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In this paper we consider the algebra of upper triangular matrices UT$_n(F)$, endowed with a $\mathbb{Z}_2$-grading (superalgebra) and equipped with a superinvolution. These structures naturally arise in the context of Lie and Jordan…
Trees have long been used as a graphical representation of species relationships. However complex evolutionary events, such as genetic reassortments or hybrid speciations which occur commonly in viruses, bacteria and plants, do not fit into…
In our article we consider some algebraical methods which may be useful in some inverse spectral problems. The reconstraction of the matrix from its minors is considered.
This is a short exposition--mostly by way of the toy models ``double logarithm'' and ``triple logarithm''--which should serve as an introduction to a forthcoming article in which we establish a connection between multiple polylogarithms,…
This paper studies the unitary diagonalization of matrices over formal power series rings. Our main result shows that a normal matrix is unitarily diagonalizable if and only if its minimal polynomial completely splits over the ring and the…
The partial success of the block renormalization group techniques is analysed in terms of a functional operator which formalizes the idea of self-replicability of a system in terms of smaller blocks which are similar to the original. The…
The algebra of fourvectors is described. The fourvectors are more appropriate than the Hamilton quaternions for its use in Physics and the sciences in general. The fourvectors embrace the 3D vectors in a natural form. It is shown the…
We show that every polycube tree can be unfolded with a 4x4 refinement of the grid faces. This is the first constant refinement unfolding result for polycube trees that are not required to be well-separated.
We propose a probabilistic interpretation of transformers as unrolled inference steps assuming a probabilistic Laplacian Eigenmaps model from the ProbDR framework. Our derivation shows that at initialisation, transformers perform "linear"…
We consider the reconciliation problem, in which the task is to find a mapping of a gene tree into a species tree, so as to maximize the likelihood of such fitting, given the available data. We describe a model for the evolution of the…
Recently, an approach to graph signal processing based on graphons was proposed. Here we show how such a graphon-driven approach to the Fourier transform can be used on graphs sampled from a stochastic block model (SBM). In particular, we…
We present some work relating to fractal transformations on masked iterated function systems and demonstrate how well known algorithms for generating fractal transformations can be modifed for these systems. We also demonstrate that these…
The supertree problem asking for a tree displaying a set of consistent input trees has been largely considered for the reconstruction of species trees. Here, we rather explore this framework for the sake of reconstructing a gene tree from a…
Motivated by a problem in complex dynamics, we examine the block structure of the natural action of monodromy groups on the tree of preimages of a generic point. We show that in many cases, including when the polynomial has prime power…
Contour trees describe the topology of level sets in scalar fields and are widely used in topological data analysis and visualization. A main challenge of utilizing contour trees for large-scale scientific data is their computation at scale…
We study the conditions under which the isometry of spaces with metrics generated by weights given on the edges of finite trees is equivalent to the isomorphism of these trees. Similar questions are studied for ultrametric spaces generated…
We introduce tree representations for $ \alpha$-determinantal point processes. The $ \alpha$-determinantal point processes is introduced as a one parameter extension of the determinantal point process. In the previous paper with H.Osada,…
In this paper we examine the relationship between hyperconvex hulls and metric trees. After providing a linking construction for hyperconvex spaces, we show that the four-point property is inherited by the hyperconvex hull, which leads to…
Over the past decade, random forest models have become widely used as a robust method for high-dimensional data regression tasks. In part, the popularity of these models arises from the fact that they require little hyperparameter tuning…
Generalized trees, we call them O-trees, are defined as hierarchical partial orders, i.e., such that the elements larger than any one are linearly ordered. Quasi-trees are, roughly speaking, undirected O-trees. For O-trees and quasi-trees,…