Hankel determinants, Pad\'e approximations, and irrationality exponents

The irrationality exponent of an irrational number $\xi$, which measures the approximation rate of $\xi$ by rationals, is in general extremely difficult to compute explicitly, unless we know the continued fraction expansion of $\xi$. Results obtained so far are rather fragmentary, and often treated case by case. In this work, we shall unify all the known results on the subject by showing that the irrationality exponents of large classes of automatic numbers and Mahler numbers (which are transcendental) are exactly equal to $2$. Our classes contain the Thue--Morse--Mahler numbers, the sum of the reciprocals of the Fermat numbers, the regular paperfolding numbers, which have been previously considered respectively by Bugeaud, Coons, and Guo, Wu and Wen, but also new classes such as the Stern numbers and so on. Among other ingredients, our proofs use results on Hankel determinants obtained recently by Han.Comment: International Mathematics Research Notices 201

Ab initio study of the giant ferroelectric distortion and pressure induced spin-state transition in BiCoO3

Using configuration-state-constrained electronic structure calculations based on the generalized gradient approximation plus Hubbard U method, we sought the origin of the giant tetragonal ferroelectric distortion in the ambient phase of the potentially multiferroic material BiCoO3 and identified the nature of the pressure induced spin-state transition. Our results show that a strong Bi-O covalency drives the giant ferroelectric distortion, which is further stabilized by an xy-type orbital ordering of the high-spin (HS) Co3+ ions. For the orthorhombic phase under 5.8 GPa, we find that a mixed HS and low-spin (LS) state is more stable than both LS and intermediate-spin (IS) states, and that the former well accounts for the available experimental results. Thus, we identify that the pressure induced spin-state transition is via a mixed HS+LS state, and we predict that the HS-to-LS transition would be complete upon a large volume decrease of about 20%.Comment: 6 pages, 6 figures, 2 table

LAMDA-SSL: Semi-Supervised Learning in Python

LAMDA-SSL is open-sourced on GitHub and its detailed usage documentation is available at https://ygzwqzd.github.io/LAMDA-SSL/. This documentation introduces LAMDA-SSL in detail from various aspects and can be divided into four parts. The first part introduces the design idea, features and functions of LAMDA-SSL. The second part shows the usage of LAMDA-SSL by abundant examples in detail. The third part introduces all algorithms implemented by LAMDA-SSL to help users quickly understand and choose SSL algorithms. The fourth part shows the APIs of LAMDA-SSL. This detailed documentation greatly reduces the cost of familiarizing users with LAMDA-SSL toolkit and SSL algorithms

Stochastic bifurcation characteristics of cantilevered piezoelectric energy harvester

Stochastic bifurcation characteristics of cantilevered piezoelectric energy harvester were studied in this paper. Von de Pol differencial item was introduced to interpret the hysteretic phenomena of piezoelectric ceramics, and then the nonlinear dynamic model of piezoelectric cantilever beam subjected to axial stochastic excitation was developed. The stochastic stability of the system was analyzed, and the steady-state probability density function and the joint probability density function of the dynamic response of the system were obtained, and then the conditions of stochastic Hopf bifurcation were analyzed. Numerical simulation shows that stochastic Hopf bifurcation appears when bifurcation parameter varies, which can increase vibration amplitude of cantilever beam system and improve the efficiency of piezoelectric energy harvester. Finally, the theoretical and numerical results were proved by experiments. The results of this paper are helpful to application of cantilevered piezoelectric energy harvester in engineering fields

2-(4-Bromo­phen­yl)quinoxaline

In the title compound, C14H9BrN2, the benzene and quinoxaline rings are almost coplanar [r.m.s. deviation = 0.0285 (3) Å and dihedral angle = 2.1 (2)°]

1,1′-(4-Oxoheptane-1,7-di­yl)bis­(2-methyl-1H-benzimidazole) penta­hydrate

The title compound, C23H26N4O·5H2O, has noncrystallographic twofold rotation symmetry in the solid state. It crystallizes with five solvent water mol­ecules in the asymmetric unit. Four of these water mol­ecules are connected with each other via hydrogen-bonding inter­actions to form two types of centrosymmetric hexa­meric (H2O)6 rings. Via edge sharing of the hexa­mers, the water clusters thus build infinite chains that stretch parallel to the a axis. The fifth water mol­ecule provides an additional connection between the two hexa­meric (H2O)6 units via hydrogen bonds to both rings. The water mol­ecules in the channels along the a axis are also bonded via O—H⋯N hydrogen bonds to the organic units, and face-to-face π–π inter­actions [with centroid-to-centroid distances of 3.656 (1) Å and average face-to-face distances of 3.431 (5) Å] between the aromatic rings of adjacent mol­ecules complete the inter­molecular inter­actions in this structure

Knowledge-driven Meta-learning for CSI Feedback

Accurate and effective channel state information (CSI) feedback is a key technology for massive multiple-input and multiple-output systems. Recently, deep learning (DL) has been introduced for CSI feedback enhancement through massive collected training data and lengthy training time, which is quite costly and impractical for realistic deployment. In this article, a knowledge-driven meta-learning approach is proposed, where the DL model initialized by the meta model obtained from meta training phase is able to achieve rapid convergence when facing a new scenario during target retraining phase. Specifically, instead of training with massive data collected from various scenarios, the meta task environment is constructed based on the intrinsic knowledge of spatial-frequency characteristics of CSI for meta training. Moreover, the target task dataset is also augmented by exploiting the knowledge of statistical characteristics of wireless channel, so that the DL model can achieve higher performance with small actually collected dataset and short training time. In addition, we provide analyses of rationale for the improvement yielded by the knowledge in both phases. Simulation results demonstrate the superiority of the proposed approach from the perspective of feedback performance and convergence speed.Comment: arXiv admin note: text overlap with arXiv:2301.1347