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Related Concept Videos

Partial Fractions01:28

Partial Fractions

294
A partial fraction is a component of a rational expression represented as the sum of simpler fractions. When a rational function is expressed as a ratio of two polynomials, it can often be decomposed into a sum of fractions whose denominators are simpler polynomials, typically linear or irreducible quadratic factors. This process is called partial fraction decomposition, and it is used to simplify complex expressions for integration, solving equations, or analysis.Partial fraction decomposition...
294
Quadratic Equations01:29

Quadratic Equations

536
A quadratic equation is an algebraic expression where a variable is raised to the second power and combined with its first power and a constant; all equated to zero. These equations are frequently used to model relationships involving area, motion, and optimization. The general representation of a quadratic equation iswhere a, b, and c are real values, and a is nonzero to ensure the presence of the squared term.One method for solving a quadratic equation involves rewriting it as a product of...
536
Application of Nonlinear Inequalities01:29

Application of Nonlinear Inequalities

286
A nonlinear inequality describes a comparison involving an expression that curves or behaves more complexly than a straight line. These inequalities often appear in forms that include squares, products, or variables in the denominator.To solve such an inequality, one starts by rewriting it so that zero appears on one side. For example, the inequality:  can be factored as: This form makes it easier to identify the values that cause the expression to equal zero. In this case, the...
286
Integration of Rational Functions Using Partial Fractions01:29

Integration of Rational Functions Using Partial Fractions

214
Rational functions are expressions written as the ratio of two polynomials, and their integrals are evaluated by simplifying the integrand into manageable parts. These functions are classified as proper or improper based on the degrees of the numerator and denominator.A rational function is proper when the degree of the numerator is less than the degree of the denominator. In this case, partial fraction decomposition is used to rewrite the function as a sum of simpler rational terms. The...
214
Slant Asymptotes01:27

Slant Asymptotes

192
A function's behavior is often guided by asymptotic constraints, where one term dominates another, defining a limiting trend. In the given scenario, the mathematical pattern follows a rational function: a cubic term in the numerator is divided by a squared term in the denominator. This results in a function with distinct characteristics, including an oblique asymptote, critical points, and undefined regions.The function's validity is determined by the denominator, which must be nonzero. This...
192
Second Derivatives of Implicit Functions01:29

Second Derivatives of Implicit Functions

121
Elliptical arches are fundamental in architectural and structural engineering, offering aesthetic appeal and structural efficiency. The shape of an elliptical arch follows a constrained geometric relationship where the height and horizontal position are implicitly related. This means that the height y cannot be explicitly expressed as a function of the horizontal position x, necessitating implicit differentiation for slope and curvature analysis.The equation of an ellipse centered at the origin...
121

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Related Experiment Video

Updated: Mar 18, 2026

Spatial Multiobjective Optimization of Agricultural Conservation Practices using a SWAT Model and an Evolutionary Algorithm
11:53

Spatial Multiobjective Optimization of Agricultural Conservation Practices using a SWAT Model and an Evolutionary Algorithm

Published on: December 9, 2012

13.6K

Second Order Duality in Multiobjective Fractional Programming with Square Root Term under Generalized Univex

Arun Kumar Tripathy1

  • 1Department of Mathematics, Trident Academy of Technology, F2/A, Chandaka Industrial Estate, Bhubaneswar, Odisha 751024, India.

International Scholarly Research Notices
|July 6, 2016
PubMed
Summary
This summary is machine-generated.

This study introduces novel duality approaches for complex multiobjective fractional programming problems involving square roots. It establishes conditions for efficient solutions and proves duality theorems using a parameterization technique.

Related Experiment Videos

Last Updated: Mar 18, 2026

Spatial Multiobjective Optimization of Agricultural Conservation Practices using a SWAT Model and an Evolutionary Algorithm
11:53

Spatial Multiobjective Optimization of Agricultural Conservation Practices using a SWAT Model and an Evolutionary Algorithm

Published on: December 9, 2012

13.6K

Area of Science:

  • Optimization Theory
  • Mathematical Programming
  • Operations Research

Background:

  • Fractional programming problems are common in various fields.
  • Multiobjective optimization involves multiple conflicting objectives.
  • Nondifferentiable functions and square roots add complexity.

Purpose of the Study:

  • To introduce new second-order mixed-type duality approaches.
  • To establish necessary and sufficient conditions for efficient solutions.
  • To extend duality results to generalized second-order ρ-univexity.

Main Methods:

  • Development of three distinct duality approaches.
  • Application of a parameterization technique.
  • Analysis of nondifferentiable multiobjective fractional programming problems.

Main Results:

  • Successful introduction of second-order mixed-type duality.
  • Establishment of conditions for efficient solutions in fractional programming.
  • Duality results are proven under generalized second-order ρ-univexity.

Conclusions:

  • The proposed duality approaches are effective for the studied problem class.
  • The findings contribute to the theory of multiobjective fractional programming.
  • The research provides a framework for analyzing complex optimization problems.