76,658 research outputs found
Cartan-Eilenberg complexes and Auslander categories
Let be a commutative noetherian ring with a semi-dualizing module .
The Auslander categories with respect to 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 complexes and C-E
-Gorenstein complexes are introduced, where denotes
a self-orthogonal class of -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 be a graph and the adjacency matrix of . The permanental
polynomial of is defined as . 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
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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 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
. 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
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