arXiv Math @arxiv_math@qoto.org

The outbreak of the New Coronavirus has significantly increased the vulnerability of medical staff. This paper addresses the safety and stress relief of medical personnel by proposing a solution to the scheduling problem of autonomous mobile robots (AMRs) in a stochastic environment. Considering the stochastic nature of travel and service times for AMRs affected by the surrounding environment, the routes of AMRs are planned to minimize the daily cost of the hospital (including the AMR fixed cost, penalty cost of violating the time window, and transportation cost). To efficiently generate high-quality solutions, we identify several properties and incorporate them into an improved Tabu Search (I-TS) algorithm for problem-solving. Experimental evaluations demonstrate that the I-TS algorithm outperforms existing methods by producing higher-quality solutions. By leveraging the characteristics of medical request environments, we intelligently allocate an appropriate number of AMRs to efficiently provide services, resulting in substantial cost reductions for hospitals and enhanced utilization of medical resources. These findings confirm the effectiveness of the proposed stochastic programming model in determining the optimal number of AMRs and their corresponding service routes across various environmental settings.

Equations of motion are derived for (visco)elastic, self-gravitating, and variably-rotating planets. The equations are written using a decomposition of the elastic motion that separates the body's elastic deformation from its net translational and rotational motion as far as possible. This separation is achieved by introducing degrees of freedom that represent the body's rigid motions; it is made precise by imposing constraints that are physically motivated and should be practically useful. In essence, a Tisserand frame is introduced exactly into the equations of solid mechanics. The necessary concepts are first introduced in the context of a solid body, motivated by symmetries and conservation laws, and the corresponding equations of motion are derived. Next, it is shown how those ideas and equations of motion can readily be extended to describe a layered fluid--solid body. A possibly new conservation law concerning inviscid fluids is then stated. Thereafter the equilibria and linearisation of the fluid--solid equations of motion are discussed, along with new equations for use within normal-mode coupling calculations and other Galerkin methods. Finally, the extension of these ideas to the description of multiple, interacting fluid--solid planets is qualitatively discussed.

Let $M$ be a commutative monoid. The monoid $M$ is called atomic if every non-invertible element of $M$ factors into atoms (i.e., irreducible elements), while $M$ is called a Furstenberg monoid if every non-invertible element of $M$ is divisible by an atom. Additive submonoids of $\mathbb{Q}$ consisting of nonnegative rationals are called Puiseux monoids, and their atomic structure has been actively studied during the past few years. The primary purpose of this paper is to investigate the property of being Furstenberg in the context of Puiseux monoids. In this direction, we consider some properties weaker than being Furstenberg, and then we connect these properties with some atomic results which have been already established for Puiseux monoids.

In this paper we develop a framework for estimating Probability of Default (PD) based on stochastic models governing an appropriate asset value processes. In particular, we build upon a Lévy-driven Ornstein-Uhlenbeck process and consider a generalized model that incorporates multiple latent variables affecting the evolution of the process. We obtain an Integral Equation (IE) formulation for the corresponding PD as a function of the initial position of the asset value process and the time until maturity, from which we then prove that the PD function satisfies an appropriate Partial Integro-Differential Equation (PIDE). These representations allow us to show that appropriate weak (viscosity) as well as strong solutions exist, and develop subsequent numerical schemes for the estimation of the PD function. Such a framework is necessary under the newly introduced International Financial Reporting Standards (IFRS) 9 regulation, which has imposed further requirements on the sophistication and rigor underlying credit modelling methodologies. We consider special cases of the generalized model that can be used for applications to credit risk modelling and provide examples specific to provisioning under IFRS 9, and more.

This paper presents a mathematical model to investigate co-infection with HIV/AIDS and zika virus (ZIKV) in Colombia and Brazil, where the first cases were reported in 2015-2016. The model considers the sexual transmission dynamics of both viruses and vector-host interactions. We begin by exploring the qualitative behaviour of each model separately. Then, we analyze the dynamics of the co-infection model using the thresholds and results defined separately for each model. The model also considers the impact of intervention strategies, such as, personal protection, antiretroviral therapy (ART), and sexual protection (condoms use). Using available parameter values for Colombia and Brazil, the model is calibrated to predict the potential effect of implementing combinations of those intervention strategies on the co-infection spread. According to these findings, transmission through sexual contact is a determining factor in the long-term behaviour of these two diseases. Furthermore, it is important to note that co-infection with HIV and ZIKV may result in higher rates of HIV transmission and an increased risk of severe congenital disabilities linked to ZIKV infection. As a result, control measures have been implemented to limit the number of infected individuals and mosquitoes, with the aim of halting disease transmission. This study provides novel insights into the dynamics of HIV/ZIKV co-infection and highlights the importance of integrated intervention strategies in controlling the spread of these viruses, which may impact public health

Tao and Vu showed that every centrally symmetric convex progression $C\subset\mathbb{Z}^d$ is contained in a generalised arithmetic progression of size $d^{O(d^2)} \# C$. Berg and Henk improved the size bound to $d^{O(d\log d)} \# C$. We obtain the bound $d^{O(d)} \# C$, which is sharp up to the implied constant, and is of the same form as the bound in the continuous setting given by John's Theorem.

We prove a quantitative convergence result of the nonlocal Allen--Cahn equation to volume-preserving mean curvature flow. The proof uses gradient flow calibrations and the relative entropy method, which has been used in the recent literature to prove weak-strong uniqueness results for mean curvature flow and convergence of the Allen--Cahn equation. A crucial difference in this work is a new notion of gradient flow calibrations. We add a tangential component to the velocity field in order to prove the Gronwall estimate for the relative energy. This allows us to derive the optimal convergence rate without having to show the closeness of the Lagrange-multipliers.

We prove the cohomological version of the Sarnak-Xue Density Hypothesis for $SO_5$ over a totally real field and for inner forms split at all finite places. The proof relies on recent lines of work in the Langlands program: (i) Arthur's Endoscopic Classification of Representations of classical groups, and (ii) the Generalized Ramanujan-Petersson Theorem, proved for cohomological self-dual cuspidal representations of general linear groups. We give applications for the growth of cohomology of arithmetic manifolds, density-Ramanujan complexes, cutoff phenomena and optimal strong approximation.

Causal inference is often complicated by post-treatment variables, which appear in many scientific problems, including noncompliance, truncation by death, mediation, and surrogate endpoint evaluation. Principal stratification is a strategy that adjusts for the potential values of the post-treatment variables, defined as the principal strata. It allows for characterizing treatment effect heterogeneity across principal strata and unveiling the mechanism of the treatment on the outcome related to post-treatment variables. However, the existing literature has primarily focused on binary post-treatment variables, leaving the case with continuous post-treatment variables largely unexplored, due to the complexity of infinitely many principal strata that challenge both the identification and estimation of causal effects. We fill this gap by providing nonparametric identification and semiparametric estimation theory for principal stratification with continuous post-treatment variables. We propose to use working models to approximate the underlying causal effect surfaces and derive the efficient influence functions of the corresponding model parameters. Based on the theory, we construct doubly robust estimators and implement them in an R package.

Let $G$ be a classical group defined over a local field $F$ of characteristic zero. For any irreducible admissible representation $π$ of $G(F)$, which is of Casselman-Wallach type if $F$ is archimedean, we extend the study of spectral decomposition of local descents in [JZ18] for special orthogonal groups over non-archimedean local fields to more general classical groups over any local field $F$. In particular, if $π$ has a generic local $L$-parameter, we introduce the spectral first occurrence index $\mathfrak{f}_{\mathfrak{s}}(π)$ and the arithmetic first occurrence index $\mathfrak{f}_{\mathfrak{a}}(π)$ of $π$ and prove in Theorem 1.4 that $\mathfrak{f}_{\mathfrak{s}}(π) = \mathfrak{f}_{\mathfrak{a}}(π)$. Based on the theory of consecutive descents of enhanced $L$-parameters developed in [JLZ22], we are able to show in Theorem 1.5 that the first descent spectrum consists of all discrete series representations, which determines explicitly the branching decomposition problem by means of the relevant arithmetic data and extends the main result ([JZ18, Theorem 1.7]) to the great generality.

In a prize winning expository article, V. Pozdnyakov and J.M. Steele gave a beautiful demonstration of the ramifications of a basic bijection for permutations. The aim of this note is to connect this correspondence to a seemingly unrelated problem concerning odd cycles and even cycles, arising in the combinatorial study of the Cayley continuants by E. Munarini and D. Torri. As extreme cases, one encounters two special classes of permutations of $2n$ elements with the same cardinality. We find an intermediate structure which leads to a linkage between these two antipodal circumstances. Our construction can be reckoned as a variant of the Pozdnyakov-Steele bijection that takes a certain parity condition into account. We conclude with a question pertaining to pairs of matchings.

Stein proved that for each simple plane triangulation H there exists a partitioning of the vertex of H into two subsets each of which induces a forest if and only if the dual H^{*} has a Hamilton cycle. We extend the Stein theorem for graphs in the family of all loopless 2-connected plane multigraphs and we prove some other equivalent results.

The viral propagation of fake posts on online social networks (OSNs) has become an alarming concern. The paper aims to design control mechanisms for fake post detection while negligibly affecting the propagation of real posts. Towards this, a warning mechanism based on crowd-signals was recently proposed, where all users actively declare the post as real or fake. In this paper, we consider a more realistic framework where users exhibit different adversarial or non-cooperative behaviour: (i) they can independently decide whether to provide their response, (ii) they can choose not to consider the warning signal while providing the response, and (iii) they can be real-coloring adversaries who deliberately declare any post as real. To analyze the post-propagation process in this complex system, we propose and study a new branching process, namely total-current population-dependent branching process with multiple death types. At first, we compare and show that the existing warning mechanism significantly under-performs in the presence of adversaries. Then, we design new mechanisms which remarkably perform better than the existing mechanism by cleverly eliminating the influence of the responses of the adversaries. Finally, we propose another enhanced mechanism which assumes minimal knowledge about the user-specific parameters. The theoretical results are validated using Monte-Carlo simulations.

We carefully bootstrap the crossing kernels of Virasoro conformal blocks from first principles. Our approach emphasizes the Hilbert space structure of the space of Virasoro conformal blocks which makes the consistency of crossing transformations obvious. We give a pedagogical explanation of the necessary background about Virasoro conformal blocks and special functions. We also explain several applications and prove new results about the crossing kernels.

Consider the unordered configuration spaces of manifolds. Knudsen, Miller and Tosteson proved that the extremal homology groups of configuration spaces of manifold are eventually quasi polynomials. In this paper, we give the precise degree of top non-trivial quasi polynomials. This shows that the upper bound of Knudsen, Miller and Tosteson for the degree of quasi polynomials is sharp for every manifold.

The aim of this paper is to formulate a {\it non--commutative geometrical} version of the Electromagnetic Theory with magnetic monopoles with complete symmetry between the electric field and the magnetic field in a non--commutative space--time. To accomplish this purpose, we will {\it dualize} the geometrical formulation of this theory by using quantum principal bundles and quantum principal connections.

Transportation networks play a critical part in the spread of infectious diseases between populations. In this work, we define a networked susceptible-exposed-infected-recovered epidemic process with loss of immunity over time (SEIRS) that explicitly models the flow of individuals between sub-populations, which serves as the propagating mechanism for infection. We provide sufficient conditions for local stability and instability of the healthy state of the system and show that no perturbation of population flows can change the local stability of any healthy state. We also provide sufficient conditions for the existence and uniqueness of an endemic state. We then develop tools and methods for applying our model to real-world data, including spreading parameter estimation and disease arrival time prediction, and apply them in a case study using both travel and infection data from counties in Minnesota during the first year of the COVID-19 pandemic.

By optimizing the predicted performance over a receding horizon, model predictive control (MPC) provides the ability to enforce state and control constraints. The present paper considers an extension of MPC for nonlinear systems that can be written in pseudo-linear form with state- and control-dependent coefficients. The main innovation is to apply quadratic programming iteratively over the horizon, where the predicted state trajectory is updated based on the updated control sequence. Output-feedback control is facilitated by using the block-observable canonical form for linear, time-varying dynamics. This control technique is illustrated on various numerical examples, including the Kapitza pendulum with slider-crank actuation, the nonholonomic integrator, the electromagnetically controlled oscillator, and the triple integrator with control-magnitude saturation.

Assuming some pointwise estimates on certain Weyl's sum, we prove the sharp estimates of the mean value associated to the following exponential sum $$ \sum_{n=1}^N e^{2πi tn^d +2πi xn}\,. $$

To any $V$ in the Grassmannian ${\rm Gr}_k({\mathbb R}^n)$ of $k$-dimensional vector subspaces in ${\mathbb R}^n$ one can associate the diagonal entries of the ($n\times n$) matrix corresponding to the orthogonal projection of ${\mathbb R}^n$ to $V$. One obtains a map ${\rm Gr}_k({\mathbb R}^n)\to {\mathbb R}^n$ (the Schur-Horn map). The main result of this paper is a criterion for pre-images of vectors in ${\mathbb R}^n$ to be connected. This will allow us to deduce connectivity criteria for a certain class of subspaces of the real Stiefel manifold which arise naturally in frame theory. We extend in this way results of Cahill, Mixon, and Strawn.