Search research articles

ABOUT JoVE

Overview Leadership Blog JoVE Help Center

AUTHORS

Publishing Process Editorial Board Scope & Policies Peer Review FAQ Submit

LIBRARIANS

Testimonials Subscriptions Access Resources Library Advisory Board FAQ

RESEARCH

JoVE Journal Methods Collections JoVE Encyclopedia of Experiments Archive

EDUCATION

JoVE Core JoVE Business JoVE Science Education JoVE Lab Manual Faculty Resource Center Faculty Site

Terms & Conditions of Use

Related Concept Videos

Linear Approximation in Frequency Domain

Linear Approximation in Frequency Domain

Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear....

Linear Approximation in Time Domain

Linear Approximation in Time Domain

Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length,...

Classification of Systems-I

Classification of Systems-I

Linearity is a system property characterized by a direct input-output relationship, combining homogeneity and additivity.
Homogeneity dictates that if an input x(t) is multiplied by a constant c, the output y(t) is multiplied by the same constant. Mathematically, this is expressed as:

Application of Nonlinear Inequalities

Application of Nonlinear Inequalities

A nonlinear inequality describes a comparison involving an expression that curves or behaves more complexly than a straight line. These inequalities often appear in forms that include squares, products, or variables in the denominator.To solve such an inequality, one starts by rewriting it so that zero appears on one side. For example, the inequality: can be factored as: This form makes it easier to identify the values that cause the expression to equal zero. In this case, the...

Introduction to Nonlinear Inequalities

Introduction to Nonlinear Inequalities

Linear and nonlinear inequalities are fundamental for analyzing variable relationships and identifying ranges satisfying specific conditions. A linear inequality involves variables raised only to the first power, resulting in a straight-line graph. This line partitions the coordinate plane into two distinct regions: one that satisfies the inequality and one that does not. Each region represents a set of solutions where the linear relationship holds true under the specified constraint.Nonlinear...

Linear Circuits

Linear Circuits

A linear circuit is characterized by its output having a direct proportionality to its input, adhering to the linearity property, which encompasses the principles of homogeneity (scaling) and additivity. Homogeneity dictates that when the input, also referred to as the excitation, is multiplied by a constant factor, the output, known as the response, is correspondingly scaled by the same constant factor. For instance, if the current is multiplied by a constant 'k,' the voltage likewise...

You might also read

Related Articles

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

Sort by

Same author

Quantitative Homogenization for the Obstacle Problem and Its Free Boundary.

Archive for rational mechanics and analysis·2024

Same author

Management of temporomandibular disorders in the primary care setting.

The Medical journal of Australia·2021

Same author

Pathological angiogenesis in retinopathy engages cellular senescence and is amenable to therapeutic elimination via BCL-xL inhibition.

Cell metabolism·2021

Same author

Distribution of a New Invasive Species, Sipha maydis (Heteroptera: Aphididae), on Cereals and Wild Grasses in the Southern Plains and Rocky Mountain States.

Journal of economic entomology·2019

Same author

Myeloid progenitor cluster formation drives emergency and leukaemic myelopoiesis.

Nature·2017

Same author

Deletion of ribosomal protein genes is a common vulnerability in human cancer, especially in concert with <i>TP53</i> mutations.

EMBO molecular medicine·2017

Search research articles

Related Experiment Video

Updated: Dec 21, 2025

The Generation of Higher-order Laguerre-Gauss Optical Beams for High-precision Interferometry

The Generation of Higher-order Laguerre-Gauss Optical Beams for High-precision Interferometry

Published on: August 12, 2013

Higher-Order Linearization and Regularity in Nonlinear Homogenization.

Scott Armstrong¹, Samuel J Ferguson¹, Tuomo Kuusi²

¹1Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York, NY 10012 USA.

Archive for Rational Mechanics and Analysis

|May 19, 2020

Summary

This summary is machine-generated.

This study establishes large-scale regularity for nonlinear elliptic equations with random coefficients, extending Hilbert's 19th problem to homogenization. It provides a complete nonlinear generalization of large-scale regularity theory in homogenization.

More Related Videos

Measurement of Scattering Nonlinearities from a Single Plasmonic Nanoparticle

Measurement of Scattering Nonlinearities from a Single Plasmonic Nanoparticle

Published on: January 3, 2016

Studying Large Amplitude Oscillatory Shear Response of Soft Materials

Studying Large Amplitude Oscillatory Shear Response of Soft Materials

Published on: April 25, 2019

Related Experiment Videos

Last Updated: Dec 21, 2025

The Generation of Higher-order Laguerre-Gauss Optical Beams for High-precision Interferometry

The Generation of Higher-order Laguerre-Gauss Optical Beams for High-precision Interferometry

Published on: August 12, 2013

Measurement of Scattering Nonlinearities from a Single Plasmonic Nanoparticle

Measurement of Scattering Nonlinearities from a Single Plasmonic Nanoparticle

Published on: January 3, 2016

Studying Large Amplitude Oscillatory Shear Response of Soft Materials

Studying Large Amplitude Oscillatory Shear Response of Soft Materials

Published on: April 25, 2019

Area of Science:

Partial Differential Equations
Homogenization Theory
Mathematical Analysis

Background:

Hilbert's 19th problem concerns the regularity of solutions to elliptic partial differential equations.
Homogenization theory studies the behavior of PDEs with rapidly oscillating coefficients.
Existing theories primarily focus on linear elliptic equations.

Purpose of the Study:

To establish large-scale regularity for solutions of nonlinear elliptic equations with random coefficients.
To extend Hilbert's 19th problem to the context of homogenization for nonlinear equations.
To develop a comprehensive nonlinear theory for large-scale regularity in homogenization.

Main Methods:

Iterative improvement of three key statements: regularity of the homogenized Lagrangian, commutation of higher-order linearization and homogenization, and large-scale regularity of linearization errors.
Development of quantitative estimates for the scaling of linearization errors.
Application of Liouville-type theorems to analyze solutions of linearized equations.

Main Results:

Proof of large-scale regularity for solutions of nonlinear elliptic equations with random coefficients.
Quantitative estimates on the scaling of linearization errors.
A Liouville-type theorem for polynomially growing solutions of higher-order linearized equations.
An explicit Taylor series expansion for solutions with optimally controlled remainder terms.

Conclusions:

The study provides a complete generalization of large-scale regularity theory to the nonlinear setting of homogenization.
The findings offer a deeper understanding of the regularity properties of solutions to complex PDEs.
This work bridges the gap between linear and nonlinear theories in homogenization and regularity.