On the Twisted q-Euler numbers and polynomials associated with basic q-l-functions

One purpose of this paper is to construct twisted q-Euler numbers by using p-adic invariant integral on Zp in the sense of fermionic. Finally, we consider twisted Euler q-zeta function and q-l-series which interpolate twisted q-Euler numbers and polynomials.Comment: 8 page

The group of causal automorphisms

The group of causal automorphisms on Minkowski space-time is given and its structure is analyzed

Timing jitter optimization of mode-locked Yb-fiber lasers toward the attosecond regime

We demonstrate ultra-low timing jitter optical pulse trains from free- running, 80 MHz repetition rate, mode-locked Yb-fiber lasers. Timing jitter of various mode-locking conditions at close-to-zero intra-cavity dispersion (-0.004 to +0.002 ps2 range at 1040 nm center wavelength) is characterized using a sub-20-attosecond-resolution balanced optical cross-correlation method. The measured lowest rms timing jitter is 175 attoseconds when integrated from 10 kHz to 40 MHz (Nyquist frequency) offset frequency range, which corresponds to the record-low timing jitter from free-running mode-locked fiber lasers so far. We also experimentally found the mode-locking conditions of fiber lasers where both ultra-low timing jitter and relative intensity noise can be achieved.Comment: 8 pages, 7 figure