In quantum mechanics, the space-fractional Schrödinger equation provides a natural extension of the standard Schrödinger equation when the Brownian trajectories in Feynman path integrals are replaced by Levy flights. Here an optical realization of the fractional Schrödinger equation, based on transverse light dynamics in aspherical optical cavities, is proposed. As an example, a laser implement...

– The porous media equation has been proposed as a phenomenological “nonextensive” generalization of classical diffusion. Here, we show that a very similar equation can be derived, in a systematic manner, for a classical fluid by assuming nonlinear response, i.e. that the diffusive flux depends on gradients of a power of the concentration. The present equation distinguishes from the porous medi...

Mikolov et al. (2013) introduced the skip-gram formulation for neural word embeddings, wherein one tries to predict the context of a given word. Their negative-sampling algorithm improved the computational feasibility of training the embeddings. Due to their state-of-the-art performance on a number of tasks, there has been much research aimed at better understanding it. Goldberg and Levy (2014)...

We introduce a class of stochastic volatility models (Xt)t≥0 for which the absolute moments of the increments exhibit anomalous scaling: E (|Xt+h −Xt|) scales as hq/2 for q < q∗, but as hA(q) with A(q) < q/2 for q > q∗, for some threshold q∗. This multi-scaling phenomenon is observed in time series of financial assets. If the dynamics of the volatility is given by a mean-reverting equation driv...

Let a, b, c be fixed positive integers satisfying a2 + ab + b2 = c with gcd(a, b) = 1. We show that the Diophantine equation a2x+axby+b2y = cz has only the positive integer solution (x, y, z) = (1, 1, 1) under some conditions. The proof is based on elementary methods and Cohn’s ones concerning the Diophantine equation x2 + C = yn. Mathematics Subject Classification: 11D61

the main contribution of the present work is to investigate the capability of exponential-type constitutive equationto model the warm deformation flow curves of a eutectoid steel in the temperature range of 620-770 °c andat the strain rates in the range of 0.01-10 s-1 conducted on a gleeble-1500 thermomechanical simulator. warmdeformation in this temperature range facilitates the occurrence of ...