ACCURATE VARIATIONAL ANALYSIS FOR MEASUREMENT OF COMPLEX DIELECTRIC CONSTANT OF THE SAMPLE ROD INSERTED IN A CAVITY

This paper derives the Lindell formula based on the generalized variational principle.For the complex dielectric constant measurement of a small lossy dielectric rod with Rayleigh-Ritz method, an accurate variational analysis is given. The concept of complex frequency isintroduced in general, and the stability of the solution is discussed. Comparing with the resultof perturbation method, it is concluded that the deviation of perturbation algorithm should betaken into consideration.

A new mixed scheme based on variation of constants for Sobolev equation with nonlinear convection term

A new mixed scheme which combines the variation of constants and the H1-Galerkin mixed finite element method is constructed for nonlinear Sobolev equation with nonlinear con- vection term. Optimal error estimates are derived for both semidiscrete and fully discrete schemes. Finally, some numerical results are given to confirm the theoretical analysis of the proposed method.

A variational formula for controlled backward stochastic partial differential equations and some application

An optimal control problem for a controlled backward stochastic partial differential equation in the abstract evolution form with a Bolza type performance functional is considered. The control domain is not assumed to be convex, and all coefficients of the system are allowed to be random. A variational formula for the functional in a given control process direction is derived, by the Hamiltonian and associated adjoint system. As an application, a global stochastic maximum principle of Pontraygins type for the optimal controls is established.

Optimizing the intraocular lens formula constant according to intraocular lens diameter

AIM:To determine whether the different diameters of a specific intraocular lens(IOL)have significantly different optimized SRK/T A constants and whether these new A constants can improve refractive outcomes.METHODS:Data were collected prospectively from Jan.2011 to Dec.2012 on all patients undergoing routine cataract surgery at a district general hospital in the UK.Patients were divided into three groups according to the size of the Akreos AO MI60 IOL used.A constants for the SRK/T formula were optimized according to the size of the IOL.These optimized A constants were then used to select future refractive outcomes.RESULTS:A total of 2398 cataract operations were performed during the study period of which 1131 met the inclusion criteria.The three optimized A constants for the different sized IOLs were 118.98,119.13,119.32.The difference between them was highly significant(P≤0.0001).Two optimized A constants for three sizes of IOL led to an improvement in refractive outcomes(from 93.4%to 94.6%of refractive outcomes within 1.00 D of predicted spherical equivalent).The optimized A constant for the largest IOL was based on a small number of cases and was not used.CONCLUSION:Optimizing the A constant for the three distinct sizes of the Bausch&Lomb Akreos MI60 lens lead to three significantly different A constants.In our practice,using two different optimized A constants for three different sized IOLs give the least refractive error,however,using three optimized A constants may give better results with a larger dataset.

A New Loop Algebra and Corresponding Computing Formula of Constant γ in Quadratic-Form Identity

A new loop algebra containing four arbitrary constants is presented, -whose commutation operation is concise, and the corresponding computing formula of constant γ in the quadratic-form identity is obtained in this paper, which can be reduced to computing formula of constant γ in the trace identity. As application, a new Liouville integrable hierarchy, which can be reduced to AKNS hierarchy is derived.

Spatial variation of reference crop evapotranspiration on Tibetan Plateau

This study is based on meteorological observation data collected at 38 weather stations on the Tibetan Plateau over several decades. Daily reference crop evapotranspiration （ETo） was calculated with the FAO-56 standard Penman-Monteith formula. A test of normality was performed with Statistica 6.0 software, isotropic and anisotropic semi-variogram analysis was conducted with the GS＋ （geostatistics for the environmental sciences） system for Windows 7.0, and the characteristics of spatial variation of daily ETo were obtained. The following results can be obtained Daily ETo for different periods on the Tibetan Plateau are distributed normally; Except for daily ETo in the E-W （east-west） direction in the summer, which showed a slight negative correlation with distance change, the Moran＇s indexes of daily ETo for different periods in all directions on the Tibetan Plateau within a 100-km distance were positive, demonstrating a positive correlation with distance change; Variograms of daily ETo in June, the dry season, the wet season, as well as annual average daily ETo fit well with the Gaussian model; A variogram of daily ETo in December fit well with the exponential model; Variograms of daily ETo for the four seasons fit well with the linear With sill model.

Accuracy of eight intraocular lens power calculation formulas for segmented multifocal intraocular lens

AIM:To evaluate the accuracy of eight different intraocular lens(IOL)power calculation formulas for a segmented multifocal IOL.METHODS:A total of 53 eyes of 41 adult cataract patients who underwent phacoemulsification and implantation with the SBL-3 segmented multifocal IOL between January 1,2017 and January 31,2019 were included in this retrospective study.Preoperative biometry measurements were obtained using an IOL Master.Manifest refraction was performed at least 4 wk postoperatively.Accuracy of the eight formulas[Barrett Universal II,Emmetropia Verifying Optical(EVO),Haigis,Hill-RBF 2.0,Hoffer Q,Holladay 1,Kane,and SRK/T]was analyzed.RESULTS:Using current lens constants,all formulas exhibited errors of slight myopic shift in refractive prediction.The Barrett Universal II formula had a significantly lower median absolute error(MedAE)than did Holladay 1(P=0.02),Kane(P=0.001)and Hill-RBF 2.0(P<0.001)formulas.The Haigis formula had a lower MedAE value than did the Hill-RBF 2.0 formula(P=0.005).Differences in MedAE values among SRK/T,EVO and Hoffer Q formulas were not significant.After optimizing lens constants,the MedAE values of all formulas were reduced;significant changes were noted for EVO(P=0.022),Haigis(P=0.048);Hill-RBF 2.0(P=0.014),Holladay 1(P=0.045)and Kane(P=0.022)formulas.All formulas performed equally well after optimization of lens constants(P=0.203).CONCLUSION:All eight formulas tend to result in a myopic shift when using current lens constants.Optimized lens constants improve the accuracy of these formulas among adult Chinese patients.

Relativistic Correction of the Rydberg Formula

The relationship E = &minus;K holds between the energy E and kinetic energy K of the electron constituting a hydrogen atom. If the kinetic energy of the electron is determined based on that relationship, then the energy levels of the hydrogen atom are also determined. In classical quantum theory, there is a formula called the Rydberg formula for calculating the wavelength of a photon emitted by an electron. In this paper, in contrast, the formula for the wavelength of a photon is derived from the relativistic energy levels of a hydrogen atom derived by the author. The results show that, although the Rydberg constant is classically a physical constant, it cannot be regarded as a fundamental physical constant if the theory of relativity is taken into account.

Multi-symplectic variational integrators for nonlinear Schrdinger equations with variable coefficients

In this paper, we propose a variational integrator for nonlinear Schrodinger equations with variable coefficients. It is shown that our variational integrator is naturally multi-symplectic. The discrete multi-symplectic structure of the integrator is presented by a multi-symplectic form formula that can be derived from the discrete Lagrangian boundary function. As two examples of nonlinear Schrodinger equations with variable coefficients, cubic nonlinear Schrodinger equations and Gross-Pitaevskii equations are extensively studied by the proposed integrator. Our numerical simulations demonstrate that the integrator is capable of preserving the mass, momentum, and energy conservation during time evolutions. Convergence tests are presented to verify that our integrator has second-order accuracy both in time and space.

Theoretical Studies on CH_3SiH_3+H→CH_3SiH_2+H_2 Reaction with the Variational Transitional State Theory

In this paper. the abstraction reaction of CH_3SiH_3 with H has been studied by using the 'direct dynamics' method of variational transition-state theory. which is based on the information on geometries. frequencies and energies calculalcd by ab inito along the minimum energy path. The rate constants of the title reaction were calculated for the range of temperature 298-1700 K. The rate constants calculated match well with the experimental values.

An Algorithm for Global Optimization Using Formula Manupulation

Constrained nonlinear optimization problems are well known as very difficult problems. In this paper, we present a new algorithm for solving such problems. Our proposed algorithm combines the Branch-and-Bound algorithm and Lipschitz constant to limit the search area effectively;this is essential for solving constrained nonlinear optimization problems. We obtain a more appropriate Lipschitz constant by applying the formula manipulation system of each divided area. Therefore, we obtain a better approximate solution without using a lot of searching points. The efficiency of our proposed algorithm has been shown by the results of some numerical experiments.

Cumulative Reaction Probabilities of Cl+H_(2) from Quantum Scattering Calculations and Evaluation of Thermal Rate Constant

We report the cumulative reaction probabilities(CRP)of the three-dimensional Cl+H_(2)→H+HCl calculated on the Stern-Persky-Klein potential energy surface,using the generalized Newton variational principle,for total angular momentum J=0,over a total reaction energy range from 9 to 19 kcal/mol.The relation between CRP and reaction rate constants,as well as the calculation for complete reaction rates(summed over all J),bypassing the direct calculation of the CRP's for J>0,are also discussed.

Viaritional Formulas for Translating Solitons with Density

In this paper,we introduce a kind of submanifold called translating solitons with density,and obtain two variational formulas for it,and show some geometric quantities of it.

General Convergence Analysis for Three-step Projection Methods and Applications to Variational Problems

First a general model for a three-step projection method is introduced, and second it has been applied to the approximation solvability of a system of nonlinear variational inequality problems in a Hilbert space setting. Let H be a real Hilbert space and K be a nonempty closed convex subset of H. For arbitrarily chosen initial points x0, y0, z0 ∈ K, compute sequences xn, yn, zn such thatT ： K→ H is a nonlinear mapping onto K. At last three-step models are applied to some variational inequality problems.

A Method to Reduce the Lehmer Measure in a Multi-Term Machin-Like Formula for π

Previously we have proposed a new method of transforming quotients into integer reciprocals in the Machin-like formulas for π. As a further development, here we show how to generate a multi-term Machin-like formula for π with a reduced Lehmer measure. The Mathematica codes validating these results are presented.

The Physical Constant Called the Rydberg Constant Does Not Exist

In classical quantum theory, the Rydberg constant is a fundamental physical constant that plays an important role. It comes into play as an indispensable physical constant in basic formulas for describing natural phenomena. However, relativity is not taken into account in this Rydberg formula for wavelength. If the special theory of relativity is taken into account, R∞ can no longer be regarded as a physical constant. That is, we have continued to conduct experiments to this day in an attempt to determine the value of a physical constant, the Rydberg constant, which does not exist in the natural world.

Theoretical and Experimental Values for the Rydberg Constant Do Not Match

In many areas of physics and chemistry, the Rydberg constant is a fundamental physical constant that plays an important role. It comes into play as an indispensable physical constant in basic equations for describing natural phenomena. The Rydberg constant appears in the formula for calculating the wavelengths in the line spectrum emitted from the hydrogen atom. However, this Rydberg wavelength formula is a nonrelativistic formula derived at the level of classical quantum theory. In this paper, the Rydberg formula is rewritten as a wavelength formula taking into account the theory of relativity. When this is done, we come to an unexpected conclusion. What we try to determine by measuring spectra wavelengths is not actually the value of the Rydberg constant R∞ but the value Rn,m of Formula (18). R∞ came into common use in the world of nonrelativistic classical quantum theory. If the theory of relativity is taken into account, R∞ can no longer be regarded as a physical constant. That is, we have continued to conduct experiments to this day in an attempt to determine the value of a physical constant, the Rydberg constant, which does not exist in the natural world.

Consideration of the Daily Variation of Gravity on the Manifestation of Gravitational Shielding

The result of mathematical and physical analysis of the daily change in gravity is presented. The subject of consideration was the manifestation of semi-daily factors in diurnal variations of gravity. The assumption is investigated, according to which the cause of the half-day factors is the gravitational shielding of the planet Earth. Gravitational shielding is considered as a function of the size and thickness of celestial bodies and growing with distance from their poles. It is concluded that the planet Earth has the property of partial gravitational shielding, and the Moon does not have enough thickness to exhibit a tangible gravitational shielding. The obtained mathematical results correspond to the existing experimental data. It is suggested that gravitational shielding is the cause of the precession of the perihelion of Mercury and the peculiarities of the manifestation of tidal processes. It is assumed that gravitational shielding is one of the main reasons for the presence of hidden substances in the Universe. It is concluded that the physical picture with mutual shielding of interaction elements corresponds to the classical ideas of Fatio and Lesage. This approach is proposed as an alternative point of view to the existing theory on the description of tidal processes. It is shown that the existing basic approach to the description of tidal forces is unsatisfactory: the factors underlying the existing approaches have values 10 times less than those observed and cannot be considered as the reason for the manifestation of half-day manifestations in the daily change in gravity. The work is a continuation of the implementation by the author of P. Dirac’s ideas about accounting for the size of microparticles in physical theory.

Association of Variations in the Dynamics of the Lithosphere with Sea Temperature

Variations in the dynamics of the oceanic lithosphere are important at the societal and research levels because geological activities are associated with these variations. At any given section of the lithosphere, the time in which its typical geophysical parameters vary is considerably smaller than section's age. The lithosphere can, therefore, be assumed to proceed from one state of dynamic equilibrium to another displaced deferentially. When these conditions are accounted for in the thermal analysis of the oceanic lithosphere, the earth's internal heat flux through the lithosphere is found to be an adiabatic invariant. Lithosphere physical parameters exhibit constant change and linearity. These findings simplify analysis of heat and work inter actions between oceanic lithosphere and continents, lithosphere dynamics, and deep mantle heat transfer. The temperature of the solid earth remains unchanged for the foreseeable future, and variations in sea temperature vary the intensity of geological activities. If sea temperature increases, the geological activities increase and vice versa. Relevant equations are derived using this thermal analysis of the lithosphere and validated based on observations and the work of others. In addition, the analysis reveals that the eleven-year solar constant cycle is capable of inducing 1.56×10^16 J yr^-1 of geological activities.