Numerical Analysis and Computational Mathematics

Matroids were introduced by Whitney in 1935 to try to capture abstractly the essence of dependence. Matroids generalize linear dependence over vector spaces, and they also abstract the properties of graphs, in the former case they are called Vector Matroids, in the latter they are called Graphic Matroids [3]. The operation of Matroid Union was introduced by Nash-Williams in 1966. A matroid is irreducible (indecomposable) if it does not decompose as the union of non-trivial matroids [3]. In 1971 Welsh posed the problem of characterizing indecomposable matroids, this problem has turned out to be extremely difficult. As a partial solution towards its progress, Cunningham characterized binary indecomposable matroids in 1977 [1]. One interesting feature of matroids, is that some elements in the matroid can be geometrically moved to a different location without changing the structure of the matroid, whereas others cannot. The former elements are called non-fixed and the latter fixed [2]. ...

Standard dynamical systems theory is based on the study of invariant sets. However, when noise is added, there are no bounded invariant sets. Our goal is then to study the fractal structure that exists even with noise. The problem we investigate is fluid flow past an array of cylinders. We study a parameter range for which there is a periodic oscillation of the fluid, represented by vortices being shed past each cylinder. Since the motion is periodic in time, we can study a time-1 Poincaré map. Then we add a small amount of noise, so that on each iteration the Poincaré map is perturbed smoothly, but differently for each time cycle. Fix an x coordinate x0 and an initial time t0. We discuss when the set of initial points at a time t0 whose trajectory (x(t),y(t)) is semibounded (i.e., x(t)>x0 for all time) has a fractal structure called an indecomposable continuum. We believe that the indecomposable continuum will become a fundamental object in the study of dynamical systems with no...

We study a linearly transformed particle method for the aggregation equation with smooth or singular interaction forces. For the smooth interaction forces, we provide convergence estimates in and norms depending on the regularity of the initial data. Moreover, we give convergence estimates in bounded Lipschitz distance for measure valued solutions. For singular interaction forces, we establish the convergence of the error between the approximated and exact flows up to the existence time of the solutions in norm.

In this paper, we introduce a Halpern iteration process for computing the common solution of split generalized equilibrium problem and fixed points of a countable family of Bregman W-mappings with multiple output sets in reflexive Banach spaces. We prove a strong convergence result for approximating the solutions of the aforementioned problems under some mild conditions. It is worth mentioning that the iterative algorithm employ in this article is designed in such a way that it does not require the prior knowledge of operator norm. We also provide some numerical examples to illustrate the performance of our proposed iterative method. The result discuss in this paper extends and complements many related results in literature.

The internet of things (IoT) has increased exponentially in connected devices
worldwide in recent years. However, this rapid growth also introduces
significant security challenges since many IoT devices have vulnerabilities
that can be exploited for cyber-attacks. Anomaly detection using machine
learning algorithms shows promise for identifying abnormal network traffic
indicative of IoT attacks. This paper proposes an ensemble learning
framework for anomaly detection in IoT networks. A systematic literature
review analyzes recent research applying machine learning for IoT security. Subsequently, a novel stacked ensemble model is presented, combining  multiple base classifiers (random forest, neural network, support vector machine (SVM)) and meta-classifiers (gradient boosting) for improved performance. The model is evaluated on the IoTID20 dataset, using network traffic features to detect anomalies across binary, multi-class, and multi-label classifications. Experimental results demonstrate that the ensemble model achieved 99.7% accuracy and F1 score for binary classification, 99.5% accuracy for multi-class, and 91.2% accuracy for multi-label classification, outperforming previous methods. The model provides an effective anomaly detection approach to identify malicious activities and mitigate IoT security threats.

Multiplier bimonoids (or bialgebras) in arbitrary braided monoidal categories are defined. They are shown to possess monoidal categories of comodules and modules. These facts are explained by the structures carried by their induced functors.