Related papers: Analysis of two step nilsequences

This paper presents new examples of elementary and non-elementary irreducible components of the Hilbert scheme of points and its nested variants. The results are achieved via a careful analysis of the deformations of a class of finite…

A key challenge in building theoretical foundations for deep learning is the complex optimization dynamics of neural networks, resulting from the high-dimensional interactions between the large number of network parameters. Such non-trivial…

We introduce and study a dimensional-like characteristic of an uniformly almost periodic function, which we call the Diophantine dimension. By definition, it is the exponent in the asymptotic behavior of the inclusio length. Diophantine…

An extension of Szemer\'edi's Theorem is proved for sets of positive density in approximate lattices in general locally compact and second countable abelian groups. As a consequence, we establish a recent conjecture of Klick, Strungaru and…

Hertzsprung patterns, recently introduced by Anders Claesson, are subsequences of a permutation contiguous in both positions and values, and can be seen as a subclass of bivincular patterns. This paper investigates Hertzsprung patterns…

We give examples of sequences defined by smooth functions of intermediate growth, and we study the Furstenberg systems that model their statistical behavior. In particular, we show that the systems are Bernoulli. We do so by studying…

A solution of two-stage stochastic generalized equations is a pair: a first stage solution which is independent of realization of the random data and a second stage solution which is a function of random variables.This paper studies…

In this paper we elaborate on the structure of the semigroup tree and the regularities on the number of descendants of each node observed earlier. These regularites admit two different types of behavior and in this work we investigate which…

This paper is a modified chapter of the author's Ph.D. thesis. We introduce the notions of sequentially approximated types and sequentially approximated Keisler measures. As the names imply, these are types which can be approximated by a…

We consider real 2-step metric nilpotent Lie algebras associated to graphs with possibly repeated edge labels as constructed by Ray in 2016. We determine how the structure of the egde labeling within the graph contributes to the abelian…

The document tries to put focus on sequences with certain properties and periods leading to the first value smaller than the starting value in the Collatz problem. With the idea that, if all starting numbers lead ultimately to a smaller…

We consider finite-dimensional complex Lie algebras admitting a periodic derivation, i.e., a nonsingular derivation which has finite multiplicative order. We show that such Lie algebras are at most two-step nilpotent and give several…

We look at the number $L(n)$ of $O$-sequences of length $n$. Recall that an $O$-sequence can be defined algebraically as the Hilbert function of a standard graded $k$-algebra, or combinatorially as the $f$-vector of a multicomplex. The…

The extension structure of the 2-dimensional current algebra of non-linear sigma models is analysed by introducing Kostant Sternberg $(L,M)$ systems. It is found that the algebra obeys a two step extension by abelian ideals. The second step…

Using a result of Behrend concerning sets without arithmetic progressions, we construct some examples of dynamical systems with slow time of multiple recurrence. Our theorem is a quatitative analog of Furstenberg's Correspondence Principle.

It has been observed by Belkin et al.\ that over-parametrized neural networks exhibit a `double descent' phenomenon. That is, as the model complexity (as reflected in the number of features) increases, the test error initially decreases,…

A key tool in recent advances in understanding arithmetic progressions and other patterns in subsets of the integers is certain norms or seminorms. One example is the norms on $\Z/N\Z$ introduced by Gowers in his proof of Szemer\'edi's…

Generalized Fibonacci-like sequences appear in finite difference approximations of the Partial Differential Equations based upon replacing partial differential equations by finite difference equations. This paper studies properties of the…

We develop a new, powerful method for counting elements in a multiset. As a first application, we use this algorithm to study the number of occurrences of patterns in a permutation. For patterns of length 3 there are two Wilf classes, and…

We give a basic treatment of lattices $\Gamma$ in these groups. Certain tori $T_F$ and $T_B$ provide the model fiber and the base for a submersion of $\Gamma\backslash N$. This submersion may not be pseudoriemannian in the usual sense,…