1. The document studies the matching domination parameters of Euler Totient Cayley graphs. It defines Euler Totient Cayley graphs and presents theorems about the domination number and matching domination number for various cases like when n is prime, a power of a prime, or neither prime nor a power of a prime.
2. Theorems and proofs are presented for the matching domination number when n is prime, a power of a prime, of the form kp where k is an odd prime, and neither prime nor a power of a prime.
3. Examples illustrating the matching dominating sets for various n are shown through figures.
INDEPENDENT DOMINATION NUMBER OF EULER TOTIENT CAYLEY GRAPHS AND ARITHMETIC G...IAEME Publication
Nathanson was the pioneer in introducing the concepts of Number Theory, particularly, the “Theory of Congruences” in Graph Theory, thus paved the way for the emergence of a new class of graphs, namely “Arithmetic Graphs”. Cayley graphs are another class of graphs associated wi th the elements of a group. If this group is associated with some arithmetic function then the Cayley graph becomes an Arithmetic graph.
American Journal of Multidisciplinary Research and Development is indexed, refereed and peer-reviewed journal, which is designed to publish research articles.
Fair Restrained Dominating Set in the Corona of GraphsDr. Amarjeet Singh
In this paper, we give the characterization of a fair restrained dominating set in the corona of two nontrivial connected graphs and give some important results.
In this paper, we extend the concept of fair secure dominating sets by characterizing the corona of two nontrivial connected graphs and give some important results.
This paper introduces the concept of h-integrability as a condition for obtaining weak laws of large numbers for arrays (WLLNFA). The paper defines various types of uniform integrability including Cesaro uniform integrability, -uniform integrability, Cesaro α-integrability, and h-integrability. It is shown that h-integrability is a weaker condition than these other forms of uniform integrability. The main results of the paper establish that if an array is h-integrable with exponent r, then the sample mean of the array converges in probability to the expected value under certain conditions. This extends previous results on obtaining weak laws of large numbers for dependent random variables.
This summary provides an overview of Bob Hough's paper that shows the least modulus of a distinct covering system is at most 1016:
1) Hough uses an iterative sieving process where moduli are grouped based on their prime factors being below increasing thresholds. This allows estimating the density of integers left uncovered at each stage.
2) At each stage, the uncovered set is shown to have nonzero density within "good" fibers over a well-distributed subset of the previous stage's uncovered set.
3) Applying the Lovász Local Lemma twice guarantees a set of good fibers is also well-distributed, ensuring the next stage's uncovered set has nonzero density and proving the minimum modulus is bounded.
This document discusses various inference methods in propositional logic, including:
- The enumeration method, which checks all possible models to determine logical implication.
- Inference rules like modus ponens and resolution.
- The resolution method, which converts formulas to conjunctive normal form and applies the resolution rule to derive contradictions.
- Forward and backward chaining for Horn clauses, which can check implication efficiently in linear time. Examples of applying forward chaining to a rule set are provided.
I am Grey N. I am a Chemistry Assignment Expert at eduassignmenthelp.com. I hold a Ph.D. in Chemistry, from Calgary, Canada. I have been helping students with their homework for the past 6 years. I solve assignments related to Chemistry.
Visit eduassignmenthelp.com or email info@eduassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Chemistry Assignments.
INDEPENDENT DOMINATION NUMBER OF EULER TOTIENT CAYLEY GRAPHS AND ARITHMETIC G...IAEME Publication
Nathanson was the pioneer in introducing the concepts of Number Theory, particularly, the “Theory of Congruences” in Graph Theory, thus paved the way for the emergence of a new class of graphs, namely “Arithmetic Graphs”. Cayley graphs are another class of graphs associated wi th the elements of a group. If this group is associated with some arithmetic function then the Cayley graph becomes an Arithmetic graph.
American Journal of Multidisciplinary Research and Development is indexed, refereed and peer-reviewed journal, which is designed to publish research articles.
Fair Restrained Dominating Set in the Corona of GraphsDr. Amarjeet Singh
In this paper, we give the characterization of a fair restrained dominating set in the corona of two nontrivial connected graphs and give some important results.
In this paper, we extend the concept of fair secure dominating sets by characterizing the corona of two nontrivial connected graphs and give some important results.
This paper introduces the concept of h-integrability as a condition for obtaining weak laws of large numbers for arrays (WLLNFA). The paper defines various types of uniform integrability including Cesaro uniform integrability, -uniform integrability, Cesaro α-integrability, and h-integrability. It is shown that h-integrability is a weaker condition than these other forms of uniform integrability. The main results of the paper establish that if an array is h-integrable with exponent r, then the sample mean of the array converges in probability to the expected value under certain conditions. This extends previous results on obtaining weak laws of large numbers for dependent random variables.
This summary provides an overview of Bob Hough's paper that shows the least modulus of a distinct covering system is at most 1016:
1) Hough uses an iterative sieving process where moduli are grouped based on their prime factors being below increasing thresholds. This allows estimating the density of integers left uncovered at each stage.
2) At each stage, the uncovered set is shown to have nonzero density within "good" fibers over a well-distributed subset of the previous stage's uncovered set.
3) Applying the Lovász Local Lemma twice guarantees a set of good fibers is also well-distributed, ensuring the next stage's uncovered set has nonzero density and proving the minimum modulus is bounded.
This document discusses various inference methods in propositional logic, including:
- The enumeration method, which checks all possible models to determine logical implication.
- Inference rules like modus ponens and resolution.
- The resolution method, which converts formulas to conjunctive normal form and applies the resolution rule to derive contradictions.
- Forward and backward chaining for Horn clauses, which can check implication efficiently in linear time. Examples of applying forward chaining to a rule set are provided.
I am Grey N. I am a Chemistry Assignment Expert at eduassignmenthelp.com. I hold a Ph.D. in Chemistry, from Calgary, Canada. I have been helping students with their homework for the past 6 years. I solve assignments related to Chemistry.
Visit eduassignmenthelp.com or email info@eduassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Chemistry Assignments.
This document discusses the axiom of choice in set theory. It provides definitions of key terms like well-ordering, partial ordering, and Zorn's lemma. It also covers some equivalents and consequences of the axiom of choice, including the well-ordering principle and Banach-Tarski paradox. The axiom of choice allows choosing one element from each nonempty set in a collection of disjoint sets and guarantees the existence of a choice set.
This chapter discusses proof by contradiction as a powerful but less direct proof technique. It can be used to prove claims about non-existence, by assuming the claim is false and showing this leads to a contradiction. Several examples are provided, including proofs that sqrt(2) is irrational, there are infinitely many prime numbers, and that lossless data compression must sometimes result in larger file sizes. While valid, contradiction proofs are less intuitive than direct proofs.
The document discusses the natural logarithm function ln(x) and the natural exponential function exp(x). It begins by defining ln(x) as the area under the curve y=1/t from 1 to x, and noting that its derivative is 1/x. It then defines exp(x) as the inverse of ln(x). It is shown that for rational r, exp(r) = er, and this definition is extended to irrational r. The derivative of exp(x) is then shown to be exp(x) itself.
I am Craig D. I am a Physical Chemistry Assignment Expert at eduassignmenthelp.com. I hold a Ph.D. in Physical Chemistry, from The University of Queensland. I have been helping students with their homework for the past 9 years. I solve assignments related to Physical Chemistry.
Visit eduassignmenthelp.com or email info@eduassignmenthelp.com. You can also call on +1 678 648 4277 for any assistance with Physical Chemistry Assignments.
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.
This document provides an introduction to probability theory. It begins by defining probability spaces, which consist of a sample space (Ω), a σ-algebra of events (F), and a probability measure (P). It then discusses examples of σ-algebras and probability measures. The document outlines the topics that will be covered, including random variables, integration theory, distributions, and limit theorems. It presents probability theory using a rigorous mathematical approach based on measure theory and σ-algebras.
This document discusses proof by contradiction, an indirect proof method. It provides examples of using proof by contradiction to prove different mathematical statements. The key steps in a proof by contradiction are: 1) assume the statement to be proved is false, 2) show that this assumption leads to a logical contradiction, and 3) conclude that the original statement must be true since the assumption was false. The document provides examples of proofs by contradiction for statements such as "there is no greatest integer" and "if n is an integer and n3 + 5 is odd, then n is even."
This document is the preface to a textbook on number theory. It discusses the goals of the textbook, which are to encourage independent thinking and problem solving rather than rote memorization. Number theory is well-suited for this purpose as patterns in the natural numbers can be discerned through observation and experimentation, but proving theorems requires rigorous demonstration. The textbook was originally written for a course at Brown University designed to attract non-science majors to mathematics. The prerequisites are few, requiring only high school algebra and a willingness to experiment, make mistakes, learn from them, and persevere.
On Various Types of Ideals of Gamma Rings and the Corresponding Operator RingsIJERA Editor
The prime objective of this paper is to prove some deep results on various types of Gamma ideals. The
characteristics of various types of Gamma ideals viz. prime/maximal/minimal/nilpotent/primary/semi-prime
ideals of a Gamma-ring are shown to be maintained in the corresponding right (left) operator rings of the
Gamma-rings. The converse problems are also investigated with some good outcomes. Further it is shown that
the projective product of two Gamma-rings cannot be simple.
AMS Mathematics subject classification: Primary 16A21, 16A48, 16A78
This document discusses deductive reasoning and logical laws. It provides examples of applying the law of detachment, law of syllogism, and identifying inductive vs deductive reasoning. Examples include writing conditional statements, making valid conclusions, and showing a conjecture is true using deductive reasoning. Guided practice questions have students applying these logical concepts.
This document discusses proof by contradiction in mathematics. It begins by defining proof by contradiction as proving the truth of a statement by showing that assuming the statement is false leads to a contradiction. The document then provides examples of proofs by contradiction, including:
1) Proving there is no greatest integer by supposing there is a greatest integer N and showing that N+1 would also be an integer, contradicting that N was the greatest.
2) Proving the square root of 2 is irrational by supposing it is rational and showing this leads to a contradiction.
3) Explaining the general steps in a proof by contradiction: assume the statement is false, show this assumption leads to a contradiction, and thus
This document provides an introduction to series and sequences. It discusses key concepts such as convergence vs divergence of sequences and series. Specifically, it defines what an infinite sum is, absolute vs conditional convergence, and common tests to determine convergence like the integral test and alternating series test. It also introduces the big-O, little-o, and big-Theta notations used to compare the growth or decay rates of sequences. The overall goal is to establish an understanding of convergence of series before delving into numerical analysis applications that make use of infinite series like Taylor and Fourier series.
Feynman diagrams are pictorial representations of particle reaction amplitudes. They allow calculations of rates and cross sections for physical processes like muon decay or electron-positron scattering to be greatly simplified. Each diagram has a strict mathematical interpretation corresponding to terms in a power series expansion of the reaction amplitude. Diagrams become more complex at higher orders but must be combined correctly while respecting conservation laws and process symmetries to obtain the total amplitude. The anomalous magnetic moment of particles like the electron and muon can be calculated order-by-order using Feynman diagrams, with remarkable agreement between theory and precise measurements.
The document discusses analyzing the average case runtime of quicksort. It shows that:
1) The average case runtime of quicksort is O(n log n) due to partitions being randomly balanced on average.
2) This is proved rigorously by modeling quicksort as alternating best and worst case partitions, showing the cost is absorbed by subsequent partitions.
3) A recurrence is written and solved to formally prove the average case runtime is O(n log n).
This document provides an introduction and solutions to problems in Modern Particle Physics. It contains 18 chapters covering topics in particle physics like the Dirac equation, decay rates, scattering processes, and the Standard Model. The preface explains that the guide gives numerical solutions and hints to help understand questions from the first edition of the textbook. Instructors can obtain fully worked solutions from the publisher.
This document introduces mathematical induction. It defines the principle of mathematical induction as having two steps: (1) the basis step, which shows a statement P(1) is true, and (2) the inductive step, which assumes P(k) is true and shows P(k+1) is also true. It provides an example of climbing an infinite ladder to illustrate these steps. It also notes some important points about mathematical induction, such as that it is expressing a rule of inference and in proofs we show P(k) implies P(k+1) rather than assuming P(k) is true for all k.
The document discusses propositions, logical operations, predicates, quantification, and mathematical induction. It provides:
1) Definitions of predicates, propositions, universal and existential quantification, and the principle of mathematical induction.
2) Examples of applying predicates, quantification, and induction to prove statements about integers and sums.
3) The process of proving statements by mathematical induction, which involves showing the basis step and inductive step. It also introduces strong mathematical induction.
This document introduces the concept of order of an element modulo n and uses it to prove theorems about when an integer n satisfies n^2 + 1 or more generally satisfies a cyclotomic polynomial modulo a prime p. It begins by stating and proving the n^2 + 1 lemma, which says a prime p satisfies p | n^2 + 1 if and only if p ≡ 1 (mod 4). It introduces the concepts of order, primitive roots, and cyclotomic polynomials to generalize this result. It concludes by stating and proving a theorem about when a cyclotomic polynomial of an integer a is divisible by a prime p.
The document discusses proofs by contraposition. It explains that a statement of the form "if p then q" can be proven by showing its contrapositive "if not q then not p" is true. It provides examples of proofs using this method, including proving if n^2 is even then n is even, if m + n is even then m and n have the same parity, and if 3n + 2 is odd then n is odd. Homework exercises are provided applying this proof technique.
This document provides a summary of a lecture on graphs and algorithms. It discusses Hall's Theorem and provides a proof using induction. It then gives several examples of how Hall's Theorem can be applied, such as scheduling tasks among machines. The document also discusses maximum flows, the Konig-Egervary Theorem relating matchings and vertex covers, and Menger's Theorem relating disjoint paths and minimum cuts.
This document explores limiting the size of topological spaces through cardinal invariants and Arhangel'skii's Theorem. It begins by introducing set theory concepts like cardinals and ordinals. It then discusses topological spaces formed by putting the order topology on ordinals, called ordinal spaces. Finally, it covers cardinal invariants, which place bounds on the size of topological spaces, and proves a particular case of Arhangel'skii's Theorem, which showed that compact, first-countable spaces have at most the cardinality of the reals. The goal is to understand Arhangel'skii's novel "closing off" proof technique for bounding cardinalities of topological spaces.
This document discusses the axiom of choice in set theory. It provides definitions of key terms like well-ordering, partial ordering, and Zorn's lemma. It also covers some equivalents and consequences of the axiom of choice, including the well-ordering principle and Banach-Tarski paradox. The axiom of choice allows choosing one element from each nonempty set in a collection of disjoint sets and guarantees the existence of a choice set.
This chapter discusses proof by contradiction as a powerful but less direct proof technique. It can be used to prove claims about non-existence, by assuming the claim is false and showing this leads to a contradiction. Several examples are provided, including proofs that sqrt(2) is irrational, there are infinitely many prime numbers, and that lossless data compression must sometimes result in larger file sizes. While valid, contradiction proofs are less intuitive than direct proofs.
The document discusses the natural logarithm function ln(x) and the natural exponential function exp(x). It begins by defining ln(x) as the area under the curve y=1/t from 1 to x, and noting that its derivative is 1/x. It then defines exp(x) as the inverse of ln(x). It is shown that for rational r, exp(r) = er, and this definition is extended to irrational r. The derivative of exp(x) is then shown to be exp(x) itself.
I am Craig D. I am a Physical Chemistry Assignment Expert at eduassignmenthelp.com. I hold a Ph.D. in Physical Chemistry, from The University of Queensland. I have been helping students with their homework for the past 9 years. I solve assignments related to Physical Chemistry.
Visit eduassignmenthelp.com or email info@eduassignmenthelp.com. You can also call on +1 678 648 4277 for any assistance with Physical Chemistry Assignments.
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.
This document provides an introduction to probability theory. It begins by defining probability spaces, which consist of a sample space (Ω), a σ-algebra of events (F), and a probability measure (P). It then discusses examples of σ-algebras and probability measures. The document outlines the topics that will be covered, including random variables, integration theory, distributions, and limit theorems. It presents probability theory using a rigorous mathematical approach based on measure theory and σ-algebras.
This document discusses proof by contradiction, an indirect proof method. It provides examples of using proof by contradiction to prove different mathematical statements. The key steps in a proof by contradiction are: 1) assume the statement to be proved is false, 2) show that this assumption leads to a logical contradiction, and 3) conclude that the original statement must be true since the assumption was false. The document provides examples of proofs by contradiction for statements such as "there is no greatest integer" and "if n is an integer and n3 + 5 is odd, then n is even."
This document is the preface to a textbook on number theory. It discusses the goals of the textbook, which are to encourage independent thinking and problem solving rather than rote memorization. Number theory is well-suited for this purpose as patterns in the natural numbers can be discerned through observation and experimentation, but proving theorems requires rigorous demonstration. The textbook was originally written for a course at Brown University designed to attract non-science majors to mathematics. The prerequisites are few, requiring only high school algebra and a willingness to experiment, make mistakes, learn from them, and persevere.
On Various Types of Ideals of Gamma Rings and the Corresponding Operator RingsIJERA Editor
The prime objective of this paper is to prove some deep results on various types of Gamma ideals. The
characteristics of various types of Gamma ideals viz. prime/maximal/minimal/nilpotent/primary/semi-prime
ideals of a Gamma-ring are shown to be maintained in the corresponding right (left) operator rings of the
Gamma-rings. The converse problems are also investigated with some good outcomes. Further it is shown that
the projective product of two Gamma-rings cannot be simple.
AMS Mathematics subject classification: Primary 16A21, 16A48, 16A78
This document discusses deductive reasoning and logical laws. It provides examples of applying the law of detachment, law of syllogism, and identifying inductive vs deductive reasoning. Examples include writing conditional statements, making valid conclusions, and showing a conjecture is true using deductive reasoning. Guided practice questions have students applying these logical concepts.
This document discusses proof by contradiction in mathematics. It begins by defining proof by contradiction as proving the truth of a statement by showing that assuming the statement is false leads to a contradiction. The document then provides examples of proofs by contradiction, including:
1) Proving there is no greatest integer by supposing there is a greatest integer N and showing that N+1 would also be an integer, contradicting that N was the greatest.
2) Proving the square root of 2 is irrational by supposing it is rational and showing this leads to a contradiction.
3) Explaining the general steps in a proof by contradiction: assume the statement is false, show this assumption leads to a contradiction, and thus
This document provides an introduction to series and sequences. It discusses key concepts such as convergence vs divergence of sequences and series. Specifically, it defines what an infinite sum is, absolute vs conditional convergence, and common tests to determine convergence like the integral test and alternating series test. It also introduces the big-O, little-o, and big-Theta notations used to compare the growth or decay rates of sequences. The overall goal is to establish an understanding of convergence of series before delving into numerical analysis applications that make use of infinite series like Taylor and Fourier series.
Feynman diagrams are pictorial representations of particle reaction amplitudes. They allow calculations of rates and cross sections for physical processes like muon decay or electron-positron scattering to be greatly simplified. Each diagram has a strict mathematical interpretation corresponding to terms in a power series expansion of the reaction amplitude. Diagrams become more complex at higher orders but must be combined correctly while respecting conservation laws and process symmetries to obtain the total amplitude. The anomalous magnetic moment of particles like the electron and muon can be calculated order-by-order using Feynman diagrams, with remarkable agreement between theory and precise measurements.
The document discusses analyzing the average case runtime of quicksort. It shows that:
1) The average case runtime of quicksort is O(n log n) due to partitions being randomly balanced on average.
2) This is proved rigorously by modeling quicksort as alternating best and worst case partitions, showing the cost is absorbed by subsequent partitions.
3) A recurrence is written and solved to formally prove the average case runtime is O(n log n).
This document provides an introduction and solutions to problems in Modern Particle Physics. It contains 18 chapters covering topics in particle physics like the Dirac equation, decay rates, scattering processes, and the Standard Model. The preface explains that the guide gives numerical solutions and hints to help understand questions from the first edition of the textbook. Instructors can obtain fully worked solutions from the publisher.
This document introduces mathematical induction. It defines the principle of mathematical induction as having two steps: (1) the basis step, which shows a statement P(1) is true, and (2) the inductive step, which assumes P(k) is true and shows P(k+1) is also true. It provides an example of climbing an infinite ladder to illustrate these steps. It also notes some important points about mathematical induction, such as that it is expressing a rule of inference and in proofs we show P(k) implies P(k+1) rather than assuming P(k) is true for all k.
The document discusses propositions, logical operations, predicates, quantification, and mathematical induction. It provides:
1) Definitions of predicates, propositions, universal and existential quantification, and the principle of mathematical induction.
2) Examples of applying predicates, quantification, and induction to prove statements about integers and sums.
3) The process of proving statements by mathematical induction, which involves showing the basis step and inductive step. It also introduces strong mathematical induction.
This document introduces the concept of order of an element modulo n and uses it to prove theorems about when an integer n satisfies n^2 + 1 or more generally satisfies a cyclotomic polynomial modulo a prime p. It begins by stating and proving the n^2 + 1 lemma, which says a prime p satisfies p | n^2 + 1 if and only if p ≡ 1 (mod 4). It introduces the concepts of order, primitive roots, and cyclotomic polynomials to generalize this result. It concludes by stating and proving a theorem about when a cyclotomic polynomial of an integer a is divisible by a prime p.
The document discusses proofs by contraposition. It explains that a statement of the form "if p then q" can be proven by showing its contrapositive "if not q then not p" is true. It provides examples of proofs using this method, including proving if n^2 is even then n is even, if m + n is even then m and n have the same parity, and if 3n + 2 is odd then n is odd. Homework exercises are provided applying this proof technique.
This document provides a summary of a lecture on graphs and algorithms. It discusses Hall's Theorem and provides a proof using induction. It then gives several examples of how Hall's Theorem can be applied, such as scheduling tasks among machines. The document also discusses maximum flows, the Konig-Egervary Theorem relating matchings and vertex covers, and Menger's Theorem relating disjoint paths and minimum cuts.
This document explores limiting the size of topological spaces through cardinal invariants and Arhangel'skii's Theorem. It begins by introducing set theory concepts like cardinals and ordinals. It then discusses topological spaces formed by putting the order topology on ordinals, called ordinal spaces. Finally, it covers cardinal invariants, which place bounds on the size of topological spaces, and proves a particular case of Arhangel'skii's Theorem, which showed that compact, first-countable spaces have at most the cardinality of the reals. The goal is to understand Arhangel'skii's novel "closing off" proof technique for bounding cardinalities of topological spaces.
Introduction to set theory by william a r weiss professormanrak
This chapter introduces a formal language for describing sets using variables, logical connectives, quantifiers, and the membership symbol. Formulas in this language are constructed recursively from atomic formulas using negation, conjunction, disjunction, implication, biconditional, universal quantification, and existential quantification. The key concepts of subformula and bound variable are also defined. This language will allow precise discussion of sets without ambiguities like those found in natural languages.
The document discusses the pigeonhole principle, countability, and cardinality. The pigeonhole principle states that if n items are placed into m pigeonholes with n > m, then at least one pigeonhole must contain more than one item. Countability refers to sets being either finite or denumerable (having the same cardinality as the natural numbers). Cardinality compares the sizes of sets based on whether a bijection exists between them. The document provides examples and proofs of these concepts.
The document summarizes research on accurate and total accurate dominating sets of interval graphs. It begins by introducing interval graphs and defining accurate and total accurate dominating sets. It then presents an algorithm to construct a minimum dominating set of an interval graph. Several theorems are provided about when the constructed dominating set is or is not an accurate dominating set or total accurate dominating set based on properties of the interval family and dominating set. The document focuses on characterizing accurate dominating sets of interval graphs.
A dominating set is a split dominating
set in . If the induced subgraph is
disconnected in The split domination number of
is denoted by , is the minimum cardinality of
a split dominating set in . In this paper, some results on
were obtained in terms of vertices, blocks, and other
different parameters of but not members of
Further, we develop its relationship with other different
domination parameters of
It gives me great comfort to visualize this universe as the surface of an ever expanding four-dimensional sphere originating from a distant, but finite, past and growing indefinitely for ever. In this idealized model it easy to calculate the age of the universe by observing the velocity of the receding stars and also to make several other interesting conclusions. For more details, continue reading the presentation.
This document discusses linear, abelian, and continuous groups and how relaxing these properties leads to more complex groups. It begins with the simplest group, the real numbers R, and progresses to integer lattices Z and Z^n, then non-abelian Lie groups like SL(n,R). Lattices in these groups like SL(n,Z) are discussed, along with properties like the congruence subgroup property. Open questions are raised regarding the irreducibility of random matrices and deciding membership in subgroups of SL(n,Z).
Archimedean principle of real numbers.pptxNallaaadmi
The document discusses the Archimedean principle of real numbers. It states that the Archimedean principle rules out the possibility of infinitesimal distances that are so small that no finite amount can exceed any finite length. It also means that there are no infinite elements in the real number line. The document proves that the natural numbers are not bounded above in the real number line using contradiction. It explains that the Archimedean principle asserts the finite nature of elements on the real line and is useful for confirming the limits that sequences converge to.
This document provides a simplified proof of Kruskal's tree theorem. It begins with preliminaries that define
well-founded orders, homeomorphic embedding, and well-partial orders. It then states Kruskal's theorem for finite
signatures and proves the general version of the theorem. The proof uses two lemmas and shows that any infinite
sequence of terms contains indices such that the terms are related by homeomorphic embedding. This implies
homeomorphic embedding is a well-partial order. The document concludes that simplification orders are
well-founded over finite signatures due to Kruskal's theorem.
This document provides a simplified proof of Kruskal's tree theorem. It begins with preliminaries that define concepts like well-founded orders, homeomorphic embedding, and well-partial orders. It then states Kruskal's theorem for finite terms and the general version. The main part of the document proves the general version of Kruskal's theorem by constructing a minimal bad sequence and reaching a contradiction. It concludes that every simplification order over a finite signature is well-founded, which has applications in proving termination of term rewriting systems.
Cantor and Infinity
- Galileo paradoxically showed that the set of natural numbers and the set of squares both have the same infinite size, despite the squares being a proper subset.
- Cantor resolved this by developing a theory of one-to-one correspondences between sets to formally define their cardinalities (sizes).
- Using this, he proved that the set of real numbers is a higher level of infinite size than the natural numbers or integers, denoted aleph-null and aleph-one. However, whether other infinities exist between these is undecidable.
This document summarizes a proof that the prime numbers contain arbitrarily long arithmetic progressions. The proof has three main ingredients: 1) Szemerédi's theorem, which states that any subset of integers with positive density contains long arithmetic progressions; 2) a transference principle showing that pseudorandom sets also contain long progressions; 3) a result of Goldston and Yıldırım showing that almost all primes lie in a pseudorandom set. The document outlines how these ingredients can be combined to prove the main theorems that prime numbers contain arbitrarily long progressions and any subset of primes with positive density contains long progressions.
This document discusses infinite sequences and series. It defines sequences as lists of numbers written in a definite order, with each term an corresponding to the nth term. A series is defined as the sum of the terms of an infinite sequence. The document introduces ways to determine if a sequence or series converges, such as using the limit of the partial sums. It presents the integral test as a way to use integrals to determine if series converge or diverge, and discusses how to estimate the sum of a convergent series using integrals and remainders.
This research article introduces the concept of rough statistical convergence for triple sequences of real-valued functions defined using a Musielak-Orlicz function. Triple sequences are functions from the natural numbers cubed to the real numbers. Rough statistical convergence generalizes both rough convergence and statistical convergence. The paper defines pointwise rough statistical convergence and rough statistically Cauchy sequences of triple sequences. It proves some inclusion results and establishes an equivalence between pointwise rough statistical convergence and being a rough statistically Cauchy sequence. The concepts are aimed to unify different notions of convergence for triple sequences.
This research article introduces the concept of rough statistical convergence for triple sequences of real-valued functions defined using a Musielak-Orlicz function metric. Triple sequences are functions with three indices, where the indices range over the natural numbers. The paper defines pointwise rough statistical convergence and rough statistically Cauchy sequences for triple sequences. Two main theorems are proved: 1) If two triple sequence spaces satisfy certain inclusion properties, then their pointwise rough statistical convergence is identical. 2) A triple sequence is pointwise roughly statistically convergent if and only if it is a rough statistically Cauchy sequence if and only if certain limit properties hold. The concepts generalize previous notions of rough convergence and rough statistical convergence for triple sequences.
IRJET- Domination and Total Domination in an Undirected GraphIRJET Journal
1. The document discusses domination and total domination in an undirected graph where the vertices are natural numbers less than m that are not divisible by m.
2. It determines the minimal dominating sets, domination number, and total domination number for this graph.
3. Key results include that the domination number is bounded above by the number of vertices, and equals the number of vertices when m is even and n is odd, making the graph complete. The total domination number also equals the number of vertices in this case.
This document claims to provide a proof of the Goldbach Conjecture using a "string of beads" model and high school level mathematics. It introduces the conjecture and establishes two lemmas: 1) Every even number greater than 14 is the sum of two numbers of the forms 3n and 3n+1, and 2) Every even number greater than 2 is represented by a string of beads with at least one "cluster" containing more than two prime numbers. It then states the theorem that every even number greater than 4 can be expressed as the sum of two odd prime numbers, proving the Goldbach Conjecture.
A Proof Of The Conjecture Due To Parisi For The Finite Random Assignment ProblemKate Campbell
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A Proof Of The Conjecture Due To Parisi For The Finite Random Assignment Problem
IJCER (www.ijceronline.com) International Journal of computational Engineering research
1. International Journal Of Computational Engineering Research (ijceronline.com) Vol. 2 Issue. 7
Matching Dominating Sets of Euler Totient Cayley Graphs
1,
M.Manjuri,2,B.Maheswari
1,2,
Department of Applied Mathematics, Sri Padmavati Women’s University,
Tirupati - 517502, Andhra Pradesh, India.
Abstract
Graph Theory has been realized as one of the most useful branches of Mathematics of recent origin, finding
widest applications in all most all branches of sciences, social sciences, computer science and engineering. Nathanson[3]
paved the way for the emergence of a new class of graphs, namely,Arithmetic Graphs by introducing the concepts of
Number Theory, particularly, the Theory of congruences in Graph Theory. Cayley graphs are another class of graphs
associated with the elements of a group. If this group is associated with some arithmetic function then the Cayley graph
becomes an arithmetic graph. The Cayley graph associated with Euler Totient function is called an Euler Totient Cayley
graph and in this paper we study the matching domination parameters of Euler Totient Cayley graphs.
Keywords: Euler Totient Cayley Graph, Matching Dominating sets.
1. Introduction
Cayley graph
Let be a group and a symmetric subset of not containing the identity element e of The
graph G whose vertex set and edge set E = is called the Cayley graph of X corresponding to
the set and it is denoted by Madhavi [2] introduced the concept of Euler totient Cayley graphs and studied
some of its properties.
Euler Totient Cayley Graph
For each positive integer n, let be the additive group of integers modulo and be the set of all numbers
less than and relatively prime to . The Euler totient Cayley graph is defined as the graph whose vertex set V
is given by and the edge set is given by Clearly as
proved by Madhavi [2], the Euler totient Cayley graph is
1. a connected, simple and undirected graph,
2. ( ) - regular and has edges,
3. Hamiltonian,
4. Eulerian for
5. bipartite if is even and
6. complete graph if is a prime.
2. Matching Dominating Sets of Euler Totient Cayley Graphs
The theory of domination in Graphs introduced by Ore [4] and Berge [1] is an emerging area of research today.
The domination parameters of Euler Totient Cayley graphs are studied by Uma Maheswari [6] and we present some of the
results without proofs and can be found in [5].
Theorem 2.1: If is a prime, then the domination number of is 1.
Theorem 2.2: If is power of a prime, then the domination number of is 2.
Theorem 2.3: The domination number of is 2, if where is an odd prime.
Theorem 2.4: Suppose is neither a prime nor Let , where , , … are primes and
are integers ≥ 1. Then the domination number of is given by where is
the length of the longest stretch of consecutive integers in , each of which shares a prime factor with A matching in a
graph is a subset of edges of such that no two edges in are adjacent. A matching in is called a
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2. International Journal Of Computational Engineering Research (ijceronline.com) Vol. 2 Issue. 7
perfect matching if every vertex of is incident with some edge in Let G (V, E) be a graph. A subset of is said
to be a dominating set of if every vertex in is adjacent to a vertex in . The minimum cardinality of a
dominating set is called the domination number of and is denoted by A dominating set of is said to be a
matching dominating set if the induced subgraph admits a perfect matching. The cardinality of the smallest matching
dominating set is called the matching domination number and is denoted by
Theorem 1: The matching domination number of is 2, if is a prime.
Proof: Let be a prime. Then is a complete graph. It is clear that is a minimal dominating set as it
dominates all other vertices of Therefore . For any
vertex is adjacent to vertex So if then the induced subgraph admits a perfect matching with
minimum cardinality. Hence is a minimal matching dominating set of Therefore ■
Theorem 2: If is power of a prime, then the matching domination number of is 2.
Proof: Consider for where is a prime. Then the vertex set of is given
by . This set falls into disjoint subsets as below.
1. The set of integers relatively prime to
2. The set M of multiples of
3. Singleton set .Let where Then becomes a minimum
dominating set of as in Theorem 2.2. Since GCD the vertices and are adjacent. This gives
that admits a perfect matching. So is a matching dominating set of of minimum cardinality. Hence it
follows that .■
Theorem 3: The matching domination number of is 4 if , where is an odd prime.
Proof: Let us consider the Euler totient Cayley graph for , p is an odd prime. Then the vertex
set falls into the following disjoint subsets.
1. The set of odd numbers which are less than and relatively prime to
2. The set of non - zero even numbers,
3. The set of numbers 0 and
Then becomes a minimum dominating set of as in Theorem 2.3. Further the vertices in are
non-adjacent because GCD This gives that does not admit a perfect matching. So is not a matching
dominating set of In order that admits a perfect matching, we need to add at least two vertices adjacent to
each of the vertex in so that .Let . Then This
implies that is a dominating set of Moreover, since GCD , vertex is adjacent to vertex and
vertex is adjacent to vertex . Further we have . Hence and are non-adjacent and similarly
and are non-adjacent. So admits a perfect matching. Hence is a matching dominating set of minimum
cardinality.Thus .■
Theorem 4: Let be neither a prime nor and , where , , … are primes and
are integers ≥ 1. Then the matching domination number of is given by
Where is the length of the longest stretch of consecutive integers in V each of which shares a prime factor with
Proof: Let us consider for where is neither a prime nor The vertex set V of
is given by . Then the set V falls into disjoint subsets as follows.
1. The set of integers relatively prime to ,
2. The set where is a collection of consecutive positive integers such that for every in ,
GCD
3. The singleton set .
Let be the largest set in with maximum cardinality Suppose
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3. International Journal Of Computational Engineering Research (ijceronline.com) Vol. 2 Issue. 7
, where GCD for Then is a dominating
set of minimum cardinality, as in Theorem 2.4. Now two cases arise.
Case 1: Suppose is an odd number.
Consider the set Each pair , is an edge of as
So is a set of edges in Obviously, no two edges in are adjacent. So
admits a perfect matching. Hence becomes a matching dominating set of Since we
have As it follows that is a minimum matching dominating set of
Therefore , if is an odd number.
Case 2: Suppose is an even number.
Let = . Since it follows that is a dominating set of . Consider the set
As each pair , is an edge of .
So is a set of edges in Again it can be seen that no two edges in are adjacent. So admits a perfect
matching. Hence is a matching dominating set of . Now is an even number implies that is an odd
number. Since the matching domination number is always even it follows that . Therefore is
a minimal matching dominating set of Hence , if is an even number. ■
3. Illustrations
0
10 1
9 2
8 3
7 4
6 5
Matching Dominating Set = {0, 1}
0
13 1
12 2
11 3
10 4
9 5
8 6
7
Matching Dominating Set = {0, 1, 7, 8}
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4. International Journal Of Computational Engineering Research (ijceronline.com) Vol. 2 Issue. 7
24 0 1
23 2
22 3
21 4
20 5
19 6
18 7
17 8
16 9
15 10
14 11
13 12
Matching Dominating Set = {0, 1}
0
28 29 1 2
27 3
26 4
25 5
24 6
23 7
22 8
21 9
20 10
19 11
18 12
17 16 13
15 14
Matching Dominating Set = {0, 1, 2, 3, 4, 5}
References
1. Berge, C. - The Theory of Graphs and its Applications, Methuen, London (1962).
2. Madhavi, L. - Studies on domination parameters and enumeration of cycles in some Arithmetic Graphs, Ph. D.
Thesis submitted to S.V.University, Tirupati, India, (2002).
3. Nathanson and Melvyn B. - Connected components of arithmetic graphs,Monat.fur.Math, 29,219 – 220 (1980).
4. Ore, O. - Theory of Graphs, Amer. Math. Soc. Colloq. Publ., 38, Providence, (1962).
5. Uma Maheswari, S and Maheswari, B. – Domination parameters of Euler Totient Cayley Graphs,
Rev.Bull.Cal.Math.Soc. 19, (2), 207-214(2011).
6. Uma Maheswari, S. - Some Studies on the Product Graphs of Euler Totient Cayley Graphs and Arithmetic Graphs,
Ph. D. Thesis submitted to S.P.Women’s University, Tirupati, India, (2012).
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