Locally constant function

Abstract: The formulation of a new analysis on a zero measure Cantor set C(⊂I = [0,1]) is presented. A non-Archimedean absolute value is introduced in C exploiting the concept of relative infinitesimals and a scale invariant ultrametric valuation of the form loge-1 (e/x) for a given scale e > 0 and infinitesimals 0 < x < e, x ∈ I\C. Using this new absolute value, a valued (metric) measure is defined on C and is shown to be equal to the finite Hausdorff measure of the set, if it exists. The formulation of a scale invariant real analysis is also outlined, when the singleton {0} of the real line R is replaced by a zero measure Cantor set. The Cantor function is realized as a locally constant function in this setting. The ordinary derivative dx/dt in R is replaced by the scale invariant logarithmic derivative d log x/d log t on the set of valued infinitesimals. As a result, the ordinary real valued functions are expected to enjoy some novel asymptotic properties, which might have important applications in number theory and in other areas of mathematics.

Abstract: Introduction. The Selberg trace formula has become a major tool in the study of automorphic forms on reductive groups. Although its underlying principle, of computing traces of representations by means of orbital integrals, is very simple, the standard expressions for this important formula are rather complicated; this makes applications hard to accomplish. The complexity of the expression for the formula may be due to the choice of truncation made in its proof. It would be advantageous to have a simple expression for the formula, at least for a set of test functions which is sufficiently large for applications. The possibility of its existence was suggested to us by some of Kazhdan's striking work on the trace formula (see, e.g., the density theorem of [Kl; Appendix], or the study of lifting in [K2]). Here we derive an asymptotic expression of this nature, in the simplest case of GL(2). For test functions with a component which is sufficiently regular with respect to all other components we obtain a simple, practical form of the trace formula. More precisely, if FU is a nonarchimedean local field and m 2 1 is an integer, we say that a locally constant function f, on Gu = GL(2, FU), which is supported on a compact-mod-center, is m-regular if it vanishes outside the open closed subset Sm = {zg-( ?)g; g in Gu; a, z in F,x with I a = 1 } (r denotes a uniformizer in F, x) of Gu, and its normalized orbital integral F(g ,fu) = A(g)4(g ,fu) is the characteristic function of Sm in G14. If F is a global field, u is a nonarchimedean place of F, and fu = ?," f, is a product over all places v ? u of F of smooth compactly supported mod-center functionsfv on Gv, such thatfv is the unit elementf v? of the Hecke algebra for almost all v, then we show that there exists mr0 = mo(fu) such that for any m 2r mo and m-regular functionfu = f (m), the "regular" test functionf = f (m) ? fu vanishes on the conjugacy classes in

Abstract: We consider low-energy configurations for the Heitmann–Radin sticky discs functional, in the limit of diverging number of discs. More precisely, we renormalize the Heitmann–Radin potential by subtracting the minimal energy per particle, i.e. the so-called kissing number. For configurations whose energy scales like the perimeter, we prove a compactness result which shows the emergence of polycrystalline structures: The empirical measure converges to a set of finite perimeter, while a microscopic variable, representing the orientation of the underlying lattice, converges to a locally constant function. Whenever the limit configuration is a single crystal, i.e. it has constant orientation, we show that the $$\varGamma $$ -limit is the anisotropic perimeter, corresponding to the Finsler metric determined by the orientation of the single crystal.

Abstract: Irreducible crystalline representations of dimension 2 of Gal(Qpbar/Qp) depend up to twist on two parameters, the weight k and the trace of frobenius a_p. We show that the reduction modulo p of such a representation is a locally constant function of a_p (with an explicit radius) and a locally constant function of the weight k if a_p 0. We then give an algorithm for computing the reductions modulo p of these representations. The main ingredient is Fontaine's theory of (phi,Gamma)-modules as well as the theory of Wach modules.