In the paper, the large time behavior of solutions of the Cauchy problem for the one dimensional fractal Burgers equation ut + (−∂ 2 x) α/2u+ uux = 0 with α ∈ (1, 2) is studied. It is shown that if the nondecreasing initial datum approaches the constant states u± (u− < u+) as x → ±∞, respectively, then the corresponding solution converges toward the rarefaction wave, i.e. the unique entropy sol...

In this paper, we discuss the oscillatory behavior of first order delay differential equations of the form: y′(t) + p(t)y(τ(t)) = 0, t ≥ T, where p and T are continuous functions defined on [T,∞), p(t) > 0, τ(t) < t for t ≥ T, τ(t) is nondecreasing and lim t→∞ τ(t) = ∞. We present best possible conditions for the oscillation of all solutions for this equation.

A class of α-potentials represented as the sum of modified Green potential and modified Poisson integral are proved to have the growth estimates Rα,l,l x o x β n|x| h |x| −1 at infinity in the upper-half space of the n-dimensional Euclidean space, where the function h |x| is a positive nondecreasing function on the interval 0,∞ satisfying certain conditions. This result generalizes the growth p...

Let φ : [0,∞) → R and let y0 be a non-negative constant. Let (λn)n∈N be a nondecreasing sequence of positive numbers which tends to infinity, let (rn)n∈N be a complex sequence, and c a real number. Assume that φ is square-integrable on [0, y0] and for y ≥ y0, φ can be expressed as φ(y) = c+ R ( ∑ λn≤X rne iλny ) + E(y,X), for any X ≥ X0 > 0 where E(y,X) satisfies

For a probability measure P on Rd and n ∈ N consider en = inf ∫ mina∈α V (‖x − a‖) dP (x) where the infimum is taken over all subsets α of Rd with card(α) n and V is a nondecreasing function. Under certain conditions on V , we derive the precise n-asymptotics of en for self-similar distributions P and we find the asymptotic performance of optimal quantizers using weighted empirical measures.  ...

We view regularized learning of a function in a Banach space from its finite samples as an optimization problem. Within the framework of reproducing kernel Banach spaces, we prove the representer theorem for the minimizer of regularized learning schemes with a general loss function and a nondecreasing regularizer. When the loss function and the regularizer are differentiable, a characterization...

Nonasymptotic risk bounds are provided for maximum likelihood-type isotonic estimators of an unknown nondecreasing regression function, with general average loss at design points. These bounds are optimal up to scale constants, and they imply uniform n−1/3-consistency of the p risk for unknown regression functions of uniformly bounded variation, under mild assumptions on the joint probability d...

Previous literature has mostly considered R&D and licensing activities separately. In this paper we examine the effect of licensing on R&D and social welfare. We show that the effect of licensing on the incentive for doing R&D is ambiguous and depends on the costs of doing R&D. We also show that the possibility of licensing can change the identity of the innovating firm. However, we find that s...

We construct a formal power series on several variables that encodes many statistics on non-decreasing Dyck paths. In particular, we use this formal power series to count peaks, pyramid weights, and indexed sums of pyramid weights for all nondecreasing Dyck paths of length 2n. We also show that an indexed sum on pyramid weights depends only on the size and maximum element of the indexing set.