An Urban Economic Model over a Continuous Plane with Spatial Characteristic Vector Field - Consideration of Heterogeneous Geographical Conditions -

Among others Beckmann (1952) firstly introduced the concept of a two dimensional continuous space into economics. This great step had unfortunately not shown further expansion in economics. Through several papers related to Beckmann's initiation, Beckmann and Puu (1985) at last reached a systematic treatment of the continuous spatial economics. Although their achievement is fascinated by employing a partial differential equations approach, Beckmann's original philosophy, that is, the gradient law has still been inherited. Following their achievement, Puu (2003) alone developed their theory by using many computer simulations to visually show the significance of their theory. Beckmann and Puu's book (1985) aims to study formation of urban configuration in a two dimensional continuous space, focusing on flows of commodities. However, consideration of households and firms location is not necessarily sufficient, resulting in reconsideration from a new urban economics point of view. As another exceptional urban economic study of a plane city, Lucus and Rossi-Hansberg (2002) is pointed out. They were inspired by Fujita and Ogawa (1982), and indicate endogenous land use pattern over a plane city. However they neglect commodity market to simplify the analysis. Well discussion about formation of urban configuration is summarized in Anas, Arnott, and Small (1998). Differing from Beckmann and Puu's studies, Miyata (2010) introduces bid rent functions (Fujita (1989)), which are familiar in the new urban economics, for land of households and firms, and then it studies how the results of Beckmann and Puu are rigorously modified by using the theory of partial differential equations (Courant and Hilbert (1953, 1962). However Miyata (2010) deals with a symmetric equilibrium which seems to be a little unrealistic. This article extends the author's previous study introducing spatial characteristic vector field in the model which stands for heterogeneity in geographical conditions in a city, and try to show asymmetry in land use pattern and endogenous formation of transport networks in a two dimensional continuous space.

Enumeration of PLCP-orientations of the 4-cube

The linear complementarity problem (LCP) provides a unified approach to many problems such as linear programs, convex quadratic programs, and bimatrix games. The general LCP is known to be NP-hard, but there are some promising results that suggest the possibility that the LCP with a P-matrix (PLCP) may be polynomial-time solvable. However, no polynomial-time algorithm for the PLCP has been found yet and the computational complexity of the PLCP remains open. Simple principal pivoting (SPP) algorithms, also known as Bard-type algorithms, are candidates for polynomial-time algorithms for the PLCP. In 1978, Stickney and Watson interpreted SPP algorithms as a family of algorithms that seek the sink of unique-sink orientations of $n$-cubes. They performed the enumeration of the arising orientations of the $3$-cube, hereafter called PLCP-orientations. In this paper, we present the enumeration of PLCP-orientations of the $4$-cube.The enumeration is done via construction of oriented matroids generalizing P-matrices and realizability classification of oriented matroids.Some insights obtained in the computational experiments are presented as well

The Plasma Structure of the Cygnus Loop from the Northeastern Rim to the Southwestern Rim

The Cygnus Loop was observed from the northeast to the southwest with XMM-Newton. We divided the observed region into two parts, the north path and the south path, and studied the X-ray spectra along two paths. The spectra can be well fitted either by a one-component non-equilibrium ionization (NEI) model or by a two-component NEI model. The rim regions can be well fitted by a one-component model with relatively low \kTe whose metal abundances are sub-solar (0.1--0.2). The major part of the paths requires a two-component model. Due to projection effects, we concluded that the low kTe (about 0.2 keV) component surrounds the high kTe (about 0.6 keV) component, with the latter having relatively high metal abundances (about 5 times solar). Since the Cygnus Loop is thought to originate in a cavity explosion, the low-kTe component originates from the cavity wall while the high-kTe component originates from the ejecta. The flux of the cavity wall component shows a large variation along our path. We found it to be very thin in the south-west region, suggesting a blowout along our line of sight. The metal distribution inside the ejecta shows non-uniformity, depending on the element. O, Ne and Mg are relatively more abundant in the outer region while Si, S and Fe are concentrated in the inner region, with all metals showing strong asymmetry. This observational evidence implies an asymmetric explosion of the progenitor star. The abundance of the ejecta also indicates the progenitor star to be about 15 M_sun.Comment: 24 pages, 9 figures, Astrophysical Journal in pres