Table of Content

25 September 2022, Volume 37 Issue 3

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The Polynomial Function Model in Born-Infeld Theory

DAI Bing-bing, ZHANG Rui-feng

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.

Homology of Bihom-Lie Algebras

CHENG Yong-sheng, WANG Meng-ping

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.


Maximum Net Benefit Indicator and Its Applications

YANG Xiao-hui, BAI Xin-yu, LI Zi-xin, HUANG Kun

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.


Simplistic Universal Protocols for Remotely Preparing Arbitrary Equatorial States

MA Song-ya LI Xiang, LI Qi

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.


The Existence of Solutions for a Class of Schrödinger Equations via Morse Index Theory

LI Jia-yang, WANG Qi

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

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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].


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

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.


Vertex Operators, Littlewood-Richardson Rule for Generalized Symplectic Schur Functions

HUANG Fang, CHU Yan-jun

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.


The Fermion Representation of the Phase Model

CUI Zhen-nan, BAI Yang, WANG Na, WU Ke

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.