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Related Concept Videos

Introduction to Limits01:30

Introduction to Limits

A limit describes the value a function approaches as its input moves closer to a particular point. Even when a function is undefined at a specific value, limits allow us to analyze its behavior near that point. This concept is fundamental in calculus and essential for understanding continuity, derivatives, and integrals.Mathematically, a function f(x) has a limit L at x = a if its values L approach x as x gets arbitrarily close to a. This is written as:This notation expresses that the function...
Evaluating Limits by Direct Substitution01:29

Evaluating Limits by Direct Substitution

In the analysis of functions that represent continuous physical phenomena, it is often necessary to determine the output value as the input approaches a specific point. When a combination of algebraic terms defines the function and exhibits no discontinuities or abrupt changes near the point of interest, the limit of the function can be evaluated directly. This process, known as direct substitution, involves replacing the variable in the expression with the value it approaches.Direct...
Limits at Infinity01:24

Limits at Infinity

The function that decreases as the input becomes very large provides a clear example of how mathematical functions can behave at extreme values. When the input increases continuously, the output becomes smaller and smaller, getting closer to a particular fixed value. Although the output never actually reaches this value, it moves nearer to it without limit. This behavior is a fundamental concept in understanding how functions behave as the input grows indefinitely. The graphical representation...
Social Traps01:41

Social Traps

Social traps are negative situations where people get caught in a direction or relationship that later proves to be unpleasant, with no easy way to back out of or avoid. The concept was orignally introduced by John Platt who applied psychology to Garrett Hardin's "Tragedy of the Commons", where in New England herd owners could let their cattle graze in the common ground. This situation seems like a good idea, but an individual could have an advantage. If they owned more cows, the larger...
Short-distance Transport of Resources02:12

Short-distance Transport of Resources

Short-distance transport refers to transport that occurs over a distance of just 2-3 cells, crossing the plasma membrane in the process. Small uncharged molecules, such as oxygen, carbon dioxide, and water, can diffuse across the plasma membrane on their own. In contrast, ions and larger molecules require the assistance of transport proteins due to their charge or size. Transport across membranes also occurs within individual cells, playing a variety of essential roles for the plant as a whole.
Transport Number01:31

Transport Number

The transport number is the fraction of the total current carried by an ion in an electrolyte solution. It is defined as the ratio of the current carried by a specific ion to the total current flowing through the solution. The transport number, t, is central to understanding ionic mobility, which describes how fast an ion moves under the influence of an electric field. This link connects the physical behavior of ions in solution to the chemical processes that occur during electrochemical...

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Related Experiment Video

Updated: May 22, 2026

Evaluating the Effect of Roadside Parking on a Dual-Direction Urban Street
14:55

Evaluating the Effect of Roadside Parking on a Dual-Direction Urban Street

Published on: January 20, 2023

A long-time limit for world subway networks.

Camille Roth1, Soong Moon Kang, Michael Batty

  • 1CAMS (CNRS/EHESS) 190, avenue de France, 75013 Paris, France. roth@ehess.fr

Journal of the Royal Society, Interface
|May 18, 2012
PubMed
Summary

Subway networks globally evolve into a universal core-and-branch structure. This finding suggests common mechanisms drive the development of large-scale transit systems.

Area of Science:

  • Network Science
  • Urban Planning
  • Transportation Engineering

Background:

  • Subway networks are critical urban infrastructure.
  • Understanding their structural evolution is key to urban development and efficiency.

Purpose of the Study:

  • To explore the temporal evolution of the structure of the world's largest subway networks.
  • To identify common structural features and underlying developmental mechanisms.

Main Methods:

  • Exploratory analysis of network topology and spatial organization.
  • Quantitative assessment of network metrics like node degree, branch scaling, and spatial distribution.

Main Results:

  • All studied subway networks converge to a similar core-and-branch structure.

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Last Updated: May 22, 2026

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  • The core exhibits specific node degree distributions (avg. degree ~2.5, >60% k=2 nodes).
  • Branch scaling and spatial distribution follow predictable patterns, explained by geometric models.

    • Conclusions:

    • Subway network evolution appears governed by universal mechanisms, leading to a convergent structural form.
    • The core-and-branch model provides a natural explanation for observed spatial and topological features.
    • Results challenge purely fractal geometry interpretations, emphasizing geometric and spatial constraints.