Remarks on a conjecture on certain integer sequences

Summary The periodicity of sequences of integers <InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"9"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"10"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>(a_{n})_{n\in\mathbb Z}$ satisfying the inequalities <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[$$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation> 0 \le a_{n-1}+\lambda a_n +a_{n+1} < 1 \ (n \in {\mathbb Z}) $$ is studied for real $ \lambda $ with $|\lambda|< 2$. Periodicity is proved in case $ \lambda $ is the golden ratio; for other values of $ \lambda $ statements on possible period lengths are given. Further interesting results on the morphology of periods are illustrated. The problem is connected to the investigation of shift radix systems and of Salem numbers.     

Three-dimensional susceptibility-weighted imaging at 3 T using various image analysis methods in the estimation of grading intracranial gliomas

Object

Although three-dimensional (3D), high-spatial resolution susceptibility-weighted imaging (SWI) appears to be valuable in the evaluation of central nervous system gliomas, several evaluation methods are proposed in the literature. The purpose of this study was to evaluate the use of 3D SWI for grading intracranial gliomas with various analysis methods.

Materials and Methods

Twenty-three patients suspected of having gliomas participated in this study. SWI was performed in addition to conventional MR sequences. In 15 cases, post-gadolinium enhanced SWI was also obtained. Imaging evaluation criteria were conventional grade, hypointensity ratio in the tumor-dominant structure of hypointensity on SWI (hemorrhage or vascular structure) and presence of abnormal enhancement surrounding the tumor.

Results

Mean grading scores of conventional grade showed no statistically significant difference among WHO grades. Mean grading scores of hypointensity ratios in the tumor were higher for WHO Grades 3 and 4 than for lower grade tumors (P=.05, Mann–Whitney U test). Hemorrhagic foci were more frequently seen in the higher grade tumor. Post-contrast susceptibility-weighted images of five of 11 WHO Grade 3 and 4 cases showed bright enhancement surrounding the tumor, suggesting a breakdown of the blood–brain barrier.

Conclusions

SWI at 3 T may be a useful method to analyze the structural characteristics of gliomas and to evaluate pathology in vivo. Assessment of hypointensity ratios in the glioma was the most preferable method in grading glioma. However, more studies, specifically concerning a suitable method for image analysis, are needed to establish SWI at 3 T as a useful tool in clinical routine.     

Performance improvement of rubrene-based organic light emitting devices with a mixed single layer

We have investigated the performance of organic light-emitting devices (OLEDs) with a rubrene-doped mixed single layer by using 4,4′-bis[N-(1-napthyl)-N-phenyl- amion] biphenyl (α-NPD) as hole transport layer. Comparing to a conventional heterostructure OLED, equal luminance vs. current density characteristics were obtained. In addition, maximum power efficiency was threefold improved, and the achieved value was 5.90 lm/W by optimizing a mixing ratio of hole and electron transport materials. By evaluating the temperature dependence of the J – V characteristics for electron-injection dominated device, the electron injection from Al/LiF to mixed organic layer is attributed to Schottky thermal emission model. And the barrier height of the electron injection from Al/LiF into mixed single layer was obtained to be 0.62 eV, which is lower than Al/Alq₃ interface. Meanwhile, the mixed single-layer device exhibited superior operational durability at a half-luminance of 2,250 h under a constant current operation mode. The reliability was improved with a factor of two compared to the heterostructure device due to the improvement of stability in mixed organic molecules and removal of the heterojunction interface in the mixed single-layer device.     

Stable 10 W Er:ZBLAN fiber laser operating at 2.71-2.88 μm

We have developed a diode-pumped tunable 3 μm fiber laser with a cw output power of the order of 10 W with the use of an erbium-doped ZBLAN fiber. A tunability range of 110 nm (2770 to 2880 nm) with an output power between 8 and 11 W was demonstrated. As the pump power was increased, the center of the wavelength range was shifted toward longer wavelengths, but the width of the wavelength range was largely unaffected. The total tunability range for various pump power levels was 170 nm (2710 to 2880 nm). To our knowledge, this is the highest performance (output power and tunability) obtained from a tunable 3 μm fiber laser.     

Low temperature fabrication of 5-10 nm SiO2/Si structure using advanced nitric acid oxidation of silicon (NAOS) method

We have developed the advanced nitric acid oxidation of Si (NAOS) method to form relatively thick (5-10 nm) SiO₂/Si structure with good electrical characteristics. This method simply involves immersion of Si in 68 wt% nitric acid aqueous solutions at 120 °C with polysilazane films. Fourier transform infrared absorption (FT-IR) measurements show that the atomic density of the NAOS SiO₂ layer is considerably high even without post-oxidation anneal (POA), i.e., 2.28 × 10²² atoms/cm², and it increases by POA at 400 °C in wet-oxygen (2.32 × 10²² atoms/cm²) or dry-oxygen (2.30 × 10²² atoms/cm²). The leakage current density is considerably low (e.g., 10⁻⁵ A/cm² at 8 MV/cm) and it is greatly decreased (10⁻⁸ A/cm² at 8 MV/cm) by POA at 400 °C in wet-oxygen. POA in wet-oxygen increases the atomic density of the SiO₂ layer, and decreases the density of oxide fixed positive charges.     

Intrinsic versus extrinsic anomalous Hall effect in ferromagnets

A unified theory of the anomalous Hall effect (AHE) is presented for multiband ferromagnetic metals with dilute impurities. In the clean limit, the AHE is mostly due to extrinsic skew scattering. When the Fermi level is located around anticrossing of band dispersions split by spin-orbit interaction, the intrinsic AHE to be calculated ab initio is resonantly enhanced by its nonperturbative nature, revealing the extrinsic-to-intrinsic crossover which occurs when the relaxation rate is comparable to the spin-orbit coupling.     

Calcium isotope fractionation in liquid chromatography with crown ether resin in methanol medium

Breakthrough mode liquid chromatography was employed to investigate calcium (Ca) isotope fractionation in methanol medium. Highly porous silica beads, the inner pores of which were embedded with a benzo-18-crown-6 ether resin, were used as column packing material. Enrichment of heavier isotopes of Ca was observed in the frontal part of the respective Ca chromatograms. The values of the isotope fractionation coefficient (?) were in the order of 10^?3 at 25 °C. Use of methanol as solvent has little advantage over the aqueous system as far as the values of ? are concerned. However, a substantial improvement was observed concerning the adsorption capacity of the crown ether resin for Ca ions. Molecular orbital calculations supported the present isotopic results in a qualitative fashion.     

Gauss Optics and Gauss Sum on an Optical Phenomena

In the previous article (Found. Phys. Lett. 16:325–341, 2003), we showed that a reciprocity of the Gauss sums is connected with the wave and particle complementary. In this article, we revise the previous investigation by considering a relation between the Gauss optics and the Gauss sum based upon the recent studies of the Weil representation for a finite group.