On Construction of Multidimensional Continued Fractions

A thorough exposition of finite simple continued fractions is given. All rational numbers have such representations. Convergents to a simple continued fraction is studied and a handy algorithm is developed which helps to readily compute them. The uniqueness of a continued fraction representation is covered. The theory is naturally extended to infinite simple continued fractions including those for $\sqrt 2 $ and for $\pi $ as well as the general importance of finite continued fractions as approximations to real numbers. The chapter culminates in a robust discussion of Christiaan Huygens’ cogwheeled planetarium which utilized continued fractions.

We consider a one-sided problem for the Barenblatt–Zheltov–Kochina pseudoparabolic operator in the one-dimensional case, supplemented with smooth initial data and homogeneous boundary conditions. This problem is formulated in the form of a variational inequality. From the physical point of view, it models a non-stationary process of filtration of a viscous fluid in a cracky-porous gallery with a restriction on the modulus of the velocity of filtration through the cracks. The existence theorem for a weak solution of this problem is known in the literature in both one-dimensional and multidimensional cases and follows from the results obtained by M. Ptashnyk (Nonlinear Anal., 2007, vol. 66, pp. 2653–2675) using the penalty method. In M. Ptashnyk’s research, the penalty operator was chosen in a standard form, following the presentation in the monograph by J.-L. Lions “Quelques m´ethodes de r´esolution des probl´emes aux limites non lin´eaires,” Paris: Dunod Gauthier-Villars, 1969 (Theorem 5.1 in Chapter 3). In this article, we consider an approximate initial-boundary value problem for the pseudoparabolic equation incorporating Kaplan’s penalty operator and study the family of its solutions. Due to the specific structure of Kaplan’s operator, we obtain higher regularity of the weak solution of the original problem as compared to the previously known regularity properties, and also we find a strengthened property of approximating this solution by a sequence of solutions to the problem with Kaplan’s operator. In addition, we establish that the one-sided condition imposed in the original problem is satisfied by the approximate solution on a set of the spatial variable which monotone grows with decrease of the small approximation parameter.

This paper presents an extension of an earlier one, which presented an improved numerical technique for solving multicomponent stable miscible flow equations. Diffusion-convection equations for each component together with a pressure equation are solved. Second-order correct approximations for individual component flows, resulting in numerical dispersion correction terms, are employed in solving the diffusion-convection equations. Backward-in-time implicit and Crank-Nicolson approximation techniques are used to solve the pressure equation. Pressure and saturation distributions using first- and second-order correct finite-difference approximations for individual component flows are compared. One- and two-dimensional systems are considered for these comparisons. The results show that in a compressible miscible displacement process, first-order correct finite-difference scheme process, first-order correct finite-difference scheme for individual component flows results in considerable numerical smearing in the saturation distribution. The pressure distribution, which depends upon the saturation distribution, is very sensitive to numerical smearing. Therefore, high-order correct approximations for individual component flours are necessary. The results also demonstrate That the earlier approach employed to reduce numerical smearing in the numerical solution of incompressible miscible flow processes can be extended to compressible miscible flow systems. Introduction Many investigators have tried to solve in some fashion the system of equations governing the miscible displacement process. In most cases, even though the system was compressible, an incompressible system was considered. One may state several reasons for this. Since the system of equations governing compressible miscible flow is nonlinear, analytical solution of such a system, even in a one-dimensional case, is not known In a multidimensional case it may be a practical impossibility. The only methods known to solve the above problem are the numerical methods. One of such methods is presented here. Basic flow equations for the individual components present in the compressible miscible flow system present in the compressible miscible flow system are assumed. These flow equations together with the individual component continuity equations are used to arrive at the pressure equation. Crank-Nicolson and backward-in-time approximation schemes are used in an iterative procedure to solve the pressure equation. Flows of individual components are calculated with first- and second-order correct finite-difference approximations. It is also shown that a semirigorous technique, which eliminates most of the numerical smearing from the numerical solution of an incompressible miscible system, can also be used for a compressible miscible flow system. It has been observed that pressure distributions in a one-dimensional case are pressure distributions in a one-dimensional case are significantly different when the individual component flow terms are approximated with first- and second-order correct finite-difference approximations. Also, the order of approximation of the flow equations has more effect on the pressure distribution than the order of approximation (time approximation) used to solve the pressure equation.

This survey is devoted to results related to metric properties of classical continued fractions and Voronoi–Minkowski three-dimensional continued fractions. The main focus is on applications of analytic methods based on estimates of Kloosterman sums. An apparatus is developed for solving problems about three-dimensional lattices. The approach is based on reduction to the preceding dimension, an idea used earlier by Linnik and Skubenko in the study of integer solutions of the determinant equation , where is a matrix with independent coefficients and is an increasing parameter. The proposed method is used for studying statistical properties of Voronoi–Minkowski three-dimensional continued fractions in lattices with a fixed determinant. In particular, an asymptotic formula with polynomial lowering in the remainder term is proved for the average number of Minkowski bases. This result can be regarded as a three-dimensional analogue of Porter's theorem on the average length of finite continued fractions. Bibliography: 127 titles.

We extend results of Y. Benoist and J.-F. Quint concerning random walks on homogeneous spaces of simple Lie groups to the case where the measure defining the random walk generates a semigroup which is not necessarily Zariski dense, but satisfies some expansion properties for the adjoint action. Using these dynamical results, we study Diophantine properties of typical points on some self-similar fractals in $${\mathbb {R}}^d$$ . As examples, we show that for any self-similar fractal $$\mathcal {K}\subseteq {\mathbb {R}}^d$$ satisfying the open set condition (for instance any translate or dilate of Cantor’s middle thirds set or of a Koch snowflake), almost every point with respect to the natural measure on $$\mathcal {K}$$ is not badly approximable. Furthermore, almost every point on the fractal is of generic type, which means (in the one-dimensional case) that its continued fraction expansion contains all finite words with the frequencies predicted by the Gauss measure. We prove analogous results for matrix approximation, and for the case of fractals defined by Mobius transformations.

We consider the averaging of local field-theoretic Poisson brackets in the multi-dimensional case. As a result, we construct a local Poisson bracket for the regular Whitham system in the multidimensional situation. The procedure is based on the procedure of averaging of local conservation laws and follows the Dubrovin–Novikov scheme of the bracket averaging suggested in one-dimensional case. However, the features of the phase space of modulated parameters in higher dimensions lead to a different natural class of the averaged brackets in comparison with the one-dimensional situation. Here we suggest a direct procedure of construction of the bracket for the Whitham system for d > 1 and discuss the conditions of applicability of the corresponding scheme. At the end, we discuss canonical forms of the averaged Poisson bracket in the multidimensional case.

In his paper "Multidimensional continued fractions" ( Ann. Univ. Sci. Budapest. Eötvös Sect. Math. , v. 13, 1970, pp. 113-140), G. Szekeres introduced a new higher dimensional analogue of the ordinary continued fraction expansion of a single real number. The Szekeres algorithm associates with each k -tuple ( α 1 , … , α k ) ({\alpha _1}, \ldots ,{\alpha _k}) of real numbers (satisfying 0 > α i > 1 0 > {\alpha _i} > 1 ) a sequence b 1 , b 2 , … {b_1}, {b_2}, \ldots of positive integers; this sequence is called a continued k -fraction, and for k = 1 it is just the sequence of partial quotients of the ordinary continued fraction for α 1 {\alpha _1} . A simple recursive procedure applied to b 1 , b 2 , … {b_1}, {b_2}, \ldots produces a sequence a ( n ) = ( A n ( 1 ) / B n , … , A n ( k ) / B n ) ( n = 1 , 2 , … ; A n ( i ) ⩾ 0 a(n) = (A_n^{(1)}/{B_n}, \ldots ,A_n^{(k)}/{B_n})\;(n = 1,2, \ldots ;A_n^{(i)} \geqslant 0 and B n > 0 {B_n} > 0 are integers) of simultaneous rational approximations to ( α 1 , … , α k ) ({\alpha _1}, \ldots ,{\alpha _k}) and a sequence c ( n ) = ( c n 0 , c n 1 , … , c n k ) ( n = 1 , 2 , … ) c(n) = ({c_{n0}},{c_{n1}}, \ldots ,{c_{nk}})\;(n = 1,2, \ldots ) of integer ( k + 1 ) (k + 1) -tuples such that the linear combination c n 0 + c n 1 α 1 + ⋯ + c n k α k {c_{n0}} + {c_{n1}}{\alpha _1} + \cdots + {c_{nk}}{\alpha _k} approximates zero. Szekeres conjectured, on the basis of extensive computations, that the sequence a ( 1 ) , a ( 2 ) , … a(1),a(2), \ldots contains all of the "best" simultaneous rational approximations to ( α 1 , … , α k ) ({\alpha _1}, \ldots ,{\alpha _k}) and that the sequence c ( 1 ) , c ( 2 ) , … c(1),c(2), \ldots contains all of the "best" approximations to zero by the linear form x 0 + x 1 α 1 + ⋯ + x n α n {x_0} + {x_1}{\alpha _1} + \cdots + {x_n}{\alpha _n} . For the special case k = 2 and α 1 = θ 2 − 1 , α 2 = θ − 1 {\alpha _1} = {\theta ^2} - 1,{\alpha _2} = \theta - 1 (where θ = 2 cos ⁡ ( 2 π / 7 ) \theta = 2\cos (2\pi /7) is the positive root of x 3 + x 2 − 2 x − 1 = 0 {x^3} + {x^2} - 2x - 1 = 0 , Szekeres further conjectured that the 2-fraction b 1 , b 2 , … {b_1},{b_2}, \ldots is "almost periodic" in a precisely defined sense. In this paper the Szekeres conjectures concerning best approximations to zero by the linear form x 0 + x 1 ( θ 2 − 1 ) + x 2 ( θ − 1 ) {x_0} + {x_1}({\theta ^2} - 1) + {x_2}(\theta - 1) and concerning almost periodicity for the 2-fraction of ( θ 2 − 1 , θ − 1 ) ({\theta ^2} - 1,\theta - 1) are proved. The method used can be applied to other pairs of cubic irrationals α 1 , α 2 {\alpha _1},{\alpha _2} .

A 2-D parallel convex hull algorithm with optimal communication phases

We discuss a number of natural problems in arithmetic, arising in completely unrelated settings, which turn out to have a common formulation involving “thin” orbits. These include the local-global problem for integral Apollonian gaskets and Zaremba’s Conjecture on finite continued fractions with absolutely bounded partial quotients. Though these problems could have been posed by the ancient Greeks, recent progress comes from a pleasant synthesis of modern techniques from a variety of fields, including harmonic analysis, algebra, geometry, combinatorics, and dynamics. We describe the problems, partial progress, and some of the tools alluded to above.

Stability aspects in using parallel algorithms

Consider a counting process {N(t)>0}. In the one-dimensional case it is shown that the set of distributions of N(t) for t>0. This result is a "marginal"version of the characterization of a point process by its stochastic intensity. It is shown that the correspondng results have to be modifed in the multi-dimensional case. These results are particularized to the case of point processes with Poisson marginals.

: In this paper we first summarize the earlier results on the slow motion in the Korteweg theory of cappilarity in the one-dimensional case and show some numerical results. In the multidimensional case we discuss the existence of local solutions to the system of equations for compressible fluids of Korteweg type.

The work investigates the problem of using continuous VaR-criterion (CC-VaR) in multidimensional financial markets, which are markets dimensions generated by several (n > 1) underliers. Fundamental theoretical results deduced for one-dimensional markets are extended on multidimensional case. The peculiarities of multidimensional markets and their instruments are debated. As in one-dimensional case, the optimization algorithm is based on continuous applying Newman-Pearson procedure. This algorithm is derived as a result of projecting the one-dimensional algorithm onto multidimensional markets. Illustrative example of a two-dimensional theoretical market demonstrates in detail using analytical methods of constructing the optimal portfolio. The exposition is accompanied by diagrams of sequential intermediate products important for such constructing.

This paper proposes a new stabilized finite element method to solve singular diffusion problems described by the modified Helmholtz operator. The Galerkin method is known to produce spurious oscillations for low diffusion and various alternatives were proposed to improve the accuracy of the solution. The mostly used methods are the well‐known Galerkin least squares and Galerkin gradient least squares (GGLS). The GGLS method yields the exact nodal solution in the one‐dimensional case and for a uniform mesh. However, the behavior of the method deteriorates slightly in the multi‐dimensional case and for non‐uniform meshes. In this work we propose a new stabilized finite element method that leads to improved accuracy for multi‐dimensional problems. For the one‐dimensional case, the new method leads to the same results as the GGLS method and hence provides exact nodal solutions to the problem on uniform meshes. The proposed method is a Galerkin discretization used to solve a modified equation that includes a term depending on the gradient of the original partial differential equation. Copyright © 2008 John Wiley & Sons, Ltd.

We introduced in (Fortin & Serghini Mounim, 2005) a new method which allows us to extend the connection between the finite volume and dual mixed hybrid (DMH) methods to advection-diffusion problems in the one-dimensional case. In the present work we propose to extend the results of (Fortin & Serghini Mounim, 2005) to multidimensional hyperbolic and parabolic problems. The numerical approximation is achieved using the Raviart-Thomas (Raviart & Thomas, 1977) finite elements of lowest degree on triangular or rectangular partitions. We show the link with numerous finite volume schemes by use of appropriate numerical integrations. This will permit a better understanding of these finite volume schemes and the large number of DMH results available could carry out their analysis in a unified fashion. Furthermore, a stabilized method is proposed. We end with some discussion on possible extensions of our schemes.