For a quadratic extension K of Q, we consider orders O in K that are not necessarily maximal and the ideal class group Cl+(O) in the narrow sense of proper ideals of O. Characters of Cl+(O) of order at most two are traditionally called genus characters. Explicit description of such characters is known classically, but explicit L-functions associated to those characters are only recently obtained partially by Chinta and Offen and completely by Kaneko and Mizuno. As remarked in the latter paper, the present author also obtained the formula of such L-functions independently. Indeed, here we will give a simple and transparent alternative proof of the formula by rewriting explicit genus characters and their values in an adelic way starting from scratch. We also add an explicit formula for the genus number in the wide sense, which is maybe known but rarely treated. As an appendix we give an ideal-theoretic characterization of isomorphism classes of maximal orders of the matrix algebras M-n(F) over a number field F up to GL(n)(F) and GL(n)(+)( F) conjugation respectively, and apply genus numbers to count them when n = 2 and F is quadratic. Relations between classes and genera of ideals and quadratic forms are explained in the appendix. To avoid any misconception, we include some easy known details.
We study whether circles can be seen as geodesics on geodesic spheres or not. If a circle on a complex projective space has complex torsion either +/- 1 or 0, it can be seen as a geodesic on some geodesic sphere. But if its complex torsion tau satisfies 0 < |tau| < 1, then it cannot be seen as geodesics on any geodesic spheres, and can be seen as trajectories for some Sasakian magnetic fields on geodesic spheres. We show that there are three kinds of such expressions up to congruency.
Given an ordered pair of Euclidean triangles with marked vertices whose angles are all acute, we find a homeomorphism with the smallest Lipschitz constant among all homeomorphisms from the first one to the second that preserve the marking and we give a formula for the Lipschitz constant of this map. We show that on the set of pairs of marked acute triangles with fixed area, the function which assigns the logarithm of the infimum of the Lipschitz constants of Lipschitz homeomorphisms between them induces a symmetric metric. We show that this metric is Finsler, we give a necessary and sufficient condition for a path in the resulting metric space to be geodesic and we determine the isometry group of this metric space. This study is motivated by Thurston's asymmetric metric on the Teichmuller space of a hyperbolic surface, and the results in this paper constitute an analysis of a basic Euclidean analogue of Thurston's hyperbolic theory. Many interesting questions in the Euclidean setting deserve further attention.
The paper investigates the asymptotic of almost surely extreme values of discrete random variables. The answers are given to some questions that remained open. Applications to the geometric distribution and the Poisson distribution are presented. The example of the birth and death process is also studied.
Let G be a finite p-solvable group. For a given p-block B of G, we define a canonical pair (K, A), refered to as a nucleus for B, where K is a subgroup of G and A is a block of K of maximal defect, defined uniquely by B up to G-conjugacy. The irreducible characters (ordinary or modular) associated with B are closely related to those associated with A. Also, not surprisingly, (K, A) is just (G, B) in case B is of maximal defect. Given a normal subgroup N of G and a block b of N, we show that there exist a nucleus (N(sic), b(sic)) for b and a subgroup G(sic) of G containing N(sic) as a normal subgroup such that the blocks of G covering b behave quite analogously to those of G(sic)covering b(sic).
