scholarly journals SUMS OF TWO SQUARES — PAIR CORRELATION AND DISTRIBUTION IN SHORT INTERVALS

2013 ◽  
Vol 09 (07) ◽  
pp. 1687-1711 ◽  
Author(s):  
YOTAM SMILANSKY
Keyword(s):  
Poisson Distribution ◽  
Quadratic Forms ◽  
Prime Power ◽  
Pair Correlation ◽  
Surprising Result ◽  
Binary Quadratic Forms ◽  
Short Intervals ◽  
Primes In Arithmetic Progressions ◽  
Sums Of Two Squares ◽  
The Mean

In this work we show that based on a conjecture for the pair correlation of integers representable as sums of two squares, which was first suggested by Connors and Keating and reformulated here, the second moment of the distribution of the number of representable integers in short intervals is consistent with a Poisson distribution, where "short" means of length comparable to the mean spacing between sums of two squares. In addition we present a method for producing such conjectures through calculations in prime power residue rings and describe how these conjectures, as well as the above stated result, may by generalized to other binary quadratic forms. While producing these pair correlation conjectures we arrive at a surprising result regarding Mertens' formula for primes in arithmetic progressions, and in order to test the validity of the conjectures, we present numerical computations which support our approach.

scholarly journals Planetary Nebulae in M31: Abundances, and comparison with bright Planetary Nebulae in the LMC

1997 ◽  
Vol 180 ◽  
pp. 475-476
Author(s):  
M. G. Richer ◽  
G. Stasińska ◽  
M. L. McCall
Keyword(s):  
Planetary Nebula ◽  
Luminosity Function ◽  
Large Population ◽  
Wavelength Region ◽  
Planetary Nebulae ◽  
Surprising Result ◽  
Population Synthesis ◽  
Abundance Distribution ◽  
Wide Range ◽  
The Mean

We have obtained spectra of 28 planetary nebulae in the bulge of M31 using the MOS spectrograph at the Canada-France-Hawaii Telescope. Typically, we observed the [O II] λ3727 to He I λ5876 wavelength region at a resolution of approximately 1.6 å/pixel. For 19 of the 21 planetary nebulae whose [OIII]λ5007 luminosities are within 1 mag of the peak of the planetary nebula luminosity function, our oxygen abundances are based upon a measured [OIII]λ4363 intensity, so they are based upon a measured electron temperature. The oxygen abundances cover a wide range, 7.85 dex < 12 + log(O/H) < 9.09 dex, but the mean abundance is surprisingly low, 12 + log(O/H)–8.64 ± 0.32 dex, i.e., roughly half the solar value (Anders & Grevesse 1989). The distribution of oxygen abundances is shown in Figure 1, where the ordinate indicates the number of planetary nebulae with abundances within ±0.1 dex of any point on the x-axis. The dashed line indicates the mean abundance, and the dotted lines indicate the ±1 σ points. The shape of this abundance distribution seems to indicate that the bulge of M31 does not contain a large population of bright, oxygen-rich planetary nebulae. This is a surprising result, for various population synthesis studies (e.g., Bica et al. 1990) have found a mean stellar metallicity approximately 0.2 dex above solar. This 0.5 dex discrepancy leads one to question whether the mean stellar metallicity is as high as the population synthesis results indicate or if such metal-rich stars produce bright planetary nebulae at all. This could be a clue concerning the mechanism responsible for the variation in the number of bright planetary nebulae observed per unit luminosity in different galaxies (e.g., Hui et al. 1993).

scholarly journals On S-class number relations of algebraic tori in Galois extensions of global fields

1991 ◽  
Vol 124 ◽  
pp. 133-144 ◽  
Author(s):  
Masanori Morishita
Keyword(s):  
Class Number ◽  
Quadratic Forms ◽  
Euler Number ◽  
Number Fields ◽  
Binary Quadratic Forms ◽  
Algebraic Number Fields ◽  
Algebraic Tori ◽  
Genus Theory ◽  
Class Number Formula ◽  
Number Formula

As an interpretation and a generalization of Gauss’ genus theory on binary quadratic forms in the language of arithmetic of algebraic tori, Ono [02] established an equality between a kind of “Euler number E(K/k)” for a finite Galois extension K/k of algebraic number fields and other arithmetical invariants associated to K/k. His proof depended on his Tamagawa number formula [01] and Shyr’s formula [Sh] which follows from the analytic class number formula of a torus. Later, two direct proofs were given by Katayama [K] and Sasaki [Sa].

THE EFFECT OF PREMELTING: STUDY FOR SEMI-INFINITE CRYSTALS AND NANO-SYSTEMS

1994 ◽  
Vol 08 (24) ◽  
pp. 3411-3422 ◽  
Author(s):  
W. SCHOMMERS
Keyword(s):  
Pair Correlation ◽  
Mean Square Displacement ◽  
Phonon Density ◽  
Mean Square ◽  
Phonon Density Of States ◽  
Molecular Dynamics Calculations ◽  
Theory Of Melting ◽  
The Mean ◽  
Dynamical Properties ◽  
Reliable Interaction

The effect of premelting is of particular interest in connection with the theory of melting. In this paper, we discuss the structural and dynamical properties of the surfaces of semi-infinite crystals as well as of nano-clusters, which show the effect of premelting. The investigations are based on molecular-dynamics calculations: different models are used for the systematic study of the effect of premelting. In particular, the behaviour of the following functions have been studied: pair correlation function, generalized phonon density of states, and the mean-square displacement as a function of time. The calculations have been done for krypton since for this substance a reliable interaction potential is available.

Representing Primes by Binary Quadratic Forms

1991 ◽  
Vol 64 (1) ◽  
pp. 34
Author(s):  
Steven Galovich ◽  
Jeremy Resnick
Keyword(s):  
Quadratic Forms ◽  
Binary Quadratic Forms

scholarly journals A FORMULA TO PREDICT THE TRANSMISSION FREQUENCY OF ACENTRIC FRAGMENTS

Genetics ◽  
1972 ◽  
Vol 72 (4) ◽  
pp. 777-782
Author(s):  
A V Carrano
Keyword(s):  
Cell Death ◽  
Poisson Distribution ◽  
Transmission Frequency ◽  
Chromosomal Deletions ◽  
Radiation Induced ◽  
Fragment Distribution ◽  
The Mean ◽  
Th 1

ABSTRACT A formula, based on the Poisson distribution of radiation-induced chromosomal deletions, was derived to predict the frequency of transmission of acentric fragments between subsequent mitoses. The frequency of deletions observed in the i  th + 1 division subsequent to fragment distribution at the i  th division anaphase is independent of the cell death resulting from fragment loss. Further, the transmission frequency of chromosome acentric fragments is mathematically equal to the fragment frequency observed in the i  th + 1 generation divided by the mean fragment frequency in the i  th generation. The formula was also extended to chromatid deletions.

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