Special Frobenius Traces in Galois Representations

This thesis studies Frobenius traces in Galois representations from two different directions. In the first problem we explore how often they vanish in Artin-type representations. We give an upper bound for the density of the set of vanishing Frobenius traces in terms of the multiplicities of the irreducible components of the adjoint representation. Towards that, we construct an infinite family of representations of finite groups with an irreducible adjoint action.

In the second problem we partially extend for Hilbert modular forms a result of Coleman and Edixhoven that the Hecke eigenvalues a_p of classical elliptical modular newforms f of weight 2 are never extremal, i.e., a_p is strictly less than 2[square root]p. The generalization currently applies only to prime ideals p of degree one, though we expect it to hold for p of any odd degree. However, an even degree prime can be extremal for f. We prove our result in each of the following instances: when one can move to a Shimura curve defined by a quaternion algebra, when f is a CM form, when the crystalline Frobenius is semi-simple, and when the strong Tate conjecture holds for a product of two Hilbert modular surfaces (or quaternionic Shimura surfaces) over a finite field.

MODIFIED DIRECT TWOS-COMPLEMENT PARALLEL ARRAY MULTIPLICATION ALGORITHM FOR COMPLEX MATRIX OPERATION

A direct twos-complement parallel array multiplication algorithm is introduced and modified for digital optical numerical computation. The modified version overcomes the problems encountered in the conventional optical twos-complement algorithm. In the array, all the summands are generated in parallel, and the relevant summands having the same weights are added simultaneously without carries, resulting in the product expressed in a mixed twos-complement system. In a two-stage array, complex multiplication is possible with using four real subarrays. Furthermore, with a three-stage array architecture, complex matrix operation is straightforwardly accomplished. In the experiment, parallel two-stage array complex multiplication with liquid-crystal panels is demonstrated.

Studies on the complex formation of iron with humic substances [Translation from: Vom Wasser 53, 243-247, 1979]

Most of the humic substances which occur in natural waters have an iron content of a few percent, indicated by the mg/1 content of organically-bonded carbon. This iron is apparently bound in a complex with the humic substances, for it quite plainly differs in its chemical and physico-chemical properties from what one would expect from the purely inorganic iron-water system. The deviations range from the solubility to the redox behaviour, and thus are frequently the basis of analytical and technical difficulties. The key to the solution of most of this problem lies in a better understanding of the aforementioned bonds between the iron and the humic substances. This paper studies the iron content of the humic substance concentration from a bog lake sample and the complexing of iron by humic substances from the surface of the bog lake.

Three-dimensional superresolution by three-zone complex pupil filters

Complex pupil filters are introduced to improve the three-dimensional resolving power of an optical imaging system. Through the design of the essential parameters of such filters, the transmittance and radius of the first zone, three-dimensional superresolution is realized. The Strehl ratio and the transverse and axial gains of such filters are analyzed in detail. A series of simulation examples of such filters are also presented that prove that three-dimensional superresolution can be realized. The advantage of such filters is that it is easy to realize three-dimensional superresolution, and the disadvantage is that the sidelobes of the axial intensity distribution are too high. But this can be overcome by the application of a confocal system. (C) 2005 Optical Society of America.

The complex angular momentum theory of the production of three particles in collisions of two strongly interacting particles at high energy

The problem of the continuation to complex values of the angular momentum of the partial wave amplitude is examined for the simplest production process, that of two particles → three particles. The presence of so-called "anomalous singularities" complicates the procedure followed relative to that used for quasi two-body scattering amplitudes. The anomalous singularities are shown to lead to exchange degenerate amplitudes with possible poles in much the same way as "normal" singularities lead to the usual signatured amplitudes. The resulting exchange-degenerate trajectories would also be expected to occur in two-body amplitudes.

The representation of the production amplitude in terms of the singularities of the partial wave amplitude is then developed and applied to the high energy region, with attention being paid to the emergence of "double Regge" terms. Certain new results are obtained for the behavior of the amplitude at zero momentum transfer, and some predictions of polarization and minima in momentum transfer distributions are made. A calculation of the polarization of the ρ^o meson in the reaction π ^- p → π ^- ρ^op at high energy with small momentum transfer to the proton is compared with data taken at 25 Gev by W. D. Walker and collaborators. The result is favorable, although limited by the statistics of the available data.

Ceratium hirundinella - ecology of a complex, mobile, and successful plant

Recent work carried out in the English Lake District (Esthwaite Water and Blelham Tarn) is reported. The seasonal growth cycle, diel growth cycle, photosynthesis, vertical distribution and migrations, horizontal distribution, and the interaction of environmental factors, were investigated.

One-shot parallel complex Fourier-domain optical coherence tomography using a spatial carrier frequency

We propose a novel method of one-shot parallel complex Fourier-domain optical coherence tomography using a spatial carrier frequency for full range imaging. The spatial carrier frequency is introduced into the 2-D spectral interferogram in the lateral direction by using a tilted reference wavefront. This spatial-carrier- contained 2-D spectral interferogram is recorded with one shot of a 2-D CCD camera, and is Fourier-transformed in the lateral direction to obtain a 2-D complex spectral interferogram by a spatial-carrier technique. A full-range tomogram is reconstructed from the 2-D complex spectral interferogram. The principle of this method is confirmed by cross-sectional imaging of a glass slip object. (c) 2008 Society of Photo-Optical Instrumentation Engineers.

Comparison between complex amplitude envelope representation and complex analytic signal representation in studying pulsed Gaussian beam

Analytic propagation expressions of pulsed Gaussian beam are deduced by using complex amplitude envelope representation and complex analytic signal representation. Numerical calculations are given to illustrate the differences between them. The results show that the major difference between them is that there exists singularity in the beam obtained by using complex amplitude envelope representation. It is also found that singularity presents near propagation axis in the case of broadband and locates far from propagation axis in the case of narrowband. The critical condition to determine what representation should be adopted in studying pulsed Gaussian beam is also given. (C) 2004 Elsevier B.V. All rights reserved.

窄带和宽带双曲正割脉冲光束的解析传输式

从近轴波动方程出发推导了窄带和宽带双曲正割脉冲光束的解析传输公式。通过数值计算比较了分别采用复振幅包络(CAE)表示式和复解析信号(CAS)表示式得到的脉冲光束，得出了选择脉冲光束研究方法的条件。结果表明复振幅包络表示式得到的解会存在空间奇异性，使光束出现不符合物理意义的非光束行为。对于宽带光束，复振幅包络解的奇异点的位置距离轴中心较近，使复包络解不能正确表示脉冲光束，而对于奇异点位置远离轴中心的窄带光束，对脉冲光束产生的影响可以忽略。因此，窄带脉冲光束可以采用复振幅包络和复解析信号两种表示式来研究，而对