Combination Of Exponential Functions Research Articles

Hidden symmetries of power-law inflation

Abstract A scalar field with an exponential potential has been proposed as a model of inflation (called power-law inflation). Although it admits an exact solution, the integrability of the system has not been shown. We uncover the hidden symmetries behind the system by utilizing the Eisenhart lift of field theories. We find that a conformal Killing vector field in the field space exists only for a particular combination of exponential functions that includes a single exponential potential. This implies the existence of additional conserved quantity and explains the integrability of the system.

Description of Acid Battery Operating Parameters

In this paper, the operating principles of the acid battery and its features are discussed. The results of voltage tests containing the measurements conducted at the terminals of a loaded battery under constant load conditions, and dependent on time, are presented. The article depicts the principles of the development of electric models of acid batteries and their various descriptions. The principles for processing the results for the purpose of the determination and description of the battery model are characterized. The characteristics under stationary and non-stationary conditions are specified using glued functions and linear combinations of exponential functions, and the electrical parameters of the battery are determined as the components of the circuit, i.e., its electromotive force, resistance, and capacity. The dynamic characteristic of the battery in the form of transmittance was determined, using the Laplace transform. Possible uses of the crankshaft driving signals as diagnostic signals of the battery, electric starter, and internal combustion engine are also indicated.

Alternative evaluation of voltage at top of transmission line tower stricken by lightning

The object of the research is a circuit that simulates a lightning strike to a tower of 220 kV power transmission line, taking into consideration the reflection of a current wave from 10 nearest towers. Computation of the voltage arising at the top of the stricken tower is necessary further to determine the lightning performance of transmission line by various methods. For Indian conditions, the average number of lightning strikes to this power line per 100 kilometers per year is about 77, which is a fairly high figure. As a rule, for the tasks of lightning protection, the lightning current is approximated by some analytical expression. In most cases, such expressions are various combinations of exponential functions. However, the waveform of real lightning currents on oscillograms differs significantly from the waveform attributed to them and approximated by relatively simple exponential expressions. For a more detailed study of transient processes caused by thunderstorm activity, there is a need to use oscillograms of real lightning currents when modeling. The problem of determining the voltage at the top of the stricken transmission line tower was solved using circuit simulation. To simulate the lightning current, digitized oscillograms of real lightning currents with peak values of –5.256 kA and –133.586 kA were applied. The article shows that the proposed approach gives a more accurate and visual representation of the transient process at the top of the stricken tower than the approximation of the lightning current by simple exponential expressions. Applying a simplified exponential description of the lightning current leads not only to a simplification of the nature of the transient process at the tower top, but also to an underestimation of the results to 8.8%. The selection of the equivalent circuit for the power line towers also affects the result. Representation of towers in the equivalent circuit with lumped inductances leads to slightly higher values compared to application of surge impedances in the circuit. In this case, the smaller the current amplitude, the greater the difference (8.6 % in the domain of low currents and 1.9 % in the domain of high currents). Since this leads to some reserve during the computations of lightning performance, it is recommended to use an equivalent circuit with lumped inductance for a transmission line tower. The conducted research contributes to the development of methods for calculating the lightning performance of power lines and extends the scope of application of circuit simulation programs.

Analysis of sandwich single cantilever beam test specimen with graded core

In previous research, an analysis model named the ‘beam on Vlasov foundation model’ was proposed to theoretically analyze the compliance and energy release rate of sandwich single cantilever beam (SCB) test specimens. The validity of the analytical model was confirmed when it was applied to sandwich specimens with homogeneous cores. In this study, the analysis model is applied to the analysis of a sandwich SCB test specimen where the Young’s modulus of the core varies along the thickness direction. The procedures to obtain the solution for two specific cases are described. One is the case that the Young’s modulus of the core varies in the thickness direction as an exponential function. The other is the case that the Young’s modulus of the core varies as a combination of exponential functions which is symmetric with respect to the midplane of the core. By comparing with finite element analysis, it is confirmed that the beam analysis model can provide reliable results even when the core is inhomogeneous in the thickness direction.

Analytical solution to one-dimensional mathematical model of flow and morphological evolution in open channels

The evolution of open-channel flow and morphology can be simulated by one-dimensional (ID) mathematical models. These models are typically solved by numerical or analytical methods. Because the behavior of variables can be explained by explicit mathematical determinations, compared to numerical solutions, analytical solutions provide fundamental and physical insights into flow and sediment transport mechanisms. The singular perturbation technique derives a hierarchical equation of waves and describes the evolutionary nature of disturbances in hyperbolic systems. The wave hierarchy consists of dynamic, diffusion, and kinetic waves. These three types of waves interact with each other in the process of propagation. Moreover, the Laplace transform is implemented to transform partial differential equations into ordinary differential equations. Analytical expressions in the wave front are subtracted by the approximation of kinetic and diffusion models. Moreover, an analytical solution consists of a linear combination of the kinetic wave front and the diffusion wave front expressions, pursuing to describe the physical mechanism of motion in open channels as completely as possible. Analytical solutions are presented as a combination of exponential functions, hyperbolic functions, and infinite series. The obtained analytical solution was further applied to the simulation of flood path and morphological evolution in the Lower Yellow River. The phenomenon of increased peak discharge in the downstream reach was successfully simulated. It was encouraging that the results were in good agreement with the observed data.

Mathematical models for COVID-19 infection estimation: essential considerations and projections in Colombia

To estimate the COVID-19 infection behavior in Colombia using mathematical models. Two mathematical models were constructed to estimate imported confirmed cases and related confirmed cases of COVID-19 infection in Colombia, respectively. The phenomenology of imported confirmed cases is modeled with sigmoidal function, while related confirmed cases are modeled using a combination of exponential functions and polynomial algebraic functions. The fitting algorithms based on least squares methods and direct search methods are used to determine the parameters of the models. The sigmodial model performs a highly convergent estimation of the reported confirmed cases of COVID-19 infection to May 28, 2020. This model achieved a prediction error of 0.5 % measured using the normalized root mean square error. The model of the confirmed cases reported as related shows a 3.5 % prediction error and a low bias of -0.01 associated with overestimation. This work shows that the mathematical models allow to predict the behavior of the infection efficiently and effectively by COVID 19 in Colombia when the imported cases and the related cases of infection are independently considered.

Laser paint removal monitoring based on time-resolved spectroscopy.

Laser paint removal is a new cleaning technology with high efficiency. Dynamic monitoring and closed-loop control of the laser paint removal process are key to reducing the risk of metal substrate damage and to achieving the best cleaning. In this paper, the time-resolved characteristics of the elemental peaks in the laser-induced breakdown spectrum of paints and substrate were studied by using the combination of a monochromator (or a bandpass filter) and a photomultiplier tube (PMT) detector. The results show that the intensity of the elemental spectrum peak of the paint has a sudden drop while the intensity of the elemental spectrum peak of the substrate has a sudden increase when the paint is removed. The time-resolved signals can be fitted by double exponential functions, which are combinations of exponential functions with a longer and a shorter lifetime, respectively. The relative ratios of the coefficients of the shorter and longer lifetimes (A s h o r t /A l o n g ) at the wavelength correspond to the elements in paint increasing suddenly while decreasing suddenly at the wavelength corresponding to the substrate. The intensity of the elemental spectrum peaks of paints and substrate and the ratio (A s h o r t /A l o n g ) can be used to monitor the laser paint removal process in real time to reduce the damage risk of the metal substrate and achieve the purpose of efficient cleaning with low cost.

Deformation Monitoring Indexes of High Concrete Arch Dam Considering Time-varying Effect

Deformation is the most intuitive reflection of comprehensive behavior of concrete dams, and it is of great significance to determine the deformation monitoring indexes for dam abnormality identifying. In this paper, deformation monitoring indexes of arch dam were determined in the form of causal components, and the time-varying effect caused by the degradation of dam material properties was considered. The hydraulic component in monitoring index was determined into three grades by water density overloading method, and the appearances of dam heel yield depth coming up to the curtain axis, the catastrophes of dam body yield volume ratio and dam radial displacement, were respectively taken as the abnormal symbols. The temperature component was determined for two operation conditions, with respect to the measured high and low temperature periods. As for the time effect component, it was initially calculated by FEM with the environmental damage-considered rheological constitutive models, and fitted by the combination of exponential function and periodic function. A case study was performed to give more detailed introductions for determining the proposed deformation monitoring indexes. Research results show that the components of temperature and time-effect occupy a large proportion in monitoring index of arch dams.

Lump Solutions and Interaction Solutions for the Dimensionally Reduced Nonlinear Evolution Equation

In this paper, by means of the Hirota bilinear method, a dimensionally reduced nonlinear evolution equation is investigated. Through its bilinear form, lump solutions are obtained. We construct interaction solutions between lump solutions and one soliton solution by choosing quadratic functions and exponential function. Interaction solutions with the combinations of exponential functions and sine function are also given. Meanwhile, the figures of these solutions are plotted. The dynamical characteristics and properties of obtained solutions are discussed, respectively. The results show that the corresponding physical quantities and properties of nonlinear waves are associated with the values of the parameters.

Increased expression diversity buffers the loss of adaptive potential caused by reduction of genetic diversity in new unfavourable environments

Mechanisms underlying adaptation to rapid environmental change are issues in evolutionary biology. It is widely accepted that reduction in genetic diversity when suddenly exposed to an unfavourable environment limits the adaptive potential of populations. With growing empirical evidence that expression diversity is likely to increase in the new environment, the role that expression diversity plays in adaptation needs to be theorized. Here, we first established a negative exponential relationship between expression diversity and genetic diversity using a phenomenological differential equation. We then derived a complex trade-off relationship between the changes of expression and genetic diversity, which followed a combination of exponential functions. Furthermore, we found the increase in expression diversity could buffer the loss of adaptive potential as genetic diversity decreased to a certain extent. These theoretical deductions were validated by transcriptomic data of Miscanthus lutarioriparius grown in two experimental fields and supported by good fit and random simulation. These results suggest that increased expression diversity may compensate the loss of genetic diversity and allow the populations to maintain a certain level of phenotypic variation to cope with sudden environmental change. This may buffer the quick diminishing of adaptive potential and consequently increases the change of adaptation to the new environment.

Novel solitary wave solutions for the [formula omitted]-dimensional extended Jimbo–Miwa equations

This paper retrieves new periodic solitary wave solutions for the (3+1)-dimensional extended Jimbo–Miwa equations, based on the Hirota bilinear method, by utilizing Maple software. As a result, the Hirota bilinear method is successfully employed and acquired several classes of solitary wave solutions in terms of a new combination of exponential function, trigonometric function and hyperbolic functions. All solutions have been verified back into its corresponding equation by Maple. We depicted the physical explanation of the extracted solutions with the free choice of the different parameters by plotting some 3D and 2D illustrations. Finally, it has been illustrated that the executed method is robust and more efficient than other methods and the obtained solutions are trustworthy in the applied sciences.

Solar Irradiance Estimation of Photovoltaic Module based on Thevenin Equivalent circuit Model

Abstract— This Paper presents an accurate method for estimation of solar irradiance of a PV module. The solar irradiance estimation method is derived using the efficient approximation of a PV model based on Thevenin equivalent circuit . The short linear combination of exponential function is used for constructing a mathematical function that is used to express the Thevenin voltage source of a PV panel. The proposed algorithm for estimation of solar irradiance use Newton Raphson method based on extensively analysis of a photovoltaic mathematical model and it is applied for a wide range of operating point variations . The proposed method has the ability to estimate the parameters, which are required for constructing the mathematical model of a PV in addition to estimate the solar irradiance . The accuracy and effectiveness of the proposed algorithm is demonstrated in MATLAB for a photovoltaic panel by using experimental data which is obtained from data sheet under different operating conditions. Index Terms— photovoltaic modeling; pyranometer; solar irradiance Photovoltaic Model; Thevenin equivalent circuit System; Exponential sum models.

Transverse magnetic field of infinite line source placed on ground surface

An exact explicit expression is derived for the Sommerfeld integral describing the surface transverse magnetic field generated by an infinite line source placed on a plane air–earth boundary. The spatial distribution of the transverse magnetic field is given in series form, as a linear combination of exponential functions. The accuracy and convergence rate of the proposed formulation are investigated through numerical simulations.

Physical essence of the multibody contact-sliding at atomic scale

Investigation the multibody contact-sliding occurred at atomic discrete contact spot will play an important role in determine the origin of tribology behavior and evaluates the micro-mechanical property of nanomaterials and thus optimizing the design of surface texture. This paper carries out large scale parallel molecular dynamics simulation on contact-sliding at atomic scale to uncover the special physical essence. The research shows that some kind of force field exists between nanodot pair and the interaction can be expressed by the linear combination of exponential function while the effective interaction distance limited in 1 angstrom for nanodot with several tens of nanometer diameter. The variation tendency about the interaction force between nanodot array is almost the same between nanodot pairs and thus the interaction between two nanodot array can be characterized by parallel mechanical spring. Multibody effect which dominates the interaction between atoms or molecules will gradually diminish with the increasing of length scales.

Albedo problem in a semi-infinite medium with pure-triplet scattering

Particle transport problem in an absorbing, scattering and homogeneous semi-infinite medium with pure-triplet scattering is considered. The medium is subjected to externally incident radiation. The problem is solved by using an efficient and accurate method of analysis which utilizes trial functions based on Case’s eigenmodes plus a linear combination of exponential functions. This trial function is used in the integral form of the transport equations, which transfer it to a system of algebraic equations in unknown expansion coefficients. The expansion coefficients are used to calculate some physical quantities as the albedo and the net flux at the boundary of the medium. Numerical results are reported and compared with that calculated by another trial functions.

Identification of power distribution mixtures through regression of exponentials

Given two independent non-degenerate positive random variables X and Y, Lukacs (Ann Math Stat 26:319–324, 1955) proved that X/(X + Y) and X + Y are independent if and only if X and Y are gamma distributed with the same scale parameter. In this work, under the assumption X/U and U are independent, and X/U has a \({{\mathcal Be}(p,\,q)}\) distribution, we characterize the distribution of (U, X) by the condition E(h(U − X)|X) = b, where h is allowed to be a linear combination of exponential functions. Since the assumption for X and U above is equivalent to X|U being \({\mathcal{B}e(p,\,1)}\) scaled by U, hence our results can be viewed as identification of a power distribution mixture.

Exact firing-rate response of the integrate-and-fire neuron receiving finite amplitude excitatory and inhibitory post-synaptic potentials

Neurons in active networks are subject to a fluctuating synaptic drive comprising excitatory and inhibitory post-synaptic potentials arriving at high rates relative to the integration time of the cell. The standard approach for treating stochastic synaptic currents and conductances has been to approximate them by a Gaussian white-noise or coloured-noise process in which the amplitudes of the underlying post-synaptic potentials are considered small. Importantly, this diffusion approximation has allowed for various firing-rate properties of populations of fluctuation-driven neurons to be calculated using Johanessma’s [1] and Ricciardi’s [2] Fokker-Planck framework. This framework in turn has proven to be a powerful theoretical tool for deriving the emergent states of coupled neuronal networks as a function of the properties of constituent neurons. The Gaussian/diffusion model of fluctuating synaptic conductance is in widespread use theoretically, computationally and experimentally and, in many cases, is an excellent approximation to the synaptic drive experienced by neurons in active networks in vitro or in vivo. The typical synaptic coupling between neurons in many brain regions, however, is often sufficiently strong such that relatively few synchronous synaptic events are required to bring a neuron from rest to the spiking threshold. For example, pairs of neocortical layer-5 pyramidal cells form synaptic connections with amplitudes in the range 0.5mV-6mV with a mean of 1.3mV [2] and, when correlations are taken into account, near-synchronous pre-synaptic activity will lead to even stronger aggregate EPSPs. Given that the difference between rest and threshold is around 10mV for this class of cell [3] it is likely that finite-amplitude effects missed by the standard Gaussian/diffusion approximation will play an important role in shaping the steady-state and response properties of neurons in vivo. Here we present a significant generalization of the standard Gaussian/diffusion approach to treating synaptic fluctuations by exactly solving the firing-rate response for a neuron receiving finite-amplitude Poissonian excitatory and inhibitory post-synaptic potentials [5]. The mathematical framework involves the solution of a master equation and can be applied to synaptic-amplitude distributions decomposable into combinations of exponential functions -similar to those seen in experiment. It will be shown that this more general mathematical description of the neuronal response to stochastic synaptic dynamics is significantly richer and qualitatively distinct from that predicted by the Gaussian/diffusion approximation. As well as providing analytical solutions for the firing-rate response of the leaky integrate-and-fire model, it will be demonstrated how an efficient numerical scheme can be used to calculate, numerically but exactly, the response properties of non-linear integrate-and-fire neurons and also to better capture the effects of synaptic reversal potentials.

Closed Form Solutions to L2-Sensitivity Minimization of Second-Order State-Space Digital Filters with Real Poles

This letter proposes closed form solutions to the L2-sensitivity minimization of second-order state-space digital filters with real poles. We consider two cases of second-order digital filters: distinct real poles and multiple real poles. In case of second-order digital filters, we can express the L2-sensitivity of second-order digital filters by a simple linear combination of exponential functions and formulate the L2-sensitivity minimization problem by a simple polynomial equation. As a result, the minimum L2-sensitivity realizations can be synthesized by only solving a fourth-degree polynomial equation, which can be analytically solved.

Moment Approach to Deriving a Unified Parallel Viscous Stress in Magnetized Plasmas

In the moment approach, a parallel heat flux and a viscous stress are derived for arbitrary collisionality, which is becoming increasingly important in emerging concept devices. This derivation improves upon previous derivations by using the full linearized collision operators instead of the pitch-angle scattering operator and also by including the ion-electron collision operator. The parallel viscous stress can be computed by integrating thermodynamic drives along a magnetic field line weighted by kernel functions that are simple linear combinations of exponential functions. The convergence of the ion viscous stress is verified for sinusoidally varying drives with increasing number of moments.

Filtering of a signal in the form of a linear combination of decaying exponents in the presence of additive noise

A new local method for filtering (smoothing) a linear combination of exponential functions with additive measurement noise is proposed. This method is based on the application of the moving geometric mean with the use of two to five neighboring samples. A numerical example of Lanczos is used to show that the proposed method allows reduction of the rms error with respect to the initial one by a value of 14–50%.