Related papers: Finite generation and geography of models

We prove that the LMMP works for projective threefolds over function fields of characteristic $p>5$ when the canonical divisor is not pseudo-effective. In the process we show that ACC for log canonical thresholds holds in complete…

Finding the correct encoding for a generic dynamical system's trajectory is a complicated task: the symbolic sequence needs to preserve the invariant properties from the system's trajectory. In theory, the solution to this problem is found…

We treat two different topics on the log minimal model program, especially for four-dimensional log canonical pairs. (a) Finite generation of the log canonical ring in dimension four. (b) Abundance theorem for irregular fourfolds. We obtain…

We consider the problem of exact low-rank matrix completion from a geometric viewpoint: given a partially filled matrix M, we keep the positions of specified and unspecified entries fixed, and study how the minimal completion rank depends…

We prove that many of the results of the LMMP hold for $3$-folds over fields of characteristic $p>5$ which are not necessarily perfect. In particular, the existence of flips, the cone theorem, the contraction theorem for birational extremal…

The recently developed theory of extended generating functions of symplectic maps are combined with methods to prove invertibility via high-order Taylor model methods to obtain rigorous lower bounds for the domains of definition of…

We construct a class of finite rank multiplicative subgroups of the complex numbers such that the expansion of the real field by such a group is model-theoretically well-behaved. As an application we show that a classification of expansions…

A ringed finite space is a ringed space whose underlying topological space is finite. The category of ringed finite spaces contains, fully faithfully, the category of finite topological spaces and the category of affine schemes. Any ringed…

We study bihomogeneous systems defining, non-zero dimensional, biprojective varieties for which the projection onto the first group of variables results in a finite set of points. To compute (with) the 0-dimensional projection and the…

In this paper it is shown how to construct a finite topological space $X$ for a given finitely presentable group $G$ such that $\pi_1(X)\cong G$. Our construction is not optimal in the sense that the cardinality of the space $X$ might not…

Casting neural networks in generative frameworks is a highly sought-after endeavor these days. Contemporary methods, such as Generative Adversarial Networks, capture some of the generative capabilities, but not all. In particular, they lack…

We give bounds on the degree of generation and relations of section rings associated to arbitrary $\mathbb{Q}$-divisors on projective spaces of all dimensions and Hirzebruch surfaces. For section rings of effective $\mathbb{Q}$-divisors on…

Heralded by the initial success in speech recognition and image classification, learning-based approaches with neural networks, commonly referred to as deep learning, have spread across various fields. A primitive form of a neural network…

Let $(R,\mathfrak{m})$ be a local Noetherian ring with residue field $k$. While much is known about the generating sets of reductions of ideals of $R$ if $k$ is infinite, the case in which $k$ is finite is less well understood. We…

There has been a lot of recent interest in trying to characterize the error surface of deep models. This stems from a long standing question. Given that deep networks are highly nonlinear systems optimized by local gradient methods, why do…

Subspace segmentation or subspace learning is a challenging and complicated task in machine learning. This paper builds a primary frame and solid theoretical bases for the minimal subspace segmentation (MSS) of finite samples. Existence and…

In this paper, we shall study finite generation of symbolic Rees rings of the defining ideal of the space monomial curves $(t^a, t^b, t^c)$ for pairwise coprime integers $a$, $b$, $c$ such that $(a,b,c) \neq (1,1,1)$. If such a ring is not…

This article considers the generative modeling of the (mixed) states of quantum systems, and an approach based on denoising diffusion model is proposed. The key contribution is an algorithmic innovation that respects the physical nature of…

Let $\mathcal G$ denote the space of finitely generated marked groups. For any finitely generated group $G$, we construct a continuous, injective map $f$ from the space of subgroups $Sub(G)$ to $\mathcal G$ that sends conjugate subgroups to…

In this paper, in the first we give definitions of some classes of division rings which strictly contain the class of centrally finite division rings. One of our main purpose is to construct non-trivial examples of rings of new defined…