Differentially flat under-actuated robots are characterized by more degrees of freedom (DOF) than actuators: this makes possible the design of lightweight cheap robots with high dexterity. The main issue of such robots is the control of the passive joint, which requires accurate dynamic modeling of the robot.

Friction is usually discarded to simplify the models, especially in the case of low-speed trajectories. However, this simplification leads to oscillations of the end-effector about the final position, which are incompatible with fast and accurate motions.

This paper focuses on planar $n$-DOF serial robotic arms with $n-1$ actuated rotational joints plus one final passive rotational joint with stiffness and friction properties. These robots, if properly balanced, are differentially flat. When the non-actuated joint can be considered frictionless, differentially flat robots can be controlled in open loop, calculating the motor torques demanded by point-to-point motions. This paper extends the open-loop control to robots with a passive joint with viscous friction adopting a Laplace transform method. This method can be adopted by exploiting the particular structure of the equations of motion of differentially flat under-actuated robots in which the last equations are linear. Analytical expressions of the motor torques are obtained. The work is enriched by an experimental validation of a $2$-DOF under-actuated robot and by numerical simulations of the $2$- and $4$-DOF robots showing the suppression of unwanted oscillations.

Let ${\mathrm{d}} X(t) = -Y(t) \, {\mathrm{d}} t$, where Y(t) is a one-dimensional diffusion process, and let $\tau(x,y)$ be the first time the process (X(t), Y(t)), starting from (x, y), leaves a subset of the first quadrant. The problem of computing the probability $p(x,y)\,:\!=\, \mathbb{P}[X(\tau(x,y))=0]$ is considered. The Laplace transform of the function p(x, y) is obtained in important particular cases, and it is shown that the transform can at least be inverted numerically. Explicit expressions for the Laplace transform of $\mathbb{E}[\tau(x,y)]$ and of the moment-generating function of $\tau(x,y)$ can also be derived.

We obtain series expansions of the q-scale functions of arbitrary spectrally negative Lévy processes, including processes with infinite jump activity, and use these to derive various new examples of explicit q-scale functions. Moreover, we study smoothness properties of the q-scale functions of spectrally negative Lévy processes with infinite jump activity. This complements previous results of Chan et al. (Prob. Theory Relat. Fields 150, 2011) for spectrally negative Lévy processes with Gaussian component or bounded variation.

Hardy kernels are a useful tool to define integral operators on Hilbertian spaces like $L^2(\mathbb R^+)$ or $H^2(\mathbb C^+)$ . These kernels entail an algebraic $L^1$ -structure which is used in this work to study the range spaces of those operators as reproducing kernel Hilbert spaces. We obtain their reproducing kernels, which in the $L^2(\mathbb R^+)$ case turn out to be Hardy kernels as well. In the $H^2(\mathbb C^+)$ scenario, the reproducing kernels are given by holomorphic extensions of Hardy kernels. Other results presented here are theorems of Paley–Wiener type, and a connection with one-sided Hilbert transforms.

Multi-compartment models described by systems of linear ordinary differential equations are considered. Catenary models are a particular class where the compartments are arranged in a chain. A unified methodology based on the Laplace transform is utilised to solve direct and inverse problems for multi-compartment models. Explicit formulas for the parameters in a catenary model are obtained in terms of the roots of elementary symmetric polynomials. A method to estimate parameters for a general multi-compartment model is also provided. Results of numerical simulations are presented to illustrate the effectiveness of the approach.

In this paper, we develop the lower–upper-bound approximation in the space of Laplace transforms for pricing American options. We construct tight lower and upper bounds for the price of a finite-maturity American option when the underlying stock is modeled by a large class of stochastic processes, e.g. a time-homogeneous diffusion process and a jump diffusion process. The novelty of the method is to first take the Laplace transform of the price of the corresponding “capped (barrier) option” with respect to the time to maturity, and then carry out optimization procedures in the Laplace space. Finally, we numerically invert the Laplace transforms to obtain the lower bound of the price of the American option and further utilize the early exercise premium representation in the Laplace space to obtain the upper bound. Numerical examples are conducted to compare the method with a variety of existing methods in the literature as benchmark to demonstrate the accuracy and efficiency.

The longest gap $L(t)$ up to time $t$ in a homogeneous Poisson process is the maximal time subinterval between epochs of arrival times up to time $t$; it has applications in the theory of reliability. We study the Laplace transform asymptotics for $L(t)$ as $t\rightarrow \infty$ and derive two natural and different large-deviation principles for $L(t)$ with two distinct rate functions and speeds.

The distribution of the sum of dependent risks is a crucial aspect in actuarial sciences, risk management and in many branches of applied probability. In this paper, we obtain analytic expressions for the probability density function (pdf) and the cumulative distribution function (cdf) of aggregated risks, modelled according to a mixture of exponential distributions. We first review the properties of the multivariate mixture of exponential distributions, to then obtain the analytical formulation for the pdf and the cdf for the aggregated distribution. We study in detail some specific families with Pareto (Sarabia et al., 2016), gamma, Weibull and inverse Gaussian mixture of exponentials (Whitmore and Lee, 1991) claims. We also discuss briefly the computation of risk measures, formulas for the ruin probability (Albrecher et al., 2011) and the collective risk model. An extension of the basic model based on mixtures of gamma distributions is proposed, which is one of the suggested directions for future research.

Taylor's law (TL) originated as an empirical pattern in ecology. In many sets of samples of population density, the variance of each sample was approximately proportional to a power of the mean of that sample. In a family of nonnegative random variables, TL asserts that the population variance is proportional to a power of the population mean. TL, sometimes called fluctuation scaling, holds widely in physics, ecology, finance, demography, epidemiology, and other sciences, and characterizes many classical probability distributions and stochastic processes such as branching processes and birth-and-death processes. We demonstrate analytically for the first time that a version of TL holds for a class of distributions with infinite mean. These distributions, a subset of stable laws, and the associated TL differ qualitatively from those of light-tailed distributions. Our results employ and contribute to the methodology of Albrecher and Teugels (2006) and Albrecher et al. (2010). This work opens a new domain of investigation for generalizations of TL.

In this paper, we propose an estimator of the survival probability for a Lévy risk model observed at low frequency. The estimator is constructed via a regularised version of the inverse of the Laplace transform. The convergence rate of the estimator in a sense of the integrated squared error is studied for large sample size. Simulation studies are also given to show the finite sample performance of our estimator.

In this work, we introduce a mathematical model for the theory of generalized thermoelasticity with fractional heat conduction equation. The presented model will be applied to an infinitely long hollow cylinder whose inner surface is traction free and subjected to a thermal and mechanical shocks, while the external surface is traction free and subjected to a constant heat flux. Some theories of thermoelasticity can extracted as limited cases from our model. Laplace transform methods are utilized to solve the problem and the inverse of the Laplace transform is done numerically using the Fourier expansion techniques. The results for the temperature, the thermal stresses and the displacement components are illustrated graphically for various values of fractional order parameter. Moreover, some particular cases of interest have also been discussed.

The longest stretch L(n) of consecutive heads in n independent and identically distributed coin tosses is seen from the prism of large deviations. We first establish precise asymptotics for the moment generating function of L(n) and then show that there are precisely two large deviation principles, one concerning the behavior of the distribution of L(n) near its nominal value log1∕p n and one away from it. We discuss applications to inference and to logarithmic asymptotics of functionals of L(n).

This paper presents an analysis of unsteady flow of incompressible fractional Maxwell fluid filled in the annular region between two infinite coaxial circular cylinders. The fluid motion is created by the inner cylinder that applies a longitudinal time-dependent shear stress and the outer cylinder that is moving at a constant velocity. The velocity field and shear stress are determined using the Laplace and finite Hankel transforms. Obtained solutions are presented in terms of the generalized G and R functions. We also obtain the solutions for ordinary Maxwell fluid and Newtonian fluid as special cases of generalized solutions. The influence of different parameters on the velocity field and shear stress are also presented using graphical illustration. Finally, a comparison is drawn between motions of fractional Maxwell fluid, ordinary Maxwell fluid and Newtonian fluid.

Inspired by the works of Landriault et al. (2011), (2014), we study the Gerber–Shiu distribution at Parisian ruin with exponential implementation delays for a spectrally negative Lévy insurance risk process. To be more specific, we study the so-called Gerber–Shiu distribution for a ruin model where at each time the surplus process goes negative, an independent exponential clock is started. If the clock rings before the surplus becomes positive again then the insurance company is ruined. Our methodology uses excursion theory for spectrally negative Lévy processes and relies on the theory of so-called scale functions. In particular, we extend the recent results of Landriault et al. (2011), (2014).