Let’s quickly review some results from previous lectures. Definition 1 Let (Enc,Dec) denote an encryption scheme where Enc : {0, 1}m×{0, 1}n → {0, 1}λ and Dec : {0, 1}m × {0, 1}λ → {0, 1}n are functions. A correct encryption scheme satisfies ∀r, x Decr(Encr(x)) = x. Let R and X denote distributions on r and x respectively. (Enc,Dec) is said to be (k, ε)-secure if it is (R, ε)-secure for every k...

We study bounded degree graph problems, mainly the problem of k-Dimensional Matching (k-DM), namely, the problem of finding a maximal matching in a k-partite k-uniform balanced hyper-graph. We prove that k-DM cannot be efficiently approximated to within a factor of O( k ln k ) unless P = NP . This improves the previous factor of k 2O( √ ln ) by Trevisan [Tre01]. For low k values we prove NP-har...

The use of the sine-Gordon equation as a model of magnetic flux propagation in Josephson junctions motivates studying the initial-value problem for this equation in the semiclassical limit in which the dispersion parameter ε tends to zero. Assuming natural initial data having the profile of a moving −2π kink at time zero, we analytically calculate the scattering data of this completely integrab...

Assuming the Unique Games Conjecture (UGC), we show optimal inapproximability results for two classic scheduling problems. We obtain a hardness of 2 − ε for the problem of minimizing the total weighted completion time in concurrent open shops. We also obtain a hardness of 2 − ε for minimizing the makespan in the assembly line problem. These results follow from a new inapproximability result for...

We consider the following singularly perturbed Neumann problem ε∆u− u+ u = 0 in Ω, u > 0 in Ω, ∂u ∂ν = 0 on ∂Ω, where p is subcritical and Ω is a smooth and bounded domain in R with its unit outward normal ν. Lin-Ni-Wei [18] proved that there exists ε0 such that for 0 < ε < ε0 and for each integer k bounded by 1 ≤ k ≤ δ(Ω, n, p) (ε| log ε|)n (0.1) where δ(Ω, n, p) is a constant depending only o...

The first point here is that convex sets can be separated by linear functionals. Second, continuous linear functionals on subspaces of a locally convex topological vectorspace have continuous extensions to the whole space. Proofs are for real vectorspaces. The complex versions are corollaries. A crucial corollary is that on locally convex topological vectorspaces continuous linear functionals s...

We prove that for every k and every ε > 0, there exists g such that every graph with tree-width at most k and odd-girth at least g has circular chromatic number at most 2 + ε.

Szemerédi’s Regularity Lemma proved to be a powerful tool in the area of extremal graph theory. Many of its applications are based on its accompanying Counting Lemma: If G is an `-partite graph with V (G) = V1 ∪ · · · ∪ V` and |Vi| = n for all i ∈ [`], and all pairs (Vi, Vj) are ε-regular of density d for 1 ≤ i < j ≤ ` and ε d, then G contains (1± f`(ε))d “ ` 2 ” × n` cliques K`, where f`(ε) → ...

We study the asymptotic behaviour, as ε goes to zero, of the entropy solution uε to a class of nonlinear elliptic problems whose prototype is −div ( (δε)p−1 |∇uε|p−2∇uε ) = f ε in Ωε , uε = 0 on ∂Ωε, where 1 < p < N , Ωε = Ω ∪ Σε, Ω ⊂ R is a bounded domain of class C1,1, surrounded along its boundary by a layer Σε of small thickness ε, δε = 1 in Ω, δε = ε in Σε, and f ε belongs to L1(Ωε). The s...