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This document discusses abstract algebra and its applications. It begins by defining algebra and abstract algebra, which deals with algebraic structures and operations between elements. It then defines some key concepts in abstract algebra like groups, subgroups, cyclic groups, and cosets. It provides examples of dihedral groups and how they appear in nature and corporate logos. Finally, it outlines other applications of abstract algebra in fields like chemistry, cryptography, and solving Rubik's cubes.

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Chapter 4 Cyclic Groups

Chapter 4 Cyclic Groups

Symmetrics groups

Symmetrics groups

Isomorphism

Isomorphism

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Chapter 4 Cyclic Groups

This document discusses cyclic groups and their properties. It begins by defining a cyclic group as a group that can be generated by one of its elements. It then provides examples of cyclic groups like the integers under addition and groups of integers modulo n. The key properties of cyclic groups are then outlined, including that cyclic groups are abelian, and the criteria for determining subgroup order and generators. Finite cyclic groups are shown to have unique subgroups for each divisor of the group order. The document concludes by discussing the classification and enumeration of subgroups in cyclic groups.

Symmetrics groups

The document defines symmetric groups and discusses their properties. Some key points:
- A symmetric group is the group of all permutations of a finite set under function composition.
- Symmetric groups of finite sets behave differently than those of infinite sets.
- The symmetric group Sn of degree n is the set of all permutations of the set {1,2,...,n}.
- Sn is a finite group under permutation composition. Subgroups include the alternating group An of even permutations.
- Examples discussed include S2, the Klein four-group, and S3, which is non-abelian with cyclic subgroups.

Isomorphism

This is a handout about Homomorphism, Isomorphism, Kernel and Cayley's Theorem.
This is a group effort.
#LoveforAbstractAlgebra

types of sets

This document defines and provides examples of different types of sets: empty sets, singleton sets, finite and infinite sets, union of sets, intersection of sets, difference of sets, subset of a set, disjoint sets, and equality of two sets. Empty sets have no elements. Singleton sets contain one element. Finite sets have a predetermined number of elements while infinite sets may be countable or uncountable. The union of sets contains all elements that are in either set. The intersection contains elements common to both sets. The difference contains elements in the first set that are not in the second. A set is a subset if all its elements are also in another set. Sets are disjoint if their intersection is empty. Two sets are equal

himpunan Fuzzy

Fuzzy

Mathematics set theory presentation.

This document provides an overview of set theory concepts including:
1. It defines what a set is and introduces some key terms like members, elements, and operations between sets.
2. It outlines several ways to represent sets including using rosters/lists and describing characteristic properties.
3. It discusses set notation where sets are denoted by capital letters and explains membership.
4. It describes different types of sets such as finite, infinite, null, singleton, and disjoint sets.

GROUP AND SUBGROUP PPT 20By SONU KUMAR.pptx

This document discusses key concepts in abstract algebra including groups, subgroups, normal subgroups, abelian groups, rings, and fields. It provides definitions and examples for each concept. Groups are defined as sets with a binary operation that satisfy closure, associativity, identity, and inverse properties. Subgroups are subsets of a group that are also groups. Normal subgroups are subgroups where applying the group operation to a subgroup element and any group element results in another subgroup element. Abelian groups are groups where the group operation is commutative. Rings are algebraic structures with two binary operations that satisfy properties including being abelian groups under addition and semi-groups under multiplication while satisfying distributivity. Fields are non-trivial rings where multiplication is also commutative.

Completeness axiom

The document defines upper and lower bounds of a set as numbers greater than or equal to and less than or equal to elements of the set, respectively. It also defines the least upper bound and greatest lower bound as the smallest upper bound and largest lower bound. Several examples of sets are given with their greatest lower and least upper bounds identified. The homework assigns finding the greatest lower and least upper bounds of additional sets.

Group Theory and Its Application: Beamer Presentation (PPT)

This document provides an overview of a seminar presentation on group theory and its applications. The presentation covers topics such as the definition of groups, order of groups and group elements, modular arithmetic, subgroups, Lagrange's theorem, and Sylow's theorems. It also discusses some examples of groups and applications of group theory in fields like algebraic topology, number theory, and physics. The presentation aims to introduce fundamental concepts in modern algebra through group theory.

PPT of Improper Integrals IMPROPER INTEGRAL

A brief explanation of Improper integrals with appropriate figures. It can also submitted to professor at the time of submission.

Inverse matrix

This document discusses inverse matrices. An inverse matrix A-1 undoes the transformation of the original matrix A such that multiplying A by A-1 or A-1 by A results in the identity matrix. The identity matrix is a square matrix with 1s on the main diagonal and 0s elsewhere. It leaves a matrix unchanged when multiplied. The determinant of a square matrix transforms it into a scalar value and is used to calculate the inverse. Sample problems demonstrate calculating the inverse of a 2x2 matrix.

Chapter 1, Sets

Set theory is a branch of mathematics that studies sets and their properties. A set is a collection of distinct objects, which can include numbers, points, or other sets. Some key concepts in set theory include:
- Membership and subsets, where an object is a member of a set and a set is a subset of another if all its members are also in the other set.
- Binary operations on sets like union, intersection, and complement/difference.
- The power set, which contains all possible subsets of a given set.
- Finite and infinite sets, with finite sets having a definite number of members and infinite sets not having an end. The empty set contains no members.

Abstract algebra - Algebraic closed field, unit - 2 , M.Sc. l semester Maths

Abstract algebra - Algebraic closed field, unit - 2 , M.Sc. l semester Maths Shri Shankaracharya College, Bhilai,Junwani

This ppt covers the topic of Abstract Algebra in M.Sc. Mathematics 1 semester topic of Algebraic closed field is Introduction , field & sub field , finite extension field , algebraic element.Types of sets

This document defines and describes several key concepts in set theory:
1. A set is a collection of well-defined objects that can be clearly distinguished from one another. Examples of sets include the set of natural numbers from 1 to 50.
2. Sets can be described using either the roster method, which lists the elements within curly brackets, or the set-builder method, which defines a property for the elements.
3. Types of sets include the empty set, singleton sets containing one element, finite sets that can be counted, infinite sets that cannot be counted, equivalent sets with the same number of elements, and subsets where all elements of one set are contained within another set.

Sets Introduction

This document introduces key concepts about sets. It defines a set as a well-defined group of objects that share a common characteristic. It discusses subsets and the universal set. Important notations and symbols used to describe sets are explained, including roster notation, verbal descriptions, and set builder notation. Examples are provided to illustrate these concepts and notations. Activities at the end ask the reader to identify well-defined sets, list subsets, provide verbal descriptions of sets, and write sets in different notations.

Dihedral groups & abeliangroups

The document discusses dihedral groups and abelian groups. It begins by defining symmetry and the different types of symmetry like line symmetry and rotational symmetry. It then discusses dihedral groups Dn which are the symmetry groups of regular n-gons, containing n rotations and n reflections. The document also discusses abelian groups, which are groups whose binary operation is commutative. It provides examples of abelian groups and properties like every subgroup of an abelian group being normal.

Infinite series & sequence lecture 2

The document discusses sequences and their properties. A sequence is a function whose domain is the positive integers. Sequences are commonly represented using subscript notation rather than standard function notation. The nth term of a sequence is denoted an. [/SUMMARY]

Graph theory discrete mathmatics

The document discusses topics in graph theory including Hamiltonian graphs, planar graphs, maps and regions, Euler's formula, nonplanar graphs, Dijkstra's algorithm, shortest paths, minimum spanning trees, Prim's algorithm, and more. It provides definitions, theorems, and examples related to these graph theory concepts. The document is a lecture on discrete mathematics focusing on graph theory.

A course on integral calculus

This document provides an overview of integral calculus. It defines integration as the reverse process of differentiation and discusses indefinite and definite integrals. Graphical representations and general integration rules are presented. Examples are provided for integrals of simple functions using substitution and integration by parts methods. The document also covers integrals of trigonometric functions and derives formulas for several integrals. It concludes with examples of evaluating definite integrals between specified limits to find the area under a curve.

CONVERGENCE.ppt

This document discusses convergence of sequences and series. It defines a sequence as an ordered list of objects and describes how to write the first few terms. A sequence converges if its terms get closer to a single value, while a divergent sequence does not. Infinite series represent the sum of the terms in a sequence. A series converges if the partial sums converge to a finite limit. Geometric series have a common ratio between terms and their sums can be determined based on this ratio. The integral test can be used to determine if a series converges by comparing it to a corresponding integral.

Chapter 4 Cyclic Groups

Chapter 4 Cyclic Groups

Symmetrics groups

Symmetrics groups

Isomorphism

Isomorphism

types of sets

types of sets

himpunan Fuzzy

himpunan Fuzzy

Mathematics set theory presentation.

Mathematics set theory presentation.

GROUP AND SUBGROUP PPT 20By SONU KUMAR.pptx

GROUP AND SUBGROUP PPT 20By SONU KUMAR.pptx

Completeness axiom

Completeness axiom

Group Theory and Its Application: Beamer Presentation (PPT)

Group Theory and Its Application: Beamer Presentation (PPT)

PPT of Improper Integrals IMPROPER INTEGRAL

PPT of Improper Integrals IMPROPER INTEGRAL

Inverse matrix

Inverse matrix

Chapter 1, Sets

Chapter 1, Sets

Abstract algebra - Algebraic closed field, unit - 2 , M.Sc. l semester Maths

Abstract algebra - Algebraic closed field, unit - 2 , M.Sc. l semester Maths

Types of sets

Types of sets

Sets Introduction

Sets Introduction

Dihedral groups & abeliangroups

Dihedral groups & abeliangroups

Infinite series & sequence lecture 2

Infinite series & sequence lecture 2

Graph theory discrete mathmatics

Graph theory discrete mathmatics

A course on integral calculus

A course on integral calculus

CONVERGENCE.ppt

CONVERGENCE.ppt

Cyclicgroup group thoery ppt (1)

This document discusses cyclic groups and their applications. It defines what a group is by outlining the four properties a set and binary operation must satisfy to be considered a group: closure, associativity, identity, and inverses. It also defines subgroups as subsets of a group that themselves satisfy the group properties under the restriction of the operation to the subset. An example of the symmetric group S3 is given to illustrate group elements and their orders.

452Paper

This document discusses the group SU(2)xU(1), which describes the electroweak interaction. It first covers relevant group theory concepts like Lie groups and representations. It then explains that SU(2) corresponds to rotations of spinors in real space, and physically represents weak isospin. Together with U(1), SU(2)xU(1) gives rise to the three weak gauge bosons through its symmetry with weak isospin. Representations of these groups relate their mathematical properties to observable physical phenomena.

Sets, functions and groups

The document defines sets, functions, and groups in mathematics. It provides examples and notation for sets, as well as definitions of subsets, proper subsets, and the empty set. Functions are defined as relations between inputs and outputs, and examples of functions are given. Groups are defined as sets with binary operations that satisfy closure, associativity, identity, and inverse properties. Examples of groups and subgroups are provided, along with Lagrange's theorem about the orders of groups and subgroups. Normal subgroups are introduced as subgroups whose left and right cosets are equal.

Algebraic Structure

The document discusses different types of algebraic structures including semigroups, monoids, groups, and abelian groups. It defines each structure based on what axioms they satisfy such as closure, associativity, identity element, and inverses. Examples are given of sets that satisfy each structure under different binary operations like addition, multiplication, subtraction and division. The properties of algebraic structures like commutativity, associativity, identity, inverses and cancellation laws are also explained.

On The Properties of Finite Nonabelian Groups with Perfect Square Roots Using...

In this paper we determine some internal properties of non abelian groups where the centre Z(G) takes its maximum size. With this restriction we discover that the group G is necessarily a group with perfect square root as shown in our results. We also show that the property of Z(G) is inherited by direct product of groups. Furthermore groups whose centre is as large as possible were constructed.

Algebraic-Structures_123456789101112.pdf

The document defines algebraic structures as collections of objects with operations that can be performed on those objects. It focuses on groups, rings, and fields. A group is defined as a set with an operation that satisfies four properties: closure, associativity, identity, and invertability. Examples of groups given include the integers under addition and invertible matrices under multiplication. The document tasks the reader with verifying that Z7 with addition and multiplication forms a group.

Presentation-Alex-20150421

The document provides an overview of modular arithmetic and its applications to finding square roots in modular arithmetic. It defines congruences and properties of modular arithmetic. It discusses cyclic groups and their relationship to integers and modular addition/multiplication. It introduces concepts like the order of an element, Lagrange's theorem, and Sylow theorems. It also defines quadratic residues, Legendre symbols, and provides an example of finding a square root in a finite field.

Alabs1 a

The document defines what a group is in mathematics. A group is a set with an operation that is associative, has a neutral element, and where each element has an inverse. Some examples of groups are the integers under addition, rational numbers under addition, and non-zero real numbers under multiplication. Finite groups with a set number of elements, like integers modulo n, are especially important for scientific applications. Not all groups are commutative, as shown by the group of matrices under multiplication.

5.pptx

The document discusses various algebraic structures including semi groups, monoids, groups, and subgroups. It provides examples of each type of structure and defines their key properties such as closure, associativity, identity, and inverse elements. Specific examples are given to demonstrate groups including the group of order 2 under multiplication ({1,-1}), the group of cube roots of unity under multiplication ({1, ω, ω^2}), and the group of order 4 under multiplication ({1, -1, i, -i}). Modular arithmetic is also discussed along with examples of groups under addition and multiplication modulo a given integer.

Algebraic-Structures_GROUPS123344tgh.pdf

The document defines algebraic structures as collections of objects with operations that can be performed on them. It focuses on groups, rings, and fields. A group is defined as a set with an operation that satisfies four properties: closure, associativity, identity, and invertability. Examples of groups given include the integers, rational numbers, and real numbers under addition. It is noted that the integers also form an abelian group under addition. The document asks if the integers modulo 7 form a group and provides examples to verify group properties.

April2012ART_01(1)

This document summarizes research on algebraic elements in group algebras. It begins by defining a group algebra k[G] over a commutative ring k. An element of k[G] is algebraic if it satisfies a non-constant polynomial. The document discusses tools for studying algebraic elements like partial augmentations corresponding to conjugacy classes. It also summarizes results on idempotents, including Kaplansky's theorem that the trace of an idempotent is real and rational. The author's past work on dimension subgroups is also briefly outlined.

Classification of Groups and Homomorphism -By-Rajesh Bandari Yadav

The document describes classification of groups up to order 8 and homomorphisms of groups. It begins with an introduction to groups and finite groups. It then provides details on groups of order 1 through 8, including their properties, subgroups, and Cayley diagrams. It describes the non-abelian group Q8 and dihedral group D4 of order 8. It also briefly introduces the computer algebra system GAP which is used for computational group theory.

A STUDY ON L-FUZZY NORMAL SUBl -GROUP

This paper contains some definitions and results of L-fuzzy normal sub l -group and its generalized
characteristics.

A Study on L-Fuzzy Normal Subl-GROUP

This paper contains some definitions and results of L-fuzzy normal sub l -group and its generalized characteristics.

A STUDY ON L-FUZZY NORMAL SUBl -GROUP

This document presents definitions and results regarding L-fuzzy normal sub l-groups. It begins with introductions and preliminaries on L-fuzzy sets, L-fuzzy subgroups, and L-fuzzy sub l-groups. It then presents 8 theorems on properties of L-fuzzy normal sub l-groups, such as conditions for an L-fuzzy subset to be an L-fuzzy normal sub l-group, the intersection of L-fuzzy normal sub l-groups also being an L-fuzzy normal sub l-group, and conditions where an L-fuzzy sub l-group is necessarily an L-fuzzy normal sub l-group. The document references 6 sources and is focused on developing the theory of L-fuzzy

A STUDY ON L-FUZZY NORMAL SUBl -GROUP

This paper contains some definitions and results of L-fuzzy normal sub l -group and its generalized
characteristics.

A STUDY ON L-FUZZY NORMAL SUBl -GROUPThis paper contains some definitions and results of L-fuzzy normal sub l -group and its generalized characteristics.

A STUDY ON L-FUZZY NORMAL SUB L -GROUPThis paper contains some definitions and results of L-fuzzy normal sub l -group and its generalized characteristics.

A STUDY ON L-FUZZY NORMAL SUBl -GROUP

This paper contains some definitions and results of L-fuzzy normal sub l -group and its generalized
characteristics

Cyclicgroup group thoery ppt (1)

Cyclicgroup group thoery ppt (1)

452Paper

452Paper

Sets, functions and groups

Sets, functions and groups

Algebraic Structure

Algebraic Structure

On The Properties of Finite Nonabelian Groups with Perfect Square Roots Using...

On The Properties of Finite Nonabelian Groups with Perfect Square Roots Using...

Algebraic-Structures_123456789101112.pdf

Algebraic-Structures_123456789101112.pdf

Presentation-Alex-20150421

Presentation-Alex-20150421

SECTION 5 subgroups.ppthhjji90097654323321

SECTION 5 subgroups.ppthhjji90097654323321

Alabs1 a

Alabs1 a

5.pptx

5.pptx

Algebraic-Structures_GROUPS123344tgh.pdf

Algebraic-Structures_GROUPS123344tgh.pdf

April2012ART_01(1)

April2012ART_01(1)

Classification of Groups and Homomorphism -By-Rajesh Bandari Yadav

Classification of Groups and Homomorphism -By-Rajesh Bandari Yadav

A STUDY ON L-FUZZY NORMAL SUBl -GROUP

A STUDY ON L-FUZZY NORMAL SUBl -GROUP

A Study on L-Fuzzy Normal Subl-GROUP

A Study on L-Fuzzy Normal Subl-GROUP

A STUDY ON L-FUZZY NORMAL SUBl -GROUP

A STUDY ON L-FUZZY NORMAL SUBl -GROUP

A STUDY ON L-FUZZY NORMAL SUBl -GROUP

A STUDY ON L-FUZZY NORMAL SUBl -GROUP

A STUDY ON L-FUZZY NORMAL SUBl -GROUP

A STUDY ON L-FUZZY NORMAL SUBl -GROUP

A STUDY ON L-FUZZY NORMAL SUB L -GROUP

A STUDY ON L-FUZZY NORMAL SUB L -GROUP

A STUDY ON L-FUZZY NORMAL SUBl -GROUP

A STUDY ON L-FUZZY NORMAL SUBl -GROUP

CONFINED SPACE ENTRY TRAINING FOR OIL INDUSTRY ppt

CONFINED SPACE SAFETY

Synthetic Test Collections for Retrieval Evaluation (Poster)

Test collections play a vital role in evaluation of information retrieval (IR) systems. Obtaining a diverse set of user queries for test collection construction can be challenging, and acquiring relevance judgments, which indicate the appropriateness of retrieved documents to a query, is often costly and resource-intensive. Generating synthetic datasets using Large Language Models (LLMs) has recently gained significant attention in various applications. In IR, while previous work exploited the capabilities of LLMs to generate synthetic queries or documents to augment training data and improve the performance of ranking models, using LLMs for constructing synthetic test collections is relatively unexplored. Previous studies demonstrate that LLMs have the potential to generate synthetic relevance judgments for use in the evaluation of IR systems. In this paper, we comprehensively investigate whether it is possible to use LLMs to construct fully synthetic test collections by generating not only synthetic judgments but also synthetic queries. In particular, we analyse whether it is possible to construct reliable synthetic test collections and the potential risks of bias such test collections may exhibit towards LLM-based models. Our experiments indicate that using LLMs it is possible to construct synthetic test collections that can reliably be used for retrieval evaluation.

Online toll plaza booking system project report.doc.pdf

In day to day life, Millions of drivers pass through the toll booth to pay toll tax. Manual
process is too much time consuming, so we go for electronic toll plaza. Toll Plaza Management
system is a web based application that can provide all the information related to toll plazas and the
passenger checks in either online or on a mobile device and pays the amount, then the passenger
will be provided a receipt. With this receipt the passenger can leave the toll booth without waiting
for any verification call. If the user selects a place from source to destination, the number of toll
gates in between will be displayed with the specified amount. The user pays the payment via online
payment gateway. This system explains the implantation of automation in toll plaza which is a
step towards improving the user to pay the amount for travelling in predetermined routes.

Natural Is The Best: Model-Agnostic Code Simplification for Pre-trained Large...

Pre-trained Large Language Models (LLM) have achieved remarkable successes in several domains. However, code-oriented LLMs are often heavy in computational complexity, and quadratically with the length of the input code sequence. Toward simplifying the input program of an LLM, the state-of-the-art approach has the strategies to filter the input code tokens based on the attention scores given by the LLM. The decision to simplify the input program should not rely on the attention patterns of an LLM, as these patterns are influenced by both the model architecture and the pre-training dataset. Since the model and dataset are part of the solution domain, not the problem domain where the input program belongs, the outcome may differ when the model is trained on a different dataset. We propose SlimCode, a model-agnostic code simplification solution for LLMs that depends on the nature of input code tokens. As an empirical study on the LLMs including CodeBERT, CodeT5, and GPT-4 for two main tasks: code search and summarization. We reported that 1) the reduction ratio of code has a linear-like relation with the saving ratio on training time, 2) the impact of categorized tokens on code simplification can vary significantly, 3) the impact of categorized tokens on code simplification is task-specific but model-agnostic, and 4) the above findings hold for the paradigm–prompt engineering and interactive in-context learning and this study can save reduce the cost of invoking GPT-4 by 24%per API query. Importantly, SlimCode simplifies the input code with its greedy strategy and can obtain at most 133 times faster than the state-of-the-art technique with a significant improvement. This paper calls for a new direction on code-based, model-agnostic code simplification solutions to further empower LLMs.

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Chlorine and Nitric Acid application, properties, impacts.pptx

Chlorine and Nitric acid

PPT_grt.pptx engineering criteria grt for accrediation

engineering criteria grt for accrediation

Introduction to IP address concept - Computer Networking

Introduction to IP address concept - Computer NetworkingMd.Shohel Rana ( M.Sc in CSE Khulna University of Engineering & Technology (KUET))

An Internet Protocol address (IP address) is a logical numeric address that is assigned to every single computer, printer, switch, router, tablets, smartphones or any other device that is part of a TCP/IP-based network.
Types of IP address-
Dynamic means "constantly changing “ .dynamic IP addresses aren't more powerful, but they can change.
Static means staying the same. Static. Stand. Stable. Yes, static IP addresses don't change.
Most IP addresses assigned today by Internet Service Providers are dynamic IP addresses. It's more cost effective for the ISP and you.
Unit 1 Information Storage and Retrieval

Introduction to information retrieval, Major challenges in IR

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21CV61- Module 3 (CONSTRUCTION MANAGEMENT AND ENTREPRENEURSHIP.pptx

Notes of Construction management and entrepreneurship

Tutorial on MySQl and its basic concepts

Tutorial on MySQl and its basic concepts

Online airline reservation system project report.pdf

Airline Reservation System is software which is helpful for ticketing manager as
well as the customers. In the later system all the activities were done manually. It was
very time consuming and costly. Our Airline Reservation System deals with the various
activities related to the Flights.
There are mainly 3 modules in this software
1. Flight Reservation module
2. Flight Cancellation Module
3. Flight Postpone Module
In the Software only user with the legal username and password can sign in. A
ticketing manager can book, cancel or postpone any flight for any customer. Flights are
booked through Flight Reservation Module in which all the details regarding customer
and his flight are entered. A receipt no. is provide to every customer which is unique for
each customer and with the help of which cancellation and postpone of flight can be
done.

IS Code SP 23: Handbook on concrete mixes

SP-23: Hand Bank on Concrete Mixes required at the time designing

OSHA LOTO training, LOTO, lock out tag out

LOTO _ OSHA

SCADAmetrics Instrumentation for Sensus Water Meters - Core and Main Training...

SCADAmetrics Instrumentation for Sensus Water Meters - Core and Main Training 2024 July 09

readers writers Problem in operating system

Readers Writers Problem

The world of Technology Management MEM 814.pptx

The world of Technology Management MEM 814.pptx

CONFINED SPACE ENTRY TRAINING FOR OIL INDUSTRY ppt

CONFINED SPACE ENTRY TRAINING FOR OIL INDUSTRY ppt

Synthetic Test Collections for Retrieval Evaluation (Poster)

Synthetic Test Collections for Retrieval Evaluation (Poster)

Online toll plaza booking system project report.doc.pdf

Online toll plaza booking system project report.doc.pdf

Natural Is The Best: Model-Agnostic Code Simplification for Pre-trained Large...

Natural Is The Best: Model-Agnostic Code Simplification for Pre-trained Large...

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Chlorine and Nitric Acid application, properties, impacts.pptx

Chlorine and Nitric Acid application, properties, impacts.pptx

PPT_grt.pptx engineering criteria grt for accrediation

PPT_grt.pptx engineering criteria grt for accrediation

Introduction to IP address concept - Computer Networking

Introduction to IP address concept - Computer Networking

IE-469-Lecture-Notes-3IE-469-Lecture-Notes-3.pptx

IE-469-Lecture-Notes-3IE-469-Lecture-Notes-3.pptx

Unit 1 Information Storage and Retrieval

Unit 1 Information Storage and Retrieval

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Presentation python programming vtu 6th sem

21CV61- Module 3 (CONSTRUCTION MANAGEMENT AND ENTREPRENEURSHIP.pptx

21CV61- Module 3 (CONSTRUCTION MANAGEMENT AND ENTREPRENEURSHIP.pptx

Tutorial on MySQl and its basic concepts

Tutorial on MySQl and its basic concepts

Online airline reservation system project report.pdf

Online airline reservation system project report.pdf

IS Code SP 23: Handbook on concrete mixes

IS Code SP 23: Handbook on concrete mixes

OSHA LOTO training, LOTO, lock out tag out

OSHA LOTO training, LOTO, lock out tag out

SCADAmetrics Instrumentation for Sensus Water Meters - Core and Main Training...

SCADAmetrics Instrumentation for Sensus Water Meters - Core and Main Training...

readers writers Problem in operating system

readers writers Problem in operating system

- 1. The Beauty of Abstract Algebra Overview - What is Algebra ? -What is Abstract Algebra ? -Group -Some definitions( Subgroup, cyclic group , order of group ,cosets) -Beauty of Abstract Algebra -Dihedral groups and its real life applications -Dihedral groups in nature -Other applications of Abstract Algebra PP14
- 2. Algebra: In mathematics , we use symbols to represent mathematical objects(numbers , matrices, transformations,…) and the study of the rules for combining these symbols under some operation is called algebra. Eg. (𝑎 + 𝑏)2= 𝑎2 + 𝑏2 + 2𝑎𝑏 Linear Algebra Advanced Algebra Abstract Algebra Elementary algebra Commutative Algebra Algebra PP14
- 3. Abstract algebra: It deals with the study of algebraic systems or structures with one or more mathematical operations associated elements with an identifiable pattern, differing from the usual number systems. In this mainly we study – Rings Vectorspaces Groups Fields PP14
- 4. GROUP- Definition : A group is a set G , together with a binary operation ∗, such that the following hold: 1. (Associativity): (a ∗ b) ∗ c = a ∗ (b ∗ c) ∀a, b, c ∈ G. 2. (Existence of identity): ∃ e ∈ G such that a ∗ e = e ∗ a = a ∀a ∈ G. 3. (Existence of inverses): Given a ∈ G, ∃ b ∈ G such that a ∗ b = b ∗ a = e Example: The set of integers z under ordinary addition is a group. Order of a Group: The number of elements of a group (finite or infinite) is called its order. We will use |G| to denote the order of G. Example: The set 𝑧8 = {0,1,2,3,4,5,6,7} is a group under ⊕8with order 8. PP14
- 5. Subgroup Definition: If a subset H of a group G is itself a group under the operation of G, we say that H is a subgroup of G. Example: H={0,2,4,6,8} is a subgroup of 𝑍10. CYCLIC GROUP A group G is called cyclic if there is an element a in G such that G={𝑎𝑛| 𝑛ϵ𝑧}. Such an element a is called a generator of G. Example: Z under ordinary addition is cyclic. Both 1 and -1 are generators. Cosets of H in G : Let G be a group and let H be a subgroup of G. For any 𝑎ϵG, the set {𝑎ℎ|ℎϵH}is denoted by aH is called the left coset of H in G containing a. Analogously, Ha is called the right coset of H in G containing a. PP14
- 6. Beauty of Abstract Algebra Theorem: If G is a finite group and H is a subgroup of G, then |H| divides |G|. The beauty embedded in this theorem is that , we didn’t even know what kind of elements do group G have whether they are numbers or matrices or any other mathematical object, what is the binary operation of G but still we can prove that G can have subgroups whose order is a divisor of |G|. Result: Groups of prime order are cyclic. Similarly , we didn’t even know anything about the elements and binary operation of G but still we can claim that if |G| is prime the G is cyclic. With out knowing much about any algebraic systems (groups , rings,…)PP14
- 7. Dihedral Group : For every integer 𝑛 ≥ 3, the regular polygon with n sides has a group of symmetries, symbolized by 𝐷𝑛. These groups are called the dihedral groups . Real life applications of Dihedral groups – Corporation logos are rich sources of dihedral symmetry. Chrysler’s logo has 𝐷5 as a symmetry group, and that of Mercedes- Benz has 𝐷3 . PP14
- 8. Dihedral groups in nature – 1.) The phylum Echinodermata contains many sea animals (such as starfish ,sea cucumbers, feather stars, and sand dollars) that exhibit patterns with 𝐷5 symmetry. 2.) Snowflakes has 𝐷6 symmetry. Starfish sea cucumbers sand dollars Snowflake PP14
- 9. Some other applications of Abstract Algebra – 1.) Chemists classify molecules according to their symmetry .For eg. The symmetry group of a pyramidal molecule such as ammonia (𝑁𝐻3) is D3 . 2.) Ring Theory has been well-used in cryptography and many others computer vision tasks. 3.) Group theory algorithms are used to solve Rubik’s cube. PP14
- 10. ThankYou PP14