Related Experiment Videos
Transformations that preserve detailed balance in Markov models
William J Bruno1, John E Pearson
1Theoretical Biology and Biophysics, Los Alamos National Laboratory, Los Alamos, New Mexico 87544, USA.
Summary
Researchers identified similarity transformations for aggregated Markov processes. This finding is crucial for understanding model identification in complex systems, especially in single-molecule studies.
Area of Science:
- * Computational Mathematics
- * Statistical Modeling
- * Systems Biology
Background:
- * Aggregated Markov processes are often indistinguishable via steady-state experiments due to similarity transformations.
- * Understanding these transformations is key to identifying unique models from observed data.
Purpose of the Study:
- * To derive an explicit formula for similarity transformations between continuous-time Markov chains.
- * To characterize the matrices that define these transformations.
Main Methods:
- * Derivation of an explicit formula for detailed-balance preserving similarity transformations.
- * Analysis of the mathematical structure of transformation matrices within the special orthogonal group.
Main Results:
- * An explicit formula for similarity transformations between aggregated Markov processes was derived.
- * Transformation matrices are a non-linear function of elements within the N-dimensional special orthogonal group.
Conclusions:
- * The derived formula provides a framework for understanding equivalencies in Markov models.
- * Results are expected to advance the theory of model identification for aggregated Markov chains, particularly in single-molecule biophysics.
Related Concept Videos
01:20Transformations of Functions III
Transformations modify the graphical representation of a function without changing its fundamental form. One common transformation is reflection, which flips the graph across a designated axis. When the vertical coordinates of all points are multiplied by the negative one, the entire graph is mirrored over the horizontal axis. This transformation reverses the vertical orientation of peaks and troughs, akin to signal inversion in electrical systems, where a waveform is flipped, but the timing of...
01:10Entropy Change in Reversible Processes
In the Carnot engine, which achieves the maximum efficiency between two reservoirs of fixed temperatures, the total change in entropy is zero. The observation can be generalized by considering any reversible cyclic process consisting of many Carnot cycles. Thus, it can be stated that the total entropy change of any ideal reversible cycle is zero.
The statement can be further generalized to prove that entropy is a state function. Take a cyclic process between any two points on a p-V diagram.
The statement can be further generalized to prove that entropy is a state function. Take a cyclic process between any two points on a p-V diagram.
01:23Stability of Equilibrium Configuration
Understanding the stability of equilibrium configurations is a fundamental part of mechanical engineering. In any system, there are three distinct types of equilibrium: stable, neutral, and unstable.
A stable equilibrium occurs when a system tends to return to its original position when given a small displacement, and the potential energy is at its minimum. An example of a stable equilibrium is when a cantilever beam is fixed at one end and a weight is attached to the other end. If the weight...
A stable equilibrium occurs when a system tends to return to its original position when given a small displacement, and the potential energy is at its minimum. An example of a stable equilibrium is when a cantilever beam is fixed at one end and a weight is attached to the other end. If the weight...
01:29Transformations of Functions II
Transformations in mathematics alter the position or orientation of a function’s graph while preserving its fundamental shape. One important type of transformation is the horizontal shift, which involves modifying the input variable within a function’s equation. This operation affects where outputs occur along the horizontal axis but does not alter the function’s overall structure.A horizontal shift is achieved by replacing the input variable x with either x + c or x - c, where c is a constant.
01:24BIBO stability of continuous and discrete -time systems
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.
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.
01:14Conservation of Mass in Moving, Nondeforming Control Volume
Stormwater detention basins are essential in managing runoff during heavy rainfall, particularly in urban areas where impervious surfaces increase the risk of flooding. Understanding the conservation of mass in these systems allows engineers to optimize basin performance, balancing inflow, outflow, and water storage.
In the context of a detention basin, the conservation of mass states that the total mass of water entering the basin must equal the mass leaving the basin plus any accumulation of...
In the context of a detention basin, the conservation of mass states that the total mass of water entering the basin must equal the mass leaving the basin plus any accumulation of...