In this paper we obtain an inequality on the unit disk B in R2, which improves the classical Moser-Trudinger inequality and the classical Hardy inequality at the same time. Namely, there exists a constant C0 > 0 such that ∫ B e 4πu2 H(u) dx ≤ C0 <∞, ∀ u ∈ C 0 (B), where H(u) := ∫

We show that the variational inequality $VI(C,A)$ has aunique solution for a relaxed $(gamma , r)$-cocoercive,$mu$-Lipschitzian mapping $A: Cto H$ with $r>gamma mu^2$, where$C$ is a nonempty closed convex subset of a Hilbert space $H$. Fromthis result, it can be derived that, for example, the recentalgorithms given in the references of this paper, despite theirbecoming more complicated, are not...

{0, 1}-valued random variables X1, . . . , Xn are drawn independently each from Bernoulli distribution with parameter p = 0.1. Define Pn := P( 1 n ∑n i=1Xi ≤ 0.2). (a) For n = 1 to 30 calculate and plot the below in the same plot (see [1, section 6.1] for definition of Hoeffding and Bernstein inequalities): i. Exact value of Pn (binomial distribution). ii. Normal approximation for Pn. iii. Hoef...

Notice that nR̂n(h) ∼ binom(n,R(h)) and so E [ R̂n(h) ] = R(h). We would like to understand how accurate R̂n(h) is as an estimate of R(h). Thankfully there are many well known concentration inequalities that provide us with quantitative answers to this question. The goal of this lecture is to establish one such bound: Hoeffding’s inequality [2]. This inequality was originally proved in the 1960’s ...

The two-level penalty finite element methods for Navier-Stokes equations with nonlinear slip boundary conditions are investigated in this paper, whose variational formulation is the Navier-Stokes type variational inequality problem of the second kind. The basic idea is to solve the Navier-Stokes type variational inequality problem on a coarse mesh with mesh size H in combining with solving a St...