Cartan-Eilenberg complexes and Auslander categories

Let $R$ be a commutative noetherian ring with a semi-dualizing module $C$. The Auslander categories with respect to $C$ are related through Foxby equivalence: \xymatrix@C=50pt{\mathcal {A}_C(R) \ar@[r]^{C\otimes^{\mathbf{L}}_{R} -} & \mathcal {B}_C(R) \ar@[l]^{\mathbf{R}\mathrm{Hom}_{R}(C, -)}}. We firstly intend to extend the Foxby equivalence to Cartan-Eilenberg complexes. To this end, C-E Auslander categories, C-E $\mathcal{W}$ complexes and C-E $\mathcal{W}$-Gorenstein complexes are introduced, where $\mathcal{W}$ denotes a self-orthogonal class of $R$-modules. Moreover, criteria for finiteness of C-E Gorenstein dimensions of complexes in terms of resolution-free characterizations are considered.Comment: 19 pages. Comments and suggestions are appreciate

Minimax Estimation of Large Precision Matrices with Bandable Cholesky Factor

Last decade witnesses significant methodological and theoretical advances in estimating large precision matrices. In particular, there are scientific applications such as longitudinal data, meteorology and spectroscopy in which the ordering of the variables can be interpreted through a bandable structure on the Cholesky factor of the precision matrix. However, the minimax theory has still been largely unknown, as opposed to the well established minimax results over the corresponding bandable covariance matrices. In this paper, we focus on two commonly used types of parameter spaces, and develop the optimal rates of convergence under both the operator norm and the Frobenius norm. A striking phenomenon is found: two types of parameter spaces are fundamentally different under the operator norm but enjoy the same rate optimality under the Frobenius norm, which is in sharp contrast to the equivalence of corresponding two types of bandable covariance matrices under both norms. This fundamental difference is established by carefully constructing the corresponding minimax lower bounds. Two new estimation procedures are developed: for the operator norm, our optimal procedure is based on a novel local cropping estimator targeting on all principle submatrices of the precision matrix while for the Frobenius norm, our optimal procedure relies on a delicate regression-based thresholding rule. Lepski's method is considered to achieve optimal adaptation. We further establish rate optimality in the nonparanormal model. Numerical studies are carried out to confirm our theoretical findings

Enumeration of copermanental graphs

Let $G$ be a graph and $A$ the adjacency matrix of $G$. The permanental polynomial of $G$ is defined as $\mathrm{per}(xI-A)$. In this paper some of the results from a numerical study of the permanental polynomials of graphs are presented. We determine the permanental polynomials for all graphs on at most 11 vertices, and count the numbers for which there is at least one other graph with the same permanental polynomial. The data give some indication that the fraction of graphs with a copermanental mate tends to zero as the number of vertices tends to infinity, and show that the permanental polynomial does be better than characteristic polynomial when we use them to characterize graphs.Comment: 14 pages, 1 figur

Liquid Metal Enabled Droplet Circuits

Conventional electrical circuits are generally rigid in their components and working styles which are not flexible and stretchable. From an alternative, liquid metal based soft electronics is offering important opportunities for innovating modern bioelectronics and electrical engineering. However, its running in wet environments such as aqueous solution, biological tissues or allied subjects still encounters many technical challenges. Here, we proposed a new conceptual electrical circuit, termed as droplet circuits, to fulfill the special needs as raised in the above mentioned areas. Such unconventional circuits are immersed in solution and composed of liquid metal droplets, conductive ions or wires such as carbon nanotubes. With specifically designed topological or directional structures/patterns, the liquid metal droplets composing the circuit can be discretely existing and disconnected from each other, while achieving the function of electron transport through conductive routes or quantum tunneling effect. The conductive wires serve as the electron transfer stations when the distance between two separate liquid metal droplets is far beyond than that quantum tunneling effects can support. The unique advantage of the current droplet circuit lies in that it allows parallel electron transport, high flexibility, self-healing, regulativity and multi-point connectivity, without needing to worry about circuit break. This would extend the category of classical electrical circuits into the newly emerging areas like realizing room temperature quantum computing, making brain-like intelligence or nerve-machine interface electronics etc. The mechanisms and potential scientific issues of the droplet circuits are interpreted. Future prospects along this direction are outlined.Comment: 15 pages, 7 figure

Diagnostic Classification Models for Ordinal Item Responses.

The purpose of this study is to develop and evaluate two diagnostic classification models (DCMs) for scoring ordinal item data. We first applied the proposed models to an operational dataset and compared their performance to an epitome of current polytomous DCMs in which the ordered data structure is ignored. Findings suggest that the much more parsimonious models that we proposed performed similarly to the current polytomous DCMs and offered useful item-level information in addition to option-level information. We then performed a small simulation study using the applied study condition and demonstrated that the proposed models can provide unbiased parameter estimates and correctly classify individuals. In practice, the proposed models can accommodate much smaller sample sizes than current polytomous DCMs and thus prove useful in many small-scale testing scenarios

Quantum Teichm\"uller space and Kashaev algebra

Kashaev algebra associated to a surface is a noncommutative deformation of the algebra of rational functions of Kashaev coordinates. For two arbitrary complex numbers, there is a generalized Kashaev algebra. The relationship between the shear coordinates and Kashaev coordinates induces a natural relationship between the quantum Teichm\"uller space and the generalized Kashaev algebra.Comment: 26 pages, 5 figure

Dark parameterization approach to Ito equation

The novel coupling Ito systems are obtained with the dark parameterization approach. By solving the coupling equations, the traveling wave solutions are constructed with the mapping and deformation method. Some novel types of exact solutions are constructed with the solutions and symmetries of the usual Ito equation. In the meanwhile, the similarity reduction solutions of the model are also studied with the Lie point symmetry theory

Berry phases of quantum trajectories in semiconductors under strong terahertz fields

Quantum evolution of particles under strong fields can be essentially captured by a small number of quantum trajectories that satisfy the stationary phase condition in the Dirac-Feynmann path integrals. The quantum trajectories are the key concept to understand extreme nonlinear optical phenomena, such as high-order harmonic generation (HHG), above-threshold ionization (ATI), and high-order terahertz sideband generation (HSG). While HHG and ATI have been mostly studied in atoms and molecules, the HSG in semiconductors can have interesting effects due to possible nontrivial "vacuum" states of band materials. We find that in a semiconductor with non-vanishing Berry curvature in its energy bands, the cyclic quantum trajectories of an electron-hole pair under a strong terahertz field can accumulate Berry phases. Taking monolayer MoS$_2$ as a model system, we show that the Berry phases appear as the Faraday rotation angles of the pulse emission from the material under short-pulse excitation. This finding reveals an interesting transport effect in the extreme nonlinear optics regime.Comment: 5 page

Dynamical decoupling for a qubit in telegraph-like noises

Based on the stochastic theory developed by Kubo and Anderson, we present an exact result of the decoherence function of a qubit in telegraph-like noises under dynamical decoupling control. We prove that for telegraph-like noises, the decoherence can be suppressed at most to the third order of the time and the periodic Carr-Purcell-Merboom-Gill sequences are the most efficient scheme in protecting the qubit coherence in the short-time limit.Comment: 4 page

Nonlinear optical response induced by non-Abelian Berry curvature in time-reversal-invariant insulators

We propose a general framework of nonlinear optics induced by non-Abelian Berry curvature in time-reversal-invariant (TRI) insulators. We find that the third-order response of a TRI insulator under optical and terahertz light fields is directly related to the integration of the non-Abelian Berry curvature over the Brillouin zone. We apply the result to insulators with rotational symmetry near the band edge. Under resonant excitations, the optical susceptibility is proportional to the flux of the Berry curvature through the iso-energy surface, which is equal to the Chern number of the surface times $2\pi$. For the III-V compound semiconductors, microscopic calculations based on the six-band model give a third-order susceptibility with the Chern number of the iso-energy surface equal to three