Arithmetic of cyclotomic fields and Fermat-type equations

Let l be any odd prime, and ζ a primitive l-th root of unity. Let C_l be the l-Sylow subgroup of the ideal class group of Q(ζ). The Teichmüller character w : Z_l → Z^*_l is given by w(x) = x (mod l), where w(x) is a p-1-st root of unity, and x ∈ Z_l. Under the action of this character, C_l decomposes as a direct sum of C^((i))_l, where C^((i))_l is the eigenspace corresponding to w^i. Let the order of C^((3))_l be l^h_3). The main result of this thesis is the following: For every n ≥ max( 1, h_3 ), the equation x^(ln) + y^(ln) + z^(ln) = 0 has no integral solutions (x,y,z) with l ≠ xyz. The same result is also proven with n ≥ max(1,h_5), under the assumption that C_l^((5)) is a cyclic group of order l^h_5. Applications of the methods used to prove the above results to the second case of Fermat's last theorem and to a Fermat-like equation in four variables are given.

The proof uses a series of ideas of H.S. Vandiver ([Vl],[V2]) along with a theorem of M. Kurihara [Ku] and some consequences of the proof of lwasawa's main conjecture for cyclotomic fields by B. Mazur and A. Wiles [MW]. In [V1] Vandiver claimed that the first case of Fermat's Last Theorem held for l if l did not divide the class number h^+ of the maximal real subfield of Q(e^(2πi/i)). The crucial gap in Vandiver's attempted proof that has been known to experts is explained, and complete proofs of all the results used from his papers are given.

Effect of cerium oxide on the precipitation of silver nanoparticles in femtosecond laser irradiated silicate glass

We investigated the effect of cerium oxide on the precipitation of Ag nanoparticles in silicate glass via a femtosecond laser irradiation and successive annealing. Absorption spectra show that Ce3+ ions may absorb part of the laser energy via multiphoton absorption and release free electrons, resulting in an increase of the concentration of Ag atoms and a decrease of the concentration of hole-trapped color centers, which influence precipitation of the Ag nanoparticles. In addition, we found that the formed Ag-0 may reduce Ce4+ ions to Ce3+ ions during the annealing process, which inhibits the growth of the Ag nanoparticles.

Some generalizations of commutativity for linear transformations on a finite dimensional vector space

Let L</i> be the algebra of all linear transformations on an n-dimensional vector space V over a field F and let A, B, ƐL</i>. Let A_i+1 = A_iB - BA_i, i = 0, 1, 2,…, with A = A_o. Let f_k (A, B; σ) = A_2K+1 - ^σ1^A2K-1 ^{+ σ}2^A2K-3 -… +(-1)^Kσ_KA₁ where σ = (σ₁, σ₂,…, σ_K), σ_i belong to F and K = k(k-1)/2. Taussky and Wielandt [Proc. Amer. Math. Soc., 13(1962), 732-735] showed that f_n(A, B; σ) = 0 if σ_i is the i^th elementary symmetric function of (β₄- β_s)², 1 ≤ r ˂ s ≤ n, i = 1, 2, …, N, with N = n(n-1)/2, where β₄ are the characteristic roots of B. In this thesis we discuss relations involving f_k(X, Y; σ) where X, Y Ɛ L</i> and 1 ≤ k ˂ n. We show: 1. If F is infinite and if for each X Ɛ L</i> there exists σ so that f_k(A, X; σ) = 0 where 1 ≤ k ˂ n, then A is a scalar transformation. 2. If F is algebraically closed, a necessary and sufficient condition that there exists a basis of V with respect to which the matrices of A and B are both in block upper triangular form, where the blocks on the diagonals are either one- or two-dimensional, is that certain products X₁, X₂…X_r belong to the radical of the algebra generated by A and B over F, where X_i has the form f₂(A, P(A,B); σ), for all polynomials P(x, y). We partially generalize this to the case where the blocks have dimensions ≤ k. 3. If A and B generate L</i>, if the characteristic of F does not divide n and if there exists σ so that f_k(A, B; σ) = 0, for some k with 1 ≤ k ˂ n, then the characteristic roots of B belong to the splitting field of g_k(w; σ) = w^2K+1 - σ₁w^2K-1 + σ₂w^2K-3 - …. +(-1)^K σ_Kw over F. We use this result to prove a theorem involving a generalized form of property L [cf. Motzkin and Taussky, Trans. Amer. Math. Soc., 73(1952), 108-114]. 4. Also we give mild generalizations of results of McCoy [Amer. Math. Soc. Bull., 42(1936), 592-600] and Drazin [Proc. London Math. Soc., 1(1951), 222-231].

Self-organized microvoid array perpendicular to the femtosecond laser beam in CaF2 crystals

Microvoid arrays were self-organized when femtosecond laser beam was tightly focused at a fixed point inside CaF2 crystal sample. Except void array grown below the focal point which had been reported before, we found another void array grown vertical to the laser propagation direction. This result has potential application in the fabrication of integrated micro-optic elements and photonic crystals. The possible mechanism of the phenomenon was proposed and verified experimentally.

Diffusive coevolution and mutualism maintenance mechanisms in a fig-fig wasp system

In reciprocal mutualism systems, the exploitation events by exploiters might disrupt the reciprocal mutualism, wherein one exploiter species might even exclude other coexisting exploiter species over an evolutionary time frame. What remains unclear is how such a community is maintained. Niche partitioning, or spatial heterogeneity among the mutualists and exploiters, is generally believed to enable stability within a mutualistic system. However, our examination of a reciprocal mutualism between a fig species (Ficus racemosa) and its pollinator wasp (Ceratosolen fusciceps) shows that spatial niche partitioning does not sufficiently prevent exploiters from overexploiting the common resource (i.e., the female flowers), because of the considerable niche overlap between the mutualists and exploiters. In response to an exploiter, our experiment shows that the fig can (1) abort syconia-containing flowers that have been galled by the exploiter, Apocryptophagus testacea, which oviposits before the pollinators do; and (2) retain syconia-containing flowers galled by Apocryptophagus mayri, which oviposit later than pollinators. However, as a result of (2), there is decreased development of adult non-pollinators or pollinator species in syconia that have not been sufficiently pollinated, but not aborted. Such discriminative abortion of figs or reduction in offspring development of exploiters while rewarding cooperative individuals with higher offspring development by the fig will increase the fitness of cooperative pollinating wasps, but decrease the fitness of exploiters. The fig fig wasp interactions are diffusively coevolved, a case in which fig wasps diversify their genotype, phenotype, or behavior as a result of competition between wasps, while figs diverge their strategies to facilitate the evolution of cooperative fig waps or lessen the detrimental behavior by associated fig wasps. In habitats or syconia that suffer overexploitation, discriminative abortion of figs or reduction in the offspring development of exploiters in syconia that are not or not sufficiently pollinated will decrease exploiter fitness and perhaps even drive the population of exploiters to local extinction, enabling the evolution and maintenance of cooperative pollinators through the movement between habitats or syconia (i.e., the metapopulations).

中国连珠藻属(串珠藻科)的分类研究

记述了中国产连珠藻属SirodotiaKylin的6个种,即黄山连珠藻S.huangshanensisZ.X.ShietS.L.Xie,威拉连珠藻S.huillensis(WelwitschexW.etG.S.West)Skuja,中华连珠藻S.sinicaJao,纤细连珠藻S.delicatulaSkuja,瑞典连珠藻S.suecicaKylin和细连珠藻S.tenuissima(Collins)SkujaexFlint。其中,黄山连珠藻S.huangshanensisZ.X.ShietS.L.Xie为

Rearing of Procypris rabaudi during early life-history stages

The early life-history of Chinese rock carp Procypris rabaudi was investigated during a 56-day rearing period: 318 artificially propagated P. rabaudi larvae were reared throughout metamorphosis in a small-scale recirculation system (345 L water volume, 10 x 18 L rearing tanks, 150 L storage and filter compartment with bioballs, 20-30 larvae L-1) at the Institute of Hydrobiology, Wuhan, Hubei Province, China. The newly hatched larvae had an initial total length of 8.93 +/- 0.35 mm SD (n = 10) at 3 days post-hatch and reached an average total length of 33.29 mm (+/- 1.88 mm SD, n = 10) 56 days after hatching. Length increment averaged 0.45 mm day(-1), resulting in a mean growth of 24.4 mm within the 56-day period. High mortality rates of up to 92% derived from an introduced fungus infection and subsequent treatment stress with malachite green. Our results indicate that Chinese rock carp can be raised successfully from artificially fertilized eggs. We therefore assume this species to be a candidate for commercial aquaculture.

Taxonomy of the Lemaneaceae (Batrachospermales, Rhodophyta) in China

Six species, belonging to two genera of Lemaneaceae in China are described. They are Lemanea Bory (including L. sinica Jao, L. crassa S. L. Xie et Z. X. Shi, L. ramosa S. L. Xie et Z. X. Shi and L. simplex Jao) and Paralemanea (Silva) Vis et Sheath (including P catenata (Kutzing) Vis et Sheath and P. parvula (Sirodot) S. L. Xie et Z. X. Shi). Among them L. crassa S. L. Xie et Z. X. Shi and L. ramosa S. L. Xie et Z. X. Shi are new species. Moreover, Paralemanea (Silva) Vis et Sheath and P. catenata (Kutzing) Vis et Sheath, P. parvula (Sirodot) S. L. Xie et Z. X. Shi are newly recorded in China.

Compensatory growth response in three-spined stickleback in relation to feed-deprivation protocols

Different protocols of food deprivation were used to bring two groups of juvenile three-spined sticklebacks Gaslerosteus aculeatus to the same reduced body mass in comparison with a control group fed daily ad libitum. One group experienced I week or deprivation then 2 weeks on maintenance rations. The second group experienced I week of ad lithium feeding followed by 2 weeks of deprivation. The deprived groups were reduced to a mean mass ore. 80% of controls. The compensatory growth response shown when ad libitum feeding was resumed was independent of the trajectory by which the three-spined sticklebacks had reached the reduced body mass. The compensatory response was Sufficient to return the deprived groups to the mass and length trajectories shown by the control group within 4 weeks. There was full compensation for dry mass and total lipid, but incomplete compensation for lipid-free dry mass. Hyperphagia and increased growth efficiency were present in the re-feeding phase, but there was a lag of a week before the hyperphagia was established. The consistency of the compensatory response of immature three-spined sticklebacks provides a potential model system for the analysis and prediction of appetite and growth in teleosts. (C) 2003 The Fisheries Society of the British isles.

ROLE OF PHOSPHORUS CYCLING IN ALGAL METABOLISM AND ALGAL SUCCESSION IN LAKE DONGHU, CHINA

The role of phosphorus cycling in algal metabolism was studied in a shallow lake, Donghu, in Wuhan using the methods of measuring cell quota C, N and P, and calculating nutrients uptake rate by algal photosynthesis. The mean daily phosphorus uptake rate of phytoplankton varied between 0.04-0.11 and 0.027-0.053 g/m2/d in station I and station II respectively. The turnover time of phosphorus in phytoplankton metabolism ranged from 0.75-5.0 days during 1979-1986. The available P was 0.176 (+/- 0.156) g/m3 (mean +/- SD) in 1982 and 0.591 (+/- 0.24) g/m3 in 1986. The relationship between P/B ratio (Y) and TP (X: mg/l) was described by the following regression equation Y = 1.163 + 0.512logX (r = 0.731, P &lt; 0.001). The dynamics of algal biomass and algal species succession were monitored as the indicators of environmental enrichment. The small-sized algae have replaced the blue-green algae as the dominant species during 1979-1986. The small-sized algae include Merismopedia glauca, Cryptomonas ovata, Cryptomonas erosa, several species Cyclotella. There has been drastic decrease in algal biomass and an obvious increase in P/B ratio. A nutrient competition hypothesis is proposed to explain the reason of the disappearance of blue-green algae bloom. The drastic change in algal size and the results in high P/B ratio (reaching a maximum mean daily ratio of 1.09 in 1986) may suggest a transition of algal species from K-selection to r-selection in Lake Donghu.