Infinity Of Order Research Articles

A Numerical Integrator for Oscillatory Problems

Presented here is a numerical integrator, with sixth order of convergence, for solving oscillatory problems. Dispersion and dissipation errors are taken into account in the course of deriving the method. As a result, the method possesses dissipation of order infinity and dispersive of order six. Validity and effectiveness of the method are tested on a number of test problems. Results obtained show that the new method is better than its equals in the scientific literature.

Persistence of Gaussian processes: non-summable correlations

Suppose the auto-correlations of real-valued, centered Gaussian process $$Z(\cdot )$$ are non-negative and decay as $$\rho (|s-t|)$$ for some $$\rho (\cdot )$$ regularly varying at infinity of order $$-\alpha \in [-1,0)$$ . With $$I_\rho (t)=\int _0^t \rho (s)ds$$ its primitive, we show that the persistence probabilities decay rate of $$ -\log \mathbb {P}(\sup _{t \in [0,T]}\{Z(t)\}<0)$$ is precisely of order $$(T/I_\rho (T)) \log I_\rho (T)$$ , thereby closing the gap between the lower and upper bounds of Newell and Rosenblatt (Ann. Math. Stat. 33:1306–1313, 1962), which stood as such for over fifty years. We demonstrate its usefulness by sharpening recent results of Sakagawa (Adv. Appl. Probab. 47:146–163, 2015) about the dependence on d of such persistence decay for the Langevin dynamics of certain $$\nabla \phi $$ -interface models on $$\mathbb {Z}^d$$ .

On solvability of homogeneous Riemann–Hilbert problem with discontinuities of coefficients and two-sided curling at infinity of a logarithmic order

We consider a homogeneous Riemann–Hilbert boundary-value problem for upper halfplane in the situation where its coefficients have countable set of discontinuities of jump type and two-side curling at infinity of a logarithmic order. We obtain general solution and describe completely its solvability in a special class of functions for the case where the index of the problem has power singularity of a logarithmic order.

Univalence of mappings from half-plane to a polygonal domains with infinite sets of vertices

We apply solutions of the Hilbert boundary value problem with discontinuous coefficients and two-side curling at the infinity of logarithmic order to the construction of the formula of conformal mapping for a half-plane onto a polygonal domain with a countable set of vertices. We prove a sufficient condition of univalence of such mappings.

Homogeneous Hilbert boundary-value problem with countable set of discontinuities and logarithmic singularity of index

We consider the Hilbert boundary-value problem for the upper half-plane with a countable set of discontinuity points of coefficient of the boundary condition and with a two-side curling at infinity of a logarithmic order. We obtain formulas for the general solution to the problem.

О разрешимости однородной задачи Гильберта со счетным множеством точек разрыва коэффициентов и двусторонним завихрением на бесконечности порядка меньше 1/2

О разрешимости однородной задачи Гильберта со счетным множеством точек разрыва коэффициентов и двусторонним завихрением на бесконечности порядка меньше 1/2

Gravity solitary waves with polynomial decay to exponentially small ripples at infinity

In this paper, we study the travelling gravity waves of velocity c in a system of two layers of perfect fluids, the bottom one being infinitely deep, the upper one having a finite thickness h. We assume that the flow is potential, and the dimensionless parameters are the ratio between densities ρ=ρ2/ρ1 and λ=gh/c2. For ε=1−λ(1−ρ) near 0+, the existence of periodic travelling waves of arbitrary small amplitude and the existence of generalized solitary waves with ripples at infinity of size larger than ε5/2 and polynomial decay rate were established in [7]. In this paper we improve this former result by showing the existence of generalized solitary waves with exponentially small ripples at infinity (of order O(e−c/ε)). We conjecture the nonexistence of true solitary waves in this case. The proof is based on a spatial dynamical formulation of the problem combined with a study of the analytic continuation of the solutions in the complex field which enables one to obtain exponentially small upper bounds of the oscillatory integrals giving the size of the oscillations at infinity.

Note on Legendre's and Bertrand's Proofs of the Parallel-Postulate by Infinite Areas

One of the most plausible of the host of “proofs” that have ever been offered for Euclid's parallel-postulate is that known as Bertrand's, which is based upon a consideration of infinite areas. The area of the whole plane being regarded as an infinity of the second order, the area of a strip of plane surface bounded by a linear segment AB and the rays AA′, BB perpendicular to AB is an infinity of the first order, since a single infinity of such strips is required to cover the plane. On the other hand, the area contained between two intersecting straight lines is an infinity of the same order as the plane, since the plane can be covered by a finite number of such sectors. Hence if AP is drawn making any angle, however small, with AA′, the area A′AP, an infinity of the second order, cannot be contained within the area A′ABB′, an infinity of the first order, and therefore AP must cut BB′. And this is just Euclid's postulate.