Related papers: Betti numbers of inductively pierced codes

We show that there exists a saturated graded ideal in a standard graded polynomial ring which has the largest total Betti numbers among all saturated graded ideals for a fixed Hilbert polynomial.

The brain processes information about the environment via neural codes. The neural ideal was introduced recently as an algebraic object that can be used to better understand the combinatorial structure of neural codes. Every neural ideal…

Image segmentation is a largely researched field where neural networks find vast applications in many facets of technology. Some of the most popular approaches to train segmentation networks employ loss functions optimizing pixel-overlap,…

This paper bridges neuroscience and artificial intelligence to propose a cortically inspired blueprint for modular perceptual AI. While current monolithic models such as GPT-4V achieve impressive performance, they often struggle to…

We present neuron embeddings, a representation that can be used to tackle polysemanticity by identifying the distinct semantic behaviours in a neuron's characteristic dataset examples, making downstream manual or automatic interpretation…

We provide a generalization of quantum polar codes to quantum channels with qudit-input, achieving the symmetric coherent information of the channel. Our scheme relies on a channel combining and splitting construction, where a two-qudit…

We provide a closed formula for the graded Betti numbers in the linear strands of all powers of binomial edge ideals $J_G$ arising from closed graphs $G$ that do not have the complete graph $K_4$ as an induced subgraph. We show that these…

Inductive Logic Programming (ILP) learns interpretable logical rules from data. Existing methods are transductive: their learned parameters are bound to specific predicates and require retraining for each new task. We introduce Neural Rule…

A class of good integers has been introduced by P. Moree in $1997$ together with the characterization of good odd integers. Such integers have shown to have nice number theoretical properties and wide applications. In this paper, a complete…

Let $\operatorname{pd}(I(G))$ and $\operatorname{reg}(I(G))$ respectively denote the projective dimension and the regularity of the edge ideal $I(G)$ of a graph $G$. For any positive integer $n$, we determine all pairs…

An ideal in a polynomial ring encodes a system of linear partial differential equations with constant coefficients. Primary decomposition organizes the solutions to the PDE. This paper develops a novel structure theory for primary ideals in…

Let $G$ be the circulant graph $C_n(S)$ with $S \subseteq \{1, 2, \dots, \lfloor \frac{n}{2} \rfloor\}$, and let $I(G)$ denote the edge ideal in the polynomial ring $R=\mathbb{K}[x_0, x_1, \dots, x_{n-1}]$ over a field $\mathbb{K}$. In this…

This article introduces patterns of ideals of numerical semigroups, thereby unifying previous definitions of patterns of numerical semigroups. Several results of general interest are proved. More precisely, this article presents results on…

We prove that for a finite set of points $X$ in the projective $n$-space over any field, the Betti number $\beta_{n,n+1}$ of the coordinate ring of $X$ is non-zero if and only if $X$ lies on the union of two planes whose sum of dimension is…

Neural codes, represented as collections of binary strings called codewords, are used to encode neural activity. A code is called convex if its codewords are represented as an arrangement of convex open sets in Euclidean space. Previous…

In the present paper, we consider upper bounds of higher linear syzygies i.e. graded Betti numbers in the first linear strand of the minimal free resolutions of projective varieties in arbitrary characteristic. For this purpose, we first…

We developed a model of cortical computation that implements key features of cortical circuitry and is capable of describing propagation of neural signals between cortical locations in response to spatially distributed stimuli. The model is…

A minor is principal means it is defined by the same row and column indices. Let $X$ be a square generic matrix, $K[X]$ the polynomial ring in entries of $X$, over an algebraically closed field, $K$. For fixed $t\leq n$, let $\mathfrak P_t$…

Predictive coding is an influential theory of cortical function which posits that the principal computation the brain performs, which underlies both perception and learning, is the minimization of prediction errors. While motivated by…

We study various ideals arising in the theory of system reliability. We use ideas from the theory of divisors, orientations and matroids on graphs to describe the minimal polyhedral cellular free resolutions of these ideals. In each case we…