In this article we show that the quotient M∞/B(Qp) of the Lubin–Tate space at infinite level M∞ by the Borel subgroup of upper triangular matrices B(Qp) ⊂ GL2(Qp) exists as a perfectoid space. As an application we show that Scholze’s functor H i ét(PCp ,Fπ ) is concentrated in degree one whenever π is an irreducible principal series representation or a twist of the Steinberg representation of G...

The variance of observables of quantum states of the Laplacian on the modular surface is calculated in the semiclassical limit. It is shown that this hermitian form is diagonalized by the irreducible representations of the modular quotient and on each of these it is equal to the classical variance of the geodesic flow after the insertion of a subtle arithmetical special value of the correspondi...

An element x of a Lie algebra L over the field F is extremal if [x, [x, L]] = Fx. Under minor assumptions, it is known that, for a simple Lie algebra L, the extremal geometry E(L) is a subspace of the projective geometry of L and either has no lines or is the root shadow space of an irreducible spherical building ∆. We prove that if ∆ is of simply-laced type, then L is a quotient of a Chevalley...

One of the main difficulties faced when analyzing Markov chains modelling evolutionary algorithms is that their cardinality grows quite fast. A reasonable way to deal with this issue is to introduce an appropriate notion of a “coarse graining” or, in mathematical language, a quotient of such a Markov chain. The main topic of the current work is the construction of such a notion. We shell introd...

The group-theoretical classification of states of identical particle pairs is presented. Then obtained states are coupled with those of an antiparticle to construct states of a three-particle system. Investigations are performed using products of irreducible projective representations of the 2D translation group. For a given BvK period N degeneracy of pair states is N , whereas three-particle s...

Although the intelligence quotient (IQ) is the most popular intelligence test in the world, little is known about the underlying biological mechanisms that lead to the differences in human. To improve our understanding of cognitive processes and identify potential biomarkers, we conducted a comprehensive investigation of 158 IQ-related genes selected from the literature. A genomic distribution ...

Let $R$ be a commutative ring. The purpose of this article is to introduce a new class of ideals of R called weakly irreducible ideals. This class could be a generalization of the families quasi-primary ideals and strongly irreducible ideals. The relationships between the notions primary, quasi-primary, weakly irreducible, strongly irreducible and irreducible ideals, in different rings, has bee...

and a group of type ln it cannot have the property that each of its subgroups of composite order is invariant. If the order of G is not a power of a prime number each of its Sylow subgroups whose order exceeds a prime number is invariant under G and hence G involves invariantly the direct product of all its Sylow subgroups of composite orders. The quotient group of G with respect to this direct...

Let (B,MB) denote a regular local ring of dimension two. Let H be a finite cyclic group acting on B, and let Z := Spec(B). Assume that the action of H on Spec(B) is free off the closed point, and thatMB is the only singular point of Z. Let f : X → Z be a resolution of the singularity, minimal with the property that the irreducible components of f(MB ) are smooth with normal crossings. Such a re...