A two-stage approximation strategy for piecewise smooth functions in two and three dimensions

2021 ◽  
Author(s):  
Sergio Amat ◽  
David Levin ◽  
Juan Ruiz-Álvarez
Keyword(s):  
Piecewise Smooth ◽  
Three Dimensions ◽  
System Of Equations ◽  
Global Approximation ◽  
Smooth Functions ◽  
Approximation Techniques ◽  
Second Stage ◽  
Principle Of Matching ◽  
The Given ◽  
Standard Approximation

Abstract Given values of a piecewise smooth function $f$ on a square grid within a domain $[0,1]^d$, $d=2,3$, we look for a piecewise adaptive approximation to $f$. Standard approximation techniques achieve reduced approximation orders near the boundary of the domain and near curves of jump singularities of the function or its derivatives. The insight used here is that the behavior near the boundaries, or near a singularity curve, is fully characterized and identified by the values of certain differences of the data across the boundary and across the singularity curve. We refer to these values as the signature of $f$. In this paper, we aim at using these values in order to define the approximation. That is, we look for an approximation whose signature is matched to the signature of $f$. Given function data on a grid, assuming the function is piecewise smooth, first, the singularity structure of the function is identified. For example, in the two-dimensional case, we find an approximation to the curves separating between smooth segments of $f$. Secondly, simultaneously, we find the approximations to the different segments of $f$. A system of equations derived from the principle of matching the signature of the approximation and the function with respect to the given grid defines a first stage approximation. A second stage improved approximation is constructed using a global approximation to the error obtained in the first stage approximation.

scholarly journals Duopoly price competition with limited capacity

2021 ◽  
Author(s):  
A. Bërdëllima
Keyword(s):  
Price Competition ◽  
Small Firms ◽  
Common Knowledge ◽  
Limited Capacity ◽  
Equilibrium Outcome ◽  
Second Stage ◽  
Duopoly Model ◽  
Full Capacity ◽  
Stage Game ◽  
The Given

AbstractWe study a variation of the duopoly model by Kreps and Scheinkman (1983). Firms limited by their capacity of production engage in a two stage game. In the first stage they commit to levels of production not exceeding their capacities which are then made common knowledge. In the second stage after production has taken place firms simultane- ously compete in prices. Solution of this sequential game shows that the unique Cournot equilibrium outcome as in Kreps and Scheinkman is not always guaranteed. However the Cournot outcome is still robust in the sense that given sufficiently large capacities this equilibrium holds. If capacities are sufficiently small, firms decide to produce at their full capacity and set a price which clears the market at the given level of output.

Opposition-based learning Multi-Verse Optimizer with disruption operator for optimization problems

2021 ◽  
Author(s):  
Mohammad Shehab ◽  
Laith Abualigah
Keyword(s):  
Real World ◽  
Optimization Problems ◽  
Second Stage ◽  
Opposition Based Learning ◽  
High Fitness ◽  
Fitness Value ◽  
Speed Up ◽  
Disruption Operator ◽  
The Given ◽  
Real World Problems

Abstract Multi-Verse Optimizer (MVO) algorithm is one of the recent metaheuristic algorithms used to solve various problems in different fields. However, MVO suffers from a lack of diversity which may trapping of local minima, and premature convergence. This paper introduces two steps of improving the basic MVO algorithm. The first step using Opposition-based learning (OBL) in MVO, called OMVO. The OBL aids to speed up the searching and improving the learning technique for selecting a better generation of candidate solutions of basic MVO. The second stage, called OMVOD, combines the disturbance operator (DO) and OMVO to improve the consistency of the chosen solution by providing a chance to solve the given problem with a high fitness value and increase diversity. To test the performance of the proposed models, fifteen CEC 2015 benchmark functions problems, thirty CEC 2017 benchmark functions problems, and seven CEC 2011 real-world problems were used in both phases of the enhancement. The second step, known as OMVOD, incorporates the disruption operator (DO) and OMVO to improve the accuracy of the chosen solution by giving a chance to solve the given problem with a high fitness value while also increasing variety. Fifteen CEC 2015 benchmark functions problems, thirty CEC 2017 benchmark functions problems and seven CEC 2011 real-world problems were used in both phases of the upgrade to assess the accuracy of the proposed models.

An accidental visualization of the Brillouin zone in an Ni–W alloyviadiffuse scattering

2013 ◽  
Vol 46 (4) ◽  
pp. 1211-1215 ◽  
Author(s):  
Sascha B. Maisel ◽  
Nils Schindzielorz ◽  
Stefan Müller ◽  
Harald Reichert ◽  
Alexei Bosak
Keyword(s):  
Fermi Surface ◽  
Brillouin Zone ◽  
Expansion Method ◽  
Reciprocal Space ◽  
Three Dimensions ◽  
X Rays ◽  
Solid State Physics ◽  
Seitz Cell ◽  
The Face ◽  
The Given

Solid state physics is built on the concept of reciprocal space. The physics of any given periodic crystal is fully defined within the Wigner–Seitz cell in reciprocal space, also known as the first Brillouin zone. It is a purely symmetry-based concept and usually does not have any eye-catching signature in the experimental data, in contrast with some other geometrical constructions like the Fermi surface. However, the particular shape of the Fermi surface of nickel allowed the visualization of the system of edges (skeleton) of the Wigner–Seitz cell of the face-centred cubic lattice in reciprocal space in three dimensions by the diffuse scattering of X-rays from Ni1−xWx(x= 0.03, 0.05, 0.08) single crystals. Employing a cluster-expansion method with first-principles input, it is possible to show that the observed scattering is inherent to the given nickel alloys and the crystal structures they form. This peculiar feature can be understood by considering the shape of the Fermi surface of pure nickel.

scholarly journals Evaluating the parameters of a mobile maize dryer in practice

2013 ◽  
Vol 61 (6) ◽  
pp. 1779-1784
Author(s):  
Josef Los ◽  
Jiří Fryč ◽  
Zdeněk Konrád
Keyword(s):  
Large Scale ◽  
Electric Energy ◽  
Operating Parameters ◽  
The Czech Republic ◽  
Maize Hybrids ◽  
Post Harvest ◽  
Second Stage ◽  
Energy Intensiveness ◽  
Two Stages ◽  
The Given

The method of drying maize for grain has been recently employed on a large scale in the Czech Republic not only thanks to new maize hybrids but also thanks to the existence of new models of drying plants. One of the new post-harvest lines is a plant in Lipoltice (mobile dryer installed in 2010, storage base in 2012) where basic operational measurements were made of the energy intensiveness of drying and operating parameters of the maize dryer were evaluated. The process of maize drying had two stages, i.e. pre-drying from the initial average grain humidity of 28.55% to 19.6% in the first stage, and the additional drying from 16.7% to a final storage grain humidity of 13.7%. Mean volumes of natural gas consumed per 1 t% for drying in the first and second stage amounted to 1.275 m3 and 1.56 m3, respectively. The total mean consumption of electric energy per 1 t% was calculated to be 1.372 kWh for the given configuration of the post-harvest line.

scholarly journals X. On Hamilton's numbers

1887 ◽  
Vol 178 ◽  
pp. 285-312
Keyword(s):  
General Equation ◽  
Minimum Degree ◽  
A Priori ◽  
Senior Author ◽  
System Of Equations ◽  
Absolute Minimum ◽  
Doctor Of Philosophy ◽  
The University ◽  
The Given ◽  
Trinomial Equation

In the year 1786 Erland Samuel Bring, Professor at the University of Lund in Sweden, showed how by an extension of the method of Tschirnhausen it was possible to deprive the general algebraical equation of the 5th degree of three of its terms without solving an equation higher than the 3rd degree. By a well-understood, however singular, academical fiction, this discovery was ascribed by him to one of his own pupils, a certain Sven Gustaf Sommelius, and embodied in a thesis humbly submitted to himself for approval by that pupil, as a preliminary to his obtaining his degree of Doctor of Philosophy in the University. The process for effecting this reduction seems to have been overlooked or forgotten, and was subsequently re-discovered many years later by Mr. Jerrard. In a report contained in the ‘Proceedings of the British Association’ for 1836, Sir William Hamilton showed that Mr. Jerrard was mistaken in supposing that the method was adequate to taking away more than three terms of the equation of the 5th degree, but supplemented this somewhat unnecessary refutation of a result, known à priori to be impossible, by an extremely valuable discussion of a question raised by Mr. Jerrard as to the number of variables required in order that any system of equations of given degrees in those variables shall admit of being satisfied without solving any equation of a degree higher than the highest of the given degrees. In the year 1886 the senior author of this memoir showed in a paper in Kronecker'e (better known as Crelle’s ) ‘Journal that the trinomial equation of the 5th degree, upon which by Bring’s method the general equation of that degree can be made to depend, has necessarily imagmaiy coefficients except in the case where four of the roots of the original equation are imaginary, and also pointed out method of obtaining the absolute minimum degree M of an equation from which an given number of specified terms can be taken away subject to the condition of no having to solve any equation of a degree higher than M. The numbers furnished be Hamilton’s method, it is to be observed, are not minima unless a more stringer condition than this is substituted, viz., that the system of equations which have to be resolved in order to take away the proposed terms shall be the simplest possible i. e ., of the lowest possible weight and not merely of the lowest order; in the memo: in ‘Crelle,’ above referred to, he has explained in what sense the words weight an order are here employed. He has given the name of Hamilton’s Numbers to these relative minima (minima, i. e ., in regard to weight) for the case where the terms to be taken away from the equation occupy consecutive places in it, beginning with the second.

scholarly journals Numerical Solutions for Multi-Term Fractional Order Differential Equations with Fractional Taylor Operational Matrix of Fractional Integration

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 96 ◽  
Author(s):  
İbrahim Avcı ◽  
Nazim I. Mahmudov
Keyword(s):  
Differential Equations ◽  
Numerical Solutions ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Main Idea ◽  
System Of Equations ◽  
Algebraic Equations ◽  
Fractional Order Differential Equations ◽  
Fractional Differential ◽  
The Given

In this article, we propose a numerical method based on the fractional Taylor vector for solving multi-term fractional differential equations. The main idea of this method is to reduce the given problems to a set of algebraic equations by utilizing the fractional Taylor operational matrix of fractional integration. This system of equations can be solved efficiently. Some numerical examples are given to demonstrate the accuracy and applicability. The results show that the presented method is efficient and applicable.

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