Search Results (166)

An optical lens focuses light and a similar device can be developed to focus surface water waves. A detailed description of such hydrodynamic lenses is given, for which the focusing is induced by shaping the bathymetry of the bottom. Classically, the Luneburg lens uses a specific radial variation of the refractive index. The modified Luneburg lens (MLL) introduces an extra degree of freedom, permitting the focal point to be tuned. It is shown how to design the MLL for water waves, and then its performance is evaluated. Compared with a simple parabolic-shaped mount, the MLL is shown to be free of spherical aberration, resulting in a focus with larger intensity and smaller size of the focal point. Moreover, the focusing properties can be tuned and enhanced thanks to the possibility of changing the position of the focal point. The focusing quality of the MLL is described in all water-depth regimes (covering dispersive and non-dispersive waves) and the focusing of linear and nonlinear waves is revealed experimentally. The option of moving the focal point outside the lens, where the water depth is constant, may be useful when locating devices for harvesting wave energy. Full article

We study the incompressible fractional viscous–resistive magnetohydrodynamic system on ${\mathbb{R}}^n$ with fractional diffusion ${(-\Delta )}^{\alpha }$, where $\alpha \in (1/2,1]$, and with positive viscosity and resistivity coefficients $\mu ,\nu >0$. The problem is treated at the scale-invariant regularity $s_c={\displaystyle \frac{n}{p}}+1-2\alpha .$ For small divergence-free initial data in the critical Triebel–Lizorkin–Lorentz space ${\dot{F}}_{p,r}^{s_c,q}$, we construct a unique global mild solution. The main contribution is the use of the single-norm time–frequency space $m_m^{\prime }{\dot{F}}_{p,r}^{s_c,q}$, built on Meyer wavelets and the parabolic gauge $t{2}^{2\alpha j}$. This space keeps the critical spatial size, the short-time behavior, and the high-frequency decay in one norm. By using a Gevrey-weighted Duhamel formulation, we prove boundedness of the corresponding fractional heat propagators and establish the bilinear paraproduct estimate required for the fixed-point argument. Consequently, $e^{{\left(t{(-\Delta )}^{\alpha }\right)}^{\gamma }}(u,b)\in {\left(m_m^{\prime }{\dot{F}}_{p,r}^{s_c,q}\right)}^{2n}$ for some $\gamma >0$ depending on the parameters. This gives a Gevrey-type spatial smoothing effect, which is stronger than ordinary analyticity in the adopted scale. The restriction $\alpha >{\displaystyle \frac{1}{2}}$ enters through the factor $2^{j(1-2\alpha )}$, which supplies the high-frequency gain needed to close the critical bilinear estimates; in this sense it is sharp for the present method. The classical viscous–resistive case is recovered when $\alpha =1$. Full article

In this study, we examine a length preserving geodesic curvature difference flow for smooth strictly horocyclically convex simple closed curves in the hyperbolic plane ${\mathbb{H}}^2$. Given an initial curve ${\gamma }_1$ and a target curve ${\gamma }_2$ of the same hyperbolic length, we evolve ${\gamma }_1$ by a normal speed given by the difference of the reciprocals of geodesic curvatures evaluated at points with the same outward unit normal, together with a time-dependent scalar term $\mathrm{\Gamma }\left(t\right)$ chosen to preserve the hyperbolic length. Using Leichtweiβ’s hyperbolic support function and Howe’s curvature formula, the flow is reformulated as a quasilinear uniformly parabolic equation on ${\mathbb{S}}^1$ with a nonlocal term $\mathrm{\Gamma }\left(t\right)$. We prove short-time existence, uniqueness, and preservation of strict horocyclic convexity. Linearizing the support function equation at the target support function yields a uniformly elliptic operator whose kernel contains the infinitesimal isometry directions. Under a spectral gap assumption on a normalized slice transverse to the isometry orbit, we prove global existence and exponential convergence for initial data sufficiently close to the target curve. In the last section, this assumption is verified explicitly when the target curve is a geodesic circle. Full article

We study one-dimensional diffusions reflected at a boundary and analyze their pathwise “episodes” away from the boundary through Itô’s excursion theory. Under a fixed height cap of $a>0$, each excursion is equipped with three natural marks: its lifetime $\zeta$, its maximum $M$, and an additive (area-type) functional $A_f={\int }_0^{\zeta }f\left(e_t\right)dt$. Our main object is the height-truncated Itô-excursion Laplace exponent ${\mathsf{\Psi }}_{\alpha ,\lambda ;af}:=n\left(1-e^{-\alpha \zeta -\lambda A_f}; M<a\right)$ which jointly characterizes episode duration and cumulative load while excluding barrier-crossing spikes. We establish a general boundary–flux representation: ${\mathsf{\Psi }}_{\alpha ,\lambda ;af}$ is obtained as a boundary flux (in scale) of the unique solution to a one-dimensional killed Feynman–Kac boundary-value problem on $(0, a)$. This transfer principle yields a unified and tractable route to explicit computation. We implement it in three solvable families—the reflected arithmetic Brownian motion, reflected Ornstein–Uhlenbeck diffusions, and squared Bessel/Bessel-type diffusions—obtaining closed forms in terms of Airy, parabolic-cylinder, and confluent hypergeometric/Whittaker functions. Using the Poisson point process structure of excursions indexed by local time, we derive explicit extreme-burst laws (maxima and order statistics) for the additive marks up to a local-time horizon, and connect tail intensities to Laplace exponents via numerical Laplace inversion. Finally, we identify the strictly truncated cumulative load in local time as a (typically infinite-activity) subordinator whose Lévy measure coincides with the excursion-mark intensity, linking cumulative-load and extreme-burst statistics through the same exponent. Full article

Recent advances in design, manufacturing and development techniques have been very relevant to making solar collectors feasible for production in a variety of applications. In the field of concentrated solar thermal technologies, several techniques have been developed to achieve high levels of radiation concentration. The generation of concave curvature geometry through the polishing of the reflective surface or through specialized machining is one of the most common methods. However, the way in which these bends are obtained can vary significantly, depending on the required quality of optical concentration for the application. This study presents a simple parametric technique to achieve a parabolic curvature for solar concentration applications. To do this, a controlled bending deformation was applied to a metal hollow profile beam supported by a pin and roller at each of the ends, and only two symmetric point loads were applied to generate a bending moment to induce a bending of a curved shape. It was found that, for a given load configuration, a parabolic geometry was generated along a partial center section of the beam. The analysis carried out showed that under the load configuration analyzed, up to 66% of the beam length adopted a fully parabolic geometry. The technique proposed in this work allows for the creation of parabolas with variable focal distances, offering versatility in the design of solar concentrating systems. It also allows corrective adjustments to be made during the assembly of the complete solar concentrator system. Full article

The dynamics of two point masses interacting in a gravitational field has been the object of several scientific works. However, the complete explicit solution of the two-body problem is, to the best of our knowledge, not always available in the scientific literature. In this work, we describe the dynamics of a two-body system with that of an equivalent single-body with a reduced mass. Then, we solve the specific problems for elliptical, circular and parabolic trajectories, starting from different initial conditions. Through detailed analytical calculations, we write the Cartesian equations of the trajectories and the equations of motion both in the reference system of the centre of mass and in the original reference system. The proposed methodology is a simple but rigorous way to analyse the two-body dynamics under gravitational interactions, and can be applied also to more complex cases, such as the motion in a perturbed Newtonian potential and/or precession problems. The treatment presented in this work is particularly suitable to undergraduate students. Full article

The magnetic field generation studies in astronomy lead to a number of interesting problems in mathematical physics. In the dynamo theory, the problem is reduced to a system of parabolic equations for the field components. Assuming that the field grows exponentially, we obtain an eigenvalue problem for the corresponding elliptic operator. The possibility of the field generation and behaviour of the system is characterized by the spectra of the operator. If all eigenvalues lie in the left half of the complex plane, the perturbations will decay. On the other hand, if some of the eigenvalues have positive real parts, the large-scale structures of the field can be generated. From the astrophysical point of view, galactic magnetic fields are very important to study. One of the main problems is connected with the peripheral regions, where the properties of the medium complicate the operator structure. We can use the perturbation theory to find the eigenvalues. However, the problem can be solved analytically by considering some specific approximations. We can find the spectra using numerical approaches in the case of the conditions that are close to the real ones. In this paper, we solve eigenvalue problems for different operators which are connected with magnetohydrodynamic processes in outer parts of galaxies. Full article

This article explores whether jerk, the derivative of acceleration, should be limited in trajectory planning for position-controlled mechanical systems or in the controller. The excess jerk excites structural resonances and increases actuator wear, motivating the use of a limited jerk. However, we question the necessity of incorporating the jerk directly in trajectory planning by comparing third-order jerk-limited trajectories with second-order trajectories with reduced controller bandwidth that regulate torque gradients. We demonstrate by a typical practical application that reducing controller bandwidth can achieve comparable or superior jerk reduction without extending overall motion time for point-to-point trajectories. As a result, second-order parabolic trajectory profiles simplify on-line implementation. This investigation relies on a detailed sensitivity analysis of a one-dimensional model, incorporating crucial elements such as signal and sensor quantisation, sampling, and modes of structural resonances. The study shows that smooth trajectories reduce resonant vibrations and wear, but the jerk limitation may be addressed more effectively within the controller rather than within the trajectory generator. We conclude that although the limitation of the jerk in the trajectories is valuable, feedback controllers can reduce the jerk more effectively by bandwidth reduction, allowing simpler point-to-point trajectory designs without compromising performance. Full article

Based on existing models, this paper incorporates some key ecological factors, thereby obtaining a class of eco-epidemiological models that can more objectively reflect natural phenomena. This model simultaneously integrates disease dynamics within the prey population and the Beddington–DeAngelis functional response, thus achieving an organic combination of ecological dynamics, epidemic transmission, and spatial movement under time-varying environmental conditions. The proposed framework significantly enhances ecological realism by simultaneously accounting for spatial dispersal, predator–prey interactions, disease transmission within prey species, and seasonal or temporal variations, providing a comprehensive mathematical tool for analyzing complex eco-epidemiological systems. The theoretical results obtained from this study can be summarized as follows: Firstly, the existence and uniqueness of globally positive solutions for any positive initial data are rigorously established, ensuring the well-posedness and biological feasibility of the model over extended temporal scales. Secondly, analytically tractable sufficient conditions for uniform population persistence are derived, which elucidate the mechanisms of species coexistence and biodiversity preservation even under sustained epidemiological pressure. Thirdly, by employing innovative applications of differential inequalities and fixed point theory, the existence and uniqueness of a positive spatially homogeneous periodic solution in the presence of time-periodic coefficients are conclusively demonstrated, capturing essential rhythmicities inherent in natural systems. Fourthly, through a sophisticated combination of the upper and lower solution method for parabolic partial differential equations and Lyapunov stability theory, the global asymptotic stability of this periodic solution is rigorously established, offering a powerful analytical guarantee for long-term predictive modeling. Beyond theoretical contributions, these research findings provide actionable insights and quantitative analytical tools to tackle pressing ecological and public health challenges. They facilitate the prediction of thresholds for maintaining ecosystem stability using real-world data, enable the analysis and assessment of disease persistence in spatially structured environments, and offer robust theoretical support for the planning and design of wildlife management and conservation strategies. The derived criteria support evidence-based decision-making in areas such as controlling zoonotic disease outbreaks, maintaining ecosystem stability, and mitigating anthropogenic impacts on ecological communities. A representative numerical case study has been integrated into the analysis to verify all of the theoretical findings. In doing so, it effectively highlights the model’s substantial theoretical value in informing policy-making and advancing sustainable ecosystem management practices. Full article

Tumor angiogenesis models based on coupled nonlinear parabolic partial differential equations require solving stiff systems where explicit time-stepping methods impose severe stability constraints on the time step size. Implicit–Explicit (IMEX) schemes relax this constraint by treating diffusion terms implicitly and reaction–chemotaxis terms explicitly, reducing each time step to a single linear system solution. However, standard Gaussian elimination with partial pivoting exhibits cubic complexity in the number of spatial grid points, dominating computational cost for realistic discretizations in the range of 400–800 grid points. This work presents a CUDA-based parallel algorithm that accelerates the IMEX scheme through GPU implementation of three core computational kernels: pivot finding via atomic operations on double-precision floating-point values, row swapping with coalesced memory access patterns, and elimination updates using optimized two-dimensional thread grids. Performance measurements on an NVIDIA H100 GPU demonstrate speedup factors, achieving speedup factors from 3.5× to 113× across spatial discretizations spanning $M\in [25,800]$ grid points relative to sequential CPU execution, approaching 94.2% of the theoretical maximum speedup predicted by Amdahl’s law. Numerical validation confirms that GPU and CPU solutions agree to within twelve digits of precision over extended time integration, with conservation properties preserved to machine precision. Performance analysis reveals that the elimination kernel accounts for nearly 90% of total execution time, justifying the focus on GPU parallelization of this component. The method enables parameter studies requiring $\sim {10}^4$ PDE solves, previously computationally prohibitive, facilitating model-driven investigation of anti-angiogenic therapy design. Full article

In this paper, we study Cauchy problems for the semilinear parabolic equations ${\partial }_{t}u-▵u=G\left(u\right)$ with initial data in grand Herz spaces. We extend previous results established for classical Herz spaces to the broader framework of grand Herz spaces. The existence, uniqueness and stablity of solutions, as well as for their behaviour at small time are obtained by empolying heat kernel estimates, fixed-point theorems and some functional space theory. Full article

As coal-fired power plants shift from being primary power sources to operating as flexible peak-shaving units, the reheater—a critical component of the boiler’s ‘four tubes’—has attracted significant attention. This study focuses on the tube wall temperature distributions of the reheater at different loads and measurement points, analyzing factors that contribute to its uneven heat distribution. The results indicate that the heat distribution across the tubes of the low temperature reheater (LRH) is uneven. From the left to the right side of the tube panel, the tube wall temperatures form two parabolic profiles. The tubes most susceptible to overheating are the first tube of the 91st panel and the first tube of the 181st panel. For the high-temperature reheater (HRH), at an electrical load of 217.7 MW, the maximum temperature difference is higher than that of LRH. At all other electrical loads, however, the maximum temperature difference of the HRH is lower than that of the LRH. The LRH is at a higher risk of tube rupture caused by uneven heating compared to the HRH. Full article

This work examines the mechanical behaviour of 3D-printed stochastic lattice structures fabricated using a semi-controlled design. A primary goal is to predict and optimize the mechanical response of these Acrylic Styrene Acrylonitrile (ASA) filament structures when subjected to compressive stress. By transitioning from a purely stochastic method to a semi-controlled tessellation approach within Rhinoceros 7 software, we effectively generated the proposed design models. This methodology results in mechanical responses that are both predictable and reliable. The design parameters, including nodal formation, strut thickness, and lattice generation based on a predefined geometric routine, are associated with the regulation of the relative density. This approach aims to minimize the effect of relative density on the actual stiffness and strength evaluation. Our findings are cantered on the compressive testing of structures, which were generated using a Voronoi population distributed along a parabolic curve. We analyzed their mechanical response to the point of failure by examining stress–strain fluctuations. Three distinct behaviour stages are observed: elastic range, plastic range, and collapse without densification. The influence of crosslink geometry on the elastic responses was highlighted, with parabolic configurations affecting the peak stresses and elastic line slopes. The structures exhibited purely brittle behaviour, characterized by abrupt local cracking and oscillatory plateau formation in the plastic stage. Full article

The term SHIP (solar heat for industrial processes) or SHIPs (solar heat for industrial plants) refers to the use of collected solar radiation for meeting industrial heat demands, rather than for electricity generation. The global thermal capacity of SHIP systems at the end of 2024 stood slightly above 1 GWth, which is comparable to the electric power capacity of a single power station. Despite this relatively small presence, SHIP systems play an important role in rendering industrial processes sustainable. There are two aims in the current study. The first aim is to cover various types of SHIP systems based on the variety of their collector designs, operational temperatures, applications, radiation concentration options, and solar tracking options. SHIP designs can be as simple as unglazed solar collectors (USCs), having a stationary structure without any radiation concentration. On the other hand, SHIP designs can be as complicated as solar power towers (SPTs), having a two-axis solar tracking mechanism with point-focused concentration of the solar radiation. The second aim is to shed some light on the status of SHIP deployment globally, particularly in 2024. This includes a drop during the COVID-19 pandemic. The findings of the current study show that more than 1300 SHIP systems were commissioned worldwide by the end of 2024 (cumulative number), constituting a cumulative thermal capacity of 1071.4 MWth, with a total collector area of 1,531,600 m². In 2024 alone, 120.3 MWth of thermal capacity was introduced in 106 SHIP systems having a total collector area of 171,874 m². In 2024, 65.9% of the installed global thermal capacity of SHIP systems belonged to the parabolic trough collectors (PTCs), and another 22% of this installed global thermal capacity was attributed to the unevacuated flat plate collectors (FPC-Us). Considering the 106 SHIP systems installed in 2024, the average collector area per system was 1621.4 m²/project. However, this area largely depends on the SHIP category, where it is much higher for parabolic trough collectors (37,740.5 m²/project) but lower for flat plate collectors (805.2 m²/project), and it is lowest for unglazed solar collectors (163.0 m²/project). The study anticipates large deployment in SHIP systems (particularly the PTC type) in 2026 in alignment with gigascale solar-steam utilization in alumina production. Several recommendations are provided with regard to the SHIP sector. Full article

This work presents a control strategy designed to reduce the energy consumption of direct current motors by implementing smooth motion trajectories in a point-to-point control system, utilizing a fuzzy logic controller based on the Tsukamoto inference method. The proposed controller’s energy performance was experimentally compared to that of a conventional PID controller, considering three motion profiles: parabolic, trapezoidal, and S-curve. The results demonstrate that the combination of the fuzzy controller with smooth trajectories effectively reduces energy consumption without compromising motion accuracy. Under no-load conditions, average energy savings of 11.77% for the parabolic profile, 9.27% for the trapezoidal profile, and 3.45% for the S-curve profile were achieved. This improvement remained consistent even when a load was introduced to the system. To validate these findings, the coefficient of variation was calculated, revealing lower dispersion in the fuzzy controller’s results, indicating greater consistency in energy efficiency. Furthermore, Welch’s t-tests were conducted for each profile and load condition, with all p-values falling below the 0.05 significance threshold, confirming the statistical relevance of the observed differences. Full article