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Related Concept Videos

State Space Representation01:27

State Space Representation

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The frequency-domain technique, commonly used in analyzing and designing feedback control systems, is effective for linear, time-invariant systems. However, it falls short when dealing with nonlinear, time-varying, and multiple-input multiple-output systems. The time-domain or state-space approach addresses these limitations by utilizing state variables to construct simultaneous, first-order differential equations, known as state equations, for an nth-order system.
Consider an RLC circuit, a...
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Space Trusses01:25

Space Trusses

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A space truss is a three-dimensional counterpart of a planar truss. These structures consist of members connected at their ends, often utilizing ball-and-socket joints to create a stable and versatile framework. The space truss is widely used in various construction projects due to its adaptability and capacity to withstand complex loads.
At the core of a space truss lies the fundamental unit known as the tetrahedron. This structure is composed of six members that form a three-dimensional shape...
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Transfer Function to State Space01:23

Transfer Function to State Space

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State-space representation is a powerful tool for simulating physical systems on digital computers, necessitating the conversion of the transfer function into state-space form. Consider an nth-order linear differential equation with constant coefficients, like those encountered in an RLC circuit. The state variables are selected as the output and its n−1 derivatives. Differentiating these variables and substituting them back into the original equation produces the state equations.
In an RLC...
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State Space to Transfer Function01:21

State Space to Transfer Function

576
The conversion of state-space representation to a transfer function is a fundamental process in system analysis. It provides a method for transitioning from a time-domain description to a frequency-domain representation, which is crucial for simplifying the analysis and design of control systems.
The transformation process begins with the state-space representation, characterized by the state equation and the output equation. These equations are typically represented as:
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Space Trusses: Problem Solving01:29

Space Trusses: Problem Solving

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A space truss is a three-dimensional counterpart of a planar truss. These structures consist of members connected at their ends, often utilizing ball-and-socket joints to create a stable and versatile framework. Due to its adaptability and capacity to withstand complex loads, the space truss is widely used in various construction projects.
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Rocket Propulsion in Empty Space - I01:13

Rocket Propulsion in Empty Space - I

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The driving force for the motion of any vehicle is friction, but in the case of rocket propulsion in space, the friction force is not present. The motion of a rocket changes its velocity (and hence its momentum) by ejecting burned fuel gases, thus causing it to accelerate in the direction opposite to the velocity of the ejected fuel. In this situation, the mass and velocity of the rocket constantly change along with the total mass of ejected gases. Due to conservation of momentum, the...
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Basics of Multivariate Analysis in Neuroimaging Data
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Multivariate box spline wavelets in higher-dimensional Sobolev spaces.

Raj Kumar1, Manish Chauhan2

  • 11Department of Mathematics, Kirori Mal College, University of Delhi, New Delhi, India.

Journal of Inequalities and Applications
|March 7, 2019
PubMed
Summary
This summary is machine-generated.

Researchers constructed wavelets and derived density conditions for Multiresolution Analysis (MRA) in higher-dimensional Sobolev spaces. They established conditions for wavelet orthonormality and created nonseparable orthonormal wavelets using multivariate box splines.

Keywords:
Box SplinesMultiresolution analysisSobolev spaceWavelets

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Area of Science:

  • Mathematics
  • Functional Analysis
  • Harmonic Analysis

Background:

  • Multiresolution Analysis (MRA) is fundamental in wavelet theory, providing a framework for approximating functions in function spaces.
  • Higher-dimensional Sobolev spaces are crucial for analyzing partial differential equations and other advanced mathematical problems.
  • Orthonormal wavelets offer desirable properties for signal processing and numerical analysis due to their efficiency and stability.

Purpose of the Study:

  • To construct wavelets and establish density conditions for Multiresolution Analysis (MRA) within higher-dimensional Sobolev spaces.
  • To determine the necessary and sufficient conditions for the orthonormality of wavelets in the Sobolev space .
  • To develop nonseparable orthonormal wavelets applicable to higher-dimensional analysis using multivariate box splines.

Main Methods:

  • Construction of wavelets tailored for higher-dimensional Sobolev spaces.
  • Derivation of density conditions specific to MRA in these function spaces.
  • Application of multivariate box splines to generate nonseparable orthonormal wavelets.

Main Results:

  • Successful construction of wavelets and derivation of MRA density conditions in higher-dimensional Sobolev spaces.
  • Identification of precise necessary and sufficient conditions for wavelet orthonormality in .
  • Development of novel nonseparable orthonormal wavelets suitable for higher-dimensional applications.

Conclusions:

  • The study provides a theoretical foundation for constructing and analyzing wavelets in complex, higher-dimensional function spaces.
  • The derived conditions for orthonormality are essential for practical applications of wavelets in numerical methods.
  • The newly constructed nonseparable orthonormal wavelets offer advanced tools for signal and data analysis in multiple dimensions.