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Related Concept Videos

Percentage Frequency Distribution00:57

Percentage Frequency Distribution

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A percentage frequency distribution, in general, is a display of data that indicates the percentage of observations for each data point or grouping of data points. It is a commonly used method for expressing the relative frequency of survey responses and other data. The percentage frequency distributions are often displayed as bar graphs, pie charts, or tables.
The process of making a percentage frequency distribution involves the following few steps: note the total number of observations;...
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What is a Frequency Distribution00:51

What is a Frequency Distribution

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A frequency is the number of times a value of the data occurs. The sum of all the frequency values represents the total number of students included in the sample. It is commonly used to group data of quantitative types. Frequency distributions can be displayed in a table, histogram, line graph, dot plot, or pie chart, just to name a few. A histogram is a graphical representation of tabulated frequencies, shown as adjacent rectangles, erected over discrete intervals (bins), with an area equal to...
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Variation: Normal Distribution, Range, and Standard Deviation02:32

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In the field of psychology, there are several ways to organize measurements of a trait, feature, or characteristic (i.e., variables). Qualitative data, such as ethnicity, can be tabulated into a frequency count to provide information about the proportion, as well as the variety of groups in a sample or population. On the other hand, researchers can perform a wider set of calculations on quantitative data. The mean, mode, and median, for instance, are central tendency measures to identify a...
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Relative Frequency Distribution00:55

Relative Frequency Distribution

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A relative frequency distribution is the proportion or fraction of times a value occurs in a data set. To find the relative frequencies, one can divide each frequency by the total number of data points in the sample. It is very similar to a regular frequency distribution, except that instead of reporting how many data values fall in a class, a relative frequency distribution reports the fraction of data values that fall in a class. These fractions or proportions are called relative frequencies...
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Applications of Normal Distribution01:22

Applications of Normal Distribution

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The normal distribution is a useful statistical tool. One of its practical applications is determining the door height after considering the normal distribution of heights of persons, such that many can pass through it easily without striking their heads. The normal distribution can also determine the probability of a person having a height less than a specific height.
The heights of 15 to 18-year-old males from Chile from 1984 to 1985 followed a normal distribution. The mean height is 172.36...
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Histogram01:05

Histogram

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The histogram is a graphical representation in the x-y form of data distribution in a data set. The horizontal x-axis is labeled with what the data represents (for instance, distance from your home to school). The vertical y-axis is labeled either frequency or relative frequency (or percent frequency or probability).
A histogram graph consists of contiguous (adjoining) boxes. The heights of the bars correspond to frequency values. The graph will have the same shape with respective labels. The...
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On Morphisms Between Connected Commutative Algebraic Groups over a Field of Characteristic 0.

Transformation groups·2024
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On the frequency of height values.

Gabriel A Dill1

  • 1University of Oxford, Mathematical Institute, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG UK.

Research in Number Theory
|November 1, 2021
PubMed
Summary

This study counts algebraic numbers and integer polynomials based on their degree, height, and the number of roots inside the unit disk. The research provides rough growth orders for these counts, offering insights into number theory and dynamical systems.

Area of Science:

  • Number Theory
  • Algebraic Geometry
  • Complex Dynamics

Background:

  • Understanding the distribution of algebraic numbers is crucial in number theory.
  • The Weil height and Mahler measure are fundamental invariants for algebraic numbers and polynomials.
  • The location of roots relative to the unit disk has implications in various mathematical fields.

Purpose of the Study:

  • To count algebraic numbers of a fixed degree and height with a specific number of conjugates inside the unit disk.
  • To determine the number of height values attained by such algebraic numbers.
  • To analyze integer polynomials based on degree, Mahler measure, and the number of roots within the unit disk.

Main Methods:

  • Utilizing asymptotic formulas for the logarithm of counting functions.
Keywords:
CountingHeightMahler measure

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  • Investigating specific cases where standard asymptotic analysis is not directly applicable.
  • Employing techniques from the theory of Diophantine equations and algebraic number theory.

    • Main Results:

    • Established rough orders of growth for the counts of algebraic numbers, with conditions on the number of conjugates inside the unit disk.
    • Detailed analysis of the height function's behavior in specific parameter regimes.
    • Provided counts for integer polynomials and initiated the study of their dynamical height behavior.

    Conclusions:

    • The distribution of algebraic numbers with specific properties can be characterized by growth orders rather than precise asymptotics in certain cases.
    • The study sheds light on the intricate relationship between algebraic properties and the location of roots.
    • Further research into the dynamical behavior of the height function for polynomials is warranted.