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The Fourier series is a powerful mathematical tool for representing periodic signals as an infinite sum of complex exponentials. In practice, this infinite series is truncated to a finite number of terms, yielding a partial sum. This truncation makes the approximation of the signal feasible but introduces certain challenges, particularly near discontinuities, known as the Gibbs phenomenon.
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Mechanistic models play a crucial role in algorithms for numerical problem-solving, particularly in nonlinear mixed effects modeling (NMEM). These models aim to minimize specific objective functions by evaluating various parameter estimates, leading to the development of systematic algorithms. In some cases, linearization techniques approximate the model using linear equations.
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In the application of the Routh-Hurwitz criterion, two specific scenarios can arise that complicate stability analysis.
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When magnetic nuclei in a sample achieve resonance and undergo relaxation, the signal detected in NMR is an approximately exponential free induction decay. Fourier transform of an exponential decay yields a Lorentzian peak in the frequency domain. Lorentzian peaks in an NMR spectrum are defined by their amplitude, full width at half maximum, and position, where the peak width is governed by the spin-spin relaxation time alone. In real experiments, however, the applied magnetic field is rendered...
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Time scaling of signals is a crucial concept in signal processing that affects the Fourier series representation without altering its coefficients. The process modifies the fundamental frequency, thereby changing how the series represents the signal over time. This principle is essential in various applications, including audio and image processing, where signal manipulation is frequent. Understanding function symmetries is fundamental to simplifying the Fourier series.
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Scattering And Absorption of Light in Planetary Regoliths
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Simple renormalization schemes for multiple scattering series expansions.

Aika Takatsu1, Sylvain Tricot2, Philippe Schieffer2

  • 1Faculty of Science, University of Toyama, Gofuku 3190, Toyama 930-8555, Japan.

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|February 21, 2022
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Summary
This summary is machine-generated.

Renormalization schemes improve multiple scattering series convergence. These methods accelerate calculations for electron spectroscopies in large atomic clusters, enhancing computational efficiency.

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Area of Science:

  • Condensed matter physics
  • Materials science
  • Computational chemistry

Background:

  • Multiple scattering (MS) theory is crucial for analyzing electron spectroscopies.
  • Convergence issues in MS series expansions can limit computational accuracy and efficiency.
  • Existing methods struggle with large atomic clusters common in modern materials analysis.

Purpose of the Study:

  • To investigate novel renormalization schemes for enhancing MS series convergence.
  • To assess the practical effectiveness of these schemes in realistic computational scenarios.
  • To improve the feasibility of MS calculations for complex systems.

Main Methods:

  • Exploration of various renormalization techniques.
  • Numerical implementation and testing on a Copper(111) cluster.
  • Comparative analysis of convergence rates with and without renormalization.

Main Results:

  • Demonstrated significant improvements in convergence rates, up to a factor of 2.
  • Successfully transformed divergent series into convergent ones.
  • Validated the effectiveness of the proposed schemes on a model system.

Conclusions:

  • Renormalization schemes offer a powerful solution to MS series convergence problems.
  • These techniques substantially facilitate MS calculations for electron spectroscopies.
  • The methods are particularly beneficial for large-scale cluster calculations in materials science.