Approximation by Algebraic Polynomials on Rectangles

A direct theorem for approximation by algebraic polynomials in two variables with different degrees in each variable in L_p-metric (1 p ) on rectangles is proved, and the dependence of the constants on various parameters is studied.     

Value‐Distribution of General Dirichlet Series. II

In the paper, a limit theorem in the sense of the weak convergence of probability measures in the space of meromorphic functions for general Dirichlet series is proved.     

On a conjecture of Erdos and Stewart

For any , let be the th prime number. In this paper, we confirm a conjecture of Erdos and Stewart concerning all the solutions of the diophantine equation , when .

     

Another Note on the Greatest Prime Factors of Fermat Numbers

For every positive integer k > 1, let P(k) be the largest prime divisor of k. In this note, we show that if F_m = 2^2m + 1 is the mth Fermat number, then P(F_m) 2^m+2(4m + 9) + 1 for all m 4. We also give a lower bound of a similar type for P(F_a,m), where F_a,m = a^2m + 1 whenever a is even and m a¹⁸.AMS Subject Classification (1991) 11A51 11J86     

Infrared spectral study of molecular vibrations in amorphous, nanocrystalline and AlO(OH) · αH2O bulk crystals

Amorphous, nanocrystalline, and bulk AlO(OH) · xH₂O crystals have six fundamental modes (FM) of vibration in a nonlinear AlO(OH) molecular structure. Most of them appear in groups of four IR and Raman bands. Their positions and relative intensities differ significantly in three specimens. The nanocrystals (monoclinic structure with z=8 molecules per unit cell) have four OH stretching bands at values enhanced by up to 360 cm⁻¹ at 3120, 3450, 3560 cm⁻¹ in comparison to those in bulk crystals or amorphous specimens. The first two bands are broad, bandwidth Δν_1/2200 to 350 cm⁻¹, while the other two are sharp, Δν_1/290 cm⁻¹. The sharp bands shift to 3525 and 3595 cm⁻¹ after heating the sample at 100°C. They no longer appear after heating at 300 or 500°C for 2 h (the specimen decomposes to Al₂O₃), leaving behind only two bands at 3100 and 3400 cm⁻¹. A Δν_1/2 value of 500 cm⁻¹ appears in the 3400 cm⁻¹ in a delocalized distribution of H atoms. Two bands also occur at 3098 and 3300 cm⁻¹ in bulk crystals (orthorhombic structure with z=4) or at 2990 and 3515 cm⁻¹ in an amorphous sample. More than one bands appear in a FM vibration in occurrence of sample in more than one conformers. The amorphous sample has approximately the same conformer structure as the bulk crystals. An amorphous surface structure exists in nanocrystals with a group of three bands at 1420, 1510 and 1635 cm⁻¹ in an interconnected network structure. It encapsulates the nanocrystals in an amorphous shell. Its volume fraction, 33% estimated from the integrated intensity in three bands, determines 2.2 nm thickness in the shell in spherical shape of nanocrystals in 35 nm diameter.     

Asymptotic Behavior of Loss Probability in GI/M/1/K Queue as K Tends to Infinity

We obtain an asymptotic behavior of the loss probability for the GI/M/1/K queue as K for cases of <1, >1 and =1.     

Mesoscopic and macroscopic dipole clusters: Structure and phase transitions

A two dimensional (2D) classical system of dipole particles confined by a quadratic potential is studied. This system can be used as a model for rare electrons in semiconductor structures near a metal electrode, indirect excitons in coupled quantum dots etc. For clusters of N ≤ 80 particles ground state configurations and appropriate eigenfrequencies and eigenvectors for the normal modes are found. Monte-Carlo and molecular dynamic methods are used to study the order-disorder transition (the “melting” of clusters). In mesoscopic clusters (N < 37) there is a hierarchy of transitions: at lower temperatures an intershell orientational disordering of pairs of shells takes place; at higher temperatures the intershell diffusion sets in and the shell structure disappears. In “macroscopic” clusters (N > 37) an orientational “melting” of only the outer shell is possible. The most stable clusters (having both maximal lowest nonzero eigenfrequencies and maximal temperatures of total melting) are those of completed crystal shells which are concentric groups of nodes of 2D hexagonal lattice with a number of nodes placed in the center of them. The picture of disordering in clusters is compared with that in an infinite 2D dipole system. The study of the radial diffusion constant, the structure factor, the local minima distribution and other quantities shows that the melting temperature is a nonmonotonic function of the number of particles in the system. The dynamical equilibrium between “solid-like” and “orientationally disordered” forms of clusters is considered.     

Efficient population transfer by delayed pulses despite coupling ambiguity

In traditional schemes of multilevel multilaser excitation, each laser pulse interacts with only one pair of states, and the rotating wave approximation (RWA) is applicable. Here we study the population transfer process in a three-state system when each of the two lasers interacts with each of the pair of states and when the Rabi frequencies characterizing the interaction strengths of the system are comparable to or larger than the difference of the transition frequencies. We show that complete and robust population transfer is possible under conditions more general than those hitherto considered necessary for stimulated Raman adiabatic passage (STIRAP) or for successive π pulses. Using adiabatic Floquet theory we show that successful population transfer can be interpreted as adiabatic passage by means of a transfer state which connects the initial and final states. The Floquet picture offers a convenient interpretation of the population transfer as accompanied by multiple absorption of photons from or emission into the laser fields.     

中国优秀物理学家的杰出代表——葛庭燧

回顾介绍了葛庭燧院士的生平和他在科学研究、人才培养方面所作出的贡献,展示了他的高尚情操,分析了他取得卓著成就的原因.