Search

The goal of this paper is to clarify the connection between certain structures from the theory of totally nonnegative Grassmannians, quiver Grassmannians for cyclic quivers and the theory of local models of Shimura varieties. More precisely, we generalize the construction from our previous paper relating the combinatorics and geometry of quiver Grassmannians to that of the totally nonnegative Grassmannians. The varieties we are interested in serve as realizations of local models of Shimura varieties. We exploit quiver representation techniques to study the quiver Grassmannians of interest and, in particular, to describe explicitly embeddings into affine flag varieties which allow us to realize our quiver Grassmannians as a union of Schubert varieties therein. [ABSTRACT FROM AUTHOR]

The concept of an abelian DG-category, introduced by the first-named author in Positselski (Exact DG-categories and fully faithful triangulated inclusion functors. [math.CT]), unites the notions of abelian categories and (curved) DG-modules in a common framework. In this paper we consider coderived and contraderived categories in the sense of Becker. Generalizing some constructions and results from the preceding papers by Becker (Adv Math 254:187–232, 2014. [math.CT]) and by the present authors (Positselski and Št’ovíček in J Pure Appl Algebra 226(#4):106883, 2022. [math.CT]), we define the contraderived category of a locally presentable abelian DG-category B with enough projective objects and the coderived category of a Grothendieck abelian DG-category A . We construct the related abelian model category structures and show that the resulting exotic derived categories are well-generated. Then we specialize to the case of a locally coherent Grothendieck abelian DG-category A , and prove that its coderived category is compactly generated by the absolute derived category of finitely presentable objects of A , thus generalizing a result from the second-named author’s preprint (Št’ovíček in On purity and applications to coderived and singularity categories. [math.CT]). In particular, the homotopy category of graded-injective left DG-modules over a DG-ring with a left coherent underlying graded ring is compactly generated by the absolute derived category of DG-modules with finitely presentable underlying graded modules. We also describe compact generators of the coderived categories of quasi-coherent matrix factorizations over coherent schemes. [ABSTRACT FROM AUTHOR]

In a seminal paper of 1917, Hardy and Ramanujan showed that the normal number of prime factors of a random natural number n is log log n . Their paper is often seen as inspiring the development of probabilistic number theory in that it led Erdös and Kac to discover, in 1940, a Gaussian law implied by their work. In this paper, we derive an all-purpose Erdös-Kac theorem that is applicable in diverse settings. In particular, we apply our theorem to show the validity of an Erdös-Kac type theorem for the study of the number of prime factors of sums of Fourier coefficients of Hecke eigenforms. [ABSTRACT FROM AUTHOR]

We correct two errors in the paper "Two weight Sobolev norm inequalities for smooth Calderón–Zygmund operators and doubling weights" published in Math. Z. Here we give a precise description of where the errors occur in the original paper, followed by an overview of the corrections, pointing to details appearing on the arXiv. We include statements of the main corrected theorems. [ABSTRACT FROM AUTHOR]

In this paper we derive a variety of functional inequalities for general homogeneous invariant hypoelliptic differential operators on nilpotent Lie groups. The obtained inequalities include Hardy, Sobolev, Rellich, Hardy–Littllewood–Sobolev, Gagliardo–Nirenberg, Caffarelli–Kohn–Nirenberg and Heisenberg–Pauli–Weyl type uncertainty inequalities. Some of these estimates have been known in the case of the sub-Laplacians, however, for more general hypoelliptic operators almost all of them appear to be new as no approaches for obtaining such estimates have been available. The approach developed in this paper relies on establishing integral versions of Hardy inequalities on homogeneous Lie groups, for which we also find necessary and sufficient conditions for the weights for such inequalities to be true. Consequently, we link such integral Hardy inequalities to different hypoelliptic inequalities by using the Riesz and Bessel kernels associated to the described hypoelliptic operators. [ABSTRACT FROM AUTHOR]

The classical theory of continued fractions has been widely studied for centuries for its important properties of good approximation, and more recently it has been generalized to p-adic numbers where it presents many differences with respect to the real case. In this paper we investigate periodicity for the p-adic continued fractions introduced by Browkin. We give some necessary and sufficient conditions for periodicity in general, although a full characterization of p-adic numbers having purely periodic Browkin continued fraction expansion is still missing. In the second part of the paper, we describe a general procedure to construct square roots of integers having periodic Browkin p-adic continued fraction expansion of prescribed even period length. As a consequence, we prove that, for every n ≥ 1 , there exist infinitely many m ∈ Q p with periodic Browkin expansion of period 2 n , extending a previous result of Bedocchi obtained for n = 1 . [ABSTRACT FROM AUTHOR]