On the Leibniz rule and Laplace transform for fractional derivatives

Taylor series is a useful mathematical tool when describing and constructing a function. With the series representation, some properties of fractional calculus can be revealed clearly. This paper investigates two typical applications: Lebiniz rule and Laplace transform. It is analytically shown that the commonly used Leibniz rule cannot be applied for Caputo derivative. Similarly, the well-known Laplace transform of Riemann-Liouville derivative is doubtful for n-th continuously differentiable function. By the aid of this series representation, the exact formula of Caputo Leibniz rule and the explanation of Riemann-Liouville Laplace transform are presented. Finally, three illustrative examples are revisited to confirm the obtained results

Kripke’s Category Error: Why There Are No Necessary A posteriori Propositions

Kripke’s main argument against descriptivism is rooted in a category error that confuses statements about the world with statements about models of the world. It is only because of the ambiguity introduced by the fact that a single sentence can frame two different propositions, one necessary and the other a posteriori, that one reaches the mistaken conclusion that there can be necessary a posteriori truths. This ambiguity from language was carried over into modal logic by Kripke. However, we must consider the two different propositions (1) and (2) separately. Doing so reveals that a given proposition is either necessary and a priori or contingent and a posteriori. It cannot be both

Motion-Induced Position Shifts Activate Early Visual Cortex

The ability to correctly determine the position of objects in space is a fundamental task of the visual system. The perceived position of briefly presented static objects can be influenced by nearby moving contours, as demonstrated by various illusions collectively known as motion-induced position shifts. Here we use a stimulus that produces a particularly strong effect of motion on perceived position. We test whether several regions-of-interest (ROIs), at different stages of visual processing, encode the perceived rather than retinotopically veridical position. Specifically, we collect functional MRI data while participants experience motion-induced position shifts and use a multivariate pattern analysis approach to compare the activation patterns evoked by illusory position shifts with those evoked by matched physical shifts. We find that the illusory perceived position is represented at the earliest stages of the visual processing stream, including primary visual cortex. Surprisingly, we found no evidence of percept-based encoding of position in visual areas beyond area V3. This result suggests that while it is likely that higher-level visual areas are involved in position encoding, early visual cortex also plays an important role

Microsaccade Rate Varies with Subjective Visibility during Motion-Induced Blindness

Motion-induced blindness (MIB) occurs when a dot embedded in a motion field subjectively vanishes. Here we report the first psychophysical data concerning effects of microsaccade/eyeblink rate upon perceptual switches during MIB. We find that the rate of microsaccades/eyeblink rises before and after perceptual transitions from not seeing to seeing the dot, and decreases before perceptual transitions from seeing it to not seeing it. In addition, event-related fMRI data reveal that, when a dot subjectively reappears during MIB, the blood oxygen-level dependent (BOLD) signal increases in V1v and V2v and decreases in contralateral hMT+. These BOLD signal changes observed upon perceptual state changes in MIB could be driven by the change of perceptual states and/or a confounding factor, such as the microsaccade/eyeblink rate

Direction-of-Change Forecasts for Asian Equity Markets Based on Conditional Variance, Skewness and Kurtosis Dynamics: Evidence from Hong Kong and Singapore

Recent theoretical work has revealed a direct connection between asset return volatility forecastability and asset return sign forecastability. This suggests that the pervasive volatility forecastability in equity returns could, via induced sign forecastability, be used to produce direction-ofchange forecasts useful for market timing. We attempt to do so in the context of two key Asian equity markets, with some success, as assessed by formal probability forecast scoring rules such as the Brier score. An important ingredient is our conditioning not only on conditional variance information, but also conditional skewness and kurtosis information, when forming direction-of-change forecasts.Volatility, variance, skewness, kurtosis, market timing, asset management, asset allocation, portfolio management.

Direction-of-Change Forecasts Based on Conditional Variance, Skewness and Kurtosis Dynamics : International Evidence

Recent theoretical work has revealed a direct connection between asset return volatility forecastability and asset return sign forecastability. This suggests that the pervasive volatility forecastability in equity returns could, via induced sign forecastability, be used to produce direction-of-change forecasts useful for market timing. We attempt to do so in an international sample of developed equity markets, with some success, as assessed by formal probability forecast scoring rules such as the Brier score. An important ingredient is our conditioning not only on conditional mean and variance information, but also conditional skewness and kurtosis information, when forming direction-of-change forecasts.Volatility, variance, skewness, kurtosis, market timing, asset management, asset allocation, portfolio management

Direction-of-Change Forecasts Based on Conditional Variance, Skewness and Kurtosis Dynamics: International Evidence

Recent theoretical work has revealed a direct connection between asset return volatility forecastability and asset return sign forecastability. This suggests that the pervasive volatility forecastability in equity returns could, via induced sign forecastability, be used to produce direction-of change forecasts useful for market timing. We attempt to do so in an international sample of developed equity markets, with some success, as assessed by formal probability forecast scoring rules such as the Brier score. An important ingredient is our conditioning not only on conditional mean and variance information, but also conditional skewness and kurtosis information, when forming direction-of-change forecasts.Volatility, variance, skewness, kurtosis, market timing, asset management, asset allocation, portfolio management

Detecting Long-Duration Narrow-Band Gravitational Wave Transients Associated with Soft Gamma Repeater Quasi-Periodic Oscillations

We have performed an in-depth concept study of a gravitational wave data analysis method which targets repeated long quasi-monochromatic transients (triggers) from cosmic sources. The algorithm concept can be applied to multi-trigger data sets in which the detector-source orientation and the statistical properties of the data stream change with time, and does not require the assumption that the data is Gaussian. Reconstructing or limiting the energetics of potential gravitational wave emissions associated with quasi-periodic oscillations (QPOs) observed in the X-ray lightcurve tails of soft gamma repeater flares might be an interesting endeavour of the future. Therefore we chose this in a simplified form to illustrate the flow, capabilities, and performance of the method. We investigate performance aspects of a multi-trigger based data analysis approach by using O(100 s) long stretches of mock data in coincidence with the times of observed QPOs, and by using the known sky location of the source. We analytically derive the PDF of the background distribution and compare to the results obtained by applying the concept to simulated Gaussian noise, as well as off-source playground data collected by the 4-km Hanford detector (H1) during LIGO's fifth science run (S5). We show that the transient glitch rejection and adaptive differential energy comparison methods we apply succeed in rejecting outliers in the S5 background data. Finally, we discuss how to extend the method to a network containing multiple detectors, and as an example, tune the method to maximize sensitivity to SGR 1806-20 flare times.Comment: 11 pages, 8 figure