A Heuristic for the Prime Number Theorem

Conclusion  Might there be a chance of proving in a simple way thatx/π(x) is asymptotic to an increasing function, thus getting another proof of PNT? This is probably wishful thinking. However, there is a natural candidate for the increasing function. LetL(x) be the upper convex hull of the full graph ofxπ(x) (precise definition to follow). The piecewise linear functionL(x) is increasing becausex/π(x) → ∞ asx → ∞. Moreover, using PNT, we can give a proof thatL(x) is indeed asymptotic tox/π(x). But the point of our work in this article is that for someone who wishes to understand why the growth of primes is governed by natural logarithms, a reasonable approach is to convince oneself via computation that the convex hull just mentioned satisfies the hypothesis of our theorem, and then use the relatively simple proof to show that this hypothesis rigorously implies the prime number theorem.     

Edge colorings of graphs embeddable in a surface of low genus

In this paper, by using the Discharging Method, we show that any graph with maximum degree Δ 8 that is embeddable in a surface Σ of characteristic χ(Σ) 0 is class one and any graph with maximum degree Δ 9 that is embeddable in a surface Σ of characteristic χ(Σ) = − 1 is class one. For surfaces of characteristic 0 or −1, these results improve earlier results of Mel'nikov.     

An infinite family of convex Brunnian links in $${mathbb{R}^n}$$

This paper proves that convex Brunnian links exist for every dimension n ≥ 3 by constructing explicit examples. These examples are three-component links which are higher-dimensional generalizations of the Borromean rings.     

A public-key cryptosystem utilizing cyclotomic fields

While it is well-known that the RSA public-key cryptosystem can be broken if its modulusN can be factored, it is not known whether there are other ways of breaking RSA. This paper presents a public-key scheme which necessarily requires knowledge of the factorization of its modulus in order to be broken. Rabin introduced the first system whose security is equivalent to the difficulty of factoring the modulus. His scheme is based on squaring (cubing) for encryption and extracting square (cube) roots for decryption. This introduces a 14 (19) ambiguity in the decryption. Various schemes which overcome this problem have been introduced for both the quadratic and cubic case. We generalize the ideas of Williams' cubic system to larger prime exponents. The cases of higher prime order introduce a number of problems not encountered in the quadratic and cubic cases, namely the existence of fundamental units in the underlying cyclotomic field, the evaluation of higher power residue symbols, and the increased difficulty of Euclidean division in the field.     

Rocky core solubility in Jupiter and giant exoplanets

Gas giants are believed to form by the accretion of hydrogen-helium gas around an initial protocore of rock and ice. The question of whether the rocky parts of the core dissolve into the fluid H-He layers following formation has significant implications for planetary structure and evolution. Here we use ab initio calculations to study rock solubility in fluid hydrogen, choosing MgO as a representative example of planetary rocky materials, and find MgO to be highly soluble in H for temperatures in excess of approximately 10,000 K, implying the potential for significant redistribution of rocky core material in Jupiter and larger exoplanets.     

Singlet and Triplet State Properties and Singlet Oxygen Yield of an Amino-Acyl-Quinoxalinone Yellow Dye

Amino-acyl-quinoxalinone yellow dyes are cyclised analogues of the yellow azomethine dyes developed for, and still used in, silver halide colour photography. Unlike image azomethine dyes, which are rapidly deactivated in their excited states by torsion about the azomethine bond, amino-acyl-quinoxalinone dyes have an interesting photophysics because torsion is not possible due to their cyclised structure. We report results from studies on singlet and triplet state properties, and singlet oxygen yields, of the yellow dye, 7-diethylamino-3-(2,2-dimethyl-propionyl)-5-methyl-1-phenyl-1H-quinoxalin-2-one, in polar and nonpolar solvents. The dye photophysics is characterised by a weak fluorescence, with a solvent dependent emission yield (Φ_F?≈?0.002–0.004), and short singlet state lifetime (τ_expt?≈?20–50 ps), both increasing by a factor of ≈2 in going from polar acetonitrile to non-polar dioxane as solvent. DFT ZINDO calculations show a transition involving significant electron transfer from the diethyl-amino group into the carbonyl region of the molecule. In solution, in the presence of oxygen, the triplet state decays almost exclusively by oxygen quenching, and singlet oxygen is produced in high yield (Φ_??≈?0.5–0.55). The triplet state absorbs across the 450–750 nm region with maxima around 480 and 650 nm, and moderate molar absorption coefficients (ca. 6000–8000 M^?1 cm^?1). In a glass at 77 K, triplet decay gives a red phosphorescence, with λ_max?≈?640–650 nm, and a ?≈?0.25 s lifetime. If singlet oxygen yields are a good indication of triplet yields, then internal conversion and intersystem crossing occur with roughly equal efficiency.

     

Pioneering O.R.

This paper describes the critical start-up period of a new Operational Research unit in industry. Four principles for success evolved and are discussed against a background of practical work.     

Muscle spindles are concentrated in the superior vocalis subcompartment of the human thyroarytenoid muscle

It is hypothesized that different parts of the thyroarytenoid muscle (TA) are functionally specialized. Specifically, the TA is divided into a lateral muscularis compartment and a medial vocalis compartment. This study examined the distribution of muscle spindles throughout the human TA as an indicator of these functional differences. Histological cross-sections from the anterior, middle, and posterior regions of five human membranous vocal folds were examined for the number and location of muscle spindles. There was an average of 6.1 muscle spindles in sections from each region with no significant variation between the different regions (p < .05). However, in sections from all three regions, the muscle spindles were always found to be concentrated in the superior medial quadrant of the TA (mean 85.9%, p < .01). The inferior medial, superior lateral, and inferior lateral quadrants of the TA contained 11.96%, 2.17%, and 0%, respectively, of the total muscle spindles. Within the superior medial quadrant, most of the muscle spindles were localized in the most superficial part of the muscle.The results of this study demonstrate that the majority of TA muscle spindlesare concentrated in its superior medial quadrant, an area we have termed the superior vocalis subcompartment (SC. This finding suggests that the superior vocalis SC is functionally distinct from the remainder of the TA. It is hypothesized that tension in the superior vocalis SC can be controlled independently from the remainder of the TA, and this capability is used to effect the biomechanics of vocal fold vibration during phonation.     

Liquid 2-oxazolidones. I. Dielectric constants,viscosities, and other physical properties of several liquid 2-oxazolidones

Eight derivatives of 2-oxazolidone,, have been synthesized by the Homeyer method and purified by fractional freezing or fractional distillation to obtain low-conducting liquids. Dielectric constants, viscosities, densities, and refractive indices of these 2-oxazolidones have been determined at several temperatures within the range 25 to 75°C. Values for activation energies of viscous flow also have been calculated. The isomeric 3-methyl and 5-methyl derivatives of 2-oxazolidone have high dielectric constants which differ only slightly from that of water. The 3-substituted (or N-substituted) 2-oxazolidones exhibit the most favorable combination of physical properties, stability, and ease of purification and appear to have excellent potential as new nonaqueous solvents.Based on a portion of the Ph. D. dissertation of H. L. Huffman, Jr., University of Kentucky, 1972. Presented in major part at the 21 st Southeastern Regional Meeting of the American Chemical Society, Richmond, Virginia, 1969.