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Related Concept Videos

Continuity of a Function01:23

Continuity of a Function

A function is continuous at a point a if three conditions are met: the function is defined at a, the limit of the function as x approaches a exists, and this limit equals the function’s value. Mathematically, this is written asThis definition ensures the graph of the function does not exhibit any breaks, holes, or jumps at that point. Discontinuities occur when any of these conditions fail. A removable discontinuity exists when the two-sided limit exists but the function is either undefined or...
Continuity Equation01:28

Continuity Equation

The continuity equation asserts that the mass flow rate must remain constant for a steady flow of an incompressible fluid within a confined system. This principle applies to systems where fluid passes through varying cross-sectional areas, such as nozzles, syringes, and pipes.
The mass flow rate is expressed as:
Continuity Equation01:20

Continuity Equation

The total amount of current flowing per unit cross-sectional area is called the current density. Hence, the current passing through a cross-sectional area can be written as the surface integral of the current density.
Limits with Oscillating Discontinuities01:19

Limits with Oscillating Discontinuities

An oscillating discontinuity is a type of discontinuity in which a function’s values fluctuate infinitely often as the input approaches a particular point. Unlike jump discontinuities, where the function suddenly shifts between two values, or infinite discontinuities, where the function diverges without bound, an oscillating discontinuity arises from rapid back-and-forth variation. Because the function never stabilizes toward a single value, no finite limit exists at that point.One of the most...
Properties of Continuous Functions01:29

Properties of Continuous Functions

Continuous functions exhibit smooth, uninterrupted behavior, and combining them through standard operations retains this continuity. If f and g are continuous at a point a, then the functions f+g, f-g, cf (where c is a constant), fg, and fg (provided g(a)a) are also continuous at a. This allows the construction of complex functions from simpler continuous parts without losing smoothness.Polynomials, which are expressions formed by sums of powers of x with constant coefficients, are continuous...
Gradually Varying Flow01:29

Gradually Varying Flow

Gradually varying flow (GVF) in open channels describes situations where water depth changes slowly along the channel due to factors like non-uniform bed slope, channel shape variations, or obstructions. This flow type occurs when the depth adjusts gradually to balance gravitational forces, shear forces, and energy requirements, resulting in a low rate of depth change.Characteristics of Gradually Varying FlowGVF is commonly observed in natural streams, rivers, and canals, where flow depth...

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Related Experiment Video

Updated: May 23, 2026

Reservoir Condition Pore-scale Imaging of Multiple Fluid Phases Using X-ray Microtomography
08:02

Reservoir Condition Pore-scale Imaging of Multiple Fluid Phases Using X-ray Microtomography

Published on: February 25, 2015

Ordinary percolation with discontinuous transitions.

Stefan Boettcher1, Vijay Singh, Robert M Ziff

  • 1Department of Physics, Emory University, Atlanta, Georgia 30322, USA. sboettc@emory.edu

Nature Communications
|April 19, 2012
PubMed
Summary

Introducing small-world bonds to 1D lattices creates explosive percolation transitions. This novel behavior occurs at a non-trivial critical point, unlike traditional models.

Area of Science:

  • Complex systems
  • Network science
  • Statistical physics

Background:

  • Percolation on 1D lattices and fractals is typically trivial, requiring full bond density for network connectivity.
  • Traditional models lack the rich phenomena observed in more complex network structures.

Purpose of the Study:

  • To introduce and rigorously analyze a novel percolation transition in a modified 1D lattice.
  • To demonstrate explosive cluster growth and a non-trivial critical point in a simplified network model.

Main Methods:

  • Constructing a small-world network by adding long-range bonds to a 1D lattice.
  • Employing mathematical analysis to rigorously characterize the percolation transition.
  • Investigating the behavior of the order parameter, representing the largest cluster's size.

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Evolution of Staircase Structures in Diffusive Convection
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Evolution of Staircase Structures in Diffusive Convection

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Last Updated: May 23, 2026

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Published on: February 25, 2015

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Evolution of Staircase Structures in Diffusive Convection
07:28

Evolution of Staircase Structures in Diffusive Convection

Published on: September 5, 2018

Main Results:

  • A novel percolation transition emerges at a non-trivial critical point with the addition of small-world bonds.
  • The system exhibits explosive cluster growth, a phenomenon not seen in standard 1D percolation.
  • The order parameter shows an instantaneous jump to a finite value at the critical point.

Conclusions:

  • Small-world networks can host complex percolation phenomena, including explosive transitions.
  • The modified 1D lattice serves as a tractable model for studying these emergent behaviors.
  • This research bridges concepts from network science and statistical physics to explain complex phenomena.