Axioms

Complexity (number of spanning trees) is an essential and significant component in the design of communication networks (graphs). To ensure strong resistance and stiffness and to enhance the probability of a connection between two vertices, improvements to a network’s quality and perfection increase the number of trees that span it. Using block matrices and linear algebra techniques, we derive explicit formulas for the number of spanning trees of new graph families that are produced from star graphs in this study. The number of spanning trees in a graph is measured by the entropy of spanning trees, also known as asymptotic complexity, a graph theory metric that assesses the network’s structural robustness and dependability. Increased flexibility, stronger diverse connections, and improved resistance to random structural changes are all indicated by higher entropy. We also investigate the entropy of spanning trees on our graphs at the end of this study. Lastly, we compare the entropy of our graphs to that of other previously studied graphs with average degrees of four and five.

Let ${\mathrm{\Phi }}_N(X,Y)$ be the N-th classical modular polynomial and let $Z_0\left(N\right)=\{(X,Y)\in {\mathbb{C}}^2\mid {\mathrm{\Phi }}_N(X,Y)=0\}$ be the plane model of the modular curve $X_0\left(N\right)$. We present an explicit procedure that, for a prime ℓ, enumerates all non-cuspidal singular points of $Z_0(\ell )$ over $\mathbb{C}$ and outputs the corresponding pairs of distinct points on $X_0(\ell )$ mapping to each node. The method relies on the arithmetic (CM) classification of self-intersections of the map and on effective computations of proper ideal classes in imaginary quadratic orders. We also provide a complete and self-contained exposition of Kara’s proof of the automorphism-group equality in the self-intersection setting, making explicit where Kolyvagin’s conductor lemma is used essentially. Finally, we discuss termination, correctness, and practical complexity issues, and we report computational evidence for larger primes using a parallel implementation; in particular, for $\ell =389$, we obtained $151,288$ output pairs in $151,017$ seconds on a 56-core machine.

Functional data are nowadays routinely collected and stored in a wide variety of fields. Their adequate use and analysis are a challenge for the scientific community. Mathematically, each function can be understood as a sequence of infinite related numbers. Therefore, for statisticians, functional data can be read as a collection of a strongly correlated infinite-dimensional variable. Most existing statistical procedures have been adapted to functional data scenarios. In this manuscript, we are interested in understanding the use of functions for constructing adequate ROC curves and, therefore, for carrying out binary classifications. In particular, we consider the problem of studying the real capacity of functions derived from tissue doppler imaging (TDI) for identifying cardiac dysfunction related to cardiotoxicity therapy (CRTCD) in breast cancer women with high levels of the protein human epidermal growth factor receptor 2 (HER2). With this goal, we use public and freely available data that has been already used for illustrating the use of functional data in the binary classification problem with very different take-home messages. This variability in the conclusions made us question the reproducibility of the results. Here, we explore five different functional approaches, and we think about the clinical use of the provided solutions and their potential overfitting. The main aim of this manuscript is identifying whether published results are excessively optimistic or if they adequately capture the actual capacity of TDI for accurately diagnostic CRTCD.