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.

Lower Bounds to the Eigenvalues in One-Dimensional Problems by a Shift in the Weight Function

1970 ◽  
Vol 37 (2) ◽  
pp. 267-270 ◽  
Author(s):  
D. Pnueli
Keyword(s):  
Lower Bound ◽  
Weight Function ◽  
Lower Bounds ◽  
Variational Formulation ◽  
Ritz Method ◽  
The Other ◽  
One Dimensional ◽  
Systematic Shift ◽  
The One ◽  
Detailed Computation

A method is presented to obtain both upper and lower bound to eigenvalues when a variational formulation of the problem exists. The method consists of a systematic shift in the weight function. A detailed procedure is offered for one-dimensional problems, which makes improvement of the bounds possible, and which involves the same order of detailed computation as the Rayleigh-Ritz method. The main contribution of this method is that it yields the “other bound;” i.e., the one which cannot be obtained by the Rayleigh-Ritz method.

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.

scholarly journals Bounds for the Collapse Load of a Beam Compressed by Three Dies

1956 ◽  
Vol 9 (4) ◽  
pp. 419
Author(s):  
W Freiberger
Keyword(s):  
Plastic Deformation ◽  
Plastic Flow ◽  
Lower Bounds ◽  
Limit Analysis ◽  
Velocity Fields ◽  
The Other ◽  
Upper And Lower Bounds ◽  
Collapse Load ◽  
Plastic Yielding ◽  
Plane Plastic

This paper deals with the problem of the plastic deformation of a beam under the action of three perfectly rough rigid dies, two dies applied to one side, one die to the other side of the beam, the single die being situated between the two others. It is treated as a problem of plane plastic flow. Discontinuous stress and velocity fields are assumed and upper and lower bounds for the pressure sufficient to cause pronounced plastic yielding determined by limit analysis.

scholarly journals Consolidation via Tacit Culpability Measures: Between Explicit and Implicit Degrees of Culpability

2021 ◽  
Author(s):  
Jandson S. Ribeiro ◽  
Matthias Thimm
Keyword(s):  
Fixed Point ◽  
Knowledge Base ◽  
Special Class ◽  
Belief Change ◽  
Minimal Change ◽  
The Other ◽  
Preference Relations ◽  
Other Hand ◽  
The One ◽  
Rationality Postulates

Restoring consistency of a knowledge base, known as consolidation, should preserve as much information as possible of the original knowledge base. On the one hand, the field of belief change captures this principle of minimal change via rationality postulates. On the other hand, within the field of inconsistency measurement, culpability measures have been developed to assess how much a formula participates in making a knowledge base inconsistent. We look at culpability measures as a tool to disclose epistemic preference relations and build rational consolidation functions. We introduce tacit culpability measures that consider semantic counterparts between conflicting formulae, and we define a special class of these culpability measures based on a fixed-point characterisation: the stable tacit culpability measures. We show that the stable tacit culpability measures yield rational consolidation functions and that these are also the only culpability measures that yield rational consolidation functions.

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.

Notes on the New Aśoka Inscription from Kandahar

1969 ◽  
Vol 101 (2) ◽  
pp. 118-122 ◽  
Author(s):  
Shaul Shaked
Keyword(s):  
Special Class ◽  
Common Type ◽  
The Other ◽  
The Common ◽  
The One ◽  
Short Section

The new inscription of Aśoka from Kandahar, published by Émile Benveniste and André Dupont-Sommer, forms, together with the one stemming from Pul-i Darunteh (Lamghān) published by W. B. Henning, a special class of Aśoka inscriptions. Both these inscriptions are bilingual, but they do not belong to the common type of bilingual documents, in which the text of each language is separately inscribed on a different part of the stone's surface. Here the two languages are mixed, each short section in one language is followed by one in the other language. The two languages involved, both written in Aramaic characters, are Aramaic and Middle Indian (Prākrit).

scholarly journals ON THE REMAK HEIGHT, THE MAHLER MEASURE AND CONJUGATE SETS OF ALGEBRAIC NUMBERS LYING ON TWO CIRCLES

2001 ◽  
Vol 44 (1) ◽  
pp. 1-17 ◽  
Author(s):  
A. Dubickas ◽  
C. J. Smyth
Keyword(s):  
Lower Bounds ◽  
Height Function ◽  
Complete Characterization ◽  
Mahler Measure ◽  
The Other ◽  
Upper And Lower Bounds ◽  
Algebraic Numbers ◽  
Salem Numbers ◽  
Primary 11R06

AbstractWe define a new height function $\mathcal{R}(\alpha)$, the Remak height of an algebraic number $\alpha$. We give sharp upper and lower bounds for $\mathcal{R}(\alpha)$ in terms of the classical Mahler measure $M(\alpha)$. Study of when one of these bounds is exact leads us to consideration of conjugate sets of algebraic numbers of norm $\pm 1$ lying on two circles centred at 0. We give a complete characterization of such conjugate sets. They turn out to be of two types: one related to certain cubic algebraic numbers, and the other related to a non-integer generalization of Salem numbers which we call extended Salem numbers.AMS 2000 Mathematics subject classification: Primary 11R06

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 Maximal prime gaps bounds

2021 ◽  
Author(s):  
Jan Feliksiak
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bound ◽  
Estimation Error ◽  
The Other ◽  
Upper And Lower Bounds ◽  
Close Approximation ◽  
Implicit Constant ◽  
Error Cost ◽  
Prime Gaps

This paper presents research results, pertinent to the maximal prime gaps bounds. Four distinct bounds are presented: Upper bound, Infimum, Supremum and finally the Lower bound. Although the Upper and Lower bounds incur a relatively high estimation error cost, the functions representing them are quite simple. This ensures, that the computation of those bounds will be straightforward and efficient. The Lower bound is essential, to address the issue of the value of the lower bound implicit constant C, in the work of Ford et al (Ford, 2016). The concluding Corollary in this paper shows, that the value of the constant C does diverge, although very slowly. The constant C, will eventually take any arbitrary value, providing that a large enough N (for p <= N) is considered. The Infimum/Supremum bounds on the other hand are computationally very demanding. Their evaluation entails computations at an extreme level of precision. In return however, we obtain bounds, which provide an extremely close approximation of the maximal prime gaps. The Infimum/Supremum estimation error gradually increases over the range of p and attains at p = 18361375334787046697 approximately the value of 0.03.

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