arXiv subjects

Pedro Patrício

Publications and source records attributed to Pedro Patrício.

8 recordsLinked to original sources

arXiv2026

On the Drazin Index of an Anti-Triangular Block Matrix

The Drazin index is a fundamental invariant in the analysis of singular matrices and their generalized inverses. While sharp results are available for block triangular matrices, the corresponding theory for anti-triangular block matrices is less developed. In this paper, we study matrices of the form \[ M=\begin{bmatrix} A & B \\ C & 0 \end{bmatrix}, \] under algebraic constraints on the blocks. Building on additive decompositions involving von Neumann inverses, we relate the Drazin index of $M$ to invariance properties of the index and minimal polynomial of expressions of the form $A^{2}A^{-}+I-AA^{-}$. This connection provides an effective mechanism to control the index of $M$ through suitable factorizations and associated block products. As a consequence, we derive explicit lower and upper bounds for $i(M)$ in terms of $i(A)$ and $i(BC)$, and characterize situations in which these bounds are attained. Under additional annihilation or orthogonality conditions on the blocks, we obtain closed-form representations for the Drazin inverse of $M$. Applications to adjacency matrices of directed graphs illustrate the sharpness of the bounds and the applicability of the results to structured matrices arising in graph-theoretic settings.

math.CO↗

arXiv2026

Real tensor factorizations and generalized inverses under the $t$-product

The algebraic theory of third-order tensors under the $t$-product is naturally formulated over the complex field via Fourier block diagonalization. However, many applications require real-valued representations. In this paper, we investigate structural conditions ensuring that tensor factorizations and generalized inverses admit real realizations. We show that these conditions can be characterized through the conjugate-pairing structure of the Fourier frontal slices, which determines when transform-domain constructions yield real tensors after inverse transformation. As applications, we obtain real versions of several tensor factorizations and analyze the existence and structure of associated generalized inverses. These results provide a framework for transferring matrix-based constructions to real tensors while preserving the algebraic constraints of the $t$-product.

math.CO↗

arXiv2026

Drazin Inverses and Walk Structure of Oriented Dutch Windmill Graphs

We investigate the Drazin invertibility of adjacency matrices associated with a class of oriented graphs known as oriented Dutch windmill graphs. By analyzing walks of prescribed lengths and exploiting the structure of the minimal polynomial, we obtain explicit expressions for the Drazin inverse and determine its index. The approach combines combinatorial enumeration with algebraic matrix analysis, offering a constructive characterization that generalizes known results for paths, cycles, and bipartite graphs. Beyond its intrinsic theoretical value, the framework provides insight into discrete models governed by cyclic feedback and may serve as a basis for symbolic computation of generalized inverses in structured networks.

math.CO↗

arXiv2026

Core EP, Dual Core EP and Composite Generalized Inverses for a Class of Structured Matrices

We study generalized inverses for matrices associated with double star digraphs. Explicit block formulas and existence criteria are obtained for core, dual core, core EP, and dual core EP inverses, expressed in terms of explicit algebraic criteria derived from the underlying block structure. Other combined outer pseudoinverses, combining Moore--Penrose and core-type inverses, are derived with existence criteria.

math.RA↗

arXiv2026

Physical Models of Embryonic Epithelial Healing

Embryonic healing in epithelial tissues is distinct from adult wound healing, as it lacks inflammatory responses or immune cell recruitments, making it ideal to test models of wound healing driven primarily by epithelial dynamics. Many models have been developed to describe this process, ranging from simple mechanistic models to more elaborate multiscale simulations. We review different classes of physical models, from discrete to continuum models, and how they address key questions about the mechanics, signaling, and coordination of cells during wound closure. We highlight tensions between model complexity and interpretability and discuss recent efforts to bridge gaps across scales. Finally, we identify directions for hybrid modeling and model-experiment integration that could push forward our understanding of epithelial repair in development and disease.

cond-mat.soft↗

arXiv2025

On generalized inverses of matrices associated with certain graph classes

We investigate generalized inverses of matrices associated with two classes of digraphs: double star digraphs and D-linked stars digraphs. For double star digraphs, we determine the Drazin index and derive explicit formulas for the Drazin inverse. We also provide necessary and sufficient conditions for the existence of the Moore-Penrose inverse and give its explicit expression whenever it exists. For D-linked stars digraphs, we characterize when the group inverse exists and obtain its explicit form. In the singular case where BC = 0, we express the Drazin index of the matrix in terms of the Drazin index of the base digraph matrix. Additionally, we establish necessary and sufficient conditions for Moore--Penrose invertibility and derive explicit formulas in that case. Our results reveal a clear connection between the algebraic structure of generalized inverses and the combinatorial properties of these graph classes, providing a unified framework for group, Drazin, and Moore-Penrose invertibility.

math.CO↗

arXiv2021

Inheritances, social classes, and wealth distribution

We consider a simple theoretical model to investigate the impact of inheritances on the wealth distribution. Wealth is described as a finite resource, which remains constant over different generations and is divided equally among offspring. All other sources of wealth are neglected. We consider different societies characterized by a different offspring probability distribution. We find that, if the population remains constant, the society reaches a stationary wealth distribution. We show that inequality emerges every time the number of children per family is not always the same. For realistic offspring distributions from developed countries, the model predicts a Gini coefficient of $G\approx 0.3$. If we divide the society into wealth classes and set the probability of getting married to depend on the distance between classes, the stationary wealth distribution crosses over from an exponential to a power-law regime as the number of wealth classes and the level of class distinction increase.

physics.soc-ph↗

arXiv2007

Complete wetting transitions of nematic liquid crystals on a structured substrate

In this article, we generalize Wenzel law, which assigns an effective contact angle for a droplet on a rough substrate, when the wetting layer has an ordered phase, like a nematic. We estimate the conditions for which the wetting behavior of an ordered fluid can be qualitatively different from the one usually found in a simple fluid. To particularize our general considerations, we will use the Landau-de Gennes mean field approach to investigate theoretically and numerically the complete wetting transition between a nematic liquid crystal and a saw-shaped structured substrate.

cond-mat.other↗