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

Types of Functions III01:28

Types of Functions III

74
Logarithmic and piecewise functions play central roles in mathematical modeling, particularly when capturing nonlinear or segmented behaviors in real-world phenomena. Although these functions differ fundamentally in structure and application, both serve to represent complex relationships in simplified mathematical terms.A logarithmic function is defined as the inverse of an exponential function, expressed as These functions grow quickly for small values of x but slow down as x increases,...
74
Laws of Logarithms I01:30

Laws of Logarithms I

76
Logarithms are fundamental mathematical operations that serve as the inverse of exponentiation. They provide a means to express how many times a base must be raised to yield a given number. For base 10, often referred to as the common logarithm, the notation is written simply as log. Thus, if 10n = x, then log⁡(x) = n. This relationship makes logarithms especially valuable in simplifying complex calculations involving multiplication, division, and exponentiation.Logarithmic expressions are...
76
Limits at Infinity01:24

Limits at Infinity

52
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...
52
Rectangular and Triangular Pulse Function01:19

Rectangular and Triangular Pulse Function

1.5K
The unit rectangular pulse function is mathematically represented by a rectangular function centered at the origin with a height of one unit. This function is defined by two parameters: T, which specifies the center location of the pulse along the time axis, and τ, which determines the pulse duration.
For example, consider a rectangular pulse with a 5V amplitude, a 3-second duration, and centered at t=2 seconds. This pulse can be expressed using the rectangular function, written as,
1.5K
Laws of Logarithms II01:28

Laws of Logarithms II

72
Logarithmic laws provide essential tools for simplifying and evaluating exponential expressions, particularly in mathematical and applied settings where powers and repeated multiplication play a central role. Two important rules are the power law and the change-of-base formula, both allowing for transforming expressions into more manageable forms.The power law of logarithms states that the logarithm of a number raised to an exponent equals the exponent multiplied by the logarithm of the base...
72
Limits with Oscillating Discontinuities01:19

Limits with Oscillating Discontinuities

75
An oscillating discontinuity is a type of discontinuity in which a function’s values fluctuate infinitely often as the input approaches a particular point. Unlike jump discontinuities, where the function suddenly shifts between two values, or infinite discontinuities, where the function diverges without bound, an oscillating discontinuity arises from rapid back-and-forth variation. Because the function never stabilizes toward a single value, no finite limit exists at that point.One of the...
75

You might also read

Related Articles

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

Sort by
Same author

Delocalisation and Continuity in 2D: Loop  <math></math> , Six-Vertex, and Random-Cluster Models.

Communications in mathematical physics·2025
Same author

Phase Diagram of the Ashkin-Teller Model.

Communications in mathematical physics·2024
Same author

On the Six-Vertex Model's Free Energy.

Communications in mathematical physics·2022
See all related articles

Related Experiment Video

Updated: Nov 15, 2025

Characterization of Surface Modifications by White Light Interferometry: Applications in Ion Sputtering, Laser Ablation, and Tribology Experiments
11:47

Characterization of Surface Modifications by White Light Interferometry: Applications in Ion Sputtering, Laser Ablation, and Tribology Experiments

Published on: February 27, 2013

15.9K

Uniform Lipschitz Functions on the Triangular Lattice Have Logarithmic Variations.

Alexander Glazman1, Ioan Manolescu2

  • 1Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria.

Communications in Mathematical Physics
|March 8, 2021
PubMed
Summary
This summary is machine-generated.

Uniform integer-valued Lipschitz functions exhibit O(2) loop model behavior on a triangular lattice. Researchers constructed a unique infinite-volume Gibbs measure, revealing scale-invariant properties with finite loops at all scales.

More Related Videos

Orientational Transition in a Liquid Crystal Triggered by the Thermodynamic Growth of Interfacial Wetting Sheets
06:26

Orientational Transition in a Liquid Crystal Triggered by the Thermodynamic Growth of Interfacial Wetting Sheets

Published on: May 15, 2017

7.4K
Visually Based Characterization of the Incipient Particle Motion in Regular Substrates: From Laminar to Turbulent Conditions
11:51

Visually Based Characterization of the Incipient Particle Motion in Regular Substrates: From Laminar to Turbulent Conditions

Published on: February 22, 2018

8.9K

Related Experiment Videos

Last Updated: Nov 15, 2025

Characterization of Surface Modifications by White Light Interferometry: Applications in Ion Sputtering, Laser Ablation, and Tribology Experiments
11:47

Characterization of Surface Modifications by White Light Interferometry: Applications in Ion Sputtering, Laser Ablation, and Tribology Experiments

Published on: February 27, 2013

15.9K
Orientational Transition in a Liquid Crystal Triggered by the Thermodynamic Growth of Interfacial Wetting Sheets
06:26

Orientational Transition in a Liquid Crystal Triggered by the Thermodynamic Growth of Interfacial Wetting Sheets

Published on: May 15, 2017

7.4K
Visually Based Characterization of the Incipient Particle Motion in Regular Substrates: From Laminar to Turbulent Conditions
11:51

Visually Based Characterization of the Incipient Particle Motion in Regular Substrates: From Laminar to Turbulent Conditions

Published on: February 22, 2018

8.9K

Area of Science:

  • Probability theory
  • Statistical mechanics
  • Discrete geometry

Background:

  • Integer-valued Lipschitz functions are crucial in discrete analysis and have connections to lattice models.
  • The O(2) model is a significant statistical mechanics model with applications in various fields.
  • Understanding the behavior of these functions and models at different scales is essential for theoretical advancements.

Purpose of the Study:

  • To analyze the variations of uniform integer-valued Lipschitz functions on a triangular lattice.
  • To establish a connection between these functions and the O(2) loop model.
  • To construct and characterize the infinite-volume Gibbs measure for the O(2) model.

Main Methods:

  • Representing the O(2) loop model using a pair of spin configurations.
  • Utilizing the FKG inequality for the spin configurations.
  • Constructing the infinite-volume Gibbs measure as a thermodynamic limit.
  • Proving RSW-type estimates for connectivity in the spin model.

Main Results:

  • Uniform integer-valued Lipschitz functions on a triangular lattice of size N show variations of order 1.
  • The level lines of these functions correspond to an O(2) loop model on a hexagonal lattice.
  • A unique infinite-volume Gibbs measure for the O(2) model was constructed, exhibiting scale-invariance with finite loops at all scales.
  • The existence of the infinite-volume measure extends to pinned height functions, though uniqueness does not.

Conclusions:

  • The study provides a rigorous framework for understanding the statistical properties of integer-valued Lipschitz functions through the lens of the O(2) loop model.
  • The constructed Gibbs measure and its scale-invariant properties offer new insights into lattice models.
  • The methods employed, including FKG inequalities and RSW-type estimates, are valuable for future research in related areas.