librium and of natural systems (whether physical or economic) and collated with the very general viewpoint of Roy ce and of C. S. Peirce (whose maturer work Keynes does not cite), might be worthy of at least a bibliographic reference by an author who is setting up a category of probability. However, it would be unreasonable to expect any discussion of categories to reach nearer the date of issu...

2 Derivative Estimates for Rectangular Lattice Topology 4 2.1 Rectangle Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Cylindrical Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Toroidal Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.4 Spherical T...

We discuss certain aspects of the combinatorial approach to the differential geometry of non-abelian gerbes due to W. Messing and the author [5], and give a more direct derivation of the associated cocycle equations. This leads us to a more restrictive definition than in [5] of the corresponding coboundary relations. We also show that the diagrammatic proofs of certain local curving and curvatu...

Groupoids provide a more appropriate framework for differential geometry than principal bundles. Synthetic differential geometry is the avantgarde branch of differential geometry, in which nilpotent infinitesimals are available in abundance. The principal objective in this paper is to show within our favorite framework of synthetic differential geometry that the tangent space of the group of bi...

This paper presents the numerical elastic solution for a real problem, functionally graded chamber of hydraulic gear pumps under internal pressure. Because of the similarity and complexity for the considering geometry, a bipolar cylindrical coordinate system is used to extract the governing equations. The material properties are considered to vary along the two tangled circles with a power-law ...