数学季刊

代兵兵, 张瑞凤

2022, 37(3): 221-236. doi:10.13371/j.cnki.chin.q.j.m.2022.03.001

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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.

Bihom-李代数的同调

程永胜, 王梦平

2022, 37(3): 237-247. doi:10.13371/j.cnki.chin.q.j.m.2022.03.002

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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.

最大净收益指标及其应用

杨晓慧, 白欣宇, 李子欣, 黄昆

2022, 37(3): 248-259. doi:10.13371/j.cnki.chin.q.j.m.2022.03.003

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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.

远程制备任意赤道态的简单通用协议

马松雅, 李翔, 李琦

2022, 37(3): 260-273. doi:10.13371/j.cnki.chin.q.j.m.2022.03.004

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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.

利用莫尔斯指标理论求解一类薛定谔方程解的存在性

李佳洋, 王琪

2022, 37(3): 274-280. doi:10.13371/j.cnki.chin.q.j.m.2022.03.005

摘要 ( 65 )

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In this paper, with the relative Morse index, we will study the existence of
solutions of (1.1) under the assumptions that V satisfies some weaker conditions than
those in [2].

基于固有时间尺度分解和自适应Huber损失的脑电特征学习模型

杨利军, 蒋淑月, 魏小鸽, 肖运海

2022, 37(3): 281-300. doi:10.13371/j.cnki.chin.q.j.m.2022.03.006

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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.

顶点算子, 广义辛Schur函数的Littlewood-Richardson规则

黄芳, 楚彦军

2022, 37(3): 301-316. doi:10.13371/j.cnki.chin.q.j.m.2022.03.007

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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.

相位模型的费米子表示

崔珍楠, 白洋, 王娜, 吴可

2022, 37(3): 317-330. doi:10.13371/j.cnki.chin.q.j.m.2022.03.008

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In this paper, we give a Fermion representation of the Phase model. We find
that the states in the phase model can be described by Maya diagrams, and operators can
be described by Fermions. We calculate the rule of multiplications of Young diagrams in
N ×M box by Fermions, and also calculate the relations in the Phase model by Fermions.

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