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Related Concept Videos

Stability01:28

Stability

192
The time response of a linear time-invariant (LTI) system can be divided into transient and steady-state responses. The transient response represents the system's initial reaction to a change in input and diminishes to zero over time. In contrast, the steady-state response is the behavior that persists after the transient effects have faded.
The stability of an LTI system is determined by the roots of its characteristic equation, known as poles. A system is stable if it produces a bounded...
192
BIBO stability of continuous and discrete -time systems01:24

BIBO stability of continuous and discrete -time systems

526
System stability is a fundamental concept in signal processing, often assessed using convolution. For a system to be considered bounded-input bounded-output (BIBO) stable, any bounded input signal must produce a bounded output signal. A bounded input signal is one where the modulus does not exceed a certain constant at any point in time.
To determine the BIBO stability, the convolution integral is utilized when a bounded continuous-time input is applied to a Linear Time-Invariant (LTI) system....
526
Pole and System Stability01:24

Pole and System Stability

426
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...
426
Traveling Waves: Lossless Lines01:27

Traveling Waves: Lossless Lines

203
The provided content explores the behavior of traveling waves on single-phase lossless transmission lines. It begins with a single-phase two-wire lossless transmission line of length Δx, characterized by a loop inductance LH/m and a line-to-line capacitance C F/m. These parameters result in a series inductance LΔx  and a shunt capacitance CΔx.
203
Noncompartmental Analysis: Mean Residence Time01:05

Noncompartmental Analysis: Mean Residence Time

281
According to statistical moment theory, mean residence time (MRT) is an important measure in pharmacokinetics. MRT can be defined as the expected mean of a probability density function distribution. It provides valuable insights into drug disposition in the body.
After the administration of a drug through intravenous bolus injection, the drug molecules are distributed throughout the body and remain there for varying periods. The MRT represents the average time these drug molecules stay in the...
281
Routh-Hurwitz Criterion II01:19

Routh-Hurwitz Criterion II

421
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...
421

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Quantifying Cytoskeleton Dynamics Using Differential Dynamic Microscopy
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Lipschitz Stability of Travel Time Data.

Joonas Ilmavirta1, Antti Kykkänen2, Matti Lassas3

  • 1Department of Mathematics and Statistics, University of Jyväskylä, Jyväskylä, Finland.

Journal of Geometric Analysis
|July 1, 2025
PubMed
Summary
This summary is machine-generated.

We show that reconstructing length spaces from travel time data is stable. This stability applies to various spaces, including Riemannian manifolds and metric trees, using boundary measurements.

Keywords:
Geometric inverse problemsGromov–Hausdorff distanceIsometric embeddingsLength spaces

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Area of Science:

  • Geometric inverse problems
  • Metric geometry
  • Topology

Background:

  • Reconstructing spaces from measurements is a key challenge in geometric inverse problems.
  • Travel time data, derived from distance functions, is crucial for this reconstruction.
  • Classical problems include Gel'fand's inverse boundary spectral problem on Riemannian manifolds.

Purpose of the Study:

  • To establish Lipschitz stability for reconstructing length spaces from travel time data.
  • To extend reconstruction stability results to a broader class of metric spaces.
  • To analyze the impact of measurement subsets on reconstruction accuracy.

Main Methods:

  • Utilizing distance functions as travel time data on a closed subset.
  • Applying techniques from geometric measure theory and analysis.
  • Investigating the properties of length spaces and their metric properties.

Main Results:

  • Proving Lipschitz stability for the reconstruction of specific length spaces.
  • Demonstrating stability using travel time data measured on a closed subset.
  • Identifying conditions under which stable reconstruction is achievable.

Conclusions:

  • The reconstruction of length spaces from travel time data is Lipschitz stable under specific conditions.
  • This stability has implications for understanding geometric inverse problems in various settings.
  • The findings are applicable to Riemannian manifolds, Euclidean domains, and metric trees.