Ideal Criteria for both Ideal Criteria for both X2-dy2 = M1 And X2-dy2 = M2 to have Primitive Solutions for any Integers M1, M2 Prime to D > 0

This article provides necessary and sufficient conditions for both of the Diophantine equations X^2 − DY^2 = m1 and x^2 − Dy^2 = m2 to have primitive solutions when m1 , m2 ∈ Z, and D ∈ N is not a perfect square. This is given in terms of the ideal theory of the underlying real quadratic order Z[√D].

Optical trapping with superfocused high-M2 laser diode beam

Many applications of high-power laser diodes demand tight focusing. This is often not possible due to the multimode nature of semiconductor laser radiation possessing beam propagation parameter M2 values in double-digits. We propose a method of 'interference' superfocusing of high-M2 diode laser beams with a technique developed for the generation of Bessel beams based on the employment of an axicon fabricated on the tip of a 100 μm diameter optical fiber with highprecision direct laser writing. Using axicons with apex angle 140º and rounded tip area as small as 10 μm diameter, we demonstrate 2-4 μm diameter focused laser 'needle' beams with approximately 20 μm propagation length generated from multimode diode laser with beam propagation parameter M2=18 and emission wavelength of 960 nm. This is a few-fold reduction compared to the minimal focal spot size of 11 μm that could be achieved if focused by an 'ideal' lens of unity numerical aperture. The same technique using a 160º axicon allowed us to demonstrate few-μm-wide laser 'needle' beams with nearly 100 μm propagation length with which to demonstrate optical trapping of 5-6 μm rat blood red cells in a water-heparin solution. Our results indicate the good potential of superfocused diode laser beams for applications relating to optical trapping and manipulation of microscopic objects including living biological objects with aspirations towards subsequent novel lab-on-chip configurations.

Superfocusing of high-M2 semiconductor laser beams:experimental demonstration

The focusing of multimode laser diode beams is probably the most significant problem that hinders the expansion of the high-power semiconductor lasers in many spatially-demanding applications. Generally, the 'quality' of laser beams is characterized by so-called 'beam propagation parameter' M2, which is defined as the ratio of the divergence of the laser beam to that of a diffraction-limited counterpart. Therefore, M2 determines the ratio of the beam focal-spot size to that of the 'ideal' Gaussian beam focused by the same optical system. Typically, M2 takes the value of 20-50 for high-power broad-stripe laser diodes thus making the focal-spot 1-2 orders of magnitude larger than the diffraction limit. The idea of 'superfocusing' for high-M2 beams relies on a technique developed for the generation of Bessel beams from laser diodes using a cone-shaped lens (axicon). With traditional focusing of multimode radiation, different curvatures of the wavefronts of the various constituent modes lead to a shift of their focal points along the optical axis that in turn implies larger focal-spot sizes with correspondingly increased values of M2. In contrast, the generation of a Bessel-type beam with an axicon relies on 'self-interference' of each mode thus eliminating the underlying reason for an increase in the focal-spot size. For an experimental demonstration of the proposed technique, we used a fiber-coupled laser diode with M2 below 20 and an emission wavelength in ~1μm range. Utilization of the axicons with apex angle of 140deg, made by direct laser writing on a fiber tip, enabled the demonstration of an order of magnitude decrease of the focal-spot size compared to that achievable using an 'ideal' lens of unity numerical aperture. © 2014 SPIE.

Measurement of scleral thickness in humans using anterior segment optical coherent tomography

Anterior segment optical coherent tomography (AS-OCT, Visante; Zeiss) is used to examine meridional variation in anterior scleral thickness (AST) and its association with refractive error, ethnicity and gender. Scleral cross-sections of 74 individuals (28 males; 46 females; aged between 18-40 years (27.7±5.3)) were sampled twice in random order in 8 meridians: [superior (S), inferior (I), nasal (N), temporal (T), superior-temporal (ST), superior-nasal (SN), inferior-temporal (IT) and inferior-nasal (IN)]. AST was measured in 1mm anterior-toposterior increments (designated the A-P distance) from the scleral spur (SS) over a 6mm distance. Axial length and refractive error were measured with a Zeiss IOLMaster biometer and an open-view binocular Shin-Nippon autorefractor. Intra- And inter-observer variability of AST was assessed for each of the 8 meridians. Mixed repeated measures ANOVAs tested meridional and A-P distance differences in AST with refractive error, gender and ethnicity. Only right eye data were analysed. AST (mean±SD) across all meridians and A-P distances was 725±46μm. Meridian SN was the thinnest (662±57μm) and I the thickest (806 ±60μm). Significant differences were found between all meridians (p<0.001), except S:ST, IT:IN, IT:N and IN:N. Significant differences between A-P distances were found except between SS and 6 mm and between 2 and 4mm. AST measurements at 1mm (682±48 μm) were the thinnest and at 6mm (818±49 μm) the thickest (p<0.001); a significant interaction occurred between meridians and A-P distances (p<0.001). AST was significantly greater (p<0.001) in male subjects but no significant differences were found between refractive error or ethnicity. Significant variations in AST occur with regard to meridian and distance from the SS and may have utility in selecting optimum sites for pharmaceutical or surgical intervention.

On strongly regular graphs with m2 = qm3 and m3 = qm2

2010 Mathematics Subject Classification: 05C50.

Client QoE-oriented segment selection for DASH

A segment selection method controlled by Quality of Experience (QoE) factors for Dynamic Adaptive Streaming over HTTP (DASH) is presented in this paper. Current rate adaption algorithms aim to eliminate buffer underrun events by significantly reducing the code rate when experiencing pauses in replay. In reality, however, viewers may choose to accept a level of buffer underrun in order to achieve an improved level of picture fidelity or to accept the degradation in picture fidelity in order to maintain the service continuity. The proposed rate adaption scheme in our work can maximize the user QoE in terms of both continuity and fidelity (picture quality) in DASH applications. It is shown that using this scheme a high level of quality for streaming services, especially at low packet loss rates, can be achieved. Our scheme can also maintain a best trade-off between continuity-based quality and fidelity-based quality, by determining proper threshold values for the level of quality intended by clients with different quality requirements. In addition, the integration of the rate adaptation mechanism with the scheduling process is investigated in the context of a mobile communication network and related performances are analyzed.

Climate variability in the Aegean Sea (marine record M2, Athos basin) during the past 1500 years

We provide new evidence on sea surface temperature (SST) variations and paleoceanographic/paleoenvironmental changes over the past 1500 years for the north Aegean Sea (NE Mediterranean). The reconstructions are based on multiproxy analyses, obtained from the high resolution (decadal to multi-decadal) marine record M2 retrieved from the Athos basin. Reconstructed SSTs show an increase from ca. 850 to 950 AD and from ca. 1100 to 1300 AD. A cooling phase of almost 1.5 °C is observed from ca. 1600 AD to 1700 AD. This seems to have been the starting point of a continuous SST warming trend until the end of the reconstructed period, interrupted by two prominent cooling events at 1832 ± 15 AD and 1995 ± 1 AD. Application of an adaptive Kernel smoothing suggests that the current warming in the reconstructed SSTs of the north Aegean might be unprecedented in the context of the past 1500 years. Internal variability in atmospheric/oceanic circulations systems as well as external forcing as solar radiation and volcanic activity could have affected temperature variations in the north Aegean Sea over the past 1500 years. The marked temperature drop of approximately ~2 °C at 1832 ± 15 yr AD could be related to the 1809 ?D 'unknown' and the 1815 AD Tambora volcanic eruptions. Paleoenvironmental proxy-indices of the M2 record show enhanced riverine/continental inputs in the northern Aegean after ca. 1450 AD. The paleoclimatic evidence derived from the M2 record is combined with a socio-environmental study of the history of the north Aegean region. We show that the cultivation of temperature-sensitive crops, i.e. walnut, vine and olive, co-occurred with stable and warmer temperatures, while its end coincided with a significant episode of cooler temperatures. Periods of agricultural growth in Macedonia coincide with periods of warmer and more stable SSTs, but further exploration is required in order to identify the causal links behind the observed phenomena. The Black Death likely caused major changes in agricultural activity in the north Aegean region, as reflected in the pollen data from land sites of Macedonia and the M2 proxy-reconstructions. Finally, we conclude that the early modern peaks in mountain vegetation in the Rhodope and Macedonia highlands, visible also in the M2 record, were very likely climate-driven.