Cyclic groups are common in everyday life and appear in patterns found in nature, geometry, and music. The document discusses several applications of cyclic groups, including:
1) Number theory, where cyclic groups are used in the division algorithm and Chinese Remainder Theorem.
2) Bell ringing methods, which form cyclic groups by permuting the order of bells rung.
3) Music, where octaves form a cyclic group through rotational symmetry.
sarminIJMA1-4-2015 forth paper after been publishing
This document discusses the probability that a group element fixes a set and its application to generalized conjugacy class graphs. It begins with background on commutativity degree and graph theory concepts. It then reviews previous work calculating the probability for various groups and defining generalized conjugacy class graphs. The main results calculate the probability for semi-dihedral and quasi-dihedral groups as 1/2 and 1/3 respectively, and determine that the corresponding generalized conjugacy class graphs are K2 and Ke (empty graph).
This document discusses the probability that an element of a metacyclic 2-group fixes a set. It begins by providing background information on metacyclic groups and related concepts such as commutativity degree. It then summarizes previous works that have studied the probability of a group element fixing a set. The main results presented calculate the probability for various metacyclic 2-groups of negative type with nilpotency class of at least two. In particular, it determines the probability for metacyclic 2-groups of negative type with nilpotency class of at least three. It also calculates the probability for other specific metacyclic 2-group structures.
On fixed point theorems in fuzzy 2 metric spaces and fuzzy 3-metric spaces
1) The document discusses fixed point theorems for mappings in fuzzy 2-metric and fuzzy 3-metric spaces.
2) It defines concepts like fuzzy metric spaces, Cauchy sequences, compatible mappings, and proves some fixed point theorems for compatible mappings.
3) The theorems show that under certain contractive conditions on the mappings, there exists a unique common fixed point for the mappings in a complete fuzzy 2-metric or fuzzy 3-metric space.
On fixed point theorems in fuzzy metric spaces in integral type
This document presents several common fixed point theorems for occasionally weakly compatible mappings in fuzzy metric spaces. It begins with definitions of key concepts such as fuzzy sets, fuzzy metric spaces, occasionally weakly compatible mappings, and Cauchy sequences in fuzzy metric spaces. It then presents four main theorems that establish the existence and uniqueness of a common fixed point for self-mappings under certain contractive conditions on the mappings and using the concept of occasionally weakly compatible pairs. The proofs of the theorems are also provided.
The document defines and discusses several key concepts relating to groups in abstract algebra:
- A group is defined as a non-empty set together with a binary operation that satisfies closure, associativity, identity, and inverse properties.
- An abelian (or commutative) group is one where the binary operation is commutative. Examples of abelian groups include integers under addition.
- The quaternion group is a non-abelian group of order 8 under multiplication.
- Theorems are presented regarding the uniqueness of identity and inverses in a group, as well as cancellation, reverse order, and inverse properties of groups.
On the k-Riemann-Liouville fractional integral and applications
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.
The document discusses orthogonal polynomials, focusing on Legendre and Chebyshev polynomials. It introduces Hilbert spaces and self-adjoint operators, describing properties like Hermitian matrices having real eigenvalues. It defines the L2 space and shows how differential operators can act as self-adjoint. Legendre polynomials are defined using Rodrigue's formula and their generating function is explored. The first few Legendre polynomials are shown.
A Common Fixed Point Theorem on Fuzzy Metric Space Using Weakly Compatible an...
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.
Fixed point theorems for four mappings in fuzzy metric space using implicit r...
This document presents theorems proving the existence and uniqueness of common fixed points for four mappings (A, B, S, T) in a fuzzy metric space using an implicit relation.
It begins with definitions of key concepts like fuzzy metric spaces, Cauchy sequences, completeness, compatibility, and occasionally weak compatibility of mappings.
The main result (Theorem 3.1) proves that if the pairs of mappings (A,S) and (B,T) are occasionally weakly compatible, and an implicit relation involving the fuzzy metric of images of x and y under the mappings is satisfied, then there exists a unique common fixed point w for A and S, and a unique common fixed point z for B and T.
This document defines and provides necessary and sufficient conditions for a pair of operators (T1, T2) on a topological vector space to be syndetically hypercyclic. It begins by introducing key concepts such as hypercyclic pairs and sequences. The main result is that a pair (T1, T2) is syndetically hypercyclic if and only if it satisfies the Hypercyclicity Criterion for syndetic sequences. Additionally, it is shown that if a pair satisfies the Hypercyclicity Criterion for a syndetic sequence, then it is topologically mixing. The proof of the main theorem utilizes the Hypercyclicity Criterion and shows topological weak mixing is equivalent to
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.
1. The document discusses groups, subgroups, cosets, normal subgroups, quotient groups, and homomorphisms.
2. It defines cosets, proves Lagrange's theorem that the order of a subgroup divides the order of the group, and provides examples of finding cosets.
3. Normal subgroups are introduced, and it is shown that the set of cosets of a normal subgroup forms a group under a defined operation, known as the quotient group. Homomorphisms between groups are defined, and examples are given.
sarminIJMA1-4-2015 forth paper after been publishingMustafa El-sanfaz
This document discusses the probability that a group element fixes a set and its application to generalized conjugacy class graphs. It begins with background on commutativity degree and graph theory concepts. It then reviews previous work calculating the probability for various groups and defining generalized conjugacy class graphs. The main results calculate the probability for semi-dihedral and quasi-dihedral groups as 1/2 and 1/3 respectively, and determine that the corresponding generalized conjugacy class graphs are K2 and Ke (empty graph).
This document discusses the probability that an element of a metacyclic 2-group fixes a set. It begins by providing background information on metacyclic groups and related concepts such as commutativity degree. It then summarizes previous works that have studied the probability of a group element fixing a set. The main results presented calculate the probability for various metacyclic 2-groups of negative type with nilpotency class of at least two. In particular, it determines the probability for metacyclic 2-groups of negative type with nilpotency class of at least three. It also calculates the probability for other specific metacyclic 2-group structures.
On fixed point theorems in fuzzy 2 metric spaces and fuzzy 3-metric spacesAlexander Decker
1) The document discusses fixed point theorems for mappings in fuzzy 2-metric and fuzzy 3-metric spaces.
2) It defines concepts like fuzzy metric spaces, Cauchy sequences, compatible mappings, and proves some fixed point theorems for compatible mappings.
3) The theorems show that under certain contractive conditions on the mappings, there exists a unique common fixed point for the mappings in a complete fuzzy 2-metric or fuzzy 3-metric space.
On fixed point theorems in fuzzy metric spaces in integral typeAlexander Decker
This document presents several common fixed point theorems for occasionally weakly compatible mappings in fuzzy metric spaces. It begins with definitions of key concepts such as fuzzy sets, fuzzy metric spaces, occasionally weakly compatible mappings, and Cauchy sequences in fuzzy metric spaces. It then presents four main theorems that establish the existence and uniqueness of a common fixed point for self-mappings under certain contractive conditions on the mappings and using the concept of occasionally weakly compatible pairs. The proofs of the theorems are also provided.
The document defines and discusses several key concepts relating to groups in abstract algebra:
- A group is defined as a non-empty set together with a binary operation that satisfies closure, associativity, identity, and inverse properties.
- An abelian (or commutative) group is one where the binary operation is commutative. Examples of abelian groups include integers under addition.
- The quaternion group is a non-abelian group of order 8 under multiplication.
- Theorems are presented regarding the uniqueness of identity and inverses in a group, as well as cancellation, reverse order, and inverse properties of groups.
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.
The document discusses orthogonal polynomials, focusing on Legendre and Chebyshev polynomials. It introduces Hilbert spaces and self-adjoint operators, describing properties like Hermitian matrices having real eigenvalues. It defines the L2 space and shows how differential operators can act as self-adjoint. Legendre polynomials are defined using Rodrigue's formula and their generating function is explored. The first few Legendre polynomials are shown.
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.
Fixed point theorems for four mappings in fuzzy metric space using implicit r...Alexander Decker
This document presents theorems proving the existence and uniqueness of common fixed points for four mappings (A, B, S, T) in a fuzzy metric space using an implicit relation.
It begins with definitions of key concepts like fuzzy metric spaces, Cauchy sequences, completeness, compatibility, and occasionally weak compatibility of mappings.
The main result (Theorem 3.1) proves that if the pairs of mappings (A,S) and (B,T) are occasionally weakly compatible, and an implicit relation involving the fuzzy metric of images of x and y under the mappings is satisfied, then there exists a unique common fixed point w for A and S, and a unique common fixed point z for B and T.
NITheP UKZN Seminar: Prof. Alexander Gorokhov (Samara State University, Russi...Rene Kotze
This document discusses the application of dynamical groups and coherent states in quantum optics and molecular spectroscopy. It provides an introduction to using Lie groups and algebras to describe quantum systems and defines coherent states. Specific applications discussed include using dynamical symmetries to calculate energy levels of systems like the harmonic oscillator and hydrogen atom. Coherent states are used to derive classical equations of motion and represent open quantum systems. Examples of coherent state dynamics are shown for two-level and three-level atoms interacting with laser fields.
Welcome to International Journal of Engineering Research and Development (IJERD)IJERD Editor
- The document investigates the action of two fuzzy translation operators, Tα+ and Tα-, on fuzzy and anti-fuzzy submodules of a module.
- It proves that if a fuzzy set is a fuzzy submodule, then applying Tα+ or Tα- results in a fuzzy submodule. However, the converse is not necessarily true.
- Some additional conditions are obtained under which the converse is also true - namely, if Tα+ and Tα- yield fuzzy submodules for all α, then the original fuzzy set must be a fuzzy submodule.
- The operators are also shown to map anti-fuzzy submodules to anti-fuzzy submodules, and the paper explores relationships
Prof. Vishnu Jejjala (Witwatersrand) TITLE: "The Geometry of Generations"Rene Kotze
This document summarizes Vishnu Jejjala's talk "The Geometry of Generations" given at the University of the witwatersrand on 23 September 2014. It discusses using the geometry of vacuum moduli spaces of supersymmetric gauge theories to understand phenomenological aspects of the Standard Model, such as the number of generations. It provides examples of calculating the vacuum moduli space for simple supersymmetric gauge theories and outlines a process for obtaining the moduli space from the F-term and D-term equations.
1. The document discusses orthogonal polynomials, which are polynomial sequences where any two different polynomials are orthogonal under some inner product.
2. Some common orthogonal polynomials are Legendre polynomials, Hermite polynomials, Laguerre polynomials, and Chebyshev polynomials.
3. It is proven that for Legendre polynomials pm and pn, the integral from -1 to 1 of pm(x)pn(x)dx is equal to 0 when m is not equal to n, and is equal to 2/(2n+1) when m is equal to n. This shows the orthogonal property of Legendre polynomials.
This document provides definitions and propositions related to abstract algebra. It begins by defining a group as a set with a binary operation that is closed, associative, has an identity element, and where each element has an inverse. It then lists several propositions about properties of groups, including that a group has a unique identity and each element has a unique inverse. The document continues defining additional algebraic structures like rings, fields, subgroups, and properties of groups like cyclic groups. It concludes by discussing matrix groups and their properties.
International Journal of Mathematics and Statistics Invention (IJMSI) inventionjournals
This document presents a common fixed point theorem for six self-maps (A, B, S, T, L, M) on a Menger space using the concept of weak compatibility. It proves that if the maps satisfy certain conditions, including being weakly compatible and their images being complete subspaces, then the maps have a unique common fixed point. The proof constructs sequences to show the maps have a coincidence point, then uses weak compatibility and lemmas to show this point is the unique common fixed point.
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.
This document provides an introduction to group theory from a physicist's perspective. It defines what a group is, including properties like closure, associativity, identity, and inverse. Examples of important groups in physics are given, including finite groups like Zn and Sn, and continuous groups like SU(n), SO(n), and the Lorentz group. The document outlines topics like discrete and finite groups, representation of groups, Lie groups and algebras, and applications of specific groups like SU(2) and SU(3) to physics.
This summary provides the key details from the document in 3 sentences:
The document investigates the structure of unital 3-fields, which are fields where addition requires 3 summands rather than the usual 2. It is shown that unital 3-fields are isomorphic to the set of invertible elements in a local ring R with Z2Z as the residual field. Pairs of elements in the 3-field are used to define binary operations that allow reducing the arity and connecting the 3-field to binary algebra. The structure of finite 3-fields is examined, proving properties like the number of elements being a power of 2.
A Study of Permutation Groups and Coherent ConfigurationsJohn Batchelor
This document provides historical background on the study of permutation groups. It discusses how Lagrange initially studied permutations when solving polynomial equations. Later, Galois connected permutation groups to field theory by introducing Galois groups. The document also mentions contributions from mathematicians like Burnside, Frobenius, and Jordan that advanced the theory of permutation groups.
Analysis and algebra on differentiable manifoldsSpringer
This chapter discusses integration on manifolds. It defines orientation of manifolds and orientation-preserving maps between manifolds. It presents Stokes' theorem and Green's theorem, which relate integrals over boundaries to integrals of differential forms. It introduces de Rham cohomology groups, which classify closed forms modulo exact forms. Examples are given of calculating the orientability and cohomology of manifolds like the cylinder, Möbius strip, and real projective plane.
PRESENTATION ON INTRODUCTION TO SEVERAL VARIABLES AND PARTIAL DERIVATIVES Mazharul Islam
This document provides an introduction to partial derivatives and several examples of calculating them. It begins by defining partial derivatives as the rate of change of a function with respect to one variable, holding other variables constant. Several examples are then provided of calculating partial derivatives of multivariable functions. The document concludes by stating the chain rule for partial derivatives, which relates the derivative of a composite function to its constituent partial derivatives.
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.
Stability criterion of periodic oscillations in a (8)Alexander Decker
This academic article discusses local cohomology modules and cofiniteness. It presents several definitions, theorems, and examples regarding local cohomology modules and systems of ideals. Specifically, it constructs an ideal system using tridiagonal matrices of tridiagonal subsets and shows that the local cohomology modules are finite in this special 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.
Group Theory and Its Application: Beamer Presentation (PPT)SIRAJAHMAD36
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.
This document describes a study of the chaotic behavior of a magnetic pendulum. It begins by defining relevant terms like chaos and magnetic pendulum. It then derives the initial value problem that approximates the motion of the magnetic pendulum based on explicit assumptions. This derives equations for the x, y, and z components of the pendulum's motion. It shows that the system exhibits two types of chaotic behavior through numerical solutions with varying parameter values presented in graphs.
This document provides an introduction to group theory with applications to quantum mechanics and solid state physics. It begins with definitions of groups and examples of groups that are important in physics. It then discusses several applications of group theory in classical mechanics, quantum mechanics, and solid state physics. Specifically, it explains how group theory can be used to evaluate matrix elements, understand degeneracies of energy eigenvalues, classify electronic states in periodic potentials, and construct models that respect crystal symmetries. It also briefly discusses the use of group theory in nuclear and particle physics.
This document describes extending the Elgamal cryptosystem to work with the second group of units of Zn and Z2[x]/<h(x)>, where h(x) is an irreducible polynomial. It first reviews the definition and construction of the second group of units U2(Zn) and U2(Z2[x]/<h(x)>). It then presents the key generation, encryption, and decryption algorithms for the Elgamal cryptosystem adapted to these new settings. The document evaluates the accuracy, efficiency and security of the modified cryptographic scheme through implementation and testing.
Here are the answers to the activity questions on page 216 of the study guide in one whole sheet of paper:
5. Kinetic energy (KE) varies jointly as mass (m) and the square of velocity (v2).
KE = kmv2
Given:
Mass (m) = 8 g
Velocity (v) = 5 cm/s
KE = 100 ergs
Find k:
100 ergs = k(8 g)(5 cm/s)2
100 ergs = k(8)(25)
100 ergs = 200k
k = 100/200 = 1/2
KE = kmv2
Given:
Mass (m) = 6 g
Monotonicity of Phaselocked Solutions in Chains and Arrays of Nearest-Neighbo...Liwei Ren任力偉
This document summarizes a paper on phaselocked solutions in chains and arrays of coupled oscillators. It introduces the topic of coupled oscillators and their importance in modeling neural activity. It describes previous work analyzing one-dimensional chains of oscillators using continuum approximations. The current paper aims to rigorously prove the existence of phaselocked solutions in chains without requiring a continuum limit. It also analyzes two-dimensional arrays by decomposing them into independent one-dimensional problems under certain frequency distributions. Key results include proving monotonicity of phaselocked solutions and spontaneous formation of target patterns in two-dimensional arrays with isotropic synaptic coupling.
TIU CET Review Math Session 6 - part 2 of 2youngeinstein
1. The document provides a review of math concepts for a college entrance exam, including functions, trigonometric functions, exponential and logarithmic functions.
2. It reviews concepts like evaluating functions, adding and composing functions, finding roots and intercepts of functions, and properties of trigonometric, exponential and logarithmic functions.
3. The document provides examples and problems to solve related to these various math concepts as a study guide for the exam.
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 ...
The document discusses using Plücker coordinates to determine if a set of homogeneous polynomials generate the vector space of higher degree polynomials when projected onto a quotient ring. It begins by defining Plücker coordinates and showing that for two quadratic generators, the cubic vector space has zero image if the Plücker quantity P1P3 - P22 is non-zero. It then aims to generalize this to three cubic generators and the quartic vector space.
Quaternions, Alexander Armstrong, Harold Baker, Owen WilliamsHarold Baker
Quaternions are a mathematical structure used to represent rotations and orientations in 3D space. The document discusses the history, theory, and applications of quaternions. It was invented in 1843 by Sir William Rowan Hamilton and has found modern applications in computer graphics, where it is used for 3D animation and rotations due to advantages over other representations like Euler angles. The theory section covers properties like multiplication and identities. Applications discussed include physics, group theory, and using quaternions in linear interpolation algorithms for smooth 3D animation.
Rissa May at 19_ A Rising Star in Entertainment and Environmental Activism.pptxashishkumarrana9
Fresh talent is usually sought for in the entertainment business, and Rissa May Age 19, surely drew its attention. This young actress and model is rapidly making herself a force to be reckoned with with her mesmerizing screen presence and varied acting range. But it’s not just her skill in front of the camera that’s drawing attention—May’s fervent environmental campaigning and dedication to sustainable development are also getting her much praise.
The Chartered Facilities Manager.PREVIEW.pdfGAFM ACADEMY
The Chartered Facilities Manager (ChFM) is a gold-standard certification exclusively from the Global Academy of Finance and Management ®. Earning this certification demonstrates that you have skills and experience in facilities management which include the maintenance of buildings, road maintenance, manufacturing plants, tools and machineries, heating, ventilation and air-conditioning systems, ensuring that the facilities meet statutory requirements and comply with occupational health and safety standards.
It forms the basis of the assessment that individuals must pass to earn the Chartered Facilities Manager status and inclusion in the Directory of The GAFM Academy of Finance and Management Certified Professionals. Individuals with several years of experience in facilities management are encouraged to acquire this certification.
https://gafm.com.my/digital-certification/gafm-book-shop/
https://gafm.com.my/digital-certification/application-for-certification/
You have been assigned to manage a project but have no clue how and where to begin. It sounds like an opportunity but it can also turn out to be a disaster if you do not possess the knowledge and skills.
You must have come across a book called The Project Management Body of Knowledge which is most commonly called PMBOK. PMBOK is about processes, tools, and techniques to manage a project. It does not talk about the art and science of executing a project from the initial phase to the end of the project life cycle. PMBOK introduces you to a bunch of processes that you may use in managing a project, initiation processes for the project initiation phase, planning group of processes that you may apply during the planning phase, and the list goes on. After reading the PMBOK guide, you still have no idea where to begin. If you do not have the time then what you need is a book that will provide a birds-eye view and content that is sufficient enough to assist you in kicking off a project. Get this book now and begin to kick off a project like a pro.
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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