The overreaching goal of this paper is to investigate the existence and uniqueness of weak solution of a semilinear parabolic equation with double nonlocality in space and in time variables that naturally arises while modeling a biological nano-sensor in the chaotic dynamics of a polymer chain. In fact, the problem under consideration involves a symmetric integrodifferential operator of Lévy type and a term called the interaction potential, that depends on the time-integral of the solution over the entire interval of solving the problem. Owing to the Galerkin approximation, the existence and uniqueness of a weak solution of the nonlocal complement value problem is proven for small time under fair conditions on the interaction potential.

We conditionally extend formulas of(Dietz and Schenzle, J. Math. Biol. 22: 117-120,1995) for the ”transmission potential” of an immunizing infection with pre-infection possible before birth and vertical transmission admitted. We look for minimum proportion to be covered to reduce basic reproduction rate below 1 by acting through vaccination. We present also a new criterion allowing the selection of an immunizing vaccination strategy by bringing the reproduction number below 1. We find that reduce vertical transmission, adds chances to eradicate disease. Moreover reduce age of vaccination reduces the minimum vaccination coverage inducing global immunization against disease by bringing down the basic reproduction number.

In this article, we aim to identify the factors that significantly influence the choice of a scientific or technological field by a female student in the case of the University of Ngaoundéré. By applying a logistic regression model (Logit) to data from a stratified survey, our results highlight the significant negative influence of gender stereotypes and the use of a career counselor on the choice of a purely scientific field by girls. Additionally, it is observed that parental influence plays an important role in the choice of fields. Furthermore, women who have excelled in scientific domains have a considerable impact on the university orientation of young girls in the case of the University of Ngaoundéré. These findings thus underscore the importance of organizing awareness sessions to present to girls the diversity of opportunities that scientific fields offer and to help them overcome the specific obstacles they face, as well as to promote successful female role models.

Motivated by (Goyal and Murray in PLoS One 9(10):e110143, 2014) we consider a partially age-structured model simulating the dynamic of two infectious diseases vertically transmitted almost independently with horizontal coinfection and a common age-structured vaccination strategy. We study influence of parameters on existence and uniqueness of solutions and epidemiological equilibria. Impact of vertical transmission on basic reproduction rate is also presented.

Persistent homology provides stable shape summaries, but stability alone does not imply usefulness for recognition. We develop a selective geometry--topology framework that combines optimal transport with dimension-specific persistence-diagram distances and activates topology only when it is both reliable and discriminative. The underlying topology-aware discrepancy is a metric on a natural descriptor quotient; we establish perturbation and contamination bounds, finite termination of the associated $K$-medoids scheme, strong consistency of empirical medoids, and a margin certificate for stable assignments. A reliability factor compares between-object topological dispersion with bootstrap variation, while a discriminability factor compares between-class with within-class distances. Their product gates each homology dimension, and a validation-selected coefficient controls the remaining contribution. A finite alignment layer minimizes sliced transport over a declared rotation bank. Experiments deliberately include positive and negative controls. On synthetic shapes, the Rips hybrid attains clean ARI $0.997\pm0.005$, while DTM filtration raises ARI from $0.497\pm0.039$ to $0.707\pm0.037$ at 10\% outliers. On ModelNet10, indiscriminate topology degrades sliced-transport clustering, motivating abstention. On all ten optical-digit classes, discriminative gating improves clean 3-NN accuracy from $66.40\%$ for sliced transport and $60.07\%$ for ungated cubical fusion to $70.93\%$. Finally, a pre-specified confirmatory study with 20 new independent splits yields gains of $37.15$ percentage points over pixel k-NN under a two-pixel translation, $7.05$ points over ungated fusion at $30^\circ$, and $8.60$ points over pixel k-NN at $40^\circ$. All three comparisons win on 20/20 splits and remain significant after Holm correction. The evidence supports a conditional conclusion: topology is useful when stability, class relevance, and the nuisance model agree.