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Elliptic Gradient Estimate for Heat Equation with Integral Ricci Curvature Condition
Jian Hong WANG
Acta Mathematica Sinica, Chinese Series. 2022, 65(5): 763774.
DOI:
10.12386/A20210040
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The main purpose of this paper is to establish the elliptic gradient estimate for the heat equation on compact Riemannian manifold with control on integral Ricci curvature. We also derived the volume comparison theorem under the new integral Ricci curvature condition which extended PetersenWei's volume comparison theorem.
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Finite Groups in Which Certain Subgroups are Between Normal and Abnormal
Peng Fei GUO, Hua Guo SHI
Acta Mathematica Sinica, Chinese Series. 2022, 65(5): 841848.
DOI:
10.12386/A20210001
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A subgroup $H$ of a finite group $G$ is a BNAsubgroup of $G$ if either $H^{x}=H$ or $x\in \langle H, H^{x}\rangle $ for all $x\in G$. A finite group $G$ is called a CBNAgroup if its all cyclic subgroups of order prime or $4$ are BNAsubgroups of $G$. The main aim of this paper is to investigate the structure of CBNAgroups, and the groups whose all proper subgroups are CBNAgroups are classified completely.
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Cotorsion Pairs and Stable Categories over Triangular Matrix Rings
Wei REN Chun, Xia ZHANG
Acta Mathematica Sinica, Chinese Series. 2022, 65(5): 849858.
DOI:
10.12386/A20210007
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In general, the properties of modules over a triangular matrix ring $T=\left(\begin{smallmatrix}A & U \\0 & B\end{smallmatrix}\right)$ are studied via modules over diagonal "small rings" $A$ and $B$. However, we use model structures on the category of $T$modules to characterize the stable categories $\underline{\mathcal{GP}}(A)$, $\underline{\mathcal{GP}}(B)$ of Gorenstein projective modules over $A$ and $B$. To this end, we introduce two subcategories of Gorenstein $T$modules, and obtain two corresponding complete cotorsion pairs. Moreover, cotorsion pairs of modules are lifted to $T$complexes, and the equivalences and recollements of homotopy categories of complexes are studied.
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Perfect Permutations and Spatially Balanced Latin Squares
Hao ZHENG, Hai Tao CAO
Acta Mathematica Sinica, Chinese Series. 2022, 65(5): 939950.
DOI:
10.12386/A20210055
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In this paper, a new conception called perfect permutation will be introduced. We focus on its algebraic properties and construction methods. The main result is that there exists a perfect permutation of order
n
when 2
n
+ 1 is a prime. Furthermore, we use perfect permutations to construct cyclic spatially balanced Latin squares and symmetric spatially balanced Latin squares both of which are widely used in experimental designs.
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Cramér's Large Deviation Expansion for a Supercritical Branching Process with Immigration in a Random Environment
Yan Qing WANG, Quan Sheng LIU, Xie Quan FAN
Acta Mathematica Sinica, Chinese Series. 2022, 65(5): 877890.
DOI:
10.12386/A20210030
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Let $(Z_n)$ be a supercritical branching process with immigration in an independent and identically distributed random environment. Based on the structure of $Z_n$, using related results on random walks and technique of measure change, we establish a Cramér's large deviation expansion for $\log Z_n$.
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$(C,\varepsilon)$Super Subdifferentials of SetValued Maps and Optimality Conditions for SetValued Optimization Problems
Zhi Ang ZHOU, Shuang YANG
Acta Mathematica Sinica, Chinese Series. 2022, 65(5): 859876.
DOI:
10.12386/A20210021
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In this paper, we study $(C,\varepsilon)$super subdifferentials of setvalued maps. First, we introduce a notion of $(C,\varepsilon)$super efficient point of a set. Some properties and equivalent characterizations of the $(C,\varepsilon)$super efficient points are presented. Scalarization theorems of the setvalued optimization problem are obtained in the sense of $(C,\varepsilon)$super efficiency. Second, we define $(C,\varepsilon)$subdifferentials of setvalued maps and research the existence conditions of $(C,\varepsilon)$subdifferentials. MoreauRockafellar type theorems characterized by $(C,\varepsilon)$subdifferentials are also established. Finally, as the applications, we establish some optimality conditions of the setvalued optimization problem involving the $(C,\varepsilon)$super subdifferentials. The results obtained in this paper unify and generalize some results characterized by the super subdifferentials or $\varepsilon$super subdifferentials of the setvalued maps in the literature.
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A New Projection Algorithm for Solving Variational Inequality Problems in Banach Spaces
Zhong Bing XIE, Gang CAI
Acta Mathematica Sinica, Chinese Series. 2022, 65(5): 907918.
DOI:
10.12386/A20210056
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We introduce a new algorithm for solving pseudomonotone variational inequality problems in Banach spaces. We prove that the sequence generated by this algorithm converges strongly an element of solutions for variational inequality problems under some suitable conditions imposed on the parameters. The results obtained in this paper extend and improve many recent ones in the literature.
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Ramanujan Expansions of Arithmetic Functions of Several Variables over $\mathbb{F}_{q}[T]$
Tian Fang QI
Acta Mathematica Sinica, Chinese Series. 2022, 65(5): 891906.
DOI:
10.12386/A20210053
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Combing Wintner, Delange, Ushiroya and Tóth's works from 1976 to 2017, we have that the multivariable arithmetic functions defined on integer ring can be expanded through the Ramanujan sums. This is an analogue of the Fourier expansion for periodic functions in the classical analysis. In this paper we further investigate the properties of Ramanujan sums in the polynomial ring $\mathbb{F}_{q}{[T]}$, and show that the multivariable arithmetic functions defined on $\mathbb{F}_{q}{[T]}$ can also be expanded through the polynomial Ramanujan sums and the unitary polynomial Ramanujan sums.
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The 2local Homogeneous Derivations of a Class of Infinite Dimensional Complete Lie Algebras
Jie HUANG, Xian Dong WANG
Acta Mathematica Sinica, Chinese Series. 2022, 65(5): 951958.
DOI:
10.12386/A20210044
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In this paper, a class of infinite dimensional complete Lie algebras is obtained by extending the one dimensional homogeneous derivations of Loop algebras over simple Lie algebras. It is proved that every 2local homogeneous derivation of this kind of infinite dimensional complete Lie algebra is a derivation.
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On the Integration by Parts and Characterization of a Fractional Diffusion Process
Xiao Xia SUN, Xuan Ming NI
Acta Mathematica Sinica, Chinese Series. 2022, 65(6): 10571066.
DOI:
10.12386/A20190028
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The relationship between a fractional diffusion process and its integration by parts formula is studied. By constructing a pull back formula, the integration by parts formula for fractional diffusion process is established. Conversely, a fractional diffusion process can be characterized through its integration by parts formula.
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Local Functional Law of the Iterated Logarithm for Increments of a Brownian Sheet
Yong Hong LIU, Qing Qing ZHANG
Acta Mathematica Sinica, Chinese Series. 2022, 65(6): 10831092.
DOI:
10.12386/A20190074
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In this paper, using large deviations for a Brownian sheet and increments of a Brownian sheet, we obtain local functional law of the iterated logarithm for a Brownian sheet and increments of a Brownian sheet.
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Spectral Collocation Method for Caputo Fractional Multipoint Value Problems
Zhen Dong GU, Li Ying SUN
Acta Mathematica Sinica, Chinese Series. 2022, 65(6): 9891002.
DOI:
10.12386/A20210034
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504
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Spectral collocation method is investigated for the nonlinear Caputo fractional multipoint value problems. The main idea of the presented method is to solve the corresponding nonlinear weakly singular VolterraFredholm integral equations obtained from the nonlinear Caputo fractional multipoint value problems. In order to carry out convergence analysis for the presented method, we investigate the Gronwall type inequality with VolterraFredholm integral terms. The provided convergence analysis shows that the presented method has spectral convergence, which is confirmed by the provided numerical experiments. At present, numerical methods for fractional multipoint value problems are rarely studied. The method and convergence analysis in this paper are useful references for the researches of related subjects.
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Superbiderivations on the Super HeisenbergVirasoro Algebras and Applications
Ya Di WU, Xiao Qing YUE
Acta Mathematica Sinica, Chinese Series. 2022, 65(4): 691698.
DOI:
10.12386/A20210045
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Let ${\mathbb L}$ be a super HeisenbergVirasoro algebra with the ${\mathbb C}$basis {$L_{n},I_{n},G_{n}$ $ n\in {\mathbb Z}$}, which satisfies the relations $[L_{m},L_{n}]=(mn)L_{m+n}$, $[L_{m},I_{n}]=\!nI_{m+n}$, $[L_{m},G_{n}]=nG_{m+n}$ and $ [G_{m},G_{n}]=I_{m+n}.$ In this paper, we prove that all superskewsymmetric superbiderivations of ${\mathbb L}$ are inner. Furthermore, we prove that every linear supercommuting map on ${\mathbb L}$ has the form $\Psi(x)=f(x)I_{0}$ for all $x\in{\mathbb L}$, where $f(x)$ is a linear map from ${\mathbb L}$ to ${\mathbb C}$.
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f
ergodicity of Markov Process by Coupling Method
Zhi Feng ZHU, Shao Yi ZHANG
Acta Mathematica Sinica, Chinese Series. 2022, 65(6): 11371142.
DOI:
10.12386/A20190048
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We first study the basic coupling and obtain an equation between total variation norm and the basic coupling. Then by use this equation we investigate the ergodicity property of continuous time Markov processes in general state space. For an ergodic continuoustime Markov processes, adding condition $\pi(f)<\infty$, by using the coupling method, there exists the full absorption set, such that the continuous time Markov processes are $f$ergodic on it.
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Dynamical Properties of Morphisms on Higher Dimensional Projective Space over Local Field
Zheng Jun ZHAO, Xiang CHEN
Acta Mathematica Sinica, Chinese Series. 2022, 65(6): 967978.
DOI:
10.12386/A20210073
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Let $\mathbb{C}_v$ be an algebraically closed nonarchimedean field, complete with respect to a valuation $v$. Let $\varphi:\mathbb{P}^N\rightarrow \mathbb{P}^N$ be a morphism of degree greater than one defined over $\mathbb{C}_v$, $\Phi$ a lift of $\varphi$. Let $\mathcal{G}_\Phi$ be the Green function of $\Phi$ and $\rho$ the chordal metric on $\mathbb{P}^N(\mathbb{C}_v)$. In this paper, we first study the properties of reduction of points in high dimensional projective space and reduction of automorphisms of $\mathbb{P}^N$ with degree one. With the help of Green function $\mathcal{G}_\Phi$ of $\Phi$, we introduce the arithmetic distance of morphisms and investigate its property. The necessary and sufficient condition which $\Phi$ has good reduction is obtained in this paper. We also describe explicitly the Filled Julia set of $\Phi$ by its Green function.
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Compound Binomial Risk Model with Random Income in Markov Chain Environment
Lin XIAO
Acta Mathematica Sinica, Chinese Series. 2022, 65(6): 10671082.
DOI:
10.12386/A20190054
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In this paper, I consider that the actuarial model is affected by the environmental process $\Theta$, premium income counting process $\eta$, claim counting process I and the claim process B, and establish a compound binomial risk model with random income in Markov chain environment, which is called MRICM, for short. The characteristic fivetuple set is given. It is proved that there exists a probabilistic space $(\Omega,\mathscr{F},P)$, and MRICM$(\Theta,\eta,I,B)$ defined on it, and its characteristic fivetuple set coincides with the given one. The recursive equations of conditional ruin probability for finite time and infinite time are obtained.
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On the Density of Transmission Eigenvalue for the Schrödinger Operator with the Robin Boundary Condition
Li Jie MA, Xiao Chuan XU
Acta Mathematica Sinica, Chinese Series. 2022, 65(6): 959966.
DOI:
10.12386/A20210064
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519
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In this paper, we study the transmission eigenvalue problem with the Robin boundary condition. Applying the related properties of entire function of exponential type, we show the relationship between the density of eigenvalues and the length of the support interval of the potential function. Meanwhile, we prove that the transmission eigenvalue problem is equivalent to a kind of SturmLiouville problem with spectral parameter in the boundary condition.
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Hausdorff Dimension of the Exceptional Set in Engel Continued Fractions
Mei Ying LÜ, Jing XIE
Acta Mathematica Sinica, Chinese Series. 2022, 65(6): 10031008.
DOI:
10.12386/A20210058
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For any real number $x\in(0,1)$, there exists a unique Engel continued fractions of $x$. In this paper, we mainly discuss the exceptional set which the logarithms of the partial quotients grow with nonlinear rate. We completely characterize the Hausdorff dimension of the relevant exceptional set.
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Autoregression Model of Time Series with Matrix CrossSection Data
Shu Jin WU, Nan HUA
Acta Mathematica Sinica, Chinese Series. 2022, 65(6): 10931104.
DOI:
10.12386/A20190057
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453
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The advantage of time series with matrix crosssection data is that multiple attributes of multiple objects can be characterized simultaneously. This paper focuses on autoregression model of time series with matrix crosssection data and presents the methods of parameter estimation, model identification and white noise test. Finally, the daily yield series and daily volume change rate series of two bank stocks are analyzed by this model.
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Schurtype Theorems about Angles and Heights
Xiao Le SU, Yi TAN, Yu Sheng WANG
Acta Mathematica Sinica, Chinese Series. 2023, 66(2): 199208.
DOI:
10.12386/A20210060
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252
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In differential geometry, there is a classical result, named Schur's Theorem, which is about the comparison of chords of two curves in $\mathbb E^3$. Inspired by it, this paper presents Schurtype theorems about the comparison of chord tangent angles of two curves, and the comparison of heights of two curves relative to their chords.
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Transformations Leaving the Maximal Eigenvalue of Quantum States Invariant
Jing YU, Kan HE, Jin Chuan HOU, Feng Yang JIA
Acta Mathematica Sinica, Chinese Series. 2022, 65(5): 775782.
DOI:
10.12386/A20210102
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Quantum states are positive operators with unit traces on a Hilbert space. The set of all quantum states is convex. In the paper, we give a characterization of maps leave the maximal eigenvalue of convex combinations of quantum states.
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The Generalized Inverses of Tensors and Generalized CayleyHamilton Theorem Based on the TProduct
Hong Xing WANG, Jie LIU
Acta Mathematica Sinica, Chinese Series. 2022, 65(5): 827840.
DOI:
10.12386/A20200226
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75
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We define the TDMP inverse and TCMP inverse of thirdorder Fsquare tensors by using the TMoorePenrose inverse, TDrazin inverse and Tcorenilpotent decomposition theorem of tensors via the Tproduct. Then, we present some characterizations and properties. Finally, the CayleyHamilton theorem of thirdorder tensors is extended to TDrazin inverses and TDMP inverses. Examples are also given to illustrate these theorems.
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Refined Esitmates of All Homogeneous Expansions for a Subclasses of Spirallike Mappings in Several Complex Variables
Xiao Song LIU
Acta Mathematica Sinica, Chinese Series. 2022, 65(5): 783796.
DOI:
10.12386/B20210044
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In this article, the refined estimates of all homogeneous expansions for a subclass of biholomorphic spirallike mappings which have a concrete parametric representation on the unit ball in complex Banach spaces and the unit polydisk in $\mathbb{C}^n$ are mainly established. In particular, the result is sharp for the ($k+1$)th homogeneous expansion. Meanwhile the estimates of all homogeneous expansions for a subclass of $k$fold symmetric biholomorphic spirallike mappings which have parametric representation on the unit ball in complex Banach spaces and the unit polydisk in $\mathbb{C}^n$ are also given, and the result is sharp for the ($k+1$)th homogeneous expansion as well. Our obtained results include many known results in some prior literatures.
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Additive Functionals for Discretetime Single Death Chains
Jia Meng TIAN, Jing WANG, Yu Hui ZHANG
Acta Mathematica Sinica, Chinese Series. 2022, 65(5): 797818.
DOI:
10.12386/A20210101
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We get an explicit and recursive representation for high order moments of additive functionals for discretetime single death chains. Meanwhile, the polynomial convergence and a central limit theorem are investigated.
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The Generators and Relations for Little $q$Schur Algebra $u_k(3,3)$
Wen Ting GAO, Ming Qiang LIU
Acta Mathematica Sinica, Chinese Series. 2022, 65(5): 819826.
DOI:
10.12386/A20200187
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65
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For $n=2$, Bian and Liu in[
Algebra Colloquium,
2017,
24
(2):297308] provided a presentation for little $q$Schur algebra in terms of generators and relations. It is more difficult for the case $n>2$. In this paper, we give a presentation for little $q$Schur algebra $u_k(3,3)$ at odd roots.
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Conformal Biderivations and Automorphism Groups of SchrödingerVirasoro Type Lie Conformal Algebras
Wei WANG, Chun Guang XIA, Ying XU
Acta Mathematica Sinica, Chinese Series. 2022, 65(5): 927938.
DOI:
10.12386/A20200224
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62
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In this paper, conformal biderivations and automorphism groups of two classes of SchrödingerVirasoro type Lie conformal algebras ${\rm TSV}(a,b)$ and ${\rm TSV}(c)$ are completely determined, respectively. The results for the Lie conformal algebras $W(a,b)$ are also obtained as a corollary.
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A PBW Basis of the Derived Hall Algebra of Type $G_2$
Yun Xia WANG, Abdukadir OBUL
Acta Mathematica Sinica, Chinese Series. 2022, 65(5): 919926.
DOI:
10.12386/A20200202
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50
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In RingelHall algebra of Dynkin type, the set $S$ of isomorphism classes of indecomposable modules forms a minimal GröbnerShirshov basis of the ideal $\hbox{Id}(S)$ generated by the set $S$, and the corresponding irreducble elements forms a PBW basis of the corresponding RingelHall algbera. Our aim is to generalize this result to the derived Hall algebra of type $G_2$. First, we compute the skew commutator relations between the isomorphism classes of indecomposable objects in the bounded derived category of type $G_2$ by using the AuslanderReiten quiver of the bounded derived category of type $G_2$. Then, we prove that the compositions between these skew commutator relations are trivial. Finally, we construct a PBW basis of the derived Hall algebra of type $G_2.$
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Closed Strongly Irreducible Operators on Banach Spaces
Li Qiong LIN, Jia Hua QUE, Yun Nan ZHANG
Acta Mathematica Sinica, Chinese Series. 2023, 66(2): 209222.
DOI:
10.12386/A20210126
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This paper firstly gives the definition of closed strongly irreducible operators on Banach spaces and gives an example of unbounded strongly irreducible operator. It obtains some properties of closed strongly irreducible operators. In particular, it obtains some equivalent descriptions of closed strongly irreducible operators. It also demonstrates some sufficient conditions for the strongly irreducibility of closed operators which have the forms of upper triangular operator matrices.
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A Simplified Dynamical Systems Method for Solving Nonlinear Equation
Jing Yue HUANG, Xing Jun LUO, Rong ZHANG
Acta Mathematica Sinica, Chinese Series. 2023, 66(2): 223238.
DOI:
10.12386/A20210092
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A simplified dynamical systems method for solving the nonlinear equation $F(u)=f$ is studied in this paper. Under certain conditions of the operator $F$ and the exact solution $y$, the error estimate of the solution of the dynamical systems equation is given, and the discrepancy principle of the posterior selection of regularized parameter is proposed to ensure the optimal rate of convergence of the solution of the dynamical systems equation. Compared with the traditional dynamical systems method, the simplified dynamical systems method reduces the computation amount of derivatives.
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Distribution Properties and Applications of Consecutive Quadratic Residues
Xiao WANG, Ai Hua LI
Acta Mathematica Sinica, Chinese Series. 2023, 66(2): 389398.
DOI:
10.12386/A20210160
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Consider any prime number $p$. In this paper, we use analytic methods, properties of Legendre's symbol modulo $p$, and estimation for character sums to study distribution properties of triples of consecutive quadratic residues (named 3CQR) and consecutive quadratic nonresidues (3CQN) modulo $p$. We provide exact formulas for the numbers $S_1(p)$ and $S_2(p)$ of 3CQRs and 3CQNs when $p\equiv 3$ or $7\pmod{8}$. Asymptotic formulas are given for $p\equiv 1$ or $5\pmod{8}$. Similarly, triples of quadratic residues with equal distance 2 are investigated and corresponding enumeration formulas are given. As an application, we further apply 3CQRs to construct magic squares of squares of full degree over $\mathbb F_p$.
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Hypothesis Testing in Partial Functional Linear Spatial Autoregressive Model
Gao Sheng LIU, Yang BAI, Ping YU
Acta Mathematica Sinica, Chinese Series. 2023, 66(2): 239252.
DOI:
10.12386/A20210176
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The paper proposes a hypothesis testing of spatial autoregressive and parametric component in partial functional linear spatial autoregressive model. The functional principle component analysis is employed to approximate the slope function. And generalized method of moments (GMM) is used to estimate parameters. Basis on consistent estimators, we construct a test statistic of the residual sums of squares under null and alternative hypothesis. In addition, we establish the asymptotic properties of the proposed test. Simulation studies show the proposed test has good size and power with finite sample size. Finally, a real data analysis of growth data is conducted to investigate the significance of spatial autoregressive and parametric coefficients with partial functional linear spatial autoregressive model.
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An Inexact NewtonLanczos Method for Solving a System of Nonlinear Equations
Chao GU, Jue Yu WANG, De Tong ZHU
Acta Mathematica Sinica, Chinese Series. 2023, 66(2): 317338.
DOI:
10.12386/A20210026
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We present an inexact NewtonKrylov subspace method with incomplete line search technique for solving symmetric nonlinear equations, in which the Krylov subspace method uses the Lanczostype decomposition technique. The iterative direction is obtained by approximately solving the Newton’s equations of the nonlinear equations using the Lanczos method. The global convergence and local superlinear convergence rate of the proposed algorithm are established under some reasonable conditions. Finally, the numerical results show the effectiveness of the proposed algorithm.
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Cohomologies of Leibniz Algebras with Higher Derivations
Shuang Jian GUO
Acta Mathematica Sinica, Chinese Series. 2022, 65(6): 979988.
DOI:
10.12386/A20200223
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We study Leibniz algebras equipped with higher derivations. We call such a tuple of a Leibniz algebra and a higher derivation by LeibHDer pair. First, we define representations of LeibHDer pairs and construct the semidirect product. Finally, we define a suitable cohomology for a LeibHDer pair with coefficients in a representation, and study central extensions and deformations of LeibHDer pairs.
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Distributed Statistical Inference for Linear Models with Multisource Massive Heterogeneous Data
Xin YANG, Mi Xia WU
Acta Mathematica Sinica, Chinese Series. 2023, 66(2): 263276.
DOI:
10.12386/A20210120
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We focus on the distributed statistical inference for linear models with multisource massive heterogeneous data. First, a communicationefficient distributed aggregation method is proposed to estimate the unknown parameter vector, and the derived estimator is proved to be best linear unbiased and asymptotically normal under some regularity conditions. Then, a distributed test method is proposed to test the heterogeneity among a large number of data sources. Finally, the simulations are conducted to illustrate the effectiveness of the proposed method.
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GröbnerShirshov Bases for the Cyclotomic Hecke Algebra of Type
A
Munayim DILXAT, Abdukadir OBUL, Dong LIU
Acta Mathematica Sinica, Chinese Series. 2023, 66(2): 309316.
DOI:
10.12386/A20210074
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In this paper, we discuss the GröbnerShirshov bases and a linear bases of the cyclotomic Hecke algebra of type $A$. First, by computing the compositions, we construct a GröbnerShirshov bases of the cyclotomic Hecke algebra of type $A$. Then using this GröbnerShirshov bases and the compositiondiamond lemma we give a linear bases of the cyclotomic Hecke algebra of type $A$.
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Linear Arboricity of 1planar Graphs
Dan Jun HUANG, Yan JIANG
Acta Mathematica Sinica, Chinese Series. 2023, 66(2): 339352.
DOI:
10.12386/A20210093
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47
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An edgepartition of a graph $G$ is a decomposition of $G$ into subgraphs $G_1, G_2,\ldots,G_m$ such that $E(G)=E(G_1)\cup\cdots\cup E(G_m)$ and $E(G_i)\cap E(G_j)=\emptyset$ for any $i\neq j$. A linear forest is forest in which each connected component is a path. The linear arboricity ${\rm la}(G)$ is the least integer $m$ such that $G$ can be edgepartitioned into $m$ linear forests. In this paper, we use the discharging method to study the linear arboricity ${\rm la}(G)$ of 1planar graphs, and prove that ${\rm la}(G)=\lceil\frac{\Delta(G)}{2}\rceil$ for each 1planar graph $G$ with $\Delta(G)\ge25$, where $\Delta(G)$ is the maximum degree of $G$.
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Spacelike Hypersurfaces of Constant
r
th
F
mean Curvature with Lightlike Boundary
Yuan Zheng ZHANG
Acta Mathematica Sinica, Chinese Series. 2023, 66(2): 277292.
DOI:
10.12386/A20210147
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Let $\overline{W}_{F(t)}$, corresponding to a rotation invariant function $F(t(\nu))$ with a convexity condition on the upper hyperboloid $\mathbb{H}_+^n$, be a compact spacelike Wulff shape bounded by a lightlike $(n1)$round sphere. By applying perturbation metric and some integral formulae, we show that the only spacelike hypersurface with constant $r$th $F$mean curvature in $\mathbb{L}^{n+1}$, which is tangent to $\overline{W}_{F(t)}$ on the boundary, is the Wulff shape.
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Softplus Beta Negative Binomial Integervalued GARCH Model
Le Le QI, Fu Kang ZHU
Acta Mathematica Sinica, Chinese Series. 2023, 66(2): 293308.
DOI:
10.12386/A20210063
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INGARCH models are often constructed based on Poisson distribution, negative binomial distribution and so on. Beta negative binomial (BNB) distribution is a flexible distribution. Recently, the related BNBINGARCH model was proposed, whose conditional mean is linear, the parameters are restricted to nonnegative and negative autocorrelation cannot be modeled. In this paper, we first propose the loglinear BNBINGARCH model to solve the above problems, but the simple form of linear mean and ARMAlike structures are lost. So we further construct softplus BNBINGARCH$(p,q)$ model by using the softplus function, which is the main research object. When $p$ and $q$ are equal to 1, the stationarity and ergodicity of the model are proved and the conditions for the existence of the second moment are given. In addition, the strong consistency and the asymptotic normality of the maximum likehood estimator are shown. Finally, the analysis of realdata examples show the usefulness of the proposed model.
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On Two Dimensional Riemann Problem with Three Constant States for Chaplygin Gas
Jie CHENG, Fang Qi CHEN, Ze Jun WANG
Acta Mathematica Sinica, Chinese Series. 2023, 66(2): 253262.
DOI:
10.12386/B20210196
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In this paper, the two dimensional Riemann problem of the Euler system for Chaplygin gas with three pieces of constant states is studied. The three states are divided by the $x$axis and the positive semiaxis of the $y$axis. Based on the assumption that each jump in initial data outside of the origin projects exactly one planar wave of shocks, centered rarefaction waves, or slip planes, and by using of the method of generalized characteristic analysis, we give the structures of the solution in detail. In fact, we divide the analysis into ten cases and among them, only four subcases are reasonable.
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Geometric Construction of Pythagorean Hodograph Ccurves
Yu Jun LI, Lin Cong FANG
Acta Mathematica Sinica, Chinese Series. 2023, 66(2): 353362.
DOI:
10.12386/A20210145
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We study the geometric characteristics of CBézier curves that possess the Pythagorean Hodograph (PH) property. Based on the algebraic necessary and sufficient conditions for PH Ccurves, we prove that a CBézier curve is a PH Ccurve if and only if the interior angles of its control polygon are equal, and the second leg length of the control polygon is the geometric mean of the first and the last ones. Our main idea is to represent a planar parametric curve in complex form. We claim that the geometric characteristics of PH Ccurves are quite similar to polynomial PH curves, which can be used to identify PH Ccurves and their constructions. As an application, we give some examples of $G^1$ Hermite interpolation using PH Ccurves. We point out that there are no more than two PH Ccurves for any given $G^1$ Hermite conditions.
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