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Related Concept Videos

Routh-Hurwitz Criterion II01:19

Routh-Hurwitz Criterion II

In the application of the Routh-Hurwitz criterion, two specific scenarios can arise that complicate stability analysis.
The first scenario occurs when a singular zero appears in the first column of the Routh table. This situation creates a division by zero issues. To resolve this, a small positive or negative number, denoted as epsilon (∈), is substituted for the zero. The stability analysis proceeds by assuming a sign for ∈. If ∈ is positive, any sign change in the first column of the Routh...
Types of Functions I01:26

Types of Functions I

Functions are fundamental mathematical tools that capture relationships between variables and describe how one quantity changes in relation to another. Their diverse forms allow them to model various real-world phenomena with precision and flexibility. Among the various categories, algebraic functions are prominent due to their formulation through basic arithmetic operations: addition, subtraction, multiplication, division, and root extraction.Algebraic functions include polynomial, rational,...
Transfer Function in Control Systems01:21

Transfer Function in Control Systems

The transfer function is a fundamental concept in the analysis and design of linear time-invariant (LTI) systems. It offers a concise way to understand how a system responds to different inputs in the frequency domain. It serves as a bridge between the time-domain differential equations that describe system dynamics and the frequency-domain representation that facilitates easier manipulation and analysis.
To derive the transfer function, consider a general nth-order linear time-invariant...
Piecewise-Defined Functions01:28

Piecewise-Defined Functions

Piecewise defined functions are mathematical models where different expressions define a function over distinct intervals of the domain. These functions are useful for representing systems with varying behaviors depending on input values.For example, the function:  uses a linear rule for inputs less than or equal to –1 and a quadratic rule for values greater than –1. Although it has two formulas, it still defines a single function.Another common type is the absolute value function, given...
Introduction to One-to-one Functions01:23

Introduction to One-to-one Functions

A one-to-one function is a mathematical function in which each element of the domain maps to a distinct and unique element in the range. This property ensures that no two different inputs result in the same output, formally expressed as f (x1) ≠ f (x2) whenever x1 ≠ x2. The graphical criterion for identifying such functions is the Horizontal Line Test, which indicates that a function is one-to-one if and only if no horizontal line intersects its graph at more than one point.A quadratic function...
Slant Asymptotes01:27

Slant Asymptotes

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...

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Related Experiment Video

Updated: May 13, 2026

Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns
13:44

Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns

Published on: August 30, 2013

The characteristic function, a method-specific alternative to the Horwitz function.

Michael Thompson1

  • 1Birkbeck University of London, School of Biological and Chemical Sciences, Malet St, London WC1E 7HX, United Kingdom. m.thompson@bbk.ac.uk

Journal of AOAC International
|March 5, 2013
PubMed
Summary
This summary is machine-generated.

The characteristic function better describes individual analytical method precision, especially near detection limits, compared to the Horwitz function. It also offers a more adaptable fitness-for-purpose criterion.

Related Experiment Videos

Last Updated: May 13, 2026

Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns
13:44

Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns

Published on: August 30, 2013

Area of Science:

  • Analytical Chemistry
  • Method Validation
  • Statistical Analysis

Background:

  • The Horwitz function models reproducibility standard deviations in food analysis across analyte concentrations.
  • It has limitations in describing individual analytical method precision.

Purpose of the Study:

  • Compare the Horwitz function and characteristic function for describing analytical method precision.
  • Evaluate their suitability for fitness-for-purpose criteria.

Main Methods:

  • Comparative analysis of the Horwitz function and characteristic function.
  • Assessment of their mathematical forms and adaptability.

Main Results:

  • The characteristic function is more adaptable for individual method precision, particularly down to detection limits.
  • The characteristic function offers a better-suited functional form for fitness-for-purpose assessment than the Horwitz ratio (HorRat).

Conclusions:

  • The characteristic function is a superior alternative for assessing analytical method precision, especially when detection limits are critical.
  • The characteristic function provides a more robust framework for fitness-for-purpose evaluations in analytical chemistry.