In 1958 this was proved independently by S. Swierczkowski [15] and by P. Erdos and V.T. Sos [12],[13]. It is often called the Steinhaus theorem or Steinhaus Conjecture. A useful way of viewing the result is to think of a circle of unit circumference with points placed around the perimeter at distances 0, 0:, ... , (N 1)0: from an arbitrary origin on the perimeter. Then the distances between adj...

The planar soap bubble problem is the mathematically analogous problem in two dimensions to search for the least-perimeter way to enclose and separate regions R1, . . . , Rm of given areas A1, . . . , Am on the plane. In this work, we study the possible configurations for perimeter minimizing enclosures for more than three regions. In particular, we focus on the case of equal pressure regions. ...

In this paper we present a newly developed tool that enables researchers interested in spatial variation of language to define a geographic perimeter of interest, collect data from the Twitter streaming API published in that perimeter, filter the obtained data by language and country, define and extract variables of interest and analyse the extracted variables by one spatial statistic and two s...

During lung expansion, the pattern of alveolar perimeter distension is likely to be an important determinant of lung functions as, for example, surfactant secretion. However, the segmental characteristics of alveolar perimeter distension remain unknown. Here, we applied real-time confocal microscopy in the isolated, perfused rat lung to determine the micromechanics of alveolar perimeter distens...

Using 100 years of fire perimeter maps, we investigate the existence of geographical patterns in fire orientation across California. We computed fire perimeter orientation, at the watershed level, using principal component analysis. Circular statistics were used to test for the existence of preferential fire perimeter orientations. Where perimeters displayed preferential orientation, we searche...

In this paper, we macroscopically describe the traffic dynamics in heterogeneous transportation urban networks by utilizing the Macroscopic Fundamental Diagram (MFD), a widely observed relation between network-wide space-mean flow and density of vehicles. A generic mathematical model for multi-reservoir networks with well-defined MFDs for each reservoir is presented first. Then, two modeling va...

A new method for automated detection of polar coronal holes is presented. This method, called perimeter tracing, uses a series of 171, 195, and 304 Å full disk images from the Extreme ultraviolet Imaging Telescope (EIT) on SOHO over solar cycle 23 to measure the perimeter of polar coronal holes as they appear on the limbs. Perimeter tracing minimizes line-of-sight obscurations caused by the emi...

A new method for automated detection of polar coronal holes is presented. This method, called perimeter tracing, uses a series of 171, 195, and 304 Å full disk images from the Extreme ultraviolet Imaging Telescope (EIT) on SOHO over solar cycle 23 to measure the perimeter of polar coronal holes as they appear on the limbs. Perimeter tracing minimizes line-of-sight obscurations caused by the emi...

In this paper we give a method to geometrically modify an open set such that the first k eigenvalues of the Dirichlet Laplacian and its perimeter are not increasing, its measure remains constant, and both perimeter and diameter decrease below a certain threshold. The key point of the analysis relies on the properties of the shape subsolutions for the torsion energy.