期刊：
Mathematical Methods in the Applied Sciences,2023年46(2):2275-2287 ISSN：0170-4214
通讯作者：
Zhiyuan Li
作者机构：
[Jiang, Daijun] Cent China Normal Univ, Hubei Key Lab Math Sci, Sch Math & Stat, Wuhan, Peoples R China.;[Li, Zhiyuan] Ningbo Univ, Sch Math & Stat, 818 Fenghua Rd, Ningbo 315211, Zhejiang, Peoples R China.;[Pauron, Matthieu] ENS Rennes, Bruz, France.;[Yamamoto, Masahiro] Univ Tokyo, Grad Sch Math Sci, Tokyo, Japan.;[Yamamoto, Masahiro] Acad Romanian Scientists, Bucharest, Romania.
通讯机构：
[Zhiyuan Li] S;School of Mathematics and Statistics, Ningbo University, 818 Fenghua Road, Zhejiang, Ningbo, China
摘要：
In this article, we discuss a solution to time-fractional diffusion equation partial differential t alpha(u-u0)+Au=0$$ {\partial}_t<^>{\alpha}\left(u-{u}_0\right)+ Au=0 $$ with the homogeneous Dirichlet boundary condition, where an elliptic operator -A$$ -A $$ is not necessarily symmetric. We prove that the solution u$$ u $$ is identically zero if its normal derivative with respect to the operator A$$ A $$ vanishes on an arbitrarily chosen subboundary of the spatial domain over a time interval. The proof is based on the Laplace transform and the spectral decomposition for a nonsymmetric elliptic operator. As a direct application, we prove the uniqueness result for an inverse problem on determining the spatial component in the source term by Neumann boundary data on subdoundary.
期刊：
Machine Learning with Applications,2023年:100479 ISSN：2666-8270
作者机构：
[Xiongjun Zhang] School of Mathematics and Statistics and Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, China;[Jin Wu] School of Biological Sciences, The University of Hong Kong, Pokfulam, Hong Kong;[Michael K. Ng] Department of Mathematics, The University of Hong Kong, Pokfulam, Hong Kong
摘要：
In this paper, we study the problem of multilinear multitask learning (MLMTL), in which all tasks are stacked into a third-order tensor for consideration. In contrast to conventional multitask learning, MLMTL can explore inherent correlations among multiple tasks in a better manner by utilizing multilinear low rank structure. Existing approaches about MLMTL are mainly based on the sum of singular values for approximating low rank matrices obtained by matricizing the third-order tensor. However, these methods are suboptimal in the Tucker rank approximation. In order to elucidate intrinsic correlations among multiple tasks, we present a new approach by the use of transformed tensor nuclear norm (TTNN) constraint in the objective function. The main advantage of the proposed approach is that it can acquire a low transformed multi-rank structure in a transformed tensor by applying suitable unitary transformations which is helpful to determine principal components in grouping multiple tasks for describing their intrinsic correlations more precisely. Furthermore, we establish an excess risk bound of the minimizer of the proposed TTNN approach. Experimental results including synthetic problems and real-world images, show that the mean-square errors of the proposed method is lower than those of the existing methods for different number of tasks and training samples in MLMTL.
期刊：
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY,2023年376(05):3063-3085 ISSN：0002-9947
作者机构：
[Yau, Shing-Tung] Harvard Univ, Dept Math, Cambridge, MA 02138 USA.;[Zhao, Quanting] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Zhao, Quanting] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.;[Zheng, Fangyang] Chongqing Normal Univ, Sch Math Sci, Chongqing 401331, Peoples R China.
摘要：
In this paper, we study a special type of compact Hermitian manifolds that are Strominger Ka.hler-like, or SKL for short. This condition means that the Strominger connection (also known as Bismut connection) is Ka.hler-like, in the sense that its curvature tensor obeys all the symmetries of the curvature of a Ka.hler manifold. Previously, we have shown that any SKL manifold (Mn, g) is always pluriclosed, and when the manifold is compact and g is not Ka.hler, it cannot admit any balanced or strongly Gauduchon (in the sense of Popovici) metric. Also, when n = 2, the SKL condition is equivalent to the Vaisman condition. In this paper, we give a classification for compact non-Ka.hler SKL manifolds in dimension 3 and those with degenerate torsion in higher dimensions. We also present some properties about SKL manifolds in general dimensions, for instance, given any compact non-Ka.hler SKL manifold, its Ka.hler form represents a non-trivial Aeppli cohomology class, the metric can never be locally conformal Ka.hler when n >= 3, and the manifold does not admit any Hermitian symplectic metric.
期刊：
Journal of Mathematical Biology,2023年86(6):1-45 ISSN：0303-6812
通讯作者：
Huang, JC;Wang, H
作者机构：
[Zhang, Yuyue; Huang, Jicai] Cent China Normal Univ, Sch Math & Stat, Minist Educ, Wuhan 430079, Hubei, Peoples R China.;[Zhang, Yuyue; Huang, Jicai] Cent China Normal Univ, Key Lab Nonlinear Anal & Applicat, Minist Educ, Wuhan 430079, Hubei, Peoples R China.;[Wang, Hao] Univ Alberta, Dept Math & Stat Sci, Edmonton, AB T6G 2G1, Canada.;[Wang, Hao] Univ Alberta, Interdisciplinary Lab Math Ecol & Epidemiol, Edmonton, AB T6G 2G1, Canada.
通讯机构：
[Wang, H ] U;[Huang, JC ] C;Cent China Normal Univ, Sch Math & Stat, Minist Educ, Wuhan 430079, Hubei, Peoples R China.;Cent China Normal Univ, Key Lab Nonlinear Anal & Applicat, Minist Educ, Wuhan 430079, Hubei, Peoples R China.;Univ Alberta, Dept Math & Stat Sci, Edmonton, AB T6G 2G1, Canada.
关键词：
Degenerate Bogdanov–Takens bifurcation;Generalist predator;Nilpotent cusp of codimension 4;Nilpotent focus of codimension 3;Predator–prey model;Specialist predator
摘要：
In this paper, we revisit a predator-prey model with specialist and generalist predators proposed by Hanski et al. (J Anim Ecol 60:353-367, 1991) , where the density of generalist predators is assumed to be a constant. It is shown that the model admits a nilpotent cusp of codimension 4 or a nilpotent focus of codimension 3 for different parameter values. As the parameters vary, the model can undergo cusp type (or focus type) degenerate Bogdanov-Takens bifurcations of codimension 4 (or 3). Our results indicate that generalist predation can induce more complex dynamical behaviors and bifurcation phenomena, such as three small-amplitude limit cycles enclosing one equilibrium, one or two large-amplitude limit cycles enclosing one or three equilibria, three limit cycles appearing in a Hopf bifurcation of codimension 3 and dying in a homoclinic bifurcation of codimension 3. In addition, we show that generalist predation stabilizes the limit cycle driven by specialist predators to a stable equilibrium, which clearly explains the famous Fennoscandia phenomenon.
作者机构：
[Ye, Jianguo] Kashi Univ, Res Ctr Modern Math & Its Applicat, Sch Math & Stat, Kashi 844000, Peoples R China.;[Yan, Guozheng] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Guozheng Yan] S;School of Mathematics and Statistics, Central China Normal University, Wuhan, China
摘要：
We consider the interior inverse scattering problem for recovering the shape of a penetrable partially coated cavity with external obstacles from the knowledge of measured scattered waves due to point sources. In the first part, we obtain the well-posedness of the direct scattering problem by the variational method. In the second part, we establish the mathematical basis of the linear sampling method to recover both the shape of the cavity, and the shape of the external obstacle, however the exterior transmission eigenvalue problem also plays a key role in the discussion of this paper.
作者机构：
[Qin, Hong; Xiao, Yao] Zhongnan Univ Econ & Law, Sch Stat & Math, Wuhan, Peoples R China.;[Qin, Hong; Wang, Shiqi; Ning, Jianhui] Cent China Normal Univ, Sch Math & Stat, Wuhan, Peoples R China.
通讯机构：
[Jianhui Ning] S;School of Mathematics and Statistics, Central China Normal University, Wuhan, People's Republic of China
关键词：
Weighted uniform designs;sequential designs;weighted discrepancy;Gaussian process model
摘要：
Uniform designs seek to distribute design points uniformly in the experimental domain. Some discrepancies have been developed to measure the uniformity by treating all factors equally. It is reason-able when there exists no prior information about the system or when the potential model is completely unclear. However, in the sit-uation of sequential designs, experimental information, such as the importance of each factor, would be obtained from previous stage experiments. With this fact, the weighted L-2-discrepancy is more suit-able than the original discrepancy for choosing follow-up designs. In this paper, the sequentially weighted uniform design is proposed, which is obtained by minimizing the weighted L-2-discrepancy. The weights, indicating the relative importance of each factor, are esti-mated through a Bayesian hierarchical Gaussian process method based on serial experimental data. Results from several classic com-puter simulator examples, as well as a real application in circuit design, demonstrate that the performance of our new method sur-passes that of its counterparts.
作者机构：
[Cui, Hengjian] Capital Normal Univ, Sch Math Sci, Beijing, Peoples R China.;[Liu, Yanyan] Wuhan Univ, Sch Math & Stat, Wuhan, Hubei, Peoples R China.;[Mao, Guangcai] Cent China Normal Univ, Sch Math & Stat, Wuhan, Hubei, Peoples R China.;[Zhang, Jing] Zhongnan Univ Econ & Law, Sch Stat & Math, Wuhan, Hubei, Peoples R China.;[Zhang, Jing] Zhongnan Univ Econ & Law, Sch Stat & Math, Wuhan 430073, Hubei, Peoples R China.
通讯机构：
[Jing Zhang] S;School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan, Hubei, China
关键词：
conditional distance correlation;model-free screening;sure screening property;ultrahigh-dimensional survival data
摘要：
How to select the active variables that have significant impact on the event of interest is a very important and meaningful problem in the statistical analysis of ultrahigh-dimensional data. In many applications, researchers often know that a certain set of covariates are active variables from some previous investigations and experiences. With the knowledge of the important prior knowledge of active variables, we propose a model-free conditional screening procedure for ultrahigh dimensional survival data based on conditional distance correlation. The proposed procedure can effectively detect the hidden active variables that are jointly important but are weakly correlated with the response. Moreover, it performs well when covariates are strongly correlated with each other. We establish the sure screening property and the ranking consistency of the proposed method and conduct extensive simulation studies, which suggests that the proposed procedure works well for practical situations. Then, we illustrate the new approach through a real dataset from the diffuse large-B-cell lymphoma study S1.
期刊：
Complex Analysis and Operator Theory,2023年17(1):1-17 ISSN：1661-8254
通讯作者：
Min-Min Zhang
作者机构：
[Zhang, Min-Min; Wei, Saidi] Cent China Normal Univ, Sch Math & Stat, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
通讯机构：
[Min-Min Zhang] H;Hubei Key Laboratory of Mathematical Sciences, School of Mathematics and Statistics, Central China Normal University, Wuhan, People’s Republic of China
摘要：
Let
$$\mu _{M,D}$$
be the self-affine measure associated with an expanding integer matrix
$$M=\left( \begin{array}{cc} p &{} 0 \\ 0 &{} q \\ \end{array}\right) $$
and
$$D=\left\{ \,\,\begin{pmatrix} 0\\ 0 \end{pmatrix},\,\,\,\begin{pmatrix} 1\\ 1 \end{pmatrix} \,\,\right\} $$
, where |p| and |q| are distinct odd bigger than 1. Such a measure is the simplest and the most important case in the study of the spectral property of self-affine measures with two-elements digit sets, which is an open problem up to now. In this paper, we first construct two classes of 4-element orthogonal exponentials in the corresponding Hilbert space
$$L^2(\mu _{M,D})$$
. Moreover, we prove that, under certain conditions, the constructed 4-element orthogonal exponentials is maximal.
作者机构：
[Yuan, Ganghua] Northeast Normal Univ, Sch Math & Stat, KLAS, Changchun 130024, Jilin, Peoples R China.;[Zhao, Yue; Zhao, Y] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Zhao, Y ] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
摘要：
In this paper, we show the stability of the inverse source problem for the Maxwell equations with conductivity. The tangential components of the electric and magnetic fields on the boundary at multiple frequencies are required as the data for the analysis. The stability estimate consists of the Lipschitz type data discrepancy and the high frequency tail of the source function, where the latter decreases as the upper bound of the frequency increases. The explicit dependence of the stability estimate on the constant conductivity is derived. The analysis employs scattering theory to obtain the holomorphic domain and an upper bound for the resolvent of the elliptic operator.& COPY; 2023 Elsevier Ltd. All rights reserved.
期刊：
Journal of Mathematical Analysis and Applications,2023年527(1):127340 ISSN：0022-247X
通讯作者：
Zhang, MM
作者机构：
[Zhang, Min-Min; Zhang, MM; Lu, Jian-Feng] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Zhang, Min-Min; Zhang, MM; Lu, Jian-Feng] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.;[Wang, Shi] Hu Bei Univ, Fac Math & Stat, Wuhan 430079, Peoples R China.;[Wang, Shi] Hu Bei Univ, Hubei Key Lab Appl Math, Wuhan 430079, Peoples R China.
通讯机构：
[Zhang, MM ] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
摘要：
Let t >= 2 be an integer and p, q be two prime numbers, D =E-p circle plus (pq)l-1p Eq be a product-form digit set with l >= 1. Let mu t,D be the self-similar measure generated by the pair (t, D). It is known that Fu et al. [4,17] characterized the spectrality of mu t,D. In this paper, given a spectrum ? of mu t,D, we study some integers b such that b is also a spectrum of mu t,D. (c) 2023 Elsevier Inc. All rights reserved.
期刊：
Calculus of Variations and Partial Differential Equations,2023年62(3):1-35 ISSN：0944-2669
通讯作者：
Peng Luo
作者机构：
[Luo, Peng; Zhou, Yang; Peng, Shuangjie] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Luo, Peng; Peng, Shuangjie] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
通讯机构：
[Peng Luo] S;School of Mathematics and Statistics and Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, China
关键词：
35A01;35B25;35J20;35J60
摘要：
We revisit the well known prescribed scalar curvature problem
$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u=\big (1+\varepsilon K(x)\big )u^{2^*-1}, u(x)>0,~~ &{}{x\in \mathbb {R}^N},\\ u\in \mathcal {D}^{1,2}(\mathbb {R}^N),\\ \end{array}\right. } \end{aligned}$$
where
$$2^*=\frac{2N}{N-2}$$
,
$$N\ge 5$$
,
$$\varepsilon >0$$
and
$$K(x)\in C^1(\mathbb {R}^N)\cap L^{\infty }(\mathbb {R}^N)$$
. It is known that there are a number of results related to the existence of solutions concentrating at the isolated critical points of K(x). However, if K(x) has non-isolated critical points with different degenerate rates along different directions, whether there exist solutions concentrating at these points is still an open problem. We give a certain positive answer to this problem via applying a blow-up argument based on local Pohozaev identities and modified finite dimensional reduction method when the dimension of critical point set of K(x) ranges from 1 to
$$N-1$$
, which generalizes some results in Cao et al. (Calc Var Partial Differ Equ 15:403–419, 2002) and Li (J Differ Equ 120:319–410, 1995; Commun Pure Appl Math 49:541–597, 1996).
关键词：
Compressible Navier-Stokes-Allen-Cahn system;Strong;classical so-;lutions;Density-dependent viscosity;Large initial data
摘要：
. This paper is concerned with a one-dimensional isentropic compressible Navier-Stokes-Allen-Cahn system with density-dependent viscosity, which models the motion of a mixture of two viscous compressible fluids. The case when the pressure p(& rho;) = & rho;-r, the viscosity & nu;(& rho;, & chi;) = & rho;& alpha;, the interface thickness & delta;(& rho;) = & rho;l3 and the relaxation time function a(& rho;, & chi;, & chi;y) = & rho;A is considered, where & rho; and & chi; are the density and the phase variable, respectively, and & gamma;, & alpha;, & beta;, & lambda; & ISIN; R are parameters. Under some suitable assumptions on the parameters & gamma;, & alpha;, & beta;, & lambda; and the initial data, we prove the global existence and large-time behavior of nonvacuum strong and classical solutions to its Cauchy problem with large initial data. This appears to be the first global existence result on the Cauchy problem of the compressible Navier-Stokes-Allen-Cahn system with density-dependent viscosity and large data.
作者机构：
[Qi, Yingfan; Cao, Rongjun; Chen, Minghua; Shi, Jiankang] Lanzhou Univ, Sch Math & Stat, Lanzhou 730000, Peoples R China.;[Yin, Xiaobo] Cent China Normal Univ, Sch Math & Stat, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
通讯机构：
[Chen, MH ] L;Lanzhou Univ, Sch Math & Stat, Lanzhou 730000, Peoples R China.
关键词：
Nonlocal model;Shifted-symmetric collocation;Asymptotically compatible scheme;Stability and convergence analysis
摘要：
The piecewise quadratic polynomial collocation is used to approximate the nonlocal model, which generally leads to a nonsymmetric indefinite system (Chen et al. (2021) [5]). In this case, the discrete maximum principle is not satisfied, which might be trickier for the stability analysis of the high-order numerical schemes (D'Elia et al. (2020) [10]; Leng et al. (2021) [26]). Here, we present a modified (shifted-symmetric) piecewise quadratic polynomial collocation for solving the linear nonlocal diffusion model, which leads to a symmetric positive definite system and satisfies the discrete maximum principle. Using Faulhaber's formula and Riemann zeta function, the perturbation error for symmetric positive definite system and nonsymmetric indefinite system are given. Then rigorous convergence analysis for the nonlocal models are provided under the general horizon parameter delta = O (h beta), with beta >= 0. More concretely, the global error is O (hmin{2,1+beta}) if delta is not set as a grid point, while it recovers O (hmax{2,4-2 beta}) when delta is set as a grid point. We also prove that the shifted-symmetric scheme is asymptotically compatible, which has the global error O (hmin{2,2 beta}) as delta, h -> 0. The numerical experiments (including two-dimensional case) are performed to verify the convergence.(c) 2022 IMACS. Published by Elsevier B.V. All rights reserved.
作者机构：
[Yu, Yuantian; Li, Shuchao] Cent China Normal Univ, Fac Math & Stat, Wuhan 430079, Peoples R China.;[Zhang, Huihui; Zhang, HH] Luoyang Normal Univ, Dept Math, Luoyang 471934, Peoples R China.
通讯机构：
[Zhang, HH ] L;Luoyang Normal Univ, Dept Math, Luoyang 471934, Peoples R China.
摘要：
Given a graph G and a real number alpha is an element of[0, 1], Nikiforov (2017) proposed the A(alpha)-matrix of Gas A(alpha)(G) = alpha D(G) +(1 - alpha) A( G), where A(G) and D( G) are the adjacency matrix and the degree diagonal matrix of G, respectively. The largest eigenvalue of A(alpha)(G), written as lambda(alpha)(G), is called the A alpha-index of G. A set of cycles in a graph G is called independent if no two cycles in it have a common vertex in G. For n > 2k - 1, let S-n,S- 2k-1 be the join of a clique on 2k - 1vertices with an independent set of n - (2k - 1) vertices. The famous Erdos-Posa theorem shows that for k >= 2 and n >= 24k, every n-vertex graph G with at least (2k - 1)(n - k) edges contains kindependent cycles, unless G congruent to S-n,S- 2k-1. In this paper, we consider an A(alpha)-spectral version of this theorem. We show that for fixed k >= 1, 0 < alpha < 1and n >= 104k(3)/alpha(a)(1- alpha), if an n-vertex graph Gsatisfies lambda(alpha)(G) >= lambda(alpha)(S-n,S- 2k-1), then it contains kindependent cycles, unless G congruent to Sn, 2k-1. This extends the result of Zhai and Liu (2022), in which they obtained the adjacency spectral version of the Erd.os-Posa theorem. (c) 2023 Elsevier B.V. All rights reserved.
摘要：
Cancer is a complex disease caused primarily by genetic variants. Reconstructing gene networks within tumors is essential for understanding the functional regulatory mechanisms of carcinogenesis. Advances in high-throughput sequencing technologies have provided tremendous opportunities for inferring gene networks via computational approaches. However, due to the heterogeneity of the same cancer type and the similarities between different cancer types, it remains a challenge to systematically investigate the commonalities and specificities between gene networks of different cancer types, which is a crucial step towards precision cancer diagnosis and treatment. In this study, we propose a new sparse regularized multi-layer decomposition graphical model to jointly estimate the gene networks of multiple cancer types. Our model can handle various types of gene expression data and decomposes each cancer-type-specific network into three components, i.e., globally shared, partially shared and cancer-type-unique components. By identifying the globally and partially shared gene network components, our model can explore the heterogeneous similarities between different cancer types, and our identified cancer-type-unique components can help to reveal the regulatory mechanisms unique to each cancer type. Extensive experiments on synthetic data illustrate the effectiveness of our model in joint estimation of multiple gene networks. We also apply our model to two real data sets to infer the gene networks of multiple cancer subtypes or cell lines. By analyzing our estimated globally shared, partially shared, and cancer-type-unique components, we identified a number of important genes associated with common and specific regulatory mechanisms across different cancer types.(c) 2023 The Author(s). Published by Elsevier B.V. on behalf of Research Network of Computational and Structural Biotechnology. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
期刊：
Finite Fields and Their Applications,2023年89:102223 ISSN：1071-5797
作者机构：
[Miao, Shujing; Wang, Junming; Li, Shuchao] Cent China Normal Univ, Fac Math & Stat, Wuhan 430079, Peoples R China.
关键词：
Skew-adjacency matrix;Cospectral graphs;Determined by generalized skew spectrum;Smith Normal Form
摘要：
Spectral characterization of graphs is an important topic in spectral graph theory. An oriented graph G(sigma) is obtained from a simple undirected graph G by assigning to every edge of G a direction so that G(sigma) becomes a directed graph. The skew-adjacency matrix of an oriented graph G(sigma) is a real skew-symmetric matrix S(G(sigma)) = (s(ij)), where s(ij) = -s(ji) = 1 if (i, j) is an arc; s(ij) = s(ji) = 0 otherwise. Let G(sigma) and H-tau be two oriented graphs whose skew-adjacency matrices are S(G(sigma)) and S(H-tau), respectively. We say G(sigma) is R-cospectral to H-tau if tJ - S(G(sigma)) and tJ - S(H-tau) have the same spectrum for any t is an element of R, where J is the all-ones matrix. An oriented graph G(sigma) is said to be determined by the generalized skew spectrum (DGSS for short), if any oriented graph which is R-cospectral to G(sigma) is isomorphic to G(sigma). Let W(G(sigma)) = [e, S(G(sigma))e, S-2(G(sigma))e, . . . , Sn-1(G(sigma))e] be the skew-walk-matrix of G(sigma), where e is the all-ones vector. A theorem of Qiu, Wang and Wang [9] states that if G(sigma) is a self-converse oriented graph and 2(-(sic)n/2(sic)) det W(G(sigma)) is odd and square-free, then G(sigma) is DGSS. In this paper, based on the Smith Normal Form of the skew-walk-matrix of G(sigma) we obtain our main result: Let q be a prime and G(sigma) be a self-converse oriented graph on n vertices with det W(G(sigma)) not equal 0. Assume that rank(q)(W(G(sigma))) = n - 1 if q is an odd prime, and rank(q)(W(G(sigma))) = (sic)n/2(sic) if q = 2. If Q is a regular rational orthogonal matrix satisfying Q(T)S(G(sigma))Q = S(H-tau) for some oriented graph H-tau, then the level of Q divides d(n)(W(G(sigma)))/q, where d(n)(W(G(sigma))) is the n-th invariant factor of q W(G(sigma)). Consequently, it leads to an easier way to prove Qiu, Wang and Wang's theorem above. (c) 2023 Elsevier Inc. All rights reserved.
期刊：
Journal of Pure and Applied Algebra,2023年227(7):107322 ISSN：0022-4049
通讯作者：
Chengkang Xu
作者机构：
[Guo, Hongyan] Cent China Normal Univ, Sch Math & Stat, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.;[Xu, Chengkang] Shangrao Normal Univ, Shangrao, Jiangxi, Peoples R China.
通讯机构：
[Chengkang Xu] S;Shangrao Normal University, Shangrao, Jiangxi, China
摘要：
This paper studies restricted modules of gap-p Virasoro algebra g(p) and their intrinsic connection to twisted modules of certain vertex algebras. We first establish an equivalence between the category of restricted g(p)-modules of level (l) under bar and the category of twisted modules of vertex algebra V-Np((l) under bar, 0), where N-p is a new Lie algebra, (l) under bar :=(l(0), 0, center dot center dot center dot, 0) is an element of C[p/2]+1, l(0) is an element of C is the action of the Virasoro center. Then we focus on the construction and classification of simple restricted g(p)-modules of level (l) under bar. More explicitly, we give a uniform construction of simple restricted g(p)-modules as induced modules. We present several equivalent characterizations of simple restricted g(p)-modules, as locally nilpotent (equivalently, locally finite) modules with respect to certain positive part of g(p). Moreover, simple restricted g(p)-modules of level (l) under bar are classified. They are either highest weight modules or simple induced modules. At the end, we exhibit several concrete examples of simple restricted g(p)-modules of level (l) under bar (including Whittaker modules). (c) 2023 Elsevier B.V. All rights reserved.
