Abstract We prove that, when n goes to infinity, the expression, with respect to the dual Kazhdan-Lusztig basis, of the product $$\hat{\underline{H}}_x\underline{H}_y$$ H ̲ ^ x H ̲ y of elements of the dual and the usual Kazhdan-Lusztig bases in the Hecke algebra of the symmetric group $$S_n$$ S n stabilizes. As an application, we define the action of projective functors on the principal block of category $$\mathcal {O}$$ O for $$\mathfrak {sl}_\infty $$ sl ∞ and show that the subcategory of finite length objects is stable under this action. As a bonus, we also prove that this latter block is Koszul, answering, for this block, a question from Math. J. 19 (4), 655–693 (2019).

Abstract Let $$N\ge 3$$ N ≥ 3 . We are concerned with a Cauchy problem of the semilinear heat equation $$\begin{aligned} \left\{ \begin{array}{ll} \partial _tu-\Delta u=f(u), & x\in \mathbb {R}^N,\ t>0,\\ u(x,0)=u_0(x), & x\in \mathbb {R}^N, \end{array} \right. \end{aligned}$$ ∂ t u - Δ u = f ( u ) , x ∈ R N , t > 0 , u ( x , 0 ) = u 0 ( x ) , x ∈ R N , where $$f(0)=0$$ f ( 0 ) = 0 , f is nonnegative, increasing and convex, $$\log f(u)$$ log f ( u ) is convex for large $$u>0$$ u > 0 and some additional assumptions are assumed. We establish a positive radial singular stationary solution $$u^*$$ u ∗ such that $$u^*(x)\rightarrow \infty $$ u ∗ ( x ) → ∞ as $$|x|\rightarrow 0$$ | x | → 0 . Then, we prove the following: The problem has a nonnegative global-in-time solution if $$0\le u_0\le u^*$$ 0 ≤ u 0 ≤ u ∗ and $$u_0\not \equiv u^*$$ u 0 ≢ u ∗ , while the problem has no nonnegative local-in-time solutions u such that $$u\ge u^*$$ u ≥ u ∗ if $$u_0\ge u^*$$ u 0 ≥ u ∗ and $$u_0\not \equiv u^*$$ u 0 ≢ u ∗ .

An important problem in computational arithmetic geometry is to find changes of coordinates to simplify a system of polynomial equations with rational coefficients. This is tackled by a combination of two techniques, called minimisation and reduction. We give an algorithm for minimising certain pairs of quadratic forms, subject to the constraint that the first quadratic form is fixed. This has applications to 2-descent on the Jacobian of a genus 2 curve.