Exponential Form Research Articles

Some Novel Fusion and Fission Phenomena for an Extended (2+1)-Dimensional Shallow Water Wave Equation

An extended (2+1)-dimensional shallow water wave (SWW) model which can describe the evolution of nonlinear shallow water wave propagation in two spatial and temporal coordinates, is systematically studied. The multi-linear variable separation approach is addressed to the extended (2+1)-dimensional SWW equation. The variable separation solution consisting of two arbitrary functions is obtained, by assumption, from a specific ansatz. By selecting these two arbitrary functions as the exponential and trigonometric forms, resonant dromion, lump, and solitoff solutions are derived. Meanwhile, some novel fission and fusion phenomena including the semifoldons, peakons, lump, dromions, and periodic waves are studied with graphical and analytical methods. The results can be used to enhance the variety of the dynamics of the nonlinear wave fields related by engineering and mathematical physics.

On Flexural Wave Dispersion of a Higher-Order Metamaterial Sandwich Composite Plate Based on a Visco-Pasternak Foundation

This paper elaborates on the wave propagation characteristics of a smart sandwich plate consisting of smart piezo composite layers on a plate’s upper and lower borders via an auxetic core. This study’s equation of motion was derived using a refined version of the Shear-deformation theory at a higher order, or HSDT. Additionally, detecting the properties of the auxetic core and piezoelectric composite layers employs a micromechanical model. The fundamental equations of the smart sandwich plates were formulated utilizing the Hamiltonian principle and Maxwell’s law. Then, the governing equations of a sandwich plate are solved using an exponential form and an analytical method. Furthermore, the phase velocity of the intelligent sandwich plate is provided based on the layer thicknesses of the auxetic core and thicknesses of intelligent piezoelectric composite layers. In addition, each figure calculates and presents the weight of the Winkler–Pasternak coefficient, the viscoelastic substances factor, the source of the outside electricity, and the volume percent of reinforcement in the matrix on the phase velocity.

The strongest aftershock in seismic models of epidemic type

SUMMARY We consider an epidemic-type aftershock model, ETAS(F), for a large class of distributions F determining the number of direct aftershocks. This class includes Poisson, Geometric, Negative Binomial distributions and many other. Assuming an exponential form of the productivity and magnitude laws, we find a limiting distribution of the strongest aftershock magnitude μa when the initial cluster event m• is large. The regime can be either subcritical or critical; the initial event can be dominant in size or not. In the subcritical regime, the mode of the limiting distribution is determined by the parameters of productivity and the magnitude laws; the shape of this distribution is not universal and is effectively determined by F. For example, the Geometric F-distribution generates the logistic law, and the Poisson distribution (studied earlier) generates the Gumbel type 1 law. The accuracy of these laws for moderate initial magnitudes is tested numerically. The limit distribution of the Båth’s difference m• − μa is independent of the initial event size only if the regime is critical, and the ratio of exponents in the laws of magnitude and productivity is contained in the interval (1,2). Previous studies of the μa-distribution have dealt with the traditional Poisson F model and with arbitrary (not necessarily dominant) initial magnitude m•.

Novel closed-form analytical solutions and modulation instability spectrum induced by the Salerno equation describing nonlinear discrete electrical lattice via symbolic computation

A nonlinear electrical lattice is an effective experimental tool for dealing with nonlinear dispersive media and creating possibilities for the realistic modeling of electrical soliton. In this work, we discuss the dynamical behavior of the governing (1+1)-dimensional time–space Salerno equation, which describes the discrete electrical lattice with nonlinear dispersion. The different kinds of solutions in the form of periodic, Jacobi elliptic, and exponential functions are obtained by the two analytical approaches, namely the modified generalized exponential rational function method (MGERFM) and the extended sinh-Gordon equation expansion method (shGEEM). The MGERF approach is applied to construct exact solitary wave solutions containing trigonometry, hyperbolic and exponential functions, while the extended shGEE technique is employed to obtain periodic wave solutions containing hyperbolic and Jacobi elliptic functions via performing the symbolic computation in the software package MATHEMATICA. Moreover, by means of standard linear-stability analysis, the modulation instability (MI) and the MI gain spectrum is analytically computed for the considered equation. By selecting suitable parametric values, solution profiles are portrayed in order to make the solutions physically relevant. These results yield periodic waves, kink waves, multi-soliton, and their interactions. By comparing the obtained solutions to the previous findings, we can reveal the novelty of the solutions. In addition, all the obtained solutions were verified by substituting them back into the governing equation using the Mathematica software. The acquired results are significant for the study of wave propagation, signal transmission, and applications of super-transmission phenomena.

Approximate Damped Oscillatory Traveling Wave in Viscous Compressible Fluid With Capillary Effect

We convert the viscous compressible fluid equations with capillary effect into the equivalent planar dynamical system and illustrate the global qualitative phase diagrams corresponding to its bounded orbits by the qualitative theory of ordinary differential equations. From the evolutionary relationship of bounded orbits in global qualitative phase diagrams and the correspondence between the bounded orbit and traveling wave, we discover the evolutionary rule of the profile of the traveling wave as the viscous coefficient varying and the forming reason for the damped oscillatory traveling wave. Further, we construct the analytical approximate explicit expression of the damped oscillatory traveling waves in accordance with the forming reason and prove its error decreases in exponential form. Finally, we compare the approximate damped oscillatory traveling wave with the numerical solution. The result shows they fit well.

Probing the diversity of soliton phenomena within conformable Estevez-Mansfield-Clarkson equation in shallow water

This study aims to employ the extended direct algebraic method (EDAM) to generate and evaluate soliton solutions to the nonlinear, space-time conformable Estevez Mansfield-Clarkson equation (CEMCE), which is utilized to simulate shallow water waves. The proposed method entails transforming nonlinear fractional partial differential equations (NFPDEs) into nonlinear ordinary differential equations (NODEs) under the assumption of a finite series solution by utilizing Riccati ordinary differential equations. Various mathematical structures/solutions for the current model are derived in the form of rational, exponential, trigonometric, and hyperbolic functions. The wide range of obtained solutions allows for a thorough analysis of their actual wave characteristics. The 3D and 2D graphs are used to illustrate that these behaviors consistently manifest as periodic, dark, and bright kink solitons. Notably, the produced soliton solutions offer new and critical insights into the intricate behaviors of the CEMCE by illuminating the basic mechanics of the wave's interaction and propagation. By analyzing these solutions, academics can better understand the model's behavior in various settings. These solutions shed light on complicated issues such as configuration dispersion in liquid drops and wave behavior in shallow water.

Exploration of nonlinear traveling wave phenomena in quintic conformable Benney-Lin equation within a liquid film

<abstract><p>In this article, we use the modified extended direct algebraic method (mEDAM) to explore and analyze the traveling wave phenomena embedded in the quintic conformable Benney-Lin equation (CBLE) that regulates liquid film dynamics. The proposed transformation-based approach developed for nonlinear partial differential equations (PDEs) and fractional PDEs (FPDEs), efficiently produces a plethora of traveling wave solutions for the targeted CBLE, capturing the system's nuanced dynamics. The methodically determined traveling wave solutions are in the form of rational, exponential, hyperbolic and trigonometric functions which include periodic waves, bell-shaped kink waves and signal and double shock waves. To accurately depict the wave phenomena linked to these solutions, we generate 2D, 3D, and contour graphs. These visualizations not only improve understanding of the CBLE model's dynamics, but also provide a detailed way to examine its behavior. Moreover, the use of the proposed techniques contributes to a better understanding of the other FPDEs' distinct characteristics, enhancing our comprehension of their underpinning dynamics.</p></abstract>

Natural frequencies of bars with variable cross-sections

This paper presents the possibility of determining the natural frequency of flexural bars treated as Bernoulli-Euler beams using the Rayleigh method. It is assumed that the shape of the bar axis during vibration is the same as the shape of the deflection line of this bar under continuous load. By comparing the potential energy in the deflected position with the kinetic energy in the undeflected position, it is possible to determine the frequency and period of natural vibration as a function of Young's modulus and material density, as well as the parameters describing the geometric shape of the bar. Examples of solutions for truncated cone and truncated wedge-shaped bars are shown, as well as a solids of revolution with the generatrix described by linear and curvilinear (parabolic, exponential) forms. It applies to both solid and hollow bars, and different types of fixing the ends of the bar. The accuracy of the obtained results (compared with the literature and with the results of FEM calculations) is sufficient for practical applications. The authors also showed the possibility of extending the method to higher frequencies of vibration.

Eternal solutions in exponential self-similar form for a quasilinear reaction-diffusion equation with critical singular potential

Eternal solutions in exponential self-similar form for a quasilinear reaction-diffusion equation with critical singular potential

Dark and bright hump solitons in the realm of the quintic Benney-Lin equation governing a liquid film

<p>This study explored and examined soliton solutions for the Quintic Benney-Lin equation (QBLE), which describes the dynamic of liquid films, using the Riccati modified extended simple equation method (RMESEM). The proposed approach, which is designed for nonlinear partial differential equations (NPDEs), effectively generates a large number of soliton solutions for the given QBLE, which basically captures the fundamental dynamics of the system. The rational, hyperbolic, rational-hyperbolic, trigonometric, and exponential forms of the scientifically specified soliton solutions are the main determinants of the hump solitons. We used 2D, 3D, and contour visualizations to offer accurate representations of the researched soliton phenomena associated with these solutions. These representations revealed the existence of dark and bright hump solitons in the framework of the QBLE and offer a thorough way to examine the model's behavioral characteristics in the liquid film by analyzing the QBLE model's soliton dynamics. Moreover, applying the suggested approach advances our knowledge of the unique features of the other similar NPDEs and the underlying dynamics.</p>

Quantitative classification of velocity pulse-like ground motions based on an adaptive response spectrum-based decomposition method

Pulse-like ground motions generally contain low-frequency velocity pulse with large amplitude, which can lead to severe damage or even collapse of near-fault structures. This study aims to proposed an adaptive decomposition method and corresponding identifying approach for pulse-like records. Firstly, an adaptive response spectrum-based decomposition (ARSD) method is proposed to decompose the records into several intrinsic mode functions (IMF) and their periods are identified by peak-point method. The decomposed results are compared with two representative methods, ensemble empirical-mode decomposition (EEMD) and variation mode decomposition (VMD). The results show that the proposed method can effectively avoid the mode mixing appears in the decomposed results by EEMD and generates similar results with those by the VMD. Then the low-frequency pulse is extracted by composing the key IMFs with the period and energy change larger than 0.5 s and 0.1. Based on the results of visual inspection method, the threshold of energy-based pulse indicator factor in general form and exponential form is calibrated to 0.26 and 0.30 respectively. The results show that the exponential form can improve identification rate of pulse-like records. The pulse extraction and classification results of the proposed method are comprehensively compared with three benchmark methods. The classification results by the proposed method are generally consistent with the three methods. Moreover, all the four methods can accurately detect the non-pulse records. And the proposed method could detect the most pulse-like records, followed by the Zhai's method, while the Feng's and Baker's methods detected the least pulse type records. The corresponding identification rates are 91.04%, 80.60%, 63.43%, and 63.06%, respectively. Finally, the periods of the records with single mode and multiple modes estimated by the proposed method and three benchmark methods are compared. The estimated periods of the records with single mode by the proposed method are highly consistent with those by the three benchmark methods, while this consistency for the estimated periods of records with multiple modes is reduced at some extent. This is due to the interference of multiple modes embedded in the records to the pulse identification.

An efficient distortion cost function design for image steganography in spatial domain using quaternion representation

It is well known that an effective steganographic distortion cost function is essential for spatial image steganography in the framework of minimal distortion embedding. To define the embedding cost by exploiting the texture complexity characterized by the high-pass filtered image residuals has been a popular approach. In this paper, by incorporating the quaternion representation of cover images, the image complexity is evaluated in terms of quaternion magnitude and phase, and a quaternion magnitude-based distortion (QMD) and a quaternion phase-based distortion (QPD) can be defined accordingly. With Hadamard product of QMD and QPD, a novel distortion cost function, namely QMP (Quaternion Magnitude-Phase), is obtained, aiming to efficiently allocate embedding modifications in complex regions of cover images with rich texture contents to improve the statistical undetectability (i.e., security). In addition, the generalized QMP (GQMP) in exponential form is further developed to boost the security performance. Experimental results demonstrate that both the proposed QMP and its generalized variant GQMP can outperform the state-of-the-art schemes, e.g., S-UNIWARD, HiLL, and MiPOD, in terms of empirical security performance against steganalysis.

Exact and Approximate Solutions for Some Classes of the Inhomogeneous Pantograph Equation

The standard pantograph delay equation (SPDDE) is one of the famous delay models. This standard model is basically homogeneous in nature and it has been extensively studied in the literature. However, the studies on the general inhomogeneous form of such a model seem rare. This paper considers the inhomogeneous pantograph delay equation (IPDDE) with a kind of arbitrary inhomogeneous term. This arbitrary inhomogeneous term is used in different forms to generate various classes of IPDDEs. The solutions of the present classes are obtained in closed series forms which satisfy the criteria of convergence. Also, the exact solutions are determined for these classes under a certain relation between the given initial condition of the model and the initial value of the inhomogeneous term. Several classes are generated and solved when the inhomogeneous term takes the form of trigonometric, exponential, and hyperbolic functions. Some existing results in the literature are recovered as special cases of the present ones. Moreover, the behaviors of the obtained solutions are demonstrated through graphs for various kinds of IPDDEs.

Excitonic Quantum Coherence in Light Emission from CsPbBr3 Metal-Halide Perovskite Nanocrystals.

The decay of excited states via radiative and nonradiative paths is well understood in molecules and bulk semiconductors but less so in nanocrystals. Here, we perform time-resolved photoluminescence (t-PL) experiments on CsPbBr3 metal-halide perovskite nanocrystals, with a time resolution of 3 ps, sufficient to observe the decay of both excitons and biexcitons as a function of temperature. The striking result is that the radiative rate constant of the single exciton increases at low temperatures with an exponential functional form, suggesting quantum coherent effects with dephasing at high temperatures. The opposing directions of the radiative and nonradiative decay rate constants enable enhanced brightening of PL from excitons to biexcitons due to quantum effects, promoting a faster approach to the quantum theoretical limits of light emission. Ab initio quantum dynamics simulations reproduce the experimental observations of radiation controlled by quantum spatial coherence enhanced at low temperatures.

Complex-Exponential-Based Bio-Inspired Neuron Model Implementation in FPGA Using Xilinx System Generator and Vivado Design Suite.

This research investigates the implementation of complex-exponential-based neurons in FPGA, which can pave the way for implementing bio-inspired spiking neural networks to compensate for the existing computational constraints in conventional artificial neural networks. The increasing use of extensive neural networks and the complexity of models in handling big data lead to higher power consumption and delays. Hence, finding solutions to reduce computational complexity is crucial for addressing power consumption challenges. The complex exponential form effectively encodes oscillating features like frequency, amplitude, and phase shift, streamlining the demanding calculations typical of conventional artificial neurons through levering the simple phase addition of complex exponential functions. The article implements such a two-neuron and a multi-neuron neural model using the Xilinx System Generator and Vivado Design Suite, employing 8-bit, 16-bit, and 32-bit fixed-point data format representations. The study evaluates the accuracy of the proposed neuron model across different FPGA implementations while also providing a detailed analysis of operating frequency, power consumption, and resource usage for the hardware implementations. BRAM-based Vivado designs outperformed Simulink regarding speed, power, and resource efficiency. Specifically, the Vivado BRAM-based approach supported up to 128 neurons, showcasing optimal LUT and FF resource utilization. Such outcomes accommodate choosing the optimal design procedure for implementing spiking neural networks on FPGAs.

Understanding step selection analysis through numerical integration

Abstract Step selection functions (SSFs) are flexible statistical models used to jointly describe animals' movement and habitat preferences. The popularity of SSFs has grown rapidly, and various extensions have been developed to increase their utility, including the ability to use multiple statistical distributions to describe movement constraints, interactions to allow movements to depend on local environmental features, and random effects and latent states to account for within‐ and among‐individual variability. Although the SSF is a relatively simple statistical model, its presentation has not been consistent in the literature, leading to confusion about model flexibility and interpretation. We believe that part of the confusion has arisen from the conflation of the SSF model with the methods used for statistical inference, and in particular, parameter estimation. Notably, conditional logistic regression (CLR) can be used to fit SSFs in exponential form, and this model fitting approach is often presented interchangeably with the actual model (the SSF itself). However, reliance on CLR reduces model flexibility, and suggests a misleading interpretation of step selection analysis as being equivalent to a case–control study. In this review, we explicitly distinguish between model formulation and inference technique, presenting a coherent framework to fit SSFs based on numerical integration and maximum likelihood estimation. We provide an overview of common numerical integration techniques (including Monte Carlo integration, importance sampling and quadrature), and explain how they relate to popular methods used in step selection analyses. This general framework unifies different model fitting techniques for SSFs, and opens the way for improved inferential methods. In this approach, it is straightforward to model movement with distributions outside the exponential family, and to apply different SSF model formulations to the same data set and compare them with AIC. By separating the model formulation from the inference technique, we hope to clarify many important concepts in step selection analysis.

A new analytical model to fit both fine and coarse grained quartz luminescence dose response curves

We address two issues: (a) the choice of suitable analytic forms for curve fitting quartz OSL-SAR dose response data and (b) possible physical interpretations of the difference between fine (4–11 μm) and coarse (63–90 μm) grain samples. We start with a phenomenological model with three levels: an OSL-active trap, a deep trap, and a radiative center. This model yields a simple analytic form for the dose response under the optically stimulated luminescence-single-aliquot regenerative dose (OSL-SAR) protocol. By comparing with experimental data from nine quartz samples, we show that this form provides excellent agreement and may, in some cases, be better than the commonly-used double-saturating-exponential form despite this form having fewer adjustable constants. Further, because this analytic form is derived from a model, physical parameters can be inferred from the curve fits. In particular, concentrations for the OSL-active and deep electron traps can be computed and the variation in concentrations of these traps between samples is discussed. Within this model, the difference between fine and coarse grain samples is consistent with the OSL-active trap being a surface trap as opposed to a volume trap.

Exponential distribution functions for positivity-preserving lattice Boltzmann schemes: Application to 2D compressible flow simulations

A family of positivity-preserving lattice Boltzmann methods (LBMs) is proposed for compressible flow simulations in the continuum regime. It relies on the efficient collide-and-stream algorithm with a collision step based on exponential distribution functions. The latter serves as a generalization of Grad's post-collision distribution functions for which here (1) the linearized non-equilibrium contributions are replaced by their exponential forms and (2) the number of these contributions can be chosen arbitrary. In practice, post-collision moments of our exponential formulation are enforced through an iterative moment-matching approach to recover any macroscopic physics of interest, with or without external forces. This methodology directly flows from the extended framework on numerical equilibria [J. Latt et al., Philos. Trans. R. Soc. A 378, 20190559 (2020)] and goes one step further by allowing for the independent relaxation of hydrodynamic and high-order modes in a given moment space, notably, making the Prandtl number freely adjustable. The model is supplemented by a shock-capturing technique, based on the deviation of non-equilibrium moments from their equilibrium counterparts, to ensure good numerical properties of the model in inviscid and under-resolved conditions. A second exponential distribution accounts for extra degrees of freedom of molecules and allows for the simulation of polyatomic gases. To validate this novel approach and to quantify the accuracy of different lattices and moment closures, several 2D benchmark tests of increasing complexity are considered: double shear layer, linear wave decay, Poiseuille flow, Riemann problem, compressible Blasius flow over a flat plate, and supersonic flow past an airfoil. Corresponding results confirm the accuracy and stability properties of our approach for the simulation of compressible flows with LBMs. Eventually, the performance analysis further highlights its efficiency on general purpose graphical processing units.

Grammar of Impact Sensitivity: An Incremental Theory.

In this paper, we report the first attempt to quantify impact sensitivity using the second-order incremental approach based on the structural features of explosives. It has been found that impact height (h50) can be expressed via a multiplicative incremental exponential form, in which the exponents are characteristic coefficients of structural increments multiplied by their numbers in the molecule. The method was developed on a large array of experimental data (450 molecules and salts) of different energetic materials, namely, nitro compounds, peroxides, nitrogen-rich salts, heterocycles, etc., while testing of the model was performed for 170 compounds. The results demonstrate a noticeable correlation with the experimental h50 values. Thus, the corresponding R2 and RMSE for the training and test sets are 0.56 (12.5 J) and 0.63 (18.8 J), respectively. In this work, we use 53 individual structural increments, but their number can be extended, and the corresponding coefficients can be refined; this allows for increasing the prediction accuracy on-the-fly. The calculation algorithm is discussed, and the corresponding examples are presented. The performed machine-based regression analysis using genetic function approximation, multiple linear regression, and artificial neural network has proven the reasonability and informativity of the proposed incremental theory. Thus, the developed approach significantly extends our understanding of the impact sensitivity phenomenon and translates it into the category of one that can be calculated by a pocket calculator.

Dynamic collision estimator for collaborative robots: A dynamic Bayesian network with Markov model for highly reliable collision detection

In this paper, we propose a highly reliable and accurate collision estimator for robot manipulators working in human–robot collaborative environments, based on the Bayesian approach for practical uses. We assume the robot collision as a dynamic Markov process, not a static event, to reflect the transient behavior of mechanical collisions. Thus, the collision estimator can integrate the prior belief on collision and the measurements on the robot state, to produce the current belief on the robot collision in a usual recursive form. An exponential form of observation model, serving as the likelihood function, is constructed on the projected observation domain by using the multi-variate statistical information of empirical models of collision and non-collision cases. The proposed method is validated by using a commercial 7 degree-of-freedom (DOF) collaborative robot arm with random impacts along the links while it is in motion. Results show that the proposed collision estimator achieves a compelling performance with collision estimation time of 8.86ms on average, an overall accuracy of 99.47%, and zero occurrence of false alarm.