Sign Ambiguity Research Articles

Optimal inequalities for the subtraction functions of proton-Compton-scattering dispersion theory

Upper and lower bounds upon the subtraction functions required in the dispersion theory of the proton Compton process are derived in a framework which optimally exploits the gauge invariance, the fixed-t analyticity, and the s-u crossing properties of the scattering amplitudes, together with the consequences of the s- and u-channel unitarity. The bounds, which are expressed only in terms of measurable s- and u-channel physical quantities, without any reference to model-dependent annihilation-channel contributions, appear to be quite restrictive for some values of the momentum transfer t. The results are significant for removing the sign ambiguity of the pion decay constant F/sub ..pi../ and for the estimation of the electromagnetic polarizabilities of the proton.

G-sum rule separation of overlapping vibrational bands

The G intensity sum rule is used to determine a separation ratio of 1.52:10 for the ν2, ν5 overlapped band of CHF3 and to resolve the sign ambiguities in the ∂p/∂Qi values of both CHF3 and CDF3. The method can be applied to molecules for which one isotopic species contains overlapping bands corresponding to normal modes of different symmetry whereas another species is free from such separation problems. This technique should prove useful as a check on computer simulation band separations and in cases where experimental separation is not possible.

Cauchy-type congruences for binomial coefficients

In 1840 Cauchy [2] showed that for a prime p = e f + 1 p = ef + 1 , e = 20 e = 20 , \[ ( 10 f f ) ≡ ± ( 10 f 3 f ) ( mod p ) , \left ( {\begin {array}{* {20}{c}} {10f} \\ f \\ \end {array} } \right ) \equiv \pm \left ( {\begin {array}{* {20}{c}} {10f} \\ {3f} \\ \end {array} } \right )\quad (\bmod p), \] and it was not until 1965 that Whiteman [6] succeeded in removing the sign ambiguity in this congruence. In this paper we show how the Davenport-Hasse relation [3] in the form given by Yamamoto [8] can be used to resolve the sign ambiguity in other Cauchy-type congruences. Details are given just for e = 8 , 12 , and 20 e = 8,12,{\text {and }}20 .

A new formulation of the law of octic reciprocity for primes ≡±3(mod8) and its consequences

Let p and q be odd primes with q ≡ ±3(mod8), p ≡ 1(mod8) = a2 + b2 = c2 + d2 and with the signs of a and c chosen so that a ≡ c ≡ 1(mod4). In this paper we show step‐by‐step how to easily obtain for large q necessary and sufficient criteria to have for j = 1, …, 8 (the cases with j odd have been treated only recently [3] in connection with the sign ambiguity in Jacobsthal sums of order 4. This is accomplished by breaking the formula of A.E. Western into three distinct parts involving two polynomials and a Legendre symbol; the latter condition restricts the validity of the method presented in section 2 to primes q ≡ 3(mod8) and significant modification is needed to obtain similar results for q ≡ ±1(mod8). Only recently the author has completely resolved the case q ≡ 5(mod8), j = 1, …, 8 and a sketch of the method appears in the closing section of this paper.Our formulation of the law of octic reciprocity makes possible a considerable extension of the results for q ≡ ±3(mod8) of earlier authors. In particular, the largest prime ≡3(mod8) treated to date is q = 19, by von Lienen [6] when j = 4 or 8 and by Hudson and Williams [3] when j = 1, 2, 3, 5, 6, or 7. For q = 19 there are 200 distinct choices relating a, b, c, d which are equivalent to (−q)(p−1)/8 ≡ ((a−b)d/ac)j(modp) for one of j = 1, …, 8. We give explicit results in this paper for primes as large as q = 83 where there are 3528 distinct choices.This paper makes several other minor contributions including a computationally efficient version of Gosset′s [2] formulation of Gauss′ law of quartic reciprocity, observations on sums ∑γi,j where the γi,j′s are the defining parameters for the distinct choices mentioned above, and proof that the results of von Lienen [6] may not only be appreciably abbreviated, but may be put into a form remarkably similar to the case in which q is a quadratic residue but a quartic non‐residue of p.An important contribution of the paper consists in showing how to use Theorems 1 and 3 of [3], in conjunction with Theorem 4 of this paper, to reduce from (q + 1/4)2 to (q − 1)/2 the number of cases which must be considered to obtain the criteria in Theorems 2 and 3.

Pauli principle in Euclidean geometry

A macroscopic toy of Dirac and Stong shows how to detect 2π rotations in our world. A theorem in geometry shows the toy is maximal; no device in three dimensions can measure bigger angles than this toy. Assuming then that the toy correctly models motion of fermions in our world, a ’’hand waving’’ derivation of the Pauli principle follows. Finally algebra is brought in to express the geometric truth. A famous sign ambiguity goes away.

Confinement in (1 + 1) dimensions and fermions

The sign ambiguity in fermions in (1 + 1) dimensions leads to half-integer gauge transformations and to non-zero vacuum expectation values of the physical fermi field which strongly suggest confinement.

Resolution of sign ambiguities in Jacobi and Jacobsthal sums

Let p be a prime = 1 (mod 16). We obtain extensions of known congruences involving parameters of bioctic Jacobi sums (modp). These extensions are used to give an elementary proof of an important congruence of Ήasse relating parameters of quartic and octic Jacobi sums (mod p). This proof leads directly to an elementary resolution of sign ambiguities of parameters of certain quartic, octic, and bioctic Jacobi and Jacobsthal sums. E. Lehmer's work on ambiguities in quartic sums is thereby extended.

10 mu m interferometry: a linear phase restitution experience

A record of the surface wave of optical systems is obtained directly without ambiguity of sign. From the setting proposed by Keilmann (1970) two variants of a same principle have been tested: a method by optical path difference interferometer mechanical modulation, a second, by laser source frequency modulation. The principal disadvantages and advantages of these methods, and some results obtained with the last one, are presented. The measurement range is better than 100 mu m and the accuracy is about 1 mu m.

The Interrelations between Moiré Patterns, Contour Fringes, Optical Surfaces and Their Sum and Difference Effects

A general relationship between potential functions and moire patterns can be realized in a number of forms. The potential function can be worked as an optical surface and combined with other potential functions similarly worked to give the solution to a complex potential problem. Contour fringes can be produced from the optical surfaces or generated on a computer graph plotter directly. When superimposed these contour lines generate moire fringes which are also solutions to potential function problems but with an essential ambiguity of sign. Two circularly symmetrical potential wells have been contoured and moire fringe solutions to simple problems are given.

Sign of theE4moment inU236by multiple Coulomb excitation withAr40ions

Multiple Coulomb excitation with $^{40}\mathrm{Ar}$ projectiles has been used to excite the ground state rotational band of $^{236}\mathrm{U}$ through the ${12}^{+}$ state. A comparison of the experimental and theoretical excitation probabilities confirms a positive value for the $E4$ moment, removing the sign ambiguity inherent in the determination of this moment by $\mathrm{alpha}$-particle Coulomb excitation.NUCLEAR REACTIONS Multiple Coulomb excitation, $^{40}\mathrm{Ar}$ +$^{236}\mathrm{U}$, ${E}_{\mathrm{lab}}=145.8, 152.4$ MeV; measured excitation probabilities, compared to theoretical predictions; deduced sign of $E4$ moment.

The application of direct methods to centrosymmetric structures containing heavy atoms. II

Direct methods are applied to the difference structure factors for a structure containing one or more heavy atoms in known positions. For those reflexions whose sign is determined by the heavy atoms, the known heavy-atom contribution is subtracted from the observed structure factor to obtain the magnitude and the sign of the light-atom contribution. The signs of the reflexions that do not have an appreciable heavy-atom contribution are found using the Σ2 sign relationship. For reflexions with an intermediate heavy-atom contribution the sign and magnitude ambiguity is also solved by the Σ2 formula. Thus it is possible to maximize the number of correct signs, and correct some of the amplitudes, before a difference Fourier map is calculated.

An effective atomic charge model for infrared intensities

Infrared intensities, in particular the dipole derivatives with respect to internal (symmetry) coordinates derived from the intensities, can be explained in terms of an effective atomic charge model which includes both the equilibrium charges and their first-order fluxes. For diatomic molecules it is found that most of the intensity arises from the equilibrium charges in the case of the hydrogen halides, but conversely, in CO, most of the intensity is due to the charge flux. The parameters found can be nicely related to elementary bonding theory. In small symmetrical polyatomic molecules the number of parameters is sufficiently restricted by symmetry and charge conservation that the parameters would be uniquely determinate except for the ambiguity in sign of the experimental dipole derivatives. The examples of AB 2 ( D ∞ h and C 2 v ), AB 3 ( D 3 h and C 3 v ), and AB 4 ( Td) are discussed in detail; a simple generalization for molecular ions is included. For nonpolar molecules, the effective equilibrium charges are determined by the motion (not the equilibrium value) of the electronic centroid.

Sign ambiguity in dipole moment derivatives of Br 2CO

Signs of the dipole moment derivatives, ∂ p x /∂ S 4 and ∂ p x /∂ S 5 where p x is the dipole moment vector along an axis perpendicular to the CO bond in the plane of the molecule and S j are the symmetry coordinates for the B 1 vibrations of Br 2CO have been re-evaluated from the reported ∂ p/∂ Q i values ( p being the dipole moment of the molecule and Q i the normal coordinates) and the L matrix elements. The new set of dipole moment derivatives fits well with similar parameters for Cl 2CO and F 2CO.

VELOCITIES UNDER PERIODIC AND RANDOM WAVES

Using the linear approximation for predicting velocities under periodic water waves, relationships are established between the velocity spectrum and the surface elevation spectrum for three-dimensional waves. The two dimensional forms of these relationships are used for comparison with velocity and elevation spectra measured in a laboratory wave tank. The predicted distortion of the spectrum shape can be observed in the experimental results. Velocity measurements were recorded using a laser velocimeter employing a rotating diffraction grating for shifting the optical frequencies. This allows measurement without sign ambiguity. The random waves were generated using a servodriven paddle with a filtered white noise input.

Spatial irradiance interferometry with sources of arbitrary symmetry*

A major limitation of irradiance interferometry in both the space and time domains has been that only sources of even symmetry could be uniquely inferred from far-field irradiance correlations, because only the modulus was derived. However, this restriction can be removed by processing, in addition, the field in a simple way before detection to form a new symmetrized function. This irradiance record corresponds to the radiation pattern derived from the pure even part of the radiance profile. The record is autocorrelated, as before, yielding a spatial power spectrum. But by the central-ordinate and hermitian properties of the spatial Fourier transform, the real part of the transformation can be specified exactly from the symmetrized record. Using the modulus of the total transform together with the real part of the transform, the phase can be inferred with only an ambiguity of sign. Thus a pair of images is derived, one erect, the other inverted. This ambiguity can be resolved, however, by moving the source in a known direction off axis.

VIBRATIONAL SPECTRA OF (t-C4H96Li)4AND (t-C4H97Li)4: Raman Intensities and the Question of Li-Li Bonding

Abstract Raman frequencies and intensities in methylcyclohexane solution and infrared frequencies in Nujol mull, have been obtained and assigned for the skeletal modes of (t-C4H9 6Li)4 and (t-C4H9 7Li)4. A normal coordinate analysis, neglecting hydrogen atoms, gave an overall frequency fit of 2% with a reasonable set of valence force constants. The calculated eigenvectors were used to transform the Raman intensities into bond polarizability derivatives. Because of coordinate mixing and the inherent sign ambiguity of molecular polarizability derivatives, there are eight sets of bond polarizability derivatives which are consistent with the measured intensities. Only one set, however, gives reasonable polarizability derivatives for CC stretching and CCC bending, and also shows the requisite invariance to isotope substitution. This set gives a low value for the Li-Li polarizability derivative, and application of the delta function potential equation suggests that the extent of Li-Li bonding is small, amountin...

Measurement and calculation of the absolute infrared intensities of PF 3

The integrated infrared intensities of the fundamental vibrations of PF 3 are experimentally determined using nitrogen broadening pressures of 550 psi. Since several sets of values arise for the molecular charge distribution parameters ∂ P/∂ S j as a consequence of the inherent sign ambiguity in the measured ∂ P/∂ Q i , a preferred set of derivatives is indicated through comparisons with semiempirical CNDO 2 molecular orbital calculations. Values for the molecular dipole derivatives ∂ P/∂ S j are as follows: A 1 species. ∂P z ∂S 1 = 3.27 ± 0.69 D/ A ̊ and ∂P z ∂S 2 = −3.16 ± 0.28 D/ A ̊ ; E species, ∂P x ∂S 1 = −7.14 ± 0.30 D/ A ̊ and ∂P x ∂S 4 = −0.94 ± 0.01 D/ A ̊ . With the molecular orbital calculations as a guide, the redistribution of the electronic charge during the fundamental vibrations is discussed as a function of net atomic charge density and atomic polarization terms. Charge reorganizations for PF 3 are compared to those for NF 3.

Adjustment costs and variable inputs in the theory of the competitive firm

This paper has focused on a number of negative results that arise when we attempt to make the theory of the competitive firm “dynamic” by introduction of adjustment costs. Such generalizations of the theory of the firm have the benefit that they do allow a theory of investment behavior, which was touched on briefly. Investment-demand can be shown to be inversely related to capital stock and to the real rental ( r + μ) g, at least in a neighborhood of a long-run equilibrium point. Furthermore, we were able to show that the long-run demand for capital was inversely related to the rental. These inferences flow from the intertemporal optimization process. However, it appears that the intertemporal optimization theory is not capable of yielding a number of other inferences that are implied by the standard atemporal model. Though all behavior functions are homogenous of degree zero in prices, we were unable to demonstrate (a) the nonpositivity of own-wage effects on the demand for the variable factor either in the short-run or the long-run, (b) the nonnegativity of supply effects of a change in product price in either the short-run or the long-run or (c) that short run effects are less elastic than long-run effects. Furthermore, the intertemporal optimization theory does not, in general, yield symmetric long-run factor-demand functions, though F 23 ≡ 0 is a sufficient condition for that property; this asymmetry, of course, excludes the possibility of obtaining these long-run factor demand functions through optimization of any surrogate static profit function. A fairly weak and plausible sufficient condition for (a) was F 12 F 23 > 0. No such “reasonable” sufficient conditions for either (b) or (c) appear to exist. The economic reasons for these problems are essentially two. The ambiguity of the results for the variable factor (i.e., the issue (a)) and the asymmetry of long-run factor demands stem from the possibility, which the author regards as entirely plausible in practise, that a variable factor may not only contribute to production activities but also to the growth activities of a firm. Labor services may facilitate plant expansion as well as being useful in the production process itself. Such a possibility is entirely excluded from the static optimization model. The second reason for many of the ambiguities of the intertemporal model arises from the dual role of the firm as both an owner of a stock of productive capital goods and a producer of a finished product. In the firm's second capacity it “rents” capital services from itself in its first capacity and this internal relationship is “monopsonistic”. This appears to lie at the root of the sign ambiguities of the supply effects of a change in product price.

ρ−ωInterference and theω−φMixing Angle

The $\ensuremath{\omega}\ensuremath{-}\ensuremath{\varphi}$ mixing angle of $\mathrm{SU}(3)$ is usually assumed to be positive, but only its magnitude is actually determined by various models of $\mathrm{SU}(3)$ breaking. The effect of this ambiguity of the relative sign of the $\mathrm{SU}(3)$ prediction of the $\ensuremath{\gamma}\ensuremath{-}\ensuremath{\rho}$ and $\ensuremath{\gamma}\ensuremath{-}\ensuremath{\omega}$ coupling constants on $\ensuremath{\rho}\ensuremath{-}\ensuremath{\omega}$ interference is investigated. The interference can be "constructive" or "destructive." In photoproduction of lepton pairs, destructive interference causes the effect of $\ensuremath{\rho}\ensuremath{-}\ensuremath{\omega}$ interference to be smaller than that of previous calculations. Two new models of $\ensuremath{\rho}\ensuremath{-}\ensuremath{\omega}$ interference in the colliding-beam reaction ${e}^{+}+{e}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}+{\ensuremath{\pi}}^{\ensuremath{-}}$ are discussed. A measurement of the cross section for the latter reaction would make possible a determination of the sign of the $\ensuremath{\omega}\ensuremath{-}\ensuremath{\varphi}$ mixing angle.

New method for the calculation of infra-red intensities

A new set of quantities called the “intensity constants”, similar to force constants and defined in terms of dipole moment derivatives, is introduced. A method is outlined for the evaluation of these constants and applied to ethylene. The calculated value of |∂µ/∂rC—H| agrees with the values reported earlier in the literature. The dipole moment derivatives of formaldehyde and methyl halides are also calculated. The method is free from sign ambiguity and gives unique value for |∂µ/∂r|.