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Related Concept Videos

Linear Approximation in Time Domain01:21

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, the...
Linear Approximation in Frequency Domain01:26

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.
IR Spectroscopy: Hooke's Law Approximation of Molecular Vibration01:16

IR Spectroscopy: Hooke's Law Approximation of Molecular Vibration

A covalently bonded heteronuclear diatomic molecule can be modeled as two vibrating masses connected by a spring. The vibrational frequency of the bond can be expressed using an equation derived from Hooke's law, which describes how the force applied to stretch or compress a spring is proportional to the displacement of the spring. In this case, the atoms behave like masses, and the bond acts like a spring.
According to Hooke's law, the vibrational frequency is directly proportional to the...
Linear Approximations01:23

Linear Approximations

For a differentiable function of two variables, linear approximation estimates values near a known point by replacing the curved surface with its tangent plane. Consider the function\begin{equation*}f(x,y)=x^2+3y^2\end{equation*}near the point (2, 1). The exact value at this point is f(2, 1) = 22 + 3(1)2 = 4 + 3 = 7.The linear approximation of f(x, y)) near (a, b) is\begin{equation*}L(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)\end{equation*}First, compute the partial derivatives: fx(x, y) = 2x and...
Vector Representation of Complex Numbers01:16

Vector Representation of Complex Numbers

Complex numbers, represented in Cartesian coordinates, can also be visualized as vectors. These vectors can be expressed in polar form, emphasizing their magnitude and angle. When a complex number is input into a function, the output is another complex number, highlighting the function's zero point from which the vector representation can originate.
Consider a function defined as the product of the complex factors in the numerator divided by the product of the complex factors in the denominator.
Vector Algebra: Method of Components01:08

Vector Algebra: Method of Components

It is cumbersome to find the magnitudes of vectors using the parallelogram rule or using the graphical method to perform mathematical operations like addition, subtraction, and multiplication. There are two ways to circumvent this algebraic complexity. One way is to draw the vectors to scale, as in navigation, and read approximate vector lengths and angles (directions) from the graphs. The other way is to use the method of components.
In many applications, the magnitudes and directions of...

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Related Experiment Video

Updated: Jun 19, 2026

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

Approximate method in coherent-mode representation.

K Kim, D Y Park

    Optics Letters
    |October 2, 2009
    PubMed
    Summary
    This summary is machine-generated.

    This study introduces an approximate method for identifying coherent-mode eigenfunctions and eigenvalues. This technique is applicable to nondegenerate cross spectral densities of a source.

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    Last Updated: Jun 19, 2026

    Generation and Coherent Control of Pulsed Quantum Frequency Combs
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    Published on: June 8, 2018

    A Photonic System for Generating Unconditional Polarization-Entangled Photons Based on Multiple Quantum Interference
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    A Photonic System for Generating Unconditional Polarization-Entangled Photons Based on Multiple Quantum Interference

    Published on: September 5, 2019

    Area of Science:

    • Physics
    • Signal Processing

    Background:

    • Cross spectral densities are crucial for analyzing signal coherence.
    • Identifying coherent modes aids in understanding source characteristics.

    Purpose of the Study:

    • To present an approximate method for calculating coherent-mode eigenfunctions and eigenvalues.
    • To address the analysis of nondegenerate cross spectral densities.

    Main Methods:

    • Development of an approximate computational approach.
    • Application to nondegenerate cross spectral density matrices.

    Main Results:

    • The method provides a way to approximate coherent-mode eigenfunctions.
    • Corresponding eigenvalues can also be approximated using this method.

    Conclusions:

    • The presented approximate method offers a practical tool for spectral density analysis.
    • This facilitates a deeper understanding of source coherence properties.