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The first passage time refers to the time required for a dynamical system to reach a target energy level for the first time, starting from a known initial state. Analytical studies of single-degree-of-freedom systems governed by the linear Mathieu equation and subjected to broadband forced and parametric excitations have revealed the existence of different regimes for the first passage time. This Master thesis aims at the experimental validation of the existence of these regimes for a real structure consisting in a strip pre-stressed by a mass. The complete process, from the structure design to the experimental validation, is conducted in this work.
A finite element model of the structure is built in Matlab and updated with various state-of-the-art techniques from the field of experimental modal analysis. A model reduction of the full multi-degree-of-freedom system is introduced to match the conditions of the analytical results. It is shown that the dynamics of the structure can be approached by a single-degree-of-freedom reduced model only when both the forced and parametric excitations are narrow-band processes. The influence of narrow-band excitations on the first passage time is therefore studied numerically. The results of this numerical preparatory study are used to define the conditions of the experimental tests. First passage time maps are reproduced experimentally in the framework of the linear single-degree-of-freedom Mathieu equation.
This work provides the first physical evidences that the first passage time of real multi-degree-of-freedom systems can be characterized with the physical properties of the structure. It also addresses for the first time the influence of narrow-band excitations. Therefore, it opens the way to broadening the scope of the first passage time theory beyond the context of one-degree-of-freedom linear systems subjected to broadband excitations considered so far. Le temps de premier passage est défini comme le temps nécessaire pour qu'un système atteigne un niveau d'énergie donné pour la première fois, partant d'un niveau initial donné. Des études analytiques d'un oscillateur à un degré de liberté dont la dynamique est gouvernée par l'équation de Mathieu linéaire et soumis à des excitations paramétriques et forcées en bande large ont révélé l'existence de différents régimes de comportement pour le temps de premier passage. Ce travail de fin d'études a comme objectif de valider expérimentalement l'existence de ces différents régimes pour une structure réelle qui consiste une bande verticale précontrainte par une masse. Le processus complet, de la conception de la structure aux validations expérimentales, est détaillé dans ce travail.
Un modèle éléments finis de la structure est construit dans Matlab et corrigé avec des techniques d'analyse modale variées. Le modèle du système à plusieurs degrés de liberté est ensuite réduit pour correspondre aux conditions des résultats analytiques. La dynamique de la structure peut être décrite par un modèle réduit à un degré de liberté si les excitations forcées et paramétriques sont définies en bande étroite. L'influence des excitations en bande étroite sur les temps de premier passage est ensuite étudiée numériquement. Les résultats de ces études sont ensuite exploités pour définir les conditions expérimentales des tests. Les cartes de temps de premier passage sont reproduites expérimentalement dans le contexte de l'équation de Mathieu linéaire et à un degré de liberté. 
Ce travail fournit les premières preuves physiques que le temps de premier passage d'un système réel à plusieurs degrés de liberté peut être caractérisé par les propriétés physiques de la structure. Il s'intéresse aussi pour la première fois à l'influence des excitations en bandes étroites. Ce travail incite ainsi à élargir la portée de la théorie des temps de premier passage au-delà des systèmes linéaires à un degré de liberté soumis à des excitations en bande large considérés jusqu'à présent.

structural dynamics --- first passage time --- stochastic Mathieu equation --- dynamique des structures --- temps de premier passage --- équation de Mathieu stochastique --- Ingénierie, informatique & technologie > Ingénierie mécanique --- Ingénierie, informatique & technologie > Ingénierie aérospatiale --- Ingénierie, informatique & technologie > Multidisciplinaire, généralités & autres

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Stochastic processes have wide relevance in mathematics both for theoretical aspects and for their numerous real-world applications in various domains. They represent a very active research field which is attracting the growing interest of scientists from a range of disciplines.This Special Issue aims to present a collection of current contributions concerning various topics related to stochastic processes and their applications. In particular, the focus here is on applications of stochastic processes as models of dynamic phenomena in research areas certain to be of interest, such as economics, statistical physics, queuing theory, biology, theoretical neurobiology, and reliability theory. Various contributions dealing with theoretical issues on stochastic processes are also included.

arithmetic progressions --- weighted quadratic variation --- fractional differential-difference equations --- small deviations --- periodic intensity functions --- realized volatility --- rate of convergence --- host-parasite interaction --- first Chebyshev function --- regularly varying functions --- Cohen and Grossberg neural networks --- mixture of Gaussian laws --- diffusion model --- transition densities --- re-service --- Strang–Marchuk splitting approach --- random delays --- nematode infection --- first-passage-time --- total variation distance --- forecast combinations --- products of primes --- discrete time stochastic model --- multiplicative noises --- slowly varying functions --- growth curves --- stochastic process --- loan interest rate regulation --- birth-death process --- non-Markovian queue --- catastrophes --- exogenous factors --- seasonal environment --- repairs --- proportional hazard rates --- structural breaks --- transient probabilities --- first passage time (FPT) --- bounds --- double-ended queues --- mixed Gaussian process --- stochastic order --- time between inspections --- busy period --- diffusion --- continuous-time Markov chains --- general bulk service --- time-non-homogeneous birth-death processes --- stand-by server --- reliability --- sensor networks --- random impulses --- scale family of distributions --- maximum likelihood estimation --- multi-state network --- totally positive of order 2 --- lognormal diffusion process --- fractional birth-death processes --- exact asymptotics --- stochastic orders --- time-non-homogeneous jump-diffusion processes --- asymptotic distribution --- inverse first-passage problem --- nonhomogeneous Poisson process --- two-dimensional signature --- multiple vacation --- first-passage time --- mean square stability --- fractional queues --- differential entropy --- random parameter matrices --- Wasserstein distance --- breakdown and repair --- fusion estimation

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Symmetry and complexity are the focus of a selection of outstanding papers, ranging from pure Mathematics and Physics to Computer Science and Engineering applications. This collection is based around fundamental problems arising from different fields, but all of them have the same task, i.e. breaking the complexity by the symmetry. In particular, in this Issue, there is an interesting paper dealing with circular multilevel systems in the frequency domain, where the analysis in the frequency domain gives a simple view of the system. Searching for symmetry in fractional oscillators or the analysis of symmetrical nanotubes are also some important contributions to this Special Issue. More papers, dealing with intelligent prognostics of degradation trajectories for rotating machinery in engineering applications or the analysis of Laplacian spectra for categorical product networks, show how this subject is interdisciplinary, i.e. ranging from theory to applications. In particular, the papers by Lee, based on the dynamics of trapped solitary waves for special differential equations, demonstrate how theory can help us to handle a practical problem. In this collection of papers, although encompassing various different fields, particular attention has been paid to the common task wherein the complexity is being broken by the search for symmetry.

History of engineering & technology --- fractional differential equations --- fractional oscillations (vibrations) --- fractional dynamical systems --- nonlinear dynamical systems --- harmonic wavelet --- filtering --- multilevel system --- forced Korteweg-de Vries equation --- trapped solitary wave solutions --- numerical stability --- two bumps or holes --- finite difference method --- Laplacian spectra --- categorical product --- Kirchhoff index --- global mean-first passage time --- spanning tree --- degradation trajectories prognostic --- asymmetric penalty sparse decomposition (APSD) --- rolling bearings --- wavelet neural network (WNN) --- recursive least squares (RLS) --- health indicators --- first multiple Zagreb index --- second multiple Zagreb index, hyper-Zagreb index --- Zagreb polynomials --- Nanotubes --- fractional differential equations --- fractional oscillations (vibrations) --- fractional dynamical systems --- nonlinear dynamical systems --- harmonic wavelet --- filtering --- multilevel system --- forced Korteweg-de Vries equation --- trapped solitary wave solutions --- numerical stability --- two bumps or holes --- finite difference method --- Laplacian spectra --- categorical product --- Kirchhoff index --- global mean-first passage time --- spanning tree --- degradation trajectories prognostic --- asymmetric penalty sparse decomposition (APSD) --- rolling bearings --- wavelet neural network (WNN) --- recursive least squares (RLS) --- health indicators --- first multiple Zagreb index --- second multiple Zagreb index, hyper-Zagreb index --- Zagreb polynomials --- Nanotubes

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This book covers a broad range of research results in the field of Markov and Semi-Markov chains, processes, systems and related emerging fields. The authors of the included research papers are well-known researchers in their field. The book presents the state-of-the-art and ideas for further research for theorists in the fields. Nonetheless, it also provides straightforwardly applicable results for diverse areas of practitioners.

Research & information: general --- Mathematics & science --- Monte Carlo --- MCMC --- Markov chains --- computational statistics --- bayesian inference --- Non-Homogeneous Markov Systems --- Markov Set Systems --- limiting set --- tail expectation --- asymptotic bound --- quasi-asymptotic independence --- heavy-tailed distribution --- dominated variation --- copula --- branching process --- migration --- continuous time --- generating function --- period-life --- reliability --- redundant systems --- preventive maintenance --- multiple vacations --- process mining --- process modelling --- phase-type models --- process target compliance --- particle filter --- missing data --- single imputation --- impoverishment --- Markov Systems --- open population Markov chain models --- Semi-Markov processes --- controllable Markov jump processes --- compound Poisson processes --- diffusion limits --- stochastic control problem with incomplete information --- novel queuing models in applications --- semi-Markov model --- Markov model --- hybrid semi-Markov model --- manpower planning --- semi-Markov modeling --- occupancy --- first passage time --- duration --- non-homogeneity --- DNA sequences --- state space model --- Kalman filter --- constrained optimization --- two-sided components --- basketball --- Markov chain --- second order --- off-ball screens --- performance --- semi-Markov --- transient analysis --- asymptotic analysis --- Monte Carlo --- MCMC --- Markov chains --- computational statistics --- bayesian inference --- Non-Homogeneous Markov Systems --- Markov Set Systems --- limiting set --- tail expectation --- asymptotic bound --- quasi-asymptotic independence --- heavy-tailed distribution --- dominated variation --- copula --- branching process --- migration --- continuous time --- generating function --- period-life --- reliability --- redundant systems --- preventive maintenance --- multiple vacations --- process mining --- process modelling --- phase-type models --- process target compliance --- particle filter --- missing data --- single imputation --- impoverishment --- Markov Systems --- open population Markov chain models --- Semi-Markov processes --- controllable Markov jump processes --- compound Poisson processes --- diffusion limits --- stochastic control problem with incomplete information --- novel queuing models in applications --- semi-Markov model --- Markov model --- hybrid semi-Markov model --- manpower planning --- semi-Markov modeling --- occupancy --- first passage time --- duration --- non-homogeneity --- DNA sequences --- state space model --- Kalman filter --- constrained optimization --- two-sided components --- basketball --- Markov chain --- second order --- off-ball screens --- performance --- semi-Markov --- transient analysis --- asymptotic analysis

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Most papers published in this Special Issue of Mathematics are written by the participants of the XXXVI International Seminar on Stability Problems for Stochastic Models, 21­25 June, 2021, Petrozavodsk, Russia. The scope of the seminar embraces the following topics: Limit theorems and stability problems; Asymptotic theory of stochastic processes; Stable distributions and processes; Asymptotic statistics; Discrete probability models; Characterization of probability distributions; Insurance and financial mathematics; Applied statistics; Queueing theory; and other fields. This Special Issue contains 12 papers by specialists who represent 6 countries: Belarus, France, Hungary, India, Italy, and Russia.

inhomogeneous continuous-time Markov chain --- weak ergodicity --- rate of convergence --- sharp bounds --- differential inequalities --- forward Kolmogorov system --- prefetching --- optimization --- Markov decision processes --- random trees --- Galton–Watson --- capacitance --- dirichlet boundary value problem --- monte carlo method --- unbiased estimator --- von-neumann-ulam scheme --- network evolution --- random graph --- multi-type branching process --- continuous-time branching process --- 2- and 3-interactions --- Malthusian parameter --- Poisson process --- life-length --- extinction --- queuing system --- elastic traffic --- inpatient claim --- non-stationary intensity --- convergence analysis --- bounds on the rate of convergence --- wireless network --- file transfer --- daily traffic profile --- blocking probability --- continuous-time ehrenfest model --- first-passage time densities --- proportional intensity functions --- asymptotic behaviors --- multi-server queueing model --- rating --- self-sufficient servers --- self-checkout --- assistants --- multi-dimensional Markov chains --- retrial queue --- negative customers --- resource heterogeneous queue --- asymptotic analysis --- discrete time functional filter --- optimal unbiased estimation --- steady state --- equilibrium arrivals --- one-server queueing system --- orbit --- retrials --- limit theorem --- sum of independent random variables --- random sum --- asymptotic expansion --- asymptotic deficiency --- kurtosis --- parameter estimation --- gamma-exponential distribution --- mixed distributions --- generalized gamma distribution --- generalized beta distribution --- method of moments --- cumulants --- asymptotic normality