New results in equal sums of like powers

This paper reports on new results for the equation

i.e., equal sums of like powers. Since the 1967 Lander, Parkin and Selfridge survey paper [4], few other numeric results have been published (see Elkies [6] and Ekl [3]). The present paper reports on several new smallest primitive solutions. Further, search limits have been extended in many cases, and tables of solutions are presented. Additionally, new solutions to the same class of problems in distinct integers have been discovered.

     

A soluble base for the copper-catalyzed imidazole N-arylations with aryl halides

[reaction: see text] CuI-catalyzed N-arylation of imidazoles with aryl bromides has been achieved in a near-homogeneous system that utilizes tetraethylammonium carbonate as base, 8-hydroxyquinoline as ligand, and H2O as cosolvent. Preliminary results with aryl chlorides are also reported.     

Effects of melt‐processing conditions on the quality of poly(ethylene terephthalate) montmorillonite clay nanocomposites

Organically modified montmorillonite was synthesized with a novel 1,2‐dimethyl‐3‐N‐alkyl imidazolium salt or a typical quaternary ammonium salt as a control. Poly(ethylene terephthalate) montmorillonite clay nanocomposites were compounded via melt‐blending in a corotating mini twin‐screw extruder operating at 285 °C. The nanocomposites were characterized with thermal analysis, X‐ray diffraction, and transmission electron microscopy to determine the extent of intercalation and/or exfoliation present in the system. Nanocomposites produced with N,N‐dimethyl‐N,N‐dioctadecylammonium treated montmorillonite (DMDODA‐MMT), which has a decomposition temperature of 250 °C, were black, brittle, and tarlike resulting from DMDODA degradation under the processing conditions. Nanocomposites compounded with 1,2‐dimethyl‐3‐N‐hexadecyl imidazolium treated MMT, which has a decomposition temperature of 350 °C, showed high levels of dispersion and delamination. © 2002 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 40: 2661–2666, 2002     

Base-triggered self-amplifying degradable polyurethanes with the ability to translate local stimulation to continuous long-range degradation

A new type of base-triggered self-amplifying degradable polyurethane is reported that degrades under mild conditions, with the release of increasing amounts of amine product leading to self-amplified degradation. The polymer incorporates a base-sensitive Fmoc-derivative into every repeating unit to enable highly sensitive amine amplified degradation. A sigmoidal degradation curve for the linear polymer was observed consistent with a self-amplifying degradation mechanism. An analogous cross-linked polyurethane gel was prepared and also found to undergo amplified breakdown. In this case, a trace amount of localized base initiates the degradation, which in turn propagates through the material in an amplified manner. The results demonstrate the potential utility of these new generation polyurethanes in enhanced disposability and as stimuli responsive materials.

A new type of base-triggered self-amplifying degradable polyurethane is reported that degrades under mild conditions, with the release of increasing amounts of amine product leading to self-amplified degradation.     

Atmospheric pressure ion trapping in a tandem FAIMS-FAIMS coupled to a TOFMS: Studies with electrospray generated gramicidin S ions

A tandem FAIMS–FAIMS system for ion trapping at room temperature and atmospheric pressure is described. The first FAIMS device consisted of a side-to-side configuration (sFAIMS) suitable for ion separation, whereas the second FAIMS device was appropriate for ion trapping (tFAIMS). Ions pre-selected by the sFAIMS entered the tFAIMS and were captured by virtual trapping fields at the hemispherical tip of the inner electrode. The use of the sFAIMS, with wider electrode diameters, and consequently better ion separation efficiency than the tFAIMS, lowered the number of background ions captured in the trapping region of tFAIMS, and thus reduced the space charge effects in the trap. This tandem device was coupled to a laboratory built time-of-flight mass spectrometer and was evaluated using the electrospray generated [M + 2H]²⁺ ion of gramicidin S. The half-time (t_1/2) of the exponential decay of the ion cloud in tFAIMS, determined by monitoring the residual intensity of ions extracted from the ion trapping region of tFAIMS after various delay times, was about 2 s.     

SU(2)×SU(2) Electroweak Theory I: The B3 Field on the Physical Vacuum

Nonlinear optics confronts the U(1) theory of electrodynamics with the dilemma of the existence of nonlinear fields. The U(1) group is completely linear and Abelian and causes consideration of an SU(2) theory of electrodynamics. An SU(2) theory of electrodynamics, with a B ³ magnetic field, means that physics is forced to consider an SU(2) × SU(2) electroweak theory. It is then demonstrated that the B ³ field exists on the physical vacuum defined by the Higgs symmetry breaking of this extended electroweak theory.     

Inconsistencies of the U(1) Theory of Electrodynamics: Stress Energy Momentum Tensor

The internal gauge space of electrodynamics considered as a U(1) gauge field theory is a scalar. This leads to the result that in free space, and for plane waves, the Poynting vector and energy vanish. This result is consistent with the fact that U(1) gauge field theory results in a null third Stokes parameter, meaning again that the field energy vanishes in free space. A self consistent definition of the stress energy momentum tensor is obtained with a Yang Mills theory applied with an O(3) symmetry internal gauge space. This theory produces the third Stokes parameter self consistently in terms of the self-dual Evans-Vigier fields B(3).     

Zeolite-fiber integrated optical chemical sensors for detection of dissolved organics in water

MFI zeolite coated optical fiber sensors have been developed for in situ detection of dissolved organics in water. The sensors operate by monitoring the optical reflectivity changes caused by the selective adsorption of organic molecules, i.e., 2-propanol or pentanoic acid in this study, from aqueous solutions in the zeolitic pores. Reversible and monotonic sensor signals were observed in response to the variation of 2-propanol concentration in water with fast response. However, the sensor exhibited a much slower response to pentanoic acid than to 2-propanol. It was also found that substitution of Si by Al in the MFI framework increased the adsorption of pentanoic acid that resulted in enhanced sensor responses.