Generalized Fractional Calculus, Special Functions and Integral Transforms: What is the Relation?

Виржиния С. Кирякова - В този обзор илюстрираме накратко наши приноси към обобщенията на дробното смятане (анализ) като теория на операторите за интегриране и диференциране от произволен (дробен) ред, на класическите специални функции и на интегралните трансформации от лапласов тип. Показано е, че тези три области на анализа са тясно свързани и взаимно индуцират своето възникване и по-нататъшно развитие. За конкретните твърдения, доказателства и примери, вж. Литературата.

Strict LpSolutions for Nonautonomous Fractional Evolution Equations

MSC 2010: 26A33, 34A08, 34K37

Solutions of Fractional Diffusion-Wave Equations in Terms of H-functions

MSC 2010: 35R11, 42A38, 26A33, 33E12

Complex Oscillations and Limit Cycles in Autonomous Two-Component Incommensurate Fractional Dynamical Systems

MSC 2010: 26A33, 34D05, 37C25

Conflict-Controlled Processes Involving Fractional Differential Equations with Impulses

MSC 2010: 34A08, 34A37, 49N70

Calculus of Variations with Classical and Fractional Derivatives

MSC 2010: 49K05, 26A33

Fractional Korovkin Theory Based on Statistical Convergence

2000 Mathematics Subject Classification: 41A25, 41A36, 40G15.

Fractional Trigonometric Korovkin Theory in Statistical Sense

2000 Mathematics Subject Classification: 41A25, 41A36.

Table 1 Core site, fractional abundance of individual isoprenoid GDGTs and TEX86 values

TEXL86 and TEXH86 are organic palaeothermometers based on the lipids of Group 1 Crenarchaeota, recently proposed as a modified version of the original TEX86 index, but with significantly improved geographical coverage. Since few data from the global core top calibration are from the Pacific, this study was carried out to assess whether the global core top calibration is regionally biased or not. The result of principal components analysis of the fractional abundance of GDGTs, an analysis of variance (ANOVA) and the comparison of the residuals of TEXH 86 derived sea surface temperature (SST) estimates of the Pacific subset with that of the global data set suggest that the Pacific subset has a similar TEXH 86-SST relationship with the global data set. However, the regression line through the Pacific data and an ANOVA on the residuals of TEXL 86 derived SST estimates suggest otherwise. The contradictory findings are likely to stem from the large scatter in the Pacific TEXL 86 values in the mid temperature range. While regionality does not seem to exert a strong bias on TEXL 86 and TEXH 86 calibration, it appears that there is a strong need to resolve the large scatter in the global data set, especially in the mid and high latitudes, in order to improve the calibration for a better SST estimation.

The spectra of linear fractional composition operators on weighted Dirichlet spaces

We completely determine the spectra of composition operators induced by linear fractional self-maps of the unit disc acting on weighted Dirichlet spaces; extending earlier results by Higdon [8] and answering the open questions in this context.

Fractional order of poling period for broadly tunable second harmonic generation

We demonstrate the possibility to use a fractional order of poling period of nonlinear crystal waveguides for tunable second harmonic generation. This approach allows one to extend wavelength coverage in the visible spectral range by frequency doubling in a single crystal waveguide.

Anodic formation and characterization of nanoporous InP in aqueous KOH electrolytes

The anodic behavior of highly doped (> 1018 cm-3) n-InP in aqueous KOH was investigated. Electrodes anodized in the absence of light in 2- 5 mol dm-3 KOH at a constant potential of 0.5- 0.75 V (SCE), or subjected to linear potential sweeps to potentials in this range, were shown to exhibit the formation of a nanoporous subsurface region. Both linear sweep voltammograms and current-time curves at constant potential showed a characteristic anodic peak, corresponding to formation of the nanoporous region. No porous region was formed during anodization in 1 mol dm-3 KOH. The nanoporous region was examined using transmission electron microscopy and found to have a thickness of some 1- 3 μm depending on the anodization conditions and to be located beneath a thin (typically ∼40 nm), dense, near-surface layer. The pores varied in width from 25 to 75 nm and both the pore width and porous region thickness were found to decrease with increasing KOH concentration. The porosity was approximately 35%. The porous layer structure is shown to form by the localized penetration of surface pits into the InP, and the dense, near-surface layer is consistent with the effect of electron depletion at the surface of the semiconductor.

Nanoporous InP: anodic formation and growth mechanism in aqueous electrolytes

Porous layers can be formed electrochemically on (100) oriented n-InP substrates in aqueous KOH. A nanoporous layer is obtained underneath a dense near-surface layer and the pores appear to propagate from holes through the near-surface layer. In the early stages of the anodization transmission electron microscopy (TEM) clearly shows individual porous domains that appear to have a square-based pyramidal shape. Each domain appears to develop from an individual surface pit which forms a channel through this near-surface layer. We suggest that the pyramidal structure arises as a result of preferential pore propagation along the <100> directions. AFM measurements show that the density of surface pits increases with time. Each of these pits acts as a source for a pyramidal porous domain. When the domains grow, the current density increases correspondingly. Eventually the domains meet, forming a continuous porous layer, the interface between the porous and bulk InP becomes relatively flat and its total effective surface area decreases resulting in a decrease in the current density. Current-time curves at constant potential exhibit a peak and porous layers are observed to form beneath the electrode surface. The density of pits formed on the surface increases with time and approaches a plateau value. Porous layers are also observed in highly doped InP but are not observed in wafers with doping densities below ~5 × 1017 cm-3. Numerical models of this process have been developed invoking a mechanism of directional selectivity of pore growth preferentially along the <100> lattice directions. Manipulation of the parameters controlling these curves shows that the fall-off in current is controlled by the rate of diffusion of electrolyte through the pore structure with the final decline in current being caused by the termination of growth at the pore tips through the formation of passivating films or some other irreversible modification of the pore tips.

Anodic oxidation of InP in KOH electrolytes

The anodic behavior of InP in 1 mol dm-3 KOH was investigated and compared with its behavior at higher concentrations of KOH. At concentrations of 2 mol dm-3 KOH or greater, selective etching of InP occurs leading to thick porous InP layers near the surface of the sustrate. In contrast, in 1 mol dm-3 KOH, no such porous layers are formed but a thin surface film is formed at potentials in the range 0.6 V to 1.3 V. The thickness of this film was determined by spectroscopic ellipsometry as a function of the upper potential and the measured film thickness corresponds to the charge passed up to a potential of 1.0 V. Anodization to potentials above 1.5 V in 1 mol dm- 3 KOH results in the growth of thick, porous oxide films (~ 1.2 µm). These films are observed to crack, ex-situ, due to shrinkage after drying in ambient air. Comparisons between the charge density and film thickness measurements indicate a porosity of approximately 77% for such films.

Ammonium excretion and respiration rate of tropical copepods and euphausiids measured during METEOR cruise M93, M97 and Maria S. Merian cruise MSM22

Respiration and ammonium excretion rates at different oxygen partial pressure were measured for calanoid copepods and euphausiids from the Eastern Tropical South Pacific and the Eastern Tropical North Atlantic. All specimens used for experiments were caught in the upper 400 m of the water column and only animals appearing unharmed and fit were used for experiments. Specimens were sorted, identified and transferred into aquaria with filtered, well-oxygenated seawater immediately after the catch and maintained for 1 to 13 hours prior to physiological experiments at the respective experimental temperature. Maintenance and physiological experiments were conducted in darkness in temperature-controlled incubators at 11, 13 or 23 degree C (±1). Before and during experiments, animals were not fed. Respiration and ammonium excretion rate measurements (both in µmol h-1 gDW-1) at varying oxygen concentrations were conducted in 12 to 60 mL gas-tight glass bottles. These were equipped with oxygen microsensors (ø 3 mm, PreSens Precision Sensing GmbH, Regensburg, Germany) attached to the inner wall of the bottles to monitor oxygen concentrations non-invasively. Read-out of oxygen concentrations was conducted using multi-channel fiber optic oxygen transmitters (Oxy-4 and Oxy-10 mini, PreSens Precision Sensing GmbH, Regensburg, Germany) that were connected via optical fibers to the outside of the bottles directly above the oxygen microsensor spots. Measurements were started at pre-adjusted oxygen and carbon dioxide levels. For this, seawater stocks with adjusted pO2 and pCO2 were prepared by equilibrating 3 to 4 L of filtered (0.2 µm filter Whatman GFF filter) and UV - sterilized (Aqua Cristal UV C 5 Watt, JBL GmbH & Co. KG, Neuhofen, Germany) water with premixed gases (certified gas mixtures from Air Liquide) for 4 hours at the respective experimental temperature. pCO2 levels were chosen to mimic the environmental pCO2 in the ETSP OMZ or the ETNA OMZ. Experimental runs were conducted with 11 to 15 trial incubations (1 or 2 animals per incubation bottle and three different treatment levels) and three animal-free control incubations (one per experimental treatment). During each run, experimental treatments comprised 100% air saturation as well as one reduced air saturation level with and without CO2. Oxygen concentrations in the incubation bottles were recorded every 5 min using the fiber-optic microsensor system and data recording for respiration rate determination was started immediately after all animals were transferred. Respiration rates were calculated from the slope of oxygen decrease over selected time intervals. Chosen time intervals were 20 to 105 min long. No respiration rate was calculated for the first 20 to 60 min after animal transfer to avoid the impact of enhanced activity of the animal or changes in the bottle water temperature during initial handling on the respiration rates and oxygen readings. Respiration rates were obtained over a maximum of 16 hours incubation time and slopes were linear at normoxia to mild hypoxia. Respiration rates in animal-free control bottles were used to correct for microbial activity. These rates were < 2% of animal respiration rates at normoxia. Samples for the measurement of ammonium concentrations were taken after 2 to 10 hours incubation time. Ammonium concentration was determined fluorimetrically (Holmes et al., 1999). Ammonium excretion was calculated as the concentration difference between incubation and animal-free control bottles. Some specimens died during the respiration and excretion rate measurements, as indicated by a cessation of respiration. No excretion rate measurements were conducted in this case, but the oxygen level at which the animal died was noted.