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Related Concept Videos

Sequence Networks of Rotating Machines01:24

Sequence Networks of Rotating Machines

A Y-connected synchronous generator, grounded through a neutral impedance, is designed to produce balanced internal phase voltages with only positive-sequence components. The generator's sequence networks include a source voltage that is exclusively in the positive-sequence network. The sequence components of line-to-ground voltages at the generator terminals illustrate this configuration.
Zero-sequence current induces a voltage drop across the generator's neutral impedance and other...
Multimachine Stability01:25

Multimachine Stability

Multimachine stability analysis is crucial for understanding the dynamics and stability of power systems with multiple synchronous machines. The objective is to solve the swing equations for a network of M machines connected to an N-bus power system.
In analyzing the system, the nodal equations represent the relationship between bus voltages, machine voltages, and machine currents. The nodal equation is given by:
Network Function of a Circuit01:25

Network Function of a Circuit

Frequency response analysis in electrical circuits provides vital insights into a circuit's behavior as the frequency of the input signal changes. The transfer function, a mathematical tool, is instrumental in understanding this behavior. It defines the relationship between phasor output and input and comes in four types: voltage gain, current gain, transfer impedance, and transfer admittance. The critical components of the transfer function are the poles and zeros.
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.
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Entropy Change in Reversible Processes01:10

Entropy Change in Reversible Processes

In the Carnot engine, which achieves the maximum efficiency between two reservoirs of fixed temperatures, the total change in entropy is zero. The observation can be generalized by considering any reversible cyclic process consisting of many Carnot cycles. Thus, it can be stated that the total entropy change of any ideal reversible cycle is zero.
The statement can be further generalized to prove that entropy is a state function. Take a cyclic process between any two points on a p-V diagram.
Woodward–Hoffmann Selection Rules and Microscopic Reversibility01:34

Woodward–Hoffmann Selection Rules and Microscopic Reversibility

Electrocyclic reactions, cycloadditions, and sigmatropic rearrangements are concerted pericyclic reactions that proceed via a cyclic transition state. These reactions are stereospecific and regioselective. The stereochemistry of the products depends on the symmetry characteristics of the interacting orbitals and the reaction conditions. Accordingly, pericyclic reactions are classified as either symmetry-allowed or symmetry-forbidden. Woodward and Hoffmann presented the selection criteria for...

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Related Experiment Video

Updated: May 16, 2026

Sealable Femtoliter Chamber Arrays for Cell-free Biology
13:44

Sealable Femtoliter Chamber Arrays for Cell-free Biology

Published on: March 11, 2015

Stochastic Turing patterns on a network.

Malbor Asslani1, Francesca Di Patti, Duccio Fanelli

  • 1Dipartimento di Scienza e Alta Tecnologia, Università degli Studi dell'Insubria, via Valleggio 11, 22100 Como, Italy.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 11, 2012
PubMed
Summary
This summary is machine-generated.

Stochastic Turing instability in scale-free networks can cause pattern formation beyond deterministic limits. This occurs due to inherent system graininess, leading to spontaneous node differentiation.

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Last Updated: May 16, 2026

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Area of Science:

  • Complex Systems
  • Chemical Kinetics
  • Network Science

Background:

  • Turing instability typically requires specific parameter ranges in deterministic systems.
  • Scale-free networks exhibit unique topological properties influencing reaction-diffusion dynamics.
  • Stochastic effects can alter classical pattern formation mechanisms.

Purpose of the Study:

  • To investigate stochastic Turing instability on scale-free networks.
  • To analyze pattern formation in the stochastic Brusselator model.
  • To understand the role of finite-size effects in stochastic pattern generation.

Main Methods:

  • Direct stochastic simulations of the Brusselator model on a scale-free network.
  • Analytical explanation of observed phenomena.
  • Identification of finite-size corrections as the underlying cause.

Main Results:

  • Spontaneous differentiation into activator-rich and activator-poor nodes observed.
  • Pattern formation occurred outside the classical deterministic Turing instability parameter region.
  • The phenomenon was analytically explained and linked to finite-size corrections.

Conclusions:

  • Stochasticity enables pattern formation in systems where deterministic models predict none.
  • Finite-size corrections in stochastic systems are crucial for understanding emergent spatial patterns.
  • Scale-free network topology combined with stochasticity provides a rich framework for studying complex phenomena.