In this paper, to solve a linear one-dimensional Volterra  integral equation of the second kind. For this purpose using the equation form, we have defined a linear transformation and by using it's conjugate and reproducing kernel functions, we obtain a basis for the functions space.Then we obtain the solution of  integral equation in terms of the basis functions. The examples presented in this ...

This is an extended account of a short presentation with this title given at the Minneapolis IMA Workshop on ‘n-categories: foundations and applications’, June 7-18, 2004, organised by John Baez and Peter May.

The theory we describe in this book was developed over a long period, starting about 1965, and always with the aim of developing groupoid methods in homotopy theory of dimension greater than 1. Algebraic work made substantial progress in the early 1970s, in work with Chris Spencer, and in 1974 by Brown and Higgins. A substantial step forward in 1974 led us over the years into many fruitful area...

In this paper the characteristics of unsteady three-dimensional incompressible flows with heat transfer are obtained along with artificial compressibility of Chorin. At first, compatibility equations and pseudo characteristics for three-dimensional flows are derived from five governing equations (continuity equation, Momentum equations in three directions, and energy equation) and then results ...

Just as knots and links can be algebraically described as certain morphisms in the category of tangles in 3 dimensions, compact surfaces smoothly embedded in R4 can be described as certain 2-morphisms in the 2-category of ‘2-tangles in 4 dimensions’. Using the work of Carter, Rieger and Saito, we prove that this 2-category is the ‘free semistrict braided monoidal 2-category with duals on one un...

A 2-Hilbert space is a category with structures and properties analogous to those of a Hilbert space. More precisely, we deene a 2-Hilbert space to be an abelian category enriched over Hilb with a-structure, conjugate-linear on the hom-sets, satisfying hf g; hi = hg; f hi = hf; hg i. We also deene monoidal, braided monoidal, and symmetric monoidal versions of 2-Hilbert spaces, which we call 2-H...