We present a new class of Lipschitz operators on Euclidean lattices that we call lattice Lipschitz maps, and we prove that the associated McShane and Whitney formulas provide the same extension result that holds for the real valued case. Essentially, these maps satisfy a (vector-valued) Lipschitz inequality involving the order of the lattice, with the peculiarity that the usual Lipschitz constant becomes a positive real function. Our main result shows that, in the case of Euclidean space, being lattice Lipschitz is equivalent to having a diagonal representation, in which the coordinate coefficients are real-valued Lipschitz functions. We also show that in the linear case the extension of a diagonalizable operator from the values in their eigenvectors coincide with the operator obtained both from the McShane and the Whitney formulae. Our work on such extension/representation formulas is intended to follow current research on the design of machine learning algorithms based on the extension of Lipschitz functions.

The axioms of symmetry and indiscernibility of a metric function can be eliminated and still define a topological space with good enough properties. However, among all the axioms that constitute th...

Heat stress is one of the main welfare and productivity problems faced by dairy cattle in Mediterranean climates. The main objective of this work is to predict heat stress in livestock from shade-seeking behavior captured by computer vision, combined with some climatic features, in a completely non-invasive way. To this end, we evaluate two soft computing algorithms—Random Forests and Neural Networks—clarifying the trade-off between accuracy and interpretability for real-world farm deployment. Data were gathered at a commercial dairy farm in Titaguas (Valencia, Spain) using overhead cameras that counted cows in the shade every 5–10 min during summer 2023. Each record contains the shaded-cow count, ambient temperature, relative humidity, and an exact timestamp. From here, three thermal indices were derived: the current THI, the previous-night mean THI, and the day-time accumulated THI. The resulting dataset covers 75 days and 6907 day-time observations. To evaluate the models’ performance a 5-fold cross-validation is also used. The results show that both soft computing models outperform a single Decision Tree baseline. The best Neural Network (3 hidden layers, 16 neurons each, learning rate $={10}^{-3}$) reaches an average RMSE of $14.78$, while a Random Forest (10 trees, depth $=5$) achieves $14.97$ and offers the best interpretability. Daily error distributions reveal a median RMSE of $13.84$ and confirm that predictions deviate less than one hour from observed shade-seeking peaks. Although the dataset came from a single farm, the results generalized well within the observed range. However, the models could not accurately predict the exact number of cows in the shade. This suggests the influence of other variables not included in the analysis (such as solar radiation or wind data), which opens the door for future research.

Consider a finite directed graph without cycles in which the arrows are weighted by positive weights. We present an algorithm for the computation of a new distance, called path-length-weighted distance, which has proven useful for graph analysis in the context of fraud detection. The idea is that the new distance explicitly takes into account the size of the paths in the calculations. It has the distinct characteristic that, when calculated along the same path, it may result in a shorter distance between far-apart vertices than between adjacent ones. This property can be particularly useful for modeling scenarios where the connections between vertices are obscured by numerous intermediate vertices, such as in cases of financial fraud. For example, to hide dirty money from financial authorities, fraudsters often use multiple institutions, banks, and intermediaries between the source of the money and its final recipient. Our distance would serve to make such situations explicit. Thus, although our algorithm is based on arguments similar to those at work for the Bellman–Ford and Dijkstra methods, it is in fact essentially different, since the calculation formula contains a weight that explicitly depends on the number of intermediate vertices. This fact totally conditions the algorithm, because longer paths could provide shorter distances—contrary to the classical algorithms mentioned above. We lay out the appropriate framework for its computation, showing the constraints and requirements for its use, along with some illustrative examples.

Here, we prove some general results that allow us to ensure that specific representations (as well as extensions) of certain Lipschitz operators exist, provided we have some additional information about the underlying space, in the context of what we call enriched metric spaces. In this conceptual framework, we introduce some new classes of Lipschitz operators whose definition depends on the notion of metric coordinate system, which are defined by specific dominance inequalities involving summations of distances between certain points in the space. We analyze “Pietsch Theorem inspired factorizations" through subspaces of ${\ell }_{\infty }$ and $L^1$, which are proved to characterize when a given metric space is Lipschitz isomorphic to a metric subspace of these spaces. As an application, extension results for Lipschitz maps that are obtained by a coordinate-wise adaptation of the McShane–Whitney formulas, are also given.

Lattice Lipschitz operators define a new class of nonlinear Banach-lattice-valued maps that can be written as diagonal functions with respect to a certain basis. In the n-dimensional case, such a map can be represented as a vector of size n of real-valued functions of one variable. In this paper we develop a method to approximate almost diagonal maps by means of lattice Lipschitz operators. The proposed technique is based on the approximation properties and error bounds obtained for these operators, together with a pointwise version of the interpolation of McShane and Whitney extension maps that can be applied to almost diagonal functions. In order to get the desired approximation, it is necessary to previously obtain an approximation to the set of eigenvectors of the original function. We focus on the explicit computation of error formulas and on illustrative examples to present our construction.