Search research articles

ABOUT JoVE

Overview Leadership Blog JoVE Help Center

AUTHORS

Publishing Process Editorial Board Scope & Policies Peer Review FAQ Submit

LIBRARIANS

Testimonials Subscriptions Access Resources Library Advisory Board FAQ

RESEARCH

JoVE Journal Methods Collections JoVE Encyclopedia of Experiments Archive

EDUCATION

JoVE Core JoVE Business JoVE Science Education JoVE Lab Manual Faculty Resource Center Faculty Site

Terms & Conditions of Use

Related Concept Videos

Critical Numbers and the Closed Interval Method

Critical Numbers and the Closed Interval Method

Understanding the maximum and minimum values of a function is essential for analyzing its overall behavior. These values, often referred to as extrema, provide insight into how a function behaves across its domain. In mathematical terms, extrema can be either local—representing peaks and valleys within a limited region—or absolute, indicating the highest or lowest points over an entire interval.A function’s extrema occur at critical numbers, which are values in the domain where the derivative...

Local Maximum and Minimum Values

Local Maximum and Minimum Values

In multivariable calculus, a function of two variables can exhibit local maximum or minimum values at certain points on its surface. A local maximum occurs when the function's value at a point is greater than at all nearby points, while a local minimum occurs when the function’s value is less than at all nearby locations. These points are referred to as local extrema and are of central importance in optimization problems.Local extrema are found at critical points, where the surface becomes...

Boundary Conditions for Current Density

Boundary Conditions for Current Density

Current density becomes discontinuous across an interface of materials with different electrical conductivities. The normal component of the current density is continuous across the boundary.

Permeability of Concrete

Permeability of Concrete

Permeability in the context of concrete refers to how easily liquids or gases can pass through the material. This quality is crucial for assessing the water-tightness and durability of concrete structures and their resistance to chemical attacks. Concrete permeability can be determined through comparative laboratory tests. These tests typically involve sealing a concrete specimen from the sides, applying water pressure to the top surface with pressure, and measuring the amount of water passing...

Colloidal precipitates

Colloidal precipitates

The high insolubility of some precipitates can result in an unfavorable relative supersaturation. This can lead to colloidal particles with a large surface-to-mass ratio, where adsorption is promoted. For instance, in the precipitation of silver chloride, silver ions are adsorbed on the surface of the colloidal particles, forming a primary layer. This layer attracts ions of opposite charge (such as nitrate ions), forming a diffuse secondary layer of adsorbed ions. This electric double layer...

The Midpoint Rule for Double Integrals

The Midpoint Rule for Double Integrals

The midpoint rule for a double integral provides a practical method for estimating volume over a rectangular region when the surface height varies continuously. In civil engineering, this method is useful for approximating the amount of soil to be moved when planning a road across uneven terrain. The road footprint is represented as a rectangle in the xy-plane. At the same time, the terrain elevation above a flat reference level is described by a continuous height function f(x,y). The objective...

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by

Same author

Threshold behavior of a social norm in response to error proneness.

Journal of theoretical biology·2025

Same author

Indirect reciprocity as a dynamics for weak balance.

Physical review. E·2025

Same author

Response of social norms to individual differences in error-proneness.

Journal of theoretical biology·2025

Same author

Exact cluster dynamics of indirect reciprocity in complete graphs.

Physical review. E·2024

Same author

Stability of twisted states in power-law-coupled Kuramoto oscillators on a circle with and without time delay.

Physical review. E·2024

Same author

Social dilemma in foraging behavior and evolution of cooperation by learning.

Scientific reports·2023

Search research articles

Related Experiment Video

Updated: Jun 12, 2026

An Inverse Analysis Approach to the Characterization of Chemical Transport in Paints

An Inverse Analysis Approach to the Characterization of Chemical Transport in Paints

Published on: August 29, 2014

Surface and bulk criticality in midpoint percolation.

Seung Ki Baek¹, Petter Minnhagen, Beom Jun Kim

¹Department of Physics, Umeå University, 901 87 Umeå, Sweden. garuda@tp.umu.se

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|May 21, 2010

Summary

Midpoint percolation characterizes double percolation transitions in hypercubic lattices. For dimensions six and above, percolation clusters have finite surface points in the infinite-size limit, approaching 2d for large dimensions.

More Related Videos

Surface Properties of Synthesized Nanoporous Carbon and Silica Matrices

Surface Properties of Synthesized Nanoporous Carbon and Silica Matrices

Published on: March 27, 2019

Visually Based Characterization of the Incipient Particle Motion in Regular Substrates: From Laminar to Turbulent Conditions

Visually Based Characterization of the Incipient Particle Motion in Regular Substrates: From Laminar to Turbulent Conditions

Published on: February 22, 2018

Related Experiment Videos

Last Updated: Jun 12, 2026

An Inverse Analysis Approach to the Characterization of Chemical Transport in Paints

An Inverse Analysis Approach to the Characterization of Chemical Transport in Paints

Published on: August 29, 2014

Surface Properties of Synthesized Nanoporous Carbon and Silica Matrices

Surface Properties of Synthesized Nanoporous Carbon and Silica Matrices

Published on: March 27, 2019

Visually Based Characterization of the Incipient Particle Motion in Regular Substrates: From Laminar to Turbulent Conditions

Visually Based Characterization of the Incipient Particle Motion in Regular Substrates: From Laminar to Turbulent Conditions

Published on: February 22, 2018

Area of Science:

Statistical Mechanics
Condensed Matter Physics
Network Science

Background:

Midpoint percolation theory applied to negatively curved structures.
Investigating percolation transitions in regular d-dimensional hypercubic lattices.

Purpose of the Study:

Characterize double percolation transitions using midpoint percolation.
Analyze site-percolation critical thresholds in hypercubic lattices up to d=10.
Examine the role of boundaries in determining critical indices.

Main Methods:

Application of the midpoint percolation concept.
Utilizing the Leath algorithm for site-percolation analysis.
Investigating critical thresholds at various dimensions (up to d=10).

Main Results:

Explicit boundary inclusion simplifies obtaining bulk and surface critical indices.
Percolation clusters contain finite surface points at and above d=6 in the infinite-size limit.
Surface points approach 2d for large dimensions; size dependence for d>=7 relates to midpoint cluster surface points.

Conclusions:

Midpoint percolation effectively characterizes percolation in hypercubic lattices.
Boundary effects are crucial for understanding critical phenomena in these systems.
Findings align with expectations from studies on negatively curved lattices.