Problem Solving in Mathematics EducationJeff Suzuki
A major focus on current mathematics education is "problem solving." But "problem solving" means something very different from "Doing the exercises at the end of the chapter." An explanation of what problem solving is, and how it can be implemented.
A presentation on famous set Cantor Set. it describes the properties of cantor set. which the most important set of early era. it is explined with proof and theorems. references are given. ppt is somewhat plane. it would not cover the area of applications of cantor set
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Problem Solving in Mathematics EducationJeff Suzuki
A major focus on current mathematics education is "problem solving." But "problem solving" means something very different from "Doing the exercises at the end of the chapter." An explanation of what problem solving is, and how it can be implemented.
A presentation on famous set Cantor Set. it describes the properties of cantor set. which the most important set of early era. it is explined with proof and theorems. references are given. ppt is somewhat plane. it would not cover the area of applications of cantor set
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
A Chinese Remainder Theorem (CRT) is explained intuitively, which helps reader to understand its connection with Abstract Algebra concepts like Groups and Rings.
Here, Author has tried to simplify the concept by explaining sub topic of Groups and Rings which is required by understanding CRT in Abstract Algebra.
Some Results on the Group of Lower Unitriangular Matrices L(3,Zp)IJERA Editor
The main objective of this paper is to find the order and its exponent, the general form of all conjugacy classes,
Artin characters table and Artin exponent for the group of lower unitriangular matrices L(3,ℤp), where p is
prime number.
The Estimation for the Eigenvalue of Quaternion Doubly Stochastic Matrices us...ijtsrd
The purpose of this paper is to locate and estimate the eigen values of quaternion doubly stochastic matrices. We present several estimation theorems about the eigen values of quaternion doubly stochastic matrices. Mean while, we obtain the distribution theorem for the eigen values of tensor products of two quaternion doubly stochastic matrices. We will conclude the paper with the distribution for the eigen values of generalized quaternion doubly stochastic matrices. Dr. Gunasekaran K | Mrs. Seethadevi R ""The Estimation for the Eigenvalue of Quaternion Doubly Stochastic Matrices using Gerschgorin Balls"" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-3 , April 2019, URL: https://www.ijtsrd.com/papers/ijtsrd22860.pdf
Paper URL: https://www.ijtsrd.com/mathemetics/other/22860/the-estimation-for-the-eigenvalue-of-quaternion-doubly-stochastic-matrices-using-gerschgorin-balls/dr-gunasekaran-k
Module 1 (Part 1)-Sets and Number Systems.pdfGaleJean
1. Fossil records of whale evolution: Search for "whale evolution fossils" or "transitional fossils of whales."2. Comparative anatomy of homologous structures: Look for images of "homologous structures in different species" or specific examples like "homologous forelimbs in vertebrates."3. Molecular biology and genetic similarities: Search for "DNA sequences in different species," "genetic similarities between primates," or "genetic code comparison."4. Biogeography and species distribution: Look for images of "marsupials in Australia and placental mammals," or "species distribution maps showing evolution and migration."5. Artificial selection examples: Search for images of "domesticated plants vs. wild ancestors" or "different dog breeds through selective breeding."By using these keywords, you should be able to find suitable images that can visually enhance your presentation and help your classmates better grasp the concepts of descent with modification and evolutionary processes.
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.
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
1. Republic of the Philippines
PANGASINAN STATE UNIVERSITY
Lingayen Campus
Cyclic Groups
2. OBJECTIVES:
Recall the meaning of cyclic groups
Determine the important characteristics of
cyclic groups
Draw a subgroup lattice of a group precisely
Find all elements and generators of a cyclic
group
Identify the relationships among the various
subgroups of a group
3. Cyclic Groups
The notion of a “group,” viewed only 30 years ago as the
epitome of sophistication, is today one of the
mathematical concepts most widely used in physics,
chemistry, biochemistry, and mathematics itself.
ALEXEY SOSINSKY , 1991
4. A Cyclic Group is a group which can be generated by one
of its elements.
That is, for some a in G,
G={an | n is an element of Z}
Or, in addition notation,
G={na |n is an element of Z}
This element a
(which need not be unique) is called a generator of G.
Alternatively, we may write G=<a>.
5. EXAMPLES
The set of integers Z under ordinary addition is cyclic.
Both 1 and −1 are generators. (Recall that, when the
operation is addition, 1n is interpreted as
1 + 1 + ∙ ∙ ∙+ 1
n terms
when n is positive and as
(-1) + (-1) + ∙ ∙ ∙+ (-1)
|n| terms
when n is negative.)
6. The set Zn = {0, 1, . . . , n−1} for n ≥ 1 is a cyclic group under
addition modulo n. Again, 1 and −1 =n−1 are generators.
Unlike Z, which has only two generators, Zn may have many
generators
(depending on which n we are given).
Z8 = <1> = <3> = <5> = <7>.
To verify, for instance, that Z8 = <3>, we note that <3> = {3, 3
+ 3, 3 + 3 + 3, . . .} is the set {3, 6, 1, 4, 7, 2, 5, 0} = Z8 . Thus,
3 is a generator of Z8 . On the other hand, 2 is not a
generator, since <2>={0, 2, 4, 6} ≠ Z 8.
7. U(10) = {1, 3, 7, 9} = {30 , 31 , 33 , 32 } = <3>. Also, {1, 3, 7,
9} = {70 , 73 , 71 , 72 } = <7>. So both 3 and 7 are
generators for U(10).
Quite often in mathematics, a “nonexample” is as helpful in
understanding a concept as an example. With regard to
cyclic groups, U(8) serves this purpose; that is, U(8) is not
a cyclic group. Note that U(8) = {1, 3, 5, 7}. But
<1> = {1}
<3> = {3, 1}
<5> = {5, 1}
<7> = {7, 1}
so U(8) ≠ <a> for any a in U(8).
8. With these examples under
our belts, we are now ready to
tackle cyclic groups in an
abstract way and state their
key properties.
9. Properties of Cyclic Groups
Theorem 4.1 Criterion for ai = aj
Let G be a group, and let a belong to G. If a has
infinite order, then aia j if and only if i=j. If a has
finite order, say, n, then <a>= {e, a, a2, . . . , an–1}
and aia j if and only if n divides i – j.
12. Corollary 1 |a| = |<a>|
For any group element a, |a| = |<a>|.
Corollary 2 ak =e Implies That |a| Divides k
Let G be a group and let a be an element of order n in G.
If ak = e, then n divides k.
13. Theorem 4.1 and its corollaries for the case |a| = 6
are illustrated in Figure 4.1.
14. Theorem 4.2
k>
<a
= <a
gcd(n,k)>
Let a be an element of order n in a group and
let k be a positive integer. Then
<ak>=<agcd(n,k)> and |ak| = n/gcd(n,k).
15. Corollary 1 Orders of Elements in Finite Cyclic Groups
In a finite cyclic group, the order of an element
divides the order of the group.
Corollary 2 Criterion for <ai> = <aj > and |ai | = |aj |
Let |a| = n. Then <ai> = <aj > if and only if
gcd(n, i) = gcd(n, j) and |ai | = |aj | if and only if
gcd(n, i) 5 gcd(n, j) .
16. Corollary 3 Generators of Finite Cyclic Groups
Let |a| = n. Then <a> = <a j > if and only if
gcd(n, j) = 1 and |a|= |<a j >| if and only if
gcd(n, j) = 1.
Corollary 4 Generators of Zn
An integer k in Zn is a generator of Zn
if and only if gcd(n, k) = 1.
17. Classification of Subgroups
of Cyclic Groups
Theorem 4.3
Fundamental Theorem of Cyclic Groups
Every subgroup of a cyclic group is cyclic. Moreover, if
|<a>| = n, then the order of any subgroup of <a> is a
divisor of n; and, for each positive divisor
k of n, the group <a> has exactly one
subgroup of order k
—namely, <an/k>.
18. Corollary Subgroups of Zn
For each positive divisor k of n, the set <n/k> is the
unique subgroup of Zn of order k; moreover, these
are the only subgroups of Zn.
19. EXAMPLE The list of subgroups of Z30 is
<1>= {0, 1, 2, . . . , 29}
<2>= {0, 2, 4, . . . , 28}
<3>= {0, 3, 6, . . . , 27}
<5>= {0, 5, 10, 15, 20, 25}
<6>= {0, 6, 12, 18, 24}
<10>= {0, 10, 20}
<15>= {0, 15}
<30>= {0}
order 30,
order 15,
order 10,
order 6,
order 5,
order 3,
order 2,
order 1.
20. Theorem 4.4
Number of Elements of Each Order in a Cyclic Group
If d is a positive divisor of n, the number of elements of
order d in a cyclic group of order n is φ(d).
21. Corollary Number of Elements of Order d in a Finite Group
In a finite group, the number of elements of order d is
divisible by φ (d).
22. The lattice diagram for Z30 is shown in Figure 4.2. Notice
that <10> is a subgroup of both <2> and <5>, but <6> is
not a subgroup of <10>.