Jove
Visualize
Contact Us
JoVE
x logofacebook logolinkedin logoyoutube logo
ABOUT JoVE
OverviewLeadershipBlogJoVE Help Center
AUTHORS
Publishing ProcessEditorial BoardScope & PoliciesPeer ReviewFAQSubmit
LIBRARIANS
TestimonialsSubscriptionsAccessResourcesLibrary Advisory BoardFAQ
RESEARCH
JoVE JournalMethods CollectionsJoVE Encyclopedia of ExperimentsArchive
EDUCATION
JoVE CoreJoVE BusinessJoVE Science EducationJoVE Lab ManualFaculty Resource CenterFaculty Site
Terms & Conditions of Use
Privacy Policy
Policies

Related Concept Videos

Chebyshev's Theorem to Interpret Standard Deviation01:15

Chebyshev's Theorem to Interpret Standard Deviation

5.0K
Chebyshev’s theorem, also known as Chebyshev’s Inequality, states that the proportion of values of a dataset for K standard deviation is calculated using the equation:
5.0K
Graphs of Polar Equations01:17

Graphs of Polar Equations

248
The polar coordinate system represents points using a distance from a central point (the pole) and an angle from a reference direction (the polar axis). Unlike rectangular coordinates, polar coordinates are ideal for graphing curves with radial symmetry or periodic behavior.Some general forms of graphs in polar coordinates include the following:Equation of a Circle (Centered at the Pole):A graph where the radius remains constant for all angles traces a circle centered at the pole:Equation of a...
248
Graphs of Functions01:30

Graphs of Functions

258
Graphs of functions provide a visual representation of how output values change in response to varying inputs. Each point on the graph corresponds to an ordered pair, where the x-coordinate (independent variable) determines the horizontal position and the y-coordinate (dependent variable) determines the vertical position. Linear functions like y = x give a straight line, indicating a constant rate of change.Nonlinear functions display more complex behaviors. Even power functions generate...
258
Graphs of Trigonometric Functions01:29

Graphs of Trigonometric Functions

288
Trigonometric functions exhibit periodic and symmetrical behavior, deeply rooted in the unit circle. The sine and cosine functions correspond to the vertical and horizontal projections, respectively, of a point rotating counterclockwise around the circle. These functions trace smooth, repeating waveforms with identical periods and bounded ranges. The tangent function is defined as the ratio of sine to cosine and produces an unbounded curve that repeats every units, with vertical asymptotes...
288
Synthetic Disvision of Polynomials01:28

Synthetic Disvision of Polynomials

147
Synthetic division is an efficient algorithmic approach for dividing a polynomial by a linear binomial of the form x - c, where c is a real number. This method is helpful due to its streamlined process, which avoids the more cumbersome steps involved in the traditional long division of polynomials. It simplifies computation and serves as a practical tool for evaluating polynomials and identifying their factors.To perform synthetic division, one begins by listing the coefficients of the...
147
Introduction to Polynomial Functions01:26

Introduction to Polynomial Functions

216
Polynomial functions are fundamental elements in algebra and calculus, defined by expressions that combine variables and constants through addition, subtraction, and multiplication, with the variable raised to nonnegative integer exponents. A general polynomial function of degree n is given byWhere an ≠ 0. The term anxn is the leading term, and an is the leading coefficient, while a0 is referred to as the constant term.Characteristics and ClassificationPolynomials are categorized by their...
216

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by
Same journal

Inverse FIP effect plasma in the solar atmosphere: a synthesis of current understanding and new insights from AR 11967.

Philosophical transactions. Series A, Mathematical, physical, and engineering sciences·2026
Same journal

Signs of sulfur fractionation under high magnetic field strength.

Philosophical transactions. Series A, Mathematical, physical, and engineering sciences·2026
Same journal

First ionization potential fractionation of sulfur observed with spectral imaging of the coronal environment.

Philosophical transactions. Series A, Mathematical, physical, and engineering sciences·2026
Same journal

Chromospheric dynamics and turbulence regulate the solar FIP effect.

Philosophical transactions. Series A, Mathematical, physical, and engineering sciences·2026
Same journal

Exploring the link between wave activity in the photospheric velocity driver and the FIP bias in the solar corona.

Philosophical transactions. Series A, Mathematical, physical, and engineering sciences·2026
Same journal

Radiative hydrodynamic simulations of first ionization potential fractionation in solar flares.

Philosophical transactions. Series A, Mathematical, physical, and engineering sciences·2026

Related Experiment Video

Updated: Jan 15, 2026

ExCYT: A Graphical User Interface for Streamlining Analysis of High-Dimensional Cytometry Data
05:12

ExCYT: A Graphical User Interface for Streamlining Analysis of High-Dimensional Cytometry Data

Published on: January 16, 2019

11.9K

Partial Chebyshev polynomials and fan graphs.

Wojciech Młotkowski1, Nobuaki Obata2

  • 1Institute of Mathematics, University of Wrocław, Wrocław, Poland.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|October 9, 2025
PubMed
Summary

Researchers introduce partial Chebyshev polynomials and derive their properties. They use these polynomials to calculate the quadratic embedding constant (QEC) for fan graphs, providing explicit values and estimates.

Keywords:
Chebyshev polynomialsdistance matrixfactorizationfan graphpartial Chebyshev polynomialsquadratic embedding constant

More Related Videos

Analysis of SEC-SAXS data via EFA deconvolution and Scatter
10:59

Analysis of SEC-SAXS data via EFA deconvolution and Scatter

Published on: January 28, 2021

9.8K
A Practical Guide to Phylogenetics for Nonexperts
12:00

A Practical Guide to Phylogenetics for Nonexperts

Published on: February 5, 2014

36.0K

Related Experiment Videos

Last Updated: Jan 15, 2026

ExCYT: A Graphical User Interface for Streamlining Analysis of High-Dimensional Cytometry Data
05:12

ExCYT: A Graphical User Interface for Streamlining Analysis of High-Dimensional Cytometry Data

Published on: January 16, 2019

11.9K
Analysis of SEC-SAXS data via EFA deconvolution and Scatter
10:59

Analysis of SEC-SAXS data via EFA deconvolution and Scatter

Published on: January 28, 2021

9.8K
A Practical Guide to Phylogenetics for Nonexperts
12:00

A Practical Guide to Phylogenetics for Nonexperts

Published on: February 5, 2014

36.0K

Area of Science:

  • Mathematics
  • Numerical Analysis
  • Spectral Graph Theory

Background:

  • Chebyshev polynomials of the second kind have a known product formula.
  • Understanding graph properties like the quadratic embedding constant (QEC) is crucial in spectral graph theory.

Purpose of the Study:

  • Introduce and define partial Chebyshev polynomials.
  • Derive basic properties and relationships of these new polynomials.
  • Apply partial Chebyshev polynomials to compute the QEC of fan graphs.

Main Methods:

  • Leveraging the product formula of Chebyshev polynomials of the second kind.
  • Developing new polynomial factorizations involving partial Chebyshev polynomials.
  • Analyzing minimal zeros of derived polynomials to determine QEC.

Main Results:

  • Introduction of partial Chebyshev polynomials P_n(x) and Q_n(x).
  • Derivation of new factorization formulas for a related polynomial.
  • Calculation of the QEC for fan graphs using partial Chebyshev polynomials, yielding explicit values for even n and estimates for odd n.

Conclusions:

  • Partial Chebyshev polynomials offer new insights into polynomial theory.
  • The derived polynomial provides a method for calculating the QEC of fan graphs.
  • The study establishes a connection between orthogonal polynomials and graph-theoretic constants.