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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...
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In the application of the Routh-Hurwitz criterion, two specific scenarios can arise that complicate stability analysis.
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Consider an electrical power grid, where stability is essential to prevent blackouts. The Routh-Hurwitz criterion is a valuable tool for assessing system stability under varying load conditions or faults. By analyzing the closed-loop transfer function, the Routh-Hurwitz criterion helps determine whether the system remains stable.
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Trigonometric and exponential functions are essential mathematical tools used to model distinct types of real-world behavior, particularly in periodic and growth-related phenomena. These functions extend the capabilities of basic algebraic models by capturing recurring cycles and rapid changes across various scientific and engineering contexts.Trigonometric functions, such as sine and cosine, are particularly effective for representing periodic phenomena. Their cyclic behavior makes them...
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Quasi-Herglotz functions and convex optimization.

Y Ivanenko1, M Nedic2, M Gustafsson3

  • 1Department of Physics and Electrical Engineering, Linnæus University, 351 95 Växjö, Sweden.

Royal Society Open Science
|March 29, 2020
PubMed
Summary
This summary is machine-generated.

We introduce quasi-Herglotz functions, an extension of Herglotz functions, for modeling non-passive systems. These functions inherit key properties, enabling new approaches in approximation theory and system modeling.

Keywords:
approximationconvex optimizationnon-passive systemsquasi-Herglotz functionssum rules

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Area of Science:

  • Mathematical Physics
  • System Modeling
  • Approximation Theory

Background:

  • Herglotz functions are crucial for modeling passive systems.
  • Non-passive systems require advanced mathematical tools for accurate representation.
  • Existing models may not fully capture the complexities of non-passive behaviors.

Purpose of the Study:

  • Introduce the novel set of quasi-Herglotz functions.
  • Demonstrate their utility in modeling non-passive systems.
  • Extend the properties and modeling perspectives of Herglotz functions.

Main Methods:

  • Define quasi-Herglotz functions as differences of Herglotz functions.
  • Analyze the inheritance of properties like integral representations and boundary values.
  • Apply these functions to model a non-passive gain medium using convex optimization.

Main Results:

  • Quasi-Herglotz functions form a linear space, extending the Herglotz function cone.
  • Key properties such as integral representations and sum rules are inherited.
  • Successful modeling of a non-passive gain medium was achieved using B-splines and point masses.

Conclusions:

  • Quasi-Herglotz functions offer a powerful framework for non-passive system modeling.
  • This extension preserves and expands upon the established theory of Herglotz functions.
  • The numerical examples validate the practical applicability of quasi-Herglotz functions.