On rectangular cartograms

A rectangular cartogram is a type of map where every region is a rectangle. The size of the rectangles is chosen such that their areas represent a geographic variable (e.g., population). Good rectangular cartograms are hard to generate: The area specifications for each rectangle may make it impossible to realize correct adjacencies between the regions and so hamper the intuitive understanding of the map.

We present the first algorithms for rectangular cartogram construction. Our algorithms depend on a precise formalization of region adjacencies and build upon existing VLSI layout algorithms. Furthermore, we characterize a non-trivial class of rectangular subdivisions for which exact cartograms can be computed efficiently. An implementation of our algorithms and various tests show that in practice, visually pleasing rectangular cartograms with small cartographic error can be generated effectively.     

Rectangle exchange transformations

We define rectangle exchange transformations analogously to interval exchange transformations. An interval exchange transformation is a mapping of the unit interval onto itself obtained by cutting the interval up into a finite number of subintervals and rearranging the pieces. A rectangle exchange transformation is a mapping of the unit square onto itself obtained by cutting the square up into a finite number of rectangular pieces and rearranging the pieces. We give a minimality condition for rectangle exchange transformations. We deal with various examples of ergodic rectangle exchange transformations. Related questions are discussed.With 2 Figures     

两连域保角映射成环域的方法

本文论述一种把型区域保角映射成环域的方法.其要点是将问题转化为Dirichlet问题,并证明该映像函数之实部应满足本文所示的边界条件,进而依据两连域上定义的调和函数的单值特性确定环域的内半径.映像函数的虚部可由Cauchy-Riemann条件得到,由此产生的积分常数仅影响映像点的幅角,并可由一一对应的映像来确定.不失其一般性,本方法可将由矩形拼成的复杂两连域保角映射成环域.笔者还对本方法作了电算,证明本方法可靠、经济、结果附有表格.     

单叶调和映射的线性极值问题

利用线性极值问题有解的必要条件，研究了从单位圆盘到自身的单叶调和映射的傅立叶系数，得到其确界估计。     

Accessory coefficients of the second-order Fuchs equation with real singular points

The author investigates the equations of Schwarz and Fuchs related with the conformal mapping of a semiplane onto a circular polygon. One investigates the question of the dependence of the accessory coefficients as functions of the singularities of the equations. As an intermediate result one constructs formulas for the Fourier coefficients of the Siegel-Selberg series and of the absolute Klein invariant. At the conclusion of the paper one considers the example of a rectangle for which the problem of the determination of the accessory coefficients is simplified.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 129, pp. 17–29, 1983.     

Poincaré's recurrence theorem and the unitarity of the S-matrix

A scattering process can be described by suitably closing the system and considering the first return map from the entrance onto itself. This scattering map may be singular and discontinuous, but it will be measured preserving as a consequence of the recurrence theorem applied to any region of a simpler map. In the case of a billiard this is the Birkhoff map. The semiclassical quantization of the Birkhoff map can be subdivided into an entrance and a repeller. The construction of a scattering operator then follows in exact analogy to the classical process. Generically, the approximate unitarity of the semiclassical Birkhoff map is inherited by the S-matrix, even for highly resonant scattering where direct quantization of the scattering map breaks down.     

On a Fourier Method of Embedding Domains Using an Optimal Distributed Control

We propose a domain embedding method to solve second order elliptic problems in arbitrary two-dimensional domains. This method can be easily extended to three-dimensional problems. The method is based on formulating the problem as an optimal distributed control problem inside a rectangle in which the arbitrary domain is embedded. A periodic solution of the equation under consideration is constructed easily by making use of Fourier series. Numerical results obtained for Dirichlet problems are presented. The numerical tests show a high accuracy of the proposed algorithm and the computed solutions are in very good agreement with the exact solutions.     

Domain-imbedding alternating direction method for linear elliptic equations on irregular regions using collocation

A new method is presented for solving elliptic partial differential equations over two-dimensional irregular regions. The scheme imbeds the irregular region in a rectangle, and then uses an alternating direction iteration to solve the resulting system of linear equations. Collocation with cubic Hermite splines is used for discretization. The method is shown to be equivalent to a multiboundary alternating direction method. A theory of convergence for a simplified case is given, details of implementation are discussed, and two numerical illustrations are presented. © 1993 John Wiley & Sons, Inc.     

A fast direct solver for elliptic problems with a divergence constraint

A fast direct solution method for a discretized vector‐valued elliptic partial differential equation with a divergence constraint is considered. Such problems are typical in many disciplines such as fluid dynamics, elasticity and electromagnetics. The method requires the problem to be posed in a rectangle and boundary conditions to be either periodic boundary conditions or the so‐called slip boundary conditions in one co‐ordinate direction. The arising saddle‐point matrix has a separable form when bilinear finite elements are used in the discretization. Based on a result for so‐called p‐circulant matrices, the saddle‐point matrix can be transformed into a block‐diagonal form by fast Fourier transformations. Thus, the fast direct solver has the same structure as methods for scalar‐valued problems which are based on Fourier analysis and, therefore, it has the same computational cost ??(N log N). Numerical experiments demonstrate the good efficiency and accuracy of the proposed method. Copyright © 2002 John Wiley & Sons, Ltd.     

弹性半空间地基上四边自由矩形板稳态振动的解析解

采用双重Fourier变换,分析得到弹性半空间地基受竖向稳态荷载作用下的积分变换解．与四边自由矩形板的振动解析解相结合,得出弹性半空间地基上四边自由矩形板稳态振动的解析解．还给出算例及参数影响分析．     

An approximate analytical solution for the stresses and strains in heterogeneous cubic materials

Continuous Fourier transforms (CFT) are used to derive analytical expressions for the stress and strain fields, respectively, in heterogeneous bodies consisting of cubic materials which additionally may be addressed to a uniform eigenstrain. The so‐called “equivalent inclusion method” (Mura 1987) builds the starting point of this analytical method. It allows to map the original problem onto an auxiliary problem, where as a simplification a homogeneous body is considered. Problems of this kind are effectively solved by means of the CFT. The application of this transformation results into an integral equation (IEQ) for the strains. For (cubic) anisotropic materials this equation can be further simplified by means of approximation techniques which have been demonstrated in [2,3]. For a particular geometry of the inhomogeneity it is illustrated how to derive a closed‐form solution of this approximated IEQ. This solution is compared with numerical results for different combinations of the matrix and the inhomogeneities.     

Sloshing frequencies

The sloshing problem is a linear eigenvalue problem for a partial differential operator that describes the small lateral oscillations of the free surface of an ideal fluid on a container subject to gravity. We consider two-dimensional problems on regions representing the cross-section of a cylindrical tank or canal. A conformal mapping transforms the sloshing problem on the given region to a weighted eigenvalue problem on a simple region such as a rectangle. The weighted problem can be treated very effectively by the powerful methods of intermediate problems. The strength and versatility of the method is illustrated with a variety of examples.     

A spectral solution method for a boundary-value problem for a mixed-type equation with two singularity lines

We consider a boundary-value problem for a mixed-type equation with two perpendicular singularity lines given in a domain whose elliptic part is a rectangle, while the hyperbolic one is a vertical half-strip. This problem differs from the Dirichlet one by the fact that at the left boundary of the rectangle and the half-strip we specify the vanishing order of the desired function rather than its value. We find a solution to the problem by a spectral method with the use of the Fourier–Bessel series and prove the uniqueness of the solution. We substantiate the uniform convergence of the corresponding series under certain requirements to the problem statement.     

Train sound radiation

The problem of sound radiation from a train is considered. This object has been simulated as sets of point sources uniformly distributed in the domain of a moving lengthened rectangle. The solution of the problem is obtained using integral Fourier transforms over the space coordinates and time. The integrals are calculated by using the stationary phase method. Numerical analysis is carried out for acoustic pressure and sound intensity.     

Optimum signal and image recovery by the method of alternating projections in fractional Fourier domains

This paper presents a signal and image recovery scheme by the method of alternating projections onto convex sets in optimum fractional Fourier domains. It is shown that the fractional Fourier domain order with minimum bandwidth is the optimum fractional Fourier domain for the method employing alternating projections in signal recovery problems. Following the estimation of optimum fractional Fourier transform orders, incomplete signal is projected onto different convex sets consecutively to restore the missing part. Using a priori information in optimum fractional Fourier domains, superior results are obtained compared to the conventional Fourier domain restoration. The algorithm is tested on 1-D linear frequency modulated signals, real biological data and 2-D signals presenting chirp-type characteristics. Better results are obtained in the matched fractional Fourier domain, compared to not only the conventional Fourier domain restoration, but also other fractional Fourier domains.     

Numerical conformal mapping methods based on Faber series

Methods are presented for approximating the conformal map from the interior of various regions to the interior of simply-connected target regions with a smooth boundary. The methods for the disk due to Fornberg (1980) and the ellipse due to DeLillo and Elcrat (1993) are reformulated so that they may be extended to other new computational regions. The case of a cross-shaped region is introduced and developed. These methods are used to circumvent the severe ill-conditioning due to the crowding phenomenon suffered by conformal maps from the unit disk to target regions with elongated sections while preserving the fast Fourier methods available on the disk. The methods are based on expanding the mapping function in the Faber series for the regions. All of these methods proceed by approximating the boundary correspondence of the map with a Newton-like iteration. At each Newton step, a system of linear equations is solved using the conjugate gradient method. The matrix-vector multiplication in this inner iteration can be implemented with fast Fourier transforms at a cost of O(N log N). It is shown that the linear systems are discretizations of the identity plus a compact operator and so the conjugate gradient method converges superlinearly. Several computational examples are given along with a discussion of the accuracy of the methods.     

Sobolev smoothing of SVD-based Fourier continuations

A method for calculating Sobolev smoothed Fourier continuations is presented. The method is based on the recently introduced singular value decomposition based Fourier continuation approach. This approach allows for highly accurate Fourier series approximations of non-periodic functions. These super-algebraically convergent approximations can be highly oscillatory in an extended region, contaminating the Fourier coefficients. It is shown that through solving a subsequent least squares problem, a Fourier continuation can be produced which has been dramatically smoothed in that the Fourier coefficients exhibit a prescribed rate of decay as the wave number increases. While the smoothing procedure has no significant negative effect on the accuracy of the Fourier series approximation, in some situations the smoothed continuations can actually yield increased accuracy in the approximation of the function and its derivatives.     

Flow in a waterfall with large Froude number

An approximate method is developed to solve the full nonlinear equations governing two-dimensional irrotational flow in a free waterfall, falling under the influence of gravity, at high Froude number based on conditions far upstream. Schwarz—Christoffel transformation is used to map the region, in the complex potential-plane, onto the upper half-plane. The Hilbert transformation as well as the perturbation technique, for large Froude number, are used as a basis for the approximate solution of the problem. A complete solution, up to second-order approximation, for the downstream free-surfaces profiles, for different Froude number, is discussed and illustrated. The obtained approximate solutions are compared with those of other authors. Favourable agreement with other results suggests that this method is effective in dealing with flow problems strongly influenced by gravity and high Froude number. The results obtained by this method are sufficiently accurate for practical purposes.     

Two-dimensional cargo overbooking models

This paper introduces two-dimensional (weight and volume) overbooking problems arising mainly in the cargo revenue management, and compares them with one-dimensional problems. It considers capacity spoilage and cargo offloading costs, and minimizes their sum. For one-dimensional problems, it shows that the optimal overbooking limit does not change with the magnitude of the booking requests. In two-dimensional problems, the overbooking limit is replaced by a curve. The curve, along with the volume and weight axes, encircles the acceptance region. The booking requests are accepted if they fall within this region. We present Curve (Cab) and Rectangle (Rab) models. The boundary of the acceptance region in the Cab (resp. Rab) model is a curve (resp. rectangle). The optimal curve for the Cab model is shown to be unique and continuous. Moreover, it can be obtained by solving a series of simple equations. Finding the optimal rectangle for the Rab model is more challenging, so we propose an approximate rectangle. The approximate rectangle is a limiting solution in the sense that it converges to the optimal rectangle as the booking requests increase. The approximate rectangle is numerically shown to yield costs that are very close to the optimal costs.