Let us consider a $3$-numerical semigroup $S=\langle{a,b,N}\rangle$. Given $m\in S$, the triple $(x,y,z)\in\mathbb{N}^3$ is a factorization of $m$ in $S$ if $xa+yb+zN=m$. This work is focused on finding the full set of factorizations of any $m\in S$ and as an application we compute the catenary degree of $S$. To this end, we relate a 2D tessellation to $S$ and we use it as a main tool.

As far as we know, usual computer algebra packages can not compute denumerants for almost medium (about a hundred digits) or almost medium-large (about a thousand digits) input data in a reasonably time cost on an ordinary computer. Implemented algorithms can manage numerical n-semigroups for small input data.Basically, the denumerant of a non-negative element s ∈ ℕ is the number of non-negative integer solutions of certain linear non-negative Diophantine equation which constant term is equal to s.Here we are interested in denumerants of numerical 3-semigroups which have almost medium input data. A new algorithm for computing denumerants is given for this task. It can manage almost medium input data in the worst case and medium-large or even large input data in some cases.

We recall L-shapes, which are minimal distance diagrams, related to weighted 2-Cayley digraphs, and we give the number and the relation between minimal distance diagrams related to the same digraph. On the other hand, we consider some classes of numerical semigroups useful in the study of curve singularity. Then, we associate L-shapes to each numerical 3-semigroup and we describe some main invariants of numerical 3-semigroups in terms of their associated L-shapes. Finally, we give a characterization of the parameters of the L-shapes associated with a numerical 3-semigroup in terms of its generators, and we use it to classify the numerical 3-semigroups of interest in curve singularity.

Denumerants of numerical semigroups are known to be difficult to obtain, even with small embedding dimension of the semigroups. In this work we give some results on denumerants of 3-semigroups S=S=. Closed expressions are obtained under certain conditions.Peer ReviewedPostprint (published version

Tailings dams represent one of the most environmentally sensitive infrastructures in the mining industry. To address the need for continuous and accurate monitoring, this paper presents an adaptive forecasting framework that combines Internet of Things (IoT) technologies with machine learning (ML) models to detect early signs of structural and ecological risks. The proposed system architecture is modular and scalable and enables the automated training, selection, and deployment of predictive models for multivariate sensor data. Each sensor data flow is independently analyzed by using a configurable set of algorithms (including linear, convolutional, recurrent, and residual models). The framework is deployed via containers with a CI/CD pipeline and includes real-time visualization through Grafana dashboards. A use case involving tiltmeters and piezometers in an operational tailing dam shows the system’s high predictive accuracy, with mean relative errors below $4\%$ across all variables (in fact, many of them have a mean relative error below $1\%$). These results highlight the potential of the proposed solution to improve structural and environmental safety in mining operations.