Avionics 738 Adaptive Filtering at Air University PAC Campus by Dr. Bilal A. Siddiqui in Spring 2018. This lecture covers background material for the course.
ME-314 Introduction to Control Engineering is a course taught to Mechanical Engineering senior undergrads. The course is taught by Dr. Bilal Siddiqui at DHA Suffa University. This lecture is about time response of systems derived by inspection of poles and zeros. Stability concepts and steady state errors are taught.
Esta es una presentacion que hice con motivo de los requisitos que exige la maestria en fisica en la Unviersidad de Bishops, en Quebec, Canada. Durante mi presentacion, hicieron incapie en un error de subindices durante el desarrollo de las ecuaciones de las ondas gravitacionales. Lamentablemente no recuerdo en que diapositiva me marcaron el error, asi que es un desafio para cualquiera que encuentre mi presentacion interesante para ser utilizada en algun proyecto. Gracias.
Avionics 738 Adaptive Filtering at Air University PAC Campus by Dr. Bilal A. Siddiqui in Spring 2018. This lecture covers background material for the course.
ME-314 Introduction to Control Engineering is a course taught to Mechanical Engineering senior undergrads. The course is taught by Dr. Bilal Siddiqui at DHA Suffa University. This lecture is about time response of systems derived by inspection of poles and zeros. Stability concepts and steady state errors are taught.
Esta es una presentacion que hice con motivo de los requisitos que exige la maestria en fisica en la Unviersidad de Bishops, en Quebec, Canada. Durante mi presentacion, hicieron incapie en un error de subindices durante el desarrollo de las ecuaciones de las ondas gravitacionales. Lamentablemente no recuerdo en que diapositiva me marcaron el error, asi que es un desafio para cualquiera que encuentre mi presentacion interesante para ser utilizada en algun proyecto. Gracias.
ME-314 Introduction to Control Engineering is a course taught to Mechanical Engineering senior undergrads. The course is taught by Dr. Bilal Siddiqui at DHA Suffa University. This lecture is about frequency domain solutions of differential equations and transfer functions.
ME-314 Introduction to Control Engineering is a course taught to Mechanical Engineering senior undergrads. The course is taught by Dr. Bilal Siddiqui at DHA Suffa University. This lecture is about basic rules of sketching root locus.
Non-linear control of a bipedal (Three-Linked) Walker using feedback Lineariz...Mike Simon
Non-linear control of a bipedal (Three-Linked) Walker using feedback Linearization is a research project for control theory subject in Robotics Master Course in the Higher Institute of Applied Science and Technology.
A brief and easy concept of Simple harmonic oscillator. How we can get simple harmonic motion equation from Lagrange's equation of motion. How can we obtain this from Lagrange's equation of motion.
Robust Fuzzy Output Feedback Controller for Affine Nonlinear Systems via T–S ...Mostafa Shokrian Zeini
This presentation concerns the design of a robust H_∞ fuzzy output feedback controller for a class of affine nonlinear systems with disturbance via Takagi-Sugeno (T–S) fuzzy bilinear model. The parallel distributed compensation (PDC) technique is utilized to design a fuzzy controller. The stability conditions of the overall closed loop T-S fuzzy bilinear model are formulated in terms of Lyapunov function via linear matrix inequality (LMI). The control law is robustified by H_∞ sense to attenuate external disturbance. Moreover, the desired controller gains can be obtained by solving a set of LMI.
Energy-Based Control of Under-Actuated Mechanical Systems - Remotely Driven A...Mostafa Shokrian Zeini
This presentation concerns the energy-based swing-up control for a remotely driven acrobot (RDA) which is a 2-link planar robot with the first link being underactuated and the second link being remotely driven by an actuator mounted at a fixed base through a belt.
The presentation presents to the reader an understanding of Scalar and Vector Spherical Harmonics, it's origin and application to various engineering fields.
The slides are designed for my guided study in MSc CUHK.
It is about the brief description on classical mechanics and quantum mechanics .
Some Slides I got from the slideshare clipboards for better illustration of the ideas in Physics. Thanks to slideshare, I make a milestone on presenting one of the prominent fields in modern physics.
ME-314 Introduction to Control Engineering is a course taught to Mechanical Engineering senior undergrads. The course is taught by Dr. Bilal Siddiqui at DHA Suffa University. This lecture is about frequency domain solutions of differential equations and transfer functions.
ME-314 Introduction to Control Engineering is a course taught to Mechanical Engineering senior undergrads. The course is taught by Dr. Bilal Siddiqui at DHA Suffa University. This lecture is about basic rules of sketching root locus.
Non-linear control of a bipedal (Three-Linked) Walker using feedback Lineariz...Mike Simon
Non-linear control of a bipedal (Three-Linked) Walker using feedback Linearization is a research project for control theory subject in Robotics Master Course in the Higher Institute of Applied Science and Technology.
A brief and easy concept of Simple harmonic oscillator. How we can get simple harmonic motion equation from Lagrange's equation of motion. How can we obtain this from Lagrange's equation of motion.
Robust Fuzzy Output Feedback Controller for Affine Nonlinear Systems via T–S ...Mostafa Shokrian Zeini
This presentation concerns the design of a robust H_∞ fuzzy output feedback controller for a class of affine nonlinear systems with disturbance via Takagi-Sugeno (T–S) fuzzy bilinear model. The parallel distributed compensation (PDC) technique is utilized to design a fuzzy controller. The stability conditions of the overall closed loop T-S fuzzy bilinear model are formulated in terms of Lyapunov function via linear matrix inequality (LMI). The control law is robustified by H_∞ sense to attenuate external disturbance. Moreover, the desired controller gains can be obtained by solving a set of LMI.
Energy-Based Control of Under-Actuated Mechanical Systems - Remotely Driven A...Mostafa Shokrian Zeini
This presentation concerns the energy-based swing-up control for a remotely driven acrobot (RDA) which is a 2-link planar robot with the first link being underactuated and the second link being remotely driven by an actuator mounted at a fixed base through a belt.
The presentation presents to the reader an understanding of Scalar and Vector Spherical Harmonics, it's origin and application to various engineering fields.
The slides are designed for my guided study in MSc CUHK.
It is about the brief description on classical mechanics and quantum mechanics .
Some Slides I got from the slideshare clipboards for better illustration of the ideas in Physics. Thanks to slideshare, I make a milestone on presenting one of the prominent fields in modern physics.
The postulates of quantum mechanics have been successfully used for deriving exact solutions to Schrodinger equation for problems like A particle in 1 Dimensional box Harmonic oscillator Rigid rotator Hydrogen atom • However for a multielectron system, the SWE cannot be solved exactly due to inter-electronic repulsion terms.
The SWE is solved by method of seperation of variables.
• However, the inter-electronic repulsion term cannot be solved because the variables cannot be seperated and the SWE cannot be solved. • Approximate methods have helped to generate solutions for such and even more complex real quantum systems. • Approximate methods have been developed for solving Schrodinger equation to find wave function and energy of the complex system under consideration. • Two widely used approximate methods are, 1. Perturbation theory 2. Variation method
Perturbation theory is an approximate method that describes a complex quantum system in terms of a simpler system for which the exact solution is known. • Perturbation theory has been categorized into, i. Time independent perturbation theory, proposed by Erwin Schrodinger, where the perturbation Hamiltonian is static. ii. Time dependent perturbation theory, proposed by Paul Dirac, which studies the effect of time dependent perturbation on a time independent Hamiltonian H0.
PERTURBATION THEOREM
FIRST ORDER PERTURBATION THEORY
FIRST ORDER ENERGY CORRECTION
FIRST ORDER WAVE FUNCTION CORRECTION
APPLICATIONS OF PERTURBATION METHOD
SIGNIFICANCE OF PERTURBATION METHOD
In tis slide, an introduction to string theory has been given. Apart from that, a simple proof of 26 dimensions of bosonic string theory is given (following Zwiebach's approach).
I explained this presentation in two parts (on my YouTube channel). Here are the links
_______________________________________________
Part 1
https://www.youtube.com/watch?v=QQA4JQ6Y-eo&list=PLDpqC3uXLZGl0cDod6g30PcjeJ4DAZWhp
_______________________________________________
Part 2
https://www.youtube.com/watch?v=vhLCtLn79jE&list=PLDpqC3uXLZGl0cDod6g30PcjeJ4DAZWhp&index=2
_______________________________________________
Navier stokes equation in coordinates binormal, tangent and normalCarlos López
The Navier-Stokes problem is a very important set of partial differential equations for analyzing fluids into the context
of the motion of fluid substances. There is no a general analytical solution related to complex fields of velocity vector
푢(푋, 푡)
, wherein the position vector is given by 푋 = (푥, 푦. 푧) and 푡 is the time variable, but there are some few solutions
associated to the simple velocity vector and the pressure 푃(푋, 푡) experienced by the fluid. However, these simple
models are not sufficient to predict the dynamic of Newtonian fluids in general. On this article is proposed an
interesting mathematical model to represent easily the equations of Navier Stokes in a TNB frame system which let
optimize the task of modeling complex equations from a Cartesian coordinate system and reducing them to a set of
equations less complex in a TNB frame whose perspective is going to be truly interesting from the physical problem.
Probabilistic Models of Time Series and SequencesZitao Liu
Tutorial on Probabilistic Models of Time Series and Sequences. Hidden Markov Models. Linear Dynamical Systems. Forward/backward algorithm. Kalman Filtering. Kalman Smoothing. Viterbi algorithm. Baum-Welch algorithm. Learning HMM. Learning LDS.
Coordinate systems
orthogonal coordinate system
Rectangular or Cartesian coordinate system
Cylindrical or circular coordinate system
Spherical coordinate system
Relationship between various coordinate system
Transformation Matrix
DIFFERENTIAL VECTOR
Curvilinear, Cartesian, Cylindrical, Spherical table
The use of Nauplii and metanauplii artemia in aquaculture (brine shrimp).pptxMAGOTI ERNEST
Although Artemia has been known to man for centuries, its use as a food for the culture of larval organisms apparently began only in the 1930s, when several investigators found that it made an excellent food for newly hatched fish larvae (Litvinenko et al., 2023). As aquaculture developed in the 1960s and ‘70s, the use of Artemia also became more widespread, due both to its convenience and to its nutritional value for larval organisms (Arenas-Pardo et al., 2024). The fact that Artemia dormant cysts can be stored for long periods in cans, and then used as an off-the-shelf food requiring only 24 h of incubation makes them the most convenient, least labor-intensive, live food available for aquaculture (Sorgeloos & Roubach, 2021). The nutritional value of Artemia, especially for marine organisms, is not constant, but varies both geographically and temporally. During the last decade, however, both the causes of Artemia nutritional variability and methods to improve poorquality Artemia have been identified (Loufi et al., 2024).
Brine shrimp (Artemia spp.) are used in marine aquaculture worldwide. Annually, more than 2,000 metric tons of dry cysts are used for cultivation of fish, crustacean, and shellfish larva. Brine shrimp are important to aquaculture because newly hatched brine shrimp nauplii (larvae) provide a food source for many fish fry (Mozanzadeh et al., 2021). Culture and harvesting of brine shrimp eggs represents another aspect of the aquaculture industry. Nauplii and metanauplii of Artemia, commonly known as brine shrimp, play a crucial role in aquaculture due to their nutritional value and suitability as live feed for many aquatic species, particularly in larval stages (Sorgeloos & Roubach, 2021).
BREEDING METHODS FOR DISEASE RESISTANCE.pptxRASHMI M G
Plant breeding for disease resistance is a strategy to reduce crop losses caused by disease. Plants have an innate immune system that allows them to recognize pathogens and provide resistance. However, breeding for long-lasting resistance often involves combining multiple resistance genes
Deep Behavioral Phenotyping in Systems Neuroscience for Functional Atlasing a...Ana Luísa Pinho
Functional Magnetic Resonance Imaging (fMRI) provides means to characterize brain activations in response to behavior. However, cognitive neuroscience has been limited to group-level effects referring to the performance of specific tasks. To obtain the functional profile of elementary cognitive mechanisms, the combination of brain responses to many tasks is required. Yet, to date, both structural atlases and parcellation-based activations do not fully account for cognitive function and still present several limitations. Further, they do not adapt overall to individual characteristics. In this talk, I will give an account of deep-behavioral phenotyping strategies, namely data-driven methods in large task-fMRI datasets, to optimize functional brain-data collection and improve inference of effects-of-interest related to mental processes. Key to this approach is the employment of fast multi-functional paradigms rich on features that can be well parametrized and, consequently, facilitate the creation of psycho-physiological constructs to be modelled with imaging data. Particular emphasis will be given to music stimuli when studying high-order cognitive mechanisms, due to their ecological nature and quality to enable complex behavior compounded by discrete entities. I will also discuss how deep-behavioral phenotyping and individualized models applied to neuroimaging data can better account for the subject-specific organization of domain-general cognitive systems in the human brain. Finally, the accumulation of functional brain signatures brings the possibility to clarify relationships among tasks and create a univocal link between brain systems and mental functions through: (1) the development of ontologies proposing an organization of cognitive processes; and (2) brain-network taxonomies describing functional specialization. To this end, tools to improve commensurability in cognitive science are necessary, such as public repositories, ontology-based platforms and automated meta-analysis tools. I will thus discuss some brain-atlasing resources currently under development, and their applicability in cognitive as well as clinical neuroscience.
Richard's aventures in two entangled wonderlandsRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
Remote Sensing and Computational, Evolutionary, Supercomputing, and Intellige...University of Maribor
Slides from talk:
Aleš Zamuda: Remote Sensing and Computational, Evolutionary, Supercomputing, and Intelligent Systems.
11th International Conference on Electrical, Electronics and Computer Engineering (IcETRAN), Niš, 3-6 June 2024
Inter-Society Networking Panel GRSS/MTT-S/CIS Panel Session: Promoting Connection and Cooperation
https://www.etran.rs/2024/en/home-english/
DERIVATION OF MODIFIED BERNOULLI EQUATION WITH VISCOUS EFFECTS AND TERMINAL V...Wasswaderrick3
In this book, we use conservation of energy techniques on a fluid element to derive the Modified Bernoulli equation of flow with viscous or friction effects. We derive the general equation of flow/ velocity and then from this we derive the Pouiselle flow equation, the transition flow equation and the turbulent flow equation. In the situations where there are no viscous effects , the equation reduces to the Bernoulli equation. From experimental results, we are able to include other terms in the Bernoulli equation. We also look at cases where pressure gradients exist. We use the Modified Bernoulli equation to derive equations of flow rate for pipes of different cross sectional areas connected together. We also extend our techniques of energy conservation to a sphere falling in a viscous medium under the effect of gravity. We demonstrate Stokes equation of terminal velocity and turbulent flow equation. We look at a way of calculating the time taken for a body to fall in a viscous medium. We also look at the general equation of terminal velocity.
Seminar of U.V. Spectroscopy by SAMIR PANDASAMIR PANDA
Spectroscopy is a branch of science dealing the study of interaction of electromagnetic radiation with matter.
Ultraviolet-visible spectroscopy refers to absorption spectroscopy or reflect spectroscopy in the UV-VIS spectral region.
Ultraviolet-visible spectroscopy is an analytical method that can measure the amount of light received by the analyte.
Phenomics assisted breeding in crop improvementIshaGoswami9
As the population is increasing and will reach about 9 billion upto 2050. Also due to climate change, it is difficult to meet the food requirement of such a large population. Facing the challenges presented by resource shortages, climate
change, and increasing global population, crop yield and quality need to be improved in a sustainable way over the coming decades. Genetic improvement by breeding is the best way to increase crop productivity. With the rapid progression of functional
genomics, an increasing number of crop genomes have been sequenced and dozens of genes influencing key agronomic traits have been identified. However, current genome sequence information has not been adequately exploited for understanding
the complex characteristics of multiple gene, owing to a lack of crop phenotypic data. Efficient, automatic, and accurate technologies and platforms that can capture phenotypic data that can
be linked to genomics information for crop improvement at all growth stages have become as important as genotyping. Thus,
high-throughput phenotyping has become the major bottleneck restricting crop breeding. Plant phenomics has been defined as the high-throughput, accurate acquisition and analysis of multi-dimensional phenotypes
during crop growing stages at the organism level, including the cell, tissue, organ, individual plant, plot, and field levels. With the rapid development of novel sensors, imaging technology,
and analysis methods, numerous infrastructure platforms have been developed for phenotyping.
Comparing Evolved Extractive Text Summary Scores of Bidirectional Encoder Rep...University of Maribor
Slides from:
11th International Conference on Electrical, Electronics and Computer Engineering (IcETRAN), Niš, 3-6 June 2024
Track: Artificial Intelligence
https://www.etran.rs/2024/en/home-english/
Professional air quality monitoring systems provide immediate, on-site data for analysis, compliance, and decision-making.
Monitor common gases, weather parameters, particulates.
ANAMOLOUS SECONDARY GROWTH IN DICOT ROOTS.pptxRASHMI M G
Abnormal or anomalous secondary growth in plants. It defines secondary growth as an increase in plant girth due to vascular cambium or cork cambium. Anomalous secondary growth does not follow the normal pattern of a single vascular cambium producing xylem internally and phloem externally.
4. Motivation
• Time reversible Markov chain (detailed balance) is known to have
symmetric probability flux and can be described by gradient system
• Symmetric flux contributes to the production of relative entropy,
whereas skew-symmetric flux doesn’t
• Skew-symmetric flux playing a very important role as circulation in
time evolution of chains is yet hardly understood
• Evolution of Markov chain can be characterized by differential
geometry, which is a powerful and indispensable tool in dynamics
6. Alpha representation
In information geometry, a probability distribution can be coded by a
parameter 𝛼 as the following,
𝑙 𝛼
=
2
1 − 𝛼
𝑝
1−𝛼
2
Important examples include:
𝛼 = −1, 𝑙(−1) = 𝑝 mixed representation
𝛼 = 1, 𝑙(1)
= log 𝑝 exponential representation
𝛼 = 0, 𝑙(0) = 2 𝑝 0-representation
9. Alpha representation
• Different representations are equipped with different geometric
structures and restrict the dynamics of Markov chains on different
manifolds.
• Particularly, 0-representation admits the flow of probability on the
(hyper)sphere, which has radical symmetry.
10. Lie group of 𝑆𝑂 𝑛
• The motion on the manifold of 𝑛-sphere 𝑀 = 𝕊 𝑛 can be seen as
continuous isometry (distance-preserving) transformation.
• Given an initial point 𝑝0 ∈ 𝑀, the trajectory of 𝑝 is given by 𝑝𝑡 = 𝑔𝑡 𝑝0,
where 𝑔0 = 𝑒, 𝑔𝑡+𝑠 = 𝑔𝑡 𝑔𝑠 form a Lie group 𝐺 = 𝑆𝑂𝑛.
• Under matrix representation, 𝐺 is the set of order-𝑛 orthogonal
matrices with determinant 1. i.e. 𝑆𝑂𝑛 = 𝑂 ∈ 𝑆𝐿 𝑛|𝑂 𝑇
𝑂 = 𝑂𝑂 𝑇
= 𝐼 𝑛
11. Lie algebra of 𝔰𝔬 𝑛
• Let 𝐺 action on the torsor (principal homogenous space) 𝑀 from the
left, we have
ሶ𝑔𝑡 = lim
𝑠→0
𝑔𝑠 − 𝑒
𝑠
𝑔𝑡 = X𝑔𝑡 ∈ 𝑇𝑔 𝑡
𝐺
𝑋 = lim
𝑠→0
𝑔s − 𝑒
𝑠
= ሶ𝑔𝑡 ∘ 𝑔𝑡
−1
∈ 𝑇𝑒 𝐺 = 𝔤
The tangent vector X is the right translation of ሶ𝑔𝑡 by 𝑔𝑡
−1
.
• Lie algebra 𝔤 can be identified as tangent space at the identity.
Given any vector 𝑋 ∈ 𝔤, there is a unique left-invariant vector field
𝑋 𝑔 = 𝑇𝐿 𝑔 𝑋 = 𝐿 𝑔∗
𝑋
12. Lie algebra of 𝔰𝔬 𝑛
• Note that for 𝑂𝑠 ∈ 𝑆𝑂𝑛 near the identity, we have
𝐼 = 𝑂𝑠
𝑇 𝑂𝑠 = 𝐼 + 𝑠Ω + 𝑜 𝑠
𝑇
𝐼 + 𝑠Ω + 𝑜 𝑠 = 𝐼 + 𝑠 Ω + Ω 𝑇 + 𝑜 𝑠
The matrix Lie algebra of 𝔰𝔬 𝑛 is the set of skew-symmetric matrices,
i.e. 𝔰𝔬 𝑛 = 𝑇𝑒 𝑆𝑂𝑛 = Ω ∈ 𝐺𝐿 𝑛| Ω + Ω 𝑇
= 0
• It can also be identified with vector space of dimension
𝑛(𝑛−1)
2
13. Adjoint and coadjoint representation of 𝔰𝔬 𝑛
• An important representation of Lie algebra, called adjoint
representation, is defined as
𝑎𝑑 ∶ 𝔤 → 𝔤𝔩 𝑛 = 𝐸𝑛𝑑 𝔤
𝑎𝑑 𝑋: 𝑌 ↦ 𝑋, 𝑌
• Choose a non-degenerate inner product , on Lie algebra 𝔤, the
coadjoint representation is defined as
𝑎𝑑 𝑍
∗
𝑋, 𝑌 = 𝑋, 𝑎𝑑 𝑍 𝑌
14. Riemannian metric
• The inner product , induces a right-invariant Riemannian metric
, 𝑔 on the whole Lie group 𝐺. Given two vectors 𝑋, 𝑌 ∈ 𝑇𝑔 𝐺, the
Riemannian metric is defined as
𝑋, 𝑌 𝑔: = 𝑇𝑅 𝑔
−1
∗
𝑋 , 𝑇𝑅 𝑔
−1
∗
𝑌
• The geodesic is defined as the extremal of the energy functional
𝐸 𝑔𝑡 = න
𝑎
𝑏
1
2
ሶ𝑔𝑡, ሶ𝑔𝑡 𝑔 𝑑𝑡
18. Geometric flow of CTMC
• Let 𝑞𝑡 be a continuous trajectory on 𝕊 𝑛 such that 𝑞𝑡 = 𝑔𝑡 𝑞0, where
𝑔𝑡 ∈ 𝑆𝑂𝑛, then
ሶ𝑞𝑡 = ሶ𝑔𝑡 𝑞0 = ሶ𝑔𝑡 𝑔𝑡
−1
𝑞𝑡 = Ω𝑞𝑡
Ω = ሶ𝑔𝑡 𝑔𝑡
−1
∈ 𝔰𝔬 𝑛
• This establishes a bijection between the trajectory on 𝕊 𝑛 and that on
𝑆𝑂𝑛. This inspires us to investigate geodesic flow on 𝑆𝑂𝑛.
21. Geometric flow of CTMC again
• Euler-Poincare equation:
ሶΩ = 𝑎𝑑Ω
∗
Ω
Note that this equation doesn’t contain 𝑔𝑡 explicitly.
• We can reconstruct the equation of the motion of 0-representation
Markov chain by
Ω = ሶ𝑔𝑡 𝑔𝑡
−1
,
ሶ𝑔𝑡 = Ω𝑔𝑡
22. Conservation law in CTMC
• By Noether’s theorem, the right-invariant geodesic flow preserves
some quantities, which can be computed by momentum map 𝜇
𝜇: 𝔤 → ℝ, 𝑋 ↦ Ω, 𝑔𝑡 𝑋𝑔𝑡
−1
• Proof
ሶ𝜇 = ሶΩ, 𝑔𝑡 𝑋𝑔𝑡
−1
+ Ω, Ω, 𝑔𝑡 𝑋𝑔𝑡
−1
= 𝑎𝑑ΩΩ, 𝑔𝑡 𝑋𝑔𝑡
−1
+ Ω, 𝑎𝑑Ω 𝑔𝑡 𝑋𝑔𝑡
−1
= 𝑎𝑑ΩΩ, 𝑔𝑡 𝑋𝑔𝑡
−1
+ 𝑎𝑑Ω
∗
𝛺, 𝑔𝑡 𝑋𝑔𝑡
−1
= 0
24. Conclusions
In summary, we give a geometric formulation of 0-representation CTMC.
This view allows us to
• Investigate the dynamics on (hyper)sphere, from both intrinsic
and extrinsic view
• Reduce the dimension of infinitesimal generator by half (from
𝑛(𝑛 − 1) to
𝑛 𝑛−1
2
)
• The time evolution of Markov chains follows Euler-Poincare
equation, whose trajectory is always geodesic flow
• Conservation quantities can be found
25. Further questions
There are many problems to be solved yet
• How to distinguish skew-symmetric flux from symmetric one in
geometric view
• Geometric formulation of CTMC in other representations
• Find master equation of geodesic flows
• Etc..