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

Plane Electromagnetic Waves I01:30

Plane Electromagnetic Waves I

4.0K
The existence of combined electric and magnetic fields that propagate through space as electromagnetic (EM) waves is the most significant prediction of Maxwell's equations. As Maxwell's equations hold in free space, the predicted electromagnetic waves do not require a medium for their propagation. An EM wave comprises an electric field, defined as the force per charge on a stationary charge, and a magnetic field, which is the force per charge on a moving charge.
The EM field is assumed to be a...
4.0K
Traveling Waves: Lossless Lines01:27

Traveling Waves: Lossless Lines

563
The provided content explores the behavior of traveling waves on single-phase lossless transmission lines. It begins with a single-phase two-wire lossless transmission line of length Δx, characterized by a loop inductance LH/m and a line-to-line capacitance C F/m. These parameters result in a series inductance LΔx and a shunt capacitance CΔx.
563
Second Derivatives and Laplace Operator01:22

Second Derivatives and Laplace Operator

2.4K
The first order operators using the del operator include the gradient, divergence and curl. Certain combinations of first order operators on a scalar or vector function yield second order expressions. Second-order expressions play a very important role in mathematics and physics. Some second order expressions include the divergence and curl of a gradient function, the divergence and curl of a curl function, and the gradient of a divergence function.
Consider a scalar function. The curl of its...
2.4K
Definition of Laplace Transform01:22

Definition of Laplace Transform

5.1K
The Laplace transform is an indispensable mathematical technique for simplifying the resolution of differential equations by converting them into more manageable algebraic expressions. The Laplace transform of a function is denoted by L[x(t)], where x(t) is the time-domain function. The laplace transform is mathematically expressed as
5.1K
Properties of Laplace Transform-I01:15

Properties of Laplace Transform-I

1.4K
The Laplace transform is a powerful mathematical tool used to convert functions from the time domain into the frequency domain, greatly simplifying the analysis and solution of linear time-invariant systems. This transformation is facilitated by several universal properties: Linearity, Time-Scaling, Time-Shifting, and Frequency Shifting.
The Linearity property is foundational to the Laplace transform. It states that the transform of a linear combination of functions is equivalent to the same...
1.4K
Region of Convergence of Laplace Tarnsform01:20

Region of Convergence of Laplace Tarnsform

1.4K
The Region of Convergence (ROC) is a fundamental concept in signal processing and system analysis, particularly associated with the Laplace transform. The ROC represents an area in the complex plane where the Laplace transform of a given signal converges, determining the transform's applicability and utility.
Consider a decaying exponential signal that begins at a specific time. When deriving its Laplace transform, the time-domain variable is replaced with a complex variable. This...
1.4K

You might also read

Related Articles

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

Sort by
Same author

Manifoldron: Direct Space Partition via Manifold Discovery.

IEEE transactions on neural networks and learning systems·2025
Same author

Design of Electronic Nose Based on MOS Gas Sensors and Its Application in Juice Identification.

Sensors (Basel, Switzerland)·2025
Same author

Real space iterative reconstruction for vector tomography (RESIRE-V).

Scientific reports·2024
Same author

On Expressivity and Trainability of Quadratic Networks.

IEEE transactions on neural networks and learning systems·2023
Same author

A unified understanding of minimum lattice thermal conductivity.

Proceedings of the National Academy of Sciences of the United States of America·2023
Same author

Accurate real space iterative reconstruction (RESIRE) algorithm for tomography.

Scientific reports·2023
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: May 3, 2026

Lens-free Video Microscopy for the Dynamic and Quantitative Analysis of Adherent Cell Culture
09:04

Lens-free Video Microscopy for the Dynamic and Quantitative Analysis of Adherent Cell Culture

Published on: February 23, 2018

9.1K

Compressed plane waves yield a compactly supported multiresolution basis for the Laplace operator.

Vidvuds Ozoliņš1, Rongjie Lai, Russel Caflisch

  • 1Departments of Materials Science and Engineering and Mathematics, University of California, Los Angeles, CA 90095-1555.

Proceedings of the National Academy of Sciences of the United States of America
|January 23, 2014
PubMed
Summary
This summary is machine-generated.

This study introduces a new L1 regularized variational framework to create compressed plane waves. This method generates an orthogonal, multiresolution basis for differential operators like the Laplace operator.

More Related Videos

Generation and Coherent Control of Pulsed Quantum Frequency Combs
06:42

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Published on: June 8, 2018

9.0K
Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns
13:44

Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns

Published on: August 30, 2013

42.6K

Related Experiment Videos

Last Updated: May 3, 2026

Lens-free Video Microscopy for the Dynamic and Quantitative Analysis of Adherent Cell Culture
09:04

Lens-free Video Microscopy for the Dynamic and Quantitative Analysis of Adherent Cell Culture

Published on: February 23, 2018

9.1K
Generation and Coherent Control of Pulsed Quantum Frequency Combs
06:42

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Published on: June 8, 2018

9.0K
Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns
13:44

Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns

Published on: August 30, 2013

42.6K

Area of Science:

  • Applied Mathematics
  • Numerical Analysis
  • Scientific Computing

Background:

  • Plane waves are fundamental in physics and engineering but lack spatial localization.
  • Existing methods for constructing basis functions for differential operators can be computationally intensive or lack multiresolution properties.

Purpose of the Study:

  • To develop a novel variational framework for generating spatially localized basis functions.
  • To generalize the concept of plane waves to create an orthogonal, real-space basis with multiresolution capabilities.
  • To apply the framework to the eigenspace of differential operators, such as the Laplace operator.

Main Methods:

  • Utilizing an L1 regularized variational approach.
  • Developing compressed plane waves as a new type of basis function.
  • Demonstrating the framework's ability to span the eigenspace of differential operators.

Main Results:

  • Successfully generated a spatially localized, orthogonal real-space basis.
  • The developed basis exhibits multiresolution capabilities.
  • The framework effectively generalizes plane waves for operator eigenspaces.

Conclusions:

  • The L1 regularized variational framework provides an effective method for constructing compressed plane waves.
  • This approach offers a powerful tool for numerical analysis and scientific computing, enabling localized and multiresolution representations.