Arithmetic Progressions,Prime Numbers,and Squarefree Integers

In this paper we establish the distribution of prime numbers in a given arithmetic progression p l for which ap + b is squarefree.     

⊕-Cofinitely Supplemented Modules

Let R be a ring and M a right R-module. M is called -cofinitely supplemented if every submodule N of M with M/N finitely generated has a supplement that is a direct summand of M. In this paper various properties of the -cofinitely supplemented modules are given. It is shown that (1) Arbitrary direct sum of -cofinitely supplemented modules is -cofinitely supplemented. (2) A ring R is semiperfect if and only if every free R-module is -cofinitely supplemented. In addition, if M has the summand sum property, then M is -cofinitely supplemented iff every maximal submodule has a supplement that is a direct summand of M.     

On the quantifier complexity of Δ<Subscript> n+1</Subscript> (T)– induction

In this paper we continue the study of the theories I _n+1 (T), initiated in [7]. We focus on the quantifier complexity of these fragments and theirs (non)finite axiomatization. A characterization is obtained for the class of theories such that I _n+1 (T) is _n+2 –axiomatizable. In particular, I _n+1 (I _n+1 ) gives an axiomatization of Th _n+2 (I _n+1 ) and is not finitely axiomatizable. This fact relates the fragment I _n+1 (I _n+1 ) to induction rule for _n+1 –formulas. Our arguments, involving a construction due to R. Kaye (see [9]), provide proofs of Parsons conservativeness theorem (see [16]) and (a weak version) of a result of L.D. Beklemishev on unnested applications of induction rules for _n+2 and _n+1 formulas (see [2]).Research partially supported by grant PB96–1345 (Spanish Goverment)Mathematics Subject Classification (2000): 03F30, 03H15     

Almost flat locally finite coverings of the sphere

For any subset A of the unit sphere of a Banach space X and for [0,2) the notion of -flatness is introduced as a measure of non-flatness of A. For any positive , construction of locally finite tilings of the unit sphere by -flat sets is carried out under suitable -renormings of X in a quite general context; moreover, a characterization of spaces having separable dual is provided in terms of the existence of such tilings. Finally, relationships between the possibility of getting such tilings of the unit sphere in the given norm and smoothness properties of the norm are discussed.     

Modeling and Simulation of Saturation Gain of IR Integrated Photonic Amplifiers

IR integrated photonic amplifiers at 1.55m operation will have good foreground in optical phasedarray radars for splitters and signal processing. The saturation gain characteristics of IR integrated photonic waveguide amplifiers (taken Er³⁺Yb³⁺ co-doped phosphate glass waveguide amplifiers as an example) are studied theoretically. For the homemade laser glass materials the calculated saturation intensities are 2.22kw /cm² for signal and 10.15kw/cm² for pump. The effects of absorption saturation of signal and pump lights on the gain of amplifiers are discussed.     

Neural Network Models for Finline Discontinuities

The radial basis network is used as the finline discontinuities electromagnetic artifical neural network(EMANN) models. EM software analysis is employed to characterize finline discontinuities. EMANN models are then trained using physical parameters and frequency as inputs and equivalent electric circuit element parameters of finline discontinuities as outputs. Once trained , the EMANN models can simulate equivalent electric circuit element parameters of finline step, notch and strip very fast and efficiently.     

Schrödinger Operators with Random Sparse Potentials. Existence of Wave Operators

We consider the connection problem for the Heun differential equation, which is a Fuchsian differential equation that has four regular singular points. We consider the case in which the parameters in this equation satisfy a certain set of conditions coming from the eigenvalue problem of the non-commutative harmonic oscillators. As an application, we describe eigenvalues with multiplicities greater than 1 and the corresponding odd eigenfunctions of the non-commutative harmonic oscillators. The existence of a rational or a certain algebraic solution of the Heun equation implies that the corresponding eigenvalues has multiplicities greater than 1.The research of the author is supported in part by a Grant-in-Aid for Scientific Research (B) (No. 15340005) from the Ministry of Education, Culture, Sports, Science and Technology.Mathematics Subject classifications (2000). primary, 34M35, secondary, 33E20.This revised version was published online in March 2005 with corrections to the cover date.     

Organization of nanoscale objects via polymer demixing

We present an extremely versatile method for the lateral organization of nano-scale objects (NOs) based on the phenomenon of polymer demixing. NOs are suspended in a solution of two immiscible polymers, which is used to form a thin polymer film by spin coating. During spin coating the two polymers separate to give a microphase structure, whose length scale depends on the experimental conditions. The NOs spontaneously partition into one or other of the polymer phases resulting in their lateral organization. In this work, the organization of CdSe nanoparticles and fluorescent organic dyes was studied by fluorescence microscopy. The NOs were organized in the polymer film in stochastic patterns or in ordered designs on substrates pre-patterned by soft-lithography techniques. Single-particle measurements, using confocal microscopy, showed that at low concentrations there was little aggregation of the particles.     

The use of synchrotron X-ray scattering coupled with in situ mechanical testing for studying deformation and structural change in isotactic polypropylene

The mechanical behaviour of semi-crystalline polymers is greatly influenced by the properties of the crystalline and the amorphous phases. As a result this topic has been the subject of extensive research. However, to date, a comprehensive relationship between the structure and mechanical properties for semi-crystalline polymers has yet to be established. This present study concerns the commissioning of a novel method for in situ data collection during the deformation of polymers. This involves the combination of three different techniques into a single experiment, namely tensile testing, synchrotron radiation wide angle X-ray scattering, and optical microscopy. For this current investigation, three isotactic polypropylene samples have been studied, produced using different thermal treatments. This enables the influence of thermal treatment on the mechanical properties and crystallographic structure to be assessed. The results indicate that tensile properties are influenced by thermal treatment via the relative fraction of -phase material in the sample. As the temperature increases at which thermal treatment takes place, iPP ductility decreases due to the greater rigidity of the increasing -phase content. Differences in crystal strain between the different iPP crystal phases are also observed although the reasons for such differences remain unclear.