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Related Concept Videos

Scale-Up Processes01:14

Scale-Up Processes

The scale-up of microbial fermentation processes is essential in industrial biotechnology, allowing the transition from laboratory-scale experiments to commercial-scale production while aiming to maintain product yield and quality. This process requires meticulous adjustment of equipment design, process parameters, and contamination control strategies to accommodate increasing culture volumes.At the laboratory scale, cultures are typically maintained in 1 to 10-liter glass or autoclavable...
Scaling01:26

Scaling

In designing and analyzing filters, resonant circuits, or circuit analysis at large, working with standard element values like 1 ohm, 1 henry, or 1 farad can be convenient before scaling these values to more realistic figures. This approach is widely utilized by not employing realistic element values in numerous examples and problems; it simplifies mastering circuit analysis through convenient component values. The complexity of calculations is thereby reduced, with the understanding that...
Introduction to Scalers01:21

Introduction to Scalers

Many familiar physical quantities can be specified completely by giving a single number and the appropriate unit. For example, "a class period lasts 50 min," or "the gas tank in my car holds 65 L," or "the distance between the two posts is 100 m." A physical quantity that can be specified completely in this manner is called a scalar quantity. The word "scalar" is a synonym for "number." Time, mass, distance, length, volume, temperature, and energy are some examples of scalar quantities.
Scalar...
Exponential Equations for Modeling Growth01:26

Exponential Equations for Modeling Growth

Exponential models are essential for describing rapid, multiplicative changes in natural systems, such as population growth. When a population doubles at regular intervals, the process can be modeled using a suitable base. For instance, a bacterial culture that doubles every three hours follows the model n(t)=n0⋅2t/3, where n(t) is the population at the time t.A more general model uses the natural base e, especially for continuous growth. This takes the form n(t)=n0⋅ert, where r is the relative...
Exponential Growth01:29

Exponential Growth

Bacterial populations exhibit exponential growth when conditions such as nutrient availability and temperature are favorable. In this phase, cells reproduce through binary fission, where each cell divides into two identical daughter cells. This process causes the population to double at regular intervals, resulting in a growth rate that is directly proportional to the current number of cells. As the population increases, the number of new cells formed during each generation also grows, creating...
Exponential Equations with Logarithms: Problem Solving01:29

Exponential Equations with Logarithms: Problem Solving

In ecological studies, exponential models are often used to predict how populations grow over time under favorable conditions. These models assume that the growth rate is proportional to the current population, leading to continuous and compounding increases.The model expresses the population as a function of time, combining the initial population with a growth factor raised to an exponent involving the growth rate and time. To estimate how long it takes for a population to reach a specific...

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Related Experiment Video

Updated: Jun 25, 2026

The Innovation Arena: A Method for Comparing Innovative Problem-Solving Across Groups
14:14

The Innovation Arena: A Method for Comparing Innovative Problem-Solving Across Groups

Published on: May 13, 2022

Superlinear scaling for innovation in cities.

Samuel Arbesman1, Jon M Kleinberg, Steven H Strogatz

  • 1Department of Health Care Policy, Harvard Medical School, 180 Longwood Avenue, Boston, Massachusetts 02115, USA. arbesman@hcp.med.harvard.edu

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|March 5, 2009
PubMed
Summary

Cities exhibit superlinear scaling in innovation, meaning productivity and creativity grow faster than population. This study presents a network model explaining this phenomenon, offering a theoretical basis for urban development and innovation dynamics.

Related Experiment Videos

Last Updated: Jun 25, 2026

The Innovation Arena: A Method for Comparing Innovative Problem-Solving Across Groups
14:14

The Innovation Arena: A Method for Comparing Innovative Problem-Solving Across Groups

Published on: May 13, 2022

Area of Science:

  • Urban studies
  • Sociology
  • Network science
  • Innovation studies

Background:

  • Superlinear scaling observed in urban sociology, where economic and creative output exceeds population growth.
  • Existing theories lack a satisfactory explanation for the observed superlinear relationship between city size and innovation.

Purpose of the Study:

  • To provide a theoretical explanation for superlinear scaling in cities.
  • To develop a network model that captures the relationship between urban population size and innovation.

Main Methods:

  • Development of a network model based on urban population dynamics.
  • Analysis of the model to derive the relationship between population size and innovation output.

Main Results:

  • The network model successfully replicates the superlinear scaling observed in cities.
  • The model provides a reasonable range for the exponent governing the superlinear relationship.

Conclusions:

  • The proposed network model offers a robust theoretical framework for understanding urban innovation.
  • This work elucidates the mechanisms driving superlinear scaling in cities, impacting urban planning and policy.