Let M be a finite or countable set of models (here we only deal with finite M but Bayes factors make sense for countable M). For each model m ∈ M we have the prior probability of the model pri(m). It does not matter if this prior on models is unnormalized. Each model m has a parameter space Θm and a prior g(θ | m), θ ∈ Θm The spaces Θm can and usually do have different dimensions. That’s the po...

In the quadratic family (the set of polynomials of degree 2), Pe-tersen and Zakeri [PZ] proved the existence of Siegel disks whose boundaries are Jordan curves, but not quasicircles. In their examples, the critical point is contained in the curve. In the first part, as an illustration of the flexibility of the tools developped in [BC], we prove the existence of examples that do not contain the ...

Let m be a positive integer and let ? finite set. The m-closure of G?Sym(?) is the largest permutation group on having same orbits as G in its induced action Cartesian product ?m. 1-closure 2-closure solvable need not solvable. We prove that always for m?3.

The Kodaira–Thurston manifold M is a compact, 4-dimensional nilmanifold which is symplectic and complex but not Kähler. We describe a construction of θ-functions associated to M which parallels the classical theory of θ-functions associated to the torus (from the point of view of representation theory and geometry), and yields pseudoperiodic complex-valued functions on R. There exists a three-s...

In this note we prove an explicit binomial formula for Jack polynomials and discuss some applications of it. 1. Jack polynomials ([M,St]). In this note we use the parameter θ = 1/α inverse to the standard parameter α for Jack polynomials. Jack symmetric polynomials Pλ(x1, . . . , xn; θ) are eigenfunctions of Sekiguchi differential operators D(u; θ) = V (x) det [ x i ( xi ∂ ∂xi + (n− j)θ + u )]

A free boundary problem arising from the optimal reinforcement of a membrane or from the reduction of traffic congestion is considered; it is of the form sup ∫ D θ dx=m inf u∈H1 0 (D) ∫ D (1 + θ 2 |∇u| − fu ) dx. We prove the existence of an optimal reinforcement θ and that it has some higher integrability properties. We also provide some numerical computations for θ and u.

where θ = 1 + √ d 2 if d mod 4 = 1 and θ = √ d if d mod 4 = 2, 3. The purpose of this paper is to compute the extended gcd of to quadratic integers in ring Z[θ]. We assume throughout that Z[θ] is principal ideal ring, but not necessarily an euclidean ring. If [a, b+ cθ] is the module {ax+(b+ cθ)y, x, y ∈ Z}, it can be shown [3] that I is an ideal of Z[θ] if and only if I = [a, b+ cθ]; where a, ...

We present improved upper and lower bounds on the spanning ratio of θgraphs with at least six cones. Given a set of points in the plane, a θ-graph partitions the plane around each vertex into m disjoint cones, each having aperture θ = 2π/m, and adds an edge to the ‘closest’ vertex in each cone. We show that for any integer k ≥ 1, θ-graphs with 4k+ 2 cones have a spanning ratio of 1 + 2 sin(θ/2)...

1 Problem A typical high-energy neutrino beam is made from the decay of π mesons that have been produced in proton interactions on a target, as sketched in the figure below. Suppose that only positively charged particles are collected by the " horn ". The main source of neutrinos is then the decay π + → µ + ν µ. 1. Give a simple estimate of the relative number of other types of neutrinos than ν...

Let φ be a 3CNF formula with n variables and m clauses. A simple nonconstructive argument shows that when m is sufficiently large compared to n, most 3CNF formulas are not satisfiable. It is an open question whether there is an efficient refutation algorithm that for most such formulas proves that they are not satisfiable. A possible approach to refute a formula φ is: first, translate it into a...