Few natural basic principles allow to extend Feynman integral over the paths to an integral over the surfaces so, that they coincide at long time scale, that is when the surface degenerates into a single particle world line. In the first approximation the loop Green functions have perimeter behavior. That corresponds to the case when free quarks interact through one gluon exchange. Quantum fluctuations of the surface generate the area low. Thus in this string theory asymptotic freedom and confinement can coexist.
We construct BPS-exact solutions of the worldvolume Born-Infeld plus WZW action of a D5-brane in the background of N D3-branes. The non-trivial background metric and RR five-form field strength play a crucial role in the solution. When a D5-brane is dragged across a stack of N D3-branes a bundle of N fundamental strings joining the two types of branes is created, as in the Hanany-Witten effect. Our solutions give a detailed description of this bundle in terms of a D5-brane wrapped on a sphere. We discuss extensions of these solutions which have an interpretation in terms of gauge theory multi-quark states via the AdS/CFT correspondence.
In this note, we give a practical solution to the problem of determining the
maximal period of matrix generators of pseudo-random numbers which are based on
an integer-valued unimodular matrix of size NxN known as MIXMAX and arithmetic
defined on a Galois field GF[p] with large prime modulus p. The existing theory
of Galois finite fields is adapted to the present case, and necessary and
sufficient condition to attain the maximum period is formulated. Three
efficient algorithms are presented. First, allowing to compute the
multiplication by the MIXMAX matrix with O(N) operations. Second, to
recursively compute the characteristic polynomial with O(N^2) operations, and
third, to apply skips of large number of steps S to the sequence in O(N^2
log(S)) operations. It is demonstrated that the dynamical properties of this
generator dramatically improve with the size of the matrix N, as compared to
the classes of generators based on sparse matrices and/or sparse characteristic
polynomials. Finally, we present the implementation details of the generator
and the results of rigorous statistical testing.Comment: 15 pages, 3 Figure
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