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EEG Feature Learning Model Based on Intrinsic Time-Scale Decomposition and Adaptive Huber Loss YANG Li-jun, JIANG Shu-yue, WEI Xiao-ge , XIAO Yun-hai Chinese Quarterly Journal of Mathematics    2022, 37 (3): 281-300.   DOI: 10.13371/j.cnki.chin.q.j.m.2022.03.006 Abstract （156）      PDF（pc） （538KB）（393）       Knowledge map    Save According to the World Health Organization, about 50 million people world- wide suffer from epilepsy. The detection and treatment of epilepsy face great challenges. Electroencephalogram (EEG) is a significant research object widely used in diagnosis and treatment of epilepsy. In this paper, an adaptive feature learning model for EEG signals is proposed, which combines Huber loss function with adaptive weight penalty term. Firstly, each EEG signal is decomposed by intrinsic time-scale decomposition. Secondly, the statistical index values are calculated from the instantaneous amplitude and frequency of every component and fed into the proposed model. Finally, the discriminative features learned by the proposed model are used to detect seizures. Our main innovation is to consider a highly flexible penalization based on Huber loss function, which can set different weights according to the influence of different features on epilepsy detection. Besides, the new model can be solved by proximal alternating direction multiplier method, which can effectively ensure the convergence of the algorithm. The performance of the proposed method is evaluated on three public EEG datasets provided by the Bonn University, Childrens Hospital Boston-Massachusetts Institute of Technology, and Neurological and Sleep Center at Hauz Khas, New Delhi(New Delhi Epilepsy data). The recognition accuracy on these two datasets is 98% and 99.05%, respectively, indicating the application value of the new model. Related Articles | Metrics

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Majority Coloring of r-Regular Digraph XIA Wei-hao, SHI Mei, XIAO Ming-yue, CAI Jian-sheng, WANG Ji-hui Chinese Quarterly Journal of Mathematics    2022, 37 (2): 142-146.   DOI: 10.13371/j.cnki.chin.q.j.m.2022.02.004 Abstract （252）      PDF（pc） （330KB）（387）       Knowledge map    Save  A majority k-coloring of a digraph D with k colors is an assignment c: V ( D ) →{ 1,2,···,k} , such that for every v∈V (D), we have c(w)=c(v) for at most half of all out-neighbors w∈N+( v ). For a natural number k≥2, a 1/k-majority coloring of a digraph is a coloring of the vertices such that each vertex receives the same color as at most a 1/k proportion of its out-neighbours. Kreutzer, Oum, Seymour, van der Zypen and Wood proved that every digraph has a majority 4-coloring and conjectured that every digraph admits a majority 3-coloring. Gir$\widetilde{a}$o, Kittipassorn and Popielarz proved that every digraph has a 1/k-majority 2k-coloring and conjectured that every digraph admits a 1\k-majority (2k−1)-coloring. We showed that every r -regular digraph D with r>36ln(2n) has a majority 3-coloring and proved that every digraph D with minimum outdegree $\delta^+>\frac{2k^2(2k-1)^2}{(k-1)^2}\ln{[(2k-1)n]}$ has a 1/k-majority (2k−1)-coloring. And we also proved that every r-regular digraph D with  $r>\frac{3k^2(2k-1)}{(k-1)^2} \ln(2n)$ has a 1/k-majority (2k−1)-coloring. Related Articles | Metrics

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Complexity on Proximal Quasi-Newton Methods for a Class of Nonconvex Composite Optimization JIN Ling-Zi Chinese Quarterly Journal of Mathematics    2023, 38 (1): 62-84.   DOI: 10.13371/j.cnki.chin.q.j.m.2023.01.005 Abstract （154）      PDF（pc） （404KB）（277）       Knowledge map    Save This paper studies a class of nonconvex composite optimization, whose objective is a summation of an average of nonconvex (weakly) smooth functions and a convex nonsmooth function, where the gradient of the former function has the Hölder continuity. By exploring the structure of such kind of problems, we first propose a proximal (quasi-)Newton algorithm wPQN (Proximal quasi-Newton algorithm for weakly smooth optimization) and investigate its theoretical complexities to find an approximate solution. Then we propose a stochastic variant algorithm wPSQN (Proximal stochastic quasi-Newton algorithm for weakly smooth optimization), which allows a random subset of component functions to be used at each iteration. Moreover, motivated by recent success of variance reduction techniques, we propose two variance reduced algorithms, wPSQN-SVRG and wPSQN-SARAH, and investigate their computational complexity separately. Related Articles | Metrics

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Dynamic Analysis of a Predator-Prey Model with State-Dependent Impulsive Effects LI Yong-feng, ZHU Cheng-zhi, LIU Yan-wei Chinese Quarterly Journal of Mathematics    2023, 38 (1): 1-19.   DOI: 10.13371/j.cnki.chin.q.j.m.2023.01.001 Abstract （141）      PDF（pc） （2367KB）（270）       Knowledge map    Save  In this paper, a pest-dependent model and integrated pest management strategy is proposed, that is, when pest populations reach levels that impair economic development, we will use a combination of strategies, such as biological, cultural and chemical control strategies reduce pests to a reasonable level. First, we investigated the system without control measures, and discussed the existence and stability of equilibria, we also proved the system has no limit cycle. Then, a state feedback impulsive model is constructed, the existence and uniqueness of the order-one periodic solution are proved by means of the successor function method to confirm the feasibility of the biological and chemical control strategy of pest management. Secondly, the stability of system is proved by the analogue of the Poincar ´ e criterion. Finally, we give an example and numerical simulations to explain the mathematical conclusions. Related Articles | Metrics

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Majorization and Fekete-Szegö Problems for Multivalent Meromorphic Functions Associated with the Mittag-Leffler Function LONG Pin-hong, GANGADHARAN Murugusundaramoorthy, HAN Hui-li, WANG Wen-shuai Chinese Quarterly Journal of Mathematics    2022, 37 (2): 111-123.   DOI: 10.13371/j.cnki.chin.q.j.m.2022.02.001 Abstract （245）      PDF（pc） （356KB）（264）       Knowledge map    Save The main object of this present paper is to investigate the problem of majorization of certain class of multivalent meromorphic functions of complex order involving Mittag-Leffler function. Meanwhile, for this subclass the corresponding coefficient estimates and some Fekete-Szeg¨ o type inequalities are obtained. Moreover we point out some new or known consequences of our main results. Related Articles | Metrics

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Space-Time Legendre Spectral Collocation Methods for Korteweg-de Vries Equation WANG Chuan, QIAO Yan Chinese Quarterly Journal of Mathematics    Accepted: 12 April 2023

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Stability and Hopf Bifurcation of an Eco-Epidemiological Model with Delay BAI Hong-fang Chinese Quarterly Journal of Mathematics    2023, 38 (2): 157-183.   DOI: 10.13371/j.cnki.chin.q.j.m.2023.02.005 Abstract （149）      PDF（pc） （1289KB）（196）       Knowledge map    Save In this paper, an eco-epidemiological model with time delay is studied. The local stability of the four equilibria, the existence of stability switches about the predation- free equilibrium and the coexistence equilibrium are discussed. It is found that Hopf bifurcations occur when the delay passes through some critical values. Formulae are obtained to determine the direction of bifurcations and the stability of bifurcating periodic solutions by using the normal form theory and center manifold theorem. Some numerical simulations are carried out to illustrate the theoretical results. Related Articles | Metrics

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Virtual Element Method of the Allen-Cahn Equation WANG Pei-zhen, TIAN Xu Chinese Quarterly Journal of Mathematics    2023, 38 (1): 20-29.   DOI: 10.13371/j.cnki.chin.q.j.m.2023.01.002 Abstract （127）      PDF（pc） （307KB）（184）       Knowledge map    Save  In this article, the virtual element method of the Allen-Cahn equation on a polygon grid is discussed in the fully discrete formulation. With the help of the energy projection operator, we give the corresponding error estimates in the L2 norm and H1 norm. Related Articles | Metrics

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Vertex Operators, Littlewood-Richardson Rule for Generalized Symplectic Schur Functions HUANG Fang, CHU Yan-jun Chinese Quarterly Journal of Mathematics    2022, 37 (3): 301-316.   DOI: 10.13371/j.cnki.chin.q.j.m.2022.03.007 Abstract （130）      PDF（pc） （375KB）（184）       Knowledge map    Save  Littlewood-Richardson rule gives the expansion formula for decomposing a product of two Schur functions as a linear sum of Schur functions, while the decomposition formula for the multiplication of two symplectic Schur function is also given by the combinatorial method. In this paper, we will construct the algebraic forms of the decomposition formula for the product of two symplectic Schur functions by using the generating functions and vertex operator realizations, and then extend these results to generalized symplectic Schur functions. Related Articles | Metrics

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Differential Identities in Prime Rings with Involution HUANG Shu-liang Chinese Quarterly Journal of Mathematics    2023, 38 (2): 134-144.   DOI: 10.13371/j.cnki.chin.q.j.m.2023.02.003 Abstract （118）      PDF（pc） （319KB）（183）       Knowledge map    Save Let R be a prime ring of characteristic different from two with the sec- ond involution ∗ and α an automorphism of R . An additive mapping F of R is called a generalized ( α,α )-derivation on R if there exists an ( α,α )-derivation d of R such that F ( xy )= F ( x ) α ( y )+ α ( x ) d ( y ) holds for all x,y∈R. The paper deals with the s- tudy of some commutativity criteria for prime rings with involution. Precisely, we describe the structure of R admitting a generalized ( α,α )-derivation F satisfying any one of the following properties: ( i ) F ( xx∗) −α ( xx∗) ∈Z ( R ). ( ii ) F ( xx∗ )+ α ( xx∗ ) ∈Z ( R ). ( iii ) F ( x ) F ( xx∗) −α ( xx∗) ∈Z ( R ). ( iv ) F ( x ) F (x∗)+ α ( xx∗) ∈Z ( R ). ( v ) F ( xx∗) −F ( x ) F (x∗ ) ∈Z ( R ). ( vi ) F ( xx∗) −F (x∗) F ( x )=0 for all x∈R . Also, some examples are given to demonstrate that the restriction of the various results is not superfluous. In fact, our results unify and extend several well known theorems in literature. Related Articles | Metrics

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On the Eigenvalues and Eigenfunctions of the Sturm-Liouville Operator with the Barrier Potential SARWAR Qanitah, HUANG Zhen-you, ZAHID Abdul Hannan, XU Xin-Jian Chinese Quarterly Journal of Mathematics    2023, 38 (1): 50-61.   DOI: 10.13371/j.cnki.chin.q.j.m.2023.01.004 Abstract （178）      PDF（pc） （279KB）（175）       Knowledge map    Save We aim to find the eigenvalues and eigenfunctions of the barrier potential case for Strum-Liouville operator on the finite interval [0 ,π ] when λ> 0. Generally, the eigenvalue problem for the Sturm-Liouville operator is often solved by using integral equations, which are sometimes complex to solve, and difficulties may arise in computing the boundary values. Considering the said complexity, we have successfully developed a technique to give the asymptotic formulae of the eigenvalue and the eigenfunction for Sturm-Liouville operator with barrier potential. The results are of significant interest in the field of quantum mechanics and atomic systems to observe discrete energy levels. Related Articles | Metrics

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Simplistic Universal Protocols for Remotely Preparing Arbitrary Equatorial States MA Song-ya LI Xiang, LI Qi Chinese Quarterly Journal of Mathematics    2022, 37 (3): 260-273.   DOI: 10.13371/j.cnki.chin.q.j.m.2022.03.004 Abstract （141）      PDF（pc） （348KB）（168）       Knowledge map    Save  We first put forward a deterministic protocol to realize the remote preparation of arbitrary multi-qubit equatorial states via EPR pairs. A set of useful measurement basis is constructed which plays a key role. The receiver just needs to perform Pauli Z operations to recover the target state. Comparing with the previous protocols, the recovery operation is simplified and expressed by a general formula. As there are no universal protocols for high-dimensional systems, we further generalize to the case of multi-qudit equatorial states by means of Fourier transformation. It is worth mentioning that the proposed schemes can be extended to multi-party controlled remote state preparation. Moreover, we consider the effect of two-type decoherence noises. Related Articles | Metrics

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Initial Boundary Value Problem for Pseudo-Parabolic p-Laplacian Type Equation with Logarithmic Nonlinearity PAN Jia-hui, FANG Shao-mei Chinese Quarterly Journal of Mathematics    2023, 38 (4): 360-369.   DOI: 10.13371/j.cnki.chin.q.j.m.2023.04.003 Abstract （77）      PDF（pc） （358KB）（162）       Knowledge map    Save  In this paper, we study the initial boundary value problem of pseudo-parabolic p-Laplacian type equation, which be use to model some important physical and biological phenomena. By using the potential well method, we obtain the global existence, asymptotic behavior and blow up results of weak solution with subcritical initial energy. Then we also extend these results to the critical initial energy. Related Articles | Metrics

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Existence and Exponential Stability of Almost Periodic Solutions to General BAM Neural Networks with Leakage Delays on Time Scales DONG Yan-shou, HAN Yan, DAI Ting-ting Chinese Quarterly Journal of Mathematics    2022, 37 (2): 189-202.   DOI: 10.13371/j.cnki.chin.q.j.m.2022.02.008 Abstract （137）      PDF（pc） （885KB）（154）       Knowledge map    Save  In this paper, the existence of almost periodic solutions to general BAM neural networks with leakage delays on time scales is first studied, by using the exponential dichotomy method of linear differential equations and fixed point theorem. Then, the exponential stability of almost periodic solutions to such BAM neural networks on time scales is discussed by utilizing differential inequality. Finally, an example is given to support our results in this paper and the results are up-to-date. Related Articles | Metrics

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Homology of Bihom-Lie Algebras CHENG Yong-sheng, WANG Meng-ping Chinese Quarterly Journal of Mathematics    2022, 37 (3): 237-247.   DOI: 10.13371/j.cnki.chin.q.j.m.2022.03.002 Abstract （107）      PDF（pc） （286KB）（151）       Knowledge map    Save  The purpose of this paper is to define Hochschild type homology of Bihom- associative algebras and Chevalley-Eilenberg type homology of Bihom-Lie algebras with non-trivial coefficients in their bimodules respectively. In particular, we give their low order homology in detail. Related Articles | Metrics

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Weighted Analytic Torsion for Weighted Digraphs REN Shi-quan, WANG Chong Chinese Quarterly Journal of Mathematics    2023, 38 (1): 30-49.   DOI: 10.13371/j.cnki.chin.q.j.m.2023.01.003 Abstract （175）      PDF（pc） （389KB）（139）       Knowledge map    Save  In 2020, Alexander Grigor’yan, Yong Lin and Shing-Tung Yau [6] introduced the Reidemeister torsion and the analytic torsion for digraphs by means of the path complex and the path homology theory. Based on the analytic torsion for digraphs introduced in [6], we consider the notion of weighted analytic torsion for vertex-weighted digraphs. For any non-vanishing real functions f and g on the vertex set, we consider the vertex-weighted digraphs with the weights ( f,g ). We calculate the ( f,g )-weighted analytic torsion by examples and prove that the ( f,g )-weighted analytic torsion only depend on the ratio f/g . In particular, if the weight is of the diagonal form ( f,f ), then the weighted analytic torsion equals to the usual (un-weighted) torsion. Related Articles | Metrics

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Induced Matching-Extendability of Halin Graphs ZHANG Qing-nan, HUI Zhi-hao, YANG Yu, WANG An, Chinese Quarterly Journal of Mathematics    2022, 37 (4): 380-385.   DOI: 10.13371/j.cnki.chin.q.j.m.2022.04.005 Abstract （185）      PDF（pc） （344KB）（138）       Knowledge map    Save  Let G be a connected graph having a perfect matching. The graph G is said to be induced matching (IM) extendable if every induced matching M of G is contained in a perfect matching of G. In this paper, we show that Halin graph G =T ∪C is IMextendable if and only if its characteristic tree T is isomorphic to K1,3, K1,5, K1,7 or S2,2. Related Articles | Metrics

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Independent Roman {2}-Domination in Trees LI Bei-bei, SHANG Wei-ping Chinese Quarterly Journal of Mathematics    2022, 37 (4): 386-393.   DOI: 10.13371/j.cnki.chin.q.j.m.2022.04.006 Abstract （167）      PDF（pc） （336KB）（137）       Knowledge map    Save For a graph G = (V,E), a Roman {2}-dominating function f :V → {0,1,2} has the property that for every vertex v ∈V with f(v) = 0, either v is adjacent to at least one vertex u for which f(u) = 2, or at least two vertices u1 and u2 for which f(u1) =f(u2) = 1. A Roman {2}-dominating function f = (V0,V1,V2) is called independent if V1∪V2 is an independent set. The weight of an independent Roman {2}-dominating function f is the value ω(f) =\sumv∈V f(v), and the independent Roman {2}-domination number i{R2}(G) is the minimum weight of an independent Roman {2}-dominating function on G. In this paper, we characterize all trees with i{R2}(T) =γ(T)+ 1, and give a linear time algorithm to compute the value of i{R2}(T) for any tree T.  Related Articles | Metrics

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Ergodicity of Bandwidth and Cutwidth on Families of Graphs and Trees LIN Yi-shu, CHANG Cai-bing, LIU Yan Chinese Quarterly Journal of Mathematics    2022, 37 (4): 355-365.   DOI: 10.13371/j.cnki.chin.q.j.m.2022.04.003 Abstract （111）      PDF（pc） （346KB）（136）       Knowledge map    Save Bandwidth, cutwidth, cyclic bandwidth, bandwidth sum and cyclic bandwidth sum are well-known indices about optimal labeling of graphs applied in VLSI design, network communications, and other areas involving the graph layout. To design the graphs with the given indices, we need to study the ergodicity. Let F be a set of graphs under consideration and ϕ an integer-valued function defined on F, namely, ϕ is an index, such as bandwidth and cutwidth. If there exists a graph G ∈ F such that ϕ(G) =x for any integer x in the interval [a,b], where a and b are the minimum and maximum of ϕ on F, respectively, then ϕ is said to have ergodicity on F. Let Gn be the set of simple connected graphs with order n and Tn the set of trees with order n. In this paper, we investigate the ergodicity of bandwidth, cutwidth, cyclic bandwidth, the bandwidth sum and cyclic bandwidth sum on Tn and Gn. Related Articles | Metrics

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A Note on Preemptive Scheduling with Multiple Maintenance Activities to Minimize the Total Late Work HE Ru-yan, YUAN Jin-jiang, ZHANG Yuan Chinese Quarterly Journal of Mathematics    2022, 37 (4): 331-342.   DOI: 10.13371/j.cnki.chin.q.j.m.2022.04.001 Abstract （119）      PDF（pc） （374KB）（132）       Knowledge map    Save We study the single-machine preemptive scheduling problem with multiple maintenance activities to minimize the total late work, in which the jobs must be processed in the time space not occupied by the maintenance intervals. For this problem, we present a polynomial algorithm to determine the optimal schedule and establish a formula expression to the optimal value. Moreover, our result is used to correct some minor errors in the literature related to the single-machine (preemptive or non-preemptive) scheduling with one maintenance activity to minimize the total late work.  Related Articles | Metrics