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Related Concept Videos

Transfer Function to State Space01:23

Transfer Function to State Space

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State-space representation is a powerful tool for simulating physical systems on digital computers, necessitating the conversion of the transfer function into state-space form. Consider an nth-order linear differential equation with constant coefficients, like those encountered in an RLC circuit. The state variables are selected as the output and its n−1 derivatives. Differentiating these variables and substituting them back into the original equation produces the state equations.
In an RLC...
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State Space to Transfer Function01:21

State Space to Transfer Function

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The conversion of state-space representation to a transfer function is a fundamental process in system analysis. It provides a method for transitioning from a time-domain description to a frequency-domain representation, which is crucial for simplifying the analysis and design of control systems.
The transformation process begins with the state-space representation, characterized by the state equation and the output equation. These equations are typically represented as:
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Orthogonal Trajectories01:26

Orthogonal Trajectories

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Orthogonal trajectories describe the geometric relationship between two families of curves that intersect each other at right angles. One illustrative case involves a family of parabolas that open sideways along the x-axis. These curves share a common shape but differ by a scaling parameter, resulting in a set of curves that all pass through the origin and widen at different rates.Determining Orthogonal TrajectoriesTo identify the orthogonal trajectories for these parabolas, the first step...
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Pole and System Stability01:24

Pole and System Stability

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The transfer function is a fundamental concept representing the ratio of two polynomials. The numerator and denominator encapsulate the system's dynamics. The zeros and poles of this transfer function are critical in determining the system's behavior and stability.
Simple poles are unique roots of the denominator polynomial. Each simple pole corresponds to a distinct solution to the system's characteristic equation, typically resulting in exponential decay terms in the system's...
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Real Zeros of Polynomials01:27

Real Zeros of Polynomials

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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...
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Properties of Fourier series I01:20

Properties of Fourier series I

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The Fourier series is a powerful tool in signal processing and communications, allowing periodic signals to be expressed as sums of sine and cosine functions. A foundational property of the Fourier series is linearity. If we consider two periodic signals, their linear combination results in a new signal whose Fourier coefficients are simply the corresponding linear combinations of the original signals' coefficients. This property is crucial in applications like frequency modulation (FM) radio,...
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Related Experiment Video

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Quantum State Engineering of Light with Continuous-wave Optical Parametric Oscillators
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Orthogonal polynomials and perfect state transfer.

Rachel Bailey1

  • 1Department of Mathematical Sciences, Bentley University, Waltham, MA, USA.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|October 9, 2025
PubMed
Summary
This summary is machine-generated.

This review explores orthogonal polynomials and their applications in quantum information processing, including quantum walks and perfect state transfer detection. It highlights connections to classical processes and advanced concepts like exceptional orthogonal polynomials.

Keywords:
Jacobi matricesKrawtchouk polynomialsorthogonal polynomialsquantum information

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Last Updated: Jan 15, 2026

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Area of Science:

  • Quantum Information Processing
  • Mathematical Physics
  • Spectral Graph Theory

Background:

  • Orthogonal polynomials are fundamental in various mathematical fields.
  • Quantum information processing leverages quantum mechanics for computation and information tasks.
  • Continuous-time quantum walks on graphs are a key area in quantum algorithms.

Purpose of the Study:

  • To provide a self-contained introduction to orthogonal polynomials and continuous-time quantum walks.
  • To discuss applications of orthogonal polynomials in quantum information processing.
  • To explore connections between orthogonal polynomials, quantum walks, and classical processes.

Main Methods:

  • Focus on Jacobi operators associated with orthogonal polynomials.
  • Analyze the detection of perfect state transfer (PST) using these operators.
  • Extend concepts to quantum walks with non-nearest-neighbor interactions using exceptional orthogonal polynomials (XOPs).

Main Results:

  • Jacobi operators can be used to detect perfect state transfer.
  • Orthogonal polynomials yield results analogous to Karlin-McGregor birth and death processes.
  • Exceptional orthogonal polynomials enable extensions to more complex quantum walks.

Conclusions:

  • Orthogonal polynomials offer powerful tools for quantum information processing.
  • The review provides a foundation for understanding these applications.
  • Open questions in the field are identified for future research.