  
Smooth Coefficients ◽  
Liouville Transformation ◽  
The One ◽  
Elastic Composites

The purpose of this paper is to present new approaches to the problem of wave frequency estimation for harmonic waves in layered elastic composites. Upper and lower bounds are obtained by adapting standard results for eigenvalue problems with smooth coefficients. The one-dimensional eigenvalue problem with discontinuous coefficients of concern here is first transformed by using an analog of the classical Liouville transformation. Upper bounds are obtained by application of a Rayleigh-Ritz technique to the transformed problem. Explicit lower bounds in terms of the coefficients are established. Results are illustrated by numerical examples.

scholarly journals Favorite-Candidate Voting for Eliminating the Least Popular Candidate in a Metric Space

2020 ◽  
Vol 34 (02) ◽  
pp. 1894-1901
Author(s):  
Xujin Chen ◽  
Minming Li ◽  
Chenhao Wang
Keyword(s):  
Lower Bounds ◽  
Metric Space ◽  
Social Value ◽  
Upper And Lower Bounds ◽  
Worst Case ◽  
The Social ◽  
Special Cases ◽  
Worst Case Ratio ◽  
The One

We study single-candidate voting embedded in a metric space, where both voters and candidates are points in the space, and the distances between voters and candidates specify the voters' preferences over candidates. In the voting, each voter is asked to submit her favorite candidate. Given the collection of favorite candidates, a mechanism for eliminating the least popular candidate finds a committee containing all candidates but the one to be eliminated. Each committee is associated with a social value that is the sum of the costs (utilities) it imposes (provides) to the voters. We design mechanisms for finding a committee to optimize the social value. We measure the quality of a mechanism by its distortion, defined as the worst-case ratio between the social value of the committee found by the mechanism and the optimal one. We establish new upper and lower bounds on the distortion of mechanisms in this single-candidate voting, for both general metrics and well-motivated special cases.

scholarly journals On the Generalized Distance Energy of Graphs

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 17 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Connected Graph ◽  
Distance Matrix ◽  
Wiener Index ◽  
Diagonal Matrix ◽  
Upper And Lower Bounds ◽  
Star Graph ◽  
Extremal Graphs ◽  
Generalized Distance

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

scholarly journals Hadamard k-fractional inequalities of Fejér type for GA-s-convex mappings and applications

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Hui Lei ◽  
Gou Hu ◽  
Zhi-Jie Cao ◽  
Ting-Song Du
Keyword(s):  
Lower Bounds ◽  
Hypergeometric Functions ◽  
Upper And Lower Bounds ◽  
Weighted Inequalities ◽  
Convex Mappings ◽  
Special Means

Abstract The main aim of this paper is to establish some Fejér-type inequalities involving hypergeometric functions in terms of GA-s-convexity. For this purpose, we construct a Hadamard k-fractional identity related to geometrically symmetric mappings. Moreover, we give the upper and lower bounds for the weighted inequalities via products of two different mappings. Some applications of the presented results to special means are also provided.

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