By Marco Brunella

The textual content offers the birational category of holomorphic foliations of surfaces.  It discusses at size the idea built through L.G. Mendes, M. McQuillan and the writer to check foliations of surfaces  within the spirit of the category of complicated algebraic surfaces.

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Extra info for Birational geometry of foliations

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5) This single operator, when combined with the some of the products defined above, constitutes the foundation of vector calculus. 6) respectively. If we consider A to be a continuous vector function of the independent variables that make up the space in which it is defined, then we may give a physical interpretation for both the divergence and curl. 3). 3 schematically shows the divergence of a vector field. In the region where the arrows of the vector field converge, the divergence is positive, implying an increase in the source of the vector field.

2), only the sign of the determinant is changed. Since the equations at each step represent a system of equations, which have the same solution as the original set, we may interchange rows and columns at any step in the procedure without altering the solution. Thus, most Gauss-Jordan programs include a search of the matrix to place the largest element on the diagonal prior to division by that element so as to minimize the effects of round off error. Should it be impossible to remove a zero from the division part of this algorithm, the one column of the matrix can be made to be completely zero.

However, before we begin any discussion of numerical methods, we must say something about the accuracy to which those calculations can be made. 1 Errors and Their Propagation One of the most reliable aspects of numerical analysis programs for the electronic digital computer is that they almost always produce numbers. As a result of the considerable reliability of the machines, it is common to regard the results of their calculations with a certain air of infallibility. However, the results can be no better than the method of analysis and implementation program utilized by the computer and these are the works of highly fallible man.