AppliedMath

Computing Shapley values for large cooperative games is an NP-hard problem. For practical applications, stochastic approximation via permutation sampling is widely used. In the context of machine learning applications of the Shapley value, the concept of antithetic sampling has become popular. The idea is to employ the reverse permutation of a sample in order to reduce variance and accelerate convergence of the algorithm. We study this approach for the Shapley and Banzhaf values, as well as for the Owen value which is a solution concept for games with precoalitions. We combine antithetic samples with established stratified sampling algorithms. Finally, we evaluate the performance of these algorithms on four different types of cooperative games. Full article

In this study, we tackle the subject of interval quadratic equations and we aim to accurately determine the root enclosures of quadratic equations, whose coefficients constitute interval variables. This study focuses on interval quadratic equations that contain only one coefficient considered as an interval variable. The four methods reviewed here in order to solve this problem are: (i) the method of classic interval analysis used by Elishakoff and Daphnis, (ii) the direct method based on minimizations and maximizations also used by the same authors, (iii) the method of quantifier elimination used by Ioakimidis, and (iv) the interval parametrization method suggested by Elishakoff and Miglis and again based on minimizations and maximizations. We will also compare the results yielded by all these methods by using the computer algebra system Mathematica for computer evaluations (including quantifier eliminations) in order to conclude which method would be the most efficient way to solve problems relevant to interval quadratic equations. Full article

We review some general aspects about the Black–Scholes equation, which is used for predicting the fair price of an option inside the stock market. Our analysis includes the symmetry properties of the equation and its solutions. We use the Hamiltonian formulation for this purpose. Taking into account that the volatility inside the Black–Scholes equation is a parameter, we then introduce the Merton–Garman equation, where the volatility is stochastic, and then it can be perceived as a field. We then show how the Black–Scholes equation and the Merton–Garman one are locally equivalent by imposing a gauge symmetry under changes in the prices over the Black–Scholes equation. This demonstrates that the stochastic volatility emerges naturally from symmetry arguments. Finally, we analyze the role of the volatility on the decisions taken by the holders of the options when they use the solution of the Black–Scholes equation as a tool for making investment decisions. Full article

In this paper, we provide a first systematic treatment of binomial sum relations involving (generalized) Fibonacci and Lucas numbers. The paper introduces various classes of relations involving (generalized) Fibonacci and Lucas numbers and different kinds of binomial coefficients. We also present some novel relations between sums with two and three binomial coefficients. In the course of exploration, we rediscover a few isolated results existing in the literature, commonly presented as problem proposals. Full article

This paper introduces the mathematical formalization of two probabilistic procedures for susceptible-infected-recovered (SIR) and susceptible-infected-susceptible (SIS) infectious diseases epidemic models, over Erdös-Rényi contact networks. In our approach, we consider the epidemic threshold, for both models, defined by the inverse of the spectral radius of the associated adjacency matrices, which expresses the network topology. The epidemic threshold dynamics are analyzed, depending on the global dynamics of the network structure. The main contribution of this work is the relationship established between the epidemic threshold and the topological entropy of the Erdös-Rényi contact networks. In addition, a relationship between the basic reproduction number and the topological entropy is also stated. The trigger of the infectious state is studied, where the probability value of the stability of the infected state after the first instant, depending on the degree of the node in the seed set, is proven. Some numerical studies are included and illustrate the implementation of the probabilistic procedures introduced, complementing the discussion on the choice of the seed set. Full article

For linear optimization problems with a parametric objective, so-called parametric linear programs (PLP), we show that the optimal decision values are, under few technical restrictions, unimodal functions of the parameter, at least in the two-degrees-of-freedom case. Assuming that the parameter is random and follows a known probability distribution, this allows for an efficient algorithm to determe the quantiles of linear combinations of the optimal decisions. The novel results are demonstrated with probabilistic economic dispatch. For an example setup with uncertain fuel costs, quantiles of the resulting inter-regional power flows are computed. The approach is compared against Monte Carlo and piecewise computation techniques, proving significantly reduced computation times for the novel procedure. This holds especially when the feasible set is complex and/or extreme quantiles are desired. This work is limited to problems with two effective degrees of freedom and a one-dimensional uncertainty. Future extensions to higher dimensions could yield a key tool for the analysis of probabilistic PLPs and, specifically, risk management in energy systems. Full article

According to the Food and Agriculture Organization, the world’s food production needs to increase by 70 percent by 2050 to feed the growing population. However, the EU agricultural workforce has declined by 35% over the last decade, and 54% of agriculture companies have cited a shortage of staff as their main challenge. These factors, among others, have led to an increased interest in advanced technologies in agriculture, such as IoT, sensors, robots, unmanned aerial vehicles (UAVs), digitalization, and artificial intelligence (AI). Artificial intelligence and machine learning have proven valuable for many agriculture tasks, including problem detection, crop health monitoring, yield prediction, price forecasting, yield mapping, pesticide, and fertilizer usage optimization. In this scoping mini review, scientific achievements regarding the main directions of agricultural technologies will be explored. Successful commercial companies, both in the Russian and international markets, that have effectively applied these technologies will be highlighted. Additionally, a concise overview of various AI approaches will be presented, and our firsthand experience in this field will be shared. Full article

Two linear algebra problems implore a solution to them, creating the themes pursued in this paper. The first problem interfaces with graph theory via binary 0-1 adjacency matrices and their Laplacian counterparts. More contemporary spatial statistics/econometrics applications motivate the second problem, which embodies approximating the eigenvalues of massively large versions of these two aforementioned matrices. The proposed solutions outlined in this paper essentially are a reformulated multiple linear regression analysis for the first problem and a matrix inertia refinement adapted to existing work for the second problem. Full article

Based on the Hodgkin–Huxley theory, this paper establishes several nonlinear system models, analyzes the models’ stability, and studies the conditions for repetitive discharge of neuronal membrane potential. Our dynamic analysis showed that the main channel currents (the fast transient sodium current, the potassium delayed rectifier current, and the fixed leak current) of a neuron determine its dynamic properties and that the GHK formula will greatly widen the stimulation current range of the repetitive discharge condition compared with the Nernst equation. The model including the change in ion concentration will lead to spreading depression (SD)-like depolarization, and the inclusion of a Na-K pump will weaken the current stimulation effect by decreasing the extracellular K accumulation. The results indicate that the Hodgkin–Huxley model is suitable for describing the response to initial stimuli, but due to changes in ion concentration, it is not suitable for describing the response to long-term stimuli. Full article

In this study, we design an algorithm to work on gate-based quantum computers. Based on the algorithm, we construct a quantum circuit that represents the surplus process of a cedant under a reinsurance agreement. This circuit takes into account a variety of factors: initial reserve, insurance premium, reinsurance premium, and specific amounts related to claims, retention, and deductibles for two different non-proportional reinsurance contracts. Additionally, we demonstrate how to perturb the actuarial stochastic process using Hadamard gates to account for unpredictable damage. We conclude by presenting graphs and numerical results to validate our capital modelling approach. Full article

Computing the solution of the Caputo fractional differential equation plays an important role in using the order of the fractional derivative as a parameter to enhance the model. In this work, we developed a power series solution method to solve a linear Caputo fractional differential equation of the order $q,$$0<q<1,$ and this solution matches with the integer solution for $q=1$. In addition, we also developed a series solution method for a linear sequential Caputo fractional differential equation with constant coefficients of order $2q,$ which is sequential for order q with Caputo fractional initial conditions. The advantage of our method is that the fractional order q can be used as a parameter to enhance the mathematical model, compared with the integer model. The methods developed here, namely, the series solution method for solving Caputo fractional differential equations of constant coefficients, can be extended to Caputo sequential differential equation with variable coefficients, such as fractional Bessel’s equation with fractional initial conditions. Full article

Motivated by results on the location of the zeros of a complex polynomial with monotonicity conditions on the coefficients (such as the classical Eneström–Kakeya theorem and its recent generalizations), we impose similar conditions and give bounds on the number of zeros in certain regions. We do so by introducing a reversal in monotonicity conditions on the real and imaginary parts of the coefficients and also on their moduli. The conditions imposed are less restrictive than many of those in the current literature and hence apply to polynomials not covered by previous results. The results presented naturally apply to certain classes of lacunary polynomials. In particular, the results apply to certain polynomials with two gaps in their coefficients. Full article

Mathematical models have been of great importance in various fields, especially for understanding the dynamical behaviour of biosystems. Several models, based on classical ordinary differential equations, delay differential equations, and stochastic processes are commonly employed to gain insights into these systems. However, there is potential to extend such models further by combining the features from the classical approaches. This work investigates stochastic delay differential equations (SDDEs)-based models to understand the behaviour of biosystems. Numerical techniques for solving these models that demonstrate a more robust representation of real-life scenarios are presented. Additionally, quantitative roles of delay and noise to gain a deeper understanding of their influence on the system’s overall behaviour are analysed. Subsequently, numerical simulations that illustrate the model’s robustness are provided and the results suggest that SDDEs provide a more comprehensive representation of many biological systems, effectively accounting for the uncertainties that arise in real-life situations. Full article

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