Anatomy of Bs→PVB_s \to PV decays and effects of next-to-leading order contributions in the perturbative QCD factorization approach

In this paper, we will make systematic calculations for the branching ratios and the CP-violating asymmetries of the twenty one $\bar{B}^0_s \to PV$ decays by employing the perturbative QCD (PQCD) factorization approach. Besides the full leading-order (LO) contributions, all currently known next-to-leading order (NLO) contributions are taken into account. We found numerically that: (a) the NLO contributions can provide $\sim 40\%$ enhancement to the LO PQCD predictions for ${\cal B}(\bar{B}_s^0 \to K^0 \bar{K}^{*0})$ and ${\cal B}(\bar{B}_s^0 \to K^{\pm}K^{*\mp})$, or a $\sim 37\%$ reduction to \calb(\bar{B}_s^0 \to \pi^{-} K^{*+}), and we confirmed that the inclusion of the known NLO contributions can improve significantly the agreement between the theory and those currently available experimental measurements, (b) the total effects on the PQCD predictions for the relevant $B\to P$ transition form factors after the inclusion of the NLO twist-2 and twist-3 contributions is generally small in magnitude: less than $10\%$ enhancement respect to the leading order result, (c) for the "tree" dominated decay $\bar B_s^0\to K^+ \rho^-$ and the "color-suppressed-tree" decay $\bar B_s^0\to \pi^0 K^{*0}$, the big difference between the PQCD predictions for their branching ratios are induced by different topological structure and by interference effects among the decay amplitude ${\cal A}_{T,C}$ and ${\cal A}_P$: constructive for the first decay but destructive for the second one, and (d) for \bar{B}_s^0 \to V(\eta, \etar) decays, the complex pattern of the PQCD predictions for their branching ratios can be understood by rather different topological structures and the interference effects between the decay amplitude \cala(V\eta_q) and \cala(V\eta_s) due to the \eta-\etar mixing.Comment: 18 pages, 2 figures, 3 tables. Some modifications of the text. Several new references are adde

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

Convergence Theory of Learning Over-parameterized ResNet: A Full Characterization

ResNet structure has achieved great empirical success since its debut. Recent work established the convergence of learning over-parameterized ResNet with a scaling factor $\tau=1/L$ on the residual branch where $L$ is the network depth. However, it is not clear how learning ResNet behaves for other values of $\tau$. In this paper, we fully characterize the convergence theory of gradient descent for learning over-parameterized ResNet with different values of $\tau$. Specifically, with hiding logarithmic factor and constant coefficients, we show that for $\tau\le 1/\sqrt{L}$ gradient descent is guaranteed to converge to the global minma, and especially when $\tau\le 1/L$ the convergence is irrelevant of the network depth. Conversely, we show that for $\tau>L^{-\frac{1}{2}+c}$, the forward output grows at least with rate $L^c$ in expectation and then the learning fails because of gradient explosion for large $L$. This means the bound $\tau\le 1/\sqrt{L}$ is sharp for learning ResNet with arbitrary depth. To the best of our knowledge, this is the first work that studies learning ResNet with full range of $\tau$.Comment: 31 page

Fractional order differentiation by integration with Jacobi polynomials

The differentiation by integration method with Jacobi polynomials was originally introduced by Mboup, Join and Fliess. This paper generalizes this method from the integer order to the fractional order for estimating the fractional order derivatives of noisy signals. The proposed fractional order differentiator is deduced from the Jacobi orthogonal polynomial filter and the Riemann-Liouville fractional order derivative definition. Exact and simple formula for this differentiator is given where an integral formula involving Jacobi polynomials and the noisy signal is used without complex mathematical deduction. Hence, it can be used both for continuous-time and discrete-time models. The comparison between our differentiator and the recently introduced digital fractional order Savitzky-Golay differentiator is given in numerical simulations so as to show its accuracy and robustness with respect to corrupting noises

Error analysis for a class of numerical differentiator: application to state observation

International audienceIn this note, firstly a modified numerical differentiation scheme is presented. The obtained scheme is rooted in the numerical differentiation method of Mboup M., Join C., Fliess M. and uses the same algebraic approach based on operational calculus. Secondly an analysis of the error due to a corrupting noise in this estimation is conducted and some upper-bounds are given on this error. Lastly a convincing simulation example gives an estimation of the state variable of a nonlinear system where the measured output is noisy

Parameters estimation of a noisy sinusoidal signal with time-varying amplitude

In this paper, we give estimators of the frequency, amplitude and phase of a noisy sinusoidal signal with time-varying amplitude by using the algebraic parametric techniques introduced by Fliess and Sira-Ramirez. We apply a similar strategy to estimate these parameters by using modulating functions method. The convergence of the noise error part due to a large class of noises is studied to show the robustness and the stability of these methods. We also show that the estimators obtained by modulating functions method are robust to "large" sampling period and to non zero-mean noises