Geometric Signals: Abstract Mathematical Model for Information Transmission

Główne pojęcia

Abstract mathematical model for transmitting information via geometric objects.

Streszczenie

• Introduction
  • Defines a signal as a submanifold of a Riemannian manifold.
  • Focuses on geometric generalization of the concept of a signal.
• Practical Applications
  • Signal processing involves various techniques like sampling and quantization.
  • Concentrates on the geometric concept of signals.
• Background Discussion
  • Illustrates how the word "dot" can be transmitted in different forms.
• Cobordisms as Signals
  • Introduces cobordisms as signals in higher dimensions.
• Riemannian Manifolds
  • Discusses Riemannian manifolds and distance functions related to signals.
• Theorems and Proofs
  • Presents theorems related to energy, noise, filtering, and composition of signals.

Statystyki

"Date: March 26, 2024."
"arXiv:2403.15978v1 [math.DG] 24 Mar 2024"

Cytaty

Kluczowe wnioski z

Geometric signals

by Tatyana Barr... o arxiv.org 03-26-2024

https://arxiv.org/pdf/2403.15978.pdf

Głębsze pytania

How does this abstract mathematical model compare to traditional signal processing methods

The abstract mathematical model presented in the context differs from traditional signal processing methods by focusing on geometric objects as higher-dimensional analogues of sound waveforms. While traditional signal processing involves techniques like sampling, quantization, and encoding to process signals for transmission and analysis, this model emphasizes treating geometric objects as information carriers without delving into digitization or further data processing steps. By defining signals as submanifolds of Riemannian manifolds and introducing concepts like cobordisms to represent the evolution of these signals intuitively, this approach offers a unique perspective compared to the more conventional function-based definitions in signal processing.

What are the practical implications of focusing solely on the geometric aspects of signals

Focusing solely on the geometric aspects of signals has practical implications that streamline the analysis and understanding of complex data structures. By abstracting signals as submanifolds within a Riemannian manifold framework, researchers can simplify signal representation and manipulation processes. This abstraction allows for a more intuitive interpretation of how information is transmitted through geometric entities rather than through numerical values alone. Additionally, concentrating on geometry enables researchers to develop novel insights into signal properties such as energy distribution across different components of a signal's Fourier transform or noise modulation within specific regions defined by metrics.

How can the concept of cobordisms be applied in other fields beyond mathematics

The concept of cobordisms introduced in mathematics can find applications beyond its original field in various disciplines such as physics, computer science, and biology. In physics, cobordism theory plays a crucial role in studying topological phases of matter where different states are connected by continuous transformations represented by cobordisms. In computer science, cobordisms can be utilized for data integration tasks where disparate datasets need to be combined seamlessly based on shared characteristics represented by manifold boundaries. Moreover, in biology, cobordism concepts could aid in analyzing evolutionary pathways between species or genetic sequences through topological mappings that capture underlying relationships effectively.