This document summarizes Ji Li's dissertation defense on counting point-determining graphs and prime graphs using Joyal's theory of species. The defense took place on May 10th, 2007 at Brandeis University, with Professor Ira Gessel serving as Ji Li's thesis advisor. The dissertation outlines the use of species theory to define and enumerate point-determining graphs, bi-point-determining graphs, and point-determining 2-colored graphs, as well as applying species theory to study prime graphs.
This document provides an introduction to functions and their representations. It defines what a function is and gives examples of functions expressed through formulas, numerically, graphically and verbally. It also covers properties of functions including monotonicity, symmetry, even and odd functions. Key examples are used throughout to illustrate concepts.
The document discusses functions and their properties. It defines what a function is - a relation where each element of the domain corresponds to exactly one element of the codomain. It also defines key properties of functions like one-to-one, onto, bijective, and inverse functions. The document discusses how to compose functions and calculate the number of possible functions between two sets. It concludes by introducing order of magnitude analysis to compare growth rates of functions.
The inverse of a function "undoes" the effect of the function. We look at the implications of that property in the derivative, as well as logarithmic functions, which are inverses of exponential functions.
This document provides an overview of key concepts in set theory, including:
1) Sets can be defined by listing elements or using predicates, and basic set operations include membership, equality, subsets, and power sets.
2) Relationships between sets such as subsets, supersets, proper subsets are defined, and examples are given to illustrate concepts like open and closed intervals.
3) Common set notations are introduced for natural numbers, integers, rational numbers, and real numbers. Binary operations on sets are defined to be well-defined and keep the set closed under the operation.
This document provides sample questions and answers related to various mathematics topics, including functions, quadratic equations/functions, simultaneous equations, indices and logarithms, and coordinate geometry.
The questions are multiple choice or short answer questions assessing skills like solving equations, graphing functions, finding inverse functions, and working with logarithmic and exponential expressions.
The answers section provides fully worked out solutions to each question in a clear format. This document appears to be a reference for teachers or students to use for exam preparation or review, with the goal of assessing conceptual understanding of core algebra and geometry topics.
The inverse of a function "undoes" the effect of the function. We look at the implications of that property in the derivative, as well as logarithmic functions, which are inverses of exponential functions.
This document provides an introduction to functions and their representations. It defines what a function is and gives examples of functions expressed through formulas, numerically, graphically and verbally. It also covers properties of functions including monotonicity, symmetry, even and odd functions. Key examples are used throughout to illustrate concepts.
The document discusses functions and their properties. It defines what a function is - a relation where each element of the domain corresponds to exactly one element of the codomain. It also defines key properties of functions like one-to-one, onto, bijective, and inverse functions. The document discusses how to compose functions and calculate the number of possible functions between two sets. It concludes by introducing order of magnitude analysis to compare growth rates of functions.
The inverse of a function "undoes" the effect of the function. We look at the implications of that property in the derivative, as well as logarithmic functions, which are inverses of exponential functions.
This document provides an overview of key concepts in set theory, including:
1) Sets can be defined by listing elements or using predicates, and basic set operations include membership, equality, subsets, and power sets.
2) Relationships between sets such as subsets, supersets, proper subsets are defined, and examples are given to illustrate concepts like open and closed intervals.
3) Common set notations are introduced for natural numbers, integers, rational numbers, and real numbers. Binary operations on sets are defined to be well-defined and keep the set closed under the operation.
This document provides sample questions and answers related to various mathematics topics, including functions, quadratic equations/functions, simultaneous equations, indices and logarithms, and coordinate geometry.
The questions are multiple choice or short answer questions assessing skills like solving equations, graphing functions, finding inverse functions, and working with logarithmic and exponential expressions.
The answers section provides fully worked out solutions to each question in a clear format. This document appears to be a reference for teachers or students to use for exam preparation or review, with the goal of assessing conceptual understanding of core algebra and geometry topics.
The inverse of a function "undoes" the effect of the function. We look at the implications of that property in the derivative, as well as logarithmic functions, which are inverses of exponential functions.
This document discusses exponents and exponential functions. It defines what exponents are, how they are read, and some of their key properties like the product rule for exponents. It also examines exponential functions where the base is raised to the power of x. The graphs of exponential functions where the base is greater than 1, between 0 and 1, and equal to 1 are explored. The graphs are always increasing, have a horizontal asymptote at y=0, and a y-intercept of (0,1). Exponential equations are also briefly covered.
This document provides an introduction to functions and their properties. It defines what a function is as a mapping from a domain set to a codomain set, and introduces related concepts like domain, codomain, range, and the notation for functions. It then gives examples of specifying functions explicitly and with formulas. The document discusses properties of functions like injectivity, surjectivity, and being increasing or decreasing. It provides examples of determining if a function has these properties. The document concludes by introducing the concept of the inverse function for bijective functions.
(1) The document discusses products of LF-topologies and separation concepts in LF-topological spaces.
(2) It defines what a GL-monoid is and introduces uniform structures on GL-monoids to characterize arbitrary products of elements in a GL-monoid.
(3) The paper then builds the LF-topology product of a family of LF-topological spaces and shows that Kolmogoroff and Hausdorff properties are inherited by the product LF-topology from the factor spaces.
The document summarizes the equilibrium search model of Burdett and Mortensen (1998). It includes:
1) Assumptions of the model including job offer rates, wages set by firms from an endogenous distribution, and job separation rates.
2) Derivation of the reservation wage equation and relationship between earnings and wage offer distributions.
3) Determination of the wage offer distribution by firms maximizing profits given the reservation wage and availability of workers.
4) Discussion of empirical issues regarding matching moments of observed wage distributions and implications for unemployment durations.
5.1 Defining and visualizing functions. A handout.Jan Plaza
This document introduces concepts related to functions including:
- Defining functions in terms of unique mappings between inputs and outputs
- Distinguishing between total, partial, and non-functions
- Specifying domains and ranges
- Using vertical line tests to identify functions from graphs
- Examples of functions defined by formulas or mappings
This document is a calculus supplement to accompany a microeconomics textbook. It introduces the concept of partial derivatives and shows how they can be used to analyze economic concepts from the textbook like demand and supply functions, substitutes and complements, and elasticities. Partial derivatives allow the slope of demand and supply functions to be determined with respect to different variables. They also allow elasticities to be defined and calculated using calculus, providing an equivalent but alternative method to the algebraic approach in the textbook. The supplement aims to illustrate the connections between calculus and microeconomic concepts.
1) The document discusses computing F-blowups, which are canonical blowups of varieties in characteristic p.
2) It provides algorithms for computing F-blowups in the toric case by using Groebner fans and in the general case by computing presentations of Frobenius pushforwards of modules.
3) Macaulay2 is used to compute the first F-blowup of a simple elliptic singularity as an example.
This document defines functions and related terminology. It discusses:
- The definition of a function, domain, codomain, range, and related terms
- Properties of functions including one-to-one, onto, and bijective functions
- Inverse functions and how they relate to bijective functions
- Examples are provided to illustrate injective, surjective and bijective functions as well as calculating inverse functions.
The computation of automorphic forms for a group Gamma is
a major problem in number theory. The only known way to approach the higher rank cases is by computing the action of Hecke operators on the cohomology.
Henceforth, we consider the explicit computation of the cohomology by using cellular complexes. We then explain how the rational elements can be made to act on the complex when it originate from perfect forms. We illustrate the results obtained for the symplectic Sp4(Z) group.
This document discusses functions and their properties. It defines a function as a relation where each input is paired with exactly one output. Functions can be represented numerically in tables, visually with graphs, algebraically with explicit formulas, or verbally. The domain is the set of inputs, the codomain is the set of all possible outputs, and the range is the set of actual outputs. Functions can be one-to-one (injective) if each input maps to a unique output, or onto (surjective) if each possible output is the image of some input.
This document is from a Calculus I class at New York University and covers continuity. It provides announcements about office hours and homework deadlines. It then discusses the objectives of understanding the definition of continuity and applying it to piecewise functions. Examples are provided to demonstrate how to show a function is continuous at a point by evaluating the limit as x approaches the point and showing it equals the function value. The students are asked to determine at which other points the example function is continuous.
We cover the inverses to the trigonometric functions sine, cosine, tangent, cotangent, secant, cosecant, and their derivatives. The remarkable fact is that although these functions and their inverses are transcendental (complicated) functions, the derivatives are algebraic functions. Also, we meet my all-time favorite function: arctan.
The document provides an overview of functions including definitions, examples, and properties. It defines a function as a relation that assigns each element in the domain to a single element in the range. Examples of functions expressed by formulas, numerically, graphically, and verbally are given. Properties like monotonicity, symmetry, evenness, and oddness are defined and illustrated with examples. The document aims to introduce the fundamental concepts of functions to readers.
This document is from a Calculus I class at New York University. It provides an overview of functions, including the definition of a function, different ways functions can be represented (formulas, tables, graphs, verbal descriptions), properties of functions like monotonicity and symmetry, and examples of determining domains and ranges of functions. It aims to help students understand functions and their representations as a foundation for calculus.
A factorization theorem for generalized exponential polynomials with infinite...Pim Piepers
The document presents a factorization theorem for a class of generalized exponential polynomials called polynomial-exponent exponential polynomials (pexponential polynomials). The theorem states that if a pexponential polynomial F(x) has infinitely many integer zeros belonging to a finite union of arithmetic progressions, then F(x) can be factorized into a product of factors corresponding to the zeros in each progression multiplied by a pexponential polynomial with only finitely many integer zeros. The proof relies on two lemmas showing that certain polynomial sums in the components of F(x) vanish for integers in the progressions.
This document discusses topics in category theory, including set-functors, adjunctions, and limits. It begins by defining set-functors and natural transformations between them. It notes that a natural transformation is uniquely determined by its value on an initial element of a functor. It then introduces adjunctions and decomposes them into left and right adjoints. It shows that a left adjoint exists if and only if certain set-functors are representable. Finally, it defines limits of diagrams (I-systems) over an index category I. It shows that a limit exists if and only if the cone functor is representable.
This document discusses categories of topological spaces and their isomorphism to categories of relational algebras for a monad. It begins with introductions to the topic and tools used, including categories, functors, natural transformations, monads, and relational algebras. The main content is divided into multiple parts, exploring the proposition that the category of topological spaces is isomorphic to the category of relational algebras. It concludes by restating the aim to formally prove this result using relational calculus.
This document summarizes Ji Li's experience tutoring mathematics students at the University of Arizona. It discusses different types of tutoring Li engaged in, including private tutoring, working with the Transitional Year Program, and tutoring in the common calculus room. The talk outlines challenges students face, best practices for tutoring, and how tutoring notes were developed to aid new tutors. It also reflects on what tutoring has taught Li about effectively communicating mathematical concepts to students.
This document discusses exponents and exponential functions. It defines what exponents are, how they are read, and some of their key properties like the product rule for exponents. It also examines exponential functions where the base is raised to the power of x. The graphs of exponential functions where the base is greater than 1, between 0 and 1, and equal to 1 are explored. The graphs are always increasing, have a horizontal asymptote at y=0, and a y-intercept of (0,1). Exponential equations are also briefly covered.
This document provides an introduction to functions and their properties. It defines what a function is as a mapping from a domain set to a codomain set, and introduces related concepts like domain, codomain, range, and the notation for functions. It then gives examples of specifying functions explicitly and with formulas. The document discusses properties of functions like injectivity, surjectivity, and being increasing or decreasing. It provides examples of determining if a function has these properties. The document concludes by introducing the concept of the inverse function for bijective functions.
(1) The document discusses products of LF-topologies and separation concepts in LF-topological spaces.
(2) It defines what a GL-monoid is and introduces uniform structures on GL-monoids to characterize arbitrary products of elements in a GL-monoid.
(3) The paper then builds the LF-topology product of a family of LF-topological spaces and shows that Kolmogoroff and Hausdorff properties are inherited by the product LF-topology from the factor spaces.
The document summarizes the equilibrium search model of Burdett and Mortensen (1998). It includes:
1) Assumptions of the model including job offer rates, wages set by firms from an endogenous distribution, and job separation rates.
2) Derivation of the reservation wage equation and relationship between earnings and wage offer distributions.
3) Determination of the wage offer distribution by firms maximizing profits given the reservation wage and availability of workers.
4) Discussion of empirical issues regarding matching moments of observed wage distributions and implications for unemployment durations.
5.1 Defining and visualizing functions. A handout.Jan Plaza
This document introduces concepts related to functions including:
- Defining functions in terms of unique mappings between inputs and outputs
- Distinguishing between total, partial, and non-functions
- Specifying domains and ranges
- Using vertical line tests to identify functions from graphs
- Examples of functions defined by formulas or mappings
This document is a calculus supplement to accompany a microeconomics textbook. It introduces the concept of partial derivatives and shows how they can be used to analyze economic concepts from the textbook like demand and supply functions, substitutes and complements, and elasticities. Partial derivatives allow the slope of demand and supply functions to be determined with respect to different variables. They also allow elasticities to be defined and calculated using calculus, providing an equivalent but alternative method to the algebraic approach in the textbook. The supplement aims to illustrate the connections between calculus and microeconomic concepts.
1) The document discusses computing F-blowups, which are canonical blowups of varieties in characteristic p.
2) It provides algorithms for computing F-blowups in the toric case by using Groebner fans and in the general case by computing presentations of Frobenius pushforwards of modules.
3) Macaulay2 is used to compute the first F-blowup of a simple elliptic singularity as an example.
This document defines functions and related terminology. It discusses:
- The definition of a function, domain, codomain, range, and related terms
- Properties of functions including one-to-one, onto, and bijective functions
- Inverse functions and how they relate to bijective functions
- Examples are provided to illustrate injective, surjective and bijective functions as well as calculating inverse functions.
The computation of automorphic forms for a group Gamma is
a major problem in number theory. The only known way to approach the higher rank cases is by computing the action of Hecke operators on the cohomology.
Henceforth, we consider the explicit computation of the cohomology by using cellular complexes. We then explain how the rational elements can be made to act on the complex when it originate from perfect forms. We illustrate the results obtained for the symplectic Sp4(Z) group.
This document discusses functions and their properties. It defines a function as a relation where each input is paired with exactly one output. Functions can be represented numerically in tables, visually with graphs, algebraically with explicit formulas, or verbally. The domain is the set of inputs, the codomain is the set of all possible outputs, and the range is the set of actual outputs. Functions can be one-to-one (injective) if each input maps to a unique output, or onto (surjective) if each possible output is the image of some input.
This document is from a Calculus I class at New York University and covers continuity. It provides announcements about office hours and homework deadlines. It then discusses the objectives of understanding the definition of continuity and applying it to piecewise functions. Examples are provided to demonstrate how to show a function is continuous at a point by evaluating the limit as x approaches the point and showing it equals the function value. The students are asked to determine at which other points the example function is continuous.
We cover the inverses to the trigonometric functions sine, cosine, tangent, cotangent, secant, cosecant, and their derivatives. The remarkable fact is that although these functions and their inverses are transcendental (complicated) functions, the derivatives are algebraic functions. Also, we meet my all-time favorite function: arctan.
The document provides an overview of functions including definitions, examples, and properties. It defines a function as a relation that assigns each element in the domain to a single element in the range. Examples of functions expressed by formulas, numerically, graphically, and verbally are given. Properties like monotonicity, symmetry, evenness, and oddness are defined and illustrated with examples. The document aims to introduce the fundamental concepts of functions to readers.
This document is from a Calculus I class at New York University. It provides an overview of functions, including the definition of a function, different ways functions can be represented (formulas, tables, graphs, verbal descriptions), properties of functions like monotonicity and symmetry, and examples of determining domains and ranges of functions. It aims to help students understand functions and their representations as a foundation for calculus.
A factorization theorem for generalized exponential polynomials with infinite...Pim Piepers
The document presents a factorization theorem for a class of generalized exponential polynomials called polynomial-exponent exponential polynomials (pexponential polynomials). The theorem states that if a pexponential polynomial F(x) has infinitely many integer zeros belonging to a finite union of arithmetic progressions, then F(x) can be factorized into a product of factors corresponding to the zeros in each progression multiplied by a pexponential polynomial with only finitely many integer zeros. The proof relies on two lemmas showing that certain polynomial sums in the components of F(x) vanish for integers in the progressions.
This document discusses topics in category theory, including set-functors, adjunctions, and limits. It begins by defining set-functors and natural transformations between them. It notes that a natural transformation is uniquely determined by its value on an initial element of a functor. It then introduces adjunctions and decomposes them into left and right adjoints. It shows that a left adjoint exists if and only if certain set-functors are representable. Finally, it defines limits of diagrams (I-systems) over an index category I. It shows that a limit exists if and only if the cone functor is representable.
This document discusses categories of topological spaces and their isomorphism to categories of relational algebras for a monad. It begins with introductions to the topic and tools used, including categories, functors, natural transformations, monads, and relational algebras. The main content is divided into multiple parts, exploring the proposition that the category of topological spaces is isomorphic to the category of relational algebras. It concludes by restating the aim to formally prove this result using relational calculus.
This document summarizes Ji Li's experience tutoring mathematics students at the University of Arizona. It discusses different types of tutoring Li engaged in, including private tutoring, working with the Transitional Year Program, and tutoring in the common calculus room. The talk outlines challenges students face, best practices for tutoring, and how tutoring notes were developed to aid new tutors. It also reflects on what tutoring has taught Li about effectively communicating mathematical concepts to students.
This document provides advice and information to parents and students about managing time, attending classes regularly, taking advantage of academic resources, getting involved on campus, meeting with advisors, embracing campus diversity, making a difference, defining goals, making choices, and helpful campus resources at Miami University. It emphasizes the importance of time management, class attendance, seeking help, getting involved, meeting mentors, being open to new experiences, allowing students to set their own path, and utilizing various campus support services.
1. The document discusses how Pokemon cards and commodities can serve as markers of social class and identity for children in Cairo, Egypt.
2. The value and status associated with Pokemon items depends on factors like where they were obtained (the US vs. Egypt), their quality (official vs. pirate copies), and their cost.
3. These material goods take on symbolic meanings and allow children to signal identities related to cosmopolitanism, foreignness, wealth, and moral values through practices like acquisition, play, and trading.
This document discusses Jean Baudrillard's theory of simulacra and simulation as it relates to media and cultural representations. Baudrillard described how representations become increasingly distant from reality, moving from serious representations, to distorted media versions, to idealized simulations that seem more real than reality. As an example, he cites Disney World, which creates a hyperreal world of representations without originals. However, the document notes that Baudrillard used provocative examples and language. Other theorists like Bruno Latour argued that reality is simultaneously real, narrated through discourse, and collective through social relations and cultural models.
1) The document investigates prime labelling of certain graphs such as the flower graph F, splitting graph of a star, bistar graph, friendship graph, and graph SF(n,1).
2) It proves that these graphs admit prime labelling by providing explicit bijective labellings of the vertices with positive integers such that the greatest common divisor of adjacent vertices is 1.
3) Examples of prime labellings are provided for specific instances of the graphs to illustrate the labellings.
This report summarizes recent work proving the fundamental lemma, which is an important step in Langlands' endoscopy theory. The fundamental lemma relates orbital integrals of a reductive group to those of its endoscopic groups. The report provides examples of how orbital integrals arise in counting problems for lattices and abelian varieties over finite fields. It also discusses how stable orbital integrals and their κ-sisters are used in the stable trace formula to relate traces of automorphic representations to orbital integrals.
This document defines key concepts related to lattices and soft sets. It begins by defining lattices and related terms like sub-lattices, ideals, distributive lattices, and modular lattices. It then introduces the concept of soft sets, providing definitions for soft set operations like union, intersection, and complement. Examples are given to illustrate soft sets and lattices. Soft lattices are then introduced, defining when a soft set is considered a soft lattice. Properties of soft lattices are discussed along with examples. The document primarily focuses on providing formal definitions for conceptual and theoretical topics related to lattices and soft sets.
Group {1, −1, i, −i} Cordial Labeling of Product Related GraphsIJASRD Journal
Let G be a (p,q) graph and A be a group. Let f : V (G) → A be a function. The order of u ∈ A is the least positive integer n such that un = e. We denote the order of u by o(u). For each edge uv assign the label 1 if (o(u), o(v)) = 1 or 0 otherwise. f is called a group A Cordial labeling if |vf (a) − vf (b)| ≤ 1 and |ef (0) − ef (1)| ≤ 1, where vf (x) and ef (n) respectively denote the number of vertices labeled with an element x and number of edges labeled with n(n = 0, 1). A graph which admits a group A Cordial labeling is called a group A Cordial graph. In this paper we define group {1,−1, i,−i} Cordial graphs and prove that Hypercube Qn = Qn−1 × K2, Book Bn = Sn × K2, n-sided prism Prn = Cn × K2 and Pn × K3 are all group {1,−1, i,−i} Cordial for all n.
This document provides an overview of character theory for finite groups and its analogy to representation theory for the infinite compact group S1. It discusses several key concepts:
1) Character theory characterizes representations of a finite group G by their characters, which are class functions that map G to complex numbers. Characters determine representations up to isomorphism.
2) The irreducible representations of S1 are 1-dimensional and in bijection with integers n, where each representation maps z to zn.
3) By analogy to Fourier analysis, the characters of S1 form an orthonormal basis for L2(S1) and decompose representations into irreducibles in the same way as for finite groups.
The document defines the operators curl and divergence for vector fields. Curl is defined as the cross product of del (the gradient operator) with the vector field and results in another vector. Divergence is defined as the dot product of del with the vector field and results in a scalar. Several examples of computing curl and divergence are worked out. Green's theorem, which relates line integrals of vector fields to surface integrals of curl and divergence, is also discussed.
This document discusses unbounded transcendent formal power series over a finite field Fq((X-1)). It presents a new transcendence criterion for continued fractions in this field. Specifically, it constructs a family of transcendental continued fractions with unbounded partial quotients obtained from algebraic elements. The main result proves that if a formal power series can be approximated by a family of algebraic series with increasingly long blocks in their continued fraction expansions, then the formal power series must be transcendental. An example is also given to illustrate the main result.
This document discusses unbounded transcendent formal power series over a finite field Fq((X-1)). It presents a new general result that establishes a transcendence criterion for continued fractions with unbounded partial quotients constructed from algebraic elements. Specifically, the theorem shows that if a formal power series can be approximated by a family of algebraic series with increasing block lengths, then the formal power series is transcendental. The proof uses previous results on continued fractions over finite fields and algebraic degree estimates. An example is also given to illustrate the main result.
1) The document discusses properties of equality and inverse functions. It provides examples of finding the inverse of functions like f(x) = 3x + 6.
2) Students are asked to find the inverse of various functions through a series of examples and activities. This includes making tables, graphing functions and their inverses, and determining domains and ranges.
3) One example finds the inverse of the function y = 150 + 50x, which describes a math tutor's hourly earnings. The inverse represents the number of students that can be assisted in an hour given a particular earnings amount.
This document discusses equilibrium of particles and free body diagrams. It provides an example of drawing a free body diagram for a cylinder suspended by two cables, and using the equations of equilibrium to solve for the unknown tensions in the cables. It also discusses 3D equilibrium, giving an example problem of finding an unknown force on a particle given its position and four other forces.
In March of 2015 I'm invited to give a presentation at Data Science program at the College of William and Mary: http://jxshix.people.wm.edu/Math410-2015/index.html. This talk is hence prepared to introduce data science to college students studying mathematics. Nonetheless I hope it is useful to a general public.
The document provides an overview of analysis of variance (ANOVA). It discusses the basic idea of comparing variability within and between treatment groups. The hypotheses aim to determine if treatment means are equal. Notations are introduced for the number of treatments, sample sizes, sums, means, and variances. An example illustrates the calculations. The theory is based on a normal model, and the treatment sum of squares captures variability between means. Proofs show the expected value of the treatment sum of squares.
The document discusses parking functions, which are sequences where cars are assigned parking spots without any car being unable to find a spot. It provides examples of parking functions and non-parking functions of different lengths. It also notes that any permutation of a parking function sequence is also a parking function.
This document appears to be from a presentation given to Tucson Middle School teachers on using dominoes and coloring techniques to teach mathematical proofs. It contains a series of examples where teachers demonstrate different domino tilings, identify whether configurations are possible or impossible, and derive a formula to calculate the number of ways to tile a 2 by n board with dominoes. The examples become increasingly complex and aim to build understanding of mathematical proofs through hands-on and visual methods.
This document outlines key concepts related to time value of money, including simple and compound interest, sinking funds, annuities, amortization schedules, and bonds. It contains examples and formulas for calculating future and present values under various interest rate scenarios. The document is a lecture on quantitative methods from Dr. Ji Li at Babson College covering topics like simple and compound interest, sinking funds, annuities, bonds, and related notations and formulas.
The document discusses Cartesian products of graphs and arithmetic products of species. It begins by defining the Cartesian product of two graphs and provides an example. Unique factorization of connected graphs into prime factors is discussed. Species are then defined in relation to graphs. The arithmetic product and exponential composition of species are introduced. The talk outlines properties of prime graphs and unique factorization of connected graphs. It establishes that the set of unlabeled connected graphs forms a commutative monoid under Cartesian product, with the set of prime graphs as generators. Finally, an equation is presented relating the number of connected graphs and prime graphs.
A bijection for counting bi-point-determining graphs using the combinatorial theory of species.
23 pages, Combinatorics Seminar, Brandeis University, 2007.
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
How to Build a Module in Odoo 17 Using the Scaffold MethodCeline George
Odoo provides an option for creating a module by using a single line command. By using this command the user can make a whole structure of a module. It is very easy for a beginner to make a module. There is no need to make each file manually. This slide will show how to create a module using the scaffold method.
Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
Walmart Business+ and Spark Good for Nonprofits.pdfTechSoup
"Learn about all the ways Walmart supports nonprofit organizations.
You will hear from Liz Willett, the Head of Nonprofits, and hear about what Walmart is doing to help nonprofits, including Walmart Business and Spark Good. Walmart Business+ is a new offer for nonprofits that offers discounts and also streamlines nonprofits order and expense tracking, saving time and money.
The webinar may also give some examples on how nonprofits can best leverage Walmart Business+.
The event will cover the following::
Walmart Business + (https://business.walmart.com/plus) is a new shopping experience for nonprofits, schools, and local business customers that connects an exclusive online shopping experience to stores. Benefits include free delivery and shipping, a 'Spend Analytics” feature, special discounts, deals and tax-exempt shopping.
Special TechSoup offer for a free 180 days membership, and up to $150 in discounts on eligible orders.
Spark Good (walmart.com/sparkgood) is a charitable platform that enables nonprofits to receive donations directly from customers and associates.
Answers about how you can do more with Walmart!"
Executive Directors Chat Leveraging AI for Diversity, Equity, and InclusionTechSoup
Let’s explore the intersection of technology and equity in the final session of our DEI series. Discover how AI tools, like ChatGPT, can be used to support and enhance your nonprofit's DEI initiatives. Participants will gain insights into practical AI applications and get tips for leveraging technology to advance their DEI goals.
Liberal Approach to the Study of Indian Politics.pdf
Thesis Defense of Ji Li
1. Counting Point-Determining Graphs
and Prime Graphs
Using Joyal’s Theory of Species
Dissertation Defense
Thesis Advisor: Professor Ira Gessel
Ji Li
Department of Mathematics
Brandeis University
415 South Street, Waltham, MA
May 10th, 2007
2. Theory of Species Point-Determining Graphs Prime Graphs
Outline
1 Theory of Species
Definition of Species
Operations of Species
2 Point-Determining Graphs
Point-Determining Graphs
Bi-Point-Determining Graphs
Point-Determining 2-colored Graphs
3 Prime Graphs
Cartesian Product of Graphs
Molecular Species and P´lya’s Cycle Index Polynomial
o
Arithmetic Product of Species
Exponentiation Group
Exponential Composition of Species
J. L. Counting Point-Determining Graphs and Prime Gra
3. Theory of Species Point-Determining Graphs Prime Graphs
Definition of Species
Let B be the category of finite sets with bijections. A species (of
structures) is a functor
F :B→B
that generates for each finite set U a finite set F [U ], the set of
F -structures on U , and for each bijection σ : U → V a bijection
F [σ] : F [U ] → F [V ],
which is called the transport of F -structures along σ.
Unlabeled F -Structures
The symmetric group Sn acts on the set F [n] = F [{1, 2, . . . , n}] by
transport of structures. The Sn -orbits under this action are called
unlabeled F -structures of order n.
J. L. Counting Point-Determining Graphs and Prime Gra
4. Theory of Species Point-Determining Graphs Prime Graphs
Species of Graphs
We denote by G the species of (simple) graphs. Then G [U ] is the set
of graphs with vertex set U
Example
1 3 5
U = {1, 2, 3, 4, 5}
2 4
σ
a c e
V = {a, b, c, d, e}
b d
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Associated Series of Species
Each species F is associated with an exponential generating series
xn
F (x) = |F [n]| ,
n!
n≥0
a type generating series
F (x) = fn xn ,
n≥0
where fn is the number of unlabeled F -structures of order n, and a
cycle index of the species F , denoted ZF , satisfying
F (x) = ZF (x, 0, 0, . . . ), F (x) = ZF (x, x2 , x3 , . . . ).
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Sum of Species
An F1 + F2 -structure on a finite set U is either an F1 -structure on U
or an F2 -structure on U .
= or
F1 F2
F1 + F2
Product of Species
An F1 F2 -structure on a finite set U is of the form (π; f1 , f2 ), where π
is an ordered partition of U with two blocks U1 and U2 , fi is an
Fi -structure on Ui for each i.
=
F1 · F2 F1 F2
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Composition of Species
An F1 (F2 )-structure on a finite set U is a tuple of the form (π, f, γ),
where
• π is a partition of U
• f is an F1 -structure on the blocks of π
• γ is a set of F2 -structures on each block of π.
F2
F2
= F2 = F2
F1 ◦ F2 F1 F2 F1 F2
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Quotient Species
We say that a group A acts naturally on a species F , if for all finite
set U , there is an A-action ρU : A × F [U ] → F [U ] so that for each
bijection σ : U → V , the following diagram commutes:
ρU
A × F [U ] − − → F [U ]
−−
idA ×F [σ]
F [σ]
ρV
A × F [V ] − − → F [V ]
−−
The quotient species of F by A, denoted F/A, is such that for any
finite set U ,
(F/A)[U ] = F [U ]/A.
In other words, the set of F/A-structures on U is the set of A-orbits
of F -structures on U .
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Composition with Ek as a Quotient Species
Let k be any positive integer. Let Ek be the species of k-element sets.
Let
F · F · ····F
Fk = .
k copies
We observe that
F F F
F F F
Sk -orbits
Ek
F k /Sk = Ek ◦ F.
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10. Theory of Species Point-Determining Graphs Prime Graphs
Outline
1 Theory of Species
Definition of Species
Operations of Species
2 Point-Determining Graphs
Point-Determining Graphs
Bi-Point-Determining Graphs
Point-Determining 2-colored Graphs
3 Prime Graphs
Cartesian Product of Graphs
Molecular Species and P´lya’s Cycle Index Polynomial
o
Arithmetic Product of Species
Exponentiation Group
Exponential Composition of Species
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Neighborhood and Augmented Neighborhood
In a graph G, the neighborhood of a vertex v is the set of vertices
adjacent to v, the augmented neighborhood of a vertex is the union of
the vertex itself and its neighborhood.
Example
v
w1 w2 w3 w4
In the above figure, the neighborhood of the vertex v is the set
{w1 , w2 , w3 , w4 }, while the augmented neighborhood of v is the set
{v, w1 , w2 , w3 , w4 }.
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Point-Determining Graphs
and Co-Point-Determining Graphs
• A graph is called point-determining if no two vertices of this
graph have the same neighborhoods.
• A graph is called co-point-determining if no two vertices of this
graph have the same augmented neighborhoods.
Example
The graph on the left is co-point-determining, and the graph on the
right is point-determining. These two graphs are complements of each
other.
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A Natural Transformation
Let P be the species of point-determining graphs, and let Q be the
species of co-point-determining graphs. There is a natural
transformation
α:P →Q
that sends each point-determining graph to its complement, which is
a co-point-determining graph on the same vertex set, such that the
following diagram commutes for any bijection σ : U → V :
P[σ]
P[U ] − − → P[V ]
−−
α
α
Q[σ]
Q[U ] − − → Q[V ]
−−
We call the species P isomorphic to the species Q, written as
P = Q.
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Transform a Graph into a Point-Determining Graph
3 3
9 2 9 2
1 5 1 5
8 6 8 6
4 7 4 7
The transformation from a graph G with vertex set
[11] = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} to a point-determining graph P
with vertex set {{1, 9, 3}, {8}, {4, 7}, {6}, {2, 5}}.
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Generating Series of Point-Determining Graphs
Let E+ be the species of non-empty sets. We get a species identity
G = P ◦ E+ ,
which enables us to enumerate point-determining graphs. For
example, we can write down the beginning terms of the exponential
generating series and the type generating series of P (previously done
by Read):
x x2 x3 x4 x5 x6 x7
P(x) = 1+ + +4 +32 +588 +21476 +1551368 +· · ·
1! 2! 3! 4! 5! 6! 7!
P(x) = 1 + x + x2 + 2 x3 + 5 x4 + 16 x5 + 78 x6 + 588 x7 + · · ·
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Bi-Point-Determining Graphs
We denote by R the species of bi-point-determining graphs, which are
graphs that are both point-determining and co-point-determining.
Example
Unlabeled bi-point-determining graphs with no more than 5 vertices.
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Alternating Phylogenetic Trees
A phylogenetic tree is a rooted tree with labeled leaves and unlabeled
internal vertices in which no vertex has exactly one child.
An alternating phylogenetic tree is either a single vertex, or a
phylogenetic tree with more than one labeled vertex whose internal
vertices are colored black or white, where no two adjacent vertices are
colored the same way.
Example
5
4
8 An alternating
6
9 phylogenetic tree on 9
3
vertices, where the root
1
7
is colored black.
2
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Transform a Graph into a Bi-Point-Determining Graph
6 3
6 3
8
8 5
5 1
2
4 7
1
2
4 7 6 8 2 5 1 4
On each step, we group vertices with the same neighborhoods
or vertices with the same augmented neighborhods.
6 3
Whenever vertices with the same neighborhods are grouped, 8
we connect the corresponding vertices/alternating phylogenetic 5
trees with a black node. 1
2
Whenever vertices with the same augmented neighborhoods 4 7
are grouped, we connected the corresponding vertices/
alternating phylogenetic trees with a white node. 7
6 8 2 5 1 4
Vertices left untouched are not colored.
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A Species Identity for Bi-Point-Determining Graphs
The species of graphs is the composition of the species of
bi-point-determining graphs and the species of alternating
phylogenetic trees
T
= T
G
R T
G =R ◦T
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Generating Series of Bi-Point-Determining Graphs
Through calculation, we write functional equations for the
exponential generating series and the type generating series of R:
x x4 x5 x6 x7 x8
R(x) = +12 +312 +13824 +1147488 +178672128 +· · ·
1! 4! 5! 6! 7! 8!
R(x) = x + x4 + 6x5 + 36x6 + 324x7 + 5280x8 + · · ·
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Multisort Species
Let Bk be the category of finite k-sets with bijective multifunctions.
A species of k sorts is a functor
F : Bk → B.
2-Colored Graphs
A 2-colored graph is a graph in which all vertices are colored either
white or black, and no two adjacent vertices are assigned the same
color.
We denote by G (X, Y ) the 2-sort species of 2-colored graphs, where
vertices colored white are of sort X, and vertices colored black are of
sort Y .
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Point-Determining 2-colored Graphs
• A 2-colored graph is called point-determining if the underlying
graph is point-determining.
• A 2-colored graph is called semi-point-determining if all vertices
of the same color have distinct neighborhoods.
• Note that the graph is semi-point-determining, but it is
not point-determining.
We denote by
• P(X, Y ) — the 2-sort species of point-determining 2-colored
graphs
• P s (X, Y ) — the 2-sort species of semi-point-determining
2-colored graph
• P c (X, Y ) — the 2-sort species of connected point-determining
2-colored graph
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Functional Equations: Part I
The idea is similar to the formula for the species of point-determining
graphs P:
G = P ◦ E+ .
We transform a 2-colored graph into a semi-point-determining
2-colored graph by grouping vertices with the same neighborhoods.
Note that if two vertices have the same neighborhoods, then they
must be colored in the same way.
G (X, Y ) = P s (E+ (X), E+ (Y )).
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Functional Equations: Part II
The observation that a semi-point-determining graph consists of
• one or none isolated vertex colored white
• one or none isolated vertex colored black
• a set (possibly empty) of connected point-determining 2-colored
graphs with at least two vertices
leads to the functional equation:
P s (X, Y ) = (1 + X)(1 + Y ) E (P≥2 (X, Y ))
c
c c
P≥2 (X, Y ) P≥2 (X, Y )
1+X
P s (X, Y ) E
1+Y
c c
P≥2 (X, Y ) P≥2 (X, Y )
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Functional Equations: Part III
Similarly, we observe that a point-determining 2-colored graph
consists of
• one or none isolated vertex, colored white or black
• a set of connected point-determining 2-colored graphs with at
least two vertices
Therefore,
c
P(X, Y ) = (1 + X + Y ) E (P≥2 (X, Y ))
c c
P≥2 (X, Y ) P≥2 (X, Y )
P(X, Y ) E
1+X +Y
c c
P≥2 (X, Y ) P≥2 (X, Y )
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Generating Series of Point-Determining 2-colored Graphs
These functional equations allow us to calculate the generating series
of the species P s (X, Y ), P c (X, Y ), and P(X, Y ).
For example,
P(x, y) = 1 + x + y+
xy + x2 y + xy 2 + 2x2 y 2
+ 3x3 y 2 + 3x2 y 3 + · · · . Unlabeled point-determining
2-colored graphs with no more
than 5 vertices.
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Outline
1 Theory of Species
Definition of Species
Operations of Species
2 Point-Determining Graphs
Point-Determining Graphs
Bi-Point-Determining Graphs
Point-Determining 2-colored Graphs
3 Prime Graphs
Cartesian Product of Graphs
Molecular Species and P´lya’s Cycle Index Polynomial
o
Arithmetic Product of Species
Exponentiation Group
Exponential Composition of Species
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Cartesian Product of Graphs
The Cartesian product of two graphs G1 and G2 , denoted G1 ⊙ G2 , is
the graph whose vertex set is
V (G1 ⊙ G2 ) = {(u, v) : u ∈ V (G1 ), v ∈ V (G2 )},
and in which the vertex (u1 , v1 ) is adjacent to the vertex (u2 , v2 ) if
either u1 = u2 and v1 is adjacent to v2 or v1 = v2 and u1 is adjacent
to u2 .
Example
1
1,1’
2 4
2,1’ 4,1’
3
3,1’ 1,2’
2,2’ 4,2’
1,3’
1’ 2,3’ 4,3’
3,2’
3’ 2’ 3,3’
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Properties of the Cartesian Product
The Cartesian product is commutative and associative. We write
n
Gn = ⊡ G.
i=1
Prime Graphs
A graph G is said to be prime with respect to Cartesian
multiplication if G is a non-trivial connected graph such that
G ∼ H1 ⊙ H2 implies that either H1 or H2 is a singleton vertex.
=
Relatively Prime
Two graphs G and H are called relatively prime with respect to
Cartesian multiplication if and only if G = G1 ⊙ J and H ∼ H1 ⊙ J
∼ =
imply that J is a singleton vertex.
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Decomposition of a Connected Graph
Any non-trivial connected graph can be decomposed into prime
factors. Sabidussi proved that such a prime factorization is unique up
to isomorphism.
Example
=
A connected graph with 24 vertices is decomposed into prime graphs
with 2 vertices 3 vertices, and 4 vertices, respectively.
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Molecular Species
• A molecular species is a species that is indecomposable under
addition.
• If M is molecular, then M = Mn for some n, i.e., M [U ] is
nonempty if and only if U is an n-element set.
• If M = Mn , then M = X n /A for some subgroup A of Sn .
• The X n /A-structures on a finite set U , where |U | = n, is the set
of A-orbits of the action A on the set of linear orders on U . In
other words, X n /A is the quotient species of X n by A.
• Each subgroup A of Sn gives rise to a molecular species X n /A.
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Cycle Index of a Group
• Let A be a subgroup of Sn . The cycle index polynomial of A,
defined by P´lya, is
o
n
1 c (σ)
Z(A) = Z(A; p1 , p2 , . . . , pn ) = pkk ,
|A|
σ∈A k=1
where for a permutation σ, ck (σ) is the number of k-cycles in σ.
• If a molecular species M = X n /A, then the cycle index of the
species M is the same as the cycle index polynomial of the group
A. That is,
Z(A) = ZX n /A .
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Species Associated to a Graph
Each graph G is associated to a species OG , where the OG -structures
on a finite set U is defined to be the set of graphs isomorphic to G
with vertex set U .
a b b c c d d e e a
d c e d a e b a c b
e a b c d
G OG [{a, b, c, d, e}]
OG is Molecular
The automorphism group of G acts on the vertex set of G. If G is a
graph with n vertices, then aut(G) may be identified with a subgroup
of Sn , and
Xn
OG = .
aut(G)
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Product Group
Let A be a subgroup of Sm , and let B be a subgroup of Sn . We
define the product group A × B to be the subgroup of Smn such that
a) the group operation is
(a1 , b1 ) · (a2 , b2 ) = (a1 a2 , b1 b2 )
b
b) an element (a, b) of
A × B acts on (i, j) for
some i ∈ [m] and j ∈ [n] by
(a, b)(i, j) = (a(i), b(j)) a
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Arithmetic Product of Species
In the above setting, we start with two molecular species X m /A and
X n /B, and get a new molecular species X mn /(A × B), which is
defined to be the arithmetic product of X m /A and X n /B:
B-orbits
Xm Xn X mn
⊡ := .
A B A×B
A-orbits
The arithmetic product of species was previously studied by Maia and
M´ndez.
e
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Properties of the Arithmetic Product
The arithmetic product has the following properties (given by Maia
and M´ndez):
e
commutativity F1 ⊡ F2 = F2 ⊡ F1 ,
associativity F1 ⊡ (F2 ⊡ F3 ) = (F1 ⊡ F2 ) ⊡ F3 ,
distributivity F1 ⊡ (F2 + F3 ) = F1 ⊡ F2 + F1 ⊡ F3 ,
unit F1 ⊡ X = X ⊡ F1 = F1 .
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Cartesian Product of Graphs
and Arithmetic Product of Species
Let G1 and G2 be two graphs that are relatively prime to each other.
Then the species associated to the Cartesian product of G1 and G2 is
equivalent to the arithmetic product of the species associated to G1
and the species associated to G2 . That is,
OG1 ⊙G2 = OG1 ⊡ OG2
Proof
Since G1 and G2 are relatively prime, a theorem of Sabidussi gives
that aut(G1 ⊙ G2 ) = aut(G1 ) × aut(G2 ). Therefore,
X mn X mn
OG1 ⊙G2 = =
aut(G1 ⊙ G2 ) aut(G1 ) × aut(G2 )
Xm Xn
= ⊡ = OG1 ⊡ OG2 .
aut(G1 ) aut(G2 )
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Exponentiation Group
Let A be a subgroup of Sm , and let B be a subgroup of Sn . The
exponentiation group B A is a subgroup of Snm , whose group
elements are of the form (α, τ ) with α ∈ A and τ : [m] → B.
a) The composition of
two elements (α, τ ) and
α
(β, η) is given by α τ (1)
(α, τ )(β, η) = (αβ, (τ ◦β)η).
τ (5 )
τ (2 )
b) The element (α, τ )
acts on the set of
functions from [m] to
α
[n] by sending each τ(
α
4) 3)
τ(
f : [m] → [n] to g,
where for all i ∈ [m],
α
g(i) = τ (i)(f (α−1 i)).
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I Operators
Let (α, τ ) be an element of the exponentiation group B A such that
• α is an m-cycle in the group A
• τ = (τ (1), τ (2), . . . , τ (m)) ∈ B m satisfies that the cycle type of
τ (m)τ (m − 1) · · · τ (2)τ (1) is λ
Palmer and Robinson defined the operators Im on the power sum
symmetric functions by
Im (pλ ) = pγ ,
where γ is the cycle type of the element (α, τ ) of B A .
More explicitly, γ = (γ1 , γ2 , . . . ) is the partition of nm with
gcd(m,l)
1 j
cj (γ) = µ ici (λ) .
j l
l|j i | l/ gcd(m,l)
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⊠ Operator
• The operation ⊠ on the symmetric functions is defined by letting
pν := pλ ⊠ pµ ,
where
ck (ν) = gcd(i, j) ci (λ)cj (µ).
lcm(i,j)=k
• If a ∈ A has cycle type λ, and b ∈ B has cycle type µ, then
(a, b) ∈ A × B has cycle type ν.
• If λ = (λ1 , λ2 , . . . ) is a partition of n, then
Iλ (pµ ) = Iλ1 (pµ ) ⊠ Iλ2 (pµ ) ⊠ · · · .
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Cycle Index of Exponentiation Group
Theorem
(Palmer and Robinson) The cycle index polynomial of B A is the
image of Z(B) under the operator obtained by substituting the
operator Ir for the variables pr in Z(A). That is,
Z(B A ) = Z(A) ∗ Z(B).
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An Example of the Exponentiation Group
Let A = S2 and B = C3 .
The element (α, τ ) of B A , with α = (1, 2), τ (1) = id and
τ (2) = (1, 2, 3), acts on the set of functions from [2] to [3].
The cycle
type of (α, τ )
is (6, 3),
which means,
I2 (p3 ) = p3 p6 .
We can calculate the cycle index of the exponentiation group using
Palmer and Robinson’s theorem:
1 9
Z(B A ) = (p + 8p3 + 3p3 p3 + 6p3 p6 ).
18 1 3 2
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Exponential Composition of Species
We define the molecular B-orbits A-
m or
species X n /B A to be the or
bit
s bit
s
A-
exponential composition of
X m /A and X n /B:
B - or
bits
B - or
bits
Xm Xn XN
:= .
A B BA
A-or
bits
A-or
bits
Or equivalently,
B- s
⊡m or bit
or
Xm Xn Xn bit
s B-
:= A.
A B B A-orbits
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Exponential Composition of a General Species
• Recall that the species of k-element sets Ek = X k /Sk . We call
Ek F the exponential composition of F of order k.
• The cycle index of the exponential composition is
ZEk X n /A = Z(Sk ) ∗ Z(A).
• Setting E0 F = X, we set
E F := Ek F .
k≥0
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Properties of Exponential Composition
The exponential composition of species satisfies the additive
properties:
k
Ek F1 + F2 = Ei F1 ⊡ Ek−i F2 ,
i=0
E F1 + F2 = E F1 ⊡ E F2 .
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Prime Power P k
Let P be any prime graph, and k any nonnegative integer.
Sabidussi showed that the automorphism group of P k is
aut(P k ) = aut(P )Sk .
Therefore,
k k
Xk Xn Xn Xn
Ek OP = = = = OP k .
Sk aut(P ) aut(P )Sk aut(P k )
E OP = X + OP + OP 2 + · · · .
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Species of Prime Graphs
Let G c be the species of connected graphs. Let P be the species of
prime graphs. We can write it in terms of the sum of all prime
graphs, i.e., P = P OP .
We then apply the additive property of the exponential composition:
E P =E OP = ⊡ E OP = ⊡(X + OP + OP 2 + · · · ).
P P
P
This means that we get all connected graphs, since each connected
graph has a unique prime factorization (Sabidussi)! Therefore,
Theorem
E P = G c.
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Cycle Index of the Species of Prime Graphs
In order to get a formula for the cycle index of the exponential
composition, we generalize Palmer and Robinson’s theorem for the
cycle index polynomial of the exponentiation group, and the cycle
index of the species of prime graphs can be then calculated, say, using
Maple:
1 2 1 2 3 1
ZP = p + p2 + p + p1 p2 + p3
2 1 2 3 1 3
35 4 7 2 7 1
+ p + p2 p2 + p1 p3 + p2 + p4
24 1 4 1 3 8 2 4
91 5 19 3 4 2 3
+ p + p p2 + p3 p3 + 5p1 p2 + p1 p4 + p2 p3 + p5
15 1 3 1 3 1 2
3 5
+ ···
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Thank you!
J. L. Counting Point-Determining Graphs and Prime Gra