The dynamical systems of trigonometric functions are explored, with a focus on sz=sin(z) and the fractal image created by iterating the Newton map, Fs(z), of s(z). The basins of attraction created from iterating Fs(z) are analyzed, and some bounds are determined for the primary basins of attraction. We further prove x and y-axis symmetry of the Newton map as well as some interesting results on periodic points on the real axis.

Minimizing all aspects of COVID-19 exposure is a high priority as universities prepare to reopen. One of those aspects includes developing protocols for interior spaces such as academic buildings. This paper applies mathematical modeling to investigate different virus exposure levels due to traffic patterns within academic buildings. The assumption used are: 1) Risk of infection is a product of exposure rate and time and 2) the exposure rate decreases with distance. One-way vs. two-way pedestrian traffic scenarios within hallways were modeled and analyzed for various configurations. The underlying assumption that a small exposure to a large number of people is similar to a large exposure to a few people is the driver to minimize exposures levels in all aspects. The analysis indicates that minimizing the time spent in passing between classes is the driving factor in minimizing risk, and one-way traffic may increase the time required to pass between classes. While the case presented is limited, the modeled approaches are intended to provoke future research that can be extended and applied to larger populations to help provide decision makers with more rigorous tools to shape future policies regarding traffic flow within buildings.

When multiple blasts occur at different times, the situation arises in which a blast wave is propagating into a medium that has already been shocked. Determining the evolution in the shape of the second shock is not trivial, as it is propagating into air that is not only non-uniform, but also non-stationary. To accomplish this task, we employ the method of Kompaneets to determine the shape of a shock in a non-uniform media. We also draw from the work of Korycansky (Astrophys J 398:184–189. https://doi.org/10.1086/171847 , 1992) on an off-center explosion in a medium with radially varying density. Extending this to treat non-stationary flow, and making use of approximations to the Sedov solution for the point blast problem, we are able to determine an analytic expression for the evolving shape of the second shock. In particular, we consider the case of a shock in air at standard ambient temperature and pressure, with the second shock occurring shortly after the original blast wave reaches it, as in a sympathetic detonation.

In this study, a mathematical analysis of a wind turbine dynamics is presented. The model represented by control blocks and transfer functions taken from recognized papers and studies, is translated to a system of nonlinear differential algebraic equations. For easier computational and numerical study, we prove the existence of a unique terminal voltage solution, which eliminates the algebraic constraint. Our study provides rigorous proofs of boundedness, existence, and uniqueness for the initial value problem of the system, allowing for the assurance that convergent numerical solutions converge to a unique solution for a given initial condition. This allows scholars to have a free simulator that will aid in dynamical studies of wind turbines without the need for software and Simulink limitations. A safe region within grid parameter space (R and X) is defined, in which existence and uniqueness are guaranteed. We presented time scale analysis and simulations to show that the system can be studied in smaller sizes. Lastly, we introduce cases of two and three time scales.

In this paper, wind turbines dynamics are considered for nonlinear behavioral modeling and simulation. The modeling part is concerned about the wind turbines exposed to lower range of wind speeds. The nonlinear model considered in this paper is derived from the models published recently. The model then is analyzed through stability, eigenvalues, sensitivity and Simulink verification versus General Electric and NREL models. The paper then introduces analysis and simulations for the wind turbines dynamics approximated to fast–slow (two) time scales and fast–medium–slow (three) time scales. The multiple time scale simulation analysis and results we present are a continuation for our previous work (Eisa et al. in Int J Dyn Control 2017.  https://doi.org/10.1007/s40435-017-0356-0) that concluded rigorous mathematical analysis for wind turbines dynamics. The paper presents full numerical simulation results for the time scale work. Finally, the paper presents a practical illustration by comparing the modeling work versus other models and real measured data from a wind farm.

In this paper, a model of type-3 Doubly Fed Asynchronous/Induction Generator (DFAG/DFIG) wind turbine is considered. We construct a system of 10 nonlinear differential equations that describes the transitional stage of dynamics when the rated speed is achieved and the rated power is not. A computation of the steady state vs. wind speed, stability in grid parameter space (resistance and reactance), and sensitivity of state variables and eigenvalues to parameters are presented in the paper. The paper recorded some sensitivity of eigenvalues to the reactance X. The paper presents simulations for a transitioning wind speed profile and a class of solutions with different wind speeds that start close to the equilibrium. These simulations are supporting the paper's conclusion.

In this paper, the model of a type-3 DFAG/DFIG wind turbine generator is considered at higher wind speeds with the pitch control activated. The main blocks of the model taken from the literature are described and translated into a system of differential equations with algebraic constraints. Eigenvalues and steady states of the model are presented with the main focus being the sensitivity analysis of state variables and eigenvalues to the parameters, and how the parameters affect the steady states. Results for sensitivities to a single parameter and two parameters are provided along with a simulation of the dynamics for time-varying wind speed.

In this paper, a type-3 Doubly Fed Asynchronous/ Induction Generator (DFAG/DFIG) wind turbine is considered for a dynamical study. The main blocks of the model taken from the literature are described and translated into a system of differential equations. By proving the possibility of eliminating the algebraic constraint, we are provided with good numerical possibilities to study wind turbine's dynamics. The paper provides a time domain analysis to emphasize and analyze the effect of adding a Q Drop function to the reactive power control dynamics. Our results, supported by simulations, suggest the important impact of the Q Drop function on the integrators blocks. The paper also provides an investigation of the system's attraction limits versus the control limits proposed by General Electric and others. The results, supported by simulations, questions the current models' validity, at least the proposed control limits.

This paper provides a theoretical stability analysis of gradual wetting fronts based on perturbation analysis. A traveling wave solution of the one‐dimensional vertical flow Richards' equation is used as the basic flow on which three‐dimensional perturbations are introduced. By locally linearizing the diffusivity form of the three‐dimensional Richards' equation a linear partial differential equation is obtained which governs the perturbation variables. The stability of each point at the wetting front is considered in a local coordinate system. The analysis of this perturbation equation at these points of the wetting front provides not only the relationship between the finger sizes and the nonponding infiltration rates at the soil surface but also the traveling speeds of the fingers rooted from these points. Once a perturbation is introduced at some point on the wetting front, there are three possibilities for the development of the perturbation. (1) The perturbation will monotonically decline with time; in this case, no fingers will form and the system is stable. (2) The perturbation does not decline with time, but its downward velocity is less than that of the stable basic wetting front; thus the distribution layer will gradually cover the fingers and the system will become stable. (3) The perturbation will increase with time and have a downward velocity greater than that of the stable wetting front; in this case, the finger will persistently grow in front of the stable wetting front and the system will become unstable. This analysis can be applied to an unsaturated homogeneous soil profile with uniform initial water content for the prediction of instability and for the estimation of finger characteristics over a wide range of infiltration rates.

Low-energy cosmic-ray neutrons play an important role in the production of the cosmogenic nuclides 36Cl and 41Ca. Previous approaches to modeling the distribution of low-energy neutrons beneath the surface of the earth have derived the thermal neutrons directly from the high-energy neutron flux. We have improved on this model by deriving the thermal neutrons from the moderation of the epithermal neutron flux, and the epithermal neutrons from the fast neutron flux. Predictions from the improved model agree well with experimental measurements of thermal and epithermal neutron fluxes both above and below the land/atmosphere interface. Recalibration of the 36Cl surface production parameters of Phillips et al. [Phillips, F.M., Zreda, M.G., Flinsch, M.R., Elmore, D., Sharma, P., 1996. A reevaluation of cosmogenic 36Cl production rates in terrestrial rocks. Geophys. Res. Lett., 23, pp. 949–952), incorporating the new approach to simulating the low-energy neutron fluxes, yielded the following values: Ps,Ca 66.8 atoms (g Ca)−1 year−1, Ps,K 137 atoms (g K)−1 year−1, and Pf(0) 626 neutrons (g air)−1 year−1 (this updated calibration also includes mugenic 36Cl production, based on independent work). Comparison of ages of three groups of samples from sites not included in the calibration data set with independently determined ages gave an average absolute error of 6.6% for all three data sets and coefficients of variation among the samples in the groups ranging from 5% to 14%.

Cosmogenic nuclides produced in situ within minerals at the surface of the Earth are proving to be an effective means of assessing geomorphic histories. The use of multiple cosmogenic nuclides permits both exposure times and erosion rates to be determined. However, if two nuclides are produced only by spallation reactions, the systematic differences in their accumulation rates depend only on the differences in their production rates and half‐lives. The relatively small differences that result require a high degree of analytical precision to yield useful results. In contrast to other spallogenic nuclides, 36Cl is also produced by low‐energy neutron absorption, which creates a different pattern of production as a function of depth. We have measured the thermal flux with depth in a concrete block using 3He‐filled neutron detectors. The measured thermal neutron profile agrees well with predictions from a simple diffusion‐ based thermal neutron distribution model. Calculations of 36Cl production using the model suggest that the use of 36Cl along with a purely spallogenic nuclide to determine erosion rates and exposure times should be less sensitive to analytical error than are determinations from two purely spallogenic nuclides.

The Laplace transform with respect to time, t, is normally used in finding analytical solutions for transient groundwater problems. The behavior of a function at large or small t is known to correspond to that of its Laplace transform counterpart at small or large p, respectively; p is the Laplace transform parameter of t. This condition is generally translated as t being inversely related to p and vice versa. By this relationship many asymptotic solutions for large or small t have been determined from the Laplace domain solutions valid only for small or large p. However, an example is given here which shows this kind of asymptotic calculation may fail to yield correct asymptotic solutions. Hence, the asymptotic calculation must be exercised with care. To deal with this possible failure, the Tauberian theorem is offered to evaluate the asymptotic behavior of functions from their Laplace transform counterparts.

A basal cell adenoma of parotid, eccrine dermal cylindromas and trichoepitheliomas occurring in the same patient were examined by light and electron microscopy and histochemistry. The eccrine and parotid adenomas were similar both structurally and histochemically except for the presence of Langerhans cells in the cutaneous adenoma and well differentiated mucinous cells in the parotid tumor. The three different hamartomas found in this individual may represent the effect of a single pleiotropic gene acting on ontogenetically related stem cells.