The document discusses enumeration and classification of musical lines in music theory. It defines key terms like tones, intervals, chords, scales and lines. It proves that there are 13,699 possible k-note lines for the diatonic set of 7 notes, using a counting argument. These lines can be classified into genera based on their diatonic length arrays. There are 1,957 possible genera that the lines can be divided into, as shown through an analysis of integer partitions.
1. The document discusses regular expressions (RE), including the recursive definition of RE, using RE to define languages, and examples of RE for various languages.
2. Key examples of RE include x* for the language of strings over {x} including the empty string, x+ for strings over {x} excluding the empty string, (a+b)* for all strings over {a,b}, and b*aab* for strings with exactly one aa.
3. The document also discusses how a single language can have multiple equivalent REs but a RE defines a unique language, and provides additional examples of REs for languages with certain characteristics like beginning/ending with a letter.
This document provides an overview of basic discrete mathematical structures including sets, functions, sequences, sums, and matrices. It begins by defining a set as an unordered collection of elements and describes various ways to represent sets, such as listing elements or using set-builder notation. It then discusses operations on sets like unions, intersections, complements, and Cartesian products. Finally, it introduces functions as assignments of elements from one set to another. The document serves as an introduction to fundamental discrete structures used throughout mathematics.
This document defines and provides examples of key concepts in set theory including:
- Sets are collections of distinct elements that can be represented using curly brackets and do not consider order (e.g. {1, 2, 3} = {3, 2, 1}).
- Subsets, unions, intersections and complements are defined along with their symbols (⊂, ∪, ∩, ).
- The number of possible subsets of a set of size n is 2n.
- A Venn diagram is used to represent a survey where subsets show those who own cats, dogs, or both.
The document discusses a thesis project that aimed to prove the non-negativity of the γ-vector for 3-dimensional polytopes. It provides background on polytopes, defining terms like faces and f-vectors. It then introduces the α-vector, which records the interior angle sums of a polytope, and the γ-vector, which is a linear transformation of the α-vector. The project focused on pyramids and prisms, proving properties of their γ-vectors like non-negativity and relationships between γ-vector entries and the polygon bases.
This document provides examples and explanations for proving triangle congruence using the Side-Side-Side (SSS) and Side-Angle-Side (SAS) postulates. It begins with definitions of key vocabulary like included angle. Example 1 uses SSS to prove two triangles congruent by showing that corresponding sides are congruent. Example 2 has students graph triangles on a coordinate plane and determine if they are congruent. Example 3 uses the midpoint theorem and vertical angles theorem to prove triangles congruent via SAS. The document concludes with practice problems for students.
This document summarizes a student paper about Ramsey theory and Ramsey numbers. It begins with an abstract, introduction, and definitions of key graph theory terms. It then defines the Ramsey number R(l1, l2, ...) as the smallest number n such that any graph of n vertices contains a complete subgraph of color 1 with l1 vertices or a complete subgraph of color 2 with l2 vertices, etc. It provides an abridged proof of Ramsey's theorem that the Ramsey number is well-defined for all natural numbers l1, l2, etc. It uses this to solve graph coloring problems like the minimum party size such that some people all know each other or all don't know each other.
The document discusses determining the angle between two lines and finding the point of intersection between two lines in 3D space. It provides the equations and process for finding the angle between two lines given their direction ratios. It also outlines the steps to find the point of intersection, which involves setting the coordinates of a point on each line equal to determine values for lambda and mu, and substituting those values back into one of the line equations. An example problem demonstrates finding the intersection point of two lines.
This document provides an introduction to the concept of sets. It begins by listing the learning objectives, which are to describe well defined sets and their properties like subsets, universal sets, and cardinality. It then provides examples of sets defined by listing, verbal description, and set builder notation. It explores subsets, universal sets, Venn diagrams, and basic set operations like intersection, union, difference, and complements. The goal is for students to understand how to represent and describe sets and their relationships, as well as solve problems involving sets.
1. The document discusses regular expressions (RE), including the recursive definition of RE, using RE to define languages, and examples of RE for various languages.
2. Key examples of RE include x* for the language of strings over {x} including the empty string, x+ for strings over {x} excluding the empty string, (a+b)* for all strings over {a,b}, and b*aab* for strings with exactly one aa.
3. The document also discusses how a single language can have multiple equivalent REs but a RE defines a unique language, and provides additional examples of REs for languages with certain characteristics like beginning/ending with a letter.
This document provides an overview of basic discrete mathematical structures including sets, functions, sequences, sums, and matrices. It begins by defining a set as an unordered collection of elements and describes various ways to represent sets, such as listing elements or using set-builder notation. It then discusses operations on sets like unions, intersections, complements, and Cartesian products. Finally, it introduces functions as assignments of elements from one set to another. The document serves as an introduction to fundamental discrete structures used throughout mathematics.
This document defines and provides examples of key concepts in set theory including:
- Sets are collections of distinct elements that can be represented using curly brackets and do not consider order (e.g. {1, 2, 3} = {3, 2, 1}).
- Subsets, unions, intersections and complements are defined along with their symbols (⊂, ∪, ∩, ).
- The number of possible subsets of a set of size n is 2n.
- A Venn diagram is used to represent a survey where subsets show those who own cats, dogs, or both.
The document discusses a thesis project that aimed to prove the non-negativity of the γ-vector for 3-dimensional polytopes. It provides background on polytopes, defining terms like faces and f-vectors. It then introduces the α-vector, which records the interior angle sums of a polytope, and the γ-vector, which is a linear transformation of the α-vector. The project focused on pyramids and prisms, proving properties of their γ-vectors like non-negativity and relationships between γ-vector entries and the polygon bases.
This document provides examples and explanations for proving triangle congruence using the Side-Side-Side (SSS) and Side-Angle-Side (SAS) postulates. It begins with definitions of key vocabulary like included angle. Example 1 uses SSS to prove two triangles congruent by showing that corresponding sides are congruent. Example 2 has students graph triangles on a coordinate plane and determine if they are congruent. Example 3 uses the midpoint theorem and vertical angles theorem to prove triangles congruent via SAS. The document concludes with practice problems for students.
This document summarizes a student paper about Ramsey theory and Ramsey numbers. It begins with an abstract, introduction, and definitions of key graph theory terms. It then defines the Ramsey number R(l1, l2, ...) as the smallest number n such that any graph of n vertices contains a complete subgraph of color 1 with l1 vertices or a complete subgraph of color 2 with l2 vertices, etc. It provides an abridged proof of Ramsey's theorem that the Ramsey number is well-defined for all natural numbers l1, l2, etc. It uses this to solve graph coloring problems like the minimum party size such that some people all know each other or all don't know each other.
The document discusses determining the angle between two lines and finding the point of intersection between two lines in 3D space. It provides the equations and process for finding the angle between two lines given their direction ratios. It also outlines the steps to find the point of intersection, which involves setting the coordinates of a point on each line equal to determine values for lambda and mu, and substituting those values back into one of the line equations. An example problem demonstrates finding the intersection point of two lines.
This document provides an introduction to the concept of sets. It begins by listing the learning objectives, which are to describe well defined sets and their properties like subsets, universal sets, and cardinality. It then provides examples of sets defined by listing, verbal description, and set builder notation. It explores subsets, universal sets, Venn diagrams, and basic set operations like intersection, union, difference, and complements. The goal is for students to understand how to represent and describe sets and their relationships, as well as solve problems involving sets.
The document discusses sequences and series, including sigma notation, arithmetic sequences and series, and geometric sequences and series. It defines sequences, finite and infinite sequences, and series. It provides the formulas for the nth term and nth partial sum of arithmetic and geometric sequences. It also gives examples of applying arithmetic and geometric series to problems involving patterns, sums, and compound interest.
Basic Concept of discrete math. I discuss a few topic of basic discrete math. I think you get a clear concept. Topics are definition,story of discrete math, Importance of discrete math, and many other basic topic.
This document discusses simplifying expressions involving surds, which are irrational numbers that cannot be written as fractions such as √2. It provides techniques for manipulating surds, including:
1) Simplifying expressions using properties of square roots and rationalizing denominators with surds to remove irrational terms;
2) Using the difference of squares formula to expand expressions like (a + b)(a - b);
3) Rationalizing fractions with surds in the denominator to obtain equivalent fractions with whole number denominators.
The document discusses Lie algebras, which are vector spaces with a non-associative multiplication called the Lie bracket. Any Lie group gives rise to a Lie algebra, and vice versa. Lie algebras allow the study of Lie groups in terms of vector spaces. A Lie subalgebra is a vector subspace of a Lie algebra that is closed under the Lie bracket, while an ideal is a subspace where the Lie bracket of any element of the Lie algebra with an element of the ideal is also in the ideal. Examples of Lie algebras and their substructures are provided.
De Morgan's Laws Proof and real world application.
De Morgan's Laws are transformational Rules for 2 Sets
1) Complement of the Union Equals the Intersection of the Complements
not (A or B) = not A and not B
2) Complement of the Intersection Equals the Union of the Complements
not (A and B) = not A or not B
Take 2 Sets A and B
Union = A U B ← Everything in A or B
Intersection = A ∩ B ← Everything in A and B
U = Universal Set (All possible elements in your defined universe)
Complement = A’ Everything not in A, but in the Universal Set
The document discusses properties of rectangles. It defines a rectangle as a parallelogram with four right angles. It lists properties such as opposite sides being parallel and congruent and opposite angles being congruent. The document also presents four examples solving problems about rectangles, finding missing side lengths and determining if a quadrilateral is a rectangle based on given information.
This document discusses properties of parallelograms. It defines a parallelogram as a quadrilateral with two pairs of parallel sides. The document lists properties of parallelograms, including: opposite sides are congruent; opposite angles are congruent; consecutive angles are supplementary; if one angle is a right angle, all angles are right angles. It also states that the diagonals of a parallelogram bisect each other and divide the parallelogram into two congruent triangles. Examples demonstrate using these properties to find missing measures and intersection points of diagonals.
This document discusses distance and midpoints between points in a coordinate plane. It defines distance as the length of a segment between two points and the Pythagorean theorem. The midpoint of a segment is the point halfway between the two endpoints. Examples are provided to demonstrate calculating distance and midpoints using formulas like the distance formula and midpoint formula.
1. The document discusses regular expressions (RE), including the recursive definition of RE, using RE to define languages, and examples of RE for various languages.
2. Key examples of RE include x* for the language of strings over {x} including the empty string, x+ for strings over {x} excluding the empty string, (a+b)* for all strings over {a,b}, and b*aab* for strings with exactly one occurrence of aa.
3. The document notes that a language can have multiple equivalent REs while a RE uniquely defines a language, and provides additional examples of RE for languages based on start/end conditions.
This document discusses the key concepts from several units in mathematics including integers, groups, finite groups, subgroups, and groups in coding theory. It then provides details on specific topics within these units, including equivalence relations, congruence relations, equivalence class partitions, the division algorithm, greatest common divisors (GCD) using division, and Euclid's lemma. The document aims to provide students with fundamental mathematical principles, methods, and tools to model, solve, and interpret a variety of problems. It also discusses enhancing students' development, problem solving skills, communication, and attitude towards mathematics.
A common fixed point theorems in menger space using occationally weakly compa...Alexander Decker
1. The document presents a theorem that proves the existence and uniqueness of a common fixed point for occasionally weakly compatible self-mappings in a Menger space.
2. It defines key concepts such as Menger space, probabilistic metric space, t-norm, occasionally weakly compatible mappings, and implicit relations.
3. The theorem shows that if four self-mappings satisfy the conditions of being occasionally weakly compatible and an implicit relation, then they have a unique common fixed point in the Menger space.
The document describes an algorithm for solving the Steiner tree problem parameterized by treewidth. The algorithm works by maintaining, for each bag of the tree decomposition, the number of solutions of size at most k that intersect the bag in various ways and have been assigned particular colors. It provides update rules for introducing, forgetting, and joining bags to combine local solutions into a global one. The algorithm isolates a particular solution by randomly assigning weights to elements and focusing on solutions of a certain total weight.
Welcome to International Journal of Engineering Research and Development (IJERD)IJERD Editor
call for paper 2012, hard copy of journal, research paper publishing, where to publish research paper,
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal,
This document defines key terms related to congruent triangles such as congruent, congruent polygons, and corresponding parts. It also defines the Third Angle Theorem. Examples are provided to demonstrate how to prove triangles are congruent by identifying corresponding congruent parts and writing congruence statements. The document also includes multi-step proofs involving congruent triangles.
This document discusses several key concepts in formal language theory including Kleene star closure, plus operation, and recursive language definitions. It provides examples of recursively defining several common languages like INTEGER, EVEN, and PALINDROME. It also solves problems involving determining if strings are members of Kleene star closures and comparing languages generated by different sets of strings.
This document discusses various topics related to formal languages including:
- Kleene star closure, which defines the set of all possible strings over a given alphabet including the empty string.
- Plus operation, which defines the set of all non-empty strings over an alphabet.
- Recursive definition of languages, which specifies basic strings, rules to generate new strings, and only allows strings generated by the rules.
- Examples of recursively defined languages such as INTEGER, EVEN, and factorial.
This document defines and provides examples of set operations including union, intersection, complement, and difference. It discusses set cardinality and set identities. Examples are provided to illustrate membership tables and Venn diagrams for sets of up to 3 elements. Methods for proving set identities such as using subsets, logic, or membership tables are also described.
This document discusses proving triangle congruence using the Side-Side-Side (SSS) and Side-Angle-Side (SAS) postulates. It begins with essential questions and vocabulary definitions, then provides examples of using SSS and SAS to prove triangles congruent. The examples illustrate setting up a logical proof by listing corresponding congruent sides and angles of the triangles based on the information given.
This document summarizes a chapter on Boolean algebra from a 2004 textbook. It includes objectives, study guides, sections on multiplying and factoring expressions, exclusive-OR and equivalence operations, the consensus theorem, and algebraic simplification of switching expressions. The chapter provides examples and exercises to illustrate different theorems and methods for manipulating Boolean expressions algebraically.
The document provides an analysis of Asian Paints' Royale Luxury Emulsion product using the 4 P's of marketing framework. It discusses the product as a premium, interior wall emulsion paint that provides a soft sheen finish. It then examines the price, which uses value-based pricing between Rs. 315-345 per liter, the extensive distribution network as the place, and the promotion strategies of celebrity endorsements, print/TV advertisements, dealer incentives and events. The analysis aims to understand Asian Paints' marketing approach for this luxury emulsion product.
This document provides information about three forms of traditional Asian theater: Japanese Kabuki theater, Chinese Peking Opera, and Indonesian Wayang Kulit shadow puppet theater. It discusses the musical elements, vocal techniques, instruments, and performance conventions of each form. Kabuki features vocal patterns like ipponchōshi and uses instruments like the shamisen. Peking Opera emphasizes meaning over accuracy and uses instruments like the jinghu and suona horn. Wayang Kulit uses shadow puppets and gamelan music to tell stories from Hindu epics.
The document discusses sequences and series, including sigma notation, arithmetic sequences and series, and geometric sequences and series. It defines sequences, finite and infinite sequences, and series. It provides the formulas for the nth term and nth partial sum of arithmetic and geometric sequences. It also gives examples of applying arithmetic and geometric series to problems involving patterns, sums, and compound interest.
Basic Concept of discrete math. I discuss a few topic of basic discrete math. I think you get a clear concept. Topics are definition,story of discrete math, Importance of discrete math, and many other basic topic.
This document discusses simplifying expressions involving surds, which are irrational numbers that cannot be written as fractions such as √2. It provides techniques for manipulating surds, including:
1) Simplifying expressions using properties of square roots and rationalizing denominators with surds to remove irrational terms;
2) Using the difference of squares formula to expand expressions like (a + b)(a - b);
3) Rationalizing fractions with surds in the denominator to obtain equivalent fractions with whole number denominators.
The document discusses Lie algebras, which are vector spaces with a non-associative multiplication called the Lie bracket. Any Lie group gives rise to a Lie algebra, and vice versa. Lie algebras allow the study of Lie groups in terms of vector spaces. A Lie subalgebra is a vector subspace of a Lie algebra that is closed under the Lie bracket, while an ideal is a subspace where the Lie bracket of any element of the Lie algebra with an element of the ideal is also in the ideal. Examples of Lie algebras and their substructures are provided.
De Morgan's Laws Proof and real world application.
De Morgan's Laws are transformational Rules for 2 Sets
1) Complement of the Union Equals the Intersection of the Complements
not (A or B) = not A and not B
2) Complement of the Intersection Equals the Union of the Complements
not (A and B) = not A or not B
Take 2 Sets A and B
Union = A U B ← Everything in A or B
Intersection = A ∩ B ← Everything in A and B
U = Universal Set (All possible elements in your defined universe)
Complement = A’ Everything not in A, but in the Universal Set
The document discusses properties of rectangles. It defines a rectangle as a parallelogram with four right angles. It lists properties such as opposite sides being parallel and congruent and opposite angles being congruent. The document also presents four examples solving problems about rectangles, finding missing side lengths and determining if a quadrilateral is a rectangle based on given information.
This document discusses properties of parallelograms. It defines a parallelogram as a quadrilateral with two pairs of parallel sides. The document lists properties of parallelograms, including: opposite sides are congruent; opposite angles are congruent; consecutive angles are supplementary; if one angle is a right angle, all angles are right angles. It also states that the diagonals of a parallelogram bisect each other and divide the parallelogram into two congruent triangles. Examples demonstrate using these properties to find missing measures and intersection points of diagonals.
This document discusses distance and midpoints between points in a coordinate plane. It defines distance as the length of a segment between two points and the Pythagorean theorem. The midpoint of a segment is the point halfway between the two endpoints. Examples are provided to demonstrate calculating distance and midpoints using formulas like the distance formula and midpoint formula.
1. The document discusses regular expressions (RE), including the recursive definition of RE, using RE to define languages, and examples of RE for various languages.
2. Key examples of RE include x* for the language of strings over {x} including the empty string, x+ for strings over {x} excluding the empty string, (a+b)* for all strings over {a,b}, and b*aab* for strings with exactly one occurrence of aa.
3. The document notes that a language can have multiple equivalent REs while a RE uniquely defines a language, and provides additional examples of RE for languages based on start/end conditions.
This document discusses the key concepts from several units in mathematics including integers, groups, finite groups, subgroups, and groups in coding theory. It then provides details on specific topics within these units, including equivalence relations, congruence relations, equivalence class partitions, the division algorithm, greatest common divisors (GCD) using division, and Euclid's lemma. The document aims to provide students with fundamental mathematical principles, methods, and tools to model, solve, and interpret a variety of problems. It also discusses enhancing students' development, problem solving skills, communication, and attitude towards mathematics.
A common fixed point theorems in menger space using occationally weakly compa...Alexander Decker
1. The document presents a theorem that proves the existence and uniqueness of a common fixed point for occasionally weakly compatible self-mappings in a Menger space.
2. It defines key concepts such as Menger space, probabilistic metric space, t-norm, occasionally weakly compatible mappings, and implicit relations.
3. The theorem shows that if four self-mappings satisfy the conditions of being occasionally weakly compatible and an implicit relation, then they have a unique common fixed point in the Menger space.
The document describes an algorithm for solving the Steiner tree problem parameterized by treewidth. The algorithm works by maintaining, for each bag of the tree decomposition, the number of solutions of size at most k that intersect the bag in various ways and have been assigned particular colors. It provides update rules for introducing, forgetting, and joining bags to combine local solutions into a global one. The algorithm isolates a particular solution by randomly assigning weights to elements and focusing on solutions of a certain total weight.
Welcome to International Journal of Engineering Research and Development (IJERD)IJERD Editor
call for paper 2012, hard copy of journal, research paper publishing, where to publish research paper,
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal,
This document defines key terms related to congruent triangles such as congruent, congruent polygons, and corresponding parts. It also defines the Third Angle Theorem. Examples are provided to demonstrate how to prove triangles are congruent by identifying corresponding congruent parts and writing congruence statements. The document also includes multi-step proofs involving congruent triangles.
This document discusses several key concepts in formal language theory including Kleene star closure, plus operation, and recursive language definitions. It provides examples of recursively defining several common languages like INTEGER, EVEN, and PALINDROME. It also solves problems involving determining if strings are members of Kleene star closures and comparing languages generated by different sets of strings.
This document discusses various topics related to formal languages including:
- Kleene star closure, which defines the set of all possible strings over a given alphabet including the empty string.
- Plus operation, which defines the set of all non-empty strings over an alphabet.
- Recursive definition of languages, which specifies basic strings, rules to generate new strings, and only allows strings generated by the rules.
- Examples of recursively defined languages such as INTEGER, EVEN, and factorial.
This document defines and provides examples of set operations including union, intersection, complement, and difference. It discusses set cardinality and set identities. Examples are provided to illustrate membership tables and Venn diagrams for sets of up to 3 elements. Methods for proving set identities such as using subsets, logic, or membership tables are also described.
This document discusses proving triangle congruence using the Side-Side-Side (SSS) and Side-Angle-Side (SAS) postulates. It begins with essential questions and vocabulary definitions, then provides examples of using SSS and SAS to prove triangles congruent. The examples illustrate setting up a logical proof by listing corresponding congruent sides and angles of the triangles based on the information given.
This document summarizes a chapter on Boolean algebra from a 2004 textbook. It includes objectives, study guides, sections on multiplying and factoring expressions, exclusive-OR and equivalence operations, the consensus theorem, and algebraic simplification of switching expressions. The chapter provides examples and exercises to illustrate different theorems and methods for manipulating Boolean expressions algebraically.
The document provides an analysis of Asian Paints' Royale Luxury Emulsion product using the 4 P's of marketing framework. It discusses the product as a premium, interior wall emulsion paint that provides a soft sheen finish. It then examines the price, which uses value-based pricing between Rs. 315-345 per liter, the extensive distribution network as the place, and the promotion strategies of celebrity endorsements, print/TV advertisements, dealer incentives and events. The analysis aims to understand Asian Paints' marketing approach for this luxury emulsion product.
This document provides information about three forms of traditional Asian theater: Japanese Kabuki theater, Chinese Peking Opera, and Indonesian Wayang Kulit shadow puppet theater. It discusses the musical elements, vocal techniques, instruments, and performance conventions of each form. Kabuki features vocal patterns like ipponchōshi and uses instruments like the shamisen. Peking Opera emphasizes meaning over accuracy and uses instruments like the jinghu and suona horn. Wayang Kulit uses shadow puppets and gamelan music to tell stories from Hindu epics.
The document discusses seven critical thinking skills: analyzing, applying standards, discriminating, information seeking, logical reasoning, predicting, and transforming knowledge. For each skill, it provides activity statements to describe that skill in more detail. It also cites two sources that discuss critical thinking habits and how to teach critical thinking.
La OMS define droga como toda sustancia que produce una alteración del sistema nervioso central y puede crear dependencia psicológica, física o ambas. Se clasifican en drogas naturales, semisintéticas y sintéticas. El consumo de drogas puede generar trastornos fisiológicos y psicológicos, afectar las relaciones personales y el rendimiento, y tener consecuencias sociales y económicas. El tratamiento incluye centros de rehabilitación, salud mental y grupos de apoyo.
Este documento trata sobre la historia y evolución del caballo, así como sobre su clasificación zoológica y los diferentes sistemas de producción relacionados con ellos. Explica la evolución del caballo a través de los años, desde Eohippus hasta Equus, y su llegada a América. También cubre la clasificación de especies de caballos y híbridos, y los diferentes sistemas de producción como explotaciones de cría, para práctica ecuestre, aptitud cárnica y asociadas a actividades agrarias
Una niña llamada Rosita escapó de su maltratadora madrastra y llegó cansada a un pueblo llamado Nueva Esperanza. Al ver una hermosa casita de piedras, pidió permiso para descansar y las puertas se abrieron, revelando que estaba habitada por hadas. Rosita se quedó a vivir con las hadas, quienes cumplían sus deseos con magia. Un día, un malvado mago intentó capturar a las hadas, pero Rosita lo convirtió en un ratón usando su deseo y lo encerró
This document describes a distributed access control solution implemented using the PiFrame framework. The system uses Raspberry Pi devices running PiFrame at each door to control access locally, with a centralized server managing user credentials and logs. The solution provides scalability, continuous operation when offline, and easy development through PiFrame. Future work may incorporate biometrics like face or voice recognition.
O documento discute a evolução da indústria do calçado em Portugal, a importância econômica do setor e apresenta a empresa Green Boots, especializada em calçado sustentável feito de materiais reciclados e naturais.
This document is a policy brief about unsafe water and inadequate sanitation. It discusses how over 800 million people lack access to clean water, despite it being recognized as a basic human right. It outlines the health issues caused by unclean water, including diseases that kill more people than war. While some progress has been made through organizations working to increase access, more efforts are needed from governments and groups collaborating to solve this global problem.
O documento apresenta a Luna Laguna, uma empresa de vendas de roupas no atacado e varejo que oferece uma oportunidade de microfranquia. A empresa não é pirâmide nem multinível e não exige compra de produtos ou pagamento de taxas. Ao invés disso, oferece a possibilidade de economizar nas compras e ganhar dinheiro revendendo produtos de qualidade.
This document contains a summary of Vinod C's career profile. It includes his 3.3 years of experience in software testing with expertise in manual testing, automation testing using UFT/QTP, test planning, test case development, and documentation. It provides details of his education qualifications and areas of expertise. It also summarizes his work experience on two projects - DHL-GEMA and BVP, describing his roles and responsibilities on each project such as requirements analysis, test case preparation, defect tracking, automation testing and reporting.
1. This document outlines a poverty reduction strategy for rural communities in South Africa's Eastern Cape province focused on benefiting from the abundant aloe ferox plant through sustainable harvesting and production of aloe-based goods.
2. The strategy aims to reduce poverty by creating jobs and economic opportunities for rural communities and establishing multisector partnerships between businesses, organizations, government, and community members.
3. If successful, the aloe beneficiation program could substantially grow the local economy and standard of living in rural Eastern Cape areas by developing sustainable enterprises around harvesting, processing, and selling aloe-based products both domestically and internationally.
Cs6660 compiler design november december 2016 Answer keyappasami
The document discusses topics related to compiler design, including:
1) The phases of a compiler include lexical analysis, syntax analysis, semantic analysis, intermediate code generation, code optimization, and code generation. Compiler construction tools help implement these phases.
2) Grouping compiler phases can improve efficiency. Errors can occur in all phases, from syntax errors to type errors.
3) Questions cover topics like symbol tables, finite automata in lexical analysis, parse trees, ambiguity, SLR parsing, syntax directed translations, code generation, and optimization techniques like loop detection.
This document introduces sequences and series, focusing on arithmetic and geometric progressions. It provides:
1) Definitions of sequences, series, and the differences between them. Arithmetic progressions have a common difference between terms, while geometric progressions multiply the previous term by a common ratio.
2) Formulas for calculating terms and sums of arithmetic and geometric progressions. The sum of an arithmetic progression can be expressed in terms of the first term, last term, number of terms, and common difference.
3) Examples of using the formulas to find specific terms and sums of progressions. Practice with these types of problems is important to master the techniques.
This document introduces arithmetic and geometric progressions. It defines a sequence as a set of numbers written in a particular order. A series is the sum of the terms in a sequence. An arithmetic progression is a sequence where each new term is obtained by adding a constant difference to the preceding term. The sum of an arithmetic progression can be found using the formula: the sum of the first n terms is equal to one-half n times the quantity of two times the first term plus (n - 1) times the common difference.
A Quest for Subexponential Time Parameterized Algorithms for Planar-k-Path: F...cseiitgn
The document summarizes a talk on obtaining subexponential time algorithms for NP-hard problems on planar graphs. It discusses using treewidth and tree decompositions to solve problems like 3-coloring in 2O(√n) time on n-vertex planar graphs. It also discusses the exponential time hypothesis and how it implies lower bounds, showing these algorithms are optimal up to constant factors in the exponent. The document outlines several chapters, including using grid minors and bidimensionality to obtain 2O(√k) algorithms for problems like k-path, even for some W[1]-hard problems parameterized by k.
The document discusses graph theory and provides definitions and examples of various graph concepts. It defines what a graph is consisting of vertices and edges. It also defines different types of graphs such as simple graphs, multigraphs, digraphs and provides examples. It discusses graph terminology, models, degree of graphs, handshaking lemma, special graphs and applications. It also provides explanations of planar graphs, Euler's formula and graph coloring.
This document provides a review for algebra sections on various topics including:
1) Solving word problems involving equations with one unknown variable.
2) Finding sums, differences, and ratios of numbers.
3) Representing word problems using tables and equations.
4) Solving uniform motion problems using tables, diagrams, and the appropriate equation based on the type of motion described.
5) Graphing linear equations and finding slopes of lines from equations or two points.
6) Writing equations of lines in different forms given information like slopes, intercepts, or two points.
This document provides a review for algebra sections on various topics including:
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2. 1 Introduction
Mathematics can be used to understand the complexity of music by defining musical terms
and aspects such as tones, intervals, scales and rhythm. The mathematical aspects of music
have a remarkable collection of number properties simply from the fact that nature in itself
holds and can be represented by mathematics. To attempt to structure new ways of com-
posing, hearing and understanding music, links between different math fields such as graph
theory, group and set theory, number theory and music have all been made which have led
to many musical applications.
Music is an art form that is all around us. Defined by its musical tones of composed
harmonies and melodies, music is an important part in our lives. One can really think of
music in two different ways, atonal music and tonal music. Atonal music can be thought
of as a lack of a key, specifically a key signature. It does not take into account and avoids
the use of major and minor chords, scales, keys, rhythm, repetition, etc. It focuses more on
individual sounds and subjectivity; this type of music is not as commonly used by composers.
This is different from tonal music which conforms to major and minor keys. It uses chords
and scales where relationships between chords and scales are most abundant.
1.1 Intervals, Chords, Scales, and Lines
2
3. A tone is an individual note, looking at the above figure [5], each individual note on the
piano is a tone. A sequence of two notes is called an interval; for example C − F is an
interval. The chromatic set is the set of tones corresponding to both the black and white
keys on a piano which is illustrated in the figure above and listed below.
{C, C#/Db, D, D#/Eb, E, F, F#/Gb, G, G#/Ab, A, A#/Bb, B}
The diatonic set is the set of all white keys on a piano {C, D, E, F, G, A, B}. The distance
between two consecutive elements of the chromatic set is a semitone. The figure above
illustrates the chromatic set and the diatonic set. Notice each black key on a piano has
two different names. Consider Eb; it is one semitone down from E. However, it can also be
thought of as being one semitone above D, also known as D#. For an interval, the diatonic
length is the number of steps needed in the diatonic scale to move from the first note of the
interval to the second. Similarly, the chromatic length is the numbers of steps needed in
the chromatic scale to move from the first note to the second. The direction of movement
from the first note to the second is always up the scale, wrapping around to the beginning, if
needed. For example, the interval (C − F) has a diatonic length of 3 and a chromatic length
of 5.
A chord is an unordered nonempty subset of the chromatic set. For example, {B, C, E}
is a chord. A scale is like a chord, except it is an ordered nonempty sequence of notes of
the chromatic set. For example, (B, C, E) and (E, B, C) are two different three note scales,
but they are the same three note chord. A scale of k distinct notes is called a k-note line;
so we can think of a line and scale as being the same thing. Because they seem the same
at face value, why do we distinguish the two apart? We distinguish a line and a scale apart
3
4. because lines can be manipulated and represented more by a mathematical standpoint. For
the main purposes of this paper, we shall restrict ourselves to the usage of k-note lines.
We can now extend the notion of intervals to k-note lines. We can represent the chromatic
and diatonic lengths of a k-note line with length arrays. Diatonic and chromatic length arrays
measure the diatonic and chromatic lengths between each note of the k-note line respectively.
So given a k-note line with notes (n1, n2..., nk), the diatonic length array is calculated
< d1, d2, ..., dk >D=< n2 − n1, n3 − n2, ..., nk − nk−1, n1 − nk >D
where all subtraction is performed modulo 7, or di = (di+1 − d1) mod 7
If all notes are in increasing order, then
k∑
i=1
d1 = 7. For example, consider the three note
line (C − E − F), it would have a diatonic length array of < 2, 1, 4 >D, and a chromatic
length array of < 4, 1, 7 >C.
1.2 Enumeration of k-note lines
How many different k-note lines are there in the musical universe?
Theorem 1.2.1: For the standard diatonic set {C, D, E, F, G, A, B}, there are a total of
13, 699 different k-note lines, 1 ≤ k ≤ 7.
Proof: A simple counting argument can be applied to find all of these different k-note lines.
There are 7 notes of the diatonic set {C, D, E, F, G, A, B}. Finding all possible k-note lines
is the same as finding all k-permutations of the set {C, D, ..., A, B} for all k, 1 ≤ k ≤ 7. For
example, for a 1-note line, there are a total of 7 different notes to choose from. For a 2-note
line, the first note can be any of the 7 notes but, since the second note must be different
4
5. from the first, there are now only 6 choices for the second note. Extending this now for
1 ≤ k ≤ 7, we can generalize this with the following formula
7∑
k=1
P(7, k)
=
7∑
k=1
7!
(7−k)!
= 7 + 7 ∗ 6 + 7 ∗ 6 ∗ 5 + ... + 7 ∗ 6 ∗ 5 ∗ 4 ∗ 3 ∗ 2 ∗ 1
= 13, 699
QED
This type of argument could be generalized to any other musical universe.
Theorem 1.2.2: For a diatonic set with d notes, there are a total of
d∑
k=1
P(d, k) different
k-note lines, 1 ≤ k ≤ d.
2 Differentiating k-note lines
2.1 Genus
So, is there a way to classify all 13, 699 k-note lines? We can, with the help of the concept
of genus and species. Two k-note lines are in the same genus if they have the same diatonic
length array. For example, the genus of (C − E − F) is the set
(C − E − F, D − F − G, E − G − A, F − A − B, G − B − C, A − C − D, B − D − E)
where this diatonic array is fixed under rotations in the diatonic set. This can be shown
pictorally; consider the figure below [7]. We fix the length array of the k-note line by drawing
5
6. lines going from each note to the next. Then, rotating about the circle, all other lines of the
same genus can be found.
Observe, that we can associate the notes of the diatonic set {C, D, E, F, G, A, B} with
the numbers {0, 1, 2, 3, 4, 5, 6} respectively. Thus we can calculate the diatonic length of an
interval using modular subtraction in Z7. For example, the length of (C − F) would be
(5 − 2) mod 7 = 3 while the length of (F − C) would be (2 − 5) mod 7 = −3 mod 7 = 4.
Similarly, for the chromatic length of an interval, we use modular subtraction in Z12.
2.2 Enumeration of genus
How many different genera are there in the musical universe? To answer this, we shall use
a helpful counting theorem.
Theorem 2.2.1: The number of positive integral solutions to the equation
x1 + x2 + ... + xk = n is
(n−1
k−1
)
.
Theorem 2.2.2: For the standard diatonic set {C, D, E, F, G, A, B, C} there are 1, 957
different genera.
6
7. The counting argument for finding these genera proves more difficult than k-note lines. To
be in the same genus, two k-note lines must have the same diatonic length array. Thus,
we can equivalently count all possible diatonic length arrays. Assuming the notes are in
increasing order,
k∑
i=1
di = 7. With all of this in mind, finding all the different genera becomes
easier. Consider the different integer partitions of 7, that is, solving the equation
x1 + x2 + x3 + ... + xk = 7 where xi ≥ 1
Each k-note line with notes in increasing order corresponds to a diatonic length array
< d1, d2, ..., dk > such that
k∑
i=1
di = 7, or equivalenty, to an integer partition of 7 into k positive
summands. Now lets extend this for the different values of k where 1 ≤ k ≤ 7 of k-note lines.
Without loss of generality, to find the arrays that are out of order, fix the first note in the
k-note line and then find the different permutations of the remaining notes to find the k-note
lines that are out of order. For example, look at < 1, 1, 5 > an array of increasing order.
Reordering the elements of this interval array gives us < 1, 1, 5 >, < 1, 5, 1 >, < 5, 1, 1 >.
Then we fix the first note, and permute the rest.
Proof:
k = 1: There is only one genus, namely < 7 >
< 7 >
= 1 genus
k = 2: There are 6 genera, the ones in order are < 1, 6 >, < 2, 5 >, < 3, 4 >, and the ones
out of order are < 6, 1 >, < 5, 2 >, < 4, 3 >.
< 1, 6 >, < 2, 5 >, < 3, 4 >, < 6, 1 >, < 5, 2 >, < 4, 3 >
7
8. = 6 genera
k = 3: Once we get to k = 3 there are 30 genera. The 4 integer partitions with summands
listed in increasing order are < 1, 1, 5 >, < 1, 2, 4 >, < 1, 3, 3 >, < 2, 2, 3 >. Look at the set
< 1, 1, 5 >, this corresponds to the diatonic array < 1, 1, 5 >D which is the three note line
(A,B,G). These notes are in clockwise order. Now we find all other 3-note lines using these
3 notes by fixing the note, A, as the first note and then permuting the remaining 2 notes,
which have 2 possible reorderings, (A,B,G) = < 1, 1, 5 >, and (A,G,B) = < 6, 2, 6 >. Again,
reordering the elements of the interval array gives the first value of the product, then fixing
the first note and permuting the remaining notes gives the second value of the product.
< 1, 1, 5 >, < 1, 2, 4 >, < 1, 3, 3 >, < 2, 2, 3 >
= 3 · 2 + 6 · 2 + 3 · 2 + 3 · 2 = 30 genera
A similar argument can now be made for the other summands and the overall argument
can be now extended for the remaining k values. The remaining different integer partitions
of 7 for the different k-note lines where 1 ≤ k ≤ 7 can be seen below.
k = 4:
< 1, 1, 1, 4 >, < 1, 1, 2, 3 >, < 1, 2, 2, 2 >
= 4 · 6 + 12 · 6 + 4 · 6 = 120 genera
k = 5:
< 1, 1, 1, 1, 3 >, < 1, 1, 1, 2, 2 >
= 5 · 4! +
(5
2
)
· 4! = 360 genera
k = 6:
8
9. < 1, 1, 1, 1, 1, 2 >
= 6 · 5! = 720 genera
k = 7:
< 1, 1, 1, 1, 1, 1, 1 >
= 1 · 6! = 720 genera
Adding up all the different genera for k-note lines where 1 ≤ k ≤ 7 gives the 1, 957 different
genera as required.
1 + 6 + 30 + 120 + 360 + 720 + 720 = 1, 957 genera
This type of counting argument where we find the diatonic arrays of consecutive order,
then reorder them and then find the remaining diatonic arrays by fixing the first note and
choosing the positions for the rest of the notes can be simplified. We can count all integer
partitions of 7, which corresponds to the following formula.
7∑
k=1
( 6
k−1
)
(k − 1)! = 1, 957
QED
2.3 Species
We can now differentiate k-note lines of the same genus even further with species. Two
k-note lines are in the same species if they have the same chromatic length arrays.
For example, consider (C, B). It has two different species.
1: (B, A), (E, D), (A, G), (D, C), (G, F)
9
10. 2: (C, B), (F, E)
Respectively, the 2-note lines in each of these species have a chromatic length of
< 10, 2 >C and < 11, 1 >C
This 2-note line contains two different species. Now consider (C, E, F), which has three
different species,
1: (C − E − F), (F − A − B)
2: (A − C − D), (B − D − E), (D − F − G), (E − G − A)
3: (F − A − B)
Respectively, the 3-note lines in each of these species have a chromatic length of
< 4, 1, 7 >C, < 3, 2, 7 >C, and < 4, 2, 6 >C
Notice, (C − B) is a 2-note line that has 2 different species. (C − E − F) is a 3-note line
that contains 3 different species. This generalizes to nice theorem proven by Clough and
Myerson [1]
Theorem 2.3.1: Given any k, 1 < k < 7, and any k-note line, the genus containing that
line contains exactly k species.
3 Circle of fifths
The proof of the above theorem makes use of the circle of fifths, discovered by German math-
ematician Johann David Heinichen [6]. In music theory, the circle of fifths is a representation
of the relationships between the twelve tones of the chromatic scale, their key signatures and
10
11. major and minor keys. It is a sequence of pitches, represented in a circle, in which the next
pitch is found seven semitones higher than the previous. Seven semitones is the same as a
musical interval referred to as a fifth. A fifth is the interval between two notes on the circle,
and the distance between two consecutive notes is seven semitones.
3.1 Proof of Theorem 2.3.1
The proof of Theorem 2.3.1 can be illustrated using a generalized version of the circle of
fifths. Since we are interested only in diatonic notes, consider only the top portion of the
circle with an edge of chromatic length six added from B to F. Consider the line (C−E−F),
with genus < 4, 1, 7 >. The other lines of the genus can be obtained by cycling clockwise
around the upper semicircle of the circle of fifths. We claim that there are three different
types of species of this 3-note line. The three species arise from the three different possible
locations of the interval B-F. The interval B-F is the only interval on the diatonic circle that
does not have chromatic length seven, thus it is the location of this interval within a line that
determines the line’s species. In general, there are, in fact, k different possible locations of
11
12. this interval and thus k different species. This argument assumes all notes are in clockwise
order and a complete proof can be found in [1]
4 Enumeration of chords
We have discussed counting k-note lines, which again, are the same as scales. Thus we
have discussed counting the scales of the musical universe. Next we ask how many d-note
diatonic pitch class sets, or chords, exist in a c-note chromatic universe? We start in the
c-note chromatic universe Zc, the group of integers mod c. We want to consider the d-
note pitch classes which are subsets of Zc with size d. Equivalently, we can count different
interval arrays with
c∑
i=1
di. Without loss of generality, assume the notes are in increasing
order. Consider D2n, which represents the symmetries of a regular n-gon with n rotations
and n reflections (n + n = 2n). Rotations and reflections can be thought of as traspositions
and inversions respectively. These operations permute groups or in this case, chords [2]. We
want to count the number of d-note pitch class sets by counting the number of equivalence
classes when D2n acts on the set of d-note pitch class sets. These sets can be represented
by interval arrays of the form < d1, d2, ..., dd >. So now the argument becomes counting the
number of equivalence classes when D2d acts of the set of interval arrays.
Theorem 4.1.1: The number of d-note pitch class sets in a c-note chromatic universe is
[4]
1
2d
T(c, d) + 1
2
I(c, d)
where T(c, d) =
∑
j|gcd(c,d)
Φ(j)
(c/j
d/j
)
12
13. and I(c,d) =
f(x) =
(c
2
−1
⌊d
2 ⌋
)
if c is even and d is odd
(⌊c
2 ⌋
⌊d
2 ⌋
)
otherwise
Where the Euler function Φ(j) is the number of positive integers that are less than or
equal to j that are also relatively prime to j [4].
The proof of this theorem uses famous results known as the Hockey Stick Theorem and
Burnside’s Lemma, tools used in combinatorics where here they help us count the number
of equivalence classes. We will state these theorems for use in the counting argument, the
complete proofs can be found in [4] [12]
Theorem 4.1.2 Burnside’s Lemma: Let G be a group acting on a set S. The number
of equivalence classes is
1
|G|
∑
g∈G
Fix(g)
where Fix(g) is the number of elements of S that are fixed by g. [10]
Theorem 4.1.3 Hockey Stick Theorem: If m and n are nonnegative integers, then
n∑
k=0
(k
m
)
=
(n+1
m+1
)
[11]
This theorem is well illustrated using Pascal’s Triangle [8]
13
14. In addition to these concepts, another theorem can be used to help rewrite some summa-
tions into neater form.
Theorem 4.1.4: Let j, k and n be integers such that 0 ≤ j ≤ k ≤ n. Then,
n−k+j∑
m=j
(m
j
)(n−m
k−j
)
=
(n+1
k+1
)
5 Conclusion
There are more generalized theorems for scales than the usual chromatic and diatonic. We
have talked about the enumeration of scales and genus. The genus of a line and any of its
permutations have the same structure of species. Because of this we may talk about the
structure and enumeration of chords. Further details about enumeration of chords are left
to the reader.
14
15. References
[1] Clough, John and Myerson, Gerald. Musical Scales and the Generalized Circle of Fifths
The American Mathematical Monthly, Vol, 93, No.9, November 1986
[2] Hook, Julian. (Used this because of mention of translations and reflections; can take out)
Uniform Triadic Transformations Hook, 2002.
[3] Silverman, Danielle and Wiseman, Jim. Noting the difference: Musical scales and Per-
mutations
[4] Case, Joshua and Koban, Lori and LeGrand, Jordan. Counting Pitch Class Sets with
Burnside’s Lemma Department of Mathematics and Computer Science University of
Maine, UMF Math Department.
[5] Image. Half steps, whole steps and scale formulas.
<http://www.bandnotes.info/tidbits/scales/half-whl.htm>. (5 May 2014).
[6] Image. <http://www.sevenstring.org/forum/music-theory-lessons-techniques/197136-
circle-fifths.htm>. (5 May 2014).
[7] Image. <http://www.electricguitarsuk.com/images/clip image002.jpg>. (5 May 2014).
[8] Image. Pascal’s Triangle and its patterns. <http://britton.disted.camosun.bc.ca/pascal/p
1h.gif>. (5 May 2014).
[9] Home page. Circle of fifths. <http://www.circleoffifths.com/>. (5 May 2014).
[10] Contemporary Abstract Algebra Gallian, Joseph A. Houghton Mifflin Company Boston.
5th edition. 2002. (p. 487)
15
16. [11] Concrete Mathematics: A Foundation for Computer Science Addison-Wesley. Reading,
MA. 2nd edidtion. 1994. (p 169,174)
[12] An Application of Burnside’s Theorem to Music Theorem to Music Theory Graham,
Jeff. The UMAP Journal. Bedford, MA. Vol. 29, No. 1 2008
16