AppliedMath, Volume 3, Issue 3 (September 2023) – 10 articles

Let X be a smooth projective variety and $f:X\to {\mathbb{P}}^r$ a morphism birational onto its image. We define the Terracini loci of the map f. Most results are only for the case $dimX=1$. With this new and more flexible definition, it is possible to prove strong nonemptiness results with the full classification of all exceptional cases. We also consider Terracini loci with restricted support (solutions not intersecting a closed set $B⊊X$ or solutions containing a prescribed $p\in X$). Our definitions work both for the Zariski and the euclidean topology and we suggest extensions to the case of real varieties. We also define Terracini loci for joins of two or more subvarieties of the same projective space. The proofs use algebro-geometric tools. Full article

The Mexico City Metrobus is one of the most popular forms of public transportation inside the city, and since its opening in 2005, it has become a vital piece of infrastructure for the city; this is why the optimal functioning of the system is of key importance to Mexico City, as it plays a crucial role in moving millions of passengers every day. This paper presents a model to simulate Line 1 of the Mexico City Metrobus, which can be adapted to simulate other bus rapid transit (BRT) systems. We give a detailed description of the model development so that the reader can replicate our model. We developed various response variables in order to evaluate the system’s performance, which focused on passenger satisfaction and measured the maximum occupancy that a passenger experiences inside the buses, as well as the time that he spends in the queues at the stations. The results of the experiments show that it is possible to increase passenger satisfaction by considering different combinations of routes while maintaining the same fuel consumption. It was shown that, by considering an appropriate combination of routes, the average passenger satisfaction could surpass the satisfaction levels obtained by a 10% increase in total fuel consumption. Full article

In this article, some prescriptions to define a distribution on the set $Q_0$ of all rational numbers in $[0,1]$ are outlined. We explored a few properties of these distributions and the possibility of making these rational numbers asymptotically equiprobable in a suitable sense. In particular, it will be shown that in the said limit—albeit no absolutely continuous uniform distribution can be properly defined in ${\mathbb{Q}}_0$—the probability allotted to every single $q\in {\mathbb{Q}}_0$ asymptotically vanishes, while that of the subset of ${\mathbb{Q}}_0$ falling in an interval $[a,b]\subseteq {\mathbb{Q}}_0$ goes to $b-a$. We finally present some hints to complete sequencing without repeating the numbers in ${\mathbb{Q}}_0$ as a prerequisite to laying down more distributions on it. Full article

This paper is concerned with the existence of solutions of a class of Cauchy problems for hyperbolic partial fractional differential inclusions (HPFD) involving the Caputo fractional derivative with an impulse whose right hand side is convex and non-convex valued. Our results are achieved within the framework of the nonlinear alternative of Leray-Schauder type and contraction multivalued maps. A detailed example was provided to support the theorem. Full article

Artificial Intelligence (AI) systems are increasingly being deployed in decentralized environments where they interact with other AI systems and humans. In these environments, each participant may have different ways of expressing the same semantics, leading to challenges in communication and collaboration. To address these challenges, this paper presents a novel hybrid model for shared conceptualization in decentralized AI systems. This model integrates ontology, epistemology, and epistemic logic, providing a formal framework for representing and reasoning about shared conceptualization. It captures both the intensional and extensional components of the conceptualization structure and incorporates epistemic logic to capture knowledge and belief relationships between agents. The model’s unique contribution lies in its ability to handle different perspectives and beliefs, making it particularly suitable for decentralized environments. To demonstrate the model’s practical application and effectiveness, it is applied to a scenario in the healthcare sector. The results show that the model has the potential to improve AI system performance in a decentralized context by enabling efficient communication and collaboration among agents. This study fills a gap in the literature concerning the representation of shared conceptualization in decentralized environments and provides a foundation for future research in this area. Full article

When there is uncertainty in the value of parameters of the input random components of a stochastic simulation model, two-level nested simulation algorithms are used to estimate the expectation of performance variables of interest. In the outer level of the algorithm n observations are generated for the parameters, and in the inner level m observations of the simulation model are generated with the values of parameters fixed at the values generated in the outer level. In this article, we consider the case in which the observations at both levels of the algorithm are independent and show how the variance of the observations can be decomposed into the sum of a parametric variance and a stochastic variance. Next, we derive central limit theorems that allow us to compute asymptotic confidence intervals to assess the accuracy of the simulation-based estimators for the point forecast and the variance components. Under this framework, we derive analytical expressions for the point forecast and the variance components of a Bayesian model to forecast sporadic demand, and we use these expressions to illustrate the validity of our theoretical results by performing simulation experiments with this forecast model. We found that, given a fixed number of total observations $nm$, the choice of only one replication in the inner level ($m=1$) is recommended to obtain a more accurate estimator for the expectation of a performance variable. Full article

The simple assembly line balancing (SALB) problem is a significant challenge faced by industries across various sectors aiming to optimise production line efficiency and resource allocation. One important issue when the decision-maker balances a line is how to keep the cycle time under a given time across all cells, even though there is variability in some parameters. When there are stochastic elements, some approaches use constraint relaxation, intervals for the stochastic parameters, and fuzzy numbers. In this paper, a three-part algorithm is proposed that first solves the balancing problem without considering stochastic parameters; then, using simulation, it measures the effect of some parameters (in this case, the inter-arrival time, processing times, speed of the material handling system which is manually performed by the workers in the cell, and the number of workers who perform the tasks on the machines); finally, the add-on OptQuest in SIMIO solves an optimisation problem to constrain the cycle time using the stochastic parameters as decision variables. A Gearbox instance from literature is solved with 15 tasks and 14 precedence rules to test the proposed approach. The deterministic balancing problem is solved optimally using the open solver GLPK and the Pyomo programming language, and, with simulation, the proposed algorithm keeps the cycle time less than or equal to 70 s in the presence of variability and deterministic inter-arrival time. Meanwhile, with stochastic inter-arrival time, the maximum cell cycle is 72.04 s. The reader can download the source code and the simulation models from the GitHub page of the authors. Full article

We propose a Quantum Neural Network (QNN) for predicting stabilization parameter for solving Singularly Perturbed Partial Differential Equations (SPDE) using the Streamline Upwind Petrov Galerkin (SUPG) stabilization technique. SPDE-Q-Net, a QNN, is proposed for approximating an optimal value of the stabilization parameter for SUPG for 2-dimensional convection-diffusion problems. Our motivation for this work stems from the recent progress made in quantum computing and the striking similarities observed between neural networks and quantum circuits. Just like how weight parameters are adjusted in traditional neural networks, the parameters of the quantum circuit, specifically the qubits’ degrees of freedom, can be fine-tuned to learn a nonlinear function. The performance of SPDE-Q-Net is found to be at par with SPDE-Net, a traditional neural network-based technique for stabilization parameter prediction in terms of the numerical error in the solution. Also, SPDE-Q-Net is found to be faster than SPDE-Net, which projects the future benefits which can be earned from the speed-up capabilities of quantum computing. Full article

As a class of non-Newtonian fluids with yield stresses, Bingham fluids possess both solid and liquid phases separated by implicitly defined non-physical yield surfaces, which makes the standard numerical discretization challenging. The variational reformulation established by Duvaut and Lions, coupled with an augmented Lagrange method (ALM), brings about a finite element approach, whereas the inevitable local mesh refinement and preconditioning of the resulting large-scaled ill-conditioned linear system can be involved. Inspired by the mesh-free feature and architecture flexibility of physics-informed neural networks (PINNs), an ALM-PINN approach to steady-state Bingham fluid flow simulation, with dynamically adaptable weights, is developed and analyzed in this work. The PINN setting enables not only a pointwise ALM formulation but also the learning of families of (physical) parameter-dependent numerical solutions through one training process, and the incorporation of ALM into a PINN induces a more feasible loss function for deep learning. Numerical results obtained via the ALM-PINN training on one- and two-dimensional benchmark models are presented to validate the proposed scheme. The efficacy and limitations of the relevant loss formulation and optimization algorithms are also discussed to motivate some directions for future research. Full article

Time series of financial data are both frequent and important in everyday practice. Numerous applications are based, for example, on time series of asset prices or market indices. In this article, the application of fractal interpolation functions in modelling financial time series is examined. Our motivation stems from the fact that financial time series often present fluctuations or abrupt changes which the fractal interpolants can inherently model. The results indicate that the use of fractal interpolation in financial applications is promising. Full article