Parameter estimation of diffusion models from discrete observations

A short review of diffusion parameter estimations methods from discrete observations is presented. The applicability of a new estimation method on inferences about a diffusion growth model is discussed

Parameter estimation of diffusion models

Parameter estimation problems of diffusion models are discussed. The problems of maximum likelihood estimation and model selections from continuous observations are illustrated through diffusion growth model which generalizes some classical ones

Ruin probabilities and decompositions for general perturbed risk processes

We study a general perturbed risk process with cumulative claims modelled by a subordinator with finite expectation, with the perturbation being a spectrally negative Levy process with zero expectation. We derive a Pollaczek-Hinchin type formula for the survival probability of that risk process, and give an interpretation of the formula based on the decomposition of the dual risk process at modified ladder epochs

Health of young athletes: aspects of sports specialization

Relevance. In modern conditions of development of sports science the process of studying the health of an athlete as a set of physiological, psychological and morphological parameters in dynamically changing conditions of extreme activity is based on the understanding that an appropriate level of health is a necessary and obligatory basis for the reliability of the athlete. The objective of the study is to analyze and systematize the current scientific and methodological knowledge and results of practical experience of domestic and foreign researchers on the health of young athletes. Results. With the proper preparation of the training process, all the adaptation processes that occur in the body of the athlete under the influence of physical activity of different nature, are the basis for achieving sports results, and training and competitive activities create the conditions for the systematic improvement of the reserve capacity of the organism. In cases where the practice of training departs from the classical principles of the theory of periodization, the body of the athlete initially develops a state of increased mobility of adaptation reserves, followed by the breakdown of the compensatory-adaptive mechanisms, which is manifested by the change in the properties of the reactivity and resistance of the organism to the action of harmful agents. Conclusions. Systematization of theoretical prerequisites, which indicate a clear increase in negative tendencies in the state of somatic health of young athletes, with insufficient study of their medical-epidemiological and ontogenetic aspects, in combination with the inability of methodological conditions to realize continuous process of physical function somatic systems and the states that precede them

A New Algorithm for Global Minimization Based on the Combination of Adaptive Random Search and Simplex Algorithm of Nelder and Mead

We propose a new general-purpose algorithm for locating global minima of differentiable and nondifferentiable multivariable functions. The algorithm is based on combination of the adaptive random search approach and the Nelder-Mead simplex minimization. We show that the new hybrid algorithm satisfies the conditions of the theorem for convergence (in probability) to global minimum. By using test functions we demonstrate that the proposed algorithm is far more efficient than the pure adaptive random search algorithm, Some of the considered test functions are related to membership set estimation method for model parameter determination which was successfully applied to kinetic problems in chemistry and biology

Period function of planar turning points

This paper is devoted to the study of the period function of planar generic and non-generic turning points. In the generic case (resp. non-generic) a non-degenerate (resp. degenerate) center disappears in the limit e → 0, where e ≥ 0 is the singular perturbation parameter. We show that, for each e > 0 and e ∼ 0, the period function is monotonously increasing (resp. has exactly one minimum). The result is valid in an e-uniform neighborhood of the turning points. We also solve a part of the conjecture about a uniform upper bound for the number of critical periods inside classical Liénard systems of fixed degree, formulated by De Maesschalck and Dumortier in 2007. We use singular perturbation theory and the family blow-up

Abelian integrals and non-generic turning points

In this paper we initiate the study of the Chebyshev property of Abelian integrals generated by a non-generic turning point in planar slow-fast systems. Such Abelian integrals generalize the Abelian integrals produced by a slow-fast Hopf point (or generic turning point), introduced in Dumortier et al. (Discrete Contin Dyn Syst Ser S 2(4):723-781, 2009), and play an important role in studying the number of limit cycles born from the non-generic turning point

Slow divergence integrals in generalized Liénard equations near centers

Using techniques from singular perturbations we show that for any $n\ge 6$ and $m\ge 2$ there are Liénard equations $\{\dot{x}=y-F(x),\ \dot{y}=G(x)\}$, with $F$ a polynomial of degree $n$ and $G$ a polynomial of degree $m$, having at least $2[\frac{n-2}{2}]+[\frac{m}{2}]$ hyperbolic limit cycles, where $[\cdot]$ denotes "the greatest integer equal or below"

The slow divergence integral and torus knots

The goal of this paper is to study global dynamics of $C^\infty$-smooth slow-fast systems on the $2$-torus of class $C^\infty$ using geometric singular perturbation theory and the notion of slow divergence integral. Given any $m\in\mathbb{N}$ and two relatively prime integers $k$ and $l$, we show that there exists a slow-fast system $Y_{\epsilon}$ on the $2$-torus that has a $2m$-link of type $(k,l)$, i.e. a (disjoint finite) union of $2m$ slow-fast limit cycles each of $(k,l)$-torus knot type, for all small $\epsilon>0$. The $(k,l)$-torus knot turns around the $2$-torus $k$ times meridionally and $l$ times longitudinally. There are exactly $m$ repelling canard limit cycles and $m$ attracting non-canard limit cycles