Some local properties of two-dimensional continued fractions

By regarding a two-dimensional continued fraction as a function of its elements and applying recursion relations for its tails, we establish formulas for the first partial derivatives of the fraction, on the basis of which we construct linear approximations of limit-periodic two-dimensional continued fractions. Translated fromMatematichni Metody i Fiziko-Mekhanichni Polya, Vol. 38, 1995.     

Some properties of a regular solution of the Helmholtz equation in a two-dimensional domain

In this paper we prove that if the function u_λ is a regular solution of the equation Δ₂u + λu = 0 in an arbitrary two-dimensional domain g and if at an arbitrary point M of the domain g we introduce polar coordinates r and?, then for an arbitrary value of the polar radius r, less than the distance of the point M from the boundary of the domain g, the following formula is valid: . Simultaneously, we show that the derivative is an n-th order trigonometric polynomial.     

Some properties of the equations governing a two-dimensional quasi-gasdynamic model of traffic flows

A recently proposed two-dimensional quasi-gasdynamic model of traffic flows is considered. Its Petrovskii parabolicity is analyzed, and the stability of small perturbations against a constant background is investigated. In a nonlinear setting, an energy equality is derived and an energy estimate of the solution is obtained.     

Topological torsion related to some recursive sequences of integers

For a recursively defined sequence u : = (u_n) of integers, we describe the subgroup t_u (??) of the elements x of the circle group ?? satisfying lim_n u_nx = 0. More attention is dedicated to the sequences satisfying a secondorder recurrence relation. In this case, we show that the size and the free‐rank of t_u (??) is determined by the asymptotic behaviour of the ratios q_n = (u_n /u_{n –1}) and we extend previous results of G. Larcher, C. Kraaikamp, and P. Liardet obtained from continued fraction expansion. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

关于DMRL序及其相关序的若干性质

证明了DMRL偏序关系在平移和尺度变换下的不变性,并获得了一个充分必要条件.同时考虑了DMRL序与IFR及IFRA序之间的关系.     

Some properties of random fields related to stochastic integrals with respect to strong martingales

We investigate the properties of random fields related to stochastic intergals with respect to a two-parameter strong square-integrable martingale in the case where the martingale and the integrands depend on the limits of integration. We prove inequalities for the moments of the uniform norm and the modulus of continuity of trajectories of modifications of such fields. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 320, 2004, pp. 80–96.     

Some double-angle formulas related to a generalized lemniscate function

The Ramanujan Journal - In this paper, we will establish some double-angle formulas related to the inverse function of $$int _0^x text {d}t/sqrt{1-t^6}$$ . This function appears in...     

Some convexity considerations for a two-dimensional traction problem

Summary A rectangular strip, consisting of homogeneous, isotropic, elastic material, is in an equilibrium state of plane stress, due to actions applied to a pair of opposite edges, the remaining pair being traction-free. The convexity of certain cross-sectional measures of stress is established; and a generalized convexity property is also established for one such measure, leading to an explicit decay estimate for the measure in the case of a semi-infinite strip, one end of which is subjected to a self-equilibrated load.     

Some results related to permanental inequalities

The FKG inequality and associated inequalities have been extensively studied in the literature. The FKG inequality has been extended to a 2m-function inequality which relates to some interesting permanental inequalities. In this paper, we prove some related inequalities and also give a simple alternative proof of the 2m-function inequality. Our proofs use tools of majorization theory and are thus based on a completely different approach than the usual one.     

Some issues related to double rounding

Double rounding is a phenomenon that may occur when different floating-point precisions are available on the same system. Although double rounding is, in general, innocuous, it may change the behavior of some useful small floating-point algorithms. We analyze the potential influence of double rounding on the Fast2Sum and 2Sum algorithms, on some summation algorithms, and Veltkamp’s splitting.     

Some inequalities related to Sobolev norms

In this paper, we study some properties related to the new characterizations of Sobolev spaces introduced in Bourgain and Nguyen (C R Acad Sci, 343:75?C80, [2006]), Nguyen (J Funct Anal 237: 689?C720, [2006]; J Eur Math Soc 10:191?C229, [2008]). More precisely, we establish variants of the Poincaré inequality, the Sobolev inequality, and the Rellich?CKondrachov compactness theorem, where ${\int_{\mathbb{R}^N} |\nabla g|^p \;dx}$ is replaced by some quantity of the type $$I_{\delta} (g) =\mathop{\int\limits_{\mathbb{R}^N}\int\limits_{\mathbb{R}^N}}_{|g(x) - g(y)| > \delta}\frac{\delta^p}{|x-y|^{N+p}}\, dx \, dy.$$      

Some polynomials related to theta functions

Summary This paper continues the study of H_n(x)=H_n(x, q)= and related polynomials discussed bySzeg? [7]. In particular, it is shown that H_n(x) has various formal properties analogous to properties of theHermite polynomials.     

Some examples related to knot sliceness

It is known that the linking form on the 2-cover of slice knots has a metabolizer. We show that several weaker conditions, or some other conditions related to sliceness, do not imply the existence of a metabolizer. We then show how the Rudolph-Bennequin inequality can be used indirectly to prove that some knots are not slice.     

Some propositions related to a dilation theorem of W. Benz

Summary This paper begins with another proof of a theorem of W. Benz [2] concerning dilations in normed linear spaces. Our proof motivates several questions which are addressed thereafter. For instance it is shown that, ifI is an open interval in ,: I ⁿ , is continuously differentiable and there exista ₁,...,a _n I such that {(a ₁,...,(a _n )} is linearly independent, then {(t): t I} contains a Hamel basis for ⁿ over .