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

Theorem of Pappus01:24

Theorem of Pappus

The Theorem of Pappus, also known as the Pappus–Guldinus Theorem, provides a geometric method for determining the volume and surface area of solids generated by the revolution of a plane region or a plane curve about an external axis. The theorem consists of two related statements. The first addresses the volume of solids formed by rotating plane areas, while the second addresses the surface area generated by rotating plane curves. Both results depend on the location of the centroid, which...
Sums of Power01:22

Sums of Power

In definite integration, Riemann sums approximate the area under a curve by dividing it into subintervals and summing the areas of rectangles. When these approximations follow predictable numerical patterns, such as arithmetic or polynomial sequences, sum formulas offer a more efficient and accurate way to compute the result. In particular, the sum of consecutive integers, squares, and cubes plays an essential role in simplifying these calculations, especially when dealing with uniform...
Singularity Functions for Bending Moment01:18

Singularity Functions for Bending Moment

Singularity functions simplify the representation of bending moments in beams subjected to discontinuous loading, allowing the use of a single mathematical expression. For a supported beam AB, with uniform loading from its midpoint M to the right side end B, the approach involves conceptual 'cuts' at specific points to determine the bending moment in each segment. By cutting the beam at a point between A and M, the bending moment for the segment before reaching midpoint M is represented using a...
The Squeeze Theorem01:30

The Squeeze Theorem

Certain mathematical functions exhibit unpredictable or highly variable behavior near specific input values, making direct evaluation of their limits challenging. This complexity may arise from rapid oscillations or irregular patterns that obscure the function’s trend. In such cases, the Squeeze Theorem offers a reliable method for determining limits.According to the Squeeze Theorem, if a function is confined between two other functions near a particular point, and both outer functions approach...
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...
The Buckingham Pi Theorem01:09

The Buckingham Pi Theorem

The Buckingham Pi theorem provides a structured method to simplify fluid dynamics problems by reducing complex systems of variables to dimensionless terms.

You might also read

Related Articles

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

Sort by
Same author

Ramanujan's partition generating functions modulo <math><mi>ℓ</mi></math>.

The Ramanujan journal·2025
Same author

Some topological genera and Jacobi forms.

Proceedings of the National Academy of Sciences of the United States of America·2025
Same author

Beneficial Effects of Enoki Mushroom Extract on Male Menopausal Symptoms in Japanese Subjects: A Randomized, Double-Blind, Placebo-Controlled Study.

Nutrients·2025
Same author

Beneficial Effects of a Formulated Supplement of Ascidiacea (Halocynthia-roretzi)-derived Plasmalogen and Tuna-derived Elastin on Memory Function in Elderly Japanese Subjects; A Randomized, Double-blind, Placebo-controlled Study.

Journal of oleo science·2024
Same author

Integer partitions detect the primes.

Proceedings of the National Academy of Sciences of the United States of America·2024
Same author

Beneficial Effects of Saw Palmetto Fruit Extract on Urinary Symptoms in Japanese Female Subjects by a Multicenter, Randomized, Double-Blind, Placebo-Controlled Study.

Nutrients·2022
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: Jun 5, 2026

Detection of Rare Genomic Variants from Pooled Sequencing Using SPLINTER
14:06

Detection of Rare Genomic Variants from Pooled Sequencing Using SPLINTER

Published on: June 23, 2012

Congruences for the Andrews spt function.

Ken Ono1

  • 1Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322, USA. ono@mathcs.emory.edu

Proceedings of the National Academy of Sciences of the United States of America
|December 24, 2010
PubMed
Summary
This summary is machine-generated.

New congruences for the Andrews spt(n) partition function are found for all prime moduli ℓ≥5. This confirms a conjecture regarding simple congruences for the spt(n) function, simplifying number theory research.

Related Experiment Videos

Last Updated: Jun 5, 2026

Detection of Rare Genomic Variants from Pooled Sequencing Using SPLINTER
14:06

Detection of Rare Genomic Variants from Pooled Sequencing Using SPLINTER

Published on: June 23, 2012

Area of Science:

  • Number Theory
  • Combinatorics
  • Algebraic Number Theory

Background:

  • Ramanujan-type congruences for the Andrews spt(n) partition function have been established for prime moduli 5 ≤ ℓ ≤ 37.
  • Previous work by Andrews and Garvan laid the groundwork for understanding these congruences.

Purpose of the Study:

  • To discover and prove unexpectedly simple Ramanujan-type congruences for the Andrews spt(n) partition function for all prime moduli ℓ≥5.
  • To confirm a conjecture proposed by Garvan regarding these congruences.

Main Methods:

  • Utilizing properties of mock theta functions and their reduction modulo prime numbers.
  • Applying Hecke operator theory, specifically the T(ℓ(2)) operator, to mock theta function reductions.
  • Demonstrating that the reduction of a specific mock theta function modulo ℓ is an eigenform of the Hecke operator T(ℓ(2)).

Main Results:

  • Unexpectedly simple Ramanujan-type congruences for the Andrews spt(n) partition function are established for all prime moduli ℓ≥5.
  • A conjecture by Garvan is confirmed: for prime ℓ≥5 with (-δ/ℓ) = 1, spt[(ℓ2(ℓn+δ)+1)/24] ≡ 0 (mod ℓ).
  • This congruence generates (ℓ - 1)/2 arithmetic progressions modulo ℓ(3) supporting a mod ℓ congruence.

Conclusions:

  • The study reveals a surprising simplicity in the modular properties of the spt(n) partition function for all prime moduli ℓ≥5.
  • The findings connect partition theory, mock theta functions, and Hecke operators, offering new insights into number theoretic congruences.
  • The confirmed conjecture and derived congruences provide a powerful tool for further research in number theory and combinatorics.