Non-planar Problems Research Articles

Duality Codes and the Integrality Gap Bound for Index Coding

This paper considers a base station that delivers packets to multiple receivers through a sequence of coded transmissions. All receivers overhear the same transmissions. Each receiver may already have some of the packets as side information, and requests another subset of the packets. This problem is known as the index coding problem and can be represented by a bipartite digraph. An integer linear program is developed that provides a lower bound on the minimum number of transmissions required for any coding algorithm. Conversely, its linear programming relaxation is shown to provide an upper bound that is achievable by a simple form of vector linear coding. Thus, the information theoretic optimum is bounded by the integrality gap between the integer program and its linear relaxation. In the special case, when the digraph has a planar structure, the integrality gap is shown to be zero, so that exact optimality is achieved. Finally, for nonplanar problems, an enhanced integer program is constructed that provides a smaller integrality gap. The dual of this problem corresponds to a more sophisticated partial clique coding strategy that time-shares between maximum distance separable codes. This paper illuminates the relationship between index coding, duality, and integrality gaps between integer programs and their linear relaxations.

Proposal of Extended Boundary Integral Equation Method for Rupture Dynamics Interacting With Medium Interfaces

The boundary integral equation method (BIEM) has been applied to the analysis of rupture propagation of nonplanar faults in an unbounded homogeneous elastic medium. Here, we propose an extended BIEM (XBIEM) that is applicable in an inhomogeneous bounded medium consisting of homogeneous sub-regions. In the formulation of the XBIEM, the interfaces of the sub-regions are regarded as extended boundaries upon which boundary integral equations are additionally derived. This has been originally known as a multiregion approach in the analysis of seismic wave propagation in the frequency domain and it is employed here for rupture dynamics interacting with medium interfaces in time domain. All of the boundary integral equations are fully coupled by imposing boundary conditions on the extended boundaries and then numerically solved after spatiotemporal discretization. This paper gives the explicit expressions of discretized stress kernels for anti-plane nonplanar problems and the numerical method for the implementation of the XBIEM, which are validated in two representative planar fault problems.

Areal velocity and angular momentum for non-planar problems in particle mechanics

The concept of areal velocity is intrinsically and historically connected with that of angular momentum. For central force fields the area swept out in the orbital plane by a particle’s radius vector is proportional to the time. For non-planar problems, it is shown that a particle’s position vector sweeps out a general conical surface and that interesting kinematical relations hold. The areal velocity vector is always perpendicular to the conical surface and is proportional to the angular momentum of the particle. In some problems, the magnitude of the areal velocity is constant while its direction changes. In others, only a component of the areal velocity vector is constant. A moving orthonormal basis associated with the areal velocity is defined and a matrix equation for the angular velocities of the basis vectors is found to have an elegant form, involving “sweeping” and “tilting” components. Illustrative examples are provided and some historical background is included.

Method of calculating planar and nonplanar problems of molecular gas flow in curvilinear channel with moving walls

The article presents a theoretical model for calculating the planar and nonplanar problems of the molecular gas flow through channels with moving walls. The useful approximations of impingent fluxes of molecule number, impulse and kinetic energy for reducing the calculating time are given. The principle equations for the mathematical description of gas flow in planar and nonplanar problems are presented. The suggested equations one can solve by iteration techniques.

Approximate solutions of radiative transfer in one-dimensional non-planar systems

An approximate method is developed for the study of radiative transfer in one-dimensional, non-planar systems. While this method can be regarded as an extension of some existing approximation techniques formulated for the one-dimensional planar problem, it does yield closed-form expressions for the radiant heat flux and the temperature profile for various non-planar problems, which have not been established before. Comparisons with the available numerical results show that the heat-flux expressions are accurate throughout the entire range of the optical thickness. Results for the temperature profile. however, have the same limitation as the various closed-form approximate solutions for the planar problem. They are not very accurate at regions near the boundary, except in the optically thick limit. Based on the closed-form expressions obtained for the non-planar radiative transfer problem, the present work establishes readily the effect of the various parameters, such as the optical thickness, the surface emissivity, the radius ratio and the heat-generation rate on the heat-transfer and the temperature profile. Differences between radiative heat-transfer characteristics of the two basic non-planar systems (concentric cylinders and concentric spheres) are discussed.

On Kirchhoff's theory in non-planar scalar diffraction

Kirchhoff's integrals for a slit aperture formed in a non-planar screen are evaluated over the plane aperture and over the extended aperture. These are compared with an interative asymptotic solution of the integral equations, and with experimental measurements for slit widths of 3, 5, 6 and 10 wavelengths. The results indicate that Kirchhoff's approximation for non-planar problems, used over the extended aperture, is poorer than for planar problems. However, the approximation used over the plane aperture gives fairly good agreement with experiment, except for certain screen interaction effects.