Archiv der Mathematik最新文献

Let (W, R) be a Coxeter system and let (w in W). We say that u is a prefix of w if there is a reduced expression for u that can be extended to one for w. That is, (w = uv) for some v in W such that (ell (w) = ell (u) + ell (v)). We say that w has the ancestor property if the set of prefixes of w contains a unique involution of maximal length. In this paper, we show that all Coxeter elements of finitely generated Coxeter groups have the ancestor property, and hence a canonical expression as a product of involutions. We conjecture that the property in fact holds for all non-identity elements of finite Coxeter groups.

We show that the (multi)function of the closest points is locally Lipschitz outside of the medial axis of a closed set (Xsubset mathbb {R}^n). With this result, we prove that the medial axis of X approaches every point where X is not Lipschitz normally embedded.

A classical theorem of Weyl states that any polynomial with an irrational coefficient other than the constant term is uniformly distributed mod 1. We prove a new function field analogue of this statement, confirming a conjecture of Lê, Liu, and Wooley.

We show the existence of a set (Asubseteq mathbb {Z}_{ge 2}) satisfying the estimates of the Bateman–Horn conjecture, Goldbach’s conjecture, and also

In this paper, we investigate the problem of which Lie algebras appear as the derived algebra of a Lie algebra. We present new results that further develop this study and address two questions raised in a paper concerned with the corresponding problem for groups.

We study n-isometric elementary operators of length one, highlighting the special case (n=1), which is fundamental due to the importance and practical relevance of classical isometries. In this case, we provide two proofs: one based on norm arguments and the other using an identification with tensor products and standard factorization properties. For arbitrary n, we furnish a direct proof of a result by Caixing Gu on n-isometric elementary operators of length one, relying solely on an antiunitary cosimilarity and a single lemma, and thereby eschewing the auxiliary arguments of the original work.

In this note, we show that the Barutello–Ortega–Verzini regularization map is scale smooth.

We use methods of direct optimization as in El Allali and Harrell (Proc Am Math Soc 150:57–587, 2022) to characterize the minimizers of the fundamental gap of Sturm–Liouville operators on an interval, under the constraint that the potential is of single-well form and that the weight function is of single-barrier form, and under similar constraints expressed in terms of convexity.

It has been shown that the set of universal functions on trees contains a linear subspace except zero, dense in the space of forward-only harmonic functions. In this paper, we show that the set of universal functions contains two linear subspaces except zero, dense in the space of forward-only harmonic functions that intersect only at zero. We work in the most general case that has been studied so far, letting our functions take values over a topological vector space.

It is conjectured that, for any group G, the Jacobson radical (J({mathbb {Z}}[G])) of the integral group ring ({mathbb {Z}}[G]) is zero. This is known to be true when G is finite. Here we show it is true for a reasonably large class of infinite groups, including finitely generated linear groups and groups which satisfy Higman’s ‘two unique products’ condition.