scholarly journals An approximate Hahn-Banach-Lagrange theorem

2016 ◽  
Vol 19 (4) ◽  
pp. 169-177
Author(s):  
Dinh Nguyen ◽  
Mo Hong Tran
Keyword(s):  
Original Version ◽  
Extended Version ◽  
General Picture ◽  
Farkas Lemma ◽  
Sandwich Theorem ◽  
Banach Theorem ◽  
Special Cases ◽  
Lagrange Theorem ◽  
Sublinear Functions ◽  
Asymptotic Forms

In this paper, we proved a new extended version of the Hahn-Banach-Lagrange theorem that is valid in the absence of a qualification condition and is called an approximate Hahn-Banach-Lagrange theorem. This result, in special cases, gives rise to approximate sandwich and approximate Hahn-Banach theorems. These results extend the Hahn-Banach-Lagrange theorem, the sandwich theorem in [18], and the celebrated Hahn-Banach theorem. The mentioned results extend the original ones into two features: Firstly, they extend the original versions to the case with extended sublinear functions (i.e., the sublinear functions that possibly possess extended real values). Secondly, they are topological versions which held without any qualification condition. Next, we showed that our approximate Hahn-Banach-Lagrange theorem was actually equivalent to the asymptotic Farkas-type results that were established recently [10]. This result, together with the results [5, 16], give us a general picture on the equivalence of the Farkas lemma and the Hahn-Banach theorem, from the original version to their corresponding extensions and in either non-asymptotic or asymptotic forms.

scholarly journals The Generalized Gielis Geometric Equation and Its Application

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 645 ◽  
Author(s):  
Peijian Shi ◽  
David A. Ratkowsky ◽  
Johan Gielis
Keyword(s):  
Power Law ◽  
Morphological Features ◽  
Original Version ◽  
Link Function ◽  
Plant Leaves ◽  
Special Cases ◽  
Geometric Equation ◽  
Special Case

Many natural shapes exhibit surprising symmetry and can be described by the Gielis equation, which has several classical geometric equations (for example, the circle, ellipse and superellipse) as special cases. However, the original Gielis equation cannot reflect some diverse shapes due to limitations of its power-law hypothesis. In the present study, we propose a generalized version by introducing a link function. Thus, the original Gielis equation can be deemed to be a special case of the generalized Gielis equation (GGE) with a power-law link function. The link function can be based on the morphological features of different objects so that the GGE is more flexible in fitting the data of the shape than its original version. The GGE is shown to be valid in depicting the shapes of some starfish and plant leaves.

Estimation of the earthquake recurrence parameters for unequal observation periods for different magnitudes

1980 ◽  
Vol 70 (4) ◽  
pp. 1337-1346
Author(s):  
Dieter H. Weichert
Keyword(s):  
Least Squares ◽  
Likelihood Function ◽  
Likelihood Estimation ◽  
Data Sets ◽  
Chi Square ◽  
Earthquake Parameters ◽  
Large Numbers ◽  
Special Cases ◽  
Weighting Methods ◽  
Asymptotic Forms

abstract Maximum likelihood estimation of the earthquake parameters No and β in the relation N = No exp (−βm) is extended to the case of events grouped in magnitude with each group observed over individual time periods. Asymptotic forms of the equation for β reduce to the estimators given for different special cases by Aki (1965), Utsu (1965, 1966), and Page (1968). The estimates of β are only approximately chi-square distributed. For sufficiently large numbers of events, they can be estimated from the curvature of the log-likelihood function. Sample calculations for three earthquake source zones in western Canada indicate that for well-constrained data sets, the often-used, least-squares estimation procedures lead to compatible results, but for less well-defined data sets, the effect of subjective plotting and weighting methods used for least-squares fitting leads to appreciably different parameters.

scholarly journals Translating the Sacred Text: A Polysystem Approach

2016 ◽  
Vol 12 (1) ◽  
pp. 13-27
Author(s):  
Abdelhamid Elewa
Keyword(s):  
Corpus Linguistics ◽  
Linguistic Analysis ◽  
Original Version ◽  
Sacred Text ◽  
Extended Version ◽  
Social Perspectives ◽  
Islamic Jurisprudence ◽  
Three Stages ◽  
Modern Context ◽  
Original Message

This paper attempts an extended version of the polysystem theory combined with corpus linguistics techniques in translating the Qur'an. The original version of the theory focuses on the network of relations around the target text, while the extended version includes the network of relations (or systems) of both the source and target text, be they linguistic, social, historical, etc. To explore the systems of both the source and target text, a model was developed for translating the Sacred Text which can serve as a framework for the translators to capture the early and late Arab contributions from linguistic, historical and social perspectives to maintain the original message and present it afresh to the modern generations. The model consists of three stages and each stage involves a number of systems, including the linguistic analysis of the original and target text, the network of relations of the early context, and the network of relations of the modern context. Three verses were selected in this paper to demonstrate the validity of the model. The verses deal with some social, historical, theological and legal debatable issues in the field of Sacred Text exegesis, theology and Islamic jurisprudence. The different readings of the verses are mirrored in the various translations of the Sacred Text.

scholarly journals Digitale Fassungsvergleiche am Beispiel von Beethovens Eigenbearbeitungen

2020 ◽  
Author(s):  
Susanne Cox ◽  
Richard Sänger
Keyword(s):  
String Quartet ◽  
Original Version ◽  
Digital Tools ◽  
Piano Sonata ◽  
Digital Edition ◽  
Special Cases ◽  
Piano Reductions

In its second module, „Beethovens Werkstatt“ deals with five of Beethoven’s compositions which exist both in their original versions and as authentic arrangements (Piano Sonata op. 14/1 arranged for string quartet, Septett op. 20 and Trio op. 38, Opferlied op. 121b and Bundeslied op. 122 as piano reductions, Große Fuge op. 133 as arrangement for piano for four hands op. 134). To demonstrate Beethoven’s arrangement practices, the original version of each work is synoptically linked with its arrangement in a digital edition called „VideApp Arr“. Through digital tools for comparison the relationships between the two versions can be investigated from different perspectives. It becomes visible how the versions are related to each other both by „invariance“ (text elements with the same structure), by „variance“ (text elements with a similar structure) and, in special cases, also by „difference“ (text elements without corresponding parameters). Each view within the „VideApp Arr“ is generated from the underlying MEI data.

scholarly journals An Extension of the Hawkins and Simon Condition Characterizing Viable Techniques

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Alberto Benítez Sánchez
Keyword(s):  
Characteristic Property ◽  
Original Version ◽  
Extended Version ◽  
Mathematical Properties ◽  
Production Models ◽  
Single Production

This paper discusses an extended version of the Hawkins and Simon condition which constitutes a synthetic formulation of the mathematical properties that viable economies must satisfy in single production models. The new version is implicit in the economic interpretations offered by them of the Hawkins and Simon condition, once a correction is introduced in one of those interpretations. Moreover, the paper details the meaning of the extended version following the interpretation of the original version proposed by Dorfman, Samuelson, and Solow. It also introduces a characteristic property of indecomposable matrices that has not previously been published.

The asymptotic behaviour of solutions of the differential equation

1975 ◽  
Vol 346 (1645) ◽  
pp. 171-207 ◽  
Keyword(s):  
Differential Equation ◽  
Asymptotic Behaviour ◽  
Bessel Function ◽  
Positive Real ◽  
Generalized Hypergeometric Functions ◽  
Asymptotic Behaviour Of Solutions ◽  
Special Cases ◽  
Transition Points ◽  
Qualitative Discussion ◽  
Asymptotic Forms

The asymptotic behaviour of solutions of the fourth order differential equation in which the constants a and b are supposed real and positive, is examined for large real values of z and b. Exact solutions are given in terms of generalized hypergeometric functions and some special cases of a and b are mentioned for which solutions may be expressed in terms of simpler known functions. The differential equation possesses four transition points and for positive values of b one of these transition points lies on the positive real 2-axis. The asymptotic discussion is centred around an integral representation which involves a modified Bessel function of the second kind whose order is purely imaginary for large positive values of b. Asymptotic forms as z-> oo and about the transition point on the positive real 2-axis are given. A qualitative discussion of the zeros of the Bessel function K^{z) for imaginary order and complex argument is presented in the appendix.

scholarly journals From the Farkas Lemma to the Hahn--Banach Theorem

2014 ◽  
Vol 24 (2) ◽  
pp. 678-701 ◽  
Author(s):  
N. Dinh ◽  
M. A. Goberna ◽  
M. A. López ◽  
T. H. Mo
Keyword(s):  
Farkas Lemma ◽  
Banach Theorem

Electron Beam Damage of Biomolecules Assessed by Energy Loss Spectroscopy

1978 ◽  
Vol 36 (3) ◽  
pp. 61-69
Author(s):  
M. Isaacson ◽  
M.L. Collins ◽  
M. Listvan
Keyword(s):  
Electron Microscopy ◽  
Radiation Damage ◽  
Structural Integrity ◽  
Analytical Electron Microscopy ◽  
High Dose ◽  
Dose Rates ◽  
Special Cases ◽  
Analytical Electron ◽  
Atom Migration ◽  
Beam Damage

Over the past five years it has become evident that radiation damage provides the fundamental limit to the study of blomolecular structure by electron microscopy. In some special cases structural determinations at very low doses can be achieved through superposition techniques to study periodic (Unwin & Henderson, 1975) and nonperiodic (Saxton & Frank, 1977) specimens. In addition, protection methods such as glucose embedding (Unwin & Henderson, 1975) and maintenance of specimen hydration at low temperatures (Taylor & Glaeser, 1976) have also shown promise. Despite these successes, the basic nature of radiation damage in the electron microscope is far from clear. In general we cannot predict exactly how different structures will behave during electron Irradiation at high dose rates. Moreover, with the rapid rise of analytical electron microscopy over the last few years, nvicroscopists are becoming concerned with questions of compositional as well as structural integrity. It is important to measure changes in elemental composition arising from atom migration in or loss from the specimen as a result of electron bombardment.

Decoration technique and surface physics

1978 ◽  
Vol 36 (1) ◽  
pp. 436-437 ◽  
Author(s):  
H. Bethge
Keyword(s):  
Diffusion Path ◽  
Special Importance ◽  
Surface Physics ◽  
Step Structure ◽  
Initial Stage ◽  
Special Cases ◽  
Atomic Surface ◽  
The Mean ◽  
Bulk Structure ◽  
Decoration Technique

Besides the atomic surface structure, diverging in special cases with respect to the bulk structure, the real structure of a surface Is determined by the step structure. Using the decoration technique /1/ it is possible to image step structures having step heights down to a single lattice plane distance electron-microscopically. For a number of problems the knowledge of the monatomic step structures is important, because numerous problems of surface physics are directly connected with processes taking place at these steps, e.g. crystal growth or evaporation, sorption and nucleatlon as initial stage of overgrowth of thin films.To demonstrate the decoration technique by means of evaporation of heavy metals Fig. 1 from our former investigations shows the monatomic step structure of an evaporated NaCI crystal. of special Importance Is the detection of the movement of steps during the growth or evaporation of a crystal. From the velocity of a step fundamental quantities for the molecular processes can be determined, e.g. the mean free diffusion path of molecules.

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