References:
Differential Geometry of Curves and Surfaces, Manfredo P. Do Carmo (2016)
Differential Geometry by Claudio Arezzo
Youtube: https://youtu.be/tKnBj7B2PSg
What is a Manifold?
Youtube: https://youtu.be/CEXSSz0gZI4
Shape analysis (MIT spring 2019) by Justin Solomon
Youtube: https://youtu.be/GEljqHZb30c
Tensor Calculus
Youtube: https://youtu.be/kGXr1SF3WmA
Manifolds: A Gentle Introduction,
Hyperbolic Geometry and Poincaré Embeddings by Brian Keng
Link: http://bjlkeng.github.io/posts/manifolds/,
http://bjlkeng.github.io/posts/hyperbolic-geometry-and-poincare-embeddings/
Statistical Learning models for Manifold-Valued measurements with application to computer vision and neuroimaging by Hyunwoo J.Kim
A NEW STUDY OF TRAPEZOIDAL, SIMPSON’S1/3 AND SIMPSON’S 3/8 RULES OF NUMERICAL...mathsjournal
The main goal of this research is to give the complete conception about numerical integration including Newton-Cotes formulas and aimed at comparing the rate of performance or the rate of accuracy of Trapezoidal, Simpson’s 1/3, and Simpson’s 3/8. To verify the accuracy, we compare each rules demonstrating the smallest error values among them. The software package MATLAB R2013a is applied to determine the best method, as well as the results, are compared. It includes graphical comparisons mentioning these methods graphically. After all, it is then emphasized that the among methods considered, Simpson’s 1/3 is more effective and accurate when the condition of the subdivision is only even for solving a definite integral.
Periodic Function, Dirichlet's Condition, Fourier series, Even & Odd functions, Euler's Formula for Fourier Coefficients, Change of Interval, Fourier series in the intervals (0,2l), (-l,l) , (-pi, pi), (0, 2pi), Half Range Cosine & Sine series Root mean square, Complex Form of Fourier series, Parseval's Identity
A short presentation on the topic Numerical Integration for Civil Engineering students.
This presentation consist of small introduction about Simpson's Rule, Trapezoidal Rule, Gaussian Quadrature and some basic Civil Engineering problems based of above methods of Numerical Integration.
Dual Spaces of Generalized Cesaro Sequence Space and Related Matrix Mappinginventionjournals
In this paper we define the generalized Cesaro sequence spaces 푐푒푠(푝, 푞, 푠). We prove the space 푐푒푠(푝, 푞, 푠) is a complete paranorm space. In section-2 we determine its Kothe-Toeplitz dual. In section-3 we establish necessary and sufficient conditions for a matrix A to map 푐푒푠 푝, 푞, 푠 to 푙∞ and 푐푒푠(푝, 푞, 푠) to c, where 푙∞ is the space of all bounded sequences and c is the space of all convergent sequences. We also get some known and unknown results as remarks.
Some Common Fixed Point Results for Expansive Mappings in a Cone Metric SpaceIOSR Journals
The purpose of this work is to extend and generalize some common fixed point theorems for Expansive type mappings in complete cone metric spaces. We are attempting to generalize the several well- known recent results. Mathematical subject classification; 54H25, 47H10
A NEW STUDY OF TRAPEZOIDAL, SIMPSON’S1/3 AND SIMPSON’S 3/8 RULES OF NUMERICAL...mathsjournal
The main goal of this research is to give the complete conception about numerical integration including Newton-Cotes formulas and aimed at comparing the rate of performance or the rate of accuracy of Trapezoidal, Simpson’s 1/3, and Simpson’s 3/8. To verify the accuracy, we compare each rules demonstrating the smallest error values among them. The software package MATLAB R2013a is applied to determine the best method, as well as the results, are compared. It includes graphical comparisons mentioning these methods graphically. After all, it is then emphasized that the among methods considered, Simpson’s 1/3 is more effective and accurate when the condition of the subdivision is only even for solving a definite integral.
Periodic Function, Dirichlet's Condition, Fourier series, Even & Odd functions, Euler's Formula for Fourier Coefficients, Change of Interval, Fourier series in the intervals (0,2l), (-l,l) , (-pi, pi), (0, 2pi), Half Range Cosine & Sine series Root mean square, Complex Form of Fourier series, Parseval's Identity
A short presentation on the topic Numerical Integration for Civil Engineering students.
This presentation consist of small introduction about Simpson's Rule, Trapezoidal Rule, Gaussian Quadrature and some basic Civil Engineering problems based of above methods of Numerical Integration.
Dual Spaces of Generalized Cesaro Sequence Space and Related Matrix Mappinginventionjournals
In this paper we define the generalized Cesaro sequence spaces 푐푒푠(푝, 푞, 푠). We prove the space 푐푒푠(푝, 푞, 푠) is a complete paranorm space. In section-2 we determine its Kothe-Toeplitz dual. In section-3 we establish necessary and sufficient conditions for a matrix A to map 푐푒푠 푝, 푞, 푠 to 푙∞ and 푐푒푠(푝, 푞, 푠) to c, where 푙∞ is the space of all bounded sequences and c is the space of all convergent sequences. We also get some known and unknown results as remarks.
Some Common Fixed Point Results for Expansive Mappings in a Cone Metric SpaceIOSR Journals
The purpose of this work is to extend and generalize some common fixed point theorems for Expansive type mappings in complete cone metric spaces. We are attempting to generalize the several well- known recent results. Mathematical subject classification; 54H25, 47H10
A Probabilistic Algorithm for Computation of Polynomial Greatest Common with ...mathsjournal
In the earlier work, Knuth present an algorithm to decrease the coefficient growth in the Euclidean algorithm of polynomials called subresultant algorithm. However, the output polynomials may have a small factor which can be removed. Then later, Brown of Bell Telephone Laboratories showed the subresultant in another way by adding a variant called 𝜏 and gave a way to compute the variant. Nevertheless, the way failed to determine every 𝜏 correctly.
In this paper, we will give a probabilistic algorithm to determine the variant 𝜏 correctly in most cases by adding a few steps instead of computing 𝑡(𝑥) when given 𝑓(𝑥) and𝑔(𝑥) ∈ ℤ[𝑥], where 𝑡(𝑥) satisfies that 𝑠(𝑥)𝑓(𝑥) + 𝑡(𝑥)𝑔(𝑥) = 𝑟(𝑥), here 𝑡(𝑥), 𝑠(𝑥) ∈ ℤ[𝑥]
A derivation of the Schwarzchild solution is presented with all relevant information. I have used this slides to teach Schwarzchild solution at my youtube channel. Here is the link
https://www.youtube.com/watch?v=ixhgvnGQZHM&t=1635s
Matrix Transformations on Some Difference Sequence SpacesIOSR Journals
The sequence spaces 𝑙∞(𝑢,𝑣,Δ), 𝑐0(𝑢,𝑣,Δ) and 𝑐(𝑢,𝑣,Δ) were recently introduced. The matrix classes (𝑐 𝑢,𝑣,Δ :𝑐) and (𝑐 𝑢,𝑣,Δ :𝑙∞) were characterized. The object of this paper is to further determine the necessary and sufficient conditions on an infinite matrix to characterize the matrix classes (𝑐 𝑢,𝑣,Δ ∶𝑏𝑠) and (𝑐 𝑢,𝑣,Δ ∶ 𝑙𝑝). It is observed that the later characterizations are additions to the existing ones
The aim of this paper is to study the existence and approximation of periodic solutions for non-linear systems of integral equations, by using the numerical-analytic method which were introduced by Samoilenko[ 10, 11]. The study of such nonlinear integral equations is more general and leads us to improve and extend the results of Butris [2].
Left and Right Folds- Comparison of a mathematical definition and a programm...Philip Schwarz
We compare typical definitions of the left and right fold functions, with their mathematical definitions in Sergei Winitzki’s upcoming book: The Science of Functional Programming.
Errata:
Slide 13: "The way 𝑓𝑜𝑙𝑑𝑙 does it is by associating to the right" - should, of course ,end in "to the left".
Basic Mathematics (non-calculus) for k-12 students in B.C. Canada. Intended as a guide for teaching basic math to young learners, and uploaded as a personal favor to my friend Oliver Cougur. This is a supplement teaching/learning material, and functions as a 'cheat sheet' for instructors and/or students.
This is not intended as curriculum material. I guarantee nothing. I claim no ownership or discovery of any of the material in this document, however I reserve my right of creative expression for materials contained. This document may not be sold, copied or altered in anyway by anyone.
Please report any errors to s.grantwilliam@ieee.org
IOSR Journal of Mathematics(IOSR-JM) is an open access international journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Main obstacles of Bayesian statistics or Bayesian machine learning is computing posterior distribution. In many contexts, computing posterior distribution is intractable. Today, there are two main stream to detour directly computing posterior distribution. One is using sampling method(ex. MCMC) and another is Variational inference. Compared to Variational inference, MCMC takes more time and vulnerable to high-dimensional parameters. However, MCMC has strength in simplicity and guarantees of convergence. I'll briefly introduce several methods people using in application.
Understanding Blackbox Prediction via Influence FunctionsSEMINARGROOT
Pang Wei Koh and Percy Liang
"Understanding Black-Box prediction via influence functions" ICML 2017 Best paper
References:
https://youtu.be/0w9fLX_T6tY
https://arxiv.org/abs/1703.04730
Attention Is All You Need (NIPS 2017)
(Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Łukasz Kaiser, Illia Polosukhin)
paper link: https://arxiv.org/pdf/1706.03762.pdf
Reference:
https://youtu.be/mxGCEWOxfe8 (by Minsuk Heo)
https://youtu.be/5vcj8kSwBCY (Stanford CS224N: NLP with Deep Learning | Winter 2019 | Lecture 14 – Transformers and Self-Attention)
Quality defects in TMT Bars, Possible causes and Potential Solutions.PrashantGoswami42
Maintaining high-quality standards in the production of TMT bars is crucial for ensuring structural integrity in construction. Addressing common defects through careful monitoring, standardized processes, and advanced technology can significantly improve the quality of TMT bars. Continuous training and adherence to quality control measures will also play a pivotal role in minimizing these defects.
Automobile Management System Project Report.pdfKamal Acharya
The proposed project is developed to manage the automobile in the automobile dealer company. The main module in this project is login, automobile management, customer management, sales, complaints and reports. The first module is the login. The automobile showroom owner should login to the project for usage. The username and password are verified and if it is correct, next form opens. If the username and password are not correct, it shows the error message.
When a customer search for a automobile, if the automobile is available, they will be taken to a page that shows the details of the automobile including automobile name, automobile ID, quantity, price etc. “Automobile Management System” is useful for maintaining automobiles, customers effectively and hence helps for establishing good relation between customer and automobile organization. It contains various customized modules for effectively maintaining automobiles and stock information accurately and safely.
When the automobile is sold to the customer, stock will be reduced automatically. When a new purchase is made, stock will be increased automatically. While selecting automobiles for sale, the proposed software will automatically check for total number of available stock of that particular item, if the total stock of that particular item is less than 5, software will notify the user to purchase the particular item.
Also when the user tries to sale items which are not in stock, the system will prompt the user that the stock is not enough. Customers of this system can search for a automobile; can purchase a automobile easily by selecting fast. On the other hand the stock of automobiles can be maintained perfectly by the automobile shop manager overcoming the drawbacks of existing system.
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
Saudi Arabia stands as a titan in the global energy landscape, renowned for its abundant oil and gas resources. It's the largest exporter of petroleum and holds some of the world's most significant reserves. Let's delve into the top 10 oil and gas projects shaping Saudi Arabia's energy future in 2024.
COLLEGE BUS MANAGEMENT SYSTEM PROJECT REPORT.pdfKamal Acharya
The College Bus Management system is completely developed by Visual Basic .NET Version. The application is connect with most secured database language MS SQL Server. The application is develop by using best combination of front-end and back-end languages. The application is totally design like flat user interface. This flat user interface is more attractive user interface in 2017. The application is gives more important to the system functionality. The application is to manage the student’s details, driver’s details, bus details, bus route details, bus fees details and more. The application has only one unit for admin. The admin can manage the entire application. The admin can login into the application by using username and password of the admin. The application is develop for big and small colleges. It is more user friendly for non-computer person. Even they can easily learn how to manage the application within hours. The application is more secure by the admin. The system will give an effective output for the VB.Net and SQL Server given as input to the system. The compiled java program given as input to the system, after scanning the program will generate different reports. The application generates the report for users. The admin can view and download the report of the data. The application deliver the excel format reports. Because, excel formatted reports is very easy to understand the income and expense of the college bus. This application is mainly develop for windows operating system users. In 2017, 73% of people enterprises are using windows operating system. So the application will easily install for all the windows operating system users. The application-developed size is very low. The application consumes very low space in disk. Therefore, the user can allocate very minimum local disk space for this application.
NO1 Uk best vashikaran specialist in delhi vashikaran baba near me online vas...Amil Baba Dawood bangali
Contact with Dawood Bhai Just call on +92322-6382012 and we'll help you. We'll solve all your problems within 12 to 24 hours and with 101% guarantee and with astrology systematic. If you want to take any personal or professional advice then also you can call us on +92322-6382012 , ONLINE LOVE PROBLEM & Other all types of Daily Life Problem's.Then CALL or WHATSAPP us on +92322-6382012 and Get all these problems solutions here by Amil Baba DAWOOD BANGALI
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CFD Simulation of By-pass Flow in a HRSG module by R&R Consult.pptxR&R Consult
CFD analysis is incredibly effective at solving mysteries and improving the performance of complex systems!
Here's a great example: At a large natural gas-fired power plant, where they use waste heat to generate steam and energy, they were puzzled that their boiler wasn't producing as much steam as expected.
R&R and Tetra Engineering Group Inc. were asked to solve the issue with reduced steam production.
An inspection had shown that a significant amount of hot flue gas was bypassing the boiler tubes, where the heat was supposed to be transferred.
R&R Consult conducted a CFD analysis, which revealed that 6.3% of the flue gas was bypassing the boiler tubes without transferring heat. The analysis also showed that the flue gas was instead being directed along the sides of the boiler and between the modules that were supposed to capture the heat. This was the cause of the reduced performance.
Based on our results, Tetra Engineering installed covering plates to reduce the bypass flow. This improved the boiler's performance and increased electricity production.
It is always satisfying when we can help solve complex challenges like this. Do your systems also need a check-up or optimization? Give us a call!
Work done in cooperation with James Malloy and David Moelling from Tetra Engineering.
More examples of our work https://www.r-r-consult.dk/en/cases-en/
TECHNICAL TRAINING MANUAL GENERAL FAMILIARIZATION COURSEDuvanRamosGarzon1
AIRCRAFT GENERAL
The Single Aisle is the most advanced family aircraft in service today, with fly-by-wire flight controls.
The A318, A319, A320 and A321 are twin-engine subsonic medium range aircraft.
The family offers a choice of engines
Industrial Training at Shahjalal Fertilizer Company Limited (SFCL)MdTanvirMahtab2
This presentation is about the working procedure of Shahjalal Fertilizer Company Limited (SFCL). A Govt. owned Company of Bangladesh Chemical Industries Corporation under Ministry of Industries.
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
Event Management System Vb Net Project Report.pdfKamal Acharya
In present era, the scopes of information technology growing with a very fast .We do not see any are untouched from this industry. The scope of information technology has become wider includes: Business and industry. Household Business, Communication, Education, Entertainment, Science, Medicine, Engineering, Distance Learning, Weather Forecasting. Carrier Searching and so on.
My project named “Event Management System” is software that store and maintained all events coordinated in college. It also helpful to print related reports. My project will help to record the events coordinated by faculties with their Name, Event subject, date & details in an efficient & effective ways.
In my system we have to make a system by which a user can record all events coordinated by a particular faculty. In our proposed system some more featured are added which differs it from the existing system such as security.
About
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Technical Specifications
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
Key Features
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface
• Compatible with MAFI CCR system
• Copatiable with IDM8000 CCR
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
Application
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
6. Topological Space
A topology on a set 𝑋 is a collection 𝜏 of subsets of 𝑋, satisfying the following axioms:
(1) The ∅ and 𝑋 itself are in 𝜏.
(2) The union of any collection of sets in 𝜏 is also in 𝜏
(3) The intersection of any finite number of sets in 𝜏 is also in 𝜏
𝑋 with its topology 𝜏 is called a topological space (𝑋, 𝜏).
8. Topological Manifolds
Hausdorff, Second countable, paracompactness
-> metrizable space
(homeomorphic to a metric space – which means measurable space)
This means that, any points in the space 𝒳 × 𝒳 → ℝ+
Locally Euclidean
9. Smooth (differentiable) Manifolds
𝑝 𝑞
Totally different ℝ 𝑛
𝜑 𝛼 𝑝
𝑢 𝛼 𝑢 𝛽
𝜑 𝛼 𝑝 = (𝑥1
, 𝑥2
, ⋯, 𝑥 𝑛
)
N-tuple
𝜑 𝛼
−1 ∙ 𝜑 𝛼 𝑝 = 𝑝
𝜑 𝛽 ∙ 𝜑 𝛼
−1 𝑢 𝛼 = 𝑢 𝛽
If 𝜑 𝛽 ∙ 𝜑 𝛼
−1 is infinitely differentiable 𝐶∞, we call it
Smooth manifolds
Def: A smooth manifold or 𝐶∞ -manifold is a differentiable manifold for which all the transition maps are
smooth. That is, derivatives of all orders exist; so it is a 𝐶 𝑘 -manifold for all 𝑘. An equivalence class of such
atlases is said to be a smooth structure.
10. Diffeomorphism
𝑋
𝑌
𝑚𝑚
𝑓: 𝑋 → 𝑌
𝛾 ∙ 𝑓 ∙ 𝜑−1: 𝑅 𝑛 → 𝑅 𝑚
𝛾
If 𝛾 ∙ 𝑓 ∙ 𝜑−1 is infinitely differentiable 𝐶∞, 𝑓 is bijection, and 𝑓−1 is also differentiable,
then 𝑋 𝑎𝑛𝑑 𝑌 are diffeomorphic and 𝑓 is diffeomorphism
13. Parametric curves
In mathematics, a parametric equation defines a group of quantities as functions of one or more
independent variables called parameters. Parametric equations are commonly used to express the
coordinates of the points that make up a geometric object such as a curve or surface, in which case
the equations are collectively called a parametric representation or parameterization (alternatively
spelled as parametrisation) of the object.
For example, the equations
𝑥 = cos 𝑡
𝑦 = sin 𝑡
𝛾 𝑡 = 𝑥 𝑡 , 𝑦 𝑡 = (cos 𝑡 , sin 𝑡)
form a parametric representation of the unit circle, where t is the parameter: A point (𝑥, 𝑦) is on the
unit circle if and only if there is a value of t such that these two equations generate that point.
14. Parametric curves
We can also find other charts to map the unit
circle. Let's take a look at another construction
using standard Euclidean coordinates and a
stereographic projection. Figure 5 shows a
picture of this construction. We can define two
charts by taking either the "north" or "south"
pole of the circle, finding any other point on the
circle and projecting the line segment onto the
𝑥 − 𝑎𝑥𝑖𝑠 . This provides the mapping from a
point on the manifold to ℝ1. The "north" pole
point is visualized in blue, while the "south" pole
point is visualized in burgundy. Note: the local
coordinates for the charts are different. The
same point on the circle mapped via the two
charts do not map to the same point in ℝ1.
𝑢1: = 𝜑1(𝑝) =
𝜑1(𝑝)
1
=
𝑥 𝑝
1 − 𝑦𝑝
32. Gaussian curvature
Starting with the center diagram (zero curvature), we see that a cylindrical surface has flat or
zero curvature in one dimension (blue) and curved around the other dimension (green), resulting
in zero curvature. Moving to the right, the sphere on has curvature along its to axis in the same
direction, resulting in a positive curvature. And to the left, we see the saddle sheet has curvature
along its axis in different directions, resulting in negative curvature. In fact, the Gaussian
curvature is the product of its two principal curvatures, which we won't get into detail here but
corresponds to our intuition of how a surface curves along its two main axes.
38. Geodesics
𝐴 𝑐𝑢𝑟𝑣𝑒 𝛾 s,t will be function of two real parameter,
t is time and s is variation of curve
if we fix s,then only t is chaging,
we give it another notation 𝛾𝑠 t , 𝑡ℎ𝑖𝑠 𝑖𝑠 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡
we call it 𝑔𝑒𝑜𝑑𝑒𝑠𝑖𝑐𝑠 if
𝑑
𝑑𝑠
ㅣ 𝑠=0 𝐿 𝛾𝑠 = 0
𝐴 𝑓𝑎𝑚𝑖𝑙𝑙𝑦 𝑜𝑓 𝛾𝑠
𝛾(𝑎) 𝛾(𝑏)
𝛾𝑠=𝑐(𝑘)
𝛾𝑠=𝑑(𝑘)
𝛾0(𝑘)
𝑇ℎ𝑒 𝑑𝑜𝑡𝑡𝑒𝑑 𝑐𝑢𝑟𝑣𝑒 𝑖𝑠 𝑠(−𝜀, 𝜀) → 𝛾𝑠 𝑡
𝑑
𝑑𝑠
𝛾𝑠 𝑘 𝑎𝑛𝑑
𝑑
𝑑𝑡
𝛾0 𝑡 𝑎𝑟𝑒 𝑖𝑛 𝑇𝛾 𝑠 𝑡
𝑤ℎ𝑒𝑛 𝑠 = 0, 𝑡 = 𝑘
𝑠 = 0
𝑡 = 𝑘
51. References
Differential Geometry of Curves and Surfaces, Manfredo P. Do Carmo (2016)
Differential Geometry by Claudio Arezzo
Youtube: https://youtu.be/tKnBj7B2PSg
What is a Manifold?
Youtube: https://youtu.be/CEXSSz0gZI4
Shape analysis (MIT spring 2019) by Justin Solomon
Youtube: https://youtu.be/GEljqHZb30c
Tensor Calculus
Youtube: https://youtu.be/kGXr1SF3WmA
Manifolds: A Gentle Introduction,
Hyperbolic Geometry and Poincaré Embeddings by Brian Keng
Link: http://bjlkeng.github.io/posts/manifolds/,
http://bjlkeng.github.io/posts/hyperbolic-geometry-and-poincare-embeddings/
Statistical Learning models for Manifold-Valued measurements with application to computer vision and
neuroimaging by Hyunwoo J.Kim