MenuEvents for 11/10/2022 from all calendars
Seminar on Banach and Metric Space Geometry
Time: 10:00AM - 11:00AM
Location: BLOC 302
Speaker: Hung Viet Chu, University of Illinois at Urbana-Champaign
Title: New Greedy-type Bases from Comparing Greedy Sums against Other Approximants of Different Sizes
Abstract: One goal of approximation theory is to approximate a vector $x$ using finite linear combinations of vectors, called approximants, in a given basis. For higher efficiency, the approximants can be made adaptive to the vector $x$ as in the thresholding greedy algorithm (TGA) introduced by Konyagin and Temlyakov in 1999. The optimality of the TGA is captured by the notion of greedy and almost greedy bases. A basis $(e_n)_{n=1}^\infty$ of a Banach space $X$ (over a field $\mathbb{F}$) is said to be greedy if there exists a constant $\mathbf C\geqslant 1$ such that $$\|x-G_m(x)\|\ \leqslant\ \mathbf C\inf_{\substack{|A|\leqslant m\\(a_n)_{n\in A}\subset \mathbb{F}}}\left\|x-\sum_{n\in A}a_ne_n\right\|.$$ Here, $G_m(x)$ is an approximant of size $m$ given by the TGA. The definition of almost greedy bases replaces the arbitrary linear combinations on the right by projections. Extending classical results, we define ($f$, greedy) bases to satisfy the condition: there exists a constant $\mathbf C\geqslant 1$ such that $$\|x-G_m(x)\|\ \leqslant\ \mathbf C\inf_{\substack{|A|\leqslant f(m)\\(a_n)_{n\in A}\subset \mathbb{F}}}\left\|x-\sum_{n\in A}a_ne_n\right\|,$$ where $f$ belongs to $\mathcal{F}$, a collection that contains functions like $f(x) = cx^{\gamma}$ for $c, \gamma\in [0,1]$. The definition of ($f$, almost greedy) is modified accordingly. We characterize these bases and establish the surprising equivalence: if $f$ is a non-identity function in $\mathcal{F}$, then a basis is ($f$, greedy) if and only if it is ($f$, almost greedy). We show that ($f$, greedy) bases form a much wider class as there exist examples of classical bases that are not almost greedy but is ($f$, greedy) for some $f\in\mathcal{F}$. Besides greedy and almost greedy bases, we also have partially greedy bases, which compare $G_m(x)$ against partially summations. Inspired by a theorem due to Dilworth, Kalton, Kutzarova, and Temlyakov, we show that for a fixed $\lambda > 1$, replacing $G_m(x)$ by $G_{\lceil\lambda m\rceil}(x)$ in the defini
Number Theory Seminar
Time: 2:30PM - 3:30PM
Location: BLOC 302
Speaker: Hung Viet Chu, University of Illinois at Urbana-Champaign
Title: Divots in the distribution of missing sums in sumsets & more-sum-than-difference sets
Abstract: We begin by talking about missing sums in sumsets. For a finite set $A$ of integers, define the sumset $A + A := \{a+b: a, b\in A\}$ and the difference set $A - A := \{a-b: a, b\in A\}$. Consider the probability model where $A$ is formed by choosing each integer in $\{0, 1, \ldots, n-1\}$ with probability $p$. Note that if $A = \{0,1,\ldots, n-1\}$, then $|A + A| = 2n-1$. Hence, the random variable $X := 2n-1 - |A+A|$ counts the number of missing sums in the sumset of $A$. Let $\mathbb{P}_{p, n}(|X| = k)$ denote the probability that we observe $k$ missing sums. This probability depends on both $p$ and $n$. In 2011, Zhao proved that $\mathbb{P}_{p}(|X| = k) := \lim_{n\rightarrow\infty}\mathbb{P}_{p, n}(|X|= k)$ exists. In 2013, Lazarev, Miller, and O’Bryant showed an interesting behavior in the distribution of missing sums in the uniform model: $$\mathbb{P}_{1/2}(|X| = 6) \ >\ \mathbb{P}_{1/2}(|X| = 7) \ <\ \mathbb{P}_{1/2}(|X| = 8).$$ This result says that in the uniform model, it is less likely to miss $7$ sums than both to miss $6$ sums and to miss $8$ sums. We call $7$ a divot when $p = 1/2$ and investigate whether a divot can happen earlier when $p$ varies. We proved $$\mathbb{P}_{p}(|X| = 0) \ >\ \mathbb{P}_{p}(|X| = 1) \ <\ \mathbb{P}_{p}(|X| = 2),\forall p \geqslant 0.68.$$ Next, we move on to talk about more-sum-than-difference (MSTD) sets, which are sets $A$ satisfying $|A + A| > |A - A|$. In 2007, Martin and O’Bryant proved a surprising result that there exists a positive constant lower bound for the proportion of MSTD subsets of $\{0, 1, \ldots, n-1\}$ as $n\rightarrow\infty$. We proved that for any $k\geqslant 2$, it is possible to partition $\{0, 1, \ldots, n-1\}$ into $k$ MSTD subsets and provide bounds on the smallest $n$ for such partitions to take place. This answered a question by Asada, Manski, Miller, and Suh in 2017. The key idea in both of the above results is fringe analysis, which centers around the observation that it is much more likely to miss a sum in the two end
Several Complex Variables Seminar
Time: 2:30PM - 3:30PM
Location: Bloc 624
Speaker: Shreedhar Bhat
Title: p-Bergman kernel
