Adaptive Exponential Synchronization of Coupled Complex Networks on General Graphs

We investigate the synchronization in complex dynamical networks, where the coupling configuration corresponds to a weighted graph. An adaptive synchronization method on general coupling configuration graphs is given. The networks may synchronize at an arbitrarily given exponential rate by enhancing the updated law of the variable coupling strength and achieve synchronization more quickly by adding edges to original graphs. Finally, numerical simulations are provided to illustrate the effectiveness of our theoretical results

Quantum Algorithm for Estimating Betti Numbers Using a Cohomology Approach

Topological data analysis has emerged as a powerful tool for analyzing large-scale data. High-dimensional data form an abstract simplicial complex, and by using tools from homology, topological features could be identified. Given a simplex, an important feature is so-called Betti numbers. Calculating Betti numbers classically is a daunting task due to the massive volume of data and its possible high-dimension. While most known quantum algorithms to estimate Betti numbers rely on homology, here we consider the `dual' approach, which is inspired by Hodge theory and de Rham cohomology, combined with recent advanced techniques in quantum algorithms. Our cohomology method offers a relatively simpler, yet more natural framework that requires exponentially less qubits, in comparison with the known homology-based quantum algorithms. Furthermore, our algorithm can calculate its $r$-th Betti number $\beta_r$ up to some multiplicative error $\delta$ with running time $\mathcal{O}\big( \log(c_r) c_r^2 / (c_r - \beta_r)^2 \delta^2 \big)$, where $c_r$ is the number of $r$-simplex. It thus works best when the $r$-th Betti number is considerably smaller than the number of the $r$-simplex in the given triangulated manifold

Characterization of polymer fiber Bragg grating with ultrafast laser micromachining

We report that the main photosensitive mechanism of poly(methyl methacrylate)-based optical fiber Bragg grating (POFBG) under ultraviolet laser micromachining is a complex process of both photodegradation and negative thermo-optic effect. We found experimentally the unique characteristics of Bragg resonance splitting and reunion during the laser micromachining process providing the evidence of photodegradation, while the mean refractive index change of POFBG was measured to be negative confirming further photodegradation of polymer fiber. The thermal-induced refractive index change of POFBG was also observed by recording the Bragg wavelength shift. Furthermore, the dynamic thermal response of the micromachined-POFBG was demonstrated under constant humidity, showing a linear and negative response of around -47.1 pm/°C

Constant-time Quantum Algorithm for Homology Detection in Closed Curves

Given a loop or more generally 1-cycle $r$ of size L on a closed two-dimensional manifold or surface, represented by a triangulated mesh, a question in computational topology asks whether or not it is homologous to zero. We frame and tackle this problem in the quantum setting. Given an oracle that one can use to query the inclusion of edges on a closed curve, we design a quantum algorithm for such a homology detection with a constant running time, with respect to the size or the number of edges on the loop $r$, requiring only a single usage of oracle. In contrast, classical algorithm requires $\Omega(L)$ oracle usage, followed by a linear time processing and can be improved to logarithmic by using a parallel algorithm. Our quantum algorithm can be extended to check whether two closed loops belong to the same homology class. Furthermore, it can be applied to a specific problem in the homotopy detection, namely, checking whether two curves are \textit{not} homotopically equivalent on a closed two-dimensional manifold