Sums over numbers with restricted prime factors

For k a non-negative integer, let P_k(n) denote the kth largest prime factor of n where P₀(n) = +∞ and if the number of prime factors of n is less than k, then P_k(n) = 1. We shall study the asymptotic behavior of the sum Ψ_k(x, y; g) = Σ_{1 ≤ n ≤ x, Pk(n) ≤ y}g(n), where g(n) is an arithmetic function satisfying certain general conditions regarding its behavior on primes. The special case where g(n) = μ(n), the Möbius function, is discussed as an application.     

Stabilization of a K+-(bis-Cage-Annulated 20-Crown-6) Complex by Bidentate Picrate

Bis-cage-annulated 18-crown-6 and 20-crown-6 macrocyclic ethers (i.e., 1 and 2, respectively) have been synthesized, and their alkali metal picrate extraction profiles have been determined. Host system 1 proved to be a significantly more avid alkali metal cation complexant than 2 and somewhat more avid than 18-crown-6. Both 1 and 18-crown-6 display modest selectivity toward K⁺ and Rb⁺. A stable host–guest complex was prepared by slow evaporation of a CH₂Cl₂–hexane solution of an equimolar mixture of 2 and potassium picrate. The X-ray crystal structure of this complex reveals that picrate anion functions as a bidentate ligand therein. The gas-phase interaction energy between the 2 K⁺ complex and picrate anion was calculated to be ca. –64.9 kcal mol^–1, thereby indicating that participation of picrate anion as an additional bidentate ligand results in significant stabilization of complex 10.     

In vivo NMR imaging and T1 measurements of water protons in the human brain

Nuclear magnetic resonance (NMR) proton density images of the human brain have been made by the FONAR method. Spin-lattice relaxation times, T₁, of water hydrogen protons have been determined at random positions within frontal and temporal regions of the human brain. The primary purpose of this ongoing research is to accumulate a large data base of normal T₁ values for water protons in normal human brain tissue. Our experience to data includes 31 measurements on 18 volunteer subjects, and the mean value ± standard deviation is 215 ± 42 msec. In addition, two metastatic lesions of the brain were studied and found to have T₁ values longer than those for normal brain tissue.     

Strength of convergence in the orbit space of a groupoid

Let G be a second-countable locally-compact Hausdorff groupoid with a Haar system, and let {xn} be a sequence in the unit space G(0) of G. We show that the notions of strength of convergence of {xn} in the orbit space G(0)/G and measure-theoretic accumulation along the orbits are equivalent ways of realising multiplicity numbers associated to a sequence of induced representation of the groupoid C^?-algebra.     

Sums over positive integers with few prime factors

The author has presented estimates for sums of multiplicative functions, satisfying certain conditions, extended over positive integers n such that n is less than or equal to x^t, the greatest prime factor of n is at most equal to x, and n is relatively prime to a natural number k. These estimates are uniform in t, x, and k.     

Evaluation of muscle degeneration in inherited muscular dystrophy by nuclear magnetic resonance techniques

Nuclear magnetic resonance (NMR) techniques were applied to study the muscular dystrophy in chicks. The water proton spin-lattice relaxation times (T₁) of fast, slow, and mixed muscles and plasma were measured. The T₁ values of dystrophic pectoralis major and posterior latissimus dorsi (PLD) were significantly higher than those of the normal pectoralis and PLD muscles. The present results establish a direct relationship between the differences in T₁ values and the severity of muscle degeneration. Consistent with this conclusion, it was also found that the T₁ values of muscles unaffected in muscular dystrophy, namely, the gastrocnemius, and anterior latissimus dorsi (ALD), were not different between the normal and dystrophic chicks. Although the affected muscles of dystrophic chicks contained higher percent water and fat than those of normal chicks, the results show that the higher T₁ values is dystrophic muscles were not solely due to variations in their water content. The increase in the T₁ values is principally a result of altered interaction between cellular water and macromolecules in the diseased muscles. These data also point out the potential use of NMR imaging in evaluating muscle degeneration.     

Effect of nuclear magnetic resonance on chromosomes of mouse bone marrow cells

Three groups of six male Balb/c mice, subjected to 30 MHz continuous wave NMR exposure in a static magnetic field of 7.05 K Gauss for one hour, were each compared to another group of ten unexposed mice with respect to chromatid and chromosomal aberrations. The exposed groups were sacrificed at two hours, 24 hours and 48 hours following NMR exposure respectively. Control mice were sacrificed 24 hours after sham-exposure. All groups had approximately 0.02 apparent aberrations per cell. These apparent aberrations were in the form of metacentric chromosomes, possibly resulting from a union of chromosomes at their centromeres or possibly simply chromosomes in association. The results are consistent with earlier in vitro findings that NMR exposure causes no adverse cytogenetic effects.     

Pricing vulnerable European options with stochastic default barriers

CF Lo and KC Ku Institute of Theoretical Physics and Department of Physics, The Chinese University of Hong Kong, Shatin, Hong Kong, China Email: cho-hoi_hui{at}hkma.gov.hk Received on 31 July 2006. Accepted on 15 March 2007. This paper develops a valuation model of European options incorporatinga stochastic default barrier, which extends a constant defaultbarrier proposed in the Hull–White model. The defaultbarrier is considered as an option writer's liability. Closed-formsolutions of vulnerable European option values based on themodel are derived to study the impact of the stochastic defaultbarriers on option values. The numerical results show that negativecorrelation between the firm values and the stochastic defaultbarriers of option writers gives material reductions in optionvalues where the options are written by firms with leverageratios corresponding to BBB or BB ratings.