Bismuth telluride topological insulator nanosheet saturable absorbers for q‐switched mode‐locked Tm:ZBLAN waveguide lasers

Nanosheets of bismuth telluride (Bi₂Te₃), a topological insulator material that exhibits broadband saturable absorption due to its non‐trivial Dirac‐cone like energy structure, are utilized to generate short pulses from Tm:ZBLAN waveguide lasers. By depositing multiple layers of a carefully prepared Bi₂Te₃ solution onto a glass substrate, the modulation depth and the saturation intensity of the fabricated devices can be controlled and optimized. This approach enables the realization of saturable absorbers that feature a modulation depth of 13% and a saturation intensity of 997 kW/cm². For the first time to our knowledge, Q‐switched mode‐locked operation of a linearly polarized mid‐IR ZBLAN waveguide chip laser was realized in an extended cavity configuration using the topological insulator Bi₂Te₃. The maximum average output power of the laser is 16.3 mW and the Q‐switched and mode‐locked repetition rates are 44 kHz and 436 MHz, respectively.     

Gears,pregears and related domains

We study conformal mappings from the unit disc to one-toothed gear-shaped planar domains from the point of view of the Schwarzian derivative. Gear-shaped (or “gearlike”) domains fit into a more general category of domains we call “pregears” (images of gears under Möbius transformations), which aid in the study of the conformal mappings for gears and which we also describe in detail. Such domains being bounded by arcs of circles, the Schwarzian derivative of the Riemann mapping is known to be a rational function of a specific form. One accessory parameter of these mappings is naturally related to the conformal modulus of the gear (or pregear) and we prove several qualitative results relating it to the principal remaining accessory parameter. The corresponding region of univalence (parameters for which the rational function is the Schwarzian derivative of a conformal mapping) is determined precisely.     

拟阵闭包算子诱导拓扑空间的分离性

拟阵是图论与代数的结合,而拓扑学是一门重要的数学基础学科.利用拟阵理论研究拓扑空间,从而将两者进行融合,通过拟阵的闭包算子定义拟阵拓扑空间,进一步利用拟阵闭包算子与拓扑闭包算子之间的关系,讨论拟阵拓扑空间的分离性.     

Two Notes on Top ological Groups

In this short paper, we firstly give a short proof of Birkhoff-Kakutani Theorem by Moore metrizable Theorem. Then we prove that G is a topological group if it is a paratopological group which is a dense G_δ-set in a locally feebly compact regular space X.     

A continuation principle for a class of periodically perturbed autonomoussystems

The paper deals with a T ‐periodically perturbed autonomous system in ℝⁿ of the form ((PS)) with ε > 0 small. The main goal of the paper is to provide conditions ensuring the existence of T ‐periodic solutions to (PS) belonging to a given open set W ⊂ C ([0, T ],ℝⁿ ). This problem is considered in the case when the boundary ∂W of W contains at most a finite number of nondegenerate T ‐periodic solutions of the autonomous system = ϕ (x). The starting point of our approach is the following property due to Malkin: if for any T ‐periodic limit cycle x 0 of = ϕ (x) belonging to ∂W the so‐called bifurcation function f (θ), θ ∈ [0, T ], associated to x₀, see (1.11), satisfies the condition f(0) ≠ 0 then the integral operator does not have fixed points on ∂W for all ε > 0 sufficiently small. By means of the Malkin's bifurcation function we then establish a formula to evaluate the Leray–Schauder topological degree of I – Q_ε on W. This formula permits to state existence results that generalize or improve several results of the existing literature. In particular, we extend a continuation principle due to Capietto, Mawhin and Zanolin where it is assumed that ∂W does not contain any T ‐periodic solutions of the unperturbed system. Moreover, we obtain generalizations or improvements of some existence results due to Malkin and Loud. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Obstruction classes of crossed modules of Lie algebroids and Lie groupoids linked to existence of principal bundles

Let K be a Lie group and P be a K-principal bundle on a manifold M. Suppose given furthermore a central extension of K. It is a classical question whether there exists a -principal bundle on M such that . Neeb (Commun. Algebra 34:991–1041, 2006) defines in this context a crossed module of topological Lie algebras whose cohomology class is an obstruction to the existence of . In the present article, we show that is up to torsion a full obstruction for this problem, and we clarify its relation to crossed modules of Lie algebroids and Lie groupoids, and finally to gerbes.      

On the topological centre problem for weighted convolution algebras and semigroup compactifications

Let be a locally compact, non-compact group (we make the non-compactness assumption, for the most part, simply to avoid trivialities). We show that under a very mild assumption on the weight function , the weighted group algebra is strongly Arens irregular in the sense of Dales and Lau; i.e., both topological centres of equal . Also, we show that the topological centre of the algebra equals the weighted measure algebra . Moreover, still in the same situation, we prove that every linear (left) -module map on is automatically bounded, and even --continuous, hence given by convolution with an element in . To this end, we derive a general factorization theorem for bounded families in the -module . Finally, using this result in the case where , we give a short proof of a theorem due to Protasov and Pym, stating that the topological centre of the semigroup is empty, where denotes the -compactification of . This sharpens an earlier result by Lau and Pym; moreover, our method of proof gives a partial answer to a problem raised by Lau and Pym in 1995.

     

Floquet engineering of topological states in realistic quantum materials via light-matter interactions

Electronic states in quantum materials can be engineered by light irradiation, which is greatly advanced by ab-initio computational predictions in realistic light-matter coupled systems. Here we review the most recent progresses from first principles computation in the light-driven Floquet steady states and transient dynamical states with topological electronic bands in real crystals. We first introduce the first-principles modeling approach, dubbed time-dependent Wannier scheme, for simulating real quantum materials under light irradiation. Then, we present a few examples of theoretically-predicted Floquet-Bloch electronic bands engineered by time-periodic light fields, which include the three types of Floquet-Dirac fermions in graphene and black phosphorus, the Floquet-Chern flat bands with an unprecedented high flatness ratio of band width over band gap in a Kagome material, and the Floquet conversion between bright and dark valley excitons in monolayer transition-metal dichalcogenides. Next, we show the ultrafast dynamical evolution of Weyl nodal points in orthorhombic WTe₂ driven by a time-aperiodic short light pulse, and discuss the connection between the Floquet and transient states engineered by light. After that, we introduce three prominent experiments, inspired by theoretical predictions, on the light-induced topological Floquet electronic bands in quantum crystalline materials. Finally, we make a brief summary and perspective on the engineering of topological electronic states through light-matter interactions.     

PI indices of pericondensed benzenoid graphs

The Padmakar–Ivan (PI) index is a graph invariant defined as the summation of the sums of n _eu (e|G) and n _ev (e|G) over all the edges e = uv of a connected graph G, i.e., , where n _eu (e|G) is the number of edges of G lying closer to u than to v and n _ev (e|G) is the number of edges of G lying closer to v than to u. An efficient formula for calculating the PI index of a class of pericondensed benzenoid graphs consisting of three rows of hexagonal of various lengths.