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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.
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An ionic compound is stable because of the electrostatic attraction between its positive and negative ions. The lattice energy of a compound is a measure of the strength of this attraction. The lattice energy (ΔHlattice) of an ionic compound is defined as the energy required to separate one mole of the solid into its component gaseous ions. For the ionic solid sodium chloride, the lattice energy is the enthalpy change of the process:
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Indirect Fabrication of Lattice Metals with Thin Sections Using Centrifugal Casting
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A space for lattice representation and clustering.

Lawrence C Andrews1, Herbert J Bernstein2, Nicholas K Sauter3

  • 1Ronin Institute, 9515 NE 137th Street, Kirkland, WA 98034-1820, USA.

Acta Crystallographica. Section A, Foundations and Advances
|May 2, 2019
PubMed
Summary

This study introduces a perturbation-stable metric for quantifying lattice differences, crucial for Bravais lattice determination and molecular replacement. This metric ensures reliable clustering in serial crystallography by satisfying the triangle inequality.

Keywords:
DelaunayDeloneNiggliSellingclusteringunit-cell reduction

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Area of Science:

  • Crystallography
  • Computational Science
  • Data Analysis

Background:

  • Algorithms quantifying lattice differences are vital for Bravais lattice determination, molecular replacement, and serial crystallography image clustering.
  • A perturbation-stable metric is desirable for reliable nearest-neighbor searches and robust data analysis.

Purpose of the Study:

  • To describe a perturbation-stable metric space for quantifying differences between lattices.
  • To present novel representations of this metric space for practical applications.

Main Methods:

  • Developed a metric space related to Selling's reduction algorithm and Delone's lattice determination methods.
  • Represented the metric space using six-dimensional real vectors and three-dimensional complex vectors.

Main Results:

  • Introduced a novel metric space that is perturbation-stable.
  • Demonstrated two equivalent vector representations for the metric space.

Conclusions:

  • The described metric provides a robust way to quantify lattice differences, enhancing applications in crystallography and data analysis.
  • The vector representations facilitate the use of standard algorithms for lattice comparison and clustering.