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Related Concept Videos

Alternative Sets of Equilibrium Equations01:31

Alternative Sets of Equilibrium Equations

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When analyzing the behavior of structures, engineers often rely on the concept of equilibrium. This refers to the state where all forces and moments acting on a system balance each other, resulting in no net movement or rotation. In many cases, equilibrium can be described by a set of standard equations. However, in some situations, alternative sets of equilibrium equations must be used to describe the system's behavior accurately.
One example of such a situation can be observed in a...
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Dynamic Equilibrium02:20

Dynamic Equilibrium

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A reversible chemical reaction represents a chemical process that proceeds in both forward (left to right) and reverse (right to left) directions. When the rates of the forward and reverse reactions are equal, the concentrations of the reactant and product species remain constant over time and the system is at equilibrium. A special double arrow is used to emphasize the reversible nature of the reaction. The relative concentrations of reactants and products in equilibrium systems vary greatly;...
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Area Computation by the Alternative Coordinate Method01:24

Area Computation by the Alternative Coordinate Method

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The alternative coordinate method, also known as the Shoelace Formula, is a technique for determining the area of a traverse using Cartesian coordinates. This method relies on the sequential arrangement of x and y coordinates for each point of the shape, ensuring accuracy and ease of application.In this approach, each corner's x and y coordinates are listed as fractions, with the x-coordinate as the numerator and the y-coordinate as the denominator. These coordinates are arranged sequentially...
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Free Energy and Equilibrium02:56

Free Energy and Equilibrium

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The free energy change for a process may be viewed as a measure of its driving force. A negative value for ΔG represents a driving force for the process in the forward direction, while a positive value represents a driving force for the process in the reverse direction. When ΔGrxn is zero, the forward and reverse driving forces are equal, and the process occurs in both directions at the same rate (the system is at equilibrium).
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Calculating the Equilibrium Constant02:46

Calculating the Equilibrium Constant

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The equilibrium constant for a reaction is calculated from the equilibrium concentrations (or pressures) of its reactants and products. If these concentrations are known, the calculation simply involves their substitution into the Kc expression.
For example, gaseous nitrogen dioxide forms dinitrogen tetroxide according to this equation:
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Solution Equilibrium and Saturation01:59

Solution Equilibrium and Saturation

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Imagine adding a small amount of sugar to a glass of water, stirring until all the sugar has dissolved, and then adding a bit more. You can repeat this process until the sugar concentration of the solution reaches its natural limit, a limit determined primarily by the relative strengths of the solute-solute, solute-solvent, and solvent-solvent attractive forces. You can be certain that you have reached this limit because, no matter how long you stir the solution, undissolved sugar remains. The...
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An alternative extragradient projection method for quasi-equilibrium problems.

Haibin Chen1, Yiju Wang1, Yi Xu2

  • 11School of Management Science, Qufu Normal University, Rizhao Shandong, China.

Journal of Inequalities and Applications
|February 9, 2018
PubMed
Summary
This summary is machine-generated.

A novel extragradient projection method was developed for quasi-equilibrium problems. This method demonstrates global convergence and converges to the nearest solution point, outperforming classical approaches in numerical tests.

Keywords:
Extragradient projection methodMulti-valued mappingPseudomonotonicityQuasi-equilibrium problems

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Area of Science:

  • Optimization
  • Game Theory
  • Applied Mathematics

Background:

  • Quasi-equilibrium problems involve complex strategic interactions.
  • Existing methods like classical extragradient projection have limitations.

Purpose of the Study:

  • To design an alternative extragradient projection method for quasi-equilibrium problems.
  • To analyze the convergence properties of the new method.

Main Methods:

  • An alternative extragradient projection method is proposed.
  • The method's expansion property relative to an initial point is analyzed.
  • Global convergence is established using pseudomonotonicity and continuity assumptions.

Main Results:

  • The new method exhibits an expansion property, unlike classical methods.
  • Global convergence is proven for the designed algorithm.
  • The generated sequence converges to the nearest point in the solution set.

Conclusions:

  • The developed extragradient projection method is efficient for solving quasi-equilibrium problems.
  • The method offers improved convergence characteristics.
  • Numerical experiments validate the method's effectiveness.