This document summarizes an algorithm to computationally prove Rota's basis conjecture for matroids of rank 4 or less. The algorithm starts with the given bases and uses basis exchange properties to infer new bases. It continues inferring bases until either finding n disjoint rainbow bases, proving the conjecture, or being unable to infer any more bases, providing a counterexample. Rank 3 is solved with one case, while rank 4 requires around 100,000 cases. The algorithm chooses 3-tuples of bases judiciously to minimize the number of cases needed at each step.
On Analytic Review of Hahn–Banach Extension Results with Some GeneralizationsBRNSS Publication Hub
The useful Hahn–Banach theorem in functional analysis has significantly been in use for many years ago. At this point in time, we discover that its domain and range of existence can be extended point wisely so as to secure a wider range of extendibility. In achieving this, we initially reviewed the existing traditional Hahn–Banach extension theorem, before we carefully and successfully used it to generate the finite extension form as in main results of section three.
A unique common fixed point theorems in generalized dAlexander Decker
This document presents some definitions and results related to generalized D*-metric spaces and establishes a unique common fixed point theorem for such spaces. It begins by introducing D*-metric spaces and generalized D*-metric spaces, along with some key definitions like convergence and Cauchy sequences. It then presents several lemmas about convergence and uniqueness of limits in generalized D*-metric spaces. The document concludes by stating that it obtains a unique common fixed point theorem for generalized D*-metric spaces.
Stability criterion of periodic oscillations in a (9)Alexander Decker
This document studies the fixed points and properties of the Duffing map, which is a 2-D discrete dynamical system. It finds that the Duffing map has three fixed points when a > b + 1. It divides the parameter space into six regions to determine whether the fixed points are attracting, repelling, or saddle points. It also determines the bifurcation points of the parameter space. Key properties of the Duffing map explored include that it is a diffeomorphism when b ≠ 0, its Jacobian is equal to b, and the eigenvalues of its derivative depend on a and b.
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.
The document discusses various topics related to sequences including:
- Definitions of sequences and different types of sequences such as arithmetic progressions, geometric progressions, and recurrence relations.
- Examples of sequences used in computer programming to determine if a number is even or odd through modulo operations.
- How the principle of mathematical induction can be used to prove statements about sequences, including the first and second principles of mathematical induction.
The document defines and explains various concepts related to sets:
- A set is a well-defined collection of objects that can be determined if an object belongs to the set or not. Sets can be defined using a roster method or set-builder notation.
- Basic set properties include sets being inherently unordered and elements being unequal. Set membership, empty sets, and common number sets are also introduced. Equality of sets, Venn diagrams, subsets, power sets, set operations, and generalized unions and intersections are further discussed.
A tutorial on the Frobenious Theorem, one of the most important results in differential geometry, with emphasis in its use in nonlinear control theory. All results are accompanied by proofs, but for a more thorough and detailed presentation refer to the book of A. Isidori.
On Analytic Review of Hahn–Banach Extension Results with Some GeneralizationsBRNSS Publication Hub
The useful Hahn–Banach theorem in functional analysis has significantly been in use for many years ago. At this point in time, we discover that its domain and range of existence can be extended point wisely so as to secure a wider range of extendibility. In achieving this, we initially reviewed the existing traditional Hahn–Banach extension theorem, before we carefully and successfully used it to generate the finite extension form as in main results of section three.
A unique common fixed point theorems in generalized dAlexander Decker
This document presents some definitions and results related to generalized D*-metric spaces and establishes a unique common fixed point theorem for such spaces. It begins by introducing D*-metric spaces and generalized D*-metric spaces, along with some key definitions like convergence and Cauchy sequences. It then presents several lemmas about convergence and uniqueness of limits in generalized D*-metric spaces. The document concludes by stating that it obtains a unique common fixed point theorem for generalized D*-metric spaces.
Stability criterion of periodic oscillations in a (9)Alexander Decker
This document studies the fixed points and properties of the Duffing map, which is a 2-D discrete dynamical system. It finds that the Duffing map has three fixed points when a > b + 1. It divides the parameter space into six regions to determine whether the fixed points are attracting, repelling, or saddle points. It also determines the bifurcation points of the parameter space. Key properties of the Duffing map explored include that it is a diffeomorphism when b ≠ 0, its Jacobian is equal to b, and the eigenvalues of its derivative depend on a and b.
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.
The document discusses various topics related to sequences including:
- Definitions of sequences and different types of sequences such as arithmetic progressions, geometric progressions, and recurrence relations.
- Examples of sequences used in computer programming to determine if a number is even or odd through modulo operations.
- How the principle of mathematical induction can be used to prove statements about sequences, including the first and second principles of mathematical induction.
The document defines and explains various concepts related to sets:
- A set is a well-defined collection of objects that can be determined if an object belongs to the set or not. Sets can be defined using a roster method or set-builder notation.
- Basic set properties include sets being inherently unordered and elements being unequal. Set membership, empty sets, and common number sets are also introduced. Equality of sets, Venn diagrams, subsets, power sets, set operations, and generalized unions and intersections are further discussed.
A tutorial on the Frobenious Theorem, one of the most important results in differential geometry, with emphasis in its use in nonlinear control theory. All results are accompanied by proofs, but for a more thorough and detailed presentation refer to the book of A. Isidori.
The document discusses Bayesian networks (BNs), which are directed acyclic graphs used to represent probabilistic relationships between variables. Each node in a BN represents a variable, and connections between nodes represent probabilistic dependencies. These dependencies are quantified with conditional probability tables (CPTs). BNs allow compact representation of joint probability distributions by exploiting conditional independence relationships. Learning BNs from data involves estimating both the network structure and the parameters of each CPT. When the structure is known and all variables are observable, parameters can be estimated with maximum likelihood.
5.3 areas, riemann sums, and the fundamental theorem of calaculusmath265
The document defines definite integrals and Riemann sums. It states that a definite integral calculates the area under a function between limits a and b by dividing the interval into subintervals and summing the areas of rectangles approximating the function over each subinterval. Riemann sums make this approximation explicit by taking the width of each subinterval times the value of the function at a sample point in the subinterval. In the limit as the subintervals approach zero width, the Riemann sum converges to the true integral value.
I am Piers L. I am a Calculus Assignment Expert at mathsassignmenthelp.com. I hold a Master's in Mathematics from, the University of Adelaide. I have been helping students with their assignments for the past 6 years. I solve assignments related to Calculus.
Visit mathsassignmenthelp.com or email info@mathsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Calculus Assignment.
This document discusses relations and cartesian products of sets. It defines a relation as a subset of the cartesian product of two sets that links elements between the sets. It provides examples of forming cartesian products of different sets to generate ordered pairs, and defines relations between the elements of those pairs based on given conditions. The key concepts covered are cartesian products, ordered pairs, domains and ranges of relations, and representing relations using arrow diagrams or set-builder and roster forms.
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 discusses various methods for solving first order differential equations, including:
1. Variable separable methods where the equation can be written as a function of x multiplied by a function of y.
2. Homogeneous equations where both sides are homogeneous functions of the same degree.
3. Exact equations where there exists an integrating factor.
4. Equations that can be transformed to an exact or separable form through substitution.
5. Linear equations that can be solved using an integrating factor that is a function of x.
A new computational methodology to find appropriateAlexander Decker
This document proposes a new computational methodology for finding appropriate solutions to fuzzy equations of the form A=B, where A and B are known fuzzy numbers and X is an unknown fuzzy number. It introduces definitions of positive, negative, and half-positive/half-negative fuzzy numbers. Propositions and theorems are presented to show that the solution to such an equation can be a positive, negative, or half-positive/half-negative fuzzy number depending on the properties of A and B. Examples are provided to illustrate solving equations where A and B are positive, negative, or half-positive/half-negative fuzzy numbers. The methodology aims to overcome limitations in solving equations where X is an unknown fuzzy number.
11.a new computational methodology to find appropriateAlexander Decker
This document proposes a new computational methodology for finding appropriate solutions to fuzzy equations of the form A=B, where A and B are known continuous triangular fuzzy numbers and X is an unknown fuzzy number. It introduces definitions of "positive fuzzy number," "negative fuzzy number," and "half-positive and half-negative fuzzy number." Several propositions and theorems are presented along with proofs to show that the solution to such a fuzzy equation can be a positive, negative, or half-positive/half-negative fuzzy number, depending on the values of A and B. Examples are provided to illustrate solving equations where A and B are positive, negative, or half-positive/half-negative fuzzy numbers. The methodology aims to overcome limitations
This document contains notes on probability theory from a course. It begins with definitions of measures, σ-algebras, Borel σ-algebras, and related concepts. It then proves some key properties, including that the Borel σ-algebra on the real line can be generated by open intervals with rational endpoints. The document also contains proofs showing when two measures are equal and the Monotone Class Theorem for sets.
On Some Types of Fuzzy Separation Axioms in Fuzzy Topological Space on Fuzzy ...IOSR Journals
This document introduces and studies various types of fuzzy open sets in fuzzy topological spaces, including fuzzy δ-open sets, fuzzy ∆-open sets, fuzzy γ-open sets, fuzzy θ-open sets, and fuzzy regular open sets. It defines each type of fuzzy open set and establishes relationships between them. The document also introduces several types of fuzzy separation axioms in fuzzy topological spaces, including fuzzy δT0 spaces, and proves some properties and theorems about these concepts. Figures and examples are provided to illustrate the concepts and show that the converse of some relationships between fuzzy open set types do not always hold.
55 addition and subtraction of rational expressions alg1testreview
The document discusses addition and subtraction of rational expressions. It states that rational expressions can only be added or subtracted if they have the same denominator. It provides the rule for adding and subtracting rational expressions with the same denominator. It also discusses converting rational expressions to have a common denominator so they can be added or subtracted, using the least common multiple of the denominators. Examples are provided to demonstrate adding and subtracting rational expressions, as well as converting them to have a common denominator.
This document provides information on number theory topics covered in Unit 1 of a mathematics course, including:
- Number systems such as natural numbers, integers, rational numbers, and irrational numbers. Rational numbers can be expressed as ratios of integers, while irrational numbers cannot.
- Representing rational and irrational numbers on the number line. Rational numbers are easy to represent precisely, while irrational numbers correspond to points.
- Properties of finite and repeating decimals, including how to write decimals as fractions.
- Absolute value, which is the distance from a number to zero on the number line.
13 multiplication and division of rational expressionselem-alg-sample
The document discusses multiplication and division of rational expressions. It presents the multiplication rule for rational expressions as the product of the numerators over the product of the denominators. It provides an example of simplifying a rational expression by factoring the top and bottom and canceling like terms. It then gives another example with two parts, simplifying and expanding the answers of rational expression operations.
This document discusses partial ordering in the context of soft sets. It begins with basic definitions of soft sets and soft set operations like complement, Cartesian product, and composition of soft set relations. It then defines what a partial order is in terms of being reflexive, antisymmetric, and transitive. A partially ordered soft set is one where the soft set elements have a partial order defined on them. Linear (total) ordering is also discussed, where all elements in the soft set are comparable. Examples are provided to illustrate these concepts of ordering in soft sets.
Special Elements of a Ternary SemiringIJERA Editor
In this paper we study the notion of some special elements such as identity, zero, absorbing, additive
idempotent, idempotent, multiplicatively sub-idempotent, regular, Intra regular, completely regular, g–regular,
invertible and the ternary semirings such as zero sum free ternary semiring, zero ternary semiring, zero divisor
free ternary semiring, ternary semi-integral domain, semi-subtractive ternary semiring, multiplicative
cancellative ternary semiring, Viterbi ternary semiring, regular ternary semiring, completely ternary semiring
and characterize these ternary semirings.
Mathematics Subject Classification : 16Y30, 16Y99.
Using Alpha-cuts and Constraint Exploration Approach on Quadratic Programming...TELKOMNIKA JOURNAL
In this paper, we propose a computational procedure to find the optimal solution of quadratic programming
problems by using fuzzy -cuts and constraint exploration approach. We solve the problems in
the original form without using any additional information such as Lagrange’s multiplier, slack, surplus and
artificial variable. In order to find the optimal solution, we divide the calculation in two stages. In the first
stage, we determine the unconstrained minimization of the quadratic programming problem (QPP) and check
its feasibility. By unconstrained minimization we identify the violated constraints and focus our searching in
these constraints. In the second stage, we explored the feasible region along side the violated constraints
until the optimal point is achieved. A numerical example is included in this paper to illustrate the capability of
-cuts and constraint exploration to find the optimal solution of QPP.
- The order of a root of a polynomial is the number of times the root repeats.
- The polynomial x5 + 2x4 + x3 has two roots, x = 0 with order 3 and x = -1 with order 2.
- In general, polynomials of the form k(x - c1)m(x - c2)m...(x - cn)m have roots x = c1 with order m1, x = c2 with order m2, and so on.
DSA (Data Structure and Algorithm) QuestionsRESHAN FARAZ
The document contains questions related to data structures and algorithms. Some questions ask to perform operations on data structures like stacks, queues, trees, graphs and hash tables. Other questions involve algorithms like topological sorting, shortest path algorithms, minimum spanning tree algorithms, and binary search tree operations. The document tests understanding of concepts taught in an algorithms course through examples and problem solving questions.
This document provides an overview of communication complexity and deterministic communication complexity. It begins with defining the problem setup, which involves two parties (Alice and Bob) trying to compute a function f based on their private inputs using the minimum communication. It then discusses protocol trees, which model communication protocols, and shows they partition the input space into rectangles. It introduces combinatorial rectangles and shows how the minimum number of rectangles needed to partition the space can provide a lower bound on communication complexity. The document also discusses fooling sets, another technique to lower bound communication complexity, and previews upcoming topics on nondeterministic and randomized communication complexity.
Este documento discute las variaciones dialectales del español hispanoamericano. En la sección 2, se describen diversos rasgos dialectales variables como la velarización no asimilatoria, el trueque de líquidas, la asibilación, la uvularización, la velarización y la anticipación, y se mencionan sus orígenes y distribuciones geográficas.
Steven Stewart has over 15 years of experience in electronics engineering, technology, and avionics. He holds a B.S. in Electronics Engineering Technology and Electronics Technology from the University of Central Missouri as well as an A.A.S. in Electronics Technology from the Community College of the Air Force. His work history includes positions as an Electronics Technician, Satellite Installation Technician, Sales Manager, Field Service Manager, and Avionics Specialist for the United States Air Force. He has extensive experience building and testing electronic systems, interpreting schematics and blueprints, providing customer service, and supervising teams.
The document discusses Bayesian networks (BNs), which are directed acyclic graphs used to represent probabilistic relationships between variables. Each node in a BN represents a variable, and connections between nodes represent probabilistic dependencies. These dependencies are quantified with conditional probability tables (CPTs). BNs allow compact representation of joint probability distributions by exploiting conditional independence relationships. Learning BNs from data involves estimating both the network structure and the parameters of each CPT. When the structure is known and all variables are observable, parameters can be estimated with maximum likelihood.
5.3 areas, riemann sums, and the fundamental theorem of calaculusmath265
The document defines definite integrals and Riemann sums. It states that a definite integral calculates the area under a function between limits a and b by dividing the interval into subintervals and summing the areas of rectangles approximating the function over each subinterval. Riemann sums make this approximation explicit by taking the width of each subinterval times the value of the function at a sample point in the subinterval. In the limit as the subintervals approach zero width, the Riemann sum converges to the true integral value.
I am Piers L. I am a Calculus Assignment Expert at mathsassignmenthelp.com. I hold a Master's in Mathematics from, the University of Adelaide. I have been helping students with their assignments for the past 6 years. I solve assignments related to Calculus.
Visit mathsassignmenthelp.com or email info@mathsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Calculus Assignment.
This document discusses relations and cartesian products of sets. It defines a relation as a subset of the cartesian product of two sets that links elements between the sets. It provides examples of forming cartesian products of different sets to generate ordered pairs, and defines relations between the elements of those pairs based on given conditions. The key concepts covered are cartesian products, ordered pairs, domains and ranges of relations, and representing relations using arrow diagrams or set-builder and roster forms.
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 discusses various methods for solving first order differential equations, including:
1. Variable separable methods where the equation can be written as a function of x multiplied by a function of y.
2. Homogeneous equations where both sides are homogeneous functions of the same degree.
3. Exact equations where there exists an integrating factor.
4. Equations that can be transformed to an exact or separable form through substitution.
5. Linear equations that can be solved using an integrating factor that is a function of x.
A new computational methodology to find appropriateAlexander Decker
This document proposes a new computational methodology for finding appropriate solutions to fuzzy equations of the form A=B, where A and B are known fuzzy numbers and X is an unknown fuzzy number. It introduces definitions of positive, negative, and half-positive/half-negative fuzzy numbers. Propositions and theorems are presented to show that the solution to such an equation can be a positive, negative, or half-positive/half-negative fuzzy number depending on the properties of A and B. Examples are provided to illustrate solving equations where A and B are positive, negative, or half-positive/half-negative fuzzy numbers. The methodology aims to overcome limitations in solving equations where X is an unknown fuzzy number.
11.a new computational methodology to find appropriateAlexander Decker
This document proposes a new computational methodology for finding appropriate solutions to fuzzy equations of the form A=B, where A and B are known continuous triangular fuzzy numbers and X is an unknown fuzzy number. It introduces definitions of "positive fuzzy number," "negative fuzzy number," and "half-positive and half-negative fuzzy number." Several propositions and theorems are presented along with proofs to show that the solution to such a fuzzy equation can be a positive, negative, or half-positive/half-negative fuzzy number, depending on the values of A and B. Examples are provided to illustrate solving equations where A and B are positive, negative, or half-positive/half-negative fuzzy numbers. The methodology aims to overcome limitations
This document contains notes on probability theory from a course. It begins with definitions of measures, σ-algebras, Borel σ-algebras, and related concepts. It then proves some key properties, including that the Borel σ-algebra on the real line can be generated by open intervals with rational endpoints. The document also contains proofs showing when two measures are equal and the Monotone Class Theorem for sets.
On Some Types of Fuzzy Separation Axioms in Fuzzy Topological Space on Fuzzy ...IOSR Journals
This document introduces and studies various types of fuzzy open sets in fuzzy topological spaces, including fuzzy δ-open sets, fuzzy ∆-open sets, fuzzy γ-open sets, fuzzy θ-open sets, and fuzzy regular open sets. It defines each type of fuzzy open set and establishes relationships between them. The document also introduces several types of fuzzy separation axioms in fuzzy topological spaces, including fuzzy δT0 spaces, and proves some properties and theorems about these concepts. Figures and examples are provided to illustrate the concepts and show that the converse of some relationships between fuzzy open set types do not always hold.
55 addition and subtraction of rational expressions alg1testreview
The document discusses addition and subtraction of rational expressions. It states that rational expressions can only be added or subtracted if they have the same denominator. It provides the rule for adding and subtracting rational expressions with the same denominator. It also discusses converting rational expressions to have a common denominator so they can be added or subtracted, using the least common multiple of the denominators. Examples are provided to demonstrate adding and subtracting rational expressions, as well as converting them to have a common denominator.
This document provides information on number theory topics covered in Unit 1 of a mathematics course, including:
- Number systems such as natural numbers, integers, rational numbers, and irrational numbers. Rational numbers can be expressed as ratios of integers, while irrational numbers cannot.
- Representing rational and irrational numbers on the number line. Rational numbers are easy to represent precisely, while irrational numbers correspond to points.
- Properties of finite and repeating decimals, including how to write decimals as fractions.
- Absolute value, which is the distance from a number to zero on the number line.
13 multiplication and division of rational expressionselem-alg-sample
The document discusses multiplication and division of rational expressions. It presents the multiplication rule for rational expressions as the product of the numerators over the product of the denominators. It provides an example of simplifying a rational expression by factoring the top and bottom and canceling like terms. It then gives another example with two parts, simplifying and expanding the answers of rational expression operations.
This document discusses partial ordering in the context of soft sets. It begins with basic definitions of soft sets and soft set operations like complement, Cartesian product, and composition of soft set relations. It then defines what a partial order is in terms of being reflexive, antisymmetric, and transitive. A partially ordered soft set is one where the soft set elements have a partial order defined on them. Linear (total) ordering is also discussed, where all elements in the soft set are comparable. Examples are provided to illustrate these concepts of ordering in soft sets.
Special Elements of a Ternary SemiringIJERA Editor
In this paper we study the notion of some special elements such as identity, zero, absorbing, additive
idempotent, idempotent, multiplicatively sub-idempotent, regular, Intra regular, completely regular, g–regular,
invertible and the ternary semirings such as zero sum free ternary semiring, zero ternary semiring, zero divisor
free ternary semiring, ternary semi-integral domain, semi-subtractive ternary semiring, multiplicative
cancellative ternary semiring, Viterbi ternary semiring, regular ternary semiring, completely ternary semiring
and characterize these ternary semirings.
Mathematics Subject Classification : 16Y30, 16Y99.
Using Alpha-cuts and Constraint Exploration Approach on Quadratic Programming...TELKOMNIKA JOURNAL
In this paper, we propose a computational procedure to find the optimal solution of quadratic programming
problems by using fuzzy -cuts and constraint exploration approach. We solve the problems in
the original form without using any additional information such as Lagrange’s multiplier, slack, surplus and
artificial variable. In order to find the optimal solution, we divide the calculation in two stages. In the first
stage, we determine the unconstrained minimization of the quadratic programming problem (QPP) and check
its feasibility. By unconstrained minimization we identify the violated constraints and focus our searching in
these constraints. In the second stage, we explored the feasible region along side the violated constraints
until the optimal point is achieved. A numerical example is included in this paper to illustrate the capability of
-cuts and constraint exploration to find the optimal solution of QPP.
- The order of a root of a polynomial is the number of times the root repeats.
- The polynomial x5 + 2x4 + x3 has two roots, x = 0 with order 3 and x = -1 with order 2.
- In general, polynomials of the form k(x - c1)m(x - c2)m...(x - cn)m have roots x = c1 with order m1, x = c2 with order m2, and so on.
DSA (Data Structure and Algorithm) QuestionsRESHAN FARAZ
The document contains questions related to data structures and algorithms. Some questions ask to perform operations on data structures like stacks, queues, trees, graphs and hash tables. Other questions involve algorithms like topological sorting, shortest path algorithms, minimum spanning tree algorithms, and binary search tree operations. The document tests understanding of concepts taught in an algorithms course through examples and problem solving questions.
This document provides an overview of communication complexity and deterministic communication complexity. It begins with defining the problem setup, which involves two parties (Alice and Bob) trying to compute a function f based on their private inputs using the minimum communication. It then discusses protocol trees, which model communication protocols, and shows they partition the input space into rectangles. It introduces combinatorial rectangles and shows how the minimum number of rectangles needed to partition the space can provide a lower bound on communication complexity. The document also discusses fooling sets, another technique to lower bound communication complexity, and previews upcoming topics on nondeterministic and randomized communication complexity.
Este documento discute las variaciones dialectales del español hispanoamericano. En la sección 2, se describen diversos rasgos dialectales variables como la velarización no asimilatoria, el trueque de líquidas, la asibilación, la uvularización, la velarización y la anticipación, y se mencionan sus orígenes y distribuciones geográficas.
Steven Stewart has over 15 years of experience in electronics engineering, technology, and avionics. He holds a B.S. in Electronics Engineering Technology and Electronics Technology from the University of Central Missouri as well as an A.A.S. in Electronics Technology from the Community College of the Air Force. His work history includes positions as an Electronics Technician, Satellite Installation Technician, Sales Manager, Field Service Manager, and Avionics Specialist for the United States Air Force. He has extensive experience building and testing electronic systems, interpreting schematics and blueprints, providing customer service, and supervising teams.
Rebekah Schilling is seeking an administrative position that allows her to utilize her organization, communication, and computer skills. She has 9 years of experience in administrative roles, including office management, customer service, and project administration for a construction company. She is proficient in Microsoft Office, Adobe, and database applications and has a typing speed of 95 wpm.
O documento discute o verdadeiro significado do Natal, que é o amor de Deus. Deus amou a humanidade e deu seu Filho Jesus como um sacrifício para pagar pelos pecados da humanidade, livrando-os da condenação. Ao aceitarmos este dom gratuito, podemos ter a eternidade com Deus.
Melissa Crain has over 10 years of experience in interior design. She has worked as an interior designer for Jacobs, Cassidy Turley, and Directions in Design. At Cassidy Turley, she specialized in designing branch offices for Edward Jones and managed the design of over 60 locations at a time. She has a Bachelor's degree in Interior Design from Southern Illinois University. Her skills include AutoCAD, Revit Architecture, Sketchup, Adobe Creative Suite, and Microsoft Office. She has also been involved in several leadership activities and volunteer programs.
Este documento presenta un mapa conceptual sobre los avances tecnológicos y nuevas técnicas en higiene y seguridad industrial. Explica cómo las nuevas tecnologías han simplificado acciones como prevenir incendios, inundaciones, y rescatar personas atrapadas, además de restablecer comunicaciones después de desastres. También describe la aplicación diaria de estas tecnologías en centros de prevención y la reorganización de la estructura de comunicación según métodos tecnológicos como Twitter, web y radio durante emergencias.
Ana Marin-Sanchez is a visual artist and creative director with over 20 years of experience in fine art, visual design, photography, video art, and installations. She has received several awards and selections for her work including the Revelation Artist Prize in 1998 and first prizes for painting and sculpture in competitions in 1987. Her areas of expertise include painting, sculpting, drawing, photography, video art, and working as an art coordinator, creative director, and technical assistant on audiovisual projects.
The candidate is applying for a senior project superintendent position and has over 30 years of experience in construction project management. He has experience managing a variety of construction projects including medical facilities, aquatic centers, colleges, and hotels. His responsibilities included scheduling, coordinating subcontractors, ensuring safety and quality standards, and using project management software. He holds several certifications and security clearances relevant to construction and project management.
This document contains 8 PowerPoint presentations about various topics related to Canada. The presentations cover resources for teaching about Canada and Canadians, a basic introduction to Canada, indigenous people of the Pacific Northwest, Canadian geography, history of western development, Canada's trade relationship with the US, and Canada's political system from a comparative perspective.
The document is a resume for Mary Ellen Bonnell. It summarizes her work experience from 2011 to present as a crossing guard and bus monitor for the Haverhill School Department, from 2009 to 2011 as a scheduling coordinator for Lahey Clinic in Burlington, and from 2008 to 2009 as a MA Health billing representative for Andover Ear, Nose and Throat Center. It also lists her education and skills.
The pictures chosen link to the abstract title sequence being filmed in a park. While some pictures may seem random, they consider elements that could be implemented, such as a night picture creating suspense. The diary on the grass photo connects to clues in the title sequence "THE LOST", creating secrecy and making the audience wonder what secrets the characters hold.
The document discusses the benefits of exercise and a healthy lifestyle. It notes that regular physical activity can help reduce the risk of diseases and improve mental health and quality of life. The document encourages readers to aim for at least 30 minutes of exercise per day to gain these advantages.
Kaisa Bowman Hall has over 30 years of experience in fundraising, communications, and administrative roles for independent schools and non-profits. She most recently worked as an independent contractor providing marketing and administrative services for realtors. Prior to that, she held director level development positions at Churchill School and Cheshire Academy where she established and managed annual giving programs, special events, databases, and donor relations. She has also served in numerous volunteer leadership roles in her community.
We manufacture finned tube heat exchanger using high quality materials, specially designed fins & good manufacturing standards to maintain high quality of manufactured finned tube heat exchanger.
IBMA 2016 - Dr. B Payne - The Great Depression in CanadaK-12 STUDY CANADA
The Great Depression had a significant impact on Canada and its political landscape in the 1930s. Canada responded conservatively without implementing programs like the New Deal. Political leaders like William Lyon Mackenzie King and Richard Bedford Bennett emphasized balanced budgets over government intervention. Radical parties led by J.S. Woodsworth and "Bible Bill" Aberhart gained popularity out of frustration with the lack of response to economic hardship, though Bennett also attempted to appeal to workers. Regional and ethnic tensions increased during this difficult time.
Shane Ledbetter Student ID 228604 Unit VIII White Paper NanomaterialShane Ledbetter
The document discusses occupational safety considerations for manufacturing nanomaterials. It notes that nanomaterials are increasingly used but little is known about their health effects. It recommends that companies establish effective safety management systems to assess hazards, implement controls like ventilation, and monitor exposures. Engineering controls are important to minimize hazards and protect workers. A full risk assessment including identifying sources of highest exposure and background information is needed to develop an appropriate safety plan for nanomaterial production. Worker health should be the top priority.
Sequencing cannabis sativa and cannabis indica, Courtagen Life Sciences, Inc,...Copenhagenomics
1. Medicinal Genomics Corporation sequenced the genomes of several cannabis cultivars including Chemdawg and LA Confidential.
2. Analysis of the genome sequences revealed high levels of polymorphisms and identified several copies of THCA synthase genes with single nucleotide variants.
3. RNA sequencing of different tissues showed tissue-specific expression of potential cannabinoid synthase genes, including a novel root-expressed synthase with a different N-terminal domain.
4. A family tree analysis grouped the potential synthase genes into families across the different cultivars.
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Paper
1. COMPUTATIONAL PROOF OF ROTA’S BASIS
CONJECTURE FOR MATROIDS OF RANK 4
MICHAEL SZE-YU CHEUNG
Abstract. We prove Rota’s basis conjecture for matroids of rank
n ≤ 4 using a computer program. The program starts with n
arbitrary bases and uses basis exchange properties to infer new
bases until it has n disjoint rainbow bases (solving the conjecture)
or no more bases can be inferred (providing a counter-example).
n = 3 is solved in one case whereas n = 4 is solved in around a
hundred thousand cases.
1. Introduction
Given n disjoint bases in an n-dimensional vector space, we wish to
show that there exist n pairwise disjoint transversals of these bases that
are also bases. If we think of the original n bases as each having a color,
then we are trying to show that there exists n disjoint “rainbow” bases.
To prove this, we know that given two bases, there are theorems that
imply the existence of other bases. Hence, we apply these theorems
to the n given bases to imply the existence of other bases. Now we
apply these theorems to the n given bases along with these other bases
to imply the existence of even more bases. This is continued until we
have implied the existence of n disjoint “rainbow” bases, or we cannot
imply any more bases (in which case the conjecture is false). Due to
the many steps involved and cases that must be considered, we use a
computer to perform this task.
The concept of independence in a vector space can be generalized
into a mathematical structure known as a matroid. This conjecture
generalizes to matroids, and all of our techniques for proving it do as
well. Hence, we use our algorithm to try and prove or disprove this
strengthened conjecture.
Definition 1.1 (matroid). A matroid is an ordered pair M = (S, B),
where S is a finite set (called the ground set) and B is a collection
of subsets of S (called the bases of M), that satisfies the following
Date: August 6, 2012.
This research was supported by NSF grant DMS-1004516.
I would like to thank my REU advisors Dr. Joshua Ducey and Dr. Minah Oh.
1
2. ROTA’S BASIS CONJECTURE 2
properties:
M1: B is nonempty.
M2: Given any two bases B1, B2 and an element x of the first basis, we
can find some element y of the second basis such that (B1 − {x}) ∪ {y}
is a basis.
Example 1.2 (matroid from vectors in a vector space). The following
four column vectors (labeled 1, 2, 3, 4)
elements: 1 2 3 4
1 0 0 1
0 1 0 1
0 0 1 0
yield the following matroid: S = {1, 2, 3, 4}, B = {{1, 2, 3}, {1, 3, 4}, {2, 3, 4}}.
Note that not all matroids arise from vector spaces. Theorems that
allow us to infer bases in a matroid from known bases are of interest
to us; we shall use the following lemma from White [5].
Lemma 1.3 (White’s lemma). Given any two bases D, E and an or-
dered partition O(D) = (D1, . . . , Dk) of D, we can find an ordered par-
tition O(E) = (E1, . . . , Ek) of E so that ∀i ∈ {1, . . . , k}, (D − Di) ∪ Ei
is a basis. Note that this requires that |Di| = |Ei| for all i.
We will simply refer to “ordered partition” by “partition” if there is
no ambiguity. Furthermore, we shall refer to each element of a partition
as a “block.” In 1.3, if we choose the first block to be D1 = D∩E, then
clearly E1 = D ∩ E = D1. Furthermore, any block containing D ∩ E
can split into D ∩ E and another block contained in D − E, so 1.3
is equivalent to the following theorem, where we replace O(D), O(E)
with O(D − E), O(E − D).
Theorem 1.4 (partition basis exchange). Given any two bases D, E
and an ordered partition O(D − E) = (D1, . . . , Dk) of D − E, we can
find an ordered partition O(E − D) = (E1, . . . , Ek) of E − D so that
∀i ∈ {1, . . . , k}, (D − Di) ∪ Ei is a basis.
Notice that we can choose two bases D, E and a partition O(D−E),
but the corresponding partition O(E −D) is unknown. Hence, we refer
to this choice (D, E, O(D−E)) as a “3-tuple,” and a partition O(E−D)
as a “candidate.” Below is an example showing why 1.3 and 1.4 are
equivalent.
3. ROTA’S BASIS CONJECTURE 3
Example 1.5 (partition basis exchange 1). If D = {a1, a2, a3} and
E = {a1, b2, b3}, then any subset of D containing a1 must be swapped
for a subset of E containing a1. Suppose not, that is, suppose some
other subset of D (a2 or a3) must be swapped for a1. If we swap say
a2 for a1 in E, then we have that (D − {a2}) ∪ {a1} = {a1, a1, a3} is a
basis, a contradiction.
Below is an example applying 1.4
Example 1.6 (partition basis exchange 2). If we are given the 3-tuple
D = {a1, a2, a3, a4}, E = {b1, b2, b3, b4}, O(D−E) = ({a1, a2}, {a3}, {a4}),
then possible candidates for O(E −D) are all partitions with one block
of order two followed by two blocks of order one:
({b1, b2}, {b3}, {b4})
({b1, b2}, {b4}, {b3})
({b1, b3}, {b2}, {b4})
({b1, b3}, {b4}, {b2})
({b1, b4}, {b2}, {b3})
({b1, b4}, {b3}, {b2})
({b2, b3}, {b1}, {b4})
({b2, b3}, {b4}, {b1})
({b2, b4}, {b1}, {b3})
({b2, b4}, {b3}, {b1})
({b3, b4}, {b1}, {b2})
({b3, b4}, {b2}, {b1})
We then have twelve cases, one for each O(E − D), each implying
the existence of three bases. Case 1 (O(E −D) = ({b1, b2}, {b3}, {b4}))
implies the existence of these bases by 1.4:
{b1, b2, a3, a4}
{a1, a2, b3, a4}
{a1, a2, a3, b4}
Notice that if the partition of D − E consists of one-element blocks
of D − E, then we have the following corollary:
Corollary 1.7 (bijective basis exchange). Given any two bases D, E,
there is a bijective function f : D − E → E − D such that ∀x ∈
D − E, (D − {x}) ∪ {f(x)} is a basis.
Notice further that if the partition of D−E contains only two blocks
{D1, D − E − D1}, then the corresponding partition of E − D contains
two blocks {E1, E − D − E1}, so 1.4 implies the existence of two bases,
4. ROTA’S BASIS CONJECTURE 4
(D−D1)∪E1 and (D−(D−E −D1))∪(E −D−E1) = D1 ∪(E −E1).
This yields the following corollary:
Corollary 1.8 (subset basis exchange). Given any two bases B1, B2
and a subset A1 of B1 − B2, there exists a subset A2 of B2 − B1 such
that (B1 − A1) ∪ A2 and (B2 − A2) ∪ A1 are bases.
Example 1.9 (subset basis exchange). For n = 3, we are given bases
D1 = {a1, a2, a3}, D2 = {b1, b2, b3}, D3 = {c1, c2, c3}. If we choose B1 =
D1, B2 = D2, A1 = {a1, a2}, A2 could be either {b1, b2}, {b1, b3}, or
{b2, b3}. Hence, we must consider three cases - one for each possible A2.
Case 1 (A2 = {b1, b2}) provides two new bases: D4 = {a3, b1, b2}, D5 =
{a1, a2, b3}. Now if we choose B1 = D5, B2 = D3, A1 = {b3}, then A2
could be either {c1}, {c2}, or {c3}. Hence, we have another three cases
(Case 1-1, 1-2, 1-3).
In 1989, Rota made a conjecture about another basis exchange prop-
erty of matroids.
Conjecture 1.10 (Rota’s basis conjecture). Let M be a matroid of
rank n and let B∗
= {B1, . . . , Bn} be a collection of n disjoint bases
in M. Then there exists n pairwise disjoint transversals of B∗
that are
bases.
If we think of each of the n given bases as having a color, we can
call these pairwise disjoint transversals “rainbow” bases. Furthermore,
we can define the rainbowness of a basis in this matroid as being the
number of bases in B∗
that it has a non-empty intersection with. For
example, all bases in B∗
have rainbowness 1 and all rainbow bases have
rainbowness n.
2. Algorithm
The n bases given by 1.10 do not fully describe the matroid, since
1.4 implies the existence of other bases. In fact, we can “complete” the
matroid by inferring new bases until no new bases can be inferred. At
this point, our collection of known bases fully describes a matroid, since
this collection along with the n2
elements from the bases in B∗
satisfies
M1 and 1.4, which implies M2. While in the process of completing this
matroid, if we ever have n disjoint rainbow bases, we have proven the
conjecture and don’t need to finish completing the matroid. On the
other hand, if we have completed the matroid yet there is no set of n
disjoint rainbow bases in it, then this matroid is a counter-example to
1.10.
5. ROTA’S BASIS CONJECTURE 5
Specifically, we start with the given n bases (assuming nothing about
these bases besides the fact that they satisfy the axioms of a matroid)
and proceed in a series of steps until we have proven the existence of
n disjoint rainbow bases (or arrived at a counter-example). At each
step, we choose a 3-tuple (D, E, O(D − E)) and apply 1.4 to prove the
existence of new bases. We will need to consider a separate case for
each valid O(E − D). It is preferable to prove 1.10 in as few cases as
possible, so we first look for 3-tuples where we can eliminate all but
one candidate O(E − D). This way, we can prove the existence of new
bases without having to branch off into more cases. If no such 3-tuple
exists, we must find the “best” 3-tuple that will branch off into the
least number of cases and prove 1.10 in as few steps as possible. To do
this, we compute a score for each 3-tuple based on the number of cases
that must be considered (one case for each candidate O(E − D)) and
the quality of the new bases resulting from each case.
2.1. 3-Tuples. Consider the 3-tuple (D, E, O(D − E)) again. If there
is some O(E − D) such that applying 1.4 will result in no new bases,
then skip this 3-tuple.
Scores for 3-tuples are computed by finding the case with the low-
est case score and dividing that score by ln |C| where |C| is the number
of cases. Case scores are computed by adding together the basis scores
for each new basis corresponding to that case. In computing the basis
scores, we require a fixed list of “element score values” and “subscore
factors.” We try to choose values and factors that minimize the number
of cases that must be considered, though this is largely experimental
and by no means rigorous. A new basis score is computed as follows.
(1) Group the elements in the new basis by which starting basis in
B∗
the element belongs to. Each group might contain anywhere
from 0 to n elements.
(2) For each existing rainbow basis, compute a subscore as follows:
(a) We first compute an element score for each of the elements
of the rainbow basis.
(b) Associate each element in the rainbow basis to a group
in the new basis based on which starting basis in B∗
the
element of the rainbow basis belongs to.
(c) For each element of the rainbow basis, find the number of
elements in its associated group that are not equal to it.
Call this the disjointness of this element of the rainbow
6. ROTA’S BASIS CONJECTURE 6
basis. The element score corresponding to each element of
the rainbow basis is a value determined by the disjointness
of that element. For instance, disjointness {0, 1, 2, 3} could
correspond to element scores of {0, 1, 3, 10}. Call this list
{0, 1, 3, 10} the “element score values.” For example, if our
rainbow basis contains an a3 and its corresponding group
is {a3, a4, a5}, then since a4, a5 are different from a3, the
element a3 of the rainbow basis has disjointness 2. This
would yield an element score of 3. Once again, the element
scores are chosen by the user, more or less experimentally.
However, we should require that the element score values
are strictly increasing, since we would like to award a higher
score to (i.e. prefer) bases that are more disjoint from
currently known rainbow bases.
(d) Average the nonzero element scores of each element of the
rainbow basis. Multiply this number by a factor deter-
mined by the number of nonzero subscores. For instance,
the factors corresponding to {0, 1, 2, 3} nonzero subscores
could be {0, 1, 2, 4}. Call this list of factors the “subscore
factors.” For instance, given element scores {0, 5, 0, 3, 4},
the average of the nonzero element scores is 4, and since
there are 3 nonzero element scores, we multiply it by the
subscore factor 4 for a subscore of 16. These subscore
factors are also chosen experimentally, but once again we
should require that the subscore factors are strictly increas-
ing.
(3) The basis score is the sum of all subscores divided by the num-
ber of rainbow bases. We divide to normalize the score across
different steps of the algorithm, since earlier steps have less rain-
bow bases whereas later steps have more. This normalization
is actually not necessary in the algorithm since within a partic-
ular step of a particular case, the number of rainbow bases will
be the same. However, it is useful for gathering statistics and
theoretical discussion.
Example 2.1 (subscore calculation). Let the element score values be
{0,2,3,4,5} and the subscore factors {0,1,3,7,15,31} Suppose our start-
ing bases are {a1, a2, a3, a4}, {b1, b2, b3, b4}, {c1, c2, c3, c4}, {d1, d2, d3, d4}
and our new basis is {a1, a2, b2, d4}. Then our four groups are {a1, a2}, {b2}, ∅, {d4}.
To calculate the subscore for the existing rainbow basis {a3, b2, c2, d2},
we compute the element scores of each element in the rainbow basis
(Figure 1). From this, the average of the nonzero subscores is 3+2
2
= 2.5.
7. ROTA’S BASIS CONJECTURE 7
Group {a1, a2} {b2} ∅ {d4}
Element a3 b2 c2 d2
Disjoint {a1, a2} ∅ ∅ {d4}
Disjoint# 2 0 0 1
Subsubscore 3 0 0 2
Figure 1. Table illustrating the calculation of a sub-
score. “Element” refers an element in the rainbow basis.
“Disjoint” refers to the bases in the group that are dis-
joint from the corresponding element.
Since we have two nonzero subscores, we multiply by the corresponding
subscore factor of 3, for a final subscore of 7.5.
This score is designed to hopefully choose the optimal 3-tuple each
time to reach n disjoint rainbow bases as quickly as possible. Of course,
this score does not rigorously determine which 3-tuple is the best, but
we believe it to be a good tradeoff between time spent computing the
score and time saved by choosing a better 3-tuple (and hence proving
1.10 in less cases and steps). The reason why we divide by ln |C| at
the end is because at each step that we apply 1.4 we get a more or less
constant number of new bases (maybe 1 to 4 for n = 4), whereas the
number of new cases is somewhat proportional to the current number
of cases. We illustrate this concept with two examples.
Example 2.2 (3-tuple scoring 1). Suppose at step s some 3-tuple
branches off into 3 cases, with the worst case (that is, the case with
the lowest case score) scoring 5. This would score 5
ln 3
= 10
ln 9
. Now if we
branch off into 3 cases with the worst case scoring 5 (again) at step s+1
for each of the 3 cases, we will have 9 cases and a combined score of
10, which corresponds with the right hand side (RHS) of the equation.
Alternatively, a 3-tuple at step s could branch off into 9 cases, with
the worst case scoring 10. By our metric, this would again yield the
same score. Hence, we believe our metric finds an appropriate trade-off
between number of cases and the combined quality of the new bases
resulting from each case.
Example 2.3 (3-tuple scoring 2). Suppose at some step in the program
we have a choice between two 3-tuples. One branches off into 3 cases at
one application of 1.4, with the worst case yielding a normalized score
of 5. Another branches off into 8 cases with the worst case yielding two
bases scoring a total of 9.5 normalized. By our metric, these 3-tuples
are about the same quality, since 9.5
ln 8
≈ 4.57 and 5
ln 3
≈ 4.55.
8. ROTA’S BASIS CONJECTURE 8
2.2. Eliminating candidates. In order to reduce the number of cases,
we try to eliminate as many of the candidate partitions O(E − D) as
possible. If the addition of the new bases proves 1.10, then we can
eliminate that candidate. If two candidates result in two similar cases,
we can eliminate one of the two candidates.
2.2.1. Permutation cases. We formalize the concept of two cases being
similar (so that only one must be considered). Define some permutation
σ of the elements of the ground set S of a matroid. Note that we can
permute all subsets of S with σ as well. Given a collection B of bases,
we can permute all the bases of B (since they are all subsets of S) to
produce a new collection of subsets of S (not necessarily all bases).
Hence, we can define the action of σ on a collection of bases in this
way.
To test two cases for similarity, let B1 and B2 be the collection of all
known bases after applying 1.4 on the first and second case. If some
permutation σ of B1 yields B2, then these cases are similar and only
one must be considered.
Example 2.4. Consider B1 = {a1, a2}, B2 = {b1, b2}, O(D) = ({a1}, {a2});
we have candidates ({b1}, {b2}), ({b2}, {b1}) that each produce two new
bases B3, B4, resulting in two collections of bases:
Collection/Basis B1 B2 B3 B4
B1 {a1, a2} {b1, b2} {b1, a2} {a1, b2}
B2 {a1, a2} {b1, b2} {b2, a2} {a1, b1}
However, B1 and B2 are unchanged under the permutation σ that
swaps b1 with b2, and the new bases B3 = {a2, b1}, {a2, b2} and B4 =
{a1, b2}, {a1, b1} are equivalent after applying σ to one or the other.
Hence, one of these cases can be eliminated.
2.2.2. Superset cases. Ultimately, we hope to prove that one can always
find n disjoint rainbow bases starting with the initial collection B∗
=
{B1, . . . , Bn}. On the way, we prove cases that start with B∗
plus some
other bases that we have inferred using 1.4. Suppose we prove a case
C1 that starts with the collection of bases B1. While proving another
case C2, we arrive at a collection of bases that contains B1. Then we
no longer need to consider C2 since we can prove the conjecture for C2
by following the same steps as C1, ignoring the additional bases that
C2 has.
In fact, we only need to consider the difference between the new
bases inferred for each case. For instance, suppose case 1 results in a
new basis X and case 2 results in a new basis Y . If the conjecture is
9. ROTA’S BASIS CONJECTURE 9
Unproven case 1-2 2-1 2-2 2-3-1 2-3-2
Proven case 1-1 1 2-1 1 2-2
Missing bases L H, I O H, I L, M
Figure 2. Details of superset cases in Figure 3
A,B,C
D,E,F,G
J
K,I
Q,R
L,P
M
H
L,M
N,O
O
H
H,I
M,N
L,P
L
Proven!
Figure 3. Case tree. Orange lines indicate a superset case
proven for case 1, and case 2 eventually infers a new basis X, then case
2 has also been proven, following the same steps as case 1. Consider
Figure 3, which illustrates an application of this concept. Each letter
represents a basis. As more bases are inferred, we find the need to
branch off into different cases. The program considers cases on the left
first, and proves the conjecture for case 1-1. However, case 1-2 infers
the basis L, meaning it is proven through case 1-1. We call case 1-2
a superset case. We list all superset cases for this example in Figure
2. Here, an “unproven case” is proven through a “proven case” by
inferring the “missing bases” that the unproven case needs to contain
all the bases of the proven case.
2.3. Exchange graph. Again from White, there is another way to
infer new bases that does not require branching off into cases [5]:
Definition 2.5 (exchange graph). An exchange graph G for a collec-
tion of bases B in a matroid M is a directed multigraph whose vertices
10. ROTA’S BASIS CONJECTURE 10
are the members of the ground set of the matroid. There is an edge
from a to b labeled B1 if and only if b ∈ B1 ∈ B and (B1−{b})∪{a} ∈ B.
Definition 2.6 (shortcut). Given a path or cycle v1
B1
−→ v2
B2
−→ . . .
Bn−1
−−−→
vn, we say this path or cycle has a shortcut if and only if we can find
i < j − 1 such that there is an edge directed from vi to vj.
Theorem 2.7 (no shortcuts). Let v1
B1
−→ v2
B2
−→ . . .
Bn−1
−−−→ vn be a path
or cycle with no shortcuts. The edge labels are not necessarily distinct
but the vertices are, except in the case of a cycle where v1 = vn. Now let
i ∈ {1, . . . , n − 1} and let a1
Bi
−→ b1, . . . , ak
Bi
−→ bk be all ordered vertex
pairs with an edge labeled by Bi. Then (Bi −{b1, . . . , bk})∪{a1, . . . , ak}
is a basis in M.
Recall that each candidate of each 3-tuple infers a number of new
bases. These new bases will introduce new edges in the exchange graph,
which may imply the existence of more new bases.
3. Results
Rota’s basis conjecture has been completely proven for n = 3 in
three cases by Chan [1], who uses 1.8 and other clever techniques. Fur-
thermore, Onn proved that when n is even, the Alon-Tarsi conjecture
implies 1.10 for all matroids arising from vector spaces (except vector
spaces of certain characteristics) [4]. The Alon-Tarsi conjecture has
been proven for n = p + 1 for all p prime by [2]. Furthermore, 1.10 has
been verified for all paving matroids [3]. Hence, n = 4 for matroids is
still open.
Using our algorithm, we have created a program that can completely
prove Rota’s basis conjecture for any n. The program generates a proof
while it runs; the full proof of n = 3 is shown below. We have proven
n = 3 in one case and n = 4 in around a hundred thousand cases. For
n ≥ 5, the program cannot complete in a reasonable amount of time.
However, we can still use the program to look for counter-examples for
these cases.
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B0 = a1 a2 a3
B1 = b1 b2 b3
B2 = c1 c2 c3
B3 = a1 a3 b1
B4 = a2 b2 b3
B5 = a1 b1 c1