This document discusses Lie algebras and the Killing form. It defines Lie algebras and introduces the Killing form, showing that it is a contravariant tensor. It then explains that the signature of the diagonalized Killing form is an invariant that can be used to classify and identify isomorphic Lie algebras. Examples are provided of low-dimensional Lie algebras classified according to their Killing form signatures.
On line BIM project execution by PLANNERLYStephen Au
BIM execution is a complex process that involve a lot of parties working together and responding instantly. How can we ensure the BIM tasks are closely aligning with construction schedule and able to deliver the right models for problem solving?
In this presentation I gave one overall overview about BIM workflow for Construction and D&B companies.
BIM is not a tool, BIM is not a software, BIM is a way of thinking about the project and put together processes to exchange information during the entire lifecycle.
This presentation is mainly focused on Autodesk platform but is applicable to many other solutions.
BIM process adoption for integrated design and constuctionReshma Philip
The document provides an overview of a BIM consultant's biography and experience, as well as a presentation on BIM and the design process. The consultant has over 20 years of experience implementing BIM and digital design systems. The presentation covers key topics like what BIM is, how it compares to traditional design processes, important terminology, software considerations, and examples of BIM implementation on government projects in KSA.
BIM Lecture Note (6/6)
Objectives
* To understand how BIM project is implemented and its challenges
Question
* How to execute a successful BIM project?
www.mtech.com.hk
BIM Level of Development Explained | LOD 100 200 300 400 500United-BIM
Level of development (LOD) is a set of specifications that gives professionals in the AEC industry the power to document, articulate and specify the content of BIM effectively and clearly. Serving as an industry standard, LOD defines the development stages of different systems in BIM. By using LOD specifications, architects, engineers, and other professionals can clearly communicate with each other without confusion for faster execution.
How BIM is able to optimise the design phase of a construction projectJacob Ostwald
This document summarizes a student report about how Building Information Modeling (BIM) can optimize the design phase of construction projects. It defines BIM as both a type of software and process that allows collaboration between different professionals involved in a project. The report discusses how BIM benefits the planning and design phases through features like clash detection and cost/time estimation. It also examines industry attitudes towards adopting BIM and its growing use in Australia and worldwide.
BIM Lecture Note (5/6)
Objectives
* The challenges of Building Construction Project
* To understand what is IPD & LEAN Construction
* To understand Asset Lifecycle Management (iBIM)
*How to apply ALM & BIM to enable LEAN Construction
Question
* How iBIM can be applied to enable IPD & LEAN Construction?
Free notes on Project Scope Management,PMP Chapter 5, PMBOK, PMP Exam Preparation training
Why Manage Scope
Plan Scope Management
Collect Requirements
Define Scope
Create WBS
Validate Scope
Control Scope
Online PMP Training,Instructor led PMP training,PMP training online,PMP Training in USA,PMP Training in California,PMP Training in Qatar,PMP training in Saudi Arabia,PMP training in India,PMP training in Mumbai,PMP Training in Bangalore
On line BIM project execution by PLANNERLYStephen Au
BIM execution is a complex process that involve a lot of parties working together and responding instantly. How can we ensure the BIM tasks are closely aligning with construction schedule and able to deliver the right models for problem solving?
In this presentation I gave one overall overview about BIM workflow for Construction and D&B companies.
BIM is not a tool, BIM is not a software, BIM is a way of thinking about the project and put together processes to exchange information during the entire lifecycle.
This presentation is mainly focused on Autodesk platform but is applicable to many other solutions.
BIM process adoption for integrated design and constuctionReshma Philip
The document provides an overview of a BIM consultant's biography and experience, as well as a presentation on BIM and the design process. The consultant has over 20 years of experience implementing BIM and digital design systems. The presentation covers key topics like what BIM is, how it compares to traditional design processes, important terminology, software considerations, and examples of BIM implementation on government projects in KSA.
BIM Lecture Note (6/6)
Objectives
* To understand how BIM project is implemented and its challenges
Question
* How to execute a successful BIM project?
www.mtech.com.hk
BIM Level of Development Explained | LOD 100 200 300 400 500United-BIM
Level of development (LOD) is a set of specifications that gives professionals in the AEC industry the power to document, articulate and specify the content of BIM effectively and clearly. Serving as an industry standard, LOD defines the development stages of different systems in BIM. By using LOD specifications, architects, engineers, and other professionals can clearly communicate with each other without confusion for faster execution.
How BIM is able to optimise the design phase of a construction projectJacob Ostwald
This document summarizes a student report about how Building Information Modeling (BIM) can optimize the design phase of construction projects. It defines BIM as both a type of software and process that allows collaboration between different professionals involved in a project. The report discusses how BIM benefits the planning and design phases through features like clash detection and cost/time estimation. It also examines industry attitudes towards adopting BIM and its growing use in Australia and worldwide.
BIM Lecture Note (5/6)
Objectives
* The challenges of Building Construction Project
* To understand what is IPD & LEAN Construction
* To understand Asset Lifecycle Management (iBIM)
*How to apply ALM & BIM to enable LEAN Construction
Question
* How iBIM can be applied to enable IPD & LEAN Construction?
Free notes on Project Scope Management,PMP Chapter 5, PMBOK, PMP Exam Preparation training
Why Manage Scope
Plan Scope Management
Collect Requirements
Define Scope
Create WBS
Validate Scope
Control Scope
Online PMP Training,Instructor led PMP training,PMP training online,PMP Training in USA,PMP Training in California,PMP Training in Qatar,PMP training in Saudi Arabia,PMP training in India,PMP training in Mumbai,PMP Training in Bangalore
5th Qatar BIM User Day, Understanding stakeholder roles in BIMBIM User Day
Author: Mohammad Obaidullah | Qatar Design Consortium
Content:
- Importance of client interests and involvements
- BIM – an adoption curve
- Case study – Walk through Qatar’s 2022 Al-Bayt Stadium (Al Khor)
About the Qatar BIM User Day: Qatar University, HOCHTIEF ViCon and Teesside University proudly take the initiative to facilitate modern and innovative methods in the Gulf construction industry. The focus is Building Information Modeling (BIM), and our aim is to establish a knowledge platform with government, research and industry experts. The User Day aims to help people to share knowledge, discuss new technologies, and identify new potentials for BIM.
3D Printing, Architectural visualization and the Future of architectural visu...Ogbuagu Kelechi Uchamma
These topics exposes you to the Digital world of Architecture right now. Architecture has grown from mere sketches drawn with paper and pencils to 3D models which can be printed or visualized graphically at all angles in the most appealing way possible. Find out more as you go through the slides.
Cheers!
The document discusses Building Information Modeling (BIM) and its potential benefits and challenges for the construction industry. BIM is a process that involves structured sharing and coordination of digital building information throughout the lifecycle. It can eliminate communication challenges and catalyze improvement. BIM is projected to reduce costs by up to 20%, improve sustainability, and increase efficiency. Potential benefits include faster processes, better design, cost control, and reduced change orders. However, adoption of new technology in construction has been slow and BIM faces challenges to widespread success.
Presentation by Mr. Lawrence K.W. CHUNG - Assistant Director of Housing (Development & Construction Division),
Housing Department, HKSAR Government
HKIPM‐HKIBIM Joint Conference: BIM in Project Management
Details of Conference
Date: 2‐Apr‐2014 (Wed)
Time: 2:00 pm – 5:00 pm
Venue: Chiang Chen Studio Theatre – The Hong Kong Polytechnic University
Organizers:
Hong Kong Institute of Project Management (HKIPM)
http://www.hkipm.org.hk/
The Hong Kong Institute of Building Information Modelling (HKIBIM)
http://www.hkibim.org
Sole Sponsor:
建造業議會 - Construction Industry Council
http://www.hkcic.org/
This Slideshare presentation is a partial preview of the full business document. To view and download the full document, please go here:
http://flevy.com/browse/business-document/pmp-exam-preparation--200-questions-3196
DOCUMENT DESCRIPTION
200 PMP questions and answers.
PMP exam very very similar questions.
You can increase your chance to pass the exam with seeing very similar questions.
You can use this document for PMP problem solving sessions, PMP preperation classes or to pass PMP.
2020년 10월 8일 가이아쓰리디(주) 주최로 개최된 [ICT/BIM/Digital Twin을 활용한 스마트 환경영향평가 웨비나]에서 발표한 자료입니다. 환경부의 [ICT기반 환경영향평가 기술개발사업]의 일환으로 연구 중인 [환경영향평가 의사결정지원 시공간 표출기술] 연구를 개략적으로 소개했습니다.
[ICT/BIM/Digital Twin을 활용한 스마트 환경영향평가 웨비나]의 전체 프로그램과 자료는 https://gaia3d.com/?p=4251에서 확인할 수 있습니다.
The document discusses common data environments (CDEs) and electronic document management systems (eDMS). It defines a CDE according to ISO 19650-1 as an agreed source of information for projects that collects, manages, and disseminates information containers through a managed process. eDMS are software systems that centrally store and organize digital documents. The document examines the advantages of eDMS and CDEs, how they improve collaboration and workflow management. It also discusses responsibilities for establishing a CDE and different CDE platforms on the market.
This document discusses a case study using combined BIM and GIS modeling for building energy conservation. It introduces the Department of Land Surveying and Geo-Informatics at PolyU, which provides training in geomatics including BIM. A 3D model was constructed of Block Z and surrounding areas using floor plans, existing survey data, and a landscape scan. Daylight and solar radiation simulations were performed on the combined model. Seasonal differences in sunlight and shadows cast by neighboring buildings were analyzed. An intelligent lighting management system was designed for the 6th floor based on daylight factor maps and lighting locations. The study demonstrated the benefits of integrating BIM and GIS for optimizing building lighting and energy conservation.
The document summarizes an introduction meeting to Unity3D game engine. It discusses what a game engine is and how Unity manages entities and subsystems. It then walks through exercises having attendees create and modify a spinning cube prefab to demonstrate core Unity concepts like hierarchies, components and scripting basics. The document stresses that the goal is to understand how to learn and explore Unity rather than fully learning game development.
DoubleVerify provides media quality and performance solutions to drive better outcomes for advertisers. It verifies advertising transactions across platforms in a privacy-friendly way. Going forward, it aims to expand its solutions to include pre-campaign activation through planning, targeting, and blocking capabilities to maximize advertiser outcomes across brand safety, viewability, and fraud prevention.
This document provides an overview of vectors, tensors, and coordinate systems in fluid mechanics. It defines scalars, vectors, and tensors, and describes how they can be represented using basis vectors. It introduces common coordinate systems like Cartesian, cylindrical, and spherical coordinates. It explains how to transform between these systems and decompose vectors into their scalar components. The document also defines tensor operations like addition, multiplication, and transpose. It describes how tensors can be represented by matrices and defines important tensors like the identity tensor.
The document discusses Lie algebras, which are vector spaces with a non-associative multiplication called the Lie bracket. Any Lie group gives rise to a Lie algebra, and vice versa. Lie algebras allow the study of Lie groups in terms of vector spaces. A Lie subalgebra is a vector subspace of a Lie algebra that is closed under the Lie bracket, while an ideal is a subspace where the Lie bracket of any element of the Lie algebra with an element of the ideal is also in the ideal. Examples of Lie algebras and their substructures are provided.
This document contains 4 physics problems involving quantum mechanics concepts:
1) Calculating the commutator of angular momentum operators and relating the time derivative of angular momentum to torque.
2) Determining spherical harmonic functions using angular momentum operators.
3) Solving for energy levels and eigenfunctions of a rigid rotor system of two particles attached at either end of a massless rod.
4) Verifying commutation relations for spin and angular momentum operators and showing that the cross product of an angular momentum operator with itself is non-zero.
This section define a level subring or level ideals obtain a set of necessary and sufficient condition for the
equality of two ideals and characterizes field in terms of its fuzzy ideals. It also presents a procedure to construct
a fuzzy subrings (fuzzy ideals) from any given ascending chain of subring ideal. We prove that the lattice of
fuzzy congruence of group G (respectively ring R) is isomorphic to the lattice of fuzzy normal subgroup of G
(respectively fuzzy ideals of R).In Yuan Boond Wu wangrning investigated the relationship between the fuzzy
ideals and the fuzzy congruences on a distributive lattice and obtained that the lattice of fuzzy ideals is
isomorphic to the lattice of fuzzy congruences on a generalized Boolean algebra. Fuzzy group theory can be
used to describe, symmetries and permutation in nature and mathematics. The fuzzy group is one of the oldest
branches of abstract algebra. For example group can be used is classify to all of the forms chemical crystal can
take. Group can be used to count the number of non-equivalent objects and permutation or symmetries. For
example, the number of different is switching functions of n, variable when permutation of the input are
allowed. Beside crystallography and combinatory group have application of quantum mechanics.
Perspectives and application of fuzzy initial value problemsRAJKRISHNA MONDAL
This document discusses fuzzy differential equations and fuzzy initial value problems. It introduces fuzzy sets and fuzzy differential equations as a way to model dynamical systems involving uncertainties. It then examines three different fuzzy initial value problems and their solutions. The solutions exhibit very different behaviors despite being fuzzy representations of equivalent crisp differential equations. This shows that different fuzzy representations of the same crisp problem can lead to different outcomes.
Perspectives and application of fuzzy initial value problemsRAJKRISHNA MONDAL
This document discusses fuzzy differential equations and fuzzy initial value problems. It introduces fuzzy sets and fuzzy differential equations as extensions of classical set theory and differential equations to account for uncertainty. Several examples of fuzzy initial value problems are analyzed, comparing their behaviors under different types of differentiability. The solutions exhibit very different properties, even though the original crisp equations were equivalent, showing that different fuzzy representations can model the same real-world problem very differently.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
5th Qatar BIM User Day, Understanding stakeholder roles in BIMBIM User Day
Author: Mohammad Obaidullah | Qatar Design Consortium
Content:
- Importance of client interests and involvements
- BIM – an adoption curve
- Case study – Walk through Qatar’s 2022 Al-Bayt Stadium (Al Khor)
About the Qatar BIM User Day: Qatar University, HOCHTIEF ViCon and Teesside University proudly take the initiative to facilitate modern and innovative methods in the Gulf construction industry. The focus is Building Information Modeling (BIM), and our aim is to establish a knowledge platform with government, research and industry experts. The User Day aims to help people to share knowledge, discuss new technologies, and identify new potentials for BIM.
3D Printing, Architectural visualization and the Future of architectural visu...Ogbuagu Kelechi Uchamma
These topics exposes you to the Digital world of Architecture right now. Architecture has grown from mere sketches drawn with paper and pencils to 3D models which can be printed or visualized graphically at all angles in the most appealing way possible. Find out more as you go through the slides.
Cheers!
The document discusses Building Information Modeling (BIM) and its potential benefits and challenges for the construction industry. BIM is a process that involves structured sharing and coordination of digital building information throughout the lifecycle. It can eliminate communication challenges and catalyze improvement. BIM is projected to reduce costs by up to 20%, improve sustainability, and increase efficiency. Potential benefits include faster processes, better design, cost control, and reduced change orders. However, adoption of new technology in construction has been slow and BIM faces challenges to widespread success.
Presentation by Mr. Lawrence K.W. CHUNG - Assistant Director of Housing (Development & Construction Division),
Housing Department, HKSAR Government
HKIPM‐HKIBIM Joint Conference: BIM in Project Management
Details of Conference
Date: 2‐Apr‐2014 (Wed)
Time: 2:00 pm – 5:00 pm
Venue: Chiang Chen Studio Theatre – The Hong Kong Polytechnic University
Organizers:
Hong Kong Institute of Project Management (HKIPM)
http://www.hkipm.org.hk/
The Hong Kong Institute of Building Information Modelling (HKIBIM)
http://www.hkibim.org
Sole Sponsor:
建造業議會 - Construction Industry Council
http://www.hkcic.org/
This Slideshare presentation is a partial preview of the full business document. To view and download the full document, please go here:
http://flevy.com/browse/business-document/pmp-exam-preparation--200-questions-3196
DOCUMENT DESCRIPTION
200 PMP questions and answers.
PMP exam very very similar questions.
You can increase your chance to pass the exam with seeing very similar questions.
You can use this document for PMP problem solving sessions, PMP preperation classes or to pass PMP.
2020년 10월 8일 가이아쓰리디(주) 주최로 개최된 [ICT/BIM/Digital Twin을 활용한 스마트 환경영향평가 웨비나]에서 발표한 자료입니다. 환경부의 [ICT기반 환경영향평가 기술개발사업]의 일환으로 연구 중인 [환경영향평가 의사결정지원 시공간 표출기술] 연구를 개략적으로 소개했습니다.
[ICT/BIM/Digital Twin을 활용한 스마트 환경영향평가 웨비나]의 전체 프로그램과 자료는 https://gaia3d.com/?p=4251에서 확인할 수 있습니다.
The document discusses common data environments (CDEs) and electronic document management systems (eDMS). It defines a CDE according to ISO 19650-1 as an agreed source of information for projects that collects, manages, and disseminates information containers through a managed process. eDMS are software systems that centrally store and organize digital documents. The document examines the advantages of eDMS and CDEs, how they improve collaboration and workflow management. It also discusses responsibilities for establishing a CDE and different CDE platforms on the market.
This document discusses a case study using combined BIM and GIS modeling for building energy conservation. It introduces the Department of Land Surveying and Geo-Informatics at PolyU, which provides training in geomatics including BIM. A 3D model was constructed of Block Z and surrounding areas using floor plans, existing survey data, and a landscape scan. Daylight and solar radiation simulations were performed on the combined model. Seasonal differences in sunlight and shadows cast by neighboring buildings were analyzed. An intelligent lighting management system was designed for the 6th floor based on daylight factor maps and lighting locations. The study demonstrated the benefits of integrating BIM and GIS for optimizing building lighting and energy conservation.
The document summarizes an introduction meeting to Unity3D game engine. It discusses what a game engine is and how Unity manages entities and subsystems. It then walks through exercises having attendees create and modify a spinning cube prefab to demonstrate core Unity concepts like hierarchies, components and scripting basics. The document stresses that the goal is to understand how to learn and explore Unity rather than fully learning game development.
DoubleVerify provides media quality and performance solutions to drive better outcomes for advertisers. It verifies advertising transactions across platforms in a privacy-friendly way. Going forward, it aims to expand its solutions to include pre-campaign activation through planning, targeting, and blocking capabilities to maximize advertiser outcomes across brand safety, viewability, and fraud prevention.
This document provides an overview of vectors, tensors, and coordinate systems in fluid mechanics. It defines scalars, vectors, and tensors, and describes how they can be represented using basis vectors. It introduces common coordinate systems like Cartesian, cylindrical, and spherical coordinates. It explains how to transform between these systems and decompose vectors into their scalar components. The document also defines tensor operations like addition, multiplication, and transpose. It describes how tensors can be represented by matrices and defines important tensors like the identity tensor.
The document discusses Lie algebras, which are vector spaces with a non-associative multiplication called the Lie bracket. Any Lie group gives rise to a Lie algebra, and vice versa. Lie algebras allow the study of Lie groups in terms of vector spaces. A Lie subalgebra is a vector subspace of a Lie algebra that is closed under the Lie bracket, while an ideal is a subspace where the Lie bracket of any element of the Lie algebra with an element of the ideal is also in the ideal. Examples of Lie algebras and their substructures are provided.
This document contains 4 physics problems involving quantum mechanics concepts:
1) Calculating the commutator of angular momentum operators and relating the time derivative of angular momentum to torque.
2) Determining spherical harmonic functions using angular momentum operators.
3) Solving for energy levels and eigenfunctions of a rigid rotor system of two particles attached at either end of a massless rod.
4) Verifying commutation relations for spin and angular momentum operators and showing that the cross product of an angular momentum operator with itself is non-zero.
This section define a level subring or level ideals obtain a set of necessary and sufficient condition for the
equality of two ideals and characterizes field in terms of its fuzzy ideals. It also presents a procedure to construct
a fuzzy subrings (fuzzy ideals) from any given ascending chain of subring ideal. We prove that the lattice of
fuzzy congruence of group G (respectively ring R) is isomorphic to the lattice of fuzzy normal subgroup of G
(respectively fuzzy ideals of R).In Yuan Boond Wu wangrning investigated the relationship between the fuzzy
ideals and the fuzzy congruences on a distributive lattice and obtained that the lattice of fuzzy ideals is
isomorphic to the lattice of fuzzy congruences on a generalized Boolean algebra. Fuzzy group theory can be
used to describe, symmetries and permutation in nature and mathematics. The fuzzy group is one of the oldest
branches of abstract algebra. For example group can be used is classify to all of the forms chemical crystal can
take. Group can be used to count the number of non-equivalent objects and permutation or symmetries. For
example, the number of different is switching functions of n, variable when permutation of the input are
allowed. Beside crystallography and combinatory group have application of quantum mechanics.
Perspectives and application of fuzzy initial value problemsRAJKRISHNA MONDAL
This document discusses fuzzy differential equations and fuzzy initial value problems. It introduces fuzzy sets and fuzzy differential equations as a way to model dynamical systems involving uncertainties. It then examines three different fuzzy initial value problems and their solutions. The solutions exhibit very different behaviors despite being fuzzy representations of equivalent crisp differential equations. This shows that different fuzzy representations of the same crisp problem can lead to different outcomes.
Perspectives and application of fuzzy initial value problemsRAJKRISHNA MONDAL
This document discusses fuzzy differential equations and fuzzy initial value problems. It introduces fuzzy sets and fuzzy differential equations as extensions of classical set theory and differential equations to account for uncertainty. Several examples of fuzzy initial value problems are analyzed, comparing their behaviors under different types of differentiability. The solutions exhibit very different properties, even though the original crisp equations were equivalent, showing that different fuzzy representations can model the same real-world problem very differently.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
class-10 Maths Ncert Book | Chapter-3 | Slides By MANAV |Slides With MANAV
This document discusses representing situations involving two related variables with pairs of linear equations and solving them graphically. It provides three examples:
1) A girl spends money at a fair based on rides and a game. This is represented as two intersecting lines, with a unique solution of 4 rides and playing the game 2 times.
2) Two friends buy pencils and erasers with different amounts. This is represented by coinciding lines, meaning the equations are equivalent and there are infinitely many solutions.
3) Two rail lines are represented by parallel lines, meaning there is no intersection and no solution to the pair of equations.
The document explains that pairs of linear equations can have a unique solution
Complex roots of the characteristic equationTarun Gehlot
The document discusses solving second-order linear differential equations when the characteristic equation has complex roots. It explains that when the roots are purely imaginary, the solutions will be sinusoidal functions like cosine and sine. When the roots are complex numbers with real and imaginary parts, the solutions will be exponential functions multiplied by trigonometric functions. The document provides examples of extracting real solutions from complex exponential solutions and checking that the solutions are linearly independent using the Wronskian. It also discusses how to determine the amplitude and period when the solution is in the form of a trigonometric function.
Rational functions are functions of the form f(x)=polynomial/polynomial. There are six key aspects to analyze in rational functions: y-intercept, x-intercepts, vertical asymptotes, horizontal/slant asymptotes, and the graph. Vertical asymptotes occur when the denominator is 0, x-intercepts when the numerator is 0, and horizontal/slant asymptotes depend on the relative degrees of the numerator and denominator polynomials.
This document discusses nonstationary covariance modeling. It begins by introducing concepts of covariance, correlation, and how correlation affects estimation and prediction. It then discusses properties of random fields like second-order stationarity and intrinsic stationarity. The difference between these two is explained. Common parametric models for isotropic covariance and variogram functions are presented, including spherical, exponential, Gaussian, rational quadratic, and Matérn models. Parameters and properties of these models like range, smoothness, and valid dimensionality are described. Examples of each model type are shown graphically.
This document discusses the convexity of the set of k-admissible functions on a compact Kähler manifold. It begins by introducing k-admissible functions and some necessary convex analysis concepts. It then proves four main results: 1) the log of elementary symmetric functions of eigenvalues is a convex function, 2) the set of matrices with eigenvalues in a convex set is convex, 3) certain functions of eigenvalues are convex, and 4) the set of k-admissible functions is convex. It uses results on conjugation of spectral functions from convex analysis to prove these results. The proofs rely on properties of convex, lower semicontinuous functions and indicator functions.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
The document presents four solutions to finding the shape of a hanging chain under its own weight. The first solution uses force balancing on small segments of the chain. This leads to two differential equations that are solved to get a hyperbolic cosine function for the shape. The other three solutions use variational arguments to minimize the chain's potential energy, subject to its total length constraint. This also results in a hyperbolic cosine shape function. The constant parameter in this function can be determined from the chain's endpoint positions and total length.
The document presents four solutions to finding the shape of a hanging chain under its own weight. The first solution uses force balancing on small segments of the chain. This leads to two differential equations that are solved to get a hyperbolic cosine function for the shape. The other three solutions use variational arguments to minimize the chain's potential energy, subject to its total length constraint. This also results in a hyperbolic cosine shape function. The constant parameter in this function can be determined from the chain's endpoints and total length.
The concept of moments of order invariant quantumAlexander Decker
The document summarizes key concepts related to moments of order invariant quantum Lévy processes. It begins by defining order equivalence and order invariant distributions for discrete random variables. It then states that if the distributions are order invariant, the moments of their sums converge as the number of variables increases. Similarly, for order invariant quantum Lévy processes, the document shows that limits exist for the moments and provides an expression involving these limits. It proves the existence of the limits for moments by induction on the length of index tuples.
I am Arcady N. I am a Computer Network Assignments Expert at computernetworkassignmenthelp.com. I hold a Master's in Computer Science from, City University, London. I have been helping students with their assignments for the past 10 years. I solve assignments related to the Computer Network.
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Gauge systems and functions, hermitian operators and clocks as conjugate func...vcuesta
This document summarizes a research article about gauge systems and constraints in physics. It discusses two key problems that can arise: 1) Clocks may not be well-defined over the entire phase space. 2) Quantum operators associated with complete observables may not be self-adjoint. The summary proposes selecting clocks such that their Poisson brackets with constraints are equal to 1. This is shown to solve the two problems for several example systems, including a free particle and a system with two constraints. Clocks and complete observables are constructed for the examples, and it is verified that the operators are self-adjoint.
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Topological Field Theories In N Dimensional Spacetimes And Cartans Equations
Lie Algebras, Killing Forms And Signatures
1. Lie algebras, Killing forms and signatures
Vladimir Cuesta †
Instituto de Ciencias Nucleares, Universidad Nacional Aut´ noma de M´ xico, 70-543, Ciudad de
o e
M´ xico, M´ xico
e e
Abstract. The killing form is a basic ingredient in abstract Lie algebra theory. If I make a linear
transformation in the set of generators of a Lie algebra, I can show that the Killing form is a contravariant
tensor, using Sylvester’s law of inertia I can diagonalize it and with this I can find all the Lie algebras
of a given dimension according to the signature or equivalently the number of negative, zero or positive
eigenvalues.
I show that there is one and only one Lie algebra with two generators, using the same method I find
that there are five possible Lie algebras with three generators, I show three of them.
With this method I can find all the Lie algebras that are isomorphic of a given dimension because
the signature is independent of the base, I present some important examples showing isomorphic Lie
algebras.
1. Introduction
Lie algebras is a branch of abstract mathematics with intense and constant development, in the
literature we can find a lot of great and concise books like [1], [2] and so on. We can study different
applications (see [3] and [4] for example), even more, we can study the subject as a infinitesimal
version of a Lie group (see [5] for instance).
Like the reader can note almost all the books are interested in a specific kind of Lie algebras:
semisimple Lie algebras. However, according to the Levi decomposition, we take an abstract real
Lie algebra with a finite number of generators and it can be divided in a solvable subalgebra and a
semisimple subalgebra and this is a first example to make research of the subject, I mean, if I take a
Lie algebra I could be interested in the Levi decomposition for different applications.
In the case of pure Lie algebras one of the problems that the researcher can find in this area of
knowledge is to determine and to classify all the semisimple subalgebras of a specific Lie algebra (see
[6] and [7] for instance), its matricial representations and so on.
As a third example, if I take a Lie algebra I can study it as a subalgebra of another algebra
with more generators (see [1]), which is the opposite of a decomposition problem. Previously I have
†
vladimir.cuesta@nucleares.unam.mx
2. Lie algebras, Killing forms and signatures 2
presented a brief list of a longer one, in the present paper I will follow the following line of reasoning:
I present basic definitions on Lie algebras, I show that the Killing form is a contravariant tensor and
like a linear algebra or gravitation specialist knows the signature is an invariant of a bilinear form
(like the Killing metric) and so that, if I find the signature of a diagonal Killing form I will obtain an
invariant of the theory. Even more, with this it is possible to characterize isomorphic Lie algebras.
That is the purpose of the present paper.
2. Lie algebras: definitions and basics
2.1. Lie algebras
I present the formal definition of a Lie algebra, along all the present work I will use the following
definition (see [2] and [5] for instance):
Let L be a vector space over the real field , L is a real Lie algebra when there exists a binary
operation denoted as [ , ] and is called commutator or Lie bracket
[ , ]:L×L→L (1)
and when the commutator obeys the following three properties
• Antisymmetry.- For all x, y ∈ L
[x, y] = −[y, x], (2)
• Bilinearity.- For all x, y ∈ L and all α, β ∈
[αx + βy, z] = α[x, z] + β[y, z], (3)
• Jacobi identity.- For all x, y, z ∈ L
[x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0, (4)
Now, Let L1 and L2 be two real Lie algebras, then L1 and L2 are isomorphic if there exist a
bijection
φ : L1 → L2 , (5)
such that
φ(αx + βy) = αφ(x) + βφ(y), φ([x, y]) = [φ(x), φ(y)], (6)
where x, y ∈ L1 and α, β ∈ .
3. Lie algebras, Killing forms and signatures 3
2.2. The Killing form as a tensor
We have the definition
[ei , ej ] = Cij k ek , (7)
for the structure constants of a general Lie algebra (see [2] and [3], for instance). With the previous
equations we can define the set of matrices Ti (adjoint representation for the ei generator) as follows:
k
(Ti )j = −Cij k , (8)
where i runs from 1 to the dimension of the Lie algebra.
Now, taking the linear transformation
ei = Ai m em , (9)
I will find the transformation law for the adjoint representations matrices. In fact, these must obey
[ei , ej ] = Cij k ek , (10)
replacing the transformation law for the Lie algebra generators in the previous equation I find
Ai m Aj n Cmn r − Cij k Ak r er = 0, (11)
and the result is
Tij s = Ai m An A−1
j
s
Tmn r , (12)
r
the following step is to replace Tij k in
gij = Tim n Tjn m , (13)
and the result is
gij = Ai m Aj n gmn , (14)
I mean, g is a tensor. In fact,
gij = T r (Ti · Tj ) = T r (Tj · Ti ) = gji , (15)
and g is a Symmetric bilinear form (see [8] for details), in the case when the Killing form is non-
degenerate we call this kind of Lie algebras as semisimple.
Now, Sylvester’s law of inertia says that when a symmetric bilinear form is written in diagonal
form the number of negative, zero and positive eigenvalues (signature) is independent of the original
set of generators (see [9] for a detailed discussion).
Let A be a Lie algebra of dimension n, then if I have the set of structure constants, I can obtain
the adjoint representation, the Killing form and I can find the set of eigenvalues and in this way I can
identify and classify a specific Lie algebra according to the signature of the diagonal Killing form.
I mean, all the Lie algebras with the same number of negative, zero and positive eigenvalues in the
diagonal Killing form are isomorphic.
4. Lie algebras, Killing forms and signatures 4
3. Examples
3.1. Lie algebras with two generators
I present the bi-dimensional Lie algebra, the general form is
[e1 , e2 ] = αe1 + βe2 , (16)
with the adjoint representations
0 0 −α −β
T1 = , T2 = ,
α β 0 0
with Killing metric
β 2 −αβ
(gij ) = , (17)
−αβ α2
and the eigenvalues that I found are λ1 = 0 and λ2 = α2 + β 2 . Then, this is the unique Lie algebra
of dimension 2 and like the reader can see, in a base where the Killing form is diagonal the number
of negative and zero eigenvalues is zero and the number of positive eigenvalues is one and the Lie
algebra is non-semisimple.
3.2. Lie algebras with three generators
In the present subsection I present three of the five possible different Lie algebras with three
generators.
Case I
The generators for this first case are (where I take the commutator between matrices)
0 0 0 0 0 1 −1 −2 0
0 0 −1 , (e2ij ) = 0
(e1ij ) = , (e3ij ) = 1
0 0 1 0 ,
1 2 0 −1 −1 0 0 0 0
the commutation relations between the three generators are,
[e1 , e2 ] = e3 , [e1 , e3 ] = −e1 − 2e2 , [e2 , e3 ] = e1 + e2 , (18)
using the Killing form definition, I find the result
−4 2 0
2 −2 0 ,
(gij ) =
0 0 −2
5. Lie algebras, Killing forms and signatures 5
√ √
and I find the eigenvalues λ1 = −3 − 5, λ2 = −2 and λ3 = −3 + 5. I mean, I have a Lie algebra
with three negative eigenvalues and it is semisimple (the Lie algebra of this example is equivalent to
a Lie algebra with three positive eigenvalues because there is a difference of global sign).
Case II
I will study the Lie algebra with commutators
[e1 , e2 ] = 2e2 , [e1 , e3 ] = 4e2 − 2e3 , [e2 , e3 ] = e1 , (19)
using this set I find the adjoint representations
0 0 0 0 2 0 0 4 −2
(e1ij ) = 0 −2 0 , (e2ij ) = 0 0 0 , (e3ij ) = 1 0 0 ,
0 −4 2 −1 0 0 0 0 0
after a straightforward calculation I find the killing form
8 0 0
(gij ) =
0 0 4 ,
0 4 8
√ √
and it has the set of eigenvalues λ1 = 4(1 + 2), λ2 = 8 and λ3 = 4(1 − 2), the number of negative
eigenvalues is one and positive eigenvalues are two, the present Lie algebra is semisimple.
Case III
The set of matrices that I will use are the following, where the Lie bracket is the commutator between
matrices
0 0 0 1 0 0 −1 0 0
(e1ij ) = −1 0 0 , (e2ij ) = 0 0 0 , (e3ij ) = −1 0 0 ,
1 0 0 1 0 0 0 0 0
The set of commutators for the three generators is
[e1 , e2 ] = e1 , [e1 , e3 ] = −e1 , [e2 , e3 ] = −e1 , (20)
and the final result for the Killing form is
0 0 0
0 1 −1 ,
(gij ) =
0 −1 1
and If I diagonalize the previous tensor I find the eigenvalues λ1 = 2, λ2 = 0 and λ3 = 0, in this case
the number of positive eigenvalues is one and two null eigenvalues, the Lie algebra is non-semisimple.
6. Lie algebras, Killing forms and signatures 6
4. Isomorphic Lie algebras
First example: su(2)
In this case the three matrices are
0 0 0 0 0 1 0 −1 0
(Sxij ) = 0 0 −1 , (Syij ) = 0 0 0 , (Szij ) = 1 0 0 ,
0 1 0 −1 0 0 0 0 0
and the commutator is the commutator between matrices, the commutation relations between the three
generators are (see [10] for instance),
[Sx , Sy ] = Sz , [Sy , Sz ] = Sx , [Sz , Sx ] = Sy , (21)
using the Killing form definition, I find the result
−2 0 0
(gij ) = 0 −2 0 ,
0 0 −2
and is a diagonal Killing form with eigenvalues λ1 = −2, λ2 = −2 and λ3 = −2. In this case all
the eigenvalues are negative and it means that this Lie algebra is isomorphic to the first Lie algebra
of the previous subsection with three positive eigenvalues because the difference between this pair of
diagonal Killing forms is a global sign.
Second example: sl(2, )
I will study the Lie algebra with generators
0 0 0 0 0 1 −2 0 0
0 0 −1 , (fij ) = 0 0 0 , (hij ) = 0 2 0 ,
(eij ) =
2 0 0 0 −2 0 0 0 0
in this case the set of commutators is (see [11] and [4] for instance)
[h, e] = 2e, [h, f ] = −2f, [e, f ] = h, (22)
and after a straightforward calculation I find the killing form
0 4 0
(gij ) =
4 0 0 ,
0 0 8
7. Lie algebras, Killing forms and signatures 7
and the set of eigenvalues for the Killing form is λ1 = 8, λ2 = −4 and λ3 = 4, the number of negative
eigenvalues is one and the number of positive eigenvalues is two and the Lie algebra is isomorphic to
the second Lie algebra of the previous subsection.
Third example: Antisymmetric 3 × 3 matrices
I will study the Lie algebra with generators
0 0 0 0 0 −1 0 1 0
0 0 1 , (e2ij ) = 0 0 0 , (e3ij ) = −1 0 0 ,
(e1ij ) =
0 −1 0 1 0 0 0 0 0
in this case the set of commutators is
[e1 , e2 ] = −e3 , [e1 , e3 ] = e2 , [e2 , e3 ] = −e1 , (23)
and after a straightforward calculation I find the killing form
−2 0 0
(gij ) = 0 −2 0 ,
0 0 −2
and the set of eigenvalues for the Killing form is λ1 = −2, λ2 = −2 and λ3 = −2, the number of
negative eigenvalues is three and the Lie algebra is isomorphic to the first Lie algebra of the previous
subsection (the difference is a global sign).
5. Conclusions and perspectives
In the present paper I have shown a general method for finding isomorphic Lie algebras (see [4], too),
all the Lie algebras with the same number of negative, null and positive eigenvalues for the Killing
form are isomorphic (see [12] for a precise discussion on signatures). I have shown that there is
one and only one Lie algebra with two generators and there are five possible Lie algebras with three
generators, altough I show only three of all. Later, I show two examples to illustrate the method to
recognize isomorphic Lie algebras in the case of three generators, I present the following table where
I resume my results
N egative Zero P ositive Lie algebra
eigenvalues eigenvalues eigenvalues type
Case I 0 0 3 semisimple
Case II 1 0 2 semisimple
Case III 0 1 2 non − semisimple
Case IV 1 1 1 non − semisimple
Case V 0 2 1 non − semisimple
8. Lie algebras, Killing forms and signatures 8
To finish the paper I show the following table, where I find all the possible signatures for the diagonal
Killing form of a Lie algebra with four generators. In fact, I found nine Lie algebras and like I said, If
I could diagonalize the Killing form I can identify isomorphic Lie algebras If the number of negative,
null and positive eigenvalues for the diagonal Killing form are the same,
N egative Zero P ositive Lie algebra
eigenvalues eigenvalues eigenvalues type
Case I 0 0 4 semisimple
Case II 1 0 3 semisimple
Case III 0 1 3 non − semisimple
Case IV 0 2 2 non − semisimple
Case V 2 0 2 semisimple
Case VI 1 1 2 non − semisimple
Case V II 0 3 1 non − semisimple
Case V III 1 2 1 non − semisimple
Case IX 0 4 0 non − semisimple
for future work, I can identify isomorphic Lie algebras with four generators or another option is to
find a classification when the generators are four, five and so on.
References
[1] M. Rausch de Traubenberg and M. J. Slupinski, Finite-dimensional Lie algebras of order F, J. Math. Phys. 43 (10),
(2002),
[2] E. van Groesen and E. M. Jager, Lie algebras, Part 1 Finite and infinite dimensional Lie algebras and applications in
physics, North-Holland, (1990),
[3] Francesco Iachello, Lie Algebras and Applications, Springer, (2006),
[4] Vladimir Cuesta, Ph.D. thesis, Cinvestav, Mexico, (2007),
[5] Roger Carter, Greame Segal and Ian Macdonald, Lectures on Lie Groups and Lie Algebras, Cambridge University
Press, (1995),
[6] M. Lorente and B. Gruber, Classification of Semisimple Subalgebras of Simple Lie Algebras, J. Math. Phys., 13 (10),
(1972),
[7] Evelyn Weimar-Woods, The three-dimensional real Lie algebras and their contractions, J. Math. Phys. 32 (8), (1991),
[8] Aleksei Ivanovich Kostrikin, Introducci´ n al algebra, Mosc´ , MIR, (1980),
o u
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[9] Elie Cartan, The theory of spinors, Dover Publications Inc., (1966),
[10] Hans Samelson, Notes on Lie Algebras, Universitext, Springer, (1990),
[11] Vladimir Cuesta, Merced Montesinos and Jose David Vergara, Gauge invariance of the action principle for gauge
systems with noncanonical symplectic structures, Phys. Rev.D 76, 025025, (2007),
[12] Steven Weinberg, Gravitation and cosmology: principles and applications of the general theory of relativity,
Massachusetts Institute of Technology, John Wiley and Sons, Inc. (1972).