Related papers: Reduction maps and minimal model theory
We consider the sigma models where the base metric is proportional to the metric of the configuration space. We show that the corresponding sigma model equation admits a Lax pair. We also show that this type of sigma models in two…
Elimination theory has many applications, in particular, it describes explicitly an image of a complex line under rational transformation and determines the number of common zeroes of two polynomials in one variable. We generalize classical…
The Minimum Description Length (MDL) principle selects the model that has the shortest code for data plus model. We show that for a countable class of models, MDL predictions are close to the true distribution in a strong sense. The result…
We give a new proof of the Semistable Reduction Theorem for curves. The main idea is to present a curve $Y$ over a local field $K$ as a finite cover of the projective line $X=\PP^1_K$. By successive blowups (and after replacing $K$ by a…
The minimum cut problem for an undirected edge-weighted graph asks us to divide its set of nodes into two blocks while minimizing the weight sum of the cut edges. Here, we introduce a linear-time algorithm to compute near-minimum cuts. Our…
A key idea in convex optimization theory is to use well-structured affine functions to approximate general functions, leading to impactful developments in conjugate functions and convex duality theory. This raises the question: what are the…
In the general submatrix detection problem, the task is to detect the presence of a small $k \times k$ submatrix with entries sampled from a distribution $\mathcal{P}$ in an $n \times n$ matrix of samples from $\mathcal{Q}$. This…
We make use of the complex implicit representation in order to provide a deterministic algorithm for checking whether or not two implicit algebraic curves are related by a similarity, a central question in Pattern Recognition and Computer…
In this paper, we study two applications of graph minor reduction. In the first part of the paper, we introduce a variant of the boxicity, called strong boxicity, where the rectangular representation satisfies an additional condition that…
We study the minimum number of minimal codewords in linear codes from the point of view of projective geometry. We derive bounds and in some cases determine the exact values. We also present an extension to minimal subcode supports.
We describe a type system for the linear-algebraic $\lambda$-calculus. The type system accounts for the linear-algebraic aspects of this extension of $\lambda$-calculus: it is able to statically describe the linear combinations of terms…
We consider the evolution of sets by nonlocal mean curvature and we discuss the preservation along the flow of two geometric properties, which are the mean convexity and the outward minimality. The main tools in our analysis are the level…
We review a range of reduction methods that have been, or may be useful for connecting models of the Earth's climate system of differing complexity. We particularly focus on methods where rigorous reduction is possible. We aim to highlight…
Let $n$ and $s$ be fixed integers such that $n\geq 2$ and $1\leq s\leq \frac{n}{2}$. Let $M_n(\mathbb{K})$ be the ring of all $n\times n$ matrices over a field $\mathbb{K}$. If a map $\delta:M_n(\mathbb{K})\rightarrow M_n(\mathbb{K})$…
If $(X, \mcF, \D)$ is a projective rank two foliated log canonical triple such that $(X,B)$ is klt for some $0 \leq B \leq \D$, we show that we can run a $(K_\mcF +\Delta)$-MMP and any such MMP terminates with either a minimal model or Mori…
We use machine learning to classify examples of braids (or flat braids) as trivial or non-trivial. Our ML takes form of supervised learning using neural networks (multilayer perceptrons). When they achieve good results in classification, we…
In this paper, we give a uniqueness theorem for the Dirichlet problem of minimal maps into general Riemannian manifolds with non-positive sectional curvature, improving Theorem 5.2 of Lee-Ooi-Tsui's paper published in J. Geom. Anal.. The…
We study the minimal resultant divisor of self-maps of the projective line over a number field or a function field and its relation to the conductor. The guiding focus is the exploration of a dynamical analog to Theorem 1.1, which bounds…
We prove that if X is any 2-regular projective scheme (in the sense of Castelnuovo-Mumford) then X is "small". This means that if L is a linear space and Y:= L\cap X is finite, then Y is "linearly independent" in the sense that the…
The stable pairs theory of local curves in 3-folds (equivariant with respect to the scaling 2-torus) is studied with stationary descendent insertions. Reduction rules are found to lower descendents when higher than the degree. Factorization…