scholarly journals On a new property of the arcs of the equilateral hyperbola

1837 ◽  
Vol 3 ◽  
pp. 258-258
Keyword(s):  
Numerical Examples ◽  
Simple Result ◽  
Analytical Process ◽  
Constant Quantity ◽  
The Difference ◽  
Experimental Researches

By an analytical process, the author arrives at the following theorem, namely, if three abscissæ of an equilateral hyperbola be materially dependent by reason of two assumed equations, which are symmetrical with respect to these three abscissæ, the sum of the arcs subtended by them is equal to three quarters of the product of the same abscissæ, or only differs therefrom by a constant quantity. In order to satisfy himself of the correctness of this theorem, the author calculated various numerical examples, which entirely confirmed it. This simple result is essentially a relation between three arcs of the equilateral hyperbola, and is by no means reducible to a relation between two; and therefore is not reducible to the celebrated theorem of Fagnani, concerning the difference of two arcs of an ellipse or hyperbola, nor to any other known property of the curve. The reading of Mr. Faraday’s Sixth Series of Experimental Researches in Electricity was commenced.

Entangled Views of Exogenous and Endogenous Change

2021 ◽  
pp. 876-887
Author(s):  
Martha S. Feldman
Keyword(s):  
Organizational Change ◽  
Analytical Process ◽  
Internal Change ◽  
Endogenous Change ◽  
The Difference

In keeping with identifying dialectics as one of four model of change, many of the chapters of this handbook identify various dualities as important to understanding organizational change. This chapter focuses particularly on the duality of exogenous and endogenous change (or external and internal change) and considers the various ways in which the two are entangled. It reflects on seven chapters of the handbook that provide a range of perspectives on the issue, and separates the range of orientations into three categories: chapters that take the difference between internal and external as an ontological fact and explore how separable entities interact with one another; chapters in which the difference between internal and external is an analytical process (and which are apparently agnostic about ontological differences); and chapters that reject an ontological distinction between exogenous and endogenous and explore the entangled nature of exogenous and endogenous within a single ontology.

scholarly journals Robust optimal usage modeling of product systems for environmental sustainability

2018 ◽  
Vol 6 (3) ◽  
pp. 429-435 ◽  
Author(s):  
Jungmok Ma
Keyword(s):  
Life Cycle ◽  
Environmental Sustainability ◽  
Product Usage ◽  
Numerical Examples ◽  
Usage Phase ◽  
Multiple Products ◽  
Product Systems ◽  
Proposed Model ◽  
Sustainable Product ◽  
The Difference

Abstract Proper modeling of the usage phase in Life Cycle Assessment (LCA) is not only critical due to its high impact among life cycle phases but also challenging due to high variations and uncertainty. Furthermore, when multiple products can be utilized, the optimal product usage should be considered together. The robust optimal usage modeling is proposed in this paper as the framework of usage modeling for LCA with consideration of the uncertainty and optimal usage. The proposed method seeks to optimal product usage in order to minimize the environmental impact of the usage phase under uncertainty. Numerical examples demonstrate the application of the robust optimal usage modeling and the difference from the previous approaches. Highlights The robust optimal usage modeling is proposed for the usage modeling of LCA. The proposed model seeks to sustainable product usage under uncertainty. Numerical examples demonstrate the difference from the previous approaches.

scholarly journals Some Order Preserving Inequalities for Cross Entropy and Kullback–Leibler Divergence

Entropy ◽  
2018 ◽  
Vol 20 (12) ◽  
pp. 959 ◽  
Author(s):  
Mateu Sbert ◽  
Min Chen ◽  
Jordi Poch ◽  
Anton Bardera
Keyword(s):  
Machine Learning ◽  
Information Theory ◽  
Geometric Mean ◽  
Cross Entropy ◽  
Rearrangement Inequality ◽  
Numerical Examples ◽  
Leibler Divergence ◽  
Recent Theorem ◽  
The Difference ◽  
New Inequalities

Cross entropy and Kullback–Leibler (K-L) divergence are fundamental quantities of information theory, and they are widely used in many fields. Since cross entropy is the negated logarithm of likelihood, minimizing cross entropy is equivalent to maximizing likelihood, and thus, cross entropy is applied for optimization in machine learning. K-L divergence also stands independently as a commonly used metric for measuring the difference between two distributions. In this paper, we introduce new inequalities regarding cross entropy and K-L divergence by using the fact that cross entropy is the negated logarithm of the weighted geometric mean. We first apply the well-known rearrangement inequality, followed by a recent theorem on weighted Kolmogorov means, and, finally, we introduce a new theorem that directly applies to inequalities between K-L divergences. To illustrate our results, we show numerical examples of distributions.

scholarly journals A Consistency Check Method for Trusted Hesitant Fuzzy Sets with Confidence Levels Based on a Distance Measure

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Huimin Xiao ◽  
Meiqi Wang ◽  
Xiaoning Xi
Keyword(s):  
Fuzzy Sets ◽  
Distance Measure ◽  
Consistency Check ◽  
Hesitant Fuzzy Sets ◽  
Numerical Examples ◽  
Confidence Levels ◽  
Check Method ◽  
The Difference

This paper proposes a consistency check method for hesitant fuzzy sets with confidence levels by employing a distance measure. Firstly, we analyze the difference between each fuzzy element and its corresponding attribute comprehensive decision value and then obtain a comprehensive distance measure for each attribute. Subsequently, by taking the relative credibility as the weight, we assess the consistency of hesitant fuzzy sets. Finally, numerical examples are put forward to verify the effectiveness and reliability of the proposed method.

Non Linear Difference Equation of Orlicz Type

2017 ◽  
Vol 14 (1) ◽  
pp. 306-313
Author(s):  
Awad. A Bakery ◽  
Afaf. R. Abou Elmatty
Keyword(s):  
Difference Equation ◽  
Equilibrium Point ◽  
Positive Solution ◽  
Orlicz Function ◽  
Sufficient Conditions ◽  
Linear Difference Equation ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Numerical Examples ◽  
The Difference

We give here the sufficient conditions on the positive solutions of the difference equation xn+1 = α+M((xn−1)/xn), n = 0, 1, …, where M is an Orlicz function, α∈ (0, ∞) with arbitrary positive initials x−1, x0 to be bounded, α-convergent and the equilibrium point to be globally asymptotically stable. Finally we present the condition for which every positive solution converges to a prime two periodic solution. Our results coincide with that known for the cases M(x) = x in Ref. [3] and M(x) = xk, where k ∈ (0, ∞) in Ref. [7]. We have given the solution of open problem proposed in Ref. [7] about the existence of the positive solution which eventually alternates above and below equilibrium and converges to the equilibrium point. Some numerical examples with figures will be given to show our results.

scholarly journals Sparse Constrained Least-Squares Reverse Time Migration Based on Kirchhoff Approximation

2021 ◽  
Vol 9 ◽  
Author(s):  
Xu Hong-Qiao ◽  
Wang Xiao-Yi ◽  
Wang Chen-Yuan ◽  
Zhang Jiang-Jie
Keyword(s):  
Least Squares ◽  
Kirchhoff Approximation ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
L1 Norm ◽  
Constrained Least Squares ◽  
Geological Structures ◽  
Numerical Examples ◽  
Time Migration ◽  
The Difference

Least-squares reverse time migration (LSRTM) is powerful for imaging complex geological structures. Most researches are based on Born modeling operator with the assumption of small perturbation. However, studies have shown that LSRTM based on Kirchhoff approximation performs better; in particular, it generates a more explicit reflected subsurface and fits large offset data well. Moreover, minimizing the difference between predicted and observed data in a least-squares sense leads to an average solution with relatively low quality. This study applies L1-norm regularization to LSRTM (L1-LSRTM) based on Kirchhoff approximation to compensate for the shortcomings of conventional LSRTM, which obtains a better reflectivity image and gets the residual and resolution in balance. Several numerical examples demonstrate that our method can effectively mitigate the deficiencies of conventional LSRTM and provide a higher resolution image profile.

scholarly journals Some Discussions on the Difference Equationxn+1=α+(xn-1m/xnk)

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Awad A. Bakery
Keyword(s):  
Periodic Solution ◽  
Difference Equation ◽  
Equilibrium Point ◽  
Positive Solution ◽  
Initial Conditions ◽  
Sufficient Conditions ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Numerical Examples ◽  
The Difference

We give in this work the sufficient conditions on the positive solutions of the difference equationxn+1=α+(xn-1m/xnk),  n=0,1,…, whereα,k, andm∈(0,∞)under positive initial conditionsx-1,  x0to be bounded,α-convergent, the equilibrium point to be globally asymptotically stable and that every positive solution converges to a prime two-periodic solution. Our results coincide with that known for the casesm=k=1of Amleh et al. (1999) andm=1of Hamza and Morsy (2009). We offer improving conditions in the case ofm=1of Gümüs and Öcalan (2012) and explain our results by some numerical examples with figures.

Responses comparison of the two discrete-time linear fractional state-space models

2016 ◽  
Vol 19 (4) ◽  
Author(s):  
Tadeusz Kaczorek ◽  
Piotr Ostalczyk
Keyword(s):  
State Space ◽  
State Space Model ◽  
State Space Models ◽  
Discrete State ◽  
Numerical Examples ◽  
Space Model ◽  
The Difference ◽  
Time Linear ◽  
Discrete State Space ◽  
Fundamental Matrices

AbstractIn this survey we consider two fractional-order discrete state-space models of linear systems. In both cases the crucial elements are the fundamental matrices. The difference between them is analyzed. A fundamental condition for the first state-space model is given. The investigations are illustrated by the numerical examples.

scholarly journals Retrospectives Irving Fisher's Appreciation and Interest (1896) and the Fisher Relation

2012 ◽  
Vol 26 (4) ◽  
pp. 185-196 ◽  
Author(s):  
Robert W Dimand ◽  
Rebeca Gomez Betancourt
Keyword(s):  
Interest Rates ◽  
20Th Century ◽  
Term Structure ◽  
Price Level ◽  
Numerical Examples ◽  
Expected Inflation ◽  
Quantitative Analyses ◽  
The Difference ◽  
Fisher Relation ◽  
Parity Condition

Irving Fisher's monograph Appreciation and Interest (1896) proposed his famous equation showing expected inflation as the difference between nominal interest and real interest rates. In addition, he drew attention to insightful remarks and numerical examples scattered through the earlier literature, and he derived results ranging from the uncovered interest arbitrage parity condition between currencies to the expectations theory of the term structure of interest rates. As J. Bradford DeLong wrote in this journal (Winter 2000), “The story of 20th century macroeconomics begins with Irving Fisher” and specifically with Appreciation and Interest because “the transformation of the quantity theory of money into a tool for making quantitative analyses and predictions of the price level, inflation, and interest rates was the creation of Irving Fisher.” I discuss the message of Appreciation and Interest, and assess how original he was.

Controlled-order multiple waveform inversion

Geophysics ◽  
2020 ◽  
Vol 85 (3) ◽  
pp. R243-R250 ◽  
Author(s):  
Yike Liu ◽  
Bin He ◽  
Yingcai Zheng
Keyword(s):  
Wave Propagation ◽  
Objective Function ◽  
Waveform Inversion ◽  
Full Waveform Inversion ◽  
Initial Model ◽  
Numerical Examples ◽  
Full Waveform ◽  
Wide Range ◽  
True Model ◽  
The Difference

Traditional full-waveform inversion (FWI) seeks to find the best model by minimizing an objective function defined as the difference between the model-predicted and observed data in amplitude and phase. In principle, FWI should fit all wave types including direct waves, diving waves, primaries, and multiples. However, when an initial model is far from the true model, FWI will encounter difficulties in matching multiples. Physically, multiples may contain more subsurface information compared to primary and diving waves. Multiples cover a wide range of reflection angles during wave propagation and offer the advantage of imaging the shadow zones that cannot be reached or are poorly illuminated by primary reflections. We have developed a new method of waveform inversion using multiples. We first separate the multiples into different orders. The objective function we seek to minimize consists of the data difference between the modeled data using a lower order multiple as the source and the higher order multiple as data. This method is called controlled-order multiple waveform inversion (CMWI). Our numerical examples determined that the CMWI is a promising method to improve velocity updates.

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