Fractional Calculus of P-transforms

MSC 2010: 44A20, 33C60, 44A10, 26A33, 33C20, 85A99

On the Quotient Function Employed in the Blind Source Separation Problem

MSC 2010: 42C40, 94A12

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.

High resolution optical time-domain reflectometry based on correlation utilizing an all-fiber chaotic source

We propose a high-resolution optical time domain reflectometry (OTDR) based on an all-fiber supercontinuum source. The source simply consists of a laser with moderate power and a section of fiber which has a zero dispersion wavelength near the laser's central wavelength. Spectrum and time domain properties of the source are investigated, showing that the source has great capability in nonlinear optics, such as correlation OTDR. We analyze one of the key factors limiting the operational range of such an OTDR, i.e., sampling time. Finally, we experimentally demonstrate a correlation OTDR with 25km sensing range and 5.3cm spatial resolution, as a verification of theoretical analysis.

Exponential approximation in the norms and semi-norms

The deviations of some entire functions of exponential type from real-valued functions and their derivatives are estimated. As approximation metrics we use the Lp-norms and power variations on R. Theorems presented here correspond to the Ganelius and Popov results concerning the one-sided trigonometric approximation of periodic functions (see [4, 5 and 8]). Some related facts were announced in [2, 3, 6 and 7].

Green's function for paraxial equation

The theory and experimental applications of optical Airy beams are in active development recently. The Airy beams are characterised by very special properties: they are non-diffractive and propagate along parabolic trajectories. Among the striking applications of the optical Airy beams are optical micro-manipulation implemented as the transport of small particles along the parabolic trajectory, Airy-Bessel linear light bullets, electron acceleration by the Airy beams, plasmonic energy routing. The detailed analysis of the mathematical aspects as well as physical interpretation of the electromagnetic Airy beams was done by considering the wave as a function of spatial coordinates only, related by the parabolic dependence between the transverse and the longitudinal coordinates. Their time dependence is assumed to be harmonic. Only a few papers consider a more general temporal dependence where such a relationship exists between the temporal and the spatial variables. This relationship is derived mostly by applying the Fourier transform to the expressions obtained for the harmonic time dependence or by a Fourier synthesis using the specific modulated spectrum near some central frequency. Spatial-temporal Airy pulses in the form of contour integrals is analysed near the caustic and the numerical solution of the nonlinear paraxial equation in time domain shows soliton shedding from the Airy pulse in Kerr medium. In this paper the explicitly time dependent solutions of the electromagnetic problem in the form of time-spatial pulses are derived in paraxial approximation through the Green's function for the paraxial equation. It is shown that a Gaussian and an Airy pulse can be obtained by applying the Green's function to a proper source current. We emphasize that the processes in time domain are directional, which leads to unexpected conclusions especially for the paraxial approximation.

Sub-cortical sources of the somatosensory pathway are hypoactive in migraine interictally:a Functional Source Separation analysis

Background: Recent morpho-functional evidence pointed out that abnormalities in the thalamus could play a major role in the expression of migraine neurophysiological and clinical correlates. Whether this phenomenon is primary or secondary to its functional disconnection from the brainstem remains to be determined. We used a Functional Source Separation algorithm of EEG signal to extract the activity of the different neuronal pools recruited at different latencies along the somatosensory pathway in interictal migraine without aura (MO) patients. Methods: Twenty MO patients and 20 healthy volunteers (HV) underwent EEG recording. Four ad-hoc functional constraints, two sub-cortical (FS14 at brainstem and FS16 at thalamic level) and two cortical (FS20 radial and FS22 tangential parietal sources), were used to extract the activity of successive stages of somatosensory information processing in response to the separate left and right median nerve electric stimulation. A band-pass digital filter (450-750 Hz) was applied offline in order to extract high-frequency oscillatory (HFO) activity from the broadband EEG signal. Results: In both stimulated sides, significant reduced sub-cortical brainstem (FS14) and thalamic (FS16) HFO activations characterized MO patients when compared with HV. No difference emerged in the two cortical HFO activations between the two groups. Conclusions: Present results are the first neurophysiological evidence supporting the hypothesis that a functional disconnection of the thalamus from the subcortical monoaminergic system may underline the interictal cortical abnormal information processing in migraine. Further studies are needed to investigate the precise directional connectivity across the entire primary subcortical and cortical somatosensory pathway in interictal MO. Written informed consent to publication was obtained from the patient(s).

Long-range and high-precision correlation optical time-domain reflectometry utilizing an all-fiber chaotic source

We propose a long range, high precision optical time domain reflectometry (OTDR) based on an all-fiber supercontinuum source. The source simply consists of a CW pump laser with moderate power and a section of fiber, which has a zero dispersion wavelength near the laser's central wavelength. Spectrum and time domain properties of the source are investigated, showing that the source has great capability in nonlinear optics, such as correlation OTDR due to its ultra-wide-band chaotic behavior, and mm-scale spatial resolution is demonstrated. Then we analyze the key factors limiting the operational range of such an OTDR, e. g., integral Rayleigh backscattering and the fiber loss, which degrades the optical signal to noise ratio at the receiver side, and then the guideline for counter-act such signal fading is discussed. Finally, we experimentally demonstrate a correlation OTDR with 100km sensing range and 8.2cm spatial resolution (1.2 million resolved points), as a verification of theoretical analysis.

Serum α-tocopherol has a nonlinear inverse association with periodontitis among US adults

Previous experimental models suggest that vitamin E may ameliorate periodontitis. However, epidemiologic studies show inconsistent evidence in supporting this plausible association. We aimed to investigate the association between serum α-tocopherol (αT) and γ-tocopherol (γT) and periodontitis in a large cross-sectional US population. This study included 4708 participants in the 1999–2001 NHANES. Serum tocopherols were measured by HPLC and values were adjusted by total cholesterol (TC). Periodontal status was assessed by mean clinical attachment loss (CAL) and probing pocket depth (PPD). Total periodontitis (TPD) was defined as the sum of mild, moderate, and severe periodontitis. All measurements were performed by NHANES. Means ± SDs of serum αT:TC ratio from low to high quartiles were 4.0 ± 0.4, 4.8 ± 0.2, 5.7 ± 0.4, and 9.1 ± 2.7 μmol/mmol. In multivariate regression models, αT:TC quartiles were inversely associated with mean CAL (P-trend = 0.06), mean PPD (P-trend < 0.001), and TPD (P-trend < 0.001) overall. Adjusted mean differences (95% CIs) between the first and fourth quartile of αT:TC were 0.12 mm (0.03, 0.20; P-difference = 0.005) for mean CAL and 0.12 mm (0.06, 0.17; P < 0.001) for mean PPD, whereas corresponding OR for TPD was 1.65 (95% CI: 1.26, 2.16; P-difference = 0.001). In a dose-response analysis, a clear inverse association between αT:TC and mean CAL, mean PPD, and TPD was observed among participants with relatively low αT:TC. No differences were seen in participants with higher αT:TC ratios. Participants with γT:TC ratio in the interquartile range showed a significantly lower mean PPD than those in the highest quartile. A nonlinear inverse association was observed between serum αT and severity of periodontitis, which was restricted to adults with normal but relatively low αT status. These findings warrant further confirmation in longitudinal or intervention settings.

A kockázat megjelenése a származtatott pénzügyi termékekben (The risk appearing in financial derivatives)

A pénzügyi kockázatok szerepe, modellezése, kezelése az utóbbi évtizedekben vált egyre hangsúlyosabbá az elméletben és a gyakorlatban egyaránt. A 2007-ben kezdődő pénzügyi válság egyik kiváltó oka a kockázatok nem megfelelő felmérése volt. A válság egyik tanulsága, hogy bár a matematika és a fizika hozzájárulása rendkívül mély módszertani apparátust biztosított a kockázatok számszerűsítésére, ezen eredmények pénzügyi alkalmazása csak akkor sikeres, ha pontosan értjük a modellek feltételeit és korlátait. Jelen cikk a pénzügyi derivatívák értékelésének alapelveit, valamint a származtatott ügyletekben megjelenő kockázatokat tekinti át, illetve bemutatja azokat a bizonytalansági tényezőket, amelyek megkérdőjelezik az értékelés objektivitását. / === / The modeling and management of financial risks became one of the most important topics of the last decade both in theory and fi nancial practice. The mismanagement of fi nancial risks can be mentioned among the reasons contributing to the eruption of the recent crisis. In order to use successfully the methodology of mathematics and physics in pricing of derivatives, we have to consider the assumptions and limits of the models. This paper introduces the main concepts – no arbitrage pricing and risk neutral valuation – in derivatives’ pricing, then presents and quantifies the risk of some derivative products. I am arguing that the assumptions of the Black–Scholes and Merton model are injured at several points, so the pricing can not be perfectly cleared from all the risk preferences. All those risks, deriving from the difference of the reality and the model are priced in the volatility parameter in the practice.

Characterization of ultra wideband antennas for personal communication devices using finite difference time domain technique

The current mobile networks don't offer sufficient data rates to support multimedia intensive applications in development for multifunctional mobile devices. Ultra wideband (UWB) wireless technology is being considered as the solution to overcome data rate bottlenecks in the current mobile networks. UWB is able to achieve such high data transmission rates because it transmits data over a very large chunk of the frequency spectrum. As currently approved by the U.S. Federal Communication Commission it utilizes 7.5 GHz of spectrum between 3.1 GHz and 10.6 GHz. ^ Successful transmission and reception of information data using UWB wireless technology in mobile devices, requires an antenna that has linear phase, low dispersion and a voltage standing wave ratio (VSWR) ≤ 2 throughout the entire frequency band. Compatibility with an integrated circuit requires an unobtrusive and electrically small design. The previous techniques that have been used to optimize the performance of UWB wireless systems, involve proper design of source pulses for optimal UWB performance. The goal of this work is directed towards the designing of antennas for personal communication devices, with optimal UWB bandwidth performance. Several techniques are proposed for optimal UWB bandwidth performance of the UWB antenna designs in this Ph.D. dissertation. ^ This Ph.D. dissertation presents novel UWB antenna designs for personal communication devices that have been characterized and optimized using the finite difference time domain (FDTD) technique. The antenna designs reported in this research are physically compact, planar for low profile use, with sufficient impedance bandwidth (>20%), antenna input impedance of 50-Ω, and an omni-directional (±1.5 dB) radiation pattern in the operating bandwidth. ^

Importance of water source in controlling leaf leaching losses in a dwarf red mangrove (Rhizophora mangle L.) wetland

The southern Everglades mangrove ecotone is characterized by extensive dwarf Rhizophora mangle L. shrub forests with a seasonally variable water source (Everglades – NE Florida Bay) and residence times ranging from short to long. We conducted a leaf leaching experiment to understand the influence that water source and its corresponding water quality have on (1) the early decay of R. mangle leaves and (2) the early exchange of total organic carbon (TOC) and total phosphorus (TP) between leaves and the water column. Newly senesced leaves collected from lower Taylor River (FL) were incubated in bottles containing water from one of three sources (Everglades, ambient mangrove, and Florida Bay) that spanned a range of salinity from 0 to 32‰, [TOC] from 710 to 1400 μM, and [TP] from 0.17 to 0.33 μM. We poisoned half the bottles in order to quantify abiotic processes (i.e., leaching) and assumed that non-poisoned bottles represented both biotic (i.e., microbial) and abiotic processes. We sacrificed bottles after 1,2, 5, 10, and 21 days of incubation and quantified changes in leaf mass and changes in water column [TOC] and [TP]. We saw 10–20% loss of leaf mass after 24 h—independent of water treatment—that leveled off by Day 21. After 3 weeks, non-poisoned leaves lost more mass than poisoned leaves, and there was only an effect of salinity on mass loss in poisoned incubations—with greatest leaching-associated losses in Everglades freshwater. Normalized concentrations of TOC in the water column increased by more than two orders of magnitude after 21 days with no effect of salinity and no difference between poisoned and non-poisoned treatments. However, normalized [TP] was lower in non-poisoned incubations as a result of immobilization by epiphytic microbes. This immobilization was greatest in Everglades freshwater and reflects the high P demand in this ecosystem. Immobilization of leached P in mangrove water and Florida Bay water was delayed by several days and may indicate an initial microbial limitation by labile C during the dry season.

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.

Impact of optical phase conjugation on the nonlinear Shannon limit

Compensation of the detrimental impacts of nonlinearity on long-haul wavelength division multiplexed system performance is discussed, and the difference between transmitter, receiver and in-line compensation analyzed. We demonstrate that ideal compensation of nonlinear noise could result in an increase in the signal-to-noise ratio (measured in dB) of 50%, and that reaches may be more than doubled for higher order modulation formats. The influence of parametric noise amplification is discussed in detail, showing how increased numbers of optical phase conjugators may further increase the received signal-tonoise ratio. Finally the impact of practical real world system imperfections, such as polarization mode dispersion, are outlined.

Source localization of reaction-diffusion models for brain tumors

We propose a mathematically well-founded approach for locating the source (initial state) of density functions evolved within a nonlinear reaction-diffusion model. The reconstruction of the initial source is an ill-posed inverse problem since the solution is highly unstable with respect to measurement noise. To address this instability problem, we introduce a regularization procedure based on the nonlinear Landweber method for the stable determination of the source location. This amounts to solving a sequence of well-posed forward reaction-diffusion problems. The developed framework is general, and as a special instance we consider the problem of source localization of brain tumors. We show numerically that the source of the initial densities of tumor cells are reconstructed well on both imaging data consisting of simple and complex geometric structures.