Predicting how a genetic change affects a given character is a major challenge in biology, and being able to tackle this problem relies on our ability to develop realistic models of gene networks. However, such models are rarely tractable mathematically. In this paper, we propose a mathematical analysis of the sigmoid variant of the Wagner gene-network model. By considering the simplest case, that is, one unique self-regulating gene, we show that numerical simulations are not the only tool available to study such models: theoretical studies can be done too, by mathematical analysis of discrete dynamical systems. It is first shown that the particular sigmoid function can be theoretically investigated. Secondly, we provide an illustration on how to apply such investigations in the case of the dynamical system representing the one self-regulating gene.

We present smooth approximations to $ \vert{x} \vert $ using sigmoid functions. In particular $ x\,erf(x/\mu) $ is proved to be better smooth approximation for $ \vert{x} \vert $ than $ x\,tanh(x/\mu) $ and $ \sqrt{x^2 + \mu} $ with respect to accuracy. To accomplish our goal we also provide sharp hyperbolic bounds for error function.

Essential oils (EO) are a great antimicrobial resource against bacterial resistance in public health. Math models are useful describing the growth, survival, and inactivation of microorganisms against antimicrobials. We evaluated the antimicrobial activity of M. armillaris EO obtained from plants placed in the province of Buenos Aires (Argentina) against Staphylococcus aureus. Minimum Inhibitory and Bactericidal Concentrations were close and decreased slightly acidifying the medium from pH 7.4 to 6.5 and 5.0. This result was also evidenced by applying a sigmoid model, where the time and EO concentration necessaries to achieve 50% of the maximum effect decreased when medium was acidified. Moreover, at pH 7.4, applying the Gompertz model, we found that subinhibitory concentrations of EO decreased the growth rate and the maximum population density, and increased the latency period respect to the control. Additionally, we established physicochemical parameters for quality control and standardization of M. armillaris EO. Mathematical modelling allowed us to estimate key parameters in the behavior of S. aureus and Melaleuca armillaris EO at different pHs. This is interesting in situations where the pH changes are relevant, such as the control of intracellular infections in public health or the development of preservatives for food industry.

Two of the most common activation functions (AF) in deep neural networks (DNN) training are Sigmoid and ReLU. Sigmoid was tend to be more popular the previous decades, but it was suffering with the common vanishing gradient problems. ReLU has resolved these problems by using zero gradient and not tiny values for negative weights and the value “1” for all positives. Although it significant resolves the vanishing of the gradients, it poses new issues with dying neurons of the zero values. Recent approaches for improvements are in a similar direction by just proposing variations of the AF, such as Leaky ReLU (LReLU), while maintaining the solution within the same unresolved gradient problems. In this paper, the combining of the Sigmoid and ReLU in one single function is proposed, as a way to take the advantages of the two. The experimental results demonstrate that by using the ReLU’s gradient solution on positive weights, and Sigmoid’s gradient solution on negatives, has a significant improvement on performance of training Neural Networks on image classification of diseases such as COVID-19, text and tabular data classification tasks on five different datasets.

Log collection and storage is a crucial process for enterprises around the globe. Log analysis helps identify potential security breaches and, in some cases, is required by law for compliance. However, enterprises often delegate these responsibilities to third-party Cloud Service Providers (CSPs), where the logs are collected and processed for anomaly detection and stored in a data warehouse for archiving. Prevalent schemes rely on plain (unencrypted) data for anomaly detection. More often, these logs can reveal sensitive information about an organization or the customers of that organization. Hence, it is best to keep it encrypted at all times. This paper presents "SigML++," an extension of work done in "SigML." We utilize Fully Homomorphic Encryption (FHE) with the Cheon-Kim-Kim-Song (CKKS) scheme for supervised log anomaly detection on encrypted data. We use an Artificial Neural Network (ANN) based probabilistic polynomial approximations using a Perceptron with linear activation. We probabilistically approximate the Sigmoid activation function (σ(x)) in the encrypted domain for the intervals [−10,10] and [−50,50]. Experiments show better approximations for Logistic Regression (LR) and Support Vector Machine (SVM) for low-order polynomials.