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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 （179）      PDF（pc） （330KB）（93）       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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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 （166）      PDF（pc） （356KB）（230）       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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Global Existence and Uniqueness of Periodic Waves for a Perturbed Combined Double-Dispersive Equation LIAO Xiao-zhao, HUANG Wen-tao, YANG Su-min Chinese Quarterly Journal of Mathematics    2022, 37 (2): 124-131.   DOI: 10.13371/j.cnki.chin.q.j.m.2022.02.002 Abstract （146）      PDF（pc） （534KB）（80）       Knowledge map    Save In this paper, we study the periodic wave propagation phenomenon in elastic waveguides modeled by a combined double-dispersive partial differential equation (PDE). The traveling wave ansazt transforms the PDE model into a perturbed integrable ordinary differential equation (ODE). The global bifurcation theory is applied for the perturbed ODE model to establish the existence and uniqueness of the limit cycle, which corresponds the periodic traveling wave for the PDE model. The main tool is the Abelian integral taken from Poincar´ e bifurcation theory. Simulation is carried out to verify the theoretical result. Related Articles | Metrics

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Volume Inequalities for Sections and Projections of Asymmetric Convex Bodies CAO Zi-xin, Li Ai-jun Chinese Quarterly Journal of Mathematics    2022, 37 (2): 178-188.   DOI: 10.13371/j.cnki.chin.q.j.m.2022.02.007 Abstract （142）      PDF（pc） （346KB）（75）       Knowledge map    Save In this paper, we establish volume inequalities for k-dimensional sections and projections of convex bodies (not necessarily symmetric) and their polars in a more general position than John’s position. 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 （131）      PDF（pc） （344KB）（80）       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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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 （122）      PDF（pc） （538KB）（316）       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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The Polynomial Function Model in Born-Infeld Theory DAI Bing-bing, ZHANG Rui-feng Chinese Quarterly Journal of Mathematics    2022, 37 (3): 221-236.   DOI: 10.13371/j.cnki.chin.q.j.m.2022.03.001 Abstract （117）      PDF（pc） （413KB）（88）       Knowledge map    Save  Based on the Lagrangian action density under Born-Infeld type dynamics and motivated by the one-dimensional prescribed mean curvature equation, we investigate the polynomial function model in Born-Infeld theory in this paper with the form of −([1−a(ϕ')2 ]ϕ' )' =λf(ϕ(x)), where λ> 0 is a real parameter, f ∈C2 (0 , + ∞ ) is a nonlinear function. We are interested in the exact number of positive solutions of the above nonlinear equation. We specifically develop for the problem combined with a careful analysis of a time-map method. 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 （112）      PDF（pc） （279KB）（131）       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 （108）      PDF（pc） （348KB）（135）       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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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 （105）      PDF（pc） （336KB）（110）       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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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 （102）      PDF（pc） （389KB）（109）       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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Orbifold Fundamental Group and Deck Translation Group LIN Yi-wu Chinese Quarterly Journal of Mathematics    2022, 37 (2): 147-161.   DOI: 10.13371/j.cnki.chin.q.j.m.2022.02.005 Abstract （98）      PDF（pc） （332KB）（101）       Knowledge map    Save In this paper, we define the notions of orbifold loop and orbifold road, with which, we reformulate the definition of orbifold fundamental group and deck translation group. We show that, for each orbifold, the orbifold fundamental group is isomorphic to the deck translation group. Related Articles | Metrics

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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 （97）      PDF（pc） （1289KB）（151）       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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A Note on Single-Machine Lot Scheduling with Splittable Jobs to Minimize the Number of Tardy Jobs SHEN Hui-jun, GENG Zhi-chao Chinese Quarterly Journal of Mathematics    2022, 37 (4): 412-421.   DOI: 10.13371/j.cnki.chin.q.j.m.2022.04.009 Abstract （93）      PDF（pc） （295KB）（43）       Knowledge map    Save The single-machine lot scheduling problem with splittable jobs to minimize the number of tardy jobs has been showed to be weakly NP-hard in the literature. In this paper, we show that a generalized version of this problem in which jobs have deadlines is strongly NP-hard, and also present the results of some related scheduling problems.  Related Articles | Metrics

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The Existence of Normalized Solution to the Kirchhoff#br# Equation with Potential#br# LIANG Yan-xia Chinese Quarterly Journal of Mathematics    2023, 38 (2): 196-209.   DOI: 10.13371/j.cnki.chin.q.j.m.2023.02.007 Abstract （90）      PDF（pc） （378KB）（61）       Knowledge map    Save  In this paper we discuss the following Kirchhoff equation \left\{ \begin{array}{lr} -\left(a+b \int_{\mathbb{R}^3}|\nabla u|^{2} d x\right) \Delta u+V(x)u+\lambda u=\mu|u|^{q-2}u+|u|^{p-2}u \ {\rm in}\ \mathbb{R}^3,&\\ \int_{\mathbb{R}^{3}}u^{2}dx=c^2, \end{array} \right. where a, b, µ and c are positive numbers, λ is unknown and appears as a Lagrange multiplier, 14/3<q<p<6 and V is a continuous non-positive function vanishing at infinity. Under some mild assumptions on V , we prove the existence of a mountain pass normalized solution. To the author’s knowledge, it is the first time to study the existence of normalized solution to Kirchhoff equation with potential via the minimax principle. 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 （85）      PDF（pc） （307KB）（112）       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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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 （84）      PDF（pc） （2367KB）（222）       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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Maximum Net Benefit Indicator and Its Applications YANG Xiao-hui, BAI Xin-yu, LI Zi-xin, HUANG Kun Chinese Quarterly Journal of Mathematics    2022, 37 (3): 248-259.   DOI: 10.13371/j.cnki.chin.q.j.m.2022.03.003 Abstract （83）      PDF（pc） （6924KB）（29）       Knowledge map    Save  Receiver operating characteristics (ROC) curve and the area under the curve (AUC) value are often used to illustrate the diagnostic ability of binary classifiers. However, both ROC and AUC focus on high accuracy in theory, which may not be effective for practical applications. In addition, it is difficult to judge which one is better when the ROC curves are intersect and the AUC values are equal. Decision curve analysis (DCA) methods improve ROC by incorporating accuracy and consequences. However, similar to ROC, DCA requires a quantitative indicator to objectively determine which one is better when DCA curves intersect. A DCA-based statistical indicator named maximum net benefit (MNB) is constructed for evaluating clinical treatment regimens rather than just accuracy as in ROC and AUC. As a simple and effective statistical indicator, the construction process of MNB is given theoretically. Moreover, the MNB can still provide effective identification when the AUC values are equal, which is proved by theory. Furthermore, the feasibility and effectiveness of the proposed MNB are verified by gene selection and classifier performance comparison on actual data. 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 （82）      PDF（pc） （885KB）（129）       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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The Boundedness of Multilinear Commutators on Grand Variable Herz Spaces PENG Shan-shan, CHEN Ji-li Chinese Quarterly Journal of Mathematics    2022, 37 (2): 203-213.   DOI: 10.13371/j.cnki.chin.q.j.m.2022.02.009 Abstract （82）      PDF（pc） （381KB）（76）       Knowledge map    Save  We consider multilinear commutators of singular integrals defined by $T_{\vec{b}}f(x) =\int_{\mathbb{R}^n}\prod^m_{i=1}(b_i(x)-b_i(y))K(x, y)f(y)dy,$ where K is a standard Calder\'{o}n-Zygmund kernel, m is a positive integer and \vec{b} b =(b1,b2,...,bm) is a family of m locally integrable functions. Based on the theory of variable exponent and on generalization of the BMO norm, we prove the boundedness of multilinear commutators T_{\vec{b}} on grand variable Herz spaces. The result is still new even in the special case of m=1. Related Articles | Metrics