2 Class field theory 2 2.1 Number fields and their completions . . . . . . . . . . . . . . . 2 2.1.1 Number fields, prime ideals . . . . . . . . . . . . . . . . 2 2.1.2 Fractional Ideals . . . . . . . . . . . . . . . . . . . . . . 3 2.1.3 Completions . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.4 Adeles and ideles . . . . . . . . . . . . . . . . . . . . . . 4 2.1.5 Cycles . . . . . . ....

The present article is devoted to a fine study of the convergence of renormalized weighted quadratic and cubic variations of a fractional Brownian motion B with Hurst index H . In the quadratic (resp. cubic) case, when H < 1/4 (resp. H < 1/6), we show by means of Malliavin calculus that the convergence holds in L2 toward an explicit limit which only depends on B. This result is somewhat surpris...

The propagation of mutual intensity through quadratic graded-index media or free space can be expressed in terms of two-dimensional fractional Fourier transforms for one-dimensional systems and in terms of fourdimensional fractional Fourier transforms for two-dimensional systems. As light propagates, its mutual intensity distribution is continually fractional Fourier transformed. These results ...

In 1991, one of the authors showed existence of quadratic transformations between Painlevé VI equations with the local monodromy differences (1/2, a, b,±1/2) and (a, a, b, b). In the present paper we give concise forms of these transformation, up to fractional-linear transformations. The transformation is related to better known quadratic transformations (due to Manin and Ramani-Grammaticos-Tam...

  Semidefinite optimization relaxations are among the widely used approaches to find global optimal or approximate solutions for many nonconvex problems. Here, we consider a specific quadratically constrained quadratic problem with an additional linear constraint. We prove that under certain conditions the semidefinite relaxation approach enables us to find a global optimal solution of the unde...

Consider an analytic map from a neighborhood of 0 in a vector space to a Euclidean space. Suppose that this map takes all germs of vector lines to germs of circles. Such map is called rounding. Two roundings are equivalent if they take the same lines to the same circles. We prove that any rounding whose differential at 0 has rank at least 2 is equivalent to a fractional quadratic rounding. The ...