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Related Concept Videos

Random Variables01:09

Random Variables

A random variable is a single numerical value that indicates the outcome of a procedure. The concept of random variables is fundamental to the probability theory and was introduced by a Russian mathematician, Pafnuty Chebyshev, in the mid-nineteenth century.
Uppercase letters such as X or Y denote a random variable. Lowercase letters like x or y denote the value of a random variable. If X is a random variable, then X is written in words, and x is given as a number.
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The randomization process involves assigning study participants randomly to experimental or control groups based on their probability of being equally assigned. Randomization is meant to eliminate selection bias and balance known and unknown confounding factors so that the control group is similar to the treatment group as much as possible. A computer program and a random number generator can be used to assign participants to groups in a way that minimizes bias.
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Routh-Hurwitz Criterion II01:19

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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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Related Experiment Video

Updated: Jun 29, 2026

2D and 3D Matrices to Study Linear Invadosome Formation and Activity
12:25

2D and 3D Matrices to Study Linear Invadosome Formation and Activity

Published on: June 2, 2017

Random cyclic matrices.

Sudhir R Jain1, Shashi C L Srivastava

  • 1Nuclear Physics Division, Van de Graaff Building, Bhabha Atomic Research Centre, Trombay, Mumbai 400 085, India. srjain@barc.gov.in

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|October 15, 2008
PubMed
Summary
This summary is machine-generated.

We studied random cyclic matrices and their eigenvalue spacing. The distribution matches Wigner

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Generation and Coherent Control of Pulsed Quantum Frequency Combs
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Generation and Coherent Control of Pulsed Quantum Frequency Combs

Published on: June 8, 2018

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Last Updated: Jun 29, 2026

2D and 3D Matrices to Study Linear Invadosome Formation and Activity
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2D and 3D Matrices to Study Linear Invadosome Formation and Activity

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Generation and Coherent Control of Pulsed Quantum Frequency Combs
06:42

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Published on: June 8, 2018

Area of Science:

  • Random Matrix Theory
  • Condensed Matter Physics
  • Quantum Chromodynamics

Background:

  • Cyclic matrices appear in diverse physical systems.
  • Understanding their spectral properties is crucial for various applications.

Purpose of the Study:

  • To analyze the spectral fluctuations of Gaussian ensembles of random cyclic matrices.
  • To compute eigenvalue distributions and spacing distributions.

Main Methods:

  • Analytical calculations of joint probability distribution functions.
  • Numerical computations for eigenvalue and spacing distributions.
  • Exploration of pseudosymmetry with respect to generalized parity.

Main Results:

  • Level spacing distribution shows Gaussian or linear forms for small spacings.
  • Spacing distribution for complex eigenvalues aligns with Wigner distribution for planar Poisson processes.
  • Demonstrated pseudosymmetry of cyclic matrices.

Conclusions:

  • Exact results for spectral fluctuations of cyclic matrices are derived.
  • Findings are applicable to disordered atomic chains, the Ising model, and statistical mechanics.
  • Provides insights into nu-parametrized quantum chromodynamics.