Mathematics

  2010 AUG 31 - (VerticalNews.com) -- According to a study from Caen, France, "With the help of continued fractions, we plan to list all the elements of the set Q(Delta) = {aX(2) + bXY + cY(2) : a, b, c is an element of Z, b(2) - 4ac = Delta with 0<= b

  "As a matter of fact, we show that for each reduced quadratic form f = aX(2) + bXY + cY(2) =< a, b, c> of discriminant Delta > 0 (and of sign sigma(f) equal to the sign of a), the quadratic forms associated with f and defined by < a + bu + cu(2), b + 2cu, c>, with 1<= + sigma(f)u <= v/2 vertical bar c vertical bar (whenever they exist), < c, -b -2cu, a + bu, cu(2) >, with with b 2/c/ <= sigma(f) u<= [omega(f)] = [b + root Delta/2 vertical bar c vertical bar], are all different from one another and build a set I(f) whose cardinality is # I(f) = {1 + [omega(f)], when (2c)/b, [omega(f)], when (2c) inverted iota b. If f and g are two different reduced quadratic forms, we show that I(f) boolean AND I(g) = (sic)," wrote E. Dubois and colleagues ...read more

  2010 AUG 31 - (VerticalNews.com) -- According to recent research from the United States, "A numerical method is developed for computing the shape of an infinite three-dimensional hydrostatic meniscus originating from interior contact lines whose projection in the horizontal plane has a specified shape. The Laplace-Young equation determining the meniscus shape is solved in orthogonal curvilinear coordinates generated by conformal mapping using a finite-difference method. ...read more

  2010 AUG 31 - (VerticalNews.com) -- "Codimension one webs are configurations of finitely many codimension one foliations in general position. Much of the classical theory evolved around the concept of abelian relation: a functional equation among the first integrals of the foliations defining the web reminiscent of Abel's Addition Theorem," scientists writing in the journal International Mathematics Research Notices report.

  "The abelian relations of a given web form a finite-dimensional vector space with dimension ( the rank of the web) bounded by Castelnuovo number pi(n, k) where n is the dimension of the ambient space and k is the number of foliations defining the web. A fundamental problem in web geometry is the classification of exceptional webs, that is, webs of maximal rank not equivalent to the dual of a projective curve. Recently, Trepreau proved that there are no exceptional k-webs for n>= 3 and k>= 2n. In dimension two, there are examples for arbitrary k>= 5 and the classification of exceptional webs is wide open. In this paper, we classify the exceptional completely decomposable quasi-linear (CDQL) webs globally defined on compact complex surfaces. By definition, these are the exceptional (k + 1)-webs on compact complex surfaces that are formed by the superposition of k ''linear'' and one non-linear foliations," wrote J.V. Pereira and colleagues ...read more