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.
Existence of Extremal Solutions of Second Order Initial Value Problemsijtsrd
In this paper existence of extremal solutions of second order initial value problems with discontinuous right hand side is obtained under certain monotonicity conditions and without assuming the existence of upper and lower solutions. Two basic differential inequalities corresponding to these initial value problems are obtained in the form of extremal solutions. And also we prove uniqueness of solutions of given initial value problems under certain conditions. A. Sreenivas ""Existence of Extremal Solutions of Second Order Initial Value Problems"" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-4 , June 2019,
URL: https://www.ijtsrd.com/papers/ijtsrd25192.pdf
Paper URL: https://www.ijtsrd.com/mathemetics/other/25192/existence-of-extremal-solutions-of-second-order-initial-value-problems/a-sreenivas
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.
Existence of Extremal Solutions of Second Order Initial Value Problemsijtsrd
In this paper existence of extremal solutions of second order initial value problems with discontinuous right hand side is obtained under certain monotonicity conditions and without assuming the existence of upper and lower solutions. Two basic differential inequalities corresponding to these initial value problems are obtained in the form of extremal solutions. And also we prove uniqueness of solutions of given initial value problems under certain conditions. A. Sreenivas ""Existence of Extremal Solutions of Second Order Initial Value Problems"" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-4 , June 2019,
URL: https://www.ijtsrd.com/papers/ijtsrd25192.pdf
Paper URL: https://www.ijtsrd.com/mathemetics/other/25192/existence-of-extremal-solutions-of-second-order-initial-value-problems/a-sreenivas
On the k-Riemann-Liouville fractional integral and applications Premier Publishers
Fractional calculus is a generalization of ordinary differentiation and integration to arbitrary non-integer order. The subject is as old as differential calculus and goes back to times when G.W. Leibniz and I. Newton invented differential calculus. Fractional integrals and derivatives arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of a complex medium. Very recently, Mubeen and Habibullah have introduced the k-Riemann-Liouville fractional integral defined by using the -Gamma function, which is a generalization of the classical Gamma function. In this paper, we presents a new fractional integration is called k-Riemann-Liouville fractional integral, which generalizes the k-Riemann-Liouville fractional integral. Then, we prove the commutativity and the semi-group properties of the -Riemann-Liouville fractional integral and we give Chebyshev inequalities for k-Riemann-Liouville fractional integral. Later, using k-Riemann-Liouville fractional integral, we establish some new integral inequalities.
A Common Fixed Point Theorem on Fuzzy Metric Space Using Weakly Compatible an...inventionjournals
The aim of this paper is to prove a fixed point theorem in a complete fuzzy metric space using six self maps. We prove our theorem with the concept of weakly compatible mappings and semi-compatible mappings in complete fuzzy metric space.
NITheP UKZN Seminar: Prof. Alexander Gorokhov (Samara State University, Russi...Rene Kotze
NITheP UKZN Seminar: Prof. Alexander Gorokhov (Samara State University, Russia)
TITLE: Dynamical Groups, Coherent States and Some of their Applications in Quantum Optics and Molecular Spectroscopy
International Journal of Mathematics and Statistics Invention (IJMSI) inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Version 2: some improvements http://arxiv.org/abs/1505.04393.
We investigate fields in which addition requires three summands. These ternary fields are shown to be isomorphic to the set of invertible elements in a local ring R having Z/2Z as a residual field. One of the important technical ingredients is to intrinsically characterize the maximal ideal of R. We include a number illustrative examples and prove that the structure of a finite 3-field is not connected to any binary field.
On the k-Riemann-Liouville fractional integral and applications Premier Publishers
Fractional calculus is a generalization of ordinary differentiation and integration to arbitrary non-integer order. The subject is as old as differential calculus and goes back to times when G.W. Leibniz and I. Newton invented differential calculus. Fractional integrals and derivatives arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of a complex medium. Very recently, Mubeen and Habibullah have introduced the k-Riemann-Liouville fractional integral defined by using the -Gamma function, which is a generalization of the classical Gamma function. In this paper, we presents a new fractional integration is called k-Riemann-Liouville fractional integral, which generalizes the k-Riemann-Liouville fractional integral. Then, we prove the commutativity and the semi-group properties of the -Riemann-Liouville fractional integral and we give Chebyshev inequalities for k-Riemann-Liouville fractional integral. Later, using k-Riemann-Liouville fractional integral, we establish some new integral inequalities.
A Common Fixed Point Theorem on Fuzzy Metric Space Using Weakly Compatible an...inventionjournals
The aim of this paper is to prove a fixed point theorem in a complete fuzzy metric space using six self maps. We prove our theorem with the concept of weakly compatible mappings and semi-compatible mappings in complete fuzzy metric space.
NITheP UKZN Seminar: Prof. Alexander Gorokhov (Samara State University, Russi...Rene Kotze
NITheP UKZN Seminar: Prof. Alexander Gorokhov (Samara State University, Russia)
TITLE: Dynamical Groups, Coherent States and Some of their Applications in Quantum Optics and Molecular Spectroscopy
International Journal of Mathematics and Statistics Invention (IJMSI) inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Version 2: some improvements http://arxiv.org/abs/1505.04393.
We investigate fields in which addition requires three summands. These ternary fields are shown to be isomorphic to the set of invertible elements in a local ring R having Z/2Z as a residual field. One of the important technical ingredients is to intrinsically characterize the maximal ideal of R. We include a number illustrative examples and prove that the structure of a finite 3-field is not connected to any binary field.
We first consider a ternary matrix group related to the von Neumann regular semigroups and to the Artin braid group (in an algebraic way). The product of a special kind of ternary matrices (idempotent and of finite order) reproduces the regular semigroups and braid groups with their binary multiplication of components. We then generalize the construction to the higher arity case, which allows us to obtain some higher degree versions (in our sense) of the regular semigroups and braid groups. The latter are connected with the generalized polyadic braid equation and R-matrix introduced by the author, which differ from any version of the well-known tetrahedron equation and higher-dimensional analogs of the Yang-Baxter equation, n-simplex equations. The higher degree (in our sense) Coxeter group and symmetry groups are then defined, and it is shown that these are connected only in the non-higher case.
On Some Notable Properties of Zero Divisors in the Ring of Integers Modulo m ...inventionjournals
The algebraic structure (m , +, ×) is a commutative ring with unity. When we examine the multiplicative structure (m , ×) we noticed that the product of some two non-zero elements is zero, thus the ring (m , +, ×) has zero divisors. In this study, we made some observations on some of the theorems regarding the zero divisors of the ring (m , +, ×).We established some of the properties of the zero divisors of (m , +, ×) . Our results showed that, For an even integer m 6 at least one of the quadratic residues modulo m in (m , +, ×) is a zero divisor also for an odd composite non-perfect square m 15 at least three of the quadratic residues are zero divisors. Furthermore, we have found that if m is composite and can be written as a power of prime p, that is m = p α where α ≥ 2 then: 1. Zero divisors in m are multiples of prime p. 2. Let D denote the set of zero divisors in (m , +, ×) and D D { 0 } , then (a) (D+ , +) is a cyclic group generated by p (b) (D+ , +, ×) is a cyclic ring (c) (D+ , +, ×) is a subring of (m , +, ×) (d) (D+ , +, ×) is an ideal of (m , +, ×) (e) (D+ , +, ×) is a principal ideal (f) (D+ , +, ×) is a prime ideal (g) (D+ , +, ×) is a maximal ideal.(h) (m , +, ×) is a local ring
On Some Notable Properties of Zero Divisors in the Ring of Integers Modulo m ...inventionjournals
The algebraic structure (m , +, ×) is a commutative ring with unity. When we examine the multiplicative structure (m , ×) we noticed that the product of some two non-zero elements is zero, thus the ring (m , +, ×) has zero divisors. In this study, we made some observations on some of the theorems regarding the zero divisors of the ring (m , +, ×).We established some of the properties of the zero divisors of (m , +, ×) . Our results showed that, For an even integer m 6 at least one of the quadratic residues modulo m in (m , +, ×) is a zero divisor also for an odd composite non-perfect square m 15 at least three of the quadratic residues are zero divisors. Furthermore, we have found that if m is composite and can be written as a power of prime p, that is m = p α where α ≥ 2 then: 1. Zero divisors in m are multiples of prime p. 2. Let D denote the set of zero divisors in (m , +, ×) and D D { 0 } , then (a) (D+ , +) is a cyclic group generated by p (b) (D+ , +, ×) is a cyclic ring (c) (D+ , +, ×) is a subring of (m , +, ×) (d) (D+ , +, ×) is an ideal of (m , +, ×) (e) (D+ , +, ×) is a principal ideal (f) (D+ , +, ×) is a prime ideal (g) (D+ , +, ×) is a maximal ideal.(h) (m , +, ×) is a local ring.
TIU CET Review Math Session 6 - part 2 of 2youngeinstein
College Entrance Test Review
Math Session 6 - part 2 of 2
FUNCTIONS
How to evaluate
Operations on functions
Composite functions
Trigonometric Functions
Pythagorean Theorem
30 60 90 triangle
45 45 90 triangle
Exponential Functions
Logarithmic Functions
MA 243 Calculus III Fall 2015 Dr. E. JacobsAssignmentsTh.docxinfantsuk
MA 243 Calculus III Fall 2015 Dr. E. Jacobs
Assignments
These assignments are keyed to Edition 7E of James Stewart’s “Calculus” (Early Transcendentals)
Assignment 1. Spheres and Other Surfaces
Read 12.1 - 12.2 and 12.6
You should be able to do the following problems:
Section 12.1/Problems 11 - 18, 20 - 22 Section 12.6/Problems 1 - 48
Hand in the following problems:
1. The following equation describes a sphere. Find the radius and the coordinates of the center.
x2 + y2 + z2 = 2(x + y + z) + 1
2. A particular sphere with center (−3, 2, 2) is tangent to both the xy-plane and the xz-plane.
It intersects the xy-plane at the point (−3, 2, 0). Find the equation of this sphere.
3. Suppose (0, 0, 0) and (0, 0, −4) are the endpoints of the diameter of a sphere. Find the
equation of this sphere.
4. Find the equation of the sphere centered around (0, 0, 4) if the sphere passes through the
origin.
5. Describe the graph of the given equation in geometric terms, using plain, clear language:
z =
√
1 − x2 − y2
Sketch each of the following surfaces
6. z = 2 − 2
√
x2 + y2
7. z = 1 − y2
8. z = 4 − x − y
9. z = 4 − x2 − y2
10. x2 + z2 = 16
Assignment 2. Dot and Cross Products
Read 12.3 and 12.4
You should be able to do the following problems:
Section 12.3/Problems 1 - 28 Section 12.4/Problems 1 - 32
Hand in the following problems:
1. Let u⃗ =
⟨
0, 1
2
,
√
3
2
⟩
and v⃗ =
⟨√
2,
√
3
2
, 1
2
⟩
a) Find the dot product b) Find the cross product
2. Let u⃗ = j⃗ + k⃗ and v⃗ = i⃗ +
√
2 j⃗.
a) Calculate the length of the projection of v⃗ in the u⃗ direction.
b) Calculate the cosine of the angle between u⃗ and v⃗
3. Consider the parallelogram with the following vertices:
(0, 0, 0) (0, 1, 1) (1, 0, 2) (1, 1, 3)
a) Find a vector perpendicular to this parallelogram.
b) Use vector methods to find the area of this parallelogram.
4. Use the dot product to find the cosine of the angle between the diagonal of a cube and one of
its edges.
5. Let L be the line that passes through the points (0, −
√
3 , −1) and (0,
√
3 , 1). Let θ be the
angle between L and the vector u⃗ = 1√
2
⟨0, 1, 1⟩. Calculate θ (to the nearest degree).
Assignment 3. Lines and Planes
Read 12.5
You should be able to do the following problems:
Section 12.5/Problems 1 - 58
Hand in the following problems:
1a. Find the equation of the line that passes through (0, 0, 1) and (1, 0, 2).
b. Find the equation of the plane that passes through (1, 0, 0) and is perpendicular to the line in
part (a).
2. The following equation describes a straight line:
r⃗(t) = ⟨1, 1, 0⟩ + t⟨0, 2, 1⟩
a. Find the angle between the given line and the vector u⃗ = ⟨1, −1, 2⟩.
b. Find the equation of the plane that passes through the point (0, 0, 4) and is perpendicular to
the given line.
3. The following two lines intersect at the point (1, 4, 4)
r⃗ = ⟨1, 4, 4⟩ + t⟨0, 1, 0⟩ r⃗ = ⟨1, 4, 4⟩ + t⟨3, 5, 4⟩
a. Find the angle between the two lines.
b. Find the equation of the plane that contains every point o ...
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MISS TEEN GONDA 2024 - WINNER ABHA VISHWAKARMADK PAGEANT
Abha Vishwakarma, a rising star from Uttar Pradesh, has been selected as the victor from Gonda for Miss High Schooler India 2024. She is a glad representative of India, having won the title through her commitment and efforts in different talent competitions conducted by DK Exhibition, where she was crowned Miss Gonda 2024.
NIDM (National Institute Of Digital Marketing) Bangalore Is One Of The Leading & best Digital Marketing Institute In Bangalore, India And We Have Brand Value For The Quality Of Education Which We Provide.
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Jill Pizzola's Tenure as Senior Talent Acquisition Partner at THOMSON REUTERS...dsnow9802
Jill Pizzola's tenure as Senior Talent Acquisition Partner at THOMSON REUTERS in Marlton, New Jersey, from 2018 to 2023, was marked by innovation and excellence.
Exploring Career Paths in Cybersecurity for Technical CommunicatorsBen Woelk, CISSP, CPTC
Brief overview of career options in cybersecurity for technical communicators. Includes discussion of my career path, certification options, NICE and NIST resources.
Resumes, Cover Letters, and Applying OnlineBruce Bennett
This webinar showcases resume styles and the elements that go into building your resume. Every job application requires unique skills, and this session will show you how to improve your resume to match the jobs to which you are applying. Additionally, we will discuss cover letters and learn about ideas to include. Every job application requires unique skills so learn ways to give you the best chance of success when applying for a new position. Learn how to take advantage of all the features when uploading a job application to a company’s applicant tracking system.
1. 1
APPLICATIONS OF CYCLIC GROUPS IN EVERYDAY LIFE
By
Koetlisi Theko Elliott
&
Ts’iliso Ramohloai
3rd
NUMERICAL ANALYSIS MINI PROJECT
BACHELOR OF SCIENCE
in
MATHEMATICS
in the
FACULTY OF SCIENCE AND TECHNOLOGY
at
NATIONAL UNIVERSITY OF LESOTHO
1.Abstract
Cyclic groups are common in our everyday life. A cyclic group is a group with an element that
has an operation applied that produces the whole set. A cyclic group is the simplest group. A cyclic
group could be a pattern found in nature, for example in a snowflake, or in a geometric pattern we
draw ourselves. Cyclic groups can also be thought of as rotations, if we rotate an object enough
times we will eventually return to the original position. Cyclic groups are used in topics such as
cryptology and number theory. In this paper we explore further applications of cyclic groups in
number theory and other applications including music and chaos theory. If someone can recognize
a cyclic group they could use the generator to find the fastest simple circuit for use in other real
world applications and in pure mathematics.
2. Introduction
A group G is called cyclic if there is an element than generates the entire set by repeatedly
applying an operation [8]. The universe and mathematics are made up of many cyclic groups. One
can think of cyclic groups as patterns that repeat until returning to the beginning.
2. 2
Figure 1. Geometric shapes and designs that are generated by the shape of design
repeating until it gets back to the origin. This shape is a knot that is being repeated three times until
it gets back to the original point.
Figure 2. natural objects that are cyclic
Figure 3. A jelly fish and urchin test that exhibit radial symmetry
Cyclic groups can be created by humans using shapes and designs. Cyclic groups are common in
the natural world. Some examples of cyclic geometries in nature are a test of an urchin, a
snowflake, a bell pepper, and flowers (Figure 2). Any organism that has radial symmetry is cyclic.
Animals are generally symmetric about an axis from the centre. Animals exhibit radial symmetry in
the phyla cnidaria (jellyfish) and echinodermata (sea stars, sea cucumber, urchin)(Figure 3). Plants
and flowers have radial symmetry(Figure 2. The petals radiate around the centre of the flower
until the centre is entirely surrounded by petals.
Cyclic groups can be thought of as rotations. An object with rotational symmetry is also known
in biological contexts as radial symmetry.
3. 3
Figure 4. 90 degree rotations of a square
We can draw a square moving 90 degrees 4 times (Figure 6). For a polygon with n sides, we can
divide 360/n to determine how may degrees each rotation will be to return to the original position.
Not all shape rotations are considered cyclic. The rotation of a circle is not cyclic. It is not like
the infinite cyclic group because it is not countable. A circle has an infinite number of sides. We
cannot map every side to the integers therefore a circle’s rotations are not countable. Rotations are
one of the common applications of cyclic groups. Cyclic groups can be used in fun puzzles such as
the Rubik cube or in protecting sensitive information such as through cryptography. Number theory
has many applications in cyclic groups. This paper will explore applications of cyclic groups in the
division algorithm and Chinese remainder theorem, bell ringing, octaves in music, and Chaos
theory.
3. Background Materials
Definition: A group (G, ) is a set G, closed under a binary operation *, such that the following∗
axioms are satisfied: For all a, b, c, G we have associativity (a b) c = a (b c).There is an∈ ∗ ∗ ∗ ∗
identity element for all x G. e x = x e = x. The inverse of every element exists in the set.∈ ∗ ∗
a a’ = a’ a = e.∗ ∗
Example 3.1. The set of integers Z
Definition: Let G be a group, and let H be a subset of G. Then H is called a subgroup of G
if H is itself a group, under the operation induced by G.
Definition: Commutative is changing of the operations does not change the result.
Example 3.2. An example of an the commutative property is 2 + 3 = 3 + 2
Definition: A function f from A to B is called onto if for all b in B there is an a in A such
4. 4
that f (a) = b. All
elements in B are
used.
Example 3.3.
Definition: A function f from A to B is called one-to-one if whenever f (a) = f (b) then a = b.
4. Applications of Cyclic Groups
.1. Number Theory. Cyclic groups are found in nature, patterns, and other fields of mathematics. A
common application of a cyclic group is in number theory. The division algorithm is a
5. 5
fundamental tool for the study of cyclic groups. Division algorithm for integers: if m is a positive
integer and n is any integer, then there exist unique integers q and r such that
(4.1)
n = mq + r and 0 ≤ r < m.
Example 4.1. Find the quotient q and remainder r when 45 is divided by 7 according to the
division algorithm. The positive multiples of 7 are 7, 14, 21, 28, 35, 42, 49 · · ·
(4.2)
45 = 42 + 3 = 7(6) + 3
The quotient is q = 6 and the remainder is r = 3.
You can use the division algorithm to show that a subgroup H of a cyclic group G is also cyclic.
Theorem 4.2. A subgroup of a cyclic group is cyclic.
Proof. Let G be a cyclic group generated by a and let H be a subgroup of G. If H = e, then
H =< e > is cyclic. If H 6 = e, then a n H for some n Z + .Let m be the smallest integer in Z +∈ ∈
such that a m H. C = a m generates H. H = <a^m > = <c>.∈
We must show that every b H is a power of c. Since b H and H ≤ G , we have b = a n∈ ∈
for some n. Find a q and r such that
(4.3) n = mq + r and 0 ≤ r < m.
Then
(4.4) a n = a mq+r = (a m ) q a r ,
So
(4.5) a r = (a^m )^−q*a^r .
Since a^n H, a^m H and H is a group, both (a^m )^−q and a n are in H. Thus (a^m )^−q n H,∈ ∈ ∈
then a r H. Since m was the smallest positive integer such that a^m H and 0 ≤ r < m, we must∈ ∈
have that r = 0.
Thus n = q^m and
(4.6) b = a^n = (a^m ) q = c^q , So b is a power of c
Definition: Let r and s be two positive integers. The positive integer d of the cyclic group
(4.7) H = rn + ms|n, m Z∈
under addition is the greatest common divisor of both r = 1r + 0s and s = 0r + 1s are in H. Since
d H we can write∈
(4.8) d = nr + ms
For some integers n and m. We see every integer dividing both r and s divides the right hand
side of the equation, and hence must be a divisor of d also. Thus, d must be the largest number
dividing both r and s.
Example 4.3. Find the gcd of 24 and 54.
The positive dividers of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. The positive dividers of 54 are 1,
6. 6
2, 3, 6, 9, 18, 27, and 54. The greatest common divisor is 6. 6 = (1)54 + (−2)24. A different result of
congruences in number theory is the Chinese remainder theorem. The Chinese remainder theorem
determines the number n that when divided by some given divisors leave given remainders.
Theorem 4.4. The Chinese remainder theorem . The system of congruences.
(4.9) x ≡ ai (mod m i ), i = 1, 2, 3, . . . k where (m i , m j ) = 1 if i 6 = j, has a unique solution
modulo m 1 m 2 m 3 . . . m k .
Proof. We first show by induction, that system (1) has a solution. The result is obvious when
k = 1. Let us consider the case k = 2. If xa1 (mod m1 ), then x = a 1 + k1 m1 for some k1 . If in
addition x ≡ a 2 (mod m2 ), then
(4.10) a1 + k 1 m 2 ≡ a 2 (mod m 2 )
or
(4.11) k1*m1 ≡ a2 − a1 (mod m2 ).
Because (m2 , m1 ) = 1, we know that this congruence, with k 1 as the unknown, has a unique
solution modulo m 2 . Call it t. Then k 1 = t + k 2 m 2 for some k 2 , and
(4.12) x ≡ ai (mod mi ), i = 1, 2, 3, . . . , r − 1.
But the system
(4.13) x ≡ s(mod m1 m2 m3 . . . m r−1 ),
(4.14) x ≡ a r (mod m r )
Has a solution modulo the product of the moduli, just as in the case k = 2, because (m 1 m 2 m 3 . . .
mk−1 , mk ) =
1. This statement is true because no prime that divides mi . The solution is unique. If r and s are
both solutions to the system then r ≡ s ≡ ai (mod mi ), i = 1, 2, 3, . . . , k,
So mi |(r − s), i = 1, 2, . . . , k. Thus r − s is a common multiple of m1 m2 m3 . . . mk , and because
the moduli are relatively prime in pairs, we have m1 m2 m3 . . . mk |(r − s). Since r and s are least
residuals modulo m1 m2 m3 . . . mk
(4.15) −m1 m2 m3 . . . mk < r − s < m1 m2 m3 . . . mk
hence r − s = 0.
4.2. Cyclic Groups in Bell Ringing. Method ringing, known as scientific ringing, is the practice
of ringing the series of bells as a series of permutations. A permutation f : 1, 2, . . . , n → 1, 2, . . . ,n,
where the domain numbers represent positions and the range numbers represent bells. f (1) would
ring the bell first and bell f (n) . The number of bells n has n! possible changes
7. 7
Plain Bob Minimus permutation
The bell ringer cannot choose to ring permutations in any order because some of the bells con-
tinue to ring up to 2 seconds. Therefore no bell must be rung twice in a row. These permutations
can all be played until it eventually returns to the original pattern of bells.
A common permutation pattern for four bells is the Plain Bob Minimus permutation (Figure 8).
The Plain Bob pattern switches the first two bells then the second set of bells. They would start
the bell ringing with 1234. The first bell would go to the second position and third would go to
the fourth; therefore the next bell combination would be 2143. The next bell switch would be the
two middle bells. Therefore the bell 2143 would turn to 2413. The bell ringers would repeat this
pattern of switching the first two and second two, followed by switching the middle until about 1/3
of the way through the permutations. At the pattern 1324, we cannot switch the middle two. If
we switched the middle two, we would get back to 1234. Therefore, the bell ringers figured out to
switch the last two bells every 8 combinations. Then after 24 moves (4!) we get back to the bell
combination of 1234. Since we made rotations of the bells and generated every combination of the
set and are now back at the beginning, we can say that the bell ringing pattern is cyclic.
9. 9
Permutation of 6 bells
There are other ways to cover all of the permutations without using the Bob Minimus
method(Figure of 4 bells). Bob Minimus method is used because it is easy for bell ringers to
accomplish because they do not have sheet music. Another common permutation method is
following the last bell and moving it over one space to the left each ring then after it is on the left
moving it back over to the right(figure of 6 bells). You can create a cyclic group with any number of
bells. However, the more bells you add the longer the cycle will take. Assuming that each bell ring
takes 2 seconds, someone can complete a set of three bells in 12 seconds. If we have 9 bells it could
take up to 8 days and 10 hours [4].
Hamiltonian graph of the permutation of 4 bells
10. 10
Hamiltonian graph of the permutation of 4 bells
The bell permutations can be expressed as a Hamiltonian graph. A Hamiltonian path is a
undirected or directed graph that visits each vertex exactly once [6]. The Hamiltonian circuit can
be drawn as a simple circuit that has a circular path back to the original vertex. Hamiltonian circuits
for the symmetric group S n mod cyclic groups Z n correspond to the change ringing principles on
n bells
4.3. Clock Arithmetic. On a clock the numbers cycle from one to twelve. After circulating
around the clock we do not go to 13 but restart at one. If it was 6 o’clock, what would it be in 9
hours? 6am + 9 = 3pm. The set of the numbers on a clock are C = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11.
This set of numbers is a group. The identity element is 0 what we will think of as 12. If we add 12
hours to anywhere on the clock we will end up in the same position.
12. 12
twelve hour clock
5. Conclusion
Human minds are designed for pattern recognition and we can find algebraic structures in common
objects and things around us. Cyclic groups are the simplest groups that have an object
that can generate the whole set. The object can generate the set by addition, multiplication, or
rotations. Cyclic groups are not only common in pure mathematics, but also in patterns, shapes,
music, and chaos. Cyclic groups are an imperative part of number theory used with the Chinese
remainder theorem and Fermats theorem. Knowing if a group is cyclic could help determine if
there can be a way to write a group as a simple circuit. This circuit could simplify the process of
generation to discover the most efficient way to generate the object for use of future applications
in mathematics and elsewhere.
References
Fraleigh, J. 2003. A first course in abstract algeabra. Pearson Education.
Fraleigh, J. 1994. A first course in abstract algeabra. Addison Wesley.
Guichard,D.R. 1999. When is U(n) cyclic? An Algebraic Approach. Mathematics Magazine.
72(2):139-142.
Polster,B.Ross,M.2009.Ringing the changes.Plus Magazine.http://plus.maths.org/content/ringing-
changes
White,A.T.1988. Ringing the cosets 2. Cambridge Philos. Soc. 105:53-65.
White,A.T.1993.Treble dodging minor methods: ringing the cosets, on six bells. Discrete
Mathematics. 122(1-
3):307-323
[7] Dougal,C.R.2008. Chaos chance money. Plus Magazine.http://plus.maths.org/content/chaos-
chance-and-money
[8] Hazewinkel,M.2001. Cyclic group.Encyclopedia of Mathematics