Proposals
Below are some proposals for talks from the past (and current). By clicking on the ID number, more details are shown. By default, these are sorted chronologically (recent first) and by then by last name. The data can be sorted by alternate means by using the links at the top right, each allowing ascending or descending orders.
Displaying 181-200 of 471 results.
Sort by:
ID:
474
Year: 2017
Name: Aqeeb Sabree
Institution: University of Iowa
Subject area(s): Advanced Calculus would be helpful
Title of Talk: Research Topics from Reproducing Kernel Hilbert Spaces
Abstract: Reproducing Kernel Hilbert Spaces (RKHS) have applications to statistics, machine learning, differential equations, and more. The goal of this presentation is to introduce the concept of a RKHS, and discuss it’s applications to many research areas. The amazing thing about this research area is that there are many research questions/topics for dissertations or undergraduate research experiences. I will give a brief history of RKHSs, highlighting where it has appeared and how it has been applied. Then I will present the theoretical foundation(s) of the subject; from here I will go into its applications. Below, you will find highlights of the theory that I will present, and some highlights of its application. You can discuss the existence of RKHSs in different ways: one, you can prove that the evaluation functional is bounded; or, two, you can prove that (given a Hilbert space) the Hilbert space has a reproducing kernel function. A nice property of the reproducing kernel is that it is unique. Thus, every RKHS has exactly one reproducing kernel; furthermore, every reproducing kernel is the reproducing kernel for a unique RKHS (Moore--Aronszajn). The process of recreating the RKHS from the kernel function is termed the it reconstruction problem, and is an interesting research area. The usefulness of the theory of RKHSs can be seen in the fact that the finite energy Fourier, Hankel, sine, and cosine transformed band-limited signals are specific realizations of the abstract reproducing kernel Hilbert space (RKHS). Sampling Theory: Sampling theory deals with the reconstruction of functions (or signals) from their values (or samples) on an appropriate set of points. When given a reproducing kernel Hilbert space, H; one asks: What are some (suitable) sets of points which reproduce (or interpolate) the full values of functions from H? And when given points in a set S, one asks: What are the RKHSs for which S is a complete set of sample points? Meaning the values of functions from H are reproduced by interpolation from S.
Year: 2017
Name: Aqeeb Sabree
Institution: University of Iowa
Subject area(s): Advanced Calculus would be helpful
Title of Talk: Research Topics from Reproducing Kernel Hilbert Spaces
Abstract: Reproducing Kernel Hilbert Spaces (RKHS) have applications to statistics, machine learning, differential equations, and more. The goal of this presentation is to introduce the concept of a RKHS, and discuss it’s applications to many research areas. The amazing thing about this research area is that there are many research questions/topics for dissertations or undergraduate research experiences. I will give a brief history of RKHSs, highlighting where it has appeared and how it has been applied. Then I will present the theoretical foundation(s) of the subject; from here I will go into its applications. Below, you will find highlights of the theory that I will present, and some highlights of its application. You can discuss the existence of RKHSs in different ways: one, you can prove that the evaluation functional is bounded; or, two, you can prove that (given a Hilbert space) the Hilbert space has a reproducing kernel function. A nice property of the reproducing kernel is that it is unique. Thus, every RKHS has exactly one reproducing kernel; furthermore, every reproducing kernel is the reproducing kernel for a unique RKHS (Moore--Aronszajn). The process of recreating the RKHS from the kernel function is termed the it reconstruction problem, and is an interesting research area. The usefulness of the theory of RKHSs can be seen in the fact that the finite energy Fourier, Hankel, sine, and cosine transformed band-limited signals are specific realizations of the abstract reproducing kernel Hilbert space (RKHS). Sampling Theory: Sampling theory deals with the reconstruction of functions (or signals) from their values (or samples) on an appropriate set of points. When given a reproducing kernel Hilbert space, H; one asks: What are some (suitable) sets of points which reproduce (or interpolate) the full values of functions from H? And when given points in a set S, one asks: What are the RKHSs for which S is a complete set of sample points? Meaning the values of functions from H are reproduced by interpolation from S.
ID:
257
Year: 2009
Name: Michael Hilgemann
Institution: Iowa State University
Subject area(s): Algebra
Title of Talk: The classification of finite-dimensional Hopf algebras
Abstract: Hopf algebras can be considered generalizations of groups, and group algebras are basic examples of such objects. In recent years there have been developments in the classification of finite-dimensional Hopf algebras over an algebraically closed field of characteristic 0, which include many examples which are neither group algebras nor the linear dual of group algebras. In this talk, we will highlight these classification results and some of the useful properties that general finite-dimensional Hopf algebras share with finite group algebras. In particular, we will discuss recent joint work with Richard Ng that completes the classification of Hopf algebras of dimension 2p^2, for p an odd prime.
Year: 2009
Name: Michael Hilgemann
Institution: Iowa State University
Subject area(s): Algebra
Title of Talk: The classification of finite-dimensional Hopf algebras
Abstract: Hopf algebras can be considered generalizations of groups, and group algebras are basic examples of such objects. In recent years there have been developments in the classification of finite-dimensional Hopf algebras over an algebraically closed field of characteristic 0, which include many examples which are neither group algebras nor the linear dual of group algebras. In this talk, we will highlight these classification results and some of the useful properties that general finite-dimensional Hopf algebras share with finite group algebras. In particular, we will discuss recent joint work with Richard Ng that completes the classification of Hopf algebras of dimension 2p^2, for p an odd prime.
ID:
52
Year: 2004
Name: Ruth Berger
Institution: Luther College
Subject area(s): algebra
Title of Talk: Fun & Games with Permutation groups
Abstract: This talk will give an introduction to the
Year: 2004
Name: Ruth Berger
Institution: Luther College
Subject area(s): algebra
Title of Talk: Fun & Games with Permutation groups
Abstract: This talk will give an introduction to the
ID:
315
Year: 2011
Name: Russell Goodman
Institution: Central College
Subject area(s): Algebra
Title of Talk: Baumslag-Solitar Groups and Their Representations
Abstract: The presenter will provide the audience with background on the Baumslag-Solitar groups and will share results from his search for simple representations of such groups. There will also be a discussion of the deformations of his simple representations and how this leads to determining the dimension of the representation variety and representation scheme at those simple representations.
Year: 2011
Name: Russell Goodman
Institution: Central College
Subject area(s): Algebra
Title of Talk: Baumslag-Solitar Groups and Their Representations
Abstract: The presenter will provide the audience with background on the Baumslag-Solitar groups and will share results from his search for simple representations of such groups. There will also be a discussion of the deformations of his simple representations and how this leads to determining the dimension of the representation variety and representation scheme at those simple representations.
ID:
418
Year: 2015
Name: Kevin Gerstle
Institution: University of Iowa
Subject area(s): Algebra
Title of Talk: Algebras and Coalgebras
Abstract: While algebra is widely recognized as an important branch of mathematics, most people do not know how the objects called algebras play a vital role in our understanding of many commonly used number systems such as the real and complex numbers. In addition, the dual notion of coalgebras give us a way to introduce a new type of structure to these systems allowing us novel, exciting ways to talk about numbers. In this talk, we will explore the interplay between algebras and coalgebras, and I will show what information these algebraic structures give us about some of our favorite number systems.
Year: 2015
Name: Kevin Gerstle
Institution: University of Iowa
Subject area(s): Algebra
Title of Talk: Algebras and Coalgebras
Abstract: While algebra is widely recognized as an important branch of mathematics, most people do not know how the objects called algebras play a vital role in our understanding of many commonly used number systems such as the real and complex numbers. In addition, the dual notion of coalgebras give us a way to introduce a new type of structure to these systems allowing us novel, exciting ways to talk about numbers. In this talk, we will explore the interplay between algebras and coalgebras, and I will show what information these algebraic structures give us about some of our favorite number systems.
ID:
503
Year: 2018
Name: Justin Hoffmeier
Institution: Northwest Missouri State University
Subject area(s): Algebra
Title of Talk: Exact Zero Divisor Graphs
Abstract: Zero divisor graphs of rings identify the elements of the annihilators. Is it possible to identify the generators of the annihilators from these graphs? We work examples for which the answer is yes. Our explanation uses exact zero divisor graphs. For this talk, rings will be only the integers modulo n and we will assume very little background knowledge.
Year: 2018
Name: Justin Hoffmeier
Institution: Northwest Missouri State University
Subject area(s): Algebra
Title of Talk: Exact Zero Divisor Graphs
Abstract: Zero divisor graphs of rings identify the elements of the annihilators. Is it possible to identify the generators of the annihilators from these graphs? We work examples for which the answer is yes. Our explanation uses exact zero divisor graphs. For this talk, rings will be only the integers modulo n and we will assume very little background knowledge.
ID:
226
Year: 2008
Name: Bokhee Im
Institution: Chonnam National University, Rep. of Korea
Subject area(s): algebra ( combitorics )
Title of Talk: Certain quasigroup homogeneous spaces
Abstract: A quasigroup is defined as a set Q equipped with a multiplication, not necessarily associative, such that in the equation x y=z, knowledge of any two of the elements x, y, z of Q specifies the third uniquely. In particular, the solution for x in terms of y and z is written as z/y. The body of the multiplication table of a finite quasigroup is a Latin quare. Nonempty associative quasigroups are groups. In this talk, we consider the usual direct product G of the symmetric group of degree 3 and the cyclic group of order 2. By changing some intercalates of the body of the multiplication table of the group G, we get various quasigroup structures on the set G. We study homogeneous spaces derived from such a quasigroup and show how each action matrix acts on an orbit contained in the homogeneous space. Action matrices show the approximate symmetry.
Year: 2008
Name: Bokhee Im
Institution: Chonnam National University, Rep. of Korea
Subject area(s): algebra ( combitorics )
Title of Talk: Certain quasigroup homogeneous spaces
Abstract: A quasigroup is defined as a set Q equipped with a multiplication, not necessarily associative, such that in the equation x y=z, knowledge of any two of the elements x, y, z of Q specifies the third uniquely. In particular, the solution for x in terms of y and z is written as z/y. The body of the multiplication table of a finite quasigroup is a Latin quare. Nonempty associative quasigroups are groups. In this talk, we consider the usual direct product G of the symmetric group of degree 3 and the cyclic group of order 2. By changing some intercalates of the body of the multiplication table of the group G, we get various quasigroup structures on the set G. We study homogeneous spaces derived from such a quasigroup and show how each action matrix acts on an orbit contained in the homogeneous space. Action matrices show the approximate symmetry.
ID:
341
Year: 2012
Name: Angela Kohlhaas
Institution: Loras College
Subject area(s): Algebra (Commutative Algebra)
Title of Talk: Cores of Monomial Ideals
Abstract: Blow-up algebras associated to an ideal I are at the center of the interplay between commutative algebra and algebraic geometry. One can study these algebras through minimal reductions of I, or simpler ideals inside of I which retain much of I
Year: 2012
Name: Angela Kohlhaas
Institution: Loras College
Subject area(s): Algebra (Commutative Algebra)
Title of Talk: Cores of Monomial Ideals
Abstract: Blow-up algebras associated to an ideal I are at the center of the interplay between commutative algebra and algebraic geometry. One can study these algebras through minimal reductions of I, or simpler ideals inside of I which retain much of I
ID:
259
Year: 2009
Name: Christian Roettger
Institution: Iowa State University
Subject area(s): Algebra, elementary number theory
Title of Talk: Sequences and their annihilators
Abstract: Annihilating polynomials have been widely used in geometry and to study sequences over fields and over the integers Z. We use the same simple ideas to study sequences over Z modulo n. There are surprising difficulties, surprisingly nice results and an open conjecture. We can demonstrate some applications to recurrence sequences like the Fibonacci and Lucas numbers, or discrete dynamical systems. Joint work with John Gillespie. Prerequisites: ring, ideal, quotient ring, Chinese Remainder theorem - suitable for undergraduates with a first course in algebra.
Year: 2009
Name: Christian Roettger
Institution: Iowa State University
Subject area(s): Algebra, elementary number theory
Title of Talk: Sequences and their annihilators
Abstract: Annihilating polynomials have been widely used in geometry and to study sequences over fields and over the integers Z. We use the same simple ideas to study sequences over Z modulo n. There are surprising difficulties, surprisingly nice results and an open conjecture. We can demonstrate some applications to recurrence sequences like the Fibonacci and Lucas numbers, or discrete dynamical systems. Joint work with John Gillespie. Prerequisites: ring, ideal, quotient ring, Chinese Remainder theorem - suitable for undergraduates with a first course in algebra.
ID:
289
Year: 2010
Name: Ruth Berger
Institution: Luther College
Subject area(s): Algebra, Group Theory
Title of Talk: Exploring Group Theory with FGB
Abstract: Finite Group Behavior (FGB) is a free windows-based program that gives beginning group theory students a chance to explore abstract group theory concepts in a very concrete setting. The heart of the software is an extensive collection of Cayley tables of groups: Cyclic groups, Dihedral groups, and groups whose structure is not immediately recognizable. Students can explore relations among the elements of a group, determine the order of each element, and even make subgroups generated by selected elements of the group. This easy to use program also includes features that allow for the investigation of isomorphisms of groups, and it gives a nice visualization of how Cosets are formed. I will share some of the worksheets that I wrote for my Abstract Algebra students to gain some hands-on experience with these otherwise abstract concepts.
Year: 2010
Name: Ruth Berger
Institution: Luther College
Subject area(s): Algebra, Group Theory
Title of Talk: Exploring Group Theory with FGB
Abstract: Finite Group Behavior (FGB) is a free windows-based program that gives beginning group theory students a chance to explore abstract group theory concepts in a very concrete setting. The heart of the software is an extensive collection of Cayley tables of groups: Cyclic groups, Dihedral groups, and groups whose structure is not immediately recognizable. Students can explore relations among the elements of a group, determine the order of each element, and even make subgroups generated by selected elements of the group. This easy to use program also includes features that allow for the investigation of isomorphisms of groups, and it gives a nice visualization of how Cosets are formed. I will share some of the worksheets that I wrote for my Abstract Algebra students to gain some hands-on experience with these otherwise abstract concepts.
ID:
120
Year: 2005
Name: Jeremy Alm
Institution: Iowa State University
Subject area(s): Algebra, Logic
Title of Talk: Don't Be So Sensitive! --On the Definition(s) of a Group
Abstract: We have all seen different variations on the definition of a group, and we all know that each one admits "the same structures". There are, however, some subtle but important differences among them. The class of groups and the properties that it has are sensitive to the signature (or similarity type) in which the groups are defined. In particular, in some signatures equational definitions are possible and in others they are not.
Year: 2005
Name: Jeremy Alm
Institution: Iowa State University
Subject area(s): Algebra, Logic
Title of Talk: Don't Be So Sensitive! --On the Definition(s) of a Group
Abstract: We have all seen different variations on the definition of a group, and we all know that each one admits "the same structures". There are, however, some subtle but important differences among them. The class of groups and the properties that it has are sensitive to the signature (or similarity type) in which the groups are defined. In particular, in some signatures equational definitions are possible and in others they are not.
ID:
395
Year: 2014
Name: Adam Case
Institution: Iowa State University
Subject area(s): Algorithmic Information Theory
Title of Talk: Mutual Dimension
Abstract: The mutual (shared) information between two random variables is a well-understood concept in Shannon information theory, but how do we think about mutual information between other kinds of objects such as strings or real numbers? In this talk, we discuss various notions of mutual information from the perspective of algorithmic information theory. First we explore the algorithmic information content of a binary string. We then discuss the notion of the dimension (density of algorithmic information) of a real number. Finally, we explain our recent solution to an open problem: the correct formulation of the mutual information between two real numbers. This is joint work with Jack Lutz. The talk will be accessible to math undergraduates.
Year: 2014
Name: Adam Case
Institution: Iowa State University
Subject area(s): Algorithmic Information Theory
Title of Talk: Mutual Dimension
Abstract: The mutual (shared) information between two random variables is a well-understood concept in Shannon information theory, but how do we think about mutual information between other kinds of objects such as strings or real numbers? In this talk, we discuss various notions of mutual information from the perspective of algorithmic information theory. First we explore the algorithmic information content of a binary string. We then discuss the notion of the dimension (density of algorithmic information) of a real number. Finally, we explain our recent solution to an open problem: the correct formulation of the mutual information between two real numbers. This is joint work with Jack Lutz. The talk will be accessible to math undergraduates.
ID:
47
Year: 2004
Name: Ronald Smith
Institution: Graceland College
Subject area(s): Algorithms
Title of Talk: The distribution of digits in consecutive integers
Abstract: The distribution of digits problem asks for the frequency of each digit (0
Year: 2004
Name: Ronald Smith
Institution: Graceland College
Subject area(s): Algorithms
Title of Talk: The distribution of digits in consecutive integers
Abstract: The distribution of digits problem asks for the frequency of each digit (0
ID:
355
Year: 2013
Name: Ronald Smith
Institution: Graceland University
Subject area(s): algorithms
Title of Talk: Beautiful Strings
Abstract: Let S and T be strings. S is more beautiful than T if (i) S is longer than T, or (ii) if S and T have the same length, then S > T lexicographically. S derives T, if T is a subsequence (not necessarily a substring) of S. T is unique if each character in T appears exactly once. The "Beautiful Strings Problem" is to find the most beautiful unique string that can be derived from a given string S. This problem appeared on the web and in at least one programming contest last year, with no correct solution known to this author. We give an efficient solution, showing the usefulness of a mathematical approach.
Year: 2013
Name: Ronald Smith
Institution: Graceland University
Subject area(s): algorithms
Title of Talk: Beautiful Strings
Abstract: Let S and T be strings. S is more beautiful than T if (i) S is longer than T, or (ii) if S and T have the same length, then S > T lexicographically. S derives T, if T is a subsequence (not necessarily a substring) of S. T is unique if each character in T appears exactly once. The "Beautiful Strings Problem" is to find the most beautiful unique string that can be derived from a given string S. This problem appeared on the web and in at least one programming contest last year, with no correct solution known to this author. We give an efficient solution, showing the usefulness of a mathematical approach.
ID:
339
Year: 2012
Name: Marc Chamberland
Institution: Grinnell College
Subject area(s): analysis
Title of Talk: A Beautiful Cantor-like Function
Abstract: Analysis students encounter various functions with exotic properties. This could include functions with infinitely many discontinuities (Dirichlet function, Thomae function, windmill functions) or continuous functions which grow in a bizarre way (Cantor function, Minkowski's question mark function). After quickly reviewing these, we introduce a new function f(x) which combines enticing properties from both of these classes: a dense set of discontinuities, fractal structure, a base-3 digital representation, satisfies f(f(x))=x, and has surprising integral properties. This function makes an excellent study to conclude a first course in analysis.
Year: 2012
Name: Marc Chamberland
Institution: Grinnell College
Subject area(s): analysis
Title of Talk: A Beautiful Cantor-like Function
Abstract: Analysis students encounter various functions with exotic properties. This could include functions with infinitely many discontinuities (Dirichlet function, Thomae function, windmill functions) or continuous functions which grow in a bizarre way (Cantor function, Minkowski's question mark function). After quickly reviewing these, we introduce a new function f(x) which combines enticing properties from both of these classes: a dense set of discontinuities, fractal structure, a base-3 digital representation, satisfies f(f(x))=x, and has surprising integral properties. This function makes an excellent study to conclude a first course in analysis.
ID:
429
Year: 2015
Name: Mariah Birgen
Institution: Wartburg College
Subject area(s): Analysis
Title of Talk: How to I keep track of classroom behavior in my IBL Classroom
Abstract: I have been teaching IBL in my upper level classes for several years now, but have struggled with keeping track of participation during class. I want to give my students credit for quality questions and answers, but sometimes (often) things go so fast, or I am so involved with the argumentation, that I can't write things down quickly. Each class starts with the best of intentions, but . . . Today I am going to talk about one nearly fool-proof method that I have discovered that works for me, along with some other ideas that I haven't course-tested, but have strong potential.
Year: 2015
Name: Mariah Birgen
Institution: Wartburg College
Subject area(s): Analysis
Title of Talk: How to I keep track of classroom behavior in my IBL Classroom
Abstract: I have been teaching IBL in my upper level classes for several years now, but have struggled with keeping track of participation during class. I want to give my students credit for quality questions and answers, but sometimes (often) things go so fast, or I am so involved with the argumentation, that I can't write things down quickly. Each class starts with the best of intentions, but . . . Today I am going to talk about one nearly fool-proof method that I have discovered that works for me, along with some other ideas that I haven't course-tested, but have strong potential.
ID:
242
Year: 2008
Name: Palle Jorgensen
Institution: University of Iowa
Subject area(s): Analysis
Title of Talk: Matrix functions
Abstract: When I was little my father, for reasons unbeknownst to me, told me about low-pass and high-pass filters. He was a telephone engineer and worked on filters in signal processing. The 'high' and 'low' part of the story refers to frequency bands. Not that this meant much to me at the time. Rather, I was fascinated by the pictures of filter designs in the EE journals stacked up on the floor. And it was only many years later I came across this stuff in mathematics: quadrature mirror filters and all that; yet the visual impression still lingered. The talk will cover some of this math, especially wavelets: Subband filters define operators in Hilbert space which satisfy all kinds of abstract relations, and they are terribly useful. They are used in math and in signal processing. Matrix functions from math are called poly-phase matrices by engineers, and they are scattering matrices in other circles, and quantum gates in physics. In fact a lot of the things we do in math are known and used in other fields, but under different names, and known in different ways.
Year: 2008
Name: Palle Jorgensen
Institution: University of Iowa
Subject area(s): Analysis
Title of Talk: Matrix functions
Abstract: When I was little my father, for reasons unbeknownst to me, told me about low-pass and high-pass filters. He was a telephone engineer and worked on filters in signal processing. The 'high' and 'low' part of the story refers to frequency bands. Not that this meant much to me at the time. Rather, I was fascinated by the pictures of filter designs in the EE journals stacked up on the floor. And it was only many years later I came across this stuff in mathematics: quadrature mirror filters and all that; yet the visual impression still lingered. The talk will cover some of this math, especially wavelets: Subband filters define operators in Hilbert space which satisfy all kinds of abstract relations, and they are terribly useful. They are used in math and in signal processing. Matrix functions from math are called poly-phase matrices by engineers, and they are scattering matrices in other circles, and quantum gates in physics. In fact a lot of the things we do in math are known and used in other fields, but under different names, and known in different ways.
ID:
68
Year: 2004
Name: Alexander Kleiner
Institution: Drake University
Subject area(s): Analysis, History
Title of Talk: "Summing" Unbounded Sequences: Some History Preliminary Report
Abstract: The question of which, if any, unbounded sequences were summed by regular methods of summation was considerd repeatedly. This talk will show how these questions were answered (over and over).
Year: 2004
Name: Alexander Kleiner
Institution: Drake University
Subject area(s): Analysis, History
Title of Talk: "Summing" Unbounded Sequences: Some History Preliminary Report
Abstract: The question of which, if any, unbounded sequences were summed by regular methods of summation was considerd repeatedly. This talk will show how these questions were answered (over and over).
ID:
170
Year: 2006
Name: Alexander Kleiner
Institution: Drake University
Subject area(s): analysis, history of mathematics
Title of Talk: The Toeplitz-Silverman Theorem Part II
Abstract: In the first two decades of the twentieth century summability developed from collection of special results used in other parts of analysis into a full-blown field. One of the main points of this transition was a collection of general results that gave conditions for a method to sum every convergent sequence. Part I of this presentation, which was given last spring, laid out the work that led to the general theory. Papers by Toeplitz, Silverman, Kojima, Schur and others established the theory. This note will look at the development of these conditions and, as time permits, the reoccurrence of these results in the early day of the "Polish" school of functional analysis
Year: 2006
Name: Alexander Kleiner
Institution: Drake University
Subject area(s): analysis, history of mathematics
Title of Talk: The Toeplitz-Silverman Theorem Part II
Abstract: In the first two decades of the twentieth century summability developed from collection of special results used in other parts of analysis into a full-blown field. One of the main points of this transition was a collection of general results that gave conditions for a method to sum every convergent sequence. Part I of this presentation, which was given last spring, laid out the work that led to the general theory. Papers by Toeplitz, Silverman, Kojima, Schur and others established the theory. This note will look at the development of these conditions and, as time permits, the reoccurrence of these results in the early day of the "Polish" school of functional analysis
ID:
230
Year: 2008
Name: Greg Ongie
Institution: Coe College
Subject area(s): Analysis, Measure Theory, Orthogonal Polynomials
Title of Talk: Orthogonal Polynomials on the Cantor Set
Abstract: The middle-thirds Cantor set is an uncountable set of Lebesgue measure zero. The Cantor measure is defined such that it assigns the Cantor set measure one, and has the Cantor set as its support. An orthogonal polynomial sequence (OPS) is traditionally defined by means of Riemann integration, but more generally an OPS can be defined by means of integration with respect to a measure. First we construct the Cantor measure and show it satisfies the properties of a measure. Then, we verify the existence of an associated OPS by examining the positivity of its moment matrix. Finally, using the Gram-Schmidt method we construct the OPS, and derive various properties of the polynomials based on results for classical orthogonal polynomials.
Year: 2008
Name: Greg Ongie
Institution: Coe College
Subject area(s): Analysis, Measure Theory, Orthogonal Polynomials
Title of Talk: Orthogonal Polynomials on the Cantor Set
Abstract: The middle-thirds Cantor set is an uncountable set of Lebesgue measure zero. The Cantor measure is defined such that it assigns the Cantor set measure one, and has the Cantor set as its support. An orthogonal polynomial sequence (OPS) is traditionally defined by means of Riemann integration, but more generally an OPS can be defined by means of integration with respect to a measure. First we construct the Cantor measure and show it satisfies the properties of a measure. Then, we verify the existence of an associated OPS by examining the positivity of its moment matrix. Finally, using the Gram-Schmidt method we construct the OPS, and derive various properties of the polynomials based on results for classical orthogonal polynomials.