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Routh-Hurwitz Criterion II01:19

Routh-Hurwitz Criterion II

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In the application of the Routh-Hurwitz criterion, two specific scenarios can arise that complicate stability analysis.
The first scenario occurs when a singular zero appears in the first column of the Routh table. This situation creates a division by zero issues. To resolve this, a small positive or negative number, denoted as epsilon (∈), is substituted for the zero. The stability analysis proceeds by assuming a sign for ∈. If ∈ is positive, any sign change in the first...
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Area Computation by the Alternative Coordinate Method01:24

Area Computation by the Alternative Coordinate Method

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The alternative coordinate method, also known as the Shoelace Formula, is a technique for determining the area of a traverse using Cartesian coordinates. This method relies on the sequential arrangement of x and y coordinates for each point of the shape, ensuring accuracy and ease of application.In this approach, each corner's x and y coordinates are listed as fractions, with the x-coordinate as the numerator and the y-coordinate as the denominator. These coordinates are arranged sequentially...
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Routh-Hurwitz Criterion I01:15

Routh-Hurwitz Criterion I

252
Consider an electrical power grid, where stability is essential to prevent blackouts. The Routh-Hurwitz criterion is a valuable tool for assessing system stability under varying load conditions or faults. By analyzing the closed-loop transfer function, the Routh-Hurwitz criterion helps determine whether the system remains stable.
To apply the Routh-Hurwitz criterion, a Routh table is constructed. The table's rows are labeled with powers of the complex frequency variable s, starting from the...
252
pV-Diagrams01:18

pV-Diagrams

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The pV diagram, which is a graph of pressure versus volume of the gas under study, is helpful in describing certain aspects of the substance. When the substance behaves like an ideal gas, the ideal gas equation describes the relationship between its pressure and volume. On a pV diagram, it is common to plot an isotherm, which is a curve showing p as a function of V with the number of molecules and the temperature fixed. Then, for an ideal gas, the product of the pressure of the gas and its...
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Compacting Factor test01:22

Compacting Factor test

164
The compacting factor test is a method used to assess the workability of concrete. It is  especially suitable for concrete mixes containing aggregates up to one and a half inches in size. This test involves specialized equipment consisting of two truncated cone-shaped hoppers and a cylinder, all with polished interior surfaces to minimize friction.
The procedure begins by placing concrete into the upper hopper without any compaction. Once filled, the bottom door of this hopper is opened,...
164
Survival Tree01:19

Survival Tree

87
Survival trees are a non-parametric method used in survival analysis to model the relationship between a set of covariates and the time until an event of interest occurs, often referred to as the "time-to-event" or "survival time." This method is particularly useful when dealing with censored data, where the event has not occurred for some individuals by the end of the study period, or when the exact time of the event is unknown.
 Building a Survival Tree
Constructing a...
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Related Experiment Video

Updated: Jul 10, 2025

Optimization of Synthetic Proteins: Identification of Interpositional Dependencies Indicating Structurally and/or Functionally Linked Residues
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Graphically Contracted Function Construction with the Recursive Pairwise Merge Algorithm.

Ron Shepard1

  • 1Chemical Sciences and Engineering Division, Argonne National Laboratory, Lemont, Illinois 60439, United States.

The Journal of Physical Chemistry. A
|November 22, 2023
PubMed
Summary
This summary is machine-generated.

A new algorithm significantly improves the efficiency of constructing Gaussian Composite Functions (GCF) from configuration state function (CSF) coefficients. This enhanced method optimizes computational speed for molecular orbital calculations.

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Area of Science:

  • Computational chemistry
  • Quantum chemistry
  • Theoretical chemistry

Background:

  • Gaussian Composite Functions (GCF) are crucial for representing molecular wave functions.
  • Existing methods for constructing GCFs from configuration state function (CSF) expansion coefficients can be computationally intensive.
  • Efficiency improvements are needed for complex molecular systems.

Purpose of the Study:

  • To present a novel, more efficient implementation of a recursive pairwise merge algorithm for GCF construction.
  • To enhance the computational performance by optimizing memory usage and reducing algorithmic complexity.
  • To evaluate the effectiveness of the new algorithm for representing GCF wave functions using different types of molecular orbitals.

Main Methods:

  • Implementation of a recursive pairwise merge algorithm for GCF construction.
  • Introduction of preallocated work arrays in intermediate merge steps.
  • Analysis of computational efficiency gains and reduction in dependence on molecular orbital dimension (n).

Main Results:

  • Achieved an order of magnitude improvement in overall computational efficiency.
  • Reduced the dependence on the molecular orbital dimension (n) from the algorithm.
  • Demonstrated successful representation of GCF wave functions using both delocalized canonical Hartree-Fock orbitals and localized molecular orbitals.
  • Observed that delocalized orbitals yield smaller errors for low wave function complexity (small average facet count, f̅), while localized orbitals are better for higher complexity (larger f̅).

Conclusions:

  • The new implementation offers a substantial performance enhancement for GCF construction.
  • The algorithm's efficiency is largely independent of the molecular orbital dimension.
  • The choice between delocalized and localized orbitals for GCF representation depends on the desired wave function complexity and acceptable error tolerance.