Form Of Analytical Functions Research Articles

Effect of the analytical form of the dipole-moment function on the rotational intensity distributions in the high-overtone vibrational bands of carbon monoxide

For modelling the line intensities of the low-overtone bands (Δv≤3 in CO), any potential-energy and dipole-moment functions (PEFs and DMFs) that describe the available data can be used. This is not true for the high-overtone bands. The Born-Oppenheimer molecular functions as functions of the interatomic separation r must obey some restrictions in the complex plane of r. In particular, the PEF and DMF must have common singularities in the complex plane and the DMF should not rapidly increase upon displacement from the real axis. Two model DMFs are considered for CO, a fully regular function over the entire complex plane and an irregular one with two branch points corresponding to complex intersections of the ground-state PEF with the excited-states ones. The predictions of the 7-0 band made with the regular DMF turn out to be unstable with respect to small variations in the database, which is associated with very rapid variations of this DMF. On the contrary, the predictions of the irregular one are stable and therefore are considered to be more reliable.

Optimisation problem for some class of hybrid differential-difference systems with delay

In the paper, the linear differential-difference dynamic systems with delayed arguments are considered. Such systems have a lot of application areas, in particular, processes with repetitive and learning structure. We apply the method of the separation hyperplane theorem for convex sets to establish optimality conditions for the control function to drive the trajectory to zero equilibrium state in the fastest possible way. For the special case of the integral control constraints, the proposed method is detailed to establish an analytical form of the optimal control function. The illustrative example is given to demonstrate the obtained results with the step-by-step calculation of the basic elements of the optimal control.

Testing momentum dependence of the nonperturbative hadron structure in a global QCD analysis

We discuss strategies for comparisons of nonperturbative QCD predictions for parton distribution functions (PDFs) with high-energy experiments in the region of large partonic momentum fractions $x$. Analytic functional forms for PDFs cannot be uniquely determined solely based on discrete experimental measurements because of a mathematical property of mimicry of PDF parametrizations that we prove using a representation based on B\'ezier curves. Predictions of nonperturbative QCD approaches for the $x$ dependence of PDFs instead should be cast in a form that enables decisive comparisons against experimental measurements. Predictions for effective power laws of $(1-x)$ dependence of PDFs may play this role. Expectations for PDFs in a proton based on quark counting rules are compared against the effective power laws of $(1-x)$ dependence satisfied by CT18 next-to-next-to-leading order parton distributions. We comment on implications for studies of PDFs in a pion, in particular on the comparison of nonperturbative approaches with phenomenological PDFs.

Analytical Models for Optimal Design of a Trapezoidal Composite Channel Cross-Section

Abstract In this paper, several analytical models are presented for the optimal design of a trapezoidal composite channel cross-section. The objective function is the cost function per unit length of the channel, which includes the excavation and lining costs. To define the system, design variables including channel depth, channel width, side slopes, freeboard, and roughness coefficients were used. The constraints include Manning’s equation, flow velocity, Froude number, and water surface width. The Simultaneous Perturbation Stochastic Approximation (SPSA) algorithm was used to solve the optimization problem. The results are presented in three parts; in the first part, the optimal values of the design variables and the objective function are presented in different discharges. In the second part, the relationship between cost and design variables in different discharges is presented in the form of conceptual and analytical models and mathematical functions. Finally, in the third part, the changes in the design variables and cost function are presented as a graph based on the discharge variations. Results indicate that the cost increases with increasing water depth, left side slope, equivalent roughness coefficient, and freeboard.

Impacts of random filling on spin squeezing via Rydberg dressing in optical clocks

We analyze spin squeezing via Rydberg dressing in optical lattice clocks with random fractional filling. We compare the achievable clock stability in different lattice geometries, including unity-filled tweezer clock arrays and fractionally filled lattice clocks with varying dimensionality. We provide practical considerations and useful tools in the form of approximate analytical expressions and fitting functions to aid in the experimental implementation of Rydberg-dressed spin squeezing. We demonstrate that spin squeezing via Rydberg dressing in one-, two-, and three-dimensional optical lattices can provide significant improvements in stability in the presence of random fractional filling.

Distributed tracking-ADMM approach for chance-constrained energy management with stochastic wind power in smart grid

This paper proposes a distributed strategy to the chance-constrained energy management for smart grid with penetration of stochastic wind power. The energy management model is constructed including chance constraints of spinning reserves for the sake of guaranteeing the maximum utilization of wind power on the basis of reliability. With the available wind power characterizing by Weibull distribution, the chance constraints can be converted into deterministic ones by the derived analytical form of inverse cumulative distribution function. Although the original problem is transformed into a typical convex optimization problem, the tight coupling of constraints presents challenges to the design of distributed strategy. As such, we formulate the problem into a compact form with each generator unit depending on individual decision variables, instead of the common form with decision vector being the collection of all local decision variables. Then, by developing a new initialization method and an adaptive weight matrix selection method, a distributed strategy based on tracking Alternating Direction Method of Multipliers (ADMM) is proposed to solve the model. The simulation results indicate that the proposed distributed strategy achieves comparable performance to the corresponding centralized scenario, and better performance than distributed consensus-based ADMM in the related literature. Moreover, the validity of the proposed distributed strategy is confirmed in one day chance-constrained energy management with stochastic wind power.

Bayesian uncertainty quantification for guided-wave-based multidamage localization in plate-like structures using Gibbs sampling

In this article, a new Bayesian approach for guided-wave-based multidamage localization by employing Gibbs sampling is proposed. By using the information of time-of-flight (ToF) embedded in guided wave signals, the posterior probability distributions of three parameter groups, that is, the horizontal and vertical coordinates of the multidamage locations (x, y) and wave velocity v, are characterized using Gibbs sampling samples. To obtain the analytical form of the conditional posterior probability density function of each parameter group conditional on the other two and the available ToF data, a first-order Taylor expansion of the nonlinear ToF-based damage localization model with respect to each parameter group is performed. Two Gibbs sampling algorithms are proposed, which differ in their strategies to address the posterior uncertainty of the prediction error parameter; however, both algorithms iteratively sample from conditional posterior probability density functions of three parameter groups. Therefore, the effective number of dimensions for Gibbs sampling is always three, regardless of the number of defects. The final damage localization results are obtained by grouping all ToFs and then comparing the posterior uncertainty of localization results of each grouping scheme to obtain the most reliable sampling results among all candidates. The proposed method not only identifies the group velocity but also localizes multiple defects by sharing the same characteristics of damage localization. Furthermore, this method can quantify the uncertainty of multidamage localization to automatically find the most reliable damage locations. The effectiveness and robustness of the proposed algorithms are validated by both numerical and experimental examples.

Analysing the topographic form of stratovolcanoes

The goal of this study is to explore “ideal” analytic functional forms that best fit the topographic shape of N = 190 isolated stratovolcanoes from around the world. Using a stochastic model for the piling of lava flows to demonstrate one set of physical processes that give rise to a stratovolcano's topography, we find that although the ideal form fits well for many stratovolcanoes, there exist deviations from this shape—particularly where the edifice is more protuberant than expected. We explore sub-surface factors—such as the emplacement of magma beneath a volcano edifice—that may be responsible for these deviations and estimate the relative contribution of dyke intrusions on the height and shape of a volcanic edifice versus the constructional piling of lava. From this comparison, we are able to gain more insight into the role of important subsurface physical factors, such as chamber depth and total intruded volume.We find that low basal ellipticity is a characteristic of nearly all volcanoes that fit the ideal stratovolcano form. By fitting along each quadrant bisected by either a semi-major or semi-minor ellipse axis, we find that the volcanoes chosen in this study fall into four groups: (1) volcanoes that are axisymmetric and fit the ideal form well, (2) volcanoes that are not axisymmetric but may be fit to the ideal form using different fitting parameters in each quadrant, (3) volcanoes that have axisymmetric protuberant deviations in all quadrants, and (4) volcanoes that have non-axisymmetric protuberant deviations that are only present in a couple of quadrants. We show that, in some cases, the non-axisymmetry of volcanoes in category (2) may be understood as the result of the piling of eruptive products along a regional planar trend. For edifices in categories (3) and (4), we model protuberant deviations as the result of a statistical distribution of elastic tensile dislocations representing dykes, which emanate from a magma reservoir and cause uplift at the surface. We support this interpretation of the “excess” topography in a few case studies where independent geologic and geophysical data corroborate our model results.

Thermodynamic analysis on two plasticity models considering the time-dependency of metal materials

Time-dependency and kinematic hardening are two important features of metal materials, and the corresponding behaviors are the emphases of numerical modeling for the safety evaluation of engineering structures. The robust models should not only capture the deformation characteristics, but also satisfy the fundamental thermodynamic laws. This paper aims at discussing the thermodynamic consistency of a unified viscoplasticity (UVP) model and a non-unified viscoplasticity (NUV) model that possess many characteristics in common. Within the thermodynamic framework, the analytical form of a dissipation potential function is defined, and thus the constitutive equations are derived with the introduction of Helmohltz energy function. Finally, a series experimental data of a Nickel based supper alloy is employed to verify the utility of the models. Different from the UVP model, the contribution of creep in a dissipation potential is considered in NUV model, which significantly improves the accuracy of modeling results in the creep and stress relaxation.

A simple fracture mechanics based Charpy-V impact energy criterion for plastic collapse

Charpy-V based design methods usually provide advice on the relation between design temperature and some Charpy-V transition criterion. Thus, the methods try to safeguard against brittle fracture. Usually, methods are limited to small plasticity where the stresses are below yield strength. Sometimes, it may be important also to ensure that the structure sustains loads up to the plastic limit load. In this case the design is based on plastic collapse. So far, no simple methods to evaluate plastic collapse based on the Charpy-V test have been developed. Here, a simple fracture mechanics based plastic collapse criterion for Charpy-V impact energy is presented. The criterion contains a requirement on impact energy and temperature, both of which must be fulfilled to ensure achievement of plastic limit load. The criterions are given in the form of simple analytical functions and they account for yield strength, strain hardening and section thickness.

Quasi-equilibrium predictions of water desorption kinetics from rapidly-heated metal oxide surfaces

Controlling sub-microsecond desorption of water and other impurities from electrode surfaces at high heating rates is crucial for pulsed power applications. Despite the short time scales involved, quasi-equilibrium ideas based on transition state theory (TST) and Arrhenius temperature dependence have been widely applied to fit desorption activation free energies. In this work, we apply molecular dynamics (MD) simulations in conjunction with equilibrium potential-of-mean-force (PMF) techniques to directly compute the activation free energies (ΔG*) associated with desorption of intact water molecules from Fe2O3 and Cr2O3 (0001) surfaces. The desorption free energy profiles are diffuse, without maxima, and have substantial dependences on temperature and surface water coverage. Incorporating the predicted ΔG* into an analytical form gives rate equations that are in reasonable agreement with non-equilibrium molecular dynamics desorption simulations. We also show that different ΔG* analytical functional forms which give similar predictions at a particular heating rate can yield desorption times that differ by up to a factor of four or more when the ramp rate is extrapolated by 8 orders of magnitude. This highlights the importance of constructing a physically-motivated ΔG* functional form to predict fast desorption kinetics.

Mid-rapidity dependence of pion production in $$p-p$$ and $$A-A$$ collisions.

The calculation of inclusive spectra of pions produced in $pp$ and $AA$ collisions as a function of rapidity $y$ is presented within the self-similarity approach. It is shown that at not large rapidities $y$ one can obtain the analytical form of the self-similarity function $\Pi(y,p_t)$ dependent of $y$ and hadron transverse momentum $p_t$. A satisfactory description of data on the rapidity spectra at $|y|\leq$ 0.3 is illustrated within a good agreement. The universal energy dependence of these spectra is also shown.

Realizable hyperuniform and nonhyperuniform particle configurations with targeted spectral functions via effective pair interactions.

The capacity to identify realizable many-body configurations associated with targeted functional forms for the pair correlation function g_{2}(r) or its corresponding structure factor S(k) is of great fundamental and practical importance. While there are obvious necessary conditions that a prescribed structure factor at number density ρ must satisfy to be configurationally realizable, sufficient conditions are generally not known due to the infinite degeneracy of configurations with different higher-order correlation functions. A major aim of this paper is to expand our theoretical knowledge of the class of pair correlation functions or structure factors that are realizable by classical disordered ensembles of particle configurations, including exotic "hyperuniform" varieties. We first introduce a theoretical formalism that provides a means to draw classical particle configurations from canonical ensembles with certain pairwise-additive potentials that could correspond to targeted analytical functional forms for the structure factor. This formulation enables us to devise an improved algorithm to construct systematically canonical-ensemble particle configurations with such targeted pair statistics, whenever realizable. As a proof of concept, we test the algorithm by targeting several different structure factors across dimensions that are known to be realizable and one hyperuniform target that is known to be nontrivially unrealizable. Our algorithm succeeds for all realizable targets and appropriately fails for the unrealizable target, demonstrating the accuracy and power of the method to numerically investigate the realizability problem. Subsequently, we also target several families of structure-factor functions that meet the known necessary realizability conditions but are not known to be realizable by disordered hyperuniform point configurations, including d-dimensional Gaussian structure factors, d-dimensional generalizations of the two-dimensional one-component plasma (OCP), and the d-dimensional Fourier duals of the previous OCP cases. Moreover, we also explore unusual nonhyperuniform targets, including "hyposurficial" and "antihyperuniform" examples. In all of these instances, the targeted structure factors are achieved with high accuracy, suggesting that they are indeed realizable by equilibrium configurations with pairwise interactions at positive temperatures. Remarkably, we also show that the structure factor of nonequilibrium perfect glass, specified by two-, three-, and four-body interactions, can also be realized by equilibrium pair interactions at positive temperatures. Our findings lead us to the conjecture that any realizable structure factor corresponding to either a translationally invariant equilibrium or nonequilibrium system can be attained by an equilibrium ensemble involving only effective pair interactions. Our investigation not only broadens our knowledge of analytical functional forms for g_{2}(r) and S(k) associated with disordered point configurations across dimensions but also deepens our understanding of many-body physics. Moreover, our work can be applied to the design of materials with desirable physical properties that can be tuned by their pair statistics.

Remaining useful lifetime prediction for equipment based on nonlinear implicit degradation modeling

Nonlinearity and implicitness are common degradation features of the stochastic degradation equipment for prognostics. These features have an uncertain effect on the remaining useful life (RUL) prediction of the equipment. The current data-driven RUL prediction method has not systematically studied the nonlinear hidden degradation modeling and the RUL distribution function. This paper uses the nonlinear Wiener process to build a dual nonlinear implicit degradation model. Based on the historical measured data of similar equipment, the maximum likelihood estimation algorithm is used to estimate the fixed coefficients and the prior distribution of a random coefficient. Using the on-site measured data of the target equipment, the posterior distribution of a random coefficient and actual degradation state are step-by-step updated based on Bayesian inference and the extended Kalman filtering algorithm. The analytical form of the RUL distribution function is derived based on the first hitting time distribution. Combined with the two case studies, the proposed method is verified to have certain advantages over the existing methods in the accuracy of prediction.

ADT: A Generalized Algorithm and Program for Beyond Born-Oppenheimer Equations of "N" Dimensional Sub-Hilbert Space.

The major bottleneck of first principle based beyond Born-Oppenheimer (BBO) treatment originates from large number and complicated expressions of adiabatic to diabatic transformation (ADT) equations for higher dimensional sub-Hilbert spaces. In order to overcome such shortcoming, we develop a generalized algorithm, "ADT" to generate the nonadiabatic equations through symbolic manipulation and to construct highly accurate diabatic surfaces for molecular processes involving excited electronic states. It is noteworthy to mention that the nonadiabatic coupling terms (NACTs) often become singular (removable) at degenerate point(s) or along a seam in the nuclear configuration space (CS) and thereby, a unitary transformation is required to convert the kinetically coupled (adiabatic) Hamiltonian to a potentially (diabatic) one to avoid such singularity(ies). The "ADT" program can be efficiently used to (a) formulate analytic functional forms of differential equations for ADT angles and diabatic potential energy matrix and (b) solve the set of coupled differential equations numerically to evaluate ADT angles, residue due to singularity(ies), ADT matrices, and finally, diabatic potential energy surfaces (PESs). For the numerical case, user can directly provide ab initio data (adiabatic PESs and NACTs) as input files to this software or can generate those input files through in-built python codes interfacing MOLPRO followed by ADT calculation. In order to establish the workability of our program package, we selectively choose six realistic molecular species, namely, NO2 radical, H3+, F + H2, NO3 radical, C6H6+ radical cation, and 1,3,5-C6H3F3+ radical cation, where two, three, five and six electronic states exhibit profound nonadiabatic interactions and are employed to compute diabatic PESs by using ab initio calculated adiabatic PESs and NACTs. The "ADT" package released under the GNU General Public License v3.0 (GPLv3) is available at https://github.com/AdhikariLAB/ADT-Program and also as the Supporting Information of this article.

Use of active synthesis in vibration reduction using an example of a four-storey building

The article presents a method of active vibration reduction of vibrating mechanical systems. This method is based on the properties of positive rational functions, which in the case of discrete dynamical systems correspond to the characteristic function describing such systems. The method formalized uses methods for decomposing positive rational functions. The advantage of this approach is taking into account the analytical form of a characteristic function of the system being tested and the vibration-reducing force, as well as the conditions that the system should meet in the event of an active vibration-reducing force. In addition, in the proposed method, the desired dynamic properties of the system and the vibration-reducing force can be defined in such a way that the determined parameters of the active force affect all forms of the natural vibrations of the examined system. Based on the formalized methodology, the force reducing the vibrations of a four-storey frame to the desired displacement amplitude was determined. The impact of the place of application of the specific active force on the reduction of vibration of the tested object was also taken into account. The vibrations of the tested structure’s model were caused by kinematic excitation with a harmonic course and an amplitude corresponding to an earthquake of a magnitude of 5 on the Richter scale. To verify the determined force reducing the vibrations of the object and to create a visualisation of the analysed phenomenon, a dynamic analysis of the building structure was carried out using PLM Siemens NX 12 software.

New approach of analytical inversion of Laplace transform for some cases

Laplace transform is a useful tool for solving of dynamic elasticity problems. However, the problem of analytical inversion of Laplace transform has not yet completely solved. Therefore, it is relevant to consider the new methods that allow to derive the analytical form of the original function by the known transform.In this paper, the new method of analytical inversion of Laplace transform for the transforms of the certain form containing exponents in the denominator that linearly depend on Laplace transform parameter is proposed. The cases of correlation between the exponential indices are considered. The theorem is proved according to which the transform is expanded into the Taylor series, and the original function is derived by term-by-term application of the inverse Laplace transform. The correctness of the term-by-term application of the inverse Laplace transform is proved. The results derived by the use of the new method are verified by comparing them with the previously known formulas. The originals of Laplace transforms that were not found in the literature are derived.

Wigner Distribution of Accelerated TripartiteW-state

In this contribution, we introduce an analytical form of the Wigner distribution function for any tripartite quantum system. These forms are extended to the accelerated tripartite quantum systems of W-state type. The quantum and classical correlations are predicted by negative/ positive values, respectively of the Wigner function. It is shown that for the non-accelerated W-state, the quantum correlations are predicted at large values of the distribution angles. However, for the accelerated W-state, the quantum correlations are predicted at different intervals of distribution angle and different initial acceleration. Due to the acceleration the quantum correlation decays, where the decay rate increases as the number of the accelerated qubits increases. We show that, the local noisy channels; bit and phase bit channels curb the decoherence of the quantum correlation. The ability of the phase-bit channel to defend the quantum correlation is stronger than that shown for the bit-channel, where it decreases if all the three qubits are accelerated. Moreover, we quantify, the amount of the survival quantum correlation by using the volume of the negativity. Our results show that, the volume of negativity doesn’t vanish even at large acceleration, which means thatW-state keeps some of its quantum correlation during the acceleration process.

A Computational Approach for Modeling the Allele Frequency Spectrum of Populations with Arbitrarily Varying Size

The allele frequency spectrum (AFS), or site frequency spectrum, is commonly used to summarize the genomic polymorphism pattern of a sample, which is informative for inferring population history and detecting natural selection. In 2013, Chen and Chen developed a method for analytically deriving the AFS for populations with temporally varying size through the coalescence time-scaling function. However, their approach is only applicable to population history scenarios in which the analytical form of the time-scaling function is tractable. In this paper, we propose a computational approach to extend the method to populations with arbitrary complex varying size by numerically approximating the time-scaling function. We demonstrate the performance of the approach by constructing the AFS for two population history scenarios: the logistic growth model and the Gompertz growth model, for which the AFS are unavailable with existing approaches. Software for implementing the algorithm can be downloaded at http://chenlab.big.ac.cn/software/.

Analytical version of the resonance coupled-channel model for D + T → 5He** → α + n reaction and its application for the description of low-energy D-T and D- 3He scattering

The purpose of the present paper is the formulation of the analytical version of the resonance coupled-channel model (RCCM) originally developed for D + T → 5He** → α + n nuclear fusion reaction. The integral in the denominator of the Breit-Wigner type is examined in the expression for S-matrix elements of binary processes in this model. Imaginary part of this integral determines the energy-dependent decay width for the near-threshold channel. It is demonstrated that this integral can be calculated explicitly with the Binet representation for the ψ-function (the logarithmic derivation of the gamma function). As the result the explicit expression for the S-matrix elements in the form of analytical functions of the channel momenta are obtained and the equivalence of the RCCM and the effective range approximation (Landau – Smorodinsky – Bethe approximation) is established on this basis. This allows expressing the parameters of the RCCM through the model independent system characteristics: the complex scattering length and the complex effective range. Several sets of model parameters of both approaches that provide a good description of the measured data on D + T → α + n reaction and D-T elastic scattering are derived. By this means we find the location of the S – matrix poles on different Riemann sheets which corresponds to Jπ = (3/2)+ state of 5He and 5Li nuclei. In particular, the location of the resonance (R) and shadow (S) poles is determined: 5He**: ZR = 46.9 – i37.2 (keV) ZS = 81.7 – i3.5 (keV) 5Li**: ZR = 205.7 – i146.8 (keV) ZS = 264.4 + i112.0 (keV). Our results agree well with previous findings. The possible generalizations of the results obtained are discussed.