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PROGRAMS AND ACTIVITIES |
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| December 23, 2024 |
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In this talk we will discuss some recents results on the Dedekind reciprocity
laws and their applications in numbers fields, finite fields and functions
fields. Many interesting number-theoretical, algebraic, geometric, topological,
differential, and combinatorial invariants are arising to theses laws will
be studied. Computation of Galois groups via permutation group theory by Hugo Chapdelaine (Université Laval) In this talk we will present a method to compute the Galois group
of certain polynomials defined over Q by combining the theory
of local fields and the theory of permutation groups. Recursions for weights of some Boolean functions in n variables by Thomas Cusick University at Buffalo Coauthors: Maxwell Bileschi, Daniel Padgett We consider degree 3 Boolean functions in n variables which are rotation symmetric, that is, invariant under any cyclic shift of the indices of the variables. We provide an algorithm for finding a recursion for the truth table of any cubic rotation symmetric Boolean function generated by a monomial, as well as a homogeneous recursion for its (Hamming) weight as n increases. Given such a function, after the weights for the base cases are computed, we can find the weight for large n by a quick computation; it is not necessary to look at the truth table at all. Thus computations which previously seemed infeasible are now practical. Back to top Stern polynomials and continued fractions We derive identities for a polynomial analogue of the Stern sequence and define two subsequences of these polynomials. We obtain various properties for these two interrelated subsequences which have 0-1 coefficients and can be seen as extensions or analogues of the Fibonacci numbers. We also define two analytic functions as limits of these sequences. As an application we obtain evaluations of certain finite and infinite continued fractions whose partial quotients are doubly exponential. In a case of particular interest, the set of convergents has exactly two limit points. Finding the square roots of graphs Graph G is the square of graph H if two vertices u and v have an edge in
G if and only if u and v are of distance at most two in H. Given H it is
easy to compute its square H^2, however Motwani and Sudan proved that it
is NP-complete to determine if a given graph G is the square of some graph
H. In this talk we discuss the characterization and recognition problems
of graphs that are squares of graphs of small girth. Special values of Dedekind zeta-functions and
motivic cohomology Back to top By definition, an integer n is said to be a congruent number
if n is the area of a right angle triangle with rational sides:
n=ab/2 where a^2+b^2 = c^2 with a, b, c being rational numbers.
It turns out that the integer n is congruent if and only the elliptic
curve E: Y^2 = X^3 - n^2 X has a rational solution (x,y) with
xy non-zero. We will give other equivalent conditions under which
n is a congruent number and we take this opportunity for exhibiting
important properties of elliptic curves. This general audience
lecture will be accessible to graduate and undergraduate students.
Back to top Let $G$ be a finite cyclic group. Every sequence $S$ over $G$ can be written in the form $S=(n_1g)\cdot\ldots\cdot(n_lg)$ where $g\in G$ and $n_1, \ldots, n_l\in[1, ord(g)]$, and the index $ind(S)$ of $S$ is defined to be the minimum of $(n_1+\cdots+n_l)/ord(g)$ over all possible $g\in G$ such that $\langle g \rangle =\langle supp(S)\rangle$. The problem regarding the index of sequences has been studied in a series papers, and a main focus is to determine sequences of index 1. In this talk, we present the most recent development in this direction and show that if $G$ is a cyclic of prime power order such that $\gcd(|G|, 6)=1$, then every minimal zero-sum sequence of length 4 has index 1. Back to top The index of an algebraic variety Back to top Norm index formula for the Tate Kernels Back to top The Fibonacci Zeta Function The Fibonacci zeta function is the sum of the reciprocals of
the k-th powers of the Fibonacci numbers. It is conjectured that
all of these values are transcendental numbers, for k=1,2,...
However, this is known only for even values of k. Suprisingly,
the problem takes us into the world of modular forms, elliptic
functions and modern transcendence theory. I will give a short
survey of this fascinating chapter of number theory. An example of modular form in three variables by Alexander Odesski (Brock University) We construct explicitely a modular form in three variables and discuss its properties. Back to top Finding the square roots of graphs Is there an objective truth hidden within great works of art that only
mathematics can explain? For hundreds of years we have revered and collected
works of art by the great masters. I'll present statistical Back to top Rational approximation to real points on
plane algebraic curves Let C be a closed algebraic curve defined over Q in projective n-space.
Assume that the set of real points of C with Q-linearly independent coordinates
is infinite, and define lambda(C) to be the supremum of the uniform exponents
of approximation of those points by rational points. Although Dirichlet's
box principle simply shows that lambda(C) is at least 1/n, it is tempting
to conjecture that it is always greater, i.e. that there always exist such
points which are constantly much better approximated by rational points
then expected from the box principle. At the moment, the only curves for
which this is known to hold are the plane curves of degree 2. In this talk,
we present heuristics which support the conjecture for all plane algebraic
curves. Back to top Well spaced integers generated by an infinite
set of primes Back to top Contributed Papers
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