Analysis of linear trade models and relation to scale economies

We discuss linear Ricardo models with a range of parameters. We show that the exact boundary of the region of equilibria of these models is obtained by solving a simple integer programming problem. We show that there is also an exact correspondence between many of the equilibria resulting from families of linear models and the multiple equilibria of economies of scale models.

Hyers–Ulam stability of functional equations with a square-symmetric operation

The stability of the functional equation f(x ○ y) = H(f(x), f(y)) (x, y ∈ S) is investigated, where H is a homogeneous function and ○ is a square-symmetric operation on the set S. The results presented include and generalize the classical theorem of Hyers obtained in 1941 on the stability of the Cauchy functional equation.

Coexistence of linear zones and pinwheels within orientation maps in cat visual cortex

Revealing the layout of cortical maps is important both for understanding the processes involved in their development and for uncovering the mechanisms underlying neural computation. The typical organization of orientation maps in the cat visual cortex is radial; complete orientation cycles are mapped around orientation singularities. In contrast, long linear zones of orientation representation have been detected in the primary visual cortex of the tree shrew. In this study, we searched for the existence of long linear sequences and wide linear zones within orientation preference maps of the cat visual cortex. Optical imaging based on intrinsic signals was used. Long linear sequences and wide linear zones of preferred orientation were occasionally detected along the border between areas 17 and 18, as well as within area 18. Adjacent zones of distinct radial and linear organizations were observed across area 18 of a single hemisphere. However, radial and linear organizations were not necessarily segregated; long (7.5 mm) linear sequences of preferred orientation were found embedded within a typical pinwheel-like organization of orientation. We conclude that, although the radial organization is dominant, perfectly linear organization may develop and perform the processing related to orientation in the cat visual cortex.

A modified gambler's ruin model of polyethylene chains in the amorphous region.

Polyethylene chains in the amorphous region between two crystalline lamellae M unit apart are modeled as random walks with one-step memory on a cubic lattice between two absorbing boundaries. These walks avoid the two preceding steps, though they are not true self-avoiding walks. Systems of difference equations are introduced to calculate the statistics of the restricted random walks. They yield that the fraction of loops is (2M - 2)/(2M + 1), the fraction of ties 3/(2M + 1), the average length of loops 2M - 0.5, the average length of ties 2/3M2 + 2/3M - 4/3, the average length of walks equals 3M - 3, the variance of the loop length 16/15M3 + O(M2), the variance of the tie length 28/45M4 + O(M3), and the variance of the walk length 2M3 + O(M2).