Contact Info
Pure MathematicsUniversity of Waterloo
200 University Avenue West
Waterloo, Ontario, Canada
N2L 3G1
Departmental office: MC 5304
Phone: 519 888 4567 x33484
Fax: 519 725 0160
Email: puremath@uwaterloo.ca
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Given natural numbers $n$ and $k$, with $n>k$, the ProuhetTarryEscott (\textsc{pte}) problem asks for distinct subsets of $\mathbb{Z}$, say $X=\{x_1,\ldots,x_n\}$ and $Y=\{y_1,\ldots,y_n\}$, such that
\[x_1^i+\ldots+x_n^i=y_1^i+\ldots+y_n^i\] for $i=1,\ldots,k$. Many
partial solutions to this problem were found in the late 19th century and early 20th century.
I will define a slick method to compute the homotopy groups of a finite reflexive digraph and then use the method to show that a motley collection of such digraphs have a nontrivial homotopy group. (Hence by Larose’s Theorem, they do not support Taylor operations.)
Please note date and time.
Motivated from the Fuglede conjecture and the discovery of exponential
orthonormal basis on the onefourth Cantor measure, but not for the
onethird one, there has been interest in understanding the kind of
measures that admit some exponential type bases and their relatives such
as Fourier frames and Riesz bases. By decomposing the measure into
Since Bhargava and Shankar's new method of counting orbits, average orders of the 2,3,4,5Selmer groups of elliptic curves over Q have been obtained. In this talk we will look at a construction of torsors of Jacobians of hyperelliptic curves using pencils of quadrics and see how they are used to compute the average order of the 2Selmer groups of Jacobians of hyperelliptic curves over Q with a rational (non)Weierstrass point.
Putting together some of the machinery developed this term, I will prove Larose’s Theorem: if X is a ﬁnite, connected reﬂexive digraph and X admits a Taylor operation, then for every k ≥ 1, the kth homotopy group of X is trivial.
In this talk, I will describe a new model for describing certain sets S of rational numbers using finite automata. We will see that it is decidable if every element of S is an integer, and that sup S is computable. However, closely related questions are still open. There are applications to combinatorics on words.
Refreshments will be served in MC 5046 at 3:30pm. All are welcome.
Abstract: Given natural numbers n and k, the ProuhetTarryEscott (PTE)
asks for integers x_1,..,x_n and y_1,...,y_n such that the sums of the
first k powers are equal. This problem has connections to combinatorics
and theoretical computer science, as well as to other areas of number
Abstract: For a locally compact group $G$, YinHei Cheng considered the closed subspace $a_0(G)$ which is generated by the pure positive definite
Abstract: Two weeks ago, we defined a digraph $F^k(X,x_0)$, whose vertices were a particular set of homomorphisms, and a group $\sigma_k(X,x_0)$, whose elements were connected components of $F^k(X,x_0)$. This week, we will begin looking at these structures in more detail.
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Departmental office: MC 5304
Phone: 519 888 4567 x33484
Fax: 519 725 0160
Email: puremath@uwaterloo.ca
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