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Related Concept Videos

Per-Unit Sequence Models01:26

Per-Unit Sequence Models

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An ideal Y-Y transformer, grounded through neutral impedances, displays per-unit sequence networks akin to those of a single-phase ideal transformer when subjected to balanced positive- or negative-sequence currents. These currents do not produce neutral currents, and their associated voltage drops.
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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.
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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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A cumulative frequency distribution is another type of frequency distribution. Instead of reporting how many data values fall in some classes, it reports how many data values are contained in either that class or any class to its left. Technically, it means the sum of frequencies of the class and all the classes below it in a frequency distribution. A cumulative frequency is calculated by adding the frequency of each class lower than the corresponding class interval or category. In general, a...
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A probability histogram is a visual representation of a probability distribution. Similar a typical histogram, the probability histogram consists of contiguous (adjoining) boxes. It has both a horizontal axis and a vertical axis. The horizontal axis is labeled with what the data represents. The vertical axis is labeled with probability. Each rectangular bar in the histogram is 1 unit wide, which suggests that the area under each bar equals the probability, P(x), where x is 1, 2, 3, and so on.
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Log concavity for unimodal sequences.

Walter Bridges1, Kathrin Bringmann1

  • 1Universität zu Köln: Universitat zu Koln, Cologne, Germany.

Research in Number Theory
|December 22, 2023
PubMed
Summary
This summary is machine-generated.

The number of unimodal sequences is proven to be log-concave, a property with implications for combinatorics and number theory. This finding stems from an exact formula for these sequences, derived from recent work on false theta functions.

Keywords:
Log-concavitySaddle-point methodUnimodal sequences

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Area of Science:

  • Combinatorics
  • Number Theory
  • Analytic Number Theory

Background:

  • Log-concavity and higher Turán inequalities are significant in the study of partitions and modular forms.
  • Existing analytic proofs for these properties often rely on precise asymptotic series with error terms.
  • Recent advancements in false theta functions have yielded exact formulas for coefficients of mixed mock/false modular objects.

Purpose of the Study:

  • To prove that the number of unimodal sequences of size n is log-concave.
  • To utilize an exact formula for unimodal sequences to perform this calculation.
  • To explore the applicability of the developed method to other related mathematical objects.

Main Methods:

  • Derivation of an exact formula for unimodal sequences, building on recent work on false theta functions.
  • Application of analytic techniques to the exact formula to establish log-concavity.
  • Leveraging properties of mixed false modular forms.

Main Results:

  • The number of unimodal sequences of size n is demonstrated to be log-concave.
  • The study provides an analytic proof for the log-concavity of these combinatorial sequences.
  • An exact formula for unimodal sequences, related to mixed false modular forms, is central to the proof.

Conclusions:

  • The log-concavity of unimodal sequences is established.
  • The methodology employed is expected to be applicable to other coefficients of mixed mock/false modular objects.
  • This work contributes to the understanding of combinatorial sequences and their connection to modular forms.