We prove that one can force over a model of AD to obtain countable cofinality of the full Galvin number.

The Ulam sequence, described by Stanisław Ulam in the 1960s, starts 1, 2 and then iteratively adds the smallest integer that can be uniquely written as the sum of two distinct earlier terms: this gives 1,2,3,4,6,8,11,⋯ Already in 1972 the great French poet Raymond Queneau wrote that it “gives an impression of great irregularity”. This irregularity appears to have a lot of structure and has inspired a great deal of work; nonetheless, very little is actually known. We improve the best upper bound on its growth and show that some small gaps have to exist: for some c>0 and all n∈ℕ,min 1≤k≤n a k+1 a k ≤1+clogn n.

We characterize atypical values at infinity of a real polynomial function of three variables by a certain sum of indices of the gradient vector field of the function restricted to a sphere with a sufficiently large radius. This is a three-variable analogue of a result of Coste and de la Puente for real polynomial functions with two variables. We also give a characterization of atypical values at infinity of a real polynomial map whose regular fibers are 2-dimensional surfaces.

Let σ = σ i | i ∈ I be a partition of the set of all primes with σ ≠ { ℙ } . Problem 19.84 of the Kourovka Notebook shows that a σ -analogue of the classical Wielandt criterion of subnormality is not true in general. In this paper, we prove a Wielandt-type σ -subnormality criterion in terms of the Baer–Suzuki width of the class of all σ 0 groups, where σ 0 is the member of σ such that 2 ∈ σ 0 .

This paper is concerned with possibly sign-changing solutions to the critical p-Laplace equation -Δ p u=|x| α |u| p α * -2 u in Ω, u=0 on ∂Ω, where 1<p<N, α>-p, p α * =(N+α)p N-p, and Ω is a bounded or unbounded domain of ℝ N . We first derive a Liouville type theorem on half-spaces. Then we classify solutions via the radial symmetry and Morse index. Moreover, we characterize the compactness of radial Palais–Smale sequences on radial domains.

Motivated by the recent results of Andreis–Iyer–Magnanini [2], we provide a short proof, revisiting the one of Escobedo–Mischler–Perthame [7], that for a large class of coagulation kernels, any weak solution to the Smoluchowski equation loses mass in finite time. The class of kernels we consider is essentially the same as the one of [2]: homogeneous kernels of degree γ>1 not vanishing on the diagonal, or homogeneous kernels of degree γ=1 not vanishing on the diagonal with some additional logarithmic factor. We also show that when γ=1, the power of the logarithmic factor ensuring gelation may depend on the shape of the kernel.

For a reductive group G over a ring A , its affine Grassmannian Gr G plays important roles in a wide range of subjects and is typically defined as the étale sheafification of the presheaf quotient L G / L + G of the loop group L G by its positive loop subgroup L + G . We show that the Zariski sheafification gives the same result. Moreover, for totally isotropic G (for instance, for quasi-split G ), we show that no sheafification is needed at all: Gr G is already the presheaf quotient L G / L + G , which seems new already in the classical case of GL n over ℂ . For totally isotropic G , we also show that the affine Grassmannian may be formed using polynomial loops. We deduce all of these results from the study of G -torsors on ℙ A 1 that is ultimately built on the geometry of Bun G .

The purpose of this paper is to study the vanishing viscosity limit for the d-dimensional Navier–Stokes equations in the whole space:∂ t u ε +u ε ·∇u ε -εΔu ε +∇p ε =0,divu ε =0.We aim to present a simple rigorous example of initial data which generates the corresponding solutions of the Navier–Stokes equations for which the dissipation rate of the kinetic energy is bounded away from zero.

We revisit the classical issue of the controllability/observability cost of linear first order evolution systems, starting with ODEs, before turning to some linear first order evolution PDEs in several space dimensions, including hyperbolic systems and pseudo-differential systems obtained by linearization in fluid mechanics. In particular we investigate the cost for localized initial data and, in the dispersive case, for initial data which are semiclassically microlocalized.

In this paper, we calculate the Hausdorff dimension of some exceptional sets that emerge from specific constraints imposed on the partial quotients of continued fractions. In particular, we calculate the Hausdorff dimension of the sets Λ 1 = x ∈ ( 0 , 1 ) : a n + 1 ( x ) ≥ ∑ i = 1 n a i ( x ) , for all n ∈ ℕ , and Λ 2 = x ∈ ( 0 , 1 ) : a n + 1 ( x ) ≥ ∑ i = 1 n a i ( x ) , for infinitely many n ∈ ℕ . We prove that the Hausdorff dimensions of Λ 1 and Λ 2 are 1 / 2 and 1 respectively. The Hausdorff dimension of some other related sets, obtained by considering different faster growth rates such as replacing the growth rate of sums of partial quotients with the product of partial quotients in the above sets, is also calculated with the dimension bounds 1 / 3 and at least 2 / 3 .