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

Chemical Formulas02:52

Chemical Formulas

61.1K
A chemical formula presents information about the proportions of atoms constituting a particular chemical compound or molecule, mainly using symbols of elements and numbers. At times other symbols, such as dashes, parentheses, brackets, commas, plus, and minus signs, are also used. A chemical formula can be one of three types – molecular, empirical, and structural.
61.1K
Ionic Compounds: Formulas and Nomenclature03:34

Ionic Compounds: Formulas and Nomenclature

86.7K
An element composed of atoms that readily lose electrons (a metal) can react with an element composed of atoms that readily gain electrons (a nonmetal) to produce ions through complete electron transfer. The compound formed by this transfer is stabilized by the electrostatic attractions (ionic bonds) between the oppositely charged ions.
86.7K
Molecular Compounds: Formulas and Nomenclature03:10

Molecular Compounds: Formulas and Nomenclature

55.6K
Molecular compounds or covalent compounds result when atoms share electrons to form covalent bonds. Since there is no electron transfer, molecular compounds do not contain ions; instead, they consist of discrete, neutral molecules. 
55.6K
Experimental Determination of Chemical Formula02:37

Experimental Determination of Chemical Formula

46.9K
The elemental makeup of a compound defines its chemical identity, and chemical formulas are the most concise way of representing this elemental makeup. When a compound’s formula is unknown, measuring the mass of its constituent elements is often the first step in determining the formula experimentally.
46.9K
Formula Mass and Mole Concepts of Compounds02:56

Formula Mass and Mole Concepts of Compounds

81.1K
Formula Mass of Covalent Compounds
81.1K
The Distance Formula01:20

The Distance Formula

637
In geometry, measuring the direct distance between two points on a plane is essential in various practical and theoretical applications. Whether in navigation, engineering, or computer graphics, determining the shortest path between two locations involves using the distance formula. This formula is derived from the Pythagorean Theorem, which relates the lengths of the sides of a right triangle. On a coordinate plane, the horizontal and vertical distances between two points serve as the legs of...
637

You might also read

Related Articles

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

Sort by
Same author

Diversity strengthens competing teams.

Royal Society open science·2022
Same author

Biodiversity of marine microbes is safeguarded by phenotypic heterogeneity in ecological traits.

PloS one·2021
Same journal

Quadratic Sparse Domination and Weighted Estimates for Non-integral Square Functions.

Journal of geometric analysis·2026
Same journal

A Schwarz Lemma for the Pentablock.

Journal of geometric analysis·2026
Same journal

Flows of Conformally Coclosed <math><msub><mi>G</mi> <mn>2</mn></msub></math> -Structures with Dilaton.

Journal of geometric analysis·2026
Same journal

Kähler-Einstein Metrics.

Journal of geometric analysis·2026
Same journal

On Shape Optimization with Large Magnetic Fields in Two Dimensions.

Journal of geometric analysis·2026
Same journal

Families of proper holomorphic maps.

Journal of geometric analysis·2026
See all related articles

Related Experiment Video

Updated: Jan 28, 2026

The Use of Induced Somatic Sector Analysis ISSA for Studying Genes and Promoters Involved in Wood Formation and Secondary Stem Development
09:54

The Use of Induced Somatic Sector Analysis ISSA for Studying Genes and Promoters Involved in Wood Formation and Secondary Stem Development

Published on: October 5, 2016

9.2K

A Polyakov Formula for Sectors.

Clara L Aldana1, Julie Rowlett2

  • 11Mathematics Research Unit, University of Luxembourg, 6, avenue de la Fonte, 4364 Esch-sur-Alzette, Luxembourg.

Journal of Geometric Analysis
|March 7, 2019
PubMed
Summary
This summary is machine-generated.

This study introduces a variational Polyakov formula for circular sectors, detailing how the zeta-regularized determinant of the Laplacian changes with the opening angle. It also identifies the square as the unique maximizer for the zeta-regularized determinant of unit-area rectangular domains.

Keywords:
Angular variationConical singularityHeat kernelLaplacianPolyakov formulaRectangleSectorSpectrumZeta-regularized determinant

More Related Videos

Author Spotlight: Standardized Herbal Decoction Protocol for Enhanced Animal Studies
03:11

Author Spotlight: Standardized Herbal Decoction Protocol for Enhanced Animal Studies

Published on: June 7, 2024

1.3K
Establishment of Hepatocarcinoma in BALB/c-nu Mice and Investigation of the Therapeutic Effect of the Sanleng Jiashen Formula
09:03

Establishment of Hepatocarcinoma in BALB/c-nu Mice and Investigation of the Therapeutic Effect of the Sanleng Jiashen Formula

Published on: January 26, 2024

688

Related Experiment Videos

Last Updated: Jan 28, 2026

The Use of Induced Somatic Sector Analysis ISSA for Studying Genes and Promoters Involved in Wood Formation and Secondary Stem Development
09:54

The Use of Induced Somatic Sector Analysis ISSA for Studying Genes and Promoters Involved in Wood Formation and Secondary Stem Development

Published on: October 5, 2016

9.2K
Author Spotlight: Standardized Herbal Decoction Protocol for Enhanced Animal Studies
03:11

Author Spotlight: Standardized Herbal Decoction Protocol for Enhanced Animal Studies

Published on: June 7, 2024

1.3K
Establishment of Hepatocarcinoma in BALB/c-nu Mice and Investigation of the Therapeutic Effect of the Sanleng Jiashen Formula
09:03

Establishment of Hepatocarcinoma in BALB/c-nu Mice and Investigation of the Therapeutic Effect of the Sanleng Jiashen Formula

Published on: January 26, 2024

688

Area of Science:

  • Mathematical Physics
  • Differential Geometry
  • Spectral Theory

Background:

  • The study of determinants of Laplacians on geometric domains is crucial in quantum field theory and spectral geometry.
  • Understanding how these determinants change under geometric deformations is a key challenge.

Purpose of the Study:

  • To derive a variational Polyakov formula for circular sectors, relating the zeta-regularized determinant of the Laplacian to the sector's opening angle.
  • To compute contributions to this formula and analyze heat kernels on sectors.
  • To determine the shape that uniquely maximizes the zeta-regularized determinant for unit-area rectangular domains.

Main Methods:

  • Utilizing variational principles to derive the Polyakov formula.
  • Employing techniques for computing zeta-regularized determinants.
  • Analyzing conformal deformations and their impact on spectral quantities.
  • Leveraging Carslaw-Sommerfeld's heat kernel for infinite area sectors.

Main Results:

  • A novel variational Polyakov formula for Euclidean circular sectors is established.
  • Explicit expressions for heat kernels on infinite area sectors are derived.
  • The unique maximization of the zeta-regularized determinant of unit-area rectangles by the square is proven.

Conclusions:

  • The variational Polyakov formula provides new insights into the spectral geometry of circular sectors.
  • The findings contribute to the understanding of determinants of Laplacians and heat kernel behavior in curved spaces.
  • The square's unique maximality property has implications for geometric inequalities.