Periodicities in cluster algebras and cluster automorphism groups

In this paper, we study the relations between groups related to cluster automorphism groups which are defined by Assem, Schiffler and Shamchenko in \cite{ASS}. We establish the relationship among (strict) direct cluster automorphism groups and those groups consisting of periodicities of respectively labeled seeds and exchange matrices in the language of short exact sequences. As an application, we characterize automorphism-finite cluster algebras in the cases with bipartite seeds or finite mutation type. Finally, we study the relation between the groups $\mathrm{Aut}\mathcal{A}$ and $\mathrm{Aut}_{M_n}S$ and give the negative answer via counter-examples to King and Pressland's a problem in \cite{KP}.Comment: 21 page

On Exchange Spectra of Valued Cluster Quivers and Cluster Algebras

Inspirited by the importance of the spectral theory of graphs, we introduce the spectral theory of valued cluster quiver of a cluster algebra. Our aim is to characterize a cluster algebra via its spectrum so as to use the spectral theory as a tool. First, we give the relations between exchange spectrum of a valued cluster quiver and adjacency spectrum of its underlying valued graph, and between exchange spectra of a valued cluster quiver and its full valued subquivers. The key point is to find some invariants from the spectrum theory under mutations of cluster algebras, which is the second part we discuss. We give a sufficient and necessary condition for a cluster quiver and its mutation to be cospectral. Following this discussion, the so-called cospectral subalgebra of a cluster algebra is introduced. We study bounds of exchange spectrum radii of cluster quivers and give a characterization of $2$-maximal cluster quivers via the classification of oriented graphs of its mutation equivalence. Finally, as an application of this result, we obtain that the preprojective algebra of a cluster quiver of Dynkin type is representation-finite if and only if the cluster quiver is $2$-maximal.Comment: 19 page

PANDA: AdaPtive Noisy Data Augmentation for Regularization of Undirected Graphical Models

We propose an AdaPtive Noise Augmentation (PANDA) technique to regularize the estimation and construction of undirected graphical models. PANDA iteratively optimizes the objective function given the noise augmented data until convergence to achieve regularization on model parameters. The augmented noises can be designed to achieve various regularization effects on graph estimation, such as the bridge (including lasso and ridge), elastic net, adaptive lasso, and SCAD penalization; it also realizes the group lasso and fused ridge. We examine the tail bound of the noise-augmented loss function and establish that the noise-augmented loss function and its minimizer converge almost surely to the expected penalized loss function and its minimizer, respectively. We derive the asymptotic distributions for the regularized parameters through PANDA in generalized linear models, based on which, inferences for the parameters can be obtained simultaneously with variable selection. We show the non-inferior performance of PANDA in constructing graphs of different types in simulation studies and apply PANDA to an autism spectrum disorder data to construct a mixed-node graph. We also show that the inferences based on the asymptotic distribution of regularized parameter estimates via PANDA achieve nominal or near-nominal coverage and are far more efficient, compared to some existing post-selection procedures. Computationally, PANDA can be easily programmed in software that implements (GLMs) without resorting to complicated optimization techniques

AdaPtive Noisy Data Augmentation (PANDA) for Simultaneous Construction of Multiple Graph Models

We extend the data augmentation technique PANDA by Li et al. (2018) that regularizes single graph estimation to jointly learning multiple graphical models with various node types in a unified framework. We design two types of noise to augment the observed data: the first type regularizes the estimation of each graph while the second type promotes either the structural similarity, referred as the \joint group lasso regularization, or the numerical similarity, referred as the joint fused ridge regularization, among the edges in the same position across graphs. The computation in PANDA is straightforward and only involves obtaining maximum likelihood estimator in generalized linear models in an iterative manner. The simulation studies demonstrate PANDA is non-inferior to existing joint estimation approaches for Gaussian graphical models, and significantly improves over the naive differencing approach for non-Gaussian graphical models. We apply PANDA to a real-life lung cancer microarray data to simultaneously construct four protein networks

Non-Gaussianity of the Cosmic Baryon Fluid: Log-Poisson Hierarchy Model

In the nonlinear regime of cosmic clustering, the mass density field of the cosmic baryon fluid is highly non-Gaussian. It shows different dynamical behavior from collisionless dark matter. Nevertheless, the evolved field of baryon fluid is scale-covariant in the range from the Jeans length to a few ten h^{-1} Mpc, in which the dynamical equations and initial perturbations are scale free. We show that in the scale-free range, the non-Gaussian features of the cosmic baryon fluid, governed by the Navier-Stokes equation in an expanding universe, can be well described by a log-Poisson hierarchical cascade. The log-Poisson scheme is a random multiplicative process (RMP), which causes non-Gaussianity and intermittency even when the original field is Gaussian. The log-Poisson RMP contains two dimensionless parameters: $\beta$ for the intermittency and $\gamma$ for the most singular structure. All the predictions given by the log-Poisson RMP model, including the hierarchical relation, the order dependence of the intermittent exponent, the moments, and the scale-scale correlation, are in good agreement with the results given by hydrodynamic simulations of the standard cold dark matter model. The intermittent parameter $\beta$ decreases slightly at low redshift and indicates that the density field of baryon fluid contains more singular structures at lower redshifts. The applicability of the model is addressed.Comment: 19 pages, 8 figures, accepted by Ap

MANTIS: Model-Augmented Neural neTwork with Incoherent k-space Sampling for efficient MR T2 mapping

Quantitative mapping of magnetic resonance (MR) parameters have been shown as valuable methods for improved assessment of a range of diseases. Due to the need to image an anatomic structure multiple times, parameter mapping usually requires long scan times compared to conventional static imaging. Therefore, accelerated parameter mapping is highly-desirable and remains a topic of great interest in the MR research community. While many recent deep learning methods have focused on highly efficient image reconstruction for conventional static MR imaging, applications of deep learning for dynamic imaging and in particular accelerated parameter mapping have been limited. The purpose of this work was to develop and evaluate a novel deep learning-based reconstruction framework called Model-Augmented Neural neTwork with Incoherent k-space Sampling (MANTIS) for efficient MR parameter mapping. Our approach combines end-to-end CNN mapping with k-space consistency using the concept of cyclic loss to further enforce data and model fidelity. Incoherent k-space sampling is used to improve reconstruction performance. A physical model is incorporated into the proposed framework, so that the parameter maps can be efficiently estimated directly from undersampled images. The performance of MANTIS was demonstrated for the spin-spin relaxation time (T2) mapping of the knee joint. Compared to conventional reconstruction approaches that exploited image sparsity, MANTIS yielded lower errors and higher similarity with respect to the reference in the T2 estimation. Our study demonstrated that the proposed MANTIS framework, with a combination of end-to-end CNN mapping, signal model-augmented data consistency, and incoherent k-space sampling, represents a promising approach for efficient MR parameter mapping. MANTIS can potentially be extended to other types of parameter mapping with appropriate models

Boltje-Maisch resolutions of Specht modules

In \cite{21}, Boltje and Maisch found a permutation complex of Specht modules in representation theory of Hecke algebras, which is the same as the Boltje-Hartmann complex appeared in the representation theory of symmetric groups and general linear groups. In this paper we prove the exactness of Boltje-Maisch complex in the dominant weight case.Comment: 17 page

Global Uniqueness of Steady Transonic Shocks in Two-Dimensional Compressible Euler Flows

We prove that for the two-dimensional steady complete compressible Euler system, with given uniform upcoming supersonic flows, the following three fundamental flow patterns (special solutions) in gas dynamics involving transonic shocks are all unique in the class of piecewise $C^1$ smooth functions, under appropriate conditions on the downstream subsonic flows: (\rmnum{1}) the normal transonic shocks in a straight duct with finite or infinite length, after fixing a point the shock-front passing through; (\rmnum{2}) the oblique transonic shocks attached to an infinite wedge; (\rmnum{3}) a flat Mach configuration containing one supersonic shock, two transonic shocks, and a contact discontinuity, after fixing the point the four discontinuities intersect. These special solutions are constructed traditionally under the assumption that they are piecewise constant, and they have played important roles in the studies of mathematical gas dynamics. Our results show that the assumption of piecewise constant can be replaced by some more weaker assumptions on the downstream subsonic flows, which are sufficient to uniquely determine these special solutions. Mathematically, these are uniqueness results on solutions of free boundary problems of a quasi-linear system of elliptic-hyperbolic composite-mixed type in bounded or unbounded planar domains, without any assumptions on smallness. The proof relies on an elliptic system of pressure $p$ and the tangent of the flow angle $w=v/u$ obtained by decomposition of the Euler system in Lagrangian coordinates, and a newly developed method for the $L^{\infty}$ estimate that is independent of the free boundaries, by combining the maximum principles of elliptic equations, and careful analysis of shock polar applied on the (maybe curved) shock-fronts.Comment: 26 pages, 9 figure

Research on Interpore Distance of Anodic Aluminum Oxide Template

The relationship between the interpore of anodic aluminum oxide (AAO) template and the influencing factors of electrolyte, temperature and oxidation voltage etc. was researched and summarized in this paper. It was pointed out that the interpore was influenced mostly by electrolyte type and oxidation voltage, and least by the electrolyte concentration and oxidation temperature. The interpore of AAO template increases with the oxidation voltage increases. By adjusting the electrolyte and oxidation voltage, a desired interpore of template can be acquired. To acquire a large interpore template can use electrolyte of phosphoric acid or chromic acid under a comparatively higher oxidation voltage. To acquire a small interpore template (with interpore within 10nm) can use sulfate electrolyte under a comparatively lower oxidation voltage or pulse voltage. Alternatively, oxidizing with lower oxidation voltage in a mixed electrolyte of H2SO4 and Al2 (SO4)3, whereafter immersion in the mixture of HCl and CuCl2 to corrode the template for some time, until the angles of cell appear six holes with small pore diameter, the interpore of AAO template decrease to about 0.6 times of the original

A Basis of the qq-Schur Module

In this paper, we construct the $q$-Schur modules as left principle ideals of the cyclotomic $q$-Schur algebras, and prove that they are isomorphic to those cell modules defined in \cite{8} and \cite{15} at any level $r$. Then we prove that these $q$-Schur modules are free modules and construct their bases. This result gives us new versions of several results about the standard basis and the branching theorem. With the help of such realizations and the new bases, we re-prove the Branch rule of Weyl modules which was first discovered and proved by Wada in \cite{23}.Comment: 14 page