arXiv subjects
Publications and source records attributed to Niny Arcila-Maya.
Let $\mathcal{A}$ be a topological Azumaya algebra of degree $mn$ with an orthogonal involution over a CW complex $X$ of dimension less than or equal to $\min\{m,n\}$. We give conditions for the positive integers $m$ and $n$ so that $\mathcal{A}$ can be decomposed as the tensor product of topological Azumaya algebras of degrees $m$ and $n$ with orthogonal involutions.
Fingerprints are the most widely deployed biometric. Verifying whether two impressions come from the same finger typically relies on minutiae, small landmarks such as skin ridge endings and bifurcations. These landmarks are extracted through a multi-stage pipeline of image enhancement, skeletonization, minutiae detection, and alignment. We investigate an alternative: using topological data analysis to represent the full pattern of skin ridges and valleys directly, bypassing minutiae detection and the downstream matching pipeline. We apply persistent homology, a topological tool that tracks how loops in the ridge pattern form and fill in across spatial scales, producing multi-scale summaries of ridge geometry. We develop and compare a range of verification methods on a standard benchmark dataset, FVC2000 DB1. Even the simplest topological summaries, with no trained parameters, substantially outperform geometry-only baselines. A trained method achieves an AUC of 0.91, while an optimal-transport method excels at the strictest false-accept thresholds, suggesting they capture different aspects of the ridge pattern. Fusing these two approaches yields the best performance at every low false-accept threshold we examine. Our results establish that these topological summaries capture substantial fingerprint identity information, far more effective for verification than raw pixel-level geometry. Because the entire pipeline is openly specified, it offers a transparent complement to minutiae-based systems, and we provide a modular framework for constructing, evaluating, and combining topological verification methods.
A graph labeling assigns values to the components of a graph (vertices, edges, etc.). In particular, distance magic labelings have been widely studied in undirected graphs. In such a labeling, the vertices are labeled with unique values from one up to the number of vertices so that the sum of labels on the neighbors of any vertex is the same across all vertices. For oriented graphs, a related concept of distance difference magic has been studied. In a distance difference magic labeling, each vertex is given a unique value from one up to the number of vertices such that for each vertex the sums of the labels of vertices in the in-neighborhood minus the sums of the labels of vertices in the out-neighborhood equals zero. In this paper, we expand on this concept by showing a connected difference distance magic oriented graph on $n$ vertices exists for each integer $n \geq 5$. We also construct arbitrarily large difference distance magic oriented graphs from smaller ones using a new graph sum and exhibit a connection between linear algebra and this type of labeling.
Let $X$ be a connected CW complex. Let $\mathcal{V}$ be a symplectic vector bundle of rank $2mn$ over $X$, and let $\mathcal{A}$ be a topological Azumaya algebra of degree $2mn$ with a symplectic involution over a $X$. We give conditions for the positive integers $m$ and $n$, and the dimension of $X$ so that $\mathcal{V}$ can be decomposed as the tensor product of a symplectic vector bundle of rank $2m$ and an orthogonal vector bundle of rank $n$; and so that $\mathcal{A}$ can be decomposed as the tensor product of topological Azumaya algebras of degrees $2m$ and $n$ with involutions of the first kind.
We show the $\mathbb{A}^{1}$-Euler characteristic of a smooth, projective scheme over a characteristic $0$ field is represented by its Hochschild complex together with a canonical bilinear form, and give an exposition of the compactly supported $\mathbb{A}^{1}$-Euler characteristic $χ^{c}_{\mathbb{A}^{1}}: K_0(\mathbf{Var}_{k}) \to \text{GW}(k)$ from the Grothendieck group of varieties to the Grothendieck--Witt group of bilinear forms. We also provide example computations.
Let $\mathcal{A}$ be a topological Azumaya algebra of degree $mn$ over a CW complex $X$. We give conditions for the positive integers $m$ and $n$, and the space $X$ so that $\mathcal{A}$ can be decomposed as the tensor product of topological Azumaya algebras of degrees $m$ and $n$. Then we prove that if $m<n$ and the dimension of $X$ is higher than $2m+1$, $\mathcal{A}$ may not have such decomposition.