Jove
Visualize
Contact Us
JoVE
x logofacebook logolinkedin logoyoutube logo
ABOUT JoVE
OverviewLeadershipBlogJoVE Help Center
AUTHORS
Publishing ProcessEditorial BoardScope & PoliciesPeer ReviewFAQSubmit
LIBRARIANS
TestimonialsSubscriptionsAccessResourcesLibrary Advisory BoardFAQ
RESEARCH
JoVE JournalMethods CollectionsJoVE Encyclopedia of ExperimentsArchive
EDUCATION
JoVE CoreJoVE BusinessJoVE Science EducationJoVE Lab ManualFaculty Resource CenterFaculty Site
Terms & Conditions of Use
Privacy Policy
Policies

Related Concept Videos

Extraction: Partition and Distribution Coefficients01:14

Extraction: Partition and Distribution Coefficients

The distribution law or Nernst's distribution law is the law that governs the distribution of a solute between two immiscible solvents. This law, also known as the partition law, states that if a solute is added to the mixture of two immiscible solvents at a constant temperature, the solute is distributed between the two solvents in such a way that the ratio of solute concentrations in the solvents remains constant at equilibrium.
For extracting a solute from an aqueous phase into an organic...
Thermal Sigmatropic Reactions: Overview01:16

Thermal Sigmatropic Reactions: Overview

Sigmatropic rearrangements are a class of pericyclic reactions in which a σ bond migrates from one part of a π system to another. These are intramolecular rearrangements where the total number of σ and π bonds remain unchanged.
Sigmatropic shifts are classified based on an order term [i, j ], where i and j indicate the number of atoms across which each end of the σ bond migrates. Below are examples of a [3,3] sigmatropic shift in 1,5-hexadiene, referred to as...
Determination of Pi Terms01:15

Determination of Pi Terms

The Buckingham Pi theorem is a valuable method in dimensional analysis, reducing complex relationships between variables into dimensionless terms. Relevant variables in analyzing the lift force on an airplane wing include lift force, air density, wing area, aircraft velocity, and air viscosity. Expressing each variable in terms of fundamental dimensions — mass, length, and time — provides a consistent foundation for constructing these dimensionless terms.
The theorem indicates that the number...
Long Division of Polynomials01:26

Long Division of Polynomials

Polynomial division is an essential algebraic process to simplify expressions and solve equations. Just as numerical division separates a number into quotient and remainder, polynomial long division partitions a polynomial into simpler components; in this context, the dividend is the polynomial being divided, the divisor is the expression dividing it, and the result is expressed in terms of a quotient and a remainder.The division begins by arranging the dividend and divisor in standard...
Synthetic Disvision of Polynomials01:28

Synthetic Disvision of Polynomials

Synthetic division is an efficient algorithmic approach for dividing a polynomial by a linear binomial of the form x - c, where c is a real number. This method is helpful due to its streamlined process, which avoids the more cumbersome steps involved in the traditional long division of polynomials. It simplifies computation and serves as a practical tool for evaluating polynomials and identifying their factors.To perform synthetic division, one begins by listing the coefficients of the...
Rationalizing Substitutions01:29

Rationalizing Substitutions

Integrals involving non-rational functions are often difficult to evaluate using standard techniques, especially when radicals appear in the integrand. Rationalizing substitution provides a systematic method for simplifying such integrals by converting them into rational forms that are easier to handle.Consider a rod whose linear mass density depends on a constant linear density, a characteristic length, and the distance from the left end of the rod. Determining the total mass requires...

You might also read

Related Articles

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

Sort by
Same author

Sequences in overpartitions.

The Ramanujan journal·2023
Same author

How Ramanujan may have discovered the mock theta functions.

Philosophical transactions. Series A, Mathematical, physical, and engineering sciences·2019
Same author

Almost partition identities.

Proceedings of the National Academy of Sciences of the United States of America·2019
Same author

Ramanujan's congruences and Dyson's crank.

Proceedings of the National Academy of Sciences of the United States of America·2005
Same journal

Tau protein as a regulator of mitochondrial function and dynamics.

Proceedings of the National Academy of Sciences of the United States of America·2026
Same journal

A scalable, dividing cell model for the robust propagation and quantification of human sporadic Creutzfeldt-Jakob disease prions.

Proceedings of the National Academy of Sciences of the United States of America·2026
Same journal

Epigenetic regulation of mesenchymal BMP signaling directs postnatal organ innervation.

Proceedings of the National Academy of Sciences of the United States of America·2026
Same journal

Single-shot wide-field biochemical imaging at 1 kHz frame rate.

Proceedings of the National Academy of Sciences of the United States of America·2026
Same journal

Morphogenesis and topological evolution of a frustrated nematic liquid crystal under confinement.

Proceedings of the National Academy of Sciences of the United States of America·2026
Same journal

B cell-intrinsic CXCR3 drives efficient generation of ectopic pulmonary germinal center responses to influenza A virus infection.

Proceedings of the National Academy of Sciences of the United States of America·2026
See all related articles

Related Experiment Video

Updated: Jul 1, 2026

Self-assembly of Complex Two-dimensional Shapes from Single-stranded DNA Tiles
10:23

Self-assembly of Complex Two-dimensional Shapes from Single-stranded DNA Tiles

Published on: May 8, 2015

Partitions with short sequences and mock theta functions.

George E Andrews1

  • 1Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA. andrews@math.psu.edu

Proceedings of the National Academy of Sciences of the United States of America
|February 18, 2005
PubMed
Summary
This summary is machine-generated.

This study explores integer partitions, specifically those without consecutive integers. These partitions have surprising applications in threshold growth models and connections to Ramanujan

More Related Videos

Robust DNA Isolation and High-throughput Sequencing Library Construction for Herbarium Specimens
13:03

Robust DNA Isolation and High-throughput Sequencing Library Construction for Herbarium Specimens

Published on: March 8, 2018

Computation of Atmospheric Concentrations of Molecular Clusters from ab initio Thermochemistry
12:11

Computation of Atmospheric Concentrations of Molecular Clusters from ab initio Thermochemistry

Published on: April 8, 2020

Related Experiment Videos

Last Updated: Jul 1, 2026

Self-assembly of Complex Two-dimensional Shapes from Single-stranded DNA Tiles
10:23

Self-assembly of Complex Two-dimensional Shapes from Single-stranded DNA Tiles

Published on: May 8, 2015

Robust DNA Isolation and High-throughput Sequencing Library Construction for Herbarium Specimens
13:03

Robust DNA Isolation and High-throughput Sequencing Library Construction for Herbarium Specimens

Published on: March 8, 2018

Computation of Atmospheric Concentrations of Molecular Clusters from ab initio Thermochemistry
12:11

Computation of Atmospheric Concentrations of Molecular Clusters from ab initio Thermochemistry

Published on: April 8, 2020

Area of Science:

  • Number Theory
  • Combinatorics
  • Mathematical Physics

Background:

  • Integer partitions are fundamental in combinatorics.
  • MacMahon's work initiated the study of partitions with restrictions on consecutive parts.
  • Recent research links these partitions to threshold growth models.

Purpose of the Study:

  • To intrinsically develop the theory of integer partitions without consecutive parts.
  • To deepen the understanding of the applications of such partitions.
  • To reveal the connection between these partitions and Ramanujan's mock theta functions.

Main Methods:

  • Combinatorial analysis of partition properties.
  • Exploration of connections to threshold growth models.
  • Investigation of relationships with mock theta functions.

Main Results:

  • Characterization of partitions with no sequences of consecutive integers of length k.
  • Demonstration of the relevance of these partitions in threshold growth models.
  • Identification of a surprising role for Ramanujan's mock theta functions.

Conclusions:

  • The study of integer partitions with restricted consecutive parts offers a rich area for theoretical development.
  • These partitions have unexpected applications in diverse fields like growth models.
  • Ramanujan's mock theta functions play a significant, previously unrecognized role in this domain.