Fractional Laplacian matrix on the finite periodic linear chain and its periodic Riesz fractional derivative continuum limit

The 1D discrete fractional Laplacian operator on a cyclically closed (periodic) linear chain with finitenumber $N$ of identical particles is introduced. We suggest a "fractional elastic harmonic potential", and obtain the $N$-periodic fractionalLaplacian operator in the form of a power law matrix function for the finite chain ($N$ arbitrary not necessarily large) in explicit form.In the limiting case $N\rightarrow \infty$ this fractional Laplacian matrix recovers the fractional Laplacian matrix ofthe infinite chain.The lattice model contains two free material constants, the particle mass $\mu$ and a frequency$\Omega\_{\alpha}$.The "periodic string continuum limit" of the fractional lattice model is analyzed where lattice constant $h\rightarrow 0$and length $L=Nh$ of the chain ("string") is kept finite: Assuming finiteness of the total mass and totalelastic energy of the chain in the continuum limit leads to asymptotic scaling behavior for $h\rightarrow 0$ of thetwo material constants,namely $\mu \sim h$ and $\Omega\_{\alpha}^2 \sim h^{-\alpha}$. In this way we obtain the $L$-periodic fractional Laplacian (Riesz fractional derivative) kernel in explicit form.This $L$-periodic fractional Laplacian kernel recovers for $L\rightarrow\infty$the well known 1D infinite space fractional Laplacian (Riesz fractional derivative) kernel. When the scaling exponentof the Laplacian takesintegers, the fractional Laplacian kernel recovers, respectively, $L$-periodic and infinite space (localized) distributionalrepresentations of integer-order Laplacians.The results of this paper appear to beuseful for the analysis of fractional finite domain problems for instance in anomalous diffusion (L\'evy flights), fractional Quantum Mechanics,and the development of fractional discrete calculus on finite lattices

Fractional stochastic differential equations satisfying fluctuation-dissipation theorem

We propose in this work a fractional stochastic differential equation (FSDE) model consistent with the over-damped limit of the generalized Langevin equation model. As a result of the `fluctuation-dissipation theorem', the differential equations driven by fractional Brownian noise to model memory effects should be paired with Caputo derivatives, and this FSDE model should be understood in an integral form. We establish the existence of strong solutions for such equations and discuss the ergodicity and convergence to Gibbs measure. In the linear forcing regime, we show rigorously the algebraic convergence to Gibbs measure when the `fluctuation-dissipation theorem' is satisfied, and this verifies that satisfying `fluctuation-dissipation theorem' indeed leads to the correct physical behavior. We further discuss possible approaches to analyze the ergodicity and convergence to Gibbs measure in the nonlinear forcing regime, while leave the rigorous analysis for future works. The FSDE model proposed is suitable for systems in contact with heat bath with power-law kernel and subdiffusion behaviors

Label-Dependencies Aware Recurrent Neural Networks

In the last few years, Recurrent Neural Networks (RNNs) have proved effective on several NLP tasks. Despite such great success, their ability to model \emph{sequence labeling} is still limited. This lead research toward solutions where RNNs are combined with models which already proved effective in this domain, such as CRFs. In this work we propose a solution far simpler but very effective: an evolution of the simple Jordan RNN, where labels are re-injected as input into the network, and converted into embeddings, in the same way as words. We compare this RNN variant to all the other RNN models, Elman and Jordan RNN, LSTM and GRU, on two well-known tasks of Spoken Language Understanding (SLU). Thanks to label embeddings and their combination at the hidden layer, the proposed variant, which uses more parameters than Elman and Jordan RNNs, but far fewer than LSTM and GRU, is more effective than other RNNs, but also outperforms sophisticated CRF models.Comment: 22 pages, 3 figures. Accepted at CICling 2017 conference. Best Verifiability, Reproducibility, and Working Description awar