Let n be a nonnegative integer and let u 2 (n; n + 1]. We say that f is u times Peano bounded in the approximate ( resp. L; 1 p 1) sense at x 2 R if there are numbers ff (x)g ; j j n such that f(x+ h) P j j n f (x)h = ! is O(h) in the approximate (resp. L) sense as h ! 0. Suppose f is u times Peano bounded in either the approximate or L sense at each point of a bounded measurable set E: Then fo...

We present a simple proof of the interior approximate controllability for the following broad class of second-order equations in the Hilbert space L2 Ω : ÿ Ay 1ωu t , t ∈ 0, τ , y 0 y0, ẏ 0 y1, where Ω is a domain in N N ≥ 1 , y0, y1 ∈ L2 Ω , ω is an open nonempty subset of Ω, 1ω denotes the characteristic function of the set ω, the distributed control u belongs to L2 0, τ ;L2 Ω , and A : D A ⊂...

An analytic result for the O(αs) corrections to the triple differential B → Xu l ν̄l decay rate is presented, to leading order in the heavy-quark expansion. This is relevant for computing partially integrated decay distributions with arbitrary cuts on kinematic variables. Several double and single differential distributions are derived, most of which generalize known results. In particular, an a...

We describe L-SR1, a new second order method to train deep neural networks. Second order methods hold great promise for distributed training of deep networks. Unfortunately, they have not proven practical. Two significant barriers to their success are inappropriate handling of saddle points, and poor conditioning of the Hessian. L-SR1 is a practical second order method that addresses these conc...

In this paper, we prove the existence of solutions for the strongly nonlinear equation of the type Au + g(x, u) = f where A is an elliptic operator of infinite order from a functional space of Sobolev type to its dual. g(x, s) is a lower order term satisfying essentially a sign condition on s and the second term f belongs to L 1 (Ω).

A subset A of a group G is called (k, l)-sumset, if A = kB−lB for some B ⊆ G, where kB − lB = {x1 + · · · + xk − xk+1 − · · · − xk+l : x1, . . . , xk+l ∈ B}. Upper and lower bounds for the number (k, l)-sumsets in groups of prime order are provided.

$\beta(x, u)$ is not monotone but has the same sign as $u$ , we prove the existence of solutions when $f(x)$ belongs to an Orlicz class arbitrarily close to $L^{1}(\Omega)$ . We also consider equation (1) with a nonlinear boundary condition. The linear case is considered by Stampacchia [18]. Our basic technique is to multiply the equation by various monotone functions of $u$ . This method was u...

This paper aims at providing a self contained introduction to notions and results connected with the L approximation power of nitely generated shift invariant spaces FSI spaces S L R d Here approx imation order refers to a scaling parameter and to the usual scaling of the L projector onto S where f ng L R d is a given set of functions the so called generators of S Special attention is given to ...