scholarly journals Difference of Some Positive Linear Approximation Operators for Higher-Order Derivatives

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 915
Author(s):  
Vijay Gupta ◽  
Ana Maria Acu ◽  
Hari Mohan Srivastava
Keyword(s):  
Linear Approximation ◽  
Approximation Theory ◽  
Classical Result ◽  
Baskakov Operators ◽  
Numerical Examples ◽  
Durrmeyer Operators ◽  
Difference Of Operators ◽  
Durrmeyer Type Operators ◽  
The Difference ◽  
Theoretical Results

In the present paper, we deal with some general estimates for the difference of operators which are associated with different fundamental functions. In order to exemplify the theoretical results presented in (for example) Theorem 2, we provide the estimates of the differences between some of the most representative operators used in Approximation Theory in especially the difference between the Baskakov and the Szász–Mirakyan operators, the difference between the Baskakov and the Szász–Mirakyan–Baskakov operators, the difference of two genuine-Durrmeyer type operators, and the difference of the Durrmeyer operators and the Lupaş–Durrmeyer operators. By means of illustrative numerical examples, we show that, for particular cases, our result improves the estimates obtained by using the classical result of Shisha and Mond. We also provide the symmetry aspects of some of these approximations operators which we have studied in this paper.

scholarly journals On difference of operators with different basis functions

Filomat ◽  
2019 ◽  
Vol 33 (10) ◽  
pp. 3023-3034 ◽  
Author(s):  
Vijay Gupta ◽  
Ana Acu
Keyword(s):  
Quantitative Estimate ◽  
Classical Result ◽  
Positive Operators ◽  
Basis Functions ◽  
Linear Positive Operators ◽  
Numerical Examples ◽  
Difference Of Operators ◽  
The Difference

In the recent years several researchers have studied problems concerning the difference of two linear positive operators, but all the available literature on this topic is for operators having same basis functions. In the present paper, we deal with the general quantitative estimate for the difference of operators having different basis functions. In the end we provide some examples. The estimates for the differences of two operators can be obtained also using classical result of Shisha and Mond. Using numerical examples we will show that for particular cases our result improves the classical one.

scholarly journals Better approximation of functions by genuine Bernstein-Durrmeyer type operators

2019 ◽  
Vol 35 (2) ◽  
pp. 125-136
Author(s):  
ANA MARIA ACU ◽  
P. N. AGRAWAL ◽  
◽  
Keyword(s):  
Asymptotic Formula ◽  
Rate Of Convergence ◽  
Modulus Of Continuity ◽  
Approximation Of Functions ◽  
Numerical Examples ◽  
Durrmeyer Operator ◽  
Durrmeyer Type Operators ◽  
Theoretical Results ◽  
Second Modulus Of Continuity

The main object of this paper is to construct a new genuine Bernstein-Durrmeyer type operators which have better features than the classical one. Some direct estimates for the modified genuine Bernstein-Durrmeyer operator by means of the first and second modulus of continuity are given. An asymptotic formula for the new operator is proved. Finally, some numerical examples with illustrative graphics have been added to validate the theoretical results and also compare the rate of convergence.

Oscillation criteria for a class of nonlinear discrete fractional order equations with damping term

2020 ◽  
Vol 70 (5) ◽  
pp. 1165-1182
Author(s):  
George E. Chatzarakis ◽  
George M. Selvam ◽  
Rajendran Janagaraj ◽  
George N. Miliaras
Keyword(s):  
Fractional Order ◽  
Difference Operator ◽  
Sufficient Conditions ◽  
Damping Term ◽  
Transformation Technique ◽  
Order Difference ◽  
Numerical Examples ◽  
Oscillation Criteria ◽  
The Difference ◽  
Theoretical Results

AbstractThe aim in this work is to investigate oscillation criteria for a class of nonlinear discrete fractional order equations with damping term of the form$$\begin{array}{} \displaystyle \Delta\left[a(t)\left[\Delta\left(r(t)g\left(\Delta^\alpha x(t)\right)\right)\right]^\beta\right]+p(t)\left[\Delta\left(r(t)g\left(\Delta^\alpha x(t)\right)\right)\right]^\beta+F(t,G(t))=0, t\in N_{t_0}. \end{array}$$In the above equation α (0 < α ≤ 1) is the fractional order, $\begin{array}{} \displaystyle G(t)=\sum\limits_{s=t_0}^{t-1+\alpha}\left(t-s-1\right)^{(-\alpha)}x(s) \end{array}$ and Δα is the difference operator of the Riemann-Liouville (R-L) derivative of order α. We establish some new sufficient conditions for the oscillation of fractional order difference equations with damping term based on a Riccati transformation technique and some inequalities. We provide numerical examples to illustrate the validity of the theoretical results.

A Finite Difference Method for Boundary Value Problems of a Caputo Fractional Differential Equation

2017 ◽  
Vol 7 (4) ◽  
pp. 752-766 ◽  
Author(s):  
Pin Lyu ◽  
Seakweng Vong ◽  
Zhibo Wang
Keyword(s):  
Finite Difference ◽  
Boundary Value ◽  
Equivalent Form ◽  
Point Boundary ◽  
Difference System ◽  
Numerical Examples ◽  
Main Equation ◽  
Fractional Differential ◽  
The Difference ◽  
Theoretical Results

AbstractIn this paper, we consider a two-point boundary value problem with Caputo fractional derivative, where the second order derivative of the exact solution is unbounded. Based on the equivalent form of the main equation, a finite difference scheme is derived. The L∞ convergence of the difference system is discussed rigorously. The convergence rate in general improves previous results. Numerical examples are provided to demonstrate the theoretical results.

scholarly journals Non-submodular model for group profit maximization problem in social networks

2021 ◽  
Vol 8 (1) ◽  
Author(s):  
Jianming Zhu ◽  
Smita Ghosh ◽  
Weili Wu ◽  
Chuangen Gao
Keyword(s):  
Social Networks ◽  
Profit Maximization ◽  
Randomized Algorithm ◽  
Political Orientation ◽  
Submodular Functions ◽  
Maximization Problem ◽  
Modular Algorithm ◽  
The Difference ◽  
Group Enterprise ◽  
Theoretical Results

AbstractIn social networks, there exist many kinds of groups in which people may have the same interests, hobbies, or political orientation. Sometimes, group decisions are made by simply majority, which means that most of the users in this group reach an agreement, such as US Presidential Elections. A group is called activated if $$\beta$$ β percent of users are influenced in the group. Enterprise will gain income from all influenced groups. Simultaneously, to propagate influence, enterprise needs pay advertisement diffusion cost. Group profit maximization (GPM) problem aims to pick k seeds to maximize the expected profit that considers the benefit of influenced groups with the diffusion cost. GPM is proved to be NP-hard and the objective function is proved to be neither submodular nor supermodular. An upper bound and a lower bound which are difference of two submodular functions are designed. We propose a submodular–modular algorithm (SMA) to solve the difference of two submodular functions and SMA is shown to converge to a local optimal. We present an randomized algorithm based on weighted group coverage maximization for GPM and apply sandwich framework to get theoretical results. Our experiments verify the efficiency of our methods.

Analogue of a theorem of Khintchine in fields of formal power series

1966 ◽  
Vol 62 (4) ◽  
pp. 637-642 ◽  
Author(s):  
T. W. Cusick
Keyword(s):  
Power Series ◽  
Positive Constant ◽  
Classical Result ◽  
Real Numbers ◽  
Linear Forms ◽  
Absolute Value ◽  
The Difference ◽  
Image Position ◽  
Formal Power ◽  
Positive Integers

For a real number λ, ‖λ‖ is the absolute value of the difference between λ and the nearest integer. Let X represent the m-tuple (x1, x2, … xm) and letbe any n linear forms in m variables, where the Θij are real numbers. The following is a classical result of Khintchine (1):For all pairs of positive integers m, n there is a positive constant Г(m, n) with the property that for any forms Lj(X) there exist real numbers α1, α2, …, αn such thatfor all integers x1, x2, …, xm not all zero.

scholarly journals Edge detection with trigonometric polynomial shearlets

2021 ◽  
Vol 47 (1) ◽  
Author(s):  
Kevin Schober ◽  
Jürgen Prestin ◽  
Serhii A. Stasyuk
Keyword(s):  
Edge Detection ◽  
Trigonometric Polynomial ◽  
Two Dimensions ◽  
Characteristic Functions ◽  
Numerical Examples ◽  
Special Cases ◽  
Periodic Characteristic ◽  
Boundary Curves ◽  
Theoretical Results ◽  
De La Vallée Poussin

AbstractIn this paper, we show that certain trigonometric polynomial shearlets which are special cases of directional de la Vallée Poussin-type wavelets are able to detect step discontinuities along boundary curves of periodic characteristic functions. Motivated by recent results for discrete shearlets in two dimensions, we provide lower and upper estimates for the magnitude of the corresponding inner products. In the proof, we use localization properties of trigonometric polynomial shearlets in the time and frequency domain and, among other things, bounds for certain Fresnel integrals. Moreover, we give numerical examples which underline the theoretical results.

scholarly journals Stability Analysis of Impulsive Control Systems with Finite and Infinite Delays

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Xuling Wang ◽  
Xiaodi Li ◽  
Gani Tr. Stamov
Keyword(s):  
Control Systems ◽  
Impulsive Control ◽  
Sufficient Conditions ◽  
Delay Systems ◽  
Stability Criteria ◽  
Numerical Examples ◽  
Infinite Delays ◽  
Impulsive Control Systems ◽  
Stable Matrix ◽  
Theoretical Results

This paper studies impulsive control systems with finite and infinite delays. Several stability criteria are established by employing the largest and smallest eigenvalue of matrix. Our sufficient conditions are less restrictive than the ones in the earlier literature. Moreover, it is shown that by using impulsive control, the delay systems can be stabilized even if it contains no stable matrix. Finally, some numerical examples are discussed to illustrate the theoretical results.

scholarly journals Approximation by Jakimovski-Leviatan-Stancu-Durrmeyer type operators

Filomat ◽  
2019 ◽  
Vol 33 (6) ◽  
pp. 1517-1530 ◽  
Author(s):  
M. Mursaleen ◽  
Shagufta Rahman ◽  
Khursheed Ansari
Keyword(s):  
Upper Bound ◽  
Simultaneous Approximation ◽  
Approximation Theorem ◽  
Approximation Properties ◽  
Statistical Approximation ◽  
Durrmeyer Operators ◽  
Durrmeyer Type Operators

In the present paper, we introduce Stancu type modification of Jakimovski-Leviatan-Durrmeyer operators. First, we estimate moments of these operators. Next, we study the problem of simultaneous approximation by these operators. An upper bound for the approximation to rth derivative of a function by these operators is established. Furthermore, we obtain A-statistical approximation properties of these operators with the help of universal korovkin type statistical approximation theorem.

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.

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