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 101-120 of 471 results.
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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:
217
Year: 2007
Name: Dennis Roseman
Institution: University of Iowa
Subject area(s):
Title of Talk: How likely is a lattice link?
Abstract: Lattice points in space are points with integer coordinates. A unit lattice edge is a line segment of unit length between lattice points. A lattice link is a finite collection union of lattice edges whose union is topologically equivalent to a union of disjoint circles. We define a notion of probability for lattice knots and links and use this to frame the question: which is more ``likely'', the square knot or the granny knot. A square knot is obtained by tying a right hand trefoil and a left had trefoil together; the granny knot is obtained by using two identical trefoils. We also discuss our progress towards calculation of these probabilities.
Year: 2007
Name: Dennis Roseman
Institution: University of Iowa
Subject area(s):
Title of Talk: How likely is a lattice link?
Abstract: Lattice points in space are points with integer coordinates. A unit lattice edge is a line segment of unit length between lattice points. A lattice link is a finite collection union of lattice edges whose union is topologically equivalent to a union of disjoint circles. We define a notion of probability for lattice knots and links and use this to frame the question: which is more ``likely'', the square knot or the granny knot. A square knot is obtained by tying a right hand trefoil and a left had trefoil together; the granny knot is obtained by using two identical trefoils. We also discuss our progress towards calculation of these probabilities.
ID:
460
Year: 2016
Name: Mark Ronnenberg
Institution: University of Northern Iowa
Subject area(s): topology
Title of Talk: Reidemeister Moves and Equivalence of Butterfly Diagrams for Links
Abstract: By a theorem of Reidemeister, two links are equivalent if and only if they have regular projections which can be related by a finite sequence of special changes called Reidemeister moves. It is an open problem to find a complete set of "butterfly moves" to turn a butterfly diagram for a given link into a butterfly diagram for an equivalent link. In this talk, we will translate the Reidemeister moves into butterfly moves and present some examples.
Year: 2016
Name: Mark Ronnenberg
Institution: University of Northern Iowa
Subject area(s): topology
Title of Talk: Reidemeister Moves and Equivalence of Butterfly Diagrams for Links
Abstract: By a theorem of Reidemeister, two links are equivalent if and only if they have regular projections which can be related by a finite sequence of special changes called Reidemeister moves. It is an open problem to find a complete set of "butterfly moves" to turn a butterfly diagram for a given link into a butterfly diagram for an equivalent link. In this talk, we will translate the Reidemeister moves into butterfly moves and present some examples.
ID:
186
Year: 2007
Name: David Romano
Institution: Grinnell College
Subject area(s): Convex geometry
Title of Talk: Connected goalies for convex polygons
Abstract: Given a compact convex body K in the plane, call a connected 1-dimensional set G in the plane a goalie if it intersects all the straight lines that intersect K. This talk is concerned with the problem of finding the minimal length goalie for polygons. For any polygon P with n sides, we prove that any shortest goalie G for P has convex hull CH(G) a polygon with at most 2n sides. For triangles T, the minimal length goalie is the Steiner minimal tree for T. This is no longer true in the case of quadrilaterals, in which case a Steiner minimal tree need not be a minimal goalie.
Year: 2007
Name: David Romano
Institution: Grinnell College
Subject area(s): Convex geometry
Title of Talk: Connected goalies for convex polygons
Abstract: Given a compact convex body K in the plane, call a connected 1-dimensional set G in the plane a goalie if it intersects all the straight lines that intersect K. This talk is concerned with the problem of finding the minimal length goalie for polygons. For any polygon P with n sides, we prove that any shortest goalie G for P has convex hull CH(G) a polygon with at most 2n sides. For triangles T, the minimal length goalie is the Steiner minimal tree for T. This is no longer true in the case of quadrilaterals, in which case a Steiner minimal tree need not be a minimal goalie.
ID:
478
Year: 2017
Name: Diego Rojas
Institution: Iowa State University
Subject area(s): Computability Theory, Analysis
Title of Talk: Differentiation of Functions on the Cantor Space and Connections to Real-Valued Functions
Abstract: The notion of online functions of the Cantor space $2^{\mathbb{N}}$, and more generally, of continuous and of computable functions on $2^{\mathbb{N}}$, have been studied recently in connection with algorithmic randomness. In this talk, we present a notion of the derivative of functions on $2^{\mathbb{N}}$, and we establish some connections between functions and their derivatives on $2^{\mathbb{N}}$ and on $\mathbb{R}$, where we can represent real-valued functions as functions acting on the dyadic representation of real numbers. This is joint work with Douglas Cenzer.
Year: 2017
Name: Diego Rojas
Institution: Iowa State University
Subject area(s): Computability Theory, Analysis
Title of Talk: Differentiation of Functions on the Cantor Space and Connections to Real-Valued Functions
Abstract: The notion of online functions of the Cantor space $2^{\mathbb{N}}$, and more generally, of continuous and of computable functions on $2^{\mathbb{N}}$, have been studied recently in connection with algorithmic randomness. In this talk, we present a notion of the derivative of functions on $2^{\mathbb{N}}$, and we establish some connections between functions and their derivatives on $2^{\mathbb{N}}$ and on $\mathbb{R}$, where we can represent real-valued functions as functions acting on the dyadic representation of real numbers. This is joint work with Douglas Cenzer.
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:
279
Year: 2010
Name: Christian Roettger
Institution: Iowa State University
Subject area(s): ODE, recurrence, power series, experimental mathematics
Title of Talk: Recurrences, power series, and ODE
Abstract: A three-term recurrence is connected to a power series, which solves a second-order ODE. The recurrence can be helpful in solving the ODE explicitly, and in approximating the power series. As is well-known, its growth rate is related to the radius of convergence of the power series. We will use a simple example straight from the textbook to investigate this in the case of a recurrence with *non-constant* coefficients. While the growth rate turns out to be surprisingly resistant to attack, it has great potential to be explored experimentally as well as theoretically - an opportunity for open-ended student projects.
Year: 2010
Name: Christian Roettger
Institution: Iowa State University
Subject area(s): ODE, recurrence, power series, experimental mathematics
Title of Talk: Recurrences, power series, and ODE
Abstract: A three-term recurrence is connected to a power series, which solves a second-order ODE. The recurrence can be helpful in solving the ODE explicitly, and in approximating the power series. As is well-known, its growth rate is related to the radius of convergence of the power series. We will use a simple example straight from the textbook to investigate this in the case of a recurrence with *non-constant* coefficients. While the growth rate turns out to be surprisingly resistant to attack, it has great potential to be explored experimentally as well as theoretically - an opportunity for open-ended student projects.
ID:
548
Year: 2019
Name: Christian Roettger
Institution: Iowa State University
Subject area(s):
Title of Talk: Balanced Numbers and Balanced Primes
Abstract: Balanced numbers are odd natural numbers n which have an equal number of 0s and 1s in the periodic part of the base-2 representation of their reciprocal 1/n. We present some insights about balanced numbers that use just elementary Number Theory like the Quadratic Reciprocity Theorem. In particular, if a prime p is congruent to 3 or 5 modulo 8, then p is balanced. If a prime p is congruent to 7 modulo 8, then p is not balanced. All powers of p are balanced iff p is. The case of primes congruent to 1 modulo 8 is much more difficult. Hasse made a breakthrough in 1966, showing that the balanced primes have a Dirichlet density of 17/24. We have refined Hasse's result slightly. Another question is how big is the set of balanced numbers (not only primes) less than x? Using a method due to Landau, we can show that this is bounded above by C x/log^(1/4) (x) and below by D x / log^(3/4)(x), with constants C, D > 0. I solemnly promise that I won't go into the gory detail, only highlight the beautiful and accessible parts of the subject. The second part of the talk is joint work with Joshua Zelinsky.
Year: 2019
Name: Christian Roettger
Institution: Iowa State University
Subject area(s):
Title of Talk: Balanced Numbers and Balanced Primes
Abstract: Balanced numbers are odd natural numbers n which have an equal number of 0s and 1s in the periodic part of the base-2 representation of their reciprocal 1/n. We present some insights about balanced numbers that use just elementary Number Theory like the Quadratic Reciprocity Theorem. In particular, if a prime p is congruent to 3 or 5 modulo 8, then p is balanced. If a prime p is congruent to 7 modulo 8, then p is not balanced. All powers of p are balanced iff p is. The case of primes congruent to 1 modulo 8 is much more difficult. Hasse made a breakthrough in 1966, showing that the balanced primes have a Dirichlet density of 17/24. We have refined Hasse's result slightly. Another question is how big is the set of balanced numbers (not only primes) less than x? Using a method due to Landau, we can show that this is bounded above by C x/log^(1/4) (x) and below by D x / log^(3/4)(x), with constants C, D > 0. I solemnly promise that I won't go into the gory detail, only highlight the beautiful and accessible parts of the subject. The second part of the talk is joint work with Joshua Zelinsky.
ID:
64
Year: 2004
Name: Christian Roettger
Institution: Iowa State University
Subject area(s): Number theory, exponential sums
Title of Talk: Uniform distribution and invertible matrices
Abstract: Uniform distribution is usually known as a property of sequences xn in the unit interval, like n alpha modulo 1, where alpha is irrational. We will present an example of uniform distribution in the unit square, explain the handy Weyl criterion used to prove uniform distribution, and conclude with an application to invertible 2x2 - matrices over the integers.
Year: 2004
Name: Christian Roettger
Institution: Iowa State University
Subject area(s): Number theory, exponential sums
Title of Talk: Uniform distribution and invertible matrices
Abstract: Uniform distribution is usually known as a property of sequences xn in the unit interval, like n alpha modulo 1, where alpha is irrational. We will present an example of uniform distribution in the unit square, explain the handy Weyl criterion used to prove uniform distribution, and conclude with an application to invertible 2x2 - matrices over the integers.
ID:
331
Year: 2012
Name: Christian Roettger
Institution: Iowa State University
Subject area(s): Multivariate Calculus, numerical mathematics
Title of Talk: Calculus III projects for Undergraduates
Abstract: Multivariate Calculus lends itself particularly well to explorations on the computer. Examples include Newton's method, Steepest Descent, two-dimensional Riemann sums, Euler's method for differential equations. Each of these can be presented in various appealing contexts and is immediately plausible for a student who understands the core concepts of the derivative of a multivariate function and Riemann sums, respectively. On the other hand, exploring the 'approximation' aspect of Calculus with paper and pencil and even with a calculator is less satisfactory than using a computer, especially if powerful mathematical software is available (eg SAGE, R, Matlab, Maple, Mathematica). Ideally, the results can be presented in an appealing graphic, and we'll show examples of student work. Finally, we do not assume any programming skills, but this kind of small project is a great opportunity to learn them.
Year: 2012
Name: Christian Roettger
Institution: Iowa State University
Subject area(s): Multivariate Calculus, numerical mathematics
Title of Talk: Calculus III projects for Undergraduates
Abstract: Multivariate Calculus lends itself particularly well to explorations on the computer. Examples include Newton's method, Steepest Descent, two-dimensional Riemann sums, Euler's method for differential equations. Each of these can be presented in various appealing contexts and is immediately plausible for a student who understands the core concepts of the derivative of a multivariate function and Riemann sums, respectively. On the other hand, exploring the 'approximation' aspect of Calculus with paper and pencil and even with a calculator is less satisfactory than using a computer, especially if powerful mathematical software is available (eg SAGE, R, Matlab, Maple, Mathematica). Ideally, the results can be presented in an appealing graphic, and we'll show examples of student work. Finally, we do not assume any programming skills, but this kind of small project is a great opportunity to learn them.
ID:
372
Year: 2013
Name: Christian Roettger
Institution: Iowa State University
Subject area(s): Number Theory, Diophantine Geometry, L-functions
Title of Talk: Geometric distribution of primes in Z[sqrt(2)]
Abstract: It all starts with the question: what can we say about integers a, b such that a^2 - 2b^2 is a prime? We will show some ways to make this question more precise - in particular, we study the distribution of the corresponding points (a,b) in the plane. The fundamental tool is the ring Z[sqrt(2)], and from there we make connections to analytic number theory (L-functions, Hecke characters) which arise very naturally - this is the context where Hecke invented 'Hecke characters', and they are much easier to understand here than when you read about them in MathWorld.
Year: 2013
Name: Christian Roettger
Institution: Iowa State University
Subject area(s): Number Theory, Diophantine Geometry, L-functions
Title of Talk: Geometric distribution of primes in Z[sqrt(2)]
Abstract: It all starts with the question: what can we say about integers a, b such that a^2 - 2b^2 is a prime? We will show some ways to make this question more precise - in particular, we study the distribution of the corresponding points (a,b) in the plane. The fundamental tool is the ring Z[sqrt(2)], and from there we make connections to analytic number theory (L-functions, Hecke characters) which arise very naturally - this is the context where Hecke invented 'Hecke characters', and they are much easier to understand here than when you read about them in MathWorld.
ID:
137
Year: 2005
Name: Christian Roettger
Institution: Iowa State University
Subject area(s): Number Theory
Title of Talk: Prime divisors of Mersenne numbers and Dirichlet series
Abstract: Mersenne numbers are the numbers 1, 3, 7, 15, ... 2^n - 1, ... It is a long-standing conjecture that this sequence contains infinitely many primes. We show how to get some asymptotic results on the 'average' prime divisor of Mersenne numbers using Dirichlet series. These series are useful for asymptotic counting, because there is a close link between their domain of convergence and the growth of their coefficients. Do not expect a big breakthrough, but a pretty result, few technicalities, and some exciting open questions.
Year: 2005
Name: Christian Roettger
Institution: Iowa State University
Subject area(s): Number Theory
Title of Talk: Prime divisors of Mersenne numbers and Dirichlet series
Abstract: Mersenne numbers are the numbers 1, 3, 7, 15, ... 2^n - 1, ... It is a long-standing conjecture that this sequence contains infinitely many primes. We show how to get some asymptotic results on the 'average' prime divisor of Mersenne numbers using Dirichlet series. These series are useful for asymptotic counting, because there is a close link between their domain of convergence and the growth of their coefficients. Do not expect a big breakthrough, but a pretty result, few technicalities, and some exciting open questions.
ID:
401
Year: 2014
Name: Christian Roettger
Institution: Iowa State University
Subject area(s): Probability
Title of Talk: Visual hypothesis testing - lineups and probability
Abstract: Police use lineups involving one suspect and several 'dummies' to get evidence that a witness can identify the suspect. In an abstract sense, we can form a hypothesis about 'suspect' data and test it in this way: literally have people looking at a lineup of plots with the task of identifying the data plot among the dummies. Repetition with several observers makes this approach surprisingly powerful. It also has potential when comparing the efficiency of different visual representations of the same data. Disclaimer: do not expect analysis of actual police lineups. But we will try out the method on the audience! This is joint work with Heike Hofmann, Di Cook, Phil Dixon, and Andreas Buja. I have investigated the underlying probability distributions. This meant evaluating some multiple integrals, and revising all the tricks from Calculus II.
Year: 2014
Name: Christian Roettger
Institution: Iowa State University
Subject area(s): Probability
Title of Talk: Visual hypothesis testing - lineups and probability
Abstract: Police use lineups involving one suspect and several 'dummies' to get evidence that a witness can identify the suspect. In an abstract sense, we can form a hypothesis about 'suspect' data and test it in this way: literally have people looking at a lineup of plots with the task of identifying the data plot among the dummies. Repetition with several observers makes this approach surprisingly powerful. It also has potential when comparing the efficiency of different visual representations of the same data. Disclaimer: do not expect analysis of actual police lineups. But we will try out the method on the audience! This is joint work with Heike Hofmann, Di Cook, Phil Dixon, and Andreas Buja. I have investigated the underlying probability distributions. This meant evaluating some multiple integrals, and revising all the tricks from Calculus II.
ID:
168
Year: 2006
Name: Christian Roettger
Institution: Iowa State University
Subject area(s): Number theory, analytic
Title of Talk: Primitive prime divisors of Mersenne numbers via Uniform Distribution
Abstract: Given a sequence a of integers, a primitive divisor of a(n) is an integer which divides a(n) but no earlier term of the sequence. Last year, we presented a result about a weighted average of primitive prime divisors of the well-known Mersenne numbers M(n) = 2^n-1. This year, we have an entirely different, simple proof of the same result, using cyclotomic polynomials and uniform distribution. We are indebted to Carl Pomerance for helpful insights. We will also mention possible applications to other sequences like the Fibonacci numbers.
Year: 2006
Name: Christian Roettger
Institution: Iowa State University
Subject area(s): Number theory, analytic
Title of Talk: Primitive prime divisors of Mersenne numbers via Uniform Distribution
Abstract: Given a sequence a of integers, a primitive divisor of a(n) is an integer which divides a(n) but no earlier term of the sequence. Last year, we presented a result about a weighted average of primitive prime divisors of the well-known Mersenne numbers M(n) = 2^n-1. This year, we have an entirely different, simple proof of the same result, using cyclotomic polynomials and uniform distribution. We are indebted to Carl Pomerance for helpful insights. We will also mention possible applications to other sequences like the Fibonacci numbers.
ID:
434
Year: 2015
Name: Christian Roettger
Institution: Iowa State University
Subject area(s):
Title of Talk: Rashomon sculptures - reconstructing 3D shapes from inexact measurements
Abstract: The art installation 'Rashomon' was displayed on the Iowa State University campus during summer 2015. It consists of 15 identical, abstract sculptures. Artist Chuck Ginnever posed the challenge whether it is possible to display the sculptures so that no two of them are in the same position (modulo translation/rotation). We investigated the related question of reconstructing such a sculpture from (ordinary tape-measure) inexact measurements. Mathematics involved are the Cayley-Menger determinant, and the gradient method / Steepest Descent. We'll explain the mathematics with some simple examples and then show the results of our reconstruction. We will only assume elementary linear algebra (matrix - vector multiplication, determinants).
Year: 2015
Name: Christian Roettger
Institution: Iowa State University
Subject area(s):
Title of Talk: Rashomon sculptures - reconstructing 3D shapes from inexact measurements
Abstract: The art installation 'Rashomon' was displayed on the Iowa State University campus during summer 2015. It consists of 15 identical, abstract sculptures. Artist Chuck Ginnever posed the challenge whether it is possible to display the sculptures so that no two of them are in the same position (modulo translation/rotation). We investigated the related question of reconstructing such a sculpture from (ordinary tape-measure) inexact measurements. Mathematics involved are the Cayley-Menger determinant, and the gradient method / Steepest Descent. We'll explain the mathematics with some simple examples and then show the results of our reconstruction. We will only assume elementary linear algebra (matrix - vector multiplication, determinants).
ID:
189
Year: 2007
Name: Christian Roettger
Institution: Iowa State University
Subject area(s): Number Theory, Dynamical Systems
Title of Talk: Pseudo-Random Walks
Abstract: In a recent Monthly article, O'Bryant, Reznick and Serbinowska [ORS] have given some fascinating new insights into the behavior of \[ S_{N}(\alpha) := \sum_{n=1}^N (-1)^{[n\alpha]} \] where [x] is the integer part of x. Since the fractional part of n*\alpha for n=1,2,3,\dots behaves 'random-ish', one can make various guesses and apply classical methods like exponential sums to explore these hypotheses. Remarkably, the guesses are often wrong and the classical methods don't seem to work very well. Instead, [ORS] use continued fractions to obtain sharp and explicit upper and lower bounds for |S_{\alpha}(N)| in terms of \log N, and as a by-product get a way of evaluating S_{\alpha}(N) for large N with amazing efficiency.\\ We will explain that last part of their work. Then we will show how to use exponential sums with a twist that gives a lower bound for |S_{\alpha}(N)| - less explicit, but more general than what the methods from [ORS] give you. And if we omit tedious computations (which we will, and which are only long, not hard), the approach is as clear-cut and beautiful as that using exponential sums to the case of the fractional part of n*\alpha. Lit.: K.~O'Bryant, B.~Reznick, M.~Serbinowska: {\em Almost alternating sums}, Monthly vol.~113/8, pp. 673-688. Prerequisites: only complex exponentials e^{it}.
Year: 2007
Name: Christian Roettger
Institution: Iowa State University
Subject area(s): Number Theory, Dynamical Systems
Title of Talk: Pseudo-Random Walks
Abstract: In a recent Monthly article, O'Bryant, Reznick and Serbinowska [ORS] have given some fascinating new insights into the behavior of \[ S_{N}(\alpha) := \sum_{n=1}^N (-1)^{[n\alpha]} \] where [x] is the integer part of x. Since the fractional part of n*\alpha for n=1,2,3,\dots behaves 'random-ish', one can make various guesses and apply classical methods like exponential sums to explore these hypotheses. Remarkably, the guesses are often wrong and the classical methods don't seem to work very well. Instead, [ORS] use continued fractions to obtain sharp and explicit upper and lower bounds for |S_{\alpha}(N)| in terms of \log N, and as a by-product get a way of evaluating S_{\alpha}(N) for large N with amazing efficiency.\\ We will explain that last part of their work. Then we will show how to use exponential sums with a twist that gives a lower bound for |S_{\alpha}(N)| - less explicit, but more general than what the methods from [ORS] give you. And if we omit tedious computations (which we will, and which are only long, not hard), the approach is as clear-cut and beautiful as that using exponential sums to the case of the fractional part of n*\alpha. Lit.: K.~O'Bryant, B.~Reznick, M.~Serbinowska: {\em Almost alternating sums}, Monthly vol.~113/8, pp. 673-688. Prerequisites: only complex exponentials e^{it}.
ID:
224
Year: 2008
Name: Christian Roettger
Institution: Iowa State University
Subject area(s): geometry, teacher education, proof
Title of Talk: Proofs in elementary geometry - what IS the sum of angles in a triangle?
Abstract: One textbook for future teachers gives no less than four 'arguments' for this theorem. It is not claimed that they are proofs, and indeed they are not (all involve some circular reasoning). But the difference between such arguments and proofs is never made clear. We'll discover the flaws in the logic here, which are not obvious at all. Then we'll look at a number of examples from standard elementary geometry - some rock-solid one-line proofs, some examples where we all skip the proof and eyeball it, and finally an example which shows how 'eyeballing it' can lead to a 'proof' of 64=65.
Year: 2008
Name: Christian Roettger
Institution: Iowa State University
Subject area(s): geometry, teacher education, proof
Title of Talk: Proofs in elementary geometry - what IS the sum of angles in a triangle?
Abstract: One textbook for future teachers gives no less than four 'arguments' for this theorem. It is not claimed that they are proofs, and indeed they are not (all involve some circular reasoning). But the difference between such arguments and proofs is never made clear. We'll discover the flaws in the logic here, which are not obvious at all. Then we'll look at a number of examples from standard elementary geometry - some rock-solid one-line proofs, some examples where we all skip the proof and eyeball it, and finally an example which shows how 'eyeballing it' can lead to a 'proof' of 64=65.
ID:
522
Year: 2018
Name: Lorenzo Riva
Institution: Creighton University
Subject area(s):
Title of Talk: Feynman Operational Calculus
Abstract: The forthcoming paper "Combining continuous and discrete phenomena for Feynman's operational calculus in the presence of a $(C_0)$ semigroup and Feynman-Kac formulas with Lebesgue-Stieltjes measures" (by L. Nielsen, to appear in Integral Equations and Operator Theory) contains, as its main result, an evolution equation which serves to describe how Feynman's operational calculus evolves with time in the presence of a $(C_0)$ semigroup of linear operators. There are several examples in this paper which give rise to so-called, Feynman-Kac formulas with Lebesgue- Stieltjes measures (first investigated from a function space integral point of view by M. L. Lapidus in the late 1980s). However, due to the different approach, the Feynman-Kac formulas obtained in the paper by Nielsen have some significant differences from those obtained by Lapidus. An associated operator differential equation (essentially a nonhomogeneous Schrodinger's equation) is also obtained in Nielsen's paper. This talk will concentrate on the explanation of the newly-found Feynman-Kac formulas and some associated results.
Year: 2018
Name: Lorenzo Riva
Institution: Creighton University
Subject area(s):
Title of Talk: Feynman Operational Calculus
Abstract: The forthcoming paper "Combining continuous and discrete phenomena for Feynman's operational calculus in the presence of a $(C_0)$ semigroup and Feynman-Kac formulas with Lebesgue-Stieltjes measures" (by L. Nielsen, to appear in Integral Equations and Operator Theory) contains, as its main result, an evolution equation which serves to describe how Feynman's operational calculus evolves with time in the presence of a $(C_0)$ semigroup of linear operators. There are several examples in this paper which give rise to so-called, Feynman-Kac formulas with Lebesgue- Stieltjes measures (first investigated from a function space integral point of view by M. L. Lapidus in the late 1980s). However, due to the different approach, the Feynman-Kac formulas obtained in the paper by Nielsen have some significant differences from those obtained by Lapidus. An associated operator differential equation (essentially a nonhomogeneous Schrodinger's equation) is also obtained in Nielsen's paper. This talk will concentrate on the explanation of the newly-found Feynman-Kac formulas and some associated results.
ID:
533
Year: 2019
Name: Lorenzo Riva
Institution: Creighton University
Subject area(s): Analysis, PDEs
Title of Talk: Low Regularity Non-$L^2(\mathbb{R}^n)$ Local Solutions to the gMHD-$\alpha$ system
Abstract: The Magneto-Hydrodynamic (MHD) system of equations governs viscous fluids subject to a magnetic field and is derived via a coupling of the Navier-Stokes equations and Maxwell's equations. It has recently become common to study generalizations of fluids-based differential equations. Here we consider the generalized Magneto-Hydrodynamic alpha (gMHD-$\alpha$) system, which differs from the original MHD system by the presence of additional non-linear terms (indexed by the choice of $\alpha$) and replacing the Laplace operators in the equations by more general Fourier multipliers with symbols of the form $-\vert \xi \vert^\gamma / g(\vert \xi \vert)$. In \cite{penn1}, the author considered the problem with initial data in Sobolev spaces of the form $H^{s,2}(\mathbb{R}^n)$ with $n \geq 3$. Here we consider the problem with initial data in $H^{s,p}(\mathbb{R}^n)$ with $n \geq 3$ and $p > 2$, with the goal of minimizing the regularity required to obtain unique existence results.
Year: 2019
Name: Lorenzo Riva
Institution: Creighton University
Subject area(s): Analysis, PDEs
Title of Talk: Low Regularity Non-$L^2(\mathbb{R}^n)$ Local Solutions to the gMHD-$\alpha$ system
Abstract: The Magneto-Hydrodynamic (MHD) system of equations governs viscous fluids subject to a magnetic field and is derived via a coupling of the Navier-Stokes equations and Maxwell's equations. It has recently become common to study generalizations of fluids-based differential equations. Here we consider the generalized Magneto-Hydrodynamic alpha (gMHD-$\alpha$) system, which differs from the original MHD system by the presence of additional non-linear terms (indexed by the choice of $\alpha$) and replacing the Laplace operators in the equations by more general Fourier multipliers with symbols of the form $-\vert \xi \vert^\gamma / g(\vert \xi \vert)$. In \cite{penn1}, the author considered the problem with initial data in Sobolev spaces of the form $H^{s,2}(\mathbb{R}^n)$ with $n \geq 3$. Here we consider the problem with initial data in $H^{s,p}(\mathbb{R}^n)$ with $n \geq 3$ and $p > 2$, with the goal of minimizing the regularity required to obtain unique existence results.
ID:
555
Year: 2019
Name: Matt Rissler
Institution: Loras College
Subject area(s):
Title of Talk: The Math of Data Science
Abstract: Data Science is one of the buzzwordiest fields right now. In this talk, I will try to define Data Science out of my work implementing it as an undergraduate major at Loras. Then I will go on to talk about where Mathematics, both from the undergraduate and graduate curricula, is integral to the development and perhaps practice of Data Science.
Year: 2019
Name: Matt Rissler
Institution: Loras College
Subject area(s):
Title of Talk: The Math of Data Science
Abstract: Data Science is one of the buzzwordiest fields right now. In this talk, I will try to define Data Science out of my work implementing it as an undergraduate major at Loras. Then I will go on to talk about where Mathematics, both from the undergraduate and graduate curricula, is integral to the development and perhaps practice of Data Science.