M@X -- Math At Xavier

Fall 2022 seminar
Spring 2023 seminar

Mathematics Department
Xavier University of Louisiana

Date Place SpeakerTitle
Tue May 2, 12:15pm NCF 574 Tewodros AmdeberhanTo count, or not to count
In this talk, we illustrate simple but effective techniques right out of undergraduate mathematics content for the purpose of handling research-level problems. To this end, we split the presentation into two parts:
  1. On combinatorial objects;
  2. On PDE items.
This way, there is something of appeal to most of the audience.

Left: T. Amdeberhan, Right: V. Moll
Dr. Tewodros Amdeberhan is an Eritrean-American mathematician and professor of mathematics at Tulane University. He obtained his PhD in mathematics in 1997 from Temple University as a student of Doron Zeilberger. He is the author of a large number of research articles and a few books in Combinatorics, Number Theory, Classical Analysis, and he is a leading expert in the computer assisted W-Z method.
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Date Place SpeakerTitle
Tue Apr 11, 12:15pm NCF 574 Victor H. MollWallis and Landen: a tale of two integrals.
One of the earliest formulas for \(\pi\) was given by John Wallis (1616--1703) in the form of an infinite product. This could also be written in the form \[\int_0^\infty \frac{dx}{(1+x^2)^{m+1}} = \frac{\pi}{2^{2m+1}}\binom{2m}{m}\] In the beginning of Integral Calculus, John Landen (1719 --1790) showed that the integral \[G(a,b)=\int_0^{\pi/2} \frac{d\phi}{\sqrt{a^2\cos^2\phi +b^2\sin^2 \phi}}\] remains invariant if you replace \(a\) by the arithmetic mean of \(a\) and \(b\), namely \(\frac{1}{2}(a+b)\) and \(b\) by their geometric mean \(\sqrt{ab}\): \[G(a,b)=G\left(\frac{a+b}{2},\sqrt{ab}\right).\] Identities of this type are behind modern calculations of \(\pi\). In this talk I will discuss some variations on this subject. Connections to Number Theory, Dynamical Systems and a new method of integration will appear.

Dr. Victor Moll is a Chilean-American mathematician and Professor of Mathematics at Tulane University, New Orleans. He obtained his PhD from the Courant Institute of Mathematical Sciences in New York under the guidance of Henry McKean, one of the leading mathematicians of the 20th century. He has been a powerhouse of mathematical activity in New Orleans ever since he moved here in the mid 1980's. His mathematical interest, expertise and contribution span a large number of areas in Classical Analysis, Special Functions, Number Theory, Combinatorics and more. He is the author of several books, and a large number of papers. He is a leading expert in computer assisted research in mathematics, or Experimental Mathematics. Dr. Moll is also an outstanding teacher and mentor, both at undergraduate and graduate level. He is a recipient of the prestigious Weiss Presidential Award for Excellence in Undergraduate Teaching, and he has supervised at least 15 PhD dissertations. Several of his former PhD students went on to become distinguished mathematicians. His collaboration with one of his especially talented students, George Boros, was particularly fruitful and a large part of Dr. Moll's current research interest is in topics that originated from that collaboration. His M@X talk this week describes some of that research. This talk has special significance for our mathematics department because George Boros was a Xavier faculty member for two years, before he untimely passed away. The talk will be at undergraduate level, and not requiring notions beyond those of the Calculus course.
Click here to access a pdf file with the slides from the talk

Date Place SpeakerTitle
Mon Mar 13, 11:00am NCF 574 Karl Dilcher Leonhard Euler, the Master of Us All
Leonhard Euler (1707-1783) was not only one of the greatest mathematicians; he was one of the most prolific and the most influential mathematicians in history. In this talk, I will give a biographical sketch and will try to put the various stages of Euler's life and career into a historical and academic perspective. I will say nothing, or very little, about Euler's mathematics.

The talk is aimed at the general audience and almost no mathematical background is required. Faculty members outside of the Mathematical and Physical Sciences Division are also encouraged to attend.

Professor Karl Dilcher is a distinguished German-Canadian mathematician working as a Professor at Dalhousie University in Halifax and a fellow of the Canadian Math Society, a receiver of several awards, was a former head of the mathematics department at Dalhousie and has been a part of the editorial board of several journals and grant committees. His huge area of interest includes classical analysis, number theory, special functions, etc.

Date Place SpeakerTitle
Tue Mar 7, 12:15pm NCF 574 John Charles Saunders The Euler Totient Function on Lucas Sequences
In 2009, Luca and Nicolae proved that the only Fibonacci numbers whose Euler totient function is another Fibonacci number are 1,2, and 3. In 2015, Faye and Luca proved that the only Pell numbers whose Euler totient function is another Pell number are 1 and 2. Here we add to these two results and prove that for any fixed natural number \(P\geq 3\), if we define the sequence \((u_n)_n\) as \(u_0 =0,u_1 =1\), and \(u_n =P u_{n−1}+u_{n−2}\) for all \(n\geq 2\), then the only solution to the Diophantine equation \(\phi(u_n) = u_m\) is \(\phi(u_1) = \phi(1) = 1 = u_1\).

Date Place SpeakerTitle
Tue Feb 7, 12:15pm NCF 574 Pranabesh Das Linear forms in Logarithms: Baker's theory
In 1934, Gelfond and Schneider proved that
Theorem 1. (Gelfond-Schneider) If \(\alpha\) and \(\beta\) are complex algebraic numbers with \(\alpha \neq 0, 1,\) and \(\beta\) not rational, then any value of \(\alpha^{\beta}\) is a transcendental number.

This immediately proves that the numbers like \(2^{\sqrt{2}}, e^{\pi}, i^i\) are transcendental.
The Theorem 1. is equivalent as saying if \(\log \alpha\) and \(\log \beta\) linearly independent over \(\mathbb Q\) for non-zero algebraic numbers \(\alpha\) and \(\beta\), then \(\log \alpha\) and \(\log \beta\) are linearly independent over the algebraic numbers. Gelfond-Schneider's method was specific to only two algebraic numbers. In 1967 Alan Baker established the following generalization of the Gelfond-Schneider theorem.

Theorem 2. (Baker1) Let \(n\) be a positive integer. Let \(\alpha_1,\cdots,\alpha_n\) be non-zero algebraic numbers and \(\log \alpha_1,\cdots,\log \alpha_n\) any determinations of their logarithms. If \(\log \alpha_1,\cdots,\log \alpha_n\) are linearly independent over \(\mathbb Q\), then \(1,\log \alpha_1,\cdots,\log \alpha_n\) are linearly independent over the algebraic numbers.

As an immediate consequence, we have

Corollary 3. \(e^{\beta_0}{\alpha_1}^{\beta_1}{\alpha_2}^{\beta_2}\cdots{\alpha_n}^{\beta_n}\) is transcendental for all non-zero algebraic numbers \(\alpha_1,\cdots,\alpha_n,\beta_0,\beta_1\cdots,\beta_n.\)

For applications to Diophantine problems, it is important that not only the above linear form is non-zero, but also that we have a strong enough lower bound for the absolute value of this linear form. We give a special case, where \(\beta_0 = 0\) and \(\beta_1,\cdots,\beta_n\) are rational integers.

Theorem 4. (Baker2)
Let \(\alpha_1,\cdots,\alpha_n\) be algebraic numbers different from \(0\) and \(1\). Further, let \(b_1,\cdots,b_m\) be rational integers such that \[b_1\log \alpha_1+\cdots +b_n\log \alpha_n \neq 0.\] Then \[\left|b_1\log \alpha_1+\cdots +b_n\log \alpha_n\right| \geq (eB)^{-C},\] where \(B:= \max(|b_1|,\cdots,|b_n|)\) and \(C\) is an effectively computable constant depending only on \(n\) and \(\alpha_1,\cdots,\alpha_n\).

Baker was awarded a Fields medal for his work on linear forms in logarithms in 1970. In a series of two-three talks, we plan to discuss some of the results of this topic and its beautiful applications in Diophantine problems.

Click here to access a pdf file with the slides from the talk