数学

The numerical reconstruction of controls for partial differential equations remains comparatively underdeveloped, despite the extensive analytical literature on controllability. This difficulty is particularly pronounced for wave equations,…

We study small--data solutions of a nonlinear scalar field equation on spatially flat $d$--dimensional FLRW spacetimes ($d\ge4$). In conformal time $\tau$ the field satisfies a damped semilinear wave/Klein--Gordon equation with…

The natural extension of the triangle map (a type of multi-dimensional continued fraction algorithm) is completely described in all possible dimensions. The motivation and inspiration for this natural extension stems from the triangle map's…

In this paper, we study a broad class of structured monotone inclusion problems in real Hilbert spaces. We propose a novel primal-dual splitting algorithm for solving such inclusions, which accommodates multiple monotone operators and…

We study wave-type equations on dynamical spacetimes that settle down to a subextremal Kerr black hole spacetime. We prove strong estimates for solutions of (tensorial) linear wave-type equations when the time-translation-invariant model…

Reid--Smith parametrised ($P$)-closed groups acting on trees using graph-based combinatorial structures known as local action diagrams. Properties of the acting (topological) group, such as being locally compact, compactly generated,…

Let \(G<Aut(X)\) be a totally disconnected locally compact group acting strongly transitively on a locally finite building \(X\) of finite-rank and minimal non-spherical type. For sufficiently large thickness, every weakly mixing strongly…

Graphs, maps on surfaces, and abstract polytopes are related combinatorial structures that tend to be studied by different communities using their own tools and databases. Maniplexes provide a unifying framework that captures all of them. A…

Data-driven dynamics often asks how to linearize a nonlinear system. We ask instead: which observables should be advanced, and where should their futures live? This leads to Petrov Regression Of Nonlinear Evolution (PRONE), a…

We consider finite groups with at least three conjugacy class sizes that are composite numbers and we prove that, in that situation, the number of prime class sizes is bounded by the number of composite class sizes. The analogous result for…

Given \(H\leq G\) finite abelian groups, a transversal \(T\subseteq G\) for \(G/H\) has fixed size \(|G/H|\), but its ambient difference support \(D(T)=T-T\) can vary with the embedding of \(H\) in \(G\). We call $ \delta(G,H)=\min_T |D(T)|…

Given a compact zero set of a Fredholm section, our theorem guarantees the existence of a perturbed compact smooth manifold nearby, leaving the original zero set unaltered wherever transversality is already achieved. Such abstract…

We normalize a first-order real planar elliptic system, by pointwise algebra, to a framed Beltrami-Vekua equation $\Phi(w_{\bar z} - \mu w_z) + \Psi(\overline{w_z} - \mu\,\overline{w_{\bar z}}) + \mathfrak{a} w + \mathfrak{b} \bar w =…

For a given Tychonoff space $X$, a point $p\in \beta(X)\setminus X$ is called {\em remote} if $p$ is not in the closure of any nowhere dense subset of $X$. In this paper, we characterize spaces with remote points in terms of certain…

The blowing-up of the projective plane at a finite set of points yields a del Pezzo surface if and only if the points lie in general position. In this note, we generalize this result to Severi--Brauer surfaces over arbitrary ground fields.…

We classify polar homogeneous foliations on rank one symmetric spaces of noncompact type up to orbit equivalence.

We study the coefficients of Ramanujan's third order mock theta function \[ \rho(q)=\sum_{m\geq 0} \frac{q^{2m(m+1)}}{(1+q+q^2)(1+q^3+q^6)\cdots(1+q^{2m+1}+q^{4m+2})} =\sum_{n\geq 0}r(n)q^n. \] Numerical evidence suggests the striking sign…

We identify the zonal and character spherical functions for quantum symmetric pairs with the symmetric Koornwinder--Macdonald polynomials. To this end, the methods of Letzter's 2004 paper are translated to modern conventions and right…

We study optimal estimation of the location parameter of a distribution known only to lie in a symmetric moment class $\mathcal C_0$: the mean-zero distributions with bounded moment $\int\phi\, d\mathbb P\le B$ for a fixed even $\phi$. Our…

Given a geometric statistic expressible as a sum of scores which depend on local data, \citet{BYY19} established central limit theorems for centered and normalized versions of these statistics, subject to the underlying point process having…