A k-arc in PG(2, q) is a set of k points no three which are collinear. hyperfocused the $$k \atopwithdelims ()2$$ secants meet some external line exactly $$k-1$$ points. Hyperfocused k-arcs can be viewed as 1-factorizations complete graph $$K_k$$ that embed q). We study 526,915,620 $$K_{12}$$ , determine embeddable q), and classify 12-arcs. Specifically we show if 12-arc $$\mathcal {K}$$ arc th...

1 Statement and history In this lecture, we prove the full-blown hypercontractivity theorem for {−1, 1} n. The idea behind the statement is that the T ρ operator smooths, or " reasonable-izes " , functions. Some examples, including two that were mentioned in the last lecture: Example 1.2 • q = 4, p = 2, ρ = 1/ √ 3: T 1/ √ 3 f 4 ≤ f 2 • q = 2, p = 4/3, ρ = 1/ √ 3: T 1/ √ 3 f 2 ≤ f 4/3 • q = q, p...

Let n, k be the positive integers (k > 1), and let Sn,q(k) be the sums of the n-th powers of positive q-integers up to k − 1: Sn,q(k) = ∑k−1 l=0 ql. Following an idea due to J. Bernoulli, we explore a formula for Sn,q(k).

We obtain q-analogues of several series for powers $$\pi $$ . For example, the identity $$\begin{aligned} \mathop {\sum }\limits _{k=0}^\infty \frac{(-1)^k}{(2k+1)^3}=\frac{\pi ^3}{32} \end{aligned}$$ has following q-analogue: (-1)^k\frac{q^{2k}(1+q^{2k+1})}{(1-q^{2k+1})^3}=\frac{(q^2;q^4)_{\infty }^2(q^4;q^4)_{\infty }^6}{(q;q^2)_{\infty }^4}, where q is any complex number with $$|q|<1$$ al...

Let c(n, q) be the number of connected labeled graphs with n vertices and q 5 N = ( ) edges. Let x = q/n and k = q n. We determine functions w k 1 , a(x ) and cp(x) such that c (n , q) w k ( z ) e n r p ( x ) e o ( x ) uniformly for all n and q 2 n. If Q > O is fixed, n+= and 4q > ( 1 + ~ ) n log n, this formula simplifies to c(n, q) ( t ) exp(-ne-zq’n). On the other hand, if k = o(n”’), this f...

We prove three results about colorings of the simplex reminiscent of Sperner’s Lemma, with applications in hardness of approximation and fair division. First, we prove a coloring lemma conjectured by [5]: Let Vk,q = {v ∈ Z+ : ∑k i=1 vi = q} and Ek,q = {{a + e1,a + e2, . . . ,a + ek} : a ∈ Z+, ∑k i=1 ai = q − 1}. Then for every Sperner-admissible labeling (` : Vk,q → [k] such that v`(v) > 0 for ...