Related Experiment Video
Updated: Jul 14, 2026
13:44
Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns
Published on: August 30, 2013
A Riemann-Hilbert approach to the Akhiezer polynomials
1Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Drive, Madison, WI 53706, USA. ychen@ic.ac.uk
Summary
This study investigates Akhiezer polynomials using a Riemann-Hilbert problem approach. Researchers solved associated differential equations, extending previous work and interpreting Hankel determinants as tau-functions.
Area of Science:
- Orthogonal Polynomials
- Integrable Systems
- Complex Analysis
Background:
- N. I. Akhiezer introduced orthogonal polynomials over disjoint intervals.
- Previous work involved meromorphic functions on hyperelliptic Riemann surfaces.
- A. Magnus previously derived Fuchsian differential equations and Schlesinger deformation equations.
Purpose of the Study:
- To study Akhiezer polynomials via a Riemann-Hilbert problem.
- To derive Fuchsian differential equations and Schlesinger deformation equations.
- To solve these equations and interpret related determinants.
Main Methods:
- Reformulation as a matrix factorization or Riemann-Hilbert problem.
- Application of the general Riemann-Hilbert scheme from integrable systems theory.
- Solving derived equations using Riemann Theta-functions.
Main Results:
- A straightforward derivation of Fuchsian differential equations for the polynomials.
- Derivation of Schlesinger deformation equations for recurrence coefficients.
- Solutions to these equations expressed via Riemann Theta-functions.
- Interpretation of Hankel determinants as tau-functions.
Conclusions:
- The Riemann-Hilbert approach provides a direct method for studying Akhiezer polynomials.
- This method extends Magnus' results by providing explicit solutions.
- The connection between Hankel determinants and tau-functions is established.
Related Concept Videos
01:28Synthetic Disvision of Polynomials
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...
01:27Real Zeros of Polynomials
Polynomials are algebraic expressions of terms with variables raised to non-negative integer powers. A central aspect of analyzing polynomial functions is determining their real zeros—values of the variable for which the polynomial evaluates to zero. These values represent the x-intercepts of the polynomial’s graph.The Rational Zeros Theorem lists possible rational solutions for a polynomial equation with integer coefficients. If f(x)=anxn+....+a0, then every rational zero is of the form p/q,...
01:15Determination of Pi Terms
The Buckingham Pi theorem is a valuable method in dimensional analysis, reducing complex relationships between variables into dimensionless terms. Relevant variables in analyzing the lift force on an airplane wing include lift force, air density, wing area, aircraft velocity, and air viscosity. Expressing each variable in terms of fundamental dimensions — mass, length, and time — provides a consistent foundation for constructing these dimensionless terms.
The theorem indicates that the number...
The theorem indicates that the number...
01:19Routh-Hurwitz Criterion II
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 column of the Routh...
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 column of the Routh...
01:29Complex Zeros
Complex zeros are the solutions to polynomial equations that include imaginary numbers, specifically, numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit defined by i2=-1. These zeros satisfy the equation P(x) = 0, where P(x) is a polynomial with real or complex coefficients. Since the complex number system includes all real numbers, it provides a complete framework for analyzing all possible roots of a polynomial.Every polynomial of degree n≥1 can be...
01:26Long Division of Polynomials
Polynomial division is an essential algebraic process to simplify expressions and solve equations. Just as numerical division separates a number into quotient and remainder, polynomial long division partitions a polynomial into simpler components; in this context, the dividend is the polynomial being divided, the divisor is the expression dividing it, and the result is expressed in terms of a quotient and a remainder.The division begins by arranging the dividend and divisor in standard...