Mathematical Inequalities Research Articles

Experimental Programs: An Experimental Study of Patterns of Presenting Transitivity Statements

The concept of mathematical inequalities is now found in most modern secondary mathematics text materials, and often in elementary school programs. The unit on mathematical inequalities usually considers the transitive property of the equality and inequality relations.

Geometry of Molecules. III. F2O, Li2O, FOH, LiOH

LC (Hartree—Fock) AO MO SCF calculations have been carried out for F2O, Li2O, FOH, and LiOH. Correlation diagrams showing the variation of orbital energies with internuclear angle have been constructed and demonstrated to be the theoretical counterparts to previous empirical correlation diagrams given by Mulliken and Walsh. Consideration of mathematical inequalities arising from the Hartree—Fock equations emphasizes the geometry-determining nature of the orbital-energy sum; at the same time such an investigation relates the fact that the sum of orbital energies does not have a minimum for the equilibrium linear configurations of Li2O and LiOH to the highly ionic character of these systems.

Interpretation of Wall-Pressure Measurements under a Turbulent Boundary Layer

Some aspects of the correlations and spectra of fluctuating wall pressure beneath a turbulent boundary layer are considered. The correlation function for the pressure field is represented by a hierarchy of mathematical models, each successive model more complex than the preceding one. These models are based on some mathematical inequalities that are applicable to a “convecting and decaying” stochastic process. Some of these models are used for estimating attenuation of the pressure spectrum measured by a transducer of finite size. The results indicate that, at high frequencies, temporal rather than spatial fluctuations of the pressure field contribute to the “fixed-microphone” spectrum as measured by a transducer of finite size. Some comments are included concerning the influence of transducer size on measured narrow-band crosscorrelations.

Geometry of Molecules. I. Wavefunctions for Some Six- and Eight-Electron Polyhydrides

Potential surfaces have been investigated for BH2+, BH2−, NH2+, BeH2, BH3, CH3+, and BeH3− by means of LC (Hartree—Fock) AO MO SCF wavefunctions. One-electron-energy-versus-angle diagrams were determined for these AH2 and AH3 molecules, and comparison with the widely used empirical diagrams of Walsh demonstrates a quantitative similarity between them. In addition, mathematical inequalities are derived relating changes in the sum of one-electron energies with angle to changes in the total energy with angle. Aided by these expressions, we are able to make direct connection between Walsh's qualitative rules and the physically well-defined ab initio solutions.

SOME DEPTH FORMULAE FOR LOCAL MAGNETIC AND GRAVITY ANOMALIES

AbstractSome mathematical inequalities are proved and their relevance to the interpretation of local magnetic and gravity anomalies is discussed. They can be applied to the readings of either a vertical magnetometer or a total field magnetometer or a gravimeter and yield a maximum possible value for the depth below the earth's surface of the top surface of the body producing the anomaly.

Inequality relations for scattering cross sections

Use is made of a lemma by Erdös and Fuchs to derive various mathematical inequalities applicable to scattering and polarization cross sections, and the conditions under which they hold. They are compared to inequalities obtained by simpler methods. One possible application of these relations is to check whether a given set of experimental cross sections obeys these inequalities or not. If it does not, we then know that the corresponding conditions are not fulfilled. The violation of these conditions would provide some information about the interaction.

Parity in the Exchange of Future Money and Future Commodities

The problem and its relation to the supply and demand control of rates of exchange involving future goods, — I. Comparative values of futures and spots in relation to available supplies from crops, 368. — A future trade is an exchange of future money for a future commodity, and does not involve present goods, 370. — II. Indications of a downward bias in grain and cotton futures call for explanation, 372. — III. Bohm-Bawerk's theory of a discount on the future is quite another matter, 375. — IV. But a difference in the relative valuation of money and any specific commodity may be expected according to whether the comparison pertains to the present or to the future, 377. — This situation as expressed in terms of mathematical inequalities, 379. — V. Commercial experience and observation of the comparative value of “immediates,” 379. — The downward bias costly to hedge sellers, 383. — VI. Conclusion, — The present article calls attention to a neglected condition, and perhaps suggests a needed addition to theory, 385.