arXiv subjects

Łukasz Sienkiewicz

Publications and source records attributed to Łukasz Sienkiewicz.

5 recordsLinked to original sources

arXiv2023

Multidimensional Hopfield Networks for clustering

We present the Multidimensional Hopfield Network (DHN), a natural generalisation of the Hopfield Network. In our theoretical investigations we focus on DHNs with a certain activation function and provide energy functions for them. We conclude that these DHNs are convergent in finite time, and are equivalent to greedy methods that aim to find graph clusterings of locally minimal cuts. We also show that the general framework of DHNs encapsulates several previously known algorithms used for generating graph embeddings and clusterings. Namely, the Cleora graph embedding algorithm, the Louvain method, and the Newmans method can be cast as DHNs with appropriate activation function and update rule. Motivated by these findings we provide a generalisation of Newmans method to the multidimensional case.

cs.NE↗

arXiv2022

Weights for Monoids and Actions of Monoids

The main objective of the paper is to define the category of monoids as a weighted limit. We also define the category of actions of monoids along the action of a monoidal category as a weighted limit.

math.CT↗

arXiv2021

Białynicki-Birula decomposition for reductive groups in positive characteristic

We prove the existence of Bialynicki-Birula decomposition for Kempf monoids, which is a large class that contains for example monoids with reductive unit group in all characteristics. This extends the existence statements from previous works of Alper-Hall-Rydh and Jelisiejew-Sienkiewicz.

math.AG↗

arXiv2018

Białynicki-Birula decomposition for reductive groups

We generalize the Białynicki-Birula decomposition from actions of $G_m$ on smooth varieties to actions of linearly reductive group ${\bf G}$ on finite type schemes and algebraic spaces. We also provide a relative version and briefly discuss the case of algebraic stacks. We define the Białynicki-Birula decomposition functorially: for a fixed ${\bf G}$-scheme $X$ and a monoid $\overline{\bf G}$ which partially compactifies ${\bf G}$, the BB decomposition parameterizes ${\bf G}$-schemes over $X$ for which the ${\bf G}$-action extends to the $\overline{\bf G}$-action. The freedom of choice of $\overline{\bf G}$ makes the theory richer than the $G_m$-case.

math.AG↗

arXiv2018

Vinberg's X_4 Revisited

The article covers the unique complex K3 surface with maximal Picard rank and discriminant four. We discuss smooth, rational curves and identify generators of its automorphism group with certain Cremona transformations of $\mathbb{P}^2$. This gives a geometric perspective of Vinberg's results.

math.AG↗