Given a Lévy process L, we consider the so-called statistical Skorohod embedding problem of recovering the distribution of an independent random time T based on i.i.d. sample from LT . Our approach is based on the genuine use of the Mellin and Laplace transforms. We propose consistent estimators for the density of T, derive their convergence rates and prove their optimality. It turns out that t...

We consider the convergence and stability property of MUSCL relaxing schemes applied to conservation laws with stii source terms. The maximum principle for the numerical schemes will be established. It will be also shown that the MUSCL relaxing schemes are uniformly l 1-and T V-stable in the sense that they are bounded by a constant independent of the relaxation parameter , the Lipschitz consta...

We present an algorithm computing, for any first order differential equation L over the affine line and any (Berkovich) point t of this affine line, the p-adic radius of convergence RL(t) of the solutions of L near t. We do explicit computations for the equation (0.1) L(f) def = xf ′ − π(px + ax)f = 0 (πp−1 = −p). where a lies in some valued extension of Qp. For a = −1 and t = 0, a solution of ...

Let C be a nonempty closed convex subset of a Hilbert spaceH, T a self-mapping of C. Recall that T is said to be nonexpansive if ‖Tx − Ty‖ ≤ ‖x − y‖, for all x, y ∈ C. Construction of fixed points of nonexpansive mappings via Mann’s iteration 1 has extensively been investigated in literature see, e.g., 2–5 and reference therein . But the convergence about Mann’s iteration and Ishikawa’s iterati...

Contents Introduction xv Chapter 1. Fourier series: some motivation 1 1.1. Some examples and key definitions 1 1.2. Main questions 5 1.3. Fourier series and Fourier coefficients 7 1.4. A little history, and motivation from the physical world 11 Chapter 2. Interlude 17 2.1. Nested classes of functions on bounded intervals 17 2.2. Modes of convergence 28 2.3. Interchanging limit operations 34 2.4...

Scientific Reports 5:9519; doi: 10.1038/srep09519; published online 27 April 2015; updated on 02 September 2015 In the Supplementary Information file originally published with this Article, there are typographical errors. In the section under ‘Convergence’, “In order to evaluate such convergence, we fit the last tγ time steps of the evolution to a linear trend, α β ( ) = +  c t t using the QR ...

This paper is concerned with the asymptotic behavior of sums of the form U(f)t = ∑[t/∆n] i=1 f(Xi∆n − X(i−1)∆n), where X is a 1-dimensional semimartingale and f a suitable test function, typically f(x) = |x|r, as ∆n → 0. We prove a variety of “laws of large numbers”, that is convergence in probability of U(f)t, sometimes after normalization. We also exhibit in many cases the rate of convergence...

The Cauchy problem for a semilinear parabolic equation is considered. Under the conditions u(x, t) = X(x)T1(t) + T2(t), ∂u ∂x = 0, it is shown that the problem is equivalent to the system of two ordinary differential equations for which exact difference scheme (EDS) with special Steklov averaging and difference schemes with arbitrary order of accuracy (ADS) are constructed on the moving mesh. T...

In this paper, we propose a new stochastic alternating direction method of multipliers (ADMM) algorithm, which incrementally approximates the full gradient in the linearized ADMM formulation. Besides having a low per-iteration complexity as existing stochastic ADMM algorithms, the proposed algorithm improves the convergence rate on convex problems from O ( 1 √ T ) to O ( 1 T ) , where T is the ...

The implicit bias of gradient descent is not fully understood even in simple linear classification tasks (e.g., logistic regression). Soudry et al. (2018) studied this bias on separable data, where there are multiple solutions that correctly classify the data. It was found that, when optimizing monotonically decreasing loss functions with exponential tails using gradient descent, the linear cla...