Mathematics Department - Experimental Mathematics Seminar - Spring 2017

Experimental Mathematics Seminar - Spring 2017



Organizer(s)

Doron Zeilberger, Nathan H Fox

Archive

Website

http://www.math.rutgers.edu/~nhf12/expmath



Upcoming Talks


Thursday, March 2nd

Adi Ben-Israel , Rutgers University

"Matrix volume and its applications "

Time: 5:00 PM
Location: Hill 705
Abstract: The volume vol(A) of an mxn matrix A of rank r is: (a) the product of the r singular values of A, or (b) the square root of the sum of squares of all r x r subdeterminants of A, or (c) the volume of the image under A of a unit cube in the range of A transpose. Definition (b) is applicable to non-numerical matrices, in particular to rectangular Jacobians. Some representative applications will be discussed.

References:

[1] The matrix volume: http://benisrael.net/VOLUME.pdf

[2] Application to change-of-variables in integration: http://benisrael.net/INTEGRAL-AMS.pdf

[3] Application to probability: http://benisrael.net/MADISON-AMS.pdf

[4] Low-rank approximation In this paper we show that the negative sample distance covariance function is a quasiconcave set function of samples of random variables that are not statistically independent. We use these properties to propose greedy algorithms to combinatorially optimize some diversity (low statistical dependence) promoting functions of distance covariance. Our greedy algorithm obtains all the inclusion-minimal maximizers of this diversity promoting objective. Inclusion-minimal maximizers are multiple solution sets of globally optimal maximizers that are not a proper subset of any other maximizing set in the solution set. We present results upon applying this approach to obtain diverse features (covariates/variables/predictors) in a feature selection setting for regression (or classification) problems. We also combine our diverse feature selection algorithm with a distance covariance based relevant feature selection algorithm of [7] to produce subsets of covariates that are both relevant yet ordered in non-increasing levels of diversity of these subsets.of matrices: A. Deshpande, L. Rademacher et al, Matrix Approximation and Projective Clustering via Volume Sampling, Theory of Computing 2(2006), 225-247


Thursday, March 9th

Praneeth Vepakomma, Motorola Solutions

"TBA"

Time: 5:00 PM
Location: Hill 705


Thursday, March 23rd

TBA

"TBA"

Time: 5:00 PM
Location: Hill 705


Thursday, March 30th

Nathan Fox, Rutgers University

"TBA"

Time: 5:00 PM
Location: Hill 705





Past Talks


Thursday, February 23rd

Matthew Russell, Rutgers University

"Fantastic identities, and where to find them "

Time: 5:00 PM
Location: Hill 705
Abstract: In this talk, I will present a veritable cornucopia of new partition identities discovered over the past calendar year. Some of these have proofs, some are still conjectures; some should break new ground in the study of affine Lie algebra; all were discovered with the use of computers (in a surprising variety of ways).

This is based on joint work in a handful of different groups, including with Terence Coelho and Jongwon Kim, with Shashank Kanade and Debajyoti Nandi, and with Shashank Kanade.


Thursday, February 16th

John Miller, Johns Hopkins University

"The distribution of extremal values of linear recurrences modulo m "

Time: 5:00 PM
Location: Hill 705
Abstract: Consider the Fibonacci sequence modulo m, e.g. modulo 10:

1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, 3, 7, 0, 7, 7, 4, 1, 5, 6, 1, 7, 8,...

We can also start with different initial values, e.g. (3,4):

3, 4, 7, 1, 8, 9, 7, 6, 3, 9, 2, 1, 3, 4, 7, 1, 8, 9, 7, 6, 3, 9, 2, 1,...

This sequence has a cycle of length 12. The cycle lengths of such sequences are a classical subject related to algebraic number theory. We may also consider the "extremal values" of this sequence. For example, the 2nd sequence has minimum 1 and maximum 9. In this talk we will discuss the properties of such extremal values as we vary the initial values and the modulus, for the Fibonacci recurrence and for other examples of linear recurrences. We will do some experiments, and we will connect the distribution of extremal values to questions of elementary geometry.

This talk arises out of work done in Professor Zeilberger's experimental mathematics class in the spring of 2012.


Thursday, February 9th

Evita Nestoridi, Princeton University

"Shuffling large decks of cards and the Bernoulli-Laplace urn model "

Time: 5:00 PM
Location: Hill 705
Abstract: In board games, in Casino games with multiple decks and cryptography, one is sometimes faced with the practical problem: how can a human (as opposed to the computer) shuffle big decks of cards. One natural procedure (used by casinos) is to break the deck into several reasonable size piles, shuffle each thoroughly, assemble, do some simple deterministic thing (like a cut) and repeat. G. White and I introduce variations of the classical Bernoulli-Laplace urn model (the first Markov chain!) involving swaps of big groups of balls. A coupling argument and spherical function theory allow the original problem to be solved.


Thursday, February 2nd

John Conway, Princeton University

"The Faulhaber triangle and the Bernoulli numbers and what they're good for "

Time: 5:00 PM
Location: Hill 705
Abstract:


Thursday, January 26th

Neil J. A. Sloane, Rutgers University and The OEIS Foundation

"Winter Fruits: New Problems from OEIS "

Time: 5:00 PM
Location: Hill 705
Abstract: New problems in number theory and combinatorics (probably not so hard) from Dec 2016 and Jan 2017. Crop circles, the number pau, and (more seriously) Fibonachos, Richard Guy, digital sums, carryless sums, Tisdale's sieve, square permutations, compact numbers, and 2-D counting.


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