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Different from the Mittag–Leffler stability or asymptotic stability, the exponential stability issue, which provides faster and explicit convergence rate, is studied in this paper for fractional‐order uncertain systems with asynchronous switching and impulses, where the impulsive functions rely on not only switching modes but also impulsive time. Instead of using the inequality Eα(t)≤1αexp(t1α)(0<α<1)$$ {E}_{\alpha }(t)\le \frac{1}{\alpha}\exp \left({t}^{\frac{1}{\alpha }}\right)\kern0.3em \left(0<\alpha <1\right) $$, by utilizing the theory of fractional‐order differential equations, the methods of Lyapunov function and mathematical induction, some novel and less‐conservative stability criteria are developed, respectively, for the case of switched stable subsystems or switched unstable subsystems. The obtained results build a tradeoff between impulsive function, impulsive interval, average dwell time and fractional order. In addition, our results with α=1$$ \alpha =1 $$ are also novel in contrast with the ones of integer‐order switched impulsive systems. Finally, five numerical examples are given to show the effectiveness of theoretical results. [ABSTRACT FROM AUTHOR]

In this work, a comparative study of four functional forms used to represent potential energy curves (PECs) is presented. The starting point is the Hulburt‐Hirschfelder, followed by the Extended Rydberg potential function, ending with two of the most recent potentials presented in the literature for diatomic systems: the Araújo‐Ballester potential and the Improved Extended Lennard‐Jones potential. The chosen potentials have in common the fact that all their parameters are algebraically calculated, without any fitting procedure, and all of them have direct dependence on Dunham's parameters. The mathematical behavior of these functions for the short‐ and long‐range regions is discussed. As study case, the diatomic system CO$$ \mathrm{CO} $$ in its ground electronic state was selected. To quantify the accuracy of the potential energy functions, the least‐squares Z‐test method, proposed by Murrell and Sorbie, is used. Furthermore, the main spectroscopic constants and vibrational energy levels are calculated and compared for all potentials. A comparative study on functions to represent the potential energy interactions of the CO diatomic system is presented. The Hulburt‐Hirschfelder, Extended Rydberg, Araújo‐Ballester and Improved Extended Lennard‐Jones potentials were chosen because they have in common the fact that all their parameters are calculated without any fitting procedure. To quantify the accuracy of potential energy functions, the least squares Z test method is used, and the main spectroscopic constants and vibrational energy levels are calculated. [ABSTRACT FROM AUTHOR]

Continental margins develop long submarine slopes, linking the shallow shelves along the continental landmasses to the deep abyssal plain. They are the results of a complex interaction between destructive and constructive processes, although by and large they are sites of deposition. There is a great amount of variation between the length, height, smoothness, gradient and variation thereof between the slope profiles; however, there is also recurring similarity in their shape. The similitude has suggested systematic relationships between the shape and the processes forming them, and led to studies on geomorphological categorisation based on curvatures. The potential for prediction of along-strike variations and connection between morphology and sedimentary process is herein approached through broadening the mathematical functions used, detailed measurement, observation and curve-fitting of over 150 passive continental margins. Previously, three functions have been used to categorise submarine slopes. The present study finds that four mathematical functions closely match the slopes: Linear, Gaussian, exponential and quadratic (positive and negative/inverse), and reveals that the fourth slope, the quadratic, is by far the most common. While exponential and quadratic slopes are similar there is a crucial difference in the way in which the angle of the slope changes. This study suggests that quadratic slopes represent systematically decreasing sediment deposition with distance, previously attributed to exponential slopes. Exponential slopes meanwhile, represent slope readjustment profiles with upper sediment bypass and lower slope aggradation. Linear slopes, which form the longest low-angle slopes, form in response to high sediment input. Abrupt shelf-edges form in shallower water and develop longer slope aprons, suggesting formation from erosional processes. This implies that the quintessential sigmoidal (s-shaped, Gaussian function) slope, with a smooth rollover, represents the fundamental depositional slope profile. [ABSTRACT FROM AUTHOR]

General‐purpose mechanisms can perform a broad range of tasks but are usually rather heavy and expensive. If only particular movements need to be executed, more efficient special‐purpose mechanisms can be employed. However, they typically require an expert to design the system based on manual inspection of simulations and experimental results. This procedure is not only time‐consuming, but the outcome also depends on the expert's experience. Hence, the design process stems from subjective criteria while only a limited number of structurally different mechanisms can be considered. In contrast, a design assistant can consider a broad range of mechanisms and leverage multi‐objective optimization to retrieve optimal designs for the given task. Due to the systems being synthesized based on mathematical functions rather than individual experience, the assistant allows a more transparent development of optimal problem‐specific mechanisms compared to the conventional process. Experts can then fine‐tune and analyze the proposed designs to compose the final system. In recent years, neural networks have been utilized to directly learn the inverse mapping from a trajectory to a mechanism design. This requires some parameterization of the trajectory to be fed into the network. In this work, we evaluate various preprocessing methods for the trajectory on a simple mechanism design model problem. We assess multiple configurations such as different neural network sizes, applying input‐output normalization, and varying the number of features. Consequently, we investigate and compare the trends and robustness of the implemented methods. [ABSTRACT FROM AUTHOR]

This article is the second part of a series dealing with the description and visualization of mathematical functions used to describe a powder diffraction pattern for teaching and education purposes. The first part dealt with the instrumental and sample contributions to the profile of a Bragg peak [Dinnebier & Scardi (2021). J. Appl. Cryst.54, 1811–1831]. The second part, here, deals with the mathematics and physics of the intensity in X‐ray powder diffraction. Scholarly scripts are again provided using the Wolfram language in Mathematica. [ABSTRACT FROM AUTHOR]

Nowadays, research on electromagnetic devices increasingly focuses on multiphysics three‐dimensional complex systems. However, this kind of simulations require huge computational resources and consequently very high CPU time. Ristagno et al. proposed a level‐set based method allowing meshing step savings in any iterative processes (as movement modeling or optimization processes). The level‐set method lies in implicit description of moving fronts as described in the study by Osher et al. (1988). Inspired by Räisänen et al., all physical parameters (material properties, supply or armature movement) are implemented by projection of mathematical functions. In this paper, the authors proposes to study and design a new kind of mechanical commutator for DC machines thanks to this coupled method. Indeed, there is a renewed interest in DC machines because they can be suitable for small mobilities. To be noted, small mobility concerns all kind of vehicles (2 wheels–4 wheels–autonomous vehicles such as drones) that have speeds below 50 km/h. However, specifications for traction applications are difficult to achieve for such a rustic machine and requires some modifications. That is why, inspired by usual synchronous machine, the commutator has been redesigned to obtain expected theoretically signal waveform. Furthermore, numerical coupled simulations have been used to design both machine and supply simultaneously. This work has led to the manufacture of a proof of concept and preliminary results are very promising. [ABSTRACT FROM AUTHOR]

We consider the highly oscillatory integral F(w):=∫−∞∞eiw(tK+2+eiθtp)g(t)dt$F(w):=\int _{-\infty }^\infty e^{iw(t^{K+2}+e^{i\theta }t^p)}g(t)dt$ for large positive values of w, −π<θ≤π$-\pi <\theta \le \pi$, K and p positive integers with 1≤p≤K$1\le p\le K$, and g(t)$g(t)$ an entire function. The standard saddle point method is complicated and we use here a simplified version of this method introduced by López et al. We derive an asymptotic approximation of this integral when w→+∞$w\rightarrow +\infty$ for general values of K and p in terms of elementary functions, and determine the Stokes lines. For p≠1$p\ne 1$, the asymptotic behavior of this integral may be classified in four different regions according to the even/odd character of the couple of parameters K and p; the special case p=1$p=1$ requires a separate analysis. As an important application, we consider the family of canonical catastrophe integrals ΨK(x1,x2,...,xK)$\Psi _K(x_1,x_2,\ldots ,x_K)$ for large values of one of its variables, say xp$x_p$, and bounded values of the remaining ones. This family of integrals may be written in the form F(w)$F(w)$ for appropriate values of the parameters w, θ and the function g(t)$g(t)$. Then, we derive an asymptotic approximation of the family of canonical catastrophe integrals for large |xp|$\vert x_p\vert$. The approximations are accompanied by several numerical experiments. The asymptotic formulas presented here fill up a gap in the NIST Handbook of Mathematical Functions by Olver et al. [ABSTRACT FROM AUTHOR]

Species' responses to broad‐scale environmental or spatial gradients are typically unimodal. Current models of species' responses along gradients tend to be overly simplistic (e.g., linear, quadratic or Gaussian GLMs), or are suitably flexible (e.g., splines, GAMs) but lack direct ecologically interpretable parameters. We describe a parametric framework for species‐environment non‐linear modelling ('senlm'). The framework has two components: (i) a non‐linear parametric mathematical function to model the mean species response along a gradient that allows asymmetry, flattening/peakedness or bimodality; and (ii) a statistical error distribution tailored for ecological data types, allowing intrinsic mean–variance relationships and zero‐inflation. We demonstrate the utility of this model framework, highlighting the flexibility of a range of possible mean functions and a broad range of potential error distributions, in analyses of fish species' abundances along a depth gradient, and how they change over time and at different latitudes. [ABSTRACT FROM AUTHOR]

Fluvial cross strata are depositional products of bedform migration that record formative flow and sediment transport conditions on planetary bodies. Bedform evolution varies with transport stage even under constant flow depths, but our understanding of how prevailing sediment transport conditions affect preserved cross strata is limited. Here, we analyzed experimental bedform evolution and preserved set thickness spanning threshold‐of‐motion to suspension‐dominated transport conditions at multiple equilibrium flow depths. Results show that bedform trough depth and mean preserved set thickness have a parabolic dependence on transport stage, with maximum values observed at intermediate transport stages. Our results indicate that transport stage is a key control on the flow‐depth‐normalized set thickness but set thickness is a poor indicator of flow depth. Thus, the dependence of bedform dimensions on transport stage should be considered in paleohydraulic reconstruction, and the analysis of set thickness may aid in the estimation of ancient fluvial sediment flux. Plain Language Summary: Alluvial rivers have ripples and dunes (i.e., bedforms) that migrate across the riverbed, resulting in characteristic signatures called cross strata found in river deposits on Earth and Mars. Cross strata encode information about the flow and sediment transport conditions under which bedforms evolved. For over a century, the relation between cross strata thickness and formative flow conditions has been studied. However, we do not understand how prevailing sediment transport conditions control set thickness distributions. We used high‐resolution topography of bedform evolution from controlled physical experiments where transport stage—a parameter that quantifies sediment transport intensity—was systematically changed across multiple flow depths. We discovered that preserved cross set thickness is primarily controlled by transport stage, and that the mean set thickness of preserved deposits is a poor indicator of flow depth. We develop a new mathematical function that describes the dependence of mean set thickness on the formative transport stage. Our work indicates that the transport‐stage dependence of set thickness should be a primary consideration in paleohydraulic reconstruction, and that analysis of cross strata may aid the estimation of paleo‐sediment‐transport conditions. Key Points: Mean bedform trough depths and set thickness show a parabolic dependence on transport stage peaking under mixed‐load transport conditionsMean set thickness is a poor indicator of flow depth, but depth‐normalized mean set thickness is strongly influenced by transport stageTransport stage exerts a key control on preserved fluvial strata and should be considered in paleo‐hydraulic reconstructions [ABSTRACT FROM AUTHOR]

To meet the challenges in resolving the complex morphologies of emergent nanoparticles, a program with a user‐friendly graphical user interface has been developed for calculating small‐angle scattering curves from custom shapes. The software allows STL‐format 3D models, models defined by mathematical functions or combinations of the two as initial input. As a transitional stage, lattice models are generated and the orientation‐averaged small‐angle scattering data can be calculated using typical spherical harmonics expansion. The validity of the protocol is verified by demonstration models with Protein Data Bank structures and known scattering functions. The software is applied to successfully calculate the scattering curves of a porous spherical shell model where traditional mathematical derivation fails. [ABSTRACT FROM AUTHOR]

The viscoelastic fluid exhibits significant elastic solid and viscous liquid responses which do not rise in Newtonian fluid because flow of viscoelastic fluid has the Markovian property. In order to describe the Markovian verses non‐Markovian properties, the dual definitions of fractional differentiations have been invoked on the Oldroyd‐B fluid model. The Oldroyd‐B fluid model is saturated by porous medium and magnet subject to no slip assumptions on the governing equations of flow. The fractional differential operators of Atangana–Baleanu and Caputo–Fabrizio are imposed on the governing equations then solved by Fourier sine and Laplace transforms. The solutions are expressed into compact form by means of elementary functions, infinite series, theorem of convolution and generalized special function so called M function. The velocity field obtained via two types of fractional approaches is depicted for disclosing the hidden impact of relaxation and retardation times on Markovian and non‐Markovian properties of viscoelastic fluid. Finally, our feasible analysis resulted that flow of viscoelastic fluid is dependent on its present state which leads to Markovian and non‐Markovian dynamics. [ABSTRACT FROM AUTHOR]

Saturated hydraulic conductivity of soil, Ksat, is a critical parameter for mathematical modelling of groundwater and soil water flow. Current empirical Ksat models, such as Kozeny–Carman model, Hazen models, or empirical approaches based on soil pedotranfer functions (PTFs) rely on the specific surface area, certain characteristic particle size(s), for example, d5, d10, d15, d50, etc., or the fractionation of sand, silt and clay. In this study, we proposed an alternative modelling framework to estimate Ksat from void ratio and arithmetic mean diameter of soil particles. We suggested to describe the particle size distribution (PSD) using a continuous mathematical distribution function, for example, the lognormal distribution function, and to obtain the Sauter mean diameter (dS) by calculating the moments of the PSD function. The proposed empirical equations were tested against the measured Ksat values for 54 sand samples with available measured PSD and void ratio as reported in Toumpanou et al. and validated using 49 sandy soil samples obtained from the Unsaturated Soil Database (UNSODA). We found that the dS was more relevant to Ksat than the number mean diameter (dN) and mass mean diameter (dM) (the RMSE was 0.486 for Ksat ‐ dN; 0.187 for Ksat ‐ dS; and 0.212 for Ksat ‐ dM, respectively). We also found that the proposed equation with dS had superior performance over the Hazen equation and PTF‐based models. Our results indicated that the Ksat calculated using dS, which was either estimated from discrete experimental data or calculated from a continuous lognormal PSD function, best matched with the measured values for coarse‐textured soils. Future studies should focus on developing empirical relationships for soil water retention curves and other soil properties based on continuous PSD functions. Highlights: A modelling framework to estimate Ksat from the void ratio and arithmetic mean diameter is proposed.Sauter mean diameter, dS, can be obtained by calculating the moments of the PSD function.Ksat model using dS performs better than that using dN, dM, d5, d10, or d50 or PTF‐based models. [ABSTRACT FROM AUTHOR]

A warming climate is associated with increasing hydroclimatic extremes, which are often interconnected through complex processes, prompting their concurrence and/or succession, and causing compound extreme events. It is critical to analyze the risks of compound events, given their disproportionately high adverse impacts. To account for the variability in two or more hydroclimatic variables (e.g., temperature and precipitation) and their dependence, a rising number of publications focuses on multivariate analysis, among which the notion of copula‐based probability distribution has attracted tremendous interest. Copula is a mathematical function that expresses the joint cumulative probability distribution of multiple variables. Our focus is to re‐emphasize the fundamental requirements and limitations of applying copulas. Confusion about these requirements may lead to misconceptions and pitfalls, which can potentially compromise the robustness of risk analyses for environmental processes and natural hazards. We conducted a systematic literature review of copulas, as a prominent tool in the arsenal of multivariate methods used for compound event analysis, and underpinned them with a hydroclimatic case study in Sweden to illustrate a practical approach to copula‐based modeling. Here, we (1) provide end‐users with a didactic overview of necessary requirements, statistical assumptions and consequential limitations of copulas, (2) synthesize common perceptions and practices, and (3) offer a user‐friendly decision support framework to employ copulas, thereby support researchers and practitioners in addressing hydroclimatic hazards, hence demystify what can be an area of confusion. This article is categorized under:Science of Water > Hydrological ProcessesScience of Water > Methods [ABSTRACT FROM AUTHOR]

This paper considers the locally and globally optimal solutions of the global optimisation problem whose objective function is a max‐plus vector‐valued function and constraint function is a real affine function. The formulas about global optimisation in the existing work are further explained and simplified. The necessary and sufficient conditions for the existence and uniqueness of locally optimal solutions of a single objective are established, which are then used to deduce these of globally optimal solutions. Furthermore, the general formulas of locally and globally optimal solutions are presented, respectively. The local optimisation is then applied to solve the load distribution problem of distributed systems with no globally optimal solution, and the optimal allocation scheme is proposed to complete the overall task at the earliest time. The proposed method is constructive, and the obtained results are illustrated by the numerical examples. [ABSTRACT FROM AUTHOR]

The small load has fuzziness on the strengthening and damage of the component under the action of random fatigue load. This paper introduces the two‐dimensional membership function in fuzzy mathematics to characterize the relationship between stress amplitude, mean value, and damage below the fatigue limit. An exponential function is used to describe the strengthening effect. Under the two‐dimensional load spectrum, the influence of various levels on the component's life is comprehensively considered, and the fatigue life prediction model under random load is proposed. Through the small sample experiment of the car swing support rod, the relevant parameters of the model are obtained. The difference between the modified two‐dimensional and the two‐dimensional load spectrum is compared in fatigue life prediction results. The rules of membership function selection and parameter setting are summarized, and it makes the estimation of component load spectrum life more reasonable. [ABSTRACT FROM AUTHOR]

The particle swarm optimization (PSO) and the whale optimization algorithm (WOA) are two admired optimization methods that have drawn various researchers' attention. The PSO implements some particles' intelligent movements in a search space, and the WOA is originated based on the hunting mechanism of humpback whales. The PSO and WOA have different strategies for moving towards the optimum solution. Nevertheless, both algorithms' performances encounter several problems, such as premature convergence and falling in local optimums. Several approaches have been proposed to enhance meta‐heuristic algorithms' performance, such as applying the chaotic maps, adding mathematical or stochastic operators or local searches, and hybridizing the algorithms. In this article, a new hybrid algorithm denoted as chaotic‐based hybrid whale and PSO has been presented by improving the WOA, combining it with PSO, and using the chaotic maps. The hybrid algorithm has significantly more diverse movements than both of the mentioned algorithms. Therefore, it explores different regions of a problem's search space more precisely and avoids local optima. The roulette wheel selection operator has also been applied based on their fitness value to select the proposed algorithm's search agents and exploit promising regions of the search space. In the hybrid algorithm, the chaotic maps have been applied to initialize the whales' population, particles of the particle swarm, and adjust motion parameters to increase population diversity. The multi‐swarm version of the proposed algorithm with higher performance than the single‐swarm version and other methods has been introduced in this article too. The proposed algorithms have been evaluated using 23 mathematical benchmark functions, including unimodal, multimodal, and composite functions and four engineering optimization problems. The obtained results and statistical tests prove that the proposed algorithms provide competitive solutions for most of the experiments, compared to the state‐of‐the‐art and well‐known optimization meta‐heuristic methods in terms of convergence towards the global optimum, local optima avoidance, exploration, and exploitation. [ABSTRACT FROM AUTHOR]

To decompose a wide‐angle X‐ray diffraction (WAXD) curve of a semi‐crystalline polymer into crystalline peaks and amorphous halos, a theoretical best‐fitted curve, i.e. a mathematical model, is constructed. In fitting the theoretical curve to the experimental one, various functions can be used to quantify and minimize the deviations between the curves. The analyses and calculations performed in this work have proved that the quality of the model, its parameters and consequently the information on the structure of the investigated polymer are considerably dependent on the shape of an objective function. It is shown that the best models are obtained employing the least‐squares method in which the sum of squared absolute errors is minimized. On the other hand, the methods in which the objective functions are based on the relative errors do not give a good fit and should not be used. The comparison and evaluation were performed using WAXD curves of seven polymers: isotactic polypropylene, polyvinylidene fluoride, cellulose I, cellulose II, polyethylene, polyethylene terephthalate and polyamide 6. The methods were compared and evaluated using statistical tests and measures of the quality of fitting. [ABSTRACT FROM AUTHOR]

As faults grow over time and become more "mature," some of their geometrical and mechanical properties evolve, and these changes modify earthquake behavior. It is thus of prime importance to know the degree of structural maturity of a fault that is likely to produce large earthquakes. Although this concept is extensively used, there is no common definition or metric to measure the structural maturity of a fault. We analyzed the heterogeneity of the surface traces of 13 large seismogenic faults whose maturity is known qualitatively. We measured the corrugations and step‐over segmentation of the traces from ∼100 m to the fault length scale. Corrugations and some properties of the segmentation are found to vary with fault structural maturity. We provide scaling relationships that quantify the structural maturity of a fault based on its surface trace. These results should help in parameterizing source faults in earthquake models. Plain Language Summary: Long‐term faults produce earthquakes. Measuring fault properties could thus help us understand earthquake behavior. However, measuring properties of large‐scale faults in particular is difficult. Here, we tackle one of the major long‐term properties of faults, their structural maturity. This property relates to the overall slip longevity of the fault (generally several million years), and it has been shown to impact earthquake behavior; mature and immature faults do not behave similarly. For 13 large seismogenic continental faults whose structural maturity was estimated qualitatively in earlier works, we examined the heterogeneity of the traces these faults form at the ground surface. Using simple tools, we measured the undulations and the discrete "stepping" segmentation of the fault traces over a broad range of scales from ∼100 m to the full fault length (up to ∼1,600 km in this study). We found that the "intensity" of undulations and the density, relative width, and size diversity of the steps separating discrete fault segments all vary with the structural maturity of the faults. These variations are described with simple mathematical functions that characterize fault structural maturity and can be used to better represent source faults in earthquake models. Key Points: We measure the corrugation and step‐over segmentation of 13 large seismogenic fault traces, at scales greater than ∼100 mThe corrugation level and the step‐over density, relative width, and size diversity vary with the structural maturity of the faultsWe provide scaling relations measuring fault structural maturity, which should help to describe source faults better in earthquake models [ABSTRACT FROM AUTHOR]

4D printing can address time‐evolving structural functions that are unattainable by conventional 3D printing. Despite the advance in materials and printing techniques, however, 4D printing of continuity of shape representation that generally characterizes 3D matters is still challenging, because the existing methodologies mostly rely on a few discrete levels of strain and their spatial distributions. Here, a 4D printing strategy of shape memory polymers (SMPs) that can program continuous levels of shape‐recovery strain is proposed. It is found that the irrecoverable state of the SMP and the corresponding recovery strain can be controlled in a continuous and precise manner by a single printing parameter. Importantly, the continuity of strain programming provides an opportunity for the translation into mathematical function representation (F‐rep), which allows the systematic derivation and implementation of 4D‐printed bilayer strain functions that are matched to the continuously varying curvatures of the target geometry. Combined with the custom‐built software, the F‐rep 4D printing strategy can produce 4D‐printed architectures that involve continuously varying strain profiles of almost any function type. The effectiveness of the framework is highlighted by a set of 3D face masks with facial feature transformation driven by a function operator. [ABSTRACT FROM AUTHOR]

Many issues involving leaf areas (LA) allometry, such as the possibility of adjusting morphometric models based on the leaves of different species and the influence model performance on intraspecific variability, are still unclear. In this work, we described allometric equations for 14 different woody tropical species. This study's novelty is to show how a generalised allometric equation accurately estimates LAs of different species with similar leaf morphologies [oblong: Ŷ = 0.6356(LW)1.0260, and elliptical: Ŷ = 0.6552(LW)1.0094]. Natural variation that occurs in the leaf development does not represent an obstacle to the implementation of generalised models. In general, simple linear models are considered ineffective to describe the full range of leaf variation. Extrapolation of these equations to other woody species of similar leaf morphology can be compared with models developed for oblong and elliptical leaves, based on the principle that in potential models, the variation of the intercept coefficient (β0) tended to 0.5 for leaves more elongated to 0.7 in less elongated leaves. While the angular coefficient (β1) tended to vary from 0.97 for less elongated leaves, up to 1 when the relationship between LA is remarkably close to a rectangle. [ABSTRACT FROM AUTHOR]

The propagation criterion is defined in cryptography in order to analyze the security of cryptographic components with respect to differential cryptanalysis. This paper explains a class of plateaued functions satisfying the propagation criterion of degree l$l$ by using the Maiorana–McFarland construction. And a variety of other desirable criteria for functions with cryptographic application could also be satisfied, such as balancedness, high non‐linearity, non‐existence of non‐zero linear structures, good global avalanche characteristics. [ABSTRACT FROM AUTHOR]

A method is proposed of calculating the geometric term of the structure factor for quasicrystals, which enables incorporation of the phason disorder. The scheme is based on the series expansion of the structure factor with moments of the distribution function as coefficients. A distribution function is a mathematical object that is constructed for reference vertices of the tiles in the quasilattice. It encloses the entire structural information of the underlying quasilattice, together with the inherent disorder, necessary to calculate the diffraction pattern. By tuning the value of the distribution moments through the refinement procedure, it is possible to obtain a very good agreement of this new model of the decagonal AlCuRh phase with the experimental data, reflected in the crystallographic R factor of 6.08%. The characteristic bias of the calculated diffraction peak intensities observed for the low‐intensity reflections is significantly diminished, confirming its origin being, to some extent, related to phason disorder. Additionally, it is no longer necessary to use the general Debye–Waller factor for phasons, as the new formula accommodates this type of structural disorder. However, the best result was obtained for the model combining the new approach with the Gaussian corrective term. [ABSTRACT FROM AUTHOR]

Training of the convolution neural network (CNN) is a problem of global optimisation. This study proposed a hybrid modified particle swarm optimisation (MPSO) and conjugate gradient (CG) algorithm for efficient training of CNN. The training involves MPSO–CG to avoid trapping in local minima. Particularly, improvements in the MPSO by introducing a novel approach for control parameters, improved parameters updating criteria, a novel parameter in the velocity update equation, and fusion of the CG allows handling the issues in training CNN. In this study, the authors validate the proposed MPSO algorithm on three benchmark mathematical test functions and also compared with three different variants of the baseline particle swarm optimisation algorithm. Furthermore, the performance of the proposed MPSO–CG is also compared with other training algorithms focusing on the analysis of computational cost, convergence, and accuracy based on a standard problem specific to classification applications on CIFAR-10 dataset and face and skin detection dataset. [ABSTRACT FROM AUTHOR]

Summary: This study aims at providing a hybrid calibration framework to estimate Hertz‐type contact parameters (particle‐scale shear modulus and Poisson ratio) for both two‐dimensional and three‐dimensional discrete element modelling (DEM). On the basis of statistically isotropic granular packings, a set of analytical formulae between macroscopic material parameters (Young modulus and Poisson ratio) and particle‐scale Hertz‐type contact parameters for granular systems are derived under small‐strain isotropic stress conditions. However, the derived analytical solutions are only estimated values for general models. By viewing each DEM modelling as an implicit mathematical function taking the particle‐level parameters as independent variables and employing the derived analytical solutions as the initial input parameters, an automatic iterative scheme is proposed to obtain the calibrated parameters with higher accuracies. Considering highly nonlinear features and discontinuities of the macro‐micro relationship in Hertz‐based discrete element models, the adaptive moment estimation algorithm is adopted in this study because of its capacity of dealing with noise gradients of cost functions. The proposed method is validated with several numerical cases including randomly distributed monodisperse and polydisperse packings. Noticeable improvements in terms of calibration efficiency and accuracy have been made. [ABSTRACT FROM AUTHOR]

Purpose: Due to differences in attenuation and the electron return effect (ERE), the presence of gas can increase the risk of toxicity in organs at risk (OAR) during magnetic resonance‐guided radiotherapy (MRgRT). Current adaptive MRgRT workflows using density overrides negate gas from the dose calculation, meaning that the effects of ERE around gas are not taken into account. In order to achieve an accurate adaptive MRgRT treatment, we should be able to quickly evaluate whether gas present during treatment causes dose constraint violation during an MRgRT fraction. We propose an analytic method for predicting dose perturbations caused by air cavities in OARs during MRgRT. Method: Ten virtual water phantoms were created: nine containing a centrally located spherical air cavity and a reference phantom without an air cavity. Monte Carlo dose calculations were produced to irradiate the phantoms with a single 7 MV photon beam under the influence of a 1.5 T transverse magnetic field (Monaco 5.19.02 Treatment Panning System (TPS) (Elekta AB, Stockholm, Sweden)). Dose distributions of the phantoms with and without air cavities were compared. We used a spherical coordinate system originating in the center of the cavity to sample the dose distributions and calculate the dose perturbation as a result of the presence of each air cavity, ∆D%(θ,Φ)calc.. Dose effects due to ERE and differences in attenuation due to density changes were considered separately. Least squared analysis was used to fit the calculated dose perturbations to mathematical functions. Effects due to ERE were fit to a modulated sinusoidal function and those due to attenuation differences were fit to a 2D Gaussian function. We used the fits to derive a single equation describing dose perturbations around spherical air cavities as a function of angles, θ, Φ, distance from cavity surface, d, and cavity radius, r. We measured the fitting error by calculating the residual error (RE); the difference between the calculated and fitted dose perturbation. Results: Both ERE and differences in attenuation contribute toward the total dose effects of air cavities in MRgRT. Whereas ERE dominates close to the surface of the cavities, attenuation effects dominate at distances >0.5 cm from the cavities. We showed that dose effects around a spherical air cavity (≤1 cm from the surface) due to ERE fit a modulated sinusoidal function with mean (RE) ≤−1.4E‐5% and root mean square error (rms) (RE) ≤4.1%. Effects due to attenuation differences fit a Gaussian function with mean (RE) ≤0.7% and rms (RE) ≤1.8%. Our general equation, which we verified using multiple sizes of spherical and cylindrical air cavity, fits Monte Carlo simulated data with mean (RE) ≤±0.9% and rms (RE) ≤6.9%. Conclusion: We show that local dose perturbations around unplanned spherical air cavities during MRgRT can be well characterized analytically. We present an equation that can be incorporated into the clinical workflow to allow for fast evaluation of dose effects of unplanned gas. We also envision this method contributing to the clinical implementation of real time adaptive radiotherapy (ART) for MRgRT using MRI planning. [ABSTRACT FROM AUTHOR]

An efficient analytical model to design wideband microstrip spurline band‐stop filter (BSF) is proposed. The geometrical parameters of the BSFs are optimized using simulation for the entire range of the desired operating frequency (1‐5 GHz) and the results indicated relatively inferior performance of the filters designed using conventional analytical model. Whereas, in the proposed model, a mathematical function is found by curve fitting of the optimal data set of geometrical parameters optimized by simulation. Therefore, the filters designed using the proposed model achieve almost optimal performance in terms of rejection ratio, bandwidth, and so on, and a better match between desired and actual operating frequency as compared with the conventional design model. To corroborate the claims, the filter prototypes designed using the proposed and conventional models are fabricated and tested. The measured results showed agreement with the simulation results and proved effectiveness of the proposed analytical model to design spurline BSF. The error limit for the proposed model is found within ±0.005% of the optimal design. [ABSTRACT FROM AUTHOR]

An approximate model called metamodel or surrogate model is a mathematical model that numerically approximates response of a system during an engineering simulation process or test. The introduction of a metamodel makes it possible to express response defined in the design problem as a simple mathematical function of design variables. A metamodel can be built with response surface method (RSM), kriging, neural network, radial basis function, and so on. Each method has its advantages and disadvantages. A combined metamodel called hybrid model, ensemble model, or multiple surrogates has been developed to maximize each metamodel's strength. The hybrid model of this research includes RSM and kriging. Besides, a strategy to refine the hybrid metamodel is implemented by reducing design space. In this process, information related to Hessian is utilized for an unconstrained optimization problem, on the contrary feasibility for a constrained optimization problem. This research presents a new hybrid metamodel‐based optimization strategy called refined hybrid metamodel. Five mathematical test problems, two‐bar design, spring design, and propeller shaft design problems are solved with the suggested method, verifying its usefulness. Most of the optimal results with the proposed method are closer to exact solutions with smaller function evaluations than existing methods. [ABSTRACT FROM AUTHOR]

New Findings: What is the central question of this study?Can the cross‐relationship between 35 rat arterial pulse waveform (APW) parameters be described by known mathematical functions and can mathematical parameters be obtained for conditions in a model of hypertension resulting from decreased NO bioavailability?What is the main finding and its importance?Mathematical functions and their parameters were obtained that approximate the cross‐relationships of 35 APW parameters to systolic blood pressure and to the augmentation index in conditions of decreased NO bioavailability. The results enable APW parameters to be assigned to decreased NO bioavailability, which may have predictive or diagnostic value. Information obtained from the arterial pulse waveform (APW) using haemodynamic parameters (HPs) is useful for characterization of the cardiovascular system in particular (patho)physiological conditions. Our goal was to find out whether the relationships between rat HPs could be described by simple mathematical functions and to find mathematical parameters for conditions of high blood pressure (BP) resulting from decreased NO bioavailability. The right jugular vein of anaesthetized Wistar rats was cannulated for i.v. administration of Nω‐nitro‐l‐arginine methyl ester (l‐NAME). The left common carotid artery was cannulated to detect the APW. From 10 points on the rat APW we defined 35 HPs (some were known already) and found 595 cross‐relationships between HPs showing unique patterns for particular cardiovascular conditions. Here we show parallel time‐dependent changes of 35 HPs and some of their cross‐relationships in condition of high BP induced by l‐NAME. We found that most of the time‐dependent changes of 35 HPs and their relationships were very well fitted by simple mathematical functions, e.g. a linear function, exponential growth, exponential decay or exponential rise to maximum. The results may enable the mathematical functions to be assigned for decreased NO bioavailability, which may have predictive or diagnostic value for conditions of high BP. Using this approach, it may be possible to find unique cross‐relationship patterns of HPs and mathematical functions between HPs for different cardiovascular (patho)physiological or drug‐modulating conditions. This knowledge can be used in studying the molecular mechanisms of particular (patho)physiological conditions or drug actions and may have predictive or diagnostic value. [ABSTRACT FROM AUTHOR]

This study deals with the numerical investigation of a mathematical model of breast cancer at the initial growth stage known as ductal carcinoma in situ. This model considered as an inverse problem and the uniqueness of solution of this inverse problem is proved. To solve this problem, a computational approach is developed based on an iterative procedure and space marching and mollification methods. The stability and convergence results are given to support the method theoretically. Two test problems are considered to demonstrate the efficiency and ability of the proposed numerical approach. [ABSTRACT FROM AUTHOR]

The lexicographic sum of graphs is defined as follows. Let G be a graph. With each v∈V(G) associate a graph Hv. The lexicographic sum of the graphs Hv over G is obtained from G by substituting each v∈V(G) by Hv. Given distinct v,w∈V(G), we have all the possible edges in the lexicographic sum between V(Hv) and V(Hw) if vw∈E(G), and none otherwise. When all the graphs Hv are isomorphic to some graph H, the lexicographic sum of the graphs Hv over G is called the lexicographic product of H by G and is denoted by G≀H. We say that a graph G is fractal if there exists a graph Γ, with at least two vertices, such that G≃Γ≀G. There is a simple way to construct fractal graphs. Let Γ be a graph with at least two vertices. The graph Γω is defined on the set V(Γ)ω of functions from ω to V(Γ) as follows. Given distinct f,g∈V(Γ)ω,fg is an edge of Γω if f(m)g(m) is an edge of Γ, where m is the smallest integer such that f(m)≠g(m). The graph Γω is fractal because Γ≀Γω≃Γ1+ω≃Γω. We prove that a fractal graph is isomorphic to a lexicographic sum over an induced subgraph of Γω, which is itself fractal. [ABSTRACT FROM AUTHOR]

An infinite cardinal λ is Magidor if and only if λ→[λ]λℵ0- bd. It is known that if λ is Magidor then λ→[λ]αℵ0- bd for some α<λ, and the first such α is denoted by αM(λ). In this paper we try to understand some of the properties of αM(λ). We prove that αM(λ) can be the successor of a supercompact cardinal, when λ is a Magidor cardinal. From this result we obtain the consistency of αM(λ) being a successor of a singular cardinal with uncountable cofinality. [ABSTRACT FROM AUTHOR]

Let M be the basic set theory that consists of the axioms of extensionality, emptyset, pair, union, powerset, infinity, transitive containment, Δ0‐separation and set foundation. This paper studies the relative strength of set theories obtained by adding fragments of the set‐theoretic collection scheme to M. We focus on two common parameterisations of the collection: Πn‐collection, which is the usual collection scheme restricted to Πn‐formulae, and strongΠn‐collection, which is equivalent to Πn‐collection plus Σn+1‐separation. The main result of this paper shows that for all n≥1, M+Πn+1− collection +Σn+2− inductionon ω proves that there exists a transitive model of Zermelo Set Theory plus Πn‐collection,the theory M+Πn+1− collection is Πn+3‐conservative over the theory M+ strong Πn− collection. It is also shown that (2) holds for n=0 when the Axiom of Choice is included in the base theory. The final section indicates how the proofs of (1) and (2) can be modified to obtain analogues of these results for theories obtained by adding fragments of collection to a base theory (Kripke‐Platek Set Theory with Infinity plus V=L) that does not include the powerset axiom. [ABSTRACT FROM AUTHOR]

Let X, Y be Polish spaces, Γ⊆℘(Y), A⊆X×Y. We say A is universal for Γ provided that each x‐section of A is in Γ and each element of Γ occurs as an x‐section of A. An equivalence relation generated by a set A⊆X×Y is denoted by EA, where xEAx′⟺Ax=Ax′. The following results are shown: (1)If A is a Σn1 set universal for all nonempty closed subsets of Y, then EA is a σ(Σn1) equivalence relation and EA≤σ(Σn1) id (2ω).(2)If A is a Σ11 set universal for all countable subsets of Y, then EA is a σ(Σ11) equivalence relation, and (i)EA≤σ(Σ11)=+ and =+≤Δ21EA;(ii)if V=L, then EA≤Δ21 id (2ω);(iii)if every Σ21 set is Lebesgue measurable or has the Baire property, then EA≰Δ21 id (2ω).(iv)for n≥2, if every Δn1 set has the Baire property, and E is any Σ30 equivalence relation, then EA≰Δn1E. [ABSTRACT FROM AUTHOR]

This article proposes a circular dual‐loop antenna for a very small Global Positioning System array with an extended cavity structure. The antenna element consists of lower and upper circular loops printed on a high‐dielectric ceramic substrate and an extended cavity structure above the ground plane to improve the isolation characteristic. To demonstrate the suitability of the proposed antenna, antenna characteristics are measured in a full‐anechoic chamber, and its performances such as bore‐sight gain, mutual coupling, and near‐field as a function of extended cavity height are analyzed. The results confirm that the proposed antenna structure can minimize the gain degradation by improving the isolation characteristic and is therefore suitable for use in very small arrays. [ABSTRACT FROM AUTHOR]

In high performance computing systems and signal processing, there is a basic set of mathematical functions that are essential. While addition, subtraction and multiplication are well understood, there is less literature on square‐rooting, which is a particularly time‐ and resource‐consuming function. Traditional non‐restoring algorithms produce a mantissa half the length of the input mantissa, causing a loss of precision. This study presents a method for increasing the accuracy of this algorithm. It is shown to work for all IEEE‐754R standard floating‐point numbers. Error analysis shows a 57‐fold (for half‐precision) and 134e6‐fold improvement (for double‐precision) in the normalised error, equivalent to at most 1 Units of Least Precision. Resource and performance optimised variants are analysed and their throughput analysed. On an Intel Stratix V device, performance optimised implementations achieve a throughput of 717 MFLOPs. Resource optimised implementations on a low‐cost device require only 127 Adaptive Logic Modules and 232 registers, with a throughput of 8.56 MFLOPs. All implementations are DSP block and memory free, saving valuable resources. The maximum throughput of the presented design is 15.5 times greater than that proposed by Pimentel et al. and two orders of magnitude greater than typical multiply‐accumulate methods. [ABSTRACT FROM AUTHOR]

We present abstract conditions under which a special flow over a probability preserving map with a non‐integrable roof function is Krickeberg mixing. Our main condition is a version of the local central limit theorem for the underlying map. We check our assumptions for independent and identically distributed random variables (renewal theorem with infinite mean) and for suspensions over Pomeau–Manneville maps. [ABSTRACT FROM AUTHOR]

Let X be a standard Markov process. We prove that a space inversion property of X implies the existence of a Kelvin transform of X‐harmonic, excessive and operator‐harmonic functions and that the inversion property is inherited by Doob h‐transforms. We determine new classes of processes having space inversion properties amongst transient processes satisfying the time inversion property. For these processes, some explicit inversions, which are often not the spherical ones, and excessive functions are given explicitly. We treat in details the examples of free scaled power Bessel processes, non‐colliding Bessel particles, Wishart processes, Gaussian Ensemble and Dyson Brownian Motion. [ABSTRACT FROM AUTHOR]

Let p(·) be a measurable function on Rn with 0

We study asymptotics of sums of the form ∑X0, where σ:=833=0.2424⋯, with an error term saving on average an arbitrary power of the logarithm over the trivial bound. This improves upon results of Mikawa and Baier–Browning–Marasingha–Zhao, who obtained statements of this form with σ replaced by 13. We obtain an analogous result for the fourth sum for most N in an interval of the form [X,X+H] with Xσ+ε⩽H⩽X1−ε. Our method starts with a variant of an argument from a paper of Zhan, using the circle method and some oscillatory integral estimates to reduce matters to establishing some mean‐value estimates for certain Dirichlet polynomials associated to 'Type d3' and 'Type d4' sums (as well as some other sums that are easier to treat). After applying Hölder's inequality to the Type d3 sum, one is left with two expressions, one of which we can control using a short interval mean value theorem of Jutila, and the other we can control using exponential sum estimates of Robert and Sargos. The Type d4 sum is treated similarly using the classical L2 mean value theorem and the classical van der Corput exponential sum estimates. In a sequel to this paper we will obtain related results for the correlations involving dk(n) for much smaller values of H but with weaker bounds. [ABSTRACT FROM AUTHOR]

Atmospheric corrosivity maps can effectively assess atmospheric corrosion risks and provide guidance for anticorrosion measures. Since the atmospheric corrosion rate of materials is affected by many natural and environmental factors, a dose‐response function may establish a quantitative relationship between the influencing factors and the atmospheric corrosion rate. In order to map the atmospheric corrosivity of Guangdong Province, corrosion rate data of 48 exposed sites and the atmospheric and environmental data of 102 monitoring stations were collected first. Then a dose‐response function suitable for Guangdong Province was established through function design and then improved through accuracy assessment. Finally, the maps were drawn by different numbers of data and an optimal integrated one was obtained. It is the first time that a corrosion map of Guangdong Province had been drawn. Atmospheric corrosion tests were undertaken at 48 different sites. A special approach is used to obtain a dose‐response function (DRF2). The maps show the overall atmospheric corrosivity of Guangdong Province. [ABSTRACT FROM AUTHOR]

We study coupled motion of a 1‐D closed elastic string immersed in a 2‐D Stokes flow, known as the Stokes immersed boundary problem in two dimensions. Using the fundamental solution of the Stokes equation and the Lagrangian coordinate of the string, we write the problem into a contour dynamic formulation, which is a nonlinear nonlocal equation solely keeping track of evolution of the string configuration. We prove existence and uniqueness of local‐in‐time solution starting from an arbitrary initial configuration that is an H5/2‐function in the Lagrangian coordinate satisfying the so‐called well‐stretched assumption. We also prove that when the initial string configuration is sufficiently close to an equilibrium, which is an evenly parametrized circular configuration, then a global‐in‐time solution uniquely exists and it will converge to an equilibrium configuration exponentially as t →  + ∞. The technique in this paper may also apply to the Stokes immersed boundary problem in three dimensions. © 2018 Wiley Periodicals, Inc. [ABSTRACT FROM AUTHOR]

In this paper, we prove the results on existence and uniqueness of the maximal solutions for measure differential equations, considering more general conditions on functions f and g by using the correspondence between the solutions of these equations and the solutions of generalized ODEs. Moreover, we prove these results for the dynamic equations on time scales, using the correspondence between the solutions of these last equations and the solutions of the measure differential equations. [ABSTRACT FROM AUTHOR]

In , Huayi Chen introduces the notion of an approximable graded algebra, which he uses to prove a Fujita‐type theorem in the arithmetic setting, and asks if any such algebra is a sub‐algebra of the graded section ring of a big line bundle on an algebraic variety. We give a counter‐example showing that this is not the case. [ABSTRACT FROM AUTHOR]

Restricted non linear approximation is a generalization of the N‐term approximation in which a measure on the index set of the approximants controls the type, instead of the number, of elements in the approximation. Thresholding is a well‐known type of non linear approximation. We relate a generalized upper and lower Temlyakov property with the decreasing rate of the thresholding approximation. This relation is in the form of a characterization through some general discrete Lorentz spaces. Thus, not only we recover some results in the literature but find new ones. As an application of these results, we compress and reduce noise of some images with wavelets and shearlets and show, at least empirically, that the L2‐norm is not necessarily the best norm to measure the approximation error. [ABSTRACT FROM AUTHOR]

Cellular automata are frequently used to model dynamics of spatial systems. This work shows how a so‐called structural adjoint sensitivity analysis may be applied to such systems. It is assumed that the cellular automaton has a continuous state, a spatiotemporal input, and is characterized by one scalar objective function specified, for example, in a given optimization or parameter optimization problem. With this analysis, it is possible to calculate efficiently a gradient of the objective function in a space of the spatiotemporal input of the automata. As an example, a model of tumor growth with an introduced spatiotemporal irradiation signal is analysed. [ABSTRACT FROM AUTHOR]

In this article, the general fractional‐order diffusion equations within the negative Prabhakar kernel are considered for the first time. With the aid of the Laplace transform, the series solutions for the problems with the general Mittag‐Leffler functions are discussed in detail. The obtained results are efficient in the description of the anomalous behaviors of the diffusive process. [ABSTRACT FROM AUTHOR]

In a recent paper of ours, we proved that a special "harmonic conjugation" operator is not bounded in weighted Bergman spaces Lα2 of quaternion‐valued functions in the 3D ball. In the present paper, we prove that, in contrast to the Bergman spaces case, the same operator is bounded in weighted Dirichlet spaces Dα2 of quaternion‐valued functions in the 3D ball. Furthermore, applying another approach for a construction of harmonic conjugates, we extend the result to weighted Dirichlet spaces Dαp with 1 < p < ∞. [ABSTRACT FROM AUTHOR]

We apply the saddle–point‐type theorems of Rabinowitz and Benci‐Rabinowitz for local Lipschitz functionals that entails an extension of the classical Palais‐Smale‐Cerami condition for a C1 functional to this setting to show the existence of new periodic solutions for second‐order Hamiltonian systems with local Lipschitz potentials, which are weaker than Rabinowitz's original conditions. The key difficulty in these arguments arises from the lack of smoothness and symmetry in the potential when extending the Cerami‐Palais‐Smale condition for the local Lipschitz case. [ABSTRACT FROM AUTHOR]

We consider a class of variational systems involving fractional Kirchhoff‐type equations of the form: M1(‖u‖X2)(−Δ)su=Fu(x,u,v)inΩ,M2(‖v‖X2)(−Δ)sv=Fv(x,u,v)inΩ,u=v=0inRN\Ω,where s ∈ (0,1), N > 2s, Ω⊂RN a smooth and bounded domain, the functions Fu, Fv, M1 and M2 are continuous and ( − Δ)s is the fractional Laplacian operator. In this paper, we show that, under appropriate growth conditions on the nonlinearities Fu and Fv and on the nonnegative functions M1 and M2, the (weak) solutions are precisely the critical points of a related functional defined on a fractional Hilbert space Y(Ω) = X(Ω) × X(Ω) and the existence infinitely many solutions can be obtained by the use of the Krasnoselskii's genus. Besides, a regularity result can also be obtained by using specific results for systems in conjunction with the growth assumptions of these functions. [ABSTRACT FROM AUTHOR]