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Related Concept Videos

Properties of the z-Transform II01:16

Properties of the z-Transform II

218
The property of Accumulation in signal processing is derived by analyzing the accumulated sum of a discrete-time signal and using the time-shifting property to determine its z-transform. This principle reveals that the z-transform of the summed signal is related to the z-transform of the original signal by a multiplicative factor.
Moreover, the convolution property indicates that the convolution of two signals in the time domain corresponds to the product of their z-transforms in the frequency...
218
Inverse z-Transform by Partial Fraction Expansion01:20

Inverse z-Transform by Partial Fraction Expansion

447
The inverse z-transform is a crucial technique for converting a function from its z-domain representation back to the time domain. One effective method for finding the inverse z-transform is the Partial Fraction Method, which involves decomposing a function into simpler fractions with distinct coefficients. These fractions correspond to known z-transform pairs, facilitating the inverse transformation process.
To begin the process, the poles of the function are identified and the function is...
447
Definition of z-Transform01:26

Definition of z-Transform

981
The z-transform is a powerful mathematical tool used in the analysis of discrete-time signals and systems. It is an essential analytical tool, analogous to the Laplace transform used in continuous-time systems. It plays a crucial role in the analysis of signals and systems, complementing the discrete-time Fourier transform. Both the z-transform and the Laplace transform convert differential or difference equations into algebraic equations, simplifying the process of solving complex problems.
981
Properties of the z-Transform I01:17

Properties of the z-Transform I

371
The z-transform is a fundamental tool in digital signal processing, enabling the analysis of discrete-time systems through its various properties. It is an invaluable tool for analyzing discrete-time systems, offering a range of properties that simplify complex signal manipulations. One fundamental property is linearity. For any two discrete-time signals, the z-transform of their linear combination equals the same linear combination of their individual z-transforms. This property is essential...
371
Relation of DFT to z-Transform01:20

Relation of DFT to z-Transform

528
The Discrete Fourier Transform (DFT) is a crucial tool for analyzing the frequency content of discrete-time signals. It converts a sequence of N samples from the time domain into its corresponding sequence in the frequency domain, where each sample represents a specific frequency component.
To understand how the DFT works, it's helpful to consider the z-transform, which is a method for representing discrete sequences in the complex frequency domain. The z-transform involves summing the...
528
Region of Convergence01:17

Region of Convergence

610
The z-transform is a powerful mathematical tool used in the analysis of discrete-time signals and systems. It is a crucial tool in the analysis of discrete-time systems, but its convergence is limited to specific values of the complex variable z. This range of values, known as the Region of Convergence (ROC), is fundamental in determining the behavior and stability of a system or signal. The ROC defines the region in the complex plane where the z-transform converges, which can take various...
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Amplitudes and the Riemann Zeta Function.

Grant N Remmen1

  • 1Kavli Institute for Theoretical Physics and Department of Physics, University of California, Santa Barbara, California 93106, USA.

Physical Review Letters
|December 24, 2021
PubMed
Summary
This summary is machine-generated.

This study maps physical properties of scattering amplitudes to the Riemann zeta function, constructing a closed-form amplitude. Real masses correspond to the Riemann hypothesis, linking fundamental physics to number theory.

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Area of Science:

  • Theoretical Physics
  • Number Theory

Background:

  • Scattering amplitudes are fundamental in quantum field theory.
  • The Riemann zeta function is a central object in number theory with deep connections to prime numbers.

Purpose of the Study:

  • To establish a direct mapping between physical properties of scattering amplitudes and the Riemann zeta function.
  • To construct a specific closed-form scattering amplitude related to the zeros of the zeta function.

Main Methods:

  • Constructing a closed-form scattering amplitude for a specific mass tower.
  • Relating physical constraints (real masses, locality, unitarity) to properties of the Riemann zeta function and its zeros.

Main Results:

  • A novel closed-form amplitude is derived, parameterized by masses related to the non-trivial zeros of the Riemann zeta function.
  • Physical requirements like real masses and locality are shown to correspond to the Riemann hypothesis and meromorphicity of the zeta function, respectively.
  • Unitarity bounds translate to positivity conditions on moments of the inverse squared masses.

Conclusions:

  • The study reveals a profound connection between scattering amplitudes in physics and the Riemann zeta function.
  • This connection provides a new perspective on the Riemann hypothesis and its physical implications.
  • The framework offers potential for new insights into both theoretical physics and number theory.