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Related Concept Videos

State Space Representation01:27

State Space Representation

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The frequency-domain technique, commonly used in analyzing and designing feedback control systems, is effective for linear, time-invariant systems. However, it falls short when dealing with nonlinear, time-varying, and multiple-input multiple-output systems. The time-domain or state-space approach addresses these limitations by utilizing state variables to construct simultaneous, first-order differential equations, known as state equations, for an nth-order system.
Consider an RLC circuit, a...
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Space Trusses01:25

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A space truss is a three-dimensional counterpart of a planar truss. These structures consist of members connected at their ends, often utilizing ball-and-socket joints to create a stable and versatile framework. The space truss is widely used in various construction projects due to its adaptability and capacity to withstand complex loads.
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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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Space Trusses: Problem Solving01:29

Space Trusses: Problem Solving

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A space truss is a three-dimensional counterpart of a planar truss. These structures consist of members connected at their ends, often utilizing ball-and-socket joints to create a stable and versatile framework. Due to its adaptability and capacity to withstand complex loads, the space truss is widely used in various construction projects.
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Rocket Propulsion in Empty Space - I01:13

Rocket Propulsion in Empty Space - I

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The driving force for the motion of any vehicle is friction, but in the case of rocket propulsion in space, the friction force is not present. The motion of a rocket changes its velocity (and hence its momentum) by ejecting burned fuel gases, thus causing it to accelerate in the direction opposite to the velocity of the ejected fuel. In this situation, the mass and velocity of the rocket constantly change along with the total mass of ejected gases. Due to conservation of momentum, the...
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Related Experiment Video

Updated: Jan 27, 2026

In vitro Synthesis of Native, Fibrous Long Spacing and Segmental Long Spacing Collagen
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In vitro Synthesis of Native, Fibrous Long Spacing and Segmental Long Spacing Collagen

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Lorentzian length spaces.

Michael Kunzinger1, Clemens Sämann1

  • 1Faculty of Mathematics, University of Vienna, Vienna, Austria.

Annals of Global Analysis and Geometry
|March 22, 2019
PubMed
Summary
This summary is machine-generated.

This study extends length space theory to Lorentzian geometry using a time separation function. This framework unifies fundamental results and introduces synthetic curvature bounds for quantum gravity applications.

Keywords:
Causality theoryLength spacesLorentzian length spacesMetric geometrySynthetic curvature boundsTriangle comparison

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Area of Science:

  • Lorentzian geometry
  • Causality theory
  • Metric spaces

Background:

  • Length spaces provide a foundation for geometric analysis.
  • Lorentzian geometry and causality theory are crucial for understanding spacetime.
  • Existing theories often require high regularity of spacetime metrics.

Purpose of the Study:

  • To develop an analogue of length space theory within Lorentzian geometry.
  • To reformulate fundamental concepts using a time separation function.
  • To introduce synthetic curvature bounds for causal spacetimes.

Main Methods:

  • Formulating basic notions using a time separation function.
  • Adapting triangle comparison methods for synthetic curvature bounds.
  • Generalizing existing results in Lorentzian geometry.

Main Results:

  • A unified framework for Lorentzian geometry and causality theory.
  • Recovery of fundamental results with broader applicability.
  • Introduction of synthetic curvature bounds applicable to low-regularity metrics.

Conclusions:

  • The time separation function offers a powerful tool for Lorentzian geometry.
  • Synthetic curvature bounds provide new insights into causal structures.
  • This work has implications for quantum gravity and spacetime analysis.