Terms Of High Power Research Articles

Dispersive derivation of the pion distribution amplitude

We derive the dependence of the leading-twist pion light-cone distribution amplitude (LCDA) on a parton momentum fraction $x$ by directly solving the dispersion relations for the moments with inputs from the operator product expansion (OPE) of the corresponding correlation function. It is noticed that these dispersion relations must be organized into those for the Gegenbauer coefficients first in order to avoid the ill-posed problem appearing in the conversion from the moments to the Gegenbauer coefficients. Given the values of various condensates in the OPE, we find solutions for the pion LCDA, which are convergent and stable in the Gegenbauer expansion. Moreover, the solution from summing contributions up to 18 Gegenbauer polynomials is smooth, and can be well approximated by a function proportional to ${x}^{p}(1\ensuremath{-}x{)}^{p}$ with $p\ensuremath{\approx}0.45$ at the scale $\ensuremath{\mu}=2\text{ }\text{ }\mathrm{GeV}$. Turning off the condensates, we get the asymptotic form for the pion LCDA as expected. We then solve for the pion LCDA at a different scale $\ensuremath{\mu}=1.5\text{ }\text{ }\mathrm{GeV}$ with the condensate inputs at this $\ensuremath{\mu}$, and demonstrate that the result is consistent with the one obtained by evolving the Gegenbauer coefficients from $\ensuremath{\mu}=2\text{ }\text{ }\mathrm{GeV}$ to 1.5 GeV. That is, our formalism is compatible with the QCD evolution. The strength of the above framework that goes beyond analyses limited to only the first few moments of a LCDA in conventional QCD sum rules is highlighted. The precision of our results can be improved systematically by including higher-order and higher-power terms in the OPE.

A Novel Online and Non-Parametric Approach for Drift Detection in Big Data

A sizable amount of current literature on online drift detection tools thrive on unrealistic parametric strictures such as normality or on non-parametric methods whose power performance is questionable. Using minimal realistic assumptions such as unimodality, we have strived to proffer an alternative, through a novel application of Bernstein’s inequality. Simulations from such parametric densities as Beta and Logit-normal as well as real-data analyses demonstrate this new method’s superiority over similar techniques relying on bounds, such as Hoeffding’s. Improvements are apparent in terms of higher power, efficient sample sizes, and sensitivity to parameter values.

Finite energy monopoles in non-Abelian gauge theories on odd-dimensional spaces

In higher-dimensional gauge theory, we need energies with higher power terms of field strength in order to realize pointwise monopoles. We consider new models with higher power terms of field strength and extraordinary kinetic terms of the scalar field. Monopole charges are computed as integrals over spheres and they are related to mapping class degree. Hedgehog solutions are investigated in these models. Every differential equation for these solutions is Abel's differential equation. A condition for the existence of a finite energy solution is shown. The spaces of 1-jets of these equations are defined as sets of zeros of polynomials. Those spaces can be interpreted as singular quartic surfaces in three-dimensional complex projective spa0008.

Pipelined IIR Filter Architecture Using Pole-Radius Minimization

An extension of a polynomial consists of the polynomial plus higher power terms. Given a polynomial with real coefficients and an integer larger than its degree, a method is given that produces a finite list of extensions of degree this larger integer such that this list necessarily contains the extension whose largest root is as small as possible. This extension is called the pole radius minimizer. The pole radius minimizer is then found by the finite check of comparing the polynomials in the list. The method is applied to obtain filter transformations that are optimal as regards throughput, but also have considerable savings in hardware overhead compared with standard methods such as Scattered Lookahead and Minimum Order Augmentation. The table in Section 5 gives an explicit comparison for various kinds of filters.

Prediction of the form of steady-state rate equations from kinetic data

The procedures of fitting kinetic data to initial steady-state rate equations are generally complex when double reciprocal plots of initial velocity vs. initial concentration of reactant (substrate, product or modifier) are not linear. For a general enzyme mechanism, if the initial enzyme concentration and all initial reactant concentrations but one are held constant, then the initial steady-state velocity can be written as a ratio of polynomials in the variable reactant concentration. An examination of some of the properties of the steady-state rate equation for a general enzyme mechanism leads to the proposal of a simple graphical procedure for carrying out preliminary analyses of experimental data. This qualitative procedure, which depends on obtaining initial steady-state velocities at very low and very high reactant concentrations provides a means for determining (1) the difference between the minimum power to which the variable reactant concentration is raised in the numerator polynomial and the minimum power to which it is raised in the denominator polynomial, and (2) the difference between the maximum power to which the variable reactant concentration is raised in the denominator polynomial and the maximum power to which it is raised in the numerator polynomial of the steady-state rate equation. For steady-state rate equations involving quadratic and higher power terms in the variable reactant concentration, this procedure should provide valuable preliminary information on the structure of the rate equation. In addition, the possibility of obtaining quantitative estimates of expressions involving some of the constants appearing in steady-state rate equations is examined.

Note on the Kerr Metric and Rotating Masses

Kerr's metric is often said to describe the geometry exterior to a body whose mass and rotation are measured by Kerr's parameters m and a, respectively, even though no interior solution is known. In this paper we give an interior solution valid in the limit when the rotation parameter a is sufficiently small so that terms of higher power than the first are negligible, but the mass parameter m is allowed to be large. This is accomplished by bringing Kerr's exterior metric into the form of the metric for a slowly rotating mass shell. Also, the connection is found between Kerr's parameters and the physical parameters characterizing the rotating body.

Nonlinear Optical Frequency Polarization in a Dielectric

A perturbation calculation of optical frequency polarization quadratic in the Maxwell field is made for a dielectric in which the electrons are localized on units of the crystal. The result is expressed in a power series in $K〈r〉\ensuremath{\sim}{10}^{\ensuremath{-}3}$ where $〈r〉$ is the size of the unit and $K$ is the wave number of the field. The term of zeroth order is the electric-dipole term which vanishes in a crystal with a center of symmetry. The term first order in $K〈r〉$ is separated into electric-quadrupole and magnetic-dipole contributions by introducing a special gauge for the electromagnetic potentials. Higher power terms are neglected.

Effect of the internal temperature of a conductor on the accuracy of determining its electrical resistance

1. When the resistance of homogeneous cylindrical conductors of 6.5 mm diameter for copper and 3.3 mm for manganin are determined at a current density of 1.5 amp/mm2, the error of measurement due to temperature differences inside the conductor and at its surface can be ignored owing to their small values. 2. When the resistance of cylindrical conductors with diameters greater than 6.5 cm for copper and 3.3 cm for manganin are determined, the errors due to the above-mentioned cause must be corrected according to formula (27). 3. Formula (27) holds for sufficiently long homogeneous cylindrical conductors with a temperature coefficient whose sum of higher-power terms is not larger than the first term.

Stabilization of Nonlinear Feedback Control Systems

Conventional methods for linear systems may be used for determining the operating characteristics of a nonlinear device near equilibrium, provided the differential equations expressing its operation can be expanded in a Taylor series about the equilibrium point. These methods are justified only for variations about equilibrium sufficiently smnall that all second and higher power terms of the expansions can be neglected, thereby reducing the expansions to linear equations. In linear feedback systems the response is independent of the equilibrium point. If a system is inherently nonlinear, it generally will not have this characteristic. It is shown that these systems can be modified so that the response of the output will also be independent of the equilibrium point, except for an amplitude scale factor, for a given incremental input variation from equilibnum.

THE CLARKE FORMULÆ FOR LATITUDE, LONGITUDE AND REVERSE AZIMUTH, WITH CORRECTIVE TERMS FOR USE ON VERY LONG LINES

Abstract25. A very complete exposition of the Clarke formulae has been made in a paper entitled “Latitudes, Longitudes and Azimuths—Clarke's Method”, by G. T. McCaw, which was cyclostyled by the G.S.G.S. in 1922. In the present article the writer carries the theme a step further by indicating more fully the maximum possible values of the various small errors, tabulating them when possible, and also giving examples of the computation of long lines which require the inclusion of the various corrective terms. The formulae for these corrective terms have been expanded to include higher power terms for investigational purposes. References are given to the page and formula number from McCaw's paper: his notation has been slightly altered, but this is fully explained in the present text. The azimuth used in the Clarke formulae is that of the geodesic and not that of the plane curve.

The Isotope Effect in Band Spectra, Part I

Theory of the isotope effect in band spectra of diatomic molecules. --- Generalizing previous discussions by Loomis and Kratzer, it is shown that each coefficient in the detailed expressions for the spectral terms (except those involved in ${\ensuremath{\nu}}^{e}$) of a band spectrum of a diatomic molecule may be written as a quantity independent of mass times some power of $\frac{1}{\ensuremath{\mu}}$, where, if $M$ and ${M}^{\ensuremath{'}}$ are the masses of the two nuclei, $\frac{1}{\ensuremath{\mu}}=\frac{1}{M}+\frac{1}{{M}^{\ensuremath{'}}}$. The frequency $\ensuremath{\nu}$ of any line may be considered as the sum of an "electronic," a (positive or negative) "vibrational," and a (positive or negative) "rotational" contribution: $\ensuremath{\nu}={\ensuremath{\nu}}^{e}+{\ensuremath{\nu}}^{n}+{\ensuremath{\nu}}^{m}$, where $n=\mathrm{vibrational}$ and $m=\mathrm{rotational}\mathrm{quantum}\mathrm{number}$. From theory and experience in line spectra, ${\ensuremath{\nu}}^{e}$ (if present) should be substantially identical for corresponding band systems of two or more isotopes, so that these should have a common "origin." This leads to the definition of the origin of a band-system: $\ensuremath{\nu}={\ensuremath{\nu}}^{e}$; ${\ensuremath{\nu}}^{n}={\ensuremath{\nu}}^{m}=0$; and the definition of the origin of a band: $\ensuremath{\nu}={\ensuremath{\nu}}^{e}+{\ensuremath{\nu}}^{n}$; ${\ensuremath{\nu}}^{m}=0$. For two isotopes, ${\ensuremath{\nu}}_{2}=\ensuremath{\nu}_{2}^{}{}_{}{}^{e}+\ensuremath{\nu}_{2}^{}{}_{}{}^{n}+\ensuremath{\nu}_{2}^{}{}_{}{}^{m}=\ensuremath{\nu}_{1}^{}{}_{}{}^{e}+\ensuremath{\rho}\ensuremath{\nu}_{1}^{}{}_{}{}^{n}+{\ensuremath{\rho}}^{2}\ensuremath{\nu}_{1}^{}{}_{}{}^{m}$, approximately, where ${\ensuremath{\rho}}^{2}=\frac{(\frac{1}{{\ensuremath{\mu}}_{2}})}{(\frac{1}{{\ensuremath{\mu}}_{1}})}$. The various bands corresponding to various values of ${\ensuremath{\nu}}^{n}$ should form for each isotope an essentially identical pattern about the origin. The scale of this pattern should, however, be greater for a lighter isotope (vibrational isotope effect), and large separations between corresponding bands of isotopes may result for bands remote from the origin. Two typical band-patterns for a mixture of isotopes are given in Fig. 1. Analogous relations hold with respect to the ${\ensuremath{\nu}}^{m}$ terms (rotational isotope effect) (cf. Fig. 2). An explicit equation for band-l eads is obtained; this contains important mixed and higher power terms in $n$ and ${n}^{\ensuremath{'}}$; the corresponding isotope effect is discussed. In studying isotope effects in band spectra, compounds of isotopic elements with other elements of high atomic weight are favorable; a low-temperature light source is very desirable; other practical factors are discussed. Quantitative and semi-quantitative confirmations of the isotope effect already obtained (BO, SiN, CuH, CuCl, CuBr, CuI) lend powerful support to the general quantum theory of band spectra.Applications of the isotope effect to the study of isotopy and the interpretation of band spectra.---Electronic band spectra give a valuable new method for the determination of the existence of isotopes. An advantage over the positive ray method is that characteristic band patterns associate without question isotopes belonging to a given element; a disadvantage is that accurate comparison of mass between different elements is less easy. Photometric intensity comparisons give a means of measuring relative abundance of isotopes. Infra-red spectra are much less favorable than electronic for the study of isotopy, largely bacause of the fewness of the bands. The vibrational isotope effect in electronic band spectra furnishes the first really reliable, and a remarkably simple, criterion for the location of the origin of a band-system. Simultaneously, it gives the first direct mode of attack on the correct absolute numbering of vibrational energy levels. Thus in BO the vibrational quantum numbers are probably 1/2, 3/2, 5/2..., not 0, 1, 2,... The rotational isotope effect should likewise aid in the analysis of band structure. The very strong dependence of the isotope coefficient ($\ensuremath{\rho}\ensuremath{-}1$) on the mass ${M}^{\ensuremath{'}}$ of the non-isotopic or second element in the molecule supplies a new and reliable method of identifying the emitters of band spectra.Isotope effects in the specific heat of gases are a necessary theoretical consequence of the success of the foregoing theory.