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On hypersurfaces of H2×H2

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science China Mathematics 2024年第2期67卷 339-366页

作者： Dong Gao Hui Ma Zeke Yao School of science Beijing University of Civil Engineering and Architecture department of mathematical sciences Tsinghua University

In this paper, we study hypersurfaces of H2× H2. We first classify the hypersurfaces with constant principal curvatures and constant product angle functions. Then we classify homogeneous hypersurfaces and isoparametric hypersurfaces, respectively. Finally, we classify the hypersurfaces with at most two distinct constant principal curvatures, as well as those with three distinct constant principal curvatures under some additional conditions.

关键词： constant principal curvature homogeneous hypersurface isoparametric hypersurface 53C42 53B25 53C40

Research on Credit Default Swaps Pricing Considering Moral Hazard Incentive under Reduce-Form Model

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Journal of Systems science and Systems Engineering 2024年第3期33卷 311-329页

作者： Liang Wu Kangjie He Zhe Guo School of mathematical science Henan Institute of Science and TechnologyXinxiang453003China Public Front Office department Industrial and Commercial Bank of ChinaZhengzhou450000China

Equilibrium pricing of credit default swaps(CDS)promotes efficient identification of credit risk in the market,which in turn leads to efficient allocation of resources.However,even when CDS have been priced in equilibrium,i.e.,when premiums are equal to anticipated payments,the moral hazard incentives of CDS buyers increase with CDS transactions.Consequentially,it becomes an interesting research direction to study the impact of moral hazard incentives on the trading mechanism or pricing of derivatives(CDS).Most of the existing literature on the impact of moral hazard incentives in CDS pricing on derivatives trading mechanisms takes a macro perspective and focuses on the agreement risk effect.The literature exploring the analysis of the impact of moral hazard on the probability of agreement default from a micro perspective is not yet available.With this in mind,this paper focuses on the mechanisms by which“fraud”,an extreme manifestation of micro-moral hazard incentives,affects the probability of default.This paper introduces for the first time the concept of“claiming fraud”by credit protection buyers,which is different from the macro perspective of moral hazard incentives,and thus defines a specific extreme form of moral hazard incentives.Meanwhile,to address the intrinsic feature of the lack of economic explanatory power of the reduce-form model,this paper introduces a moral hazard incentive factor into the reduce-form model,and proposes a moral hazard state variable as a function of the asset value of the reference entity,which gives the reduce-form model strong economic explanatory power,and the default predictability is reduced by the description of the reduce-form model.In terms of the object of study,this paper considers the issue of moral hazard incentives in the presence of claiming fraud in two reference entities to further explore the impact of moral hazard incentives on default protection at the micro level in terms of cyclic default.Finally,based on the analysis of the results of the numerical simulat

关键词： Financial engineering moral hazard incentive claiming fraud reduce-form model probability of default calculation CDS pricing

A chaotic hierarchical encryption/watermark embedding scheme for multi-medical images based on row-column confusion and closed-loop bi-directional diffusion

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Chinese Physics B 2024年第2期33卷 228-237页

作者：张哲祎牟俊 Santo Banerjee 曹颖鸿 School of Information science and Engineering Dalian Polytechnic UniversityDalian 116034China department of mathematical sciences Politecnico di TorinoTorino 10129Italy

Security during remote transmission has been an important concern for researchers in recent years.In this paper,a hierarchical encryption multi-image encryption scheme for people with different security levels is designed,and a multiimage encryption(MIE)algorithm with row and column confusion and closed-loop bi-directional diffusion is adopted in the paper.While ensuring secure communication of medical image information,people with different security levels have different levels of decryption keys,and differentiated visual effects can be obtained by using the strong sensitivity of chaotic keys.The highest security level can obtain decrypted images without watermarks,and at the same time,patient information and copyright attribution can be verified by obtaining watermark images.The experimental results show that the scheme is sufficiently secure as an MIE scheme with visualized differences and the encryption and decryption efficiency is significantly improved compared to other works.

关键词： chaotic hierarchical encryption multi-medical image encryption differentiated visual effects row-column confusion closed-loop bi-directional diffusion transform domain watermark embedding

Small Cycles Property of Some Cremer Rational Maps and Polynomials

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Chinese Annals of Mathematics,Series B 2024年第1期45卷 123-136页

作者： Rong FU Ji ZHOU Faculty of science Yibin UniversityYibin 644000SichuanChina department of mathematical sciences Sichuan Normal UniversityChengdu 610066China

This paper concerns the linearization problem on rational maps of degree d≥2 and polynomials of degree d>2 from the perspective of non-linearizability.The authors introduce a set l_(∞) of irrational numbers and show that if α∈l_(∞),then any rational map is not linearizable and has infinitely many cycles in every neighborhood of the fixed point with multiplier λ=e^(2πiα),Adding more constraints to cubic polynomials,they discuss the above problems by polynomial-like maps.For the family of polynomials,with the help of Yoccoz's method,they obtain its maximum dimension of the set in which the polynomials are non-linearizable.

关键词： Irrationally indifferent fixed point Linearization problem Small cycles property

Markovian Quadratic BSDEs with an Unbounded Sub-quadratic Growth

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Chinese Annals of Mathematics,Series B 2024年第3期45卷 441-462页

作者： Jingnan JU Shanjian TANG Shanghai Center for mathematical sciences Fudan University department of Finance and Control sciences School of Mathematical Sciences Fudan University

This paper is devoted to the solvability of Markovian quadratic backward stochastic differential equations(BSDEs for short) with bounded terminal conditions. The generator is allowed to have an unbounded sub-quadratic growth in the second unknown variable z. The existence and uniqueness results are given to these BSDEs. As an application, an existence result is given to a system of coupled forward-backward stochastic differential equations with measurable coefficients.

关键词： Markovian BSDE Quadratic growth Unbounded sub-quadratic term coefficients Coupled FBSDE 60H10

Equivariant Tautological Integrals on Flag Varieties

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Chinese Annals of Mathematics,Series B 2024年第3期45卷 463-486页

作者： Xiaobo ZHUANG department of mathematical sciences Zhejiang Sci-Tech University

The author apples the Atiyah-Bott-Berline-Vergne formula to the equivariant tautological integrals over flag varieties of types A, B, C, D, and recovers the formulas expressing the integrals as iterated residues at in... 详细信息

关键词： Localization Flag variety Residue 55N91 14M15

Orbits inside Fatou sets

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Acta Mathematica Sinica,English Series 2024年

作者： John Erik FORN?SS Mi HU department of mathematical sciences Norwegian University of Science and Technology department of Mathematics University of Oslo department of mathematical Physical and Computer Sciences University of Parma

In this paper,we investigate the precise behavior of orbits inside attracting basins of rational functions on P1and entire functions f in C.Let R（z） be a rational function on P1,A（p） be the basin of attraction of an attracting fixed point p of R,and?i（i=1,2,···） be the connected components of A（p）.Assume?1contains p.Let p0∈?1close to p.Then there exists a constant C so that for any z0∈?i,there is a point q∈∪kR-k（p0）,k≥0 so that the Kobayashi distance d?i（z0,q）≤C.For entire functions f,we generally can not have similar results as for rational functions.However,if f has finitely many critical points,then similar results hold.

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NSNO:Neumann Series Neural Operator for Solving Helmholtz Equations in Inhomogeneous Medium

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Journal of Systems science & Complexity 2024年第2期37卷 413-440页

作者： CHEN Fukai LIU Ziyang LIN Guochang CHEN Junqing SHI Zuoqiang department of mathematical sciences Tsinghua UniversityBeijing 100084China Yau mathematical sciences Center Tsinghua UniversityBeijing 100084China Yanqi Lake Beijing Institute of mathematical sciences and Applications Beijing 101408China

In this paper,the authors propose Neumann series neural operator(NSNO)to learn the solution operator of Helmholtz equation from inhomogeneity coefficients and source terms to solutions.Helmholtz equation is a crucial partial differential equation(PDE)with applications in various scientific and engineering fields.However,efficient solver of Helmholtz equation is still a big challenge especially in the case of high wavenumber.Recently,deep learning has shown great potential in solving PDEs especially in learning solution operators.Inspired by Neumann series in Helmholtz equation,the authors design a novel network architecture in which U-Net is embedded inside to capture the multiscale feature.Extensive experiments show that the proposed NSNO significantly outperforms the state-of-the-art FNO with at least 60%lower relative L^(2)-error,especially in the large wavenumber case,and has 50%lower computational cost and less data requirement.Moreover,NSNO can be used as the surrogate model in inverse scattering problems.Numerical tests show that NSNO is able to give comparable results with traditional finite difference forward solver while the computational cost is reduced tremendously.

关键词： Helmholtz equation inverse problem neumann series neural network solution operator

Model of fractured medium and nondestructive control of composite materials

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Chinese Journal of Aeronautics 2024年第2期37卷 93-99页

作者： Vasily GOLUBEV Ilia NIKITIN Katerina BEKLEMYSHEVA department of Computers science and Numerical Mathematics Moscow Institute of Physics and TechnologyDolgoprudny 141700Russia mathematical Modeling and Management Department of InformatizationInstitute of Computer Aided Design RASMoscow 123056Russia

Non-destructive testing of composites is an important issue in the modern aircraft industry.Composites are susceptible to the barely visible impact damage which can affect the residual strength of the material and occurs both during production and operation.The continuum model for describing the damaged zone is presented.The slip theory relations used for a continuous distribution of slip planes are applied.At the initial stage,the isotropic background model is used.This model allows the material slippage along the fractures based on the Coulomb friction law with the small viscous addition.In this regime,the govern system of equations becomes rigid.To overcome this difficulty,the explicit-implicit grid-characteristic scheme is proposed.The standard ultrasound diagnostic procedure of damaged composite materials is successfully simulated.Compared with the trivial free-surface fracture model,different reactions on the compression and stretch waves are registered.This approach provided an effective way for the simulation of complex dynamic behavior of damage zones.

关键词： Non-destructive testing Composite materials Numerical modeling Continuum model Grid-characteristic method