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• Xi Sheng YU, Yu Wei YAO
Acta Mathematica Sinica, Chinese Series. 2023, 66(5): 801-814. https://doi.org/10.12386/A20200164
Option pricing with discrete dividend payments is still a challenge. This paper proposes a novel model by taking the dividends into consideration, and establishes the option price theorem for obtaining the option price. Theoretical analysis shows that the proposed new model can fully take the impact of dividend payments on option price such as the dividend paying time, amount and number, and hence it can produce an accurate price for option. We also conduct a theoretical comparison of the pricing between the newly-proposed model and classic/benchmark, with which the relation and pricing differences between the new model and these models are deeply detected. The numerical results also show that the proposed model can produce highly accurate prices for options and has strong pricing robustness. Based on this, our model can be an excellent alternative of pricing European options written on the underlying asset paying discrete dividends.
• Ping XI, Jun Ren ZHENG
Acta Mathematica Sinica, Chinese Series. 2024, 67(2): 220-226. https://doi.org/10.12386/A20220113
It is conjectured by Professor Zhi-Wei Sun that for each given odd prime $p>100,$ there always exists an solution $(x,y,z)\in[1,p]^3$ to the Pythagoras equation $x^2+y^2=z^2$ such that $x,y,z$ are quadratic residues or non-residues modulo $p$ respectively (eight cases in total). In this paper, we are able to prove the above assertion for all sufficiently large primes $p$, and the method is based on the recent Burgess bound for character sums of forms in many variables due to Lillian B. Pierce and Junyan Xu.
• Wei Ning LAI, Tao CHEN, Chun Yuan DENG
Acta Mathematica Sinica, Chinese Series. 2024, 67(1): 1-20. https://doi.org/10.12386/B20220687
Let $T\in \mathcal{B(H)}$ be a bounded linear operator on a complex Hilbert space $\mathcal{H}$. The properties of the generalized pencil $T=P +\alpha Q+\beta PQ$ of pair $(P, Q)$ of projections at $(\alpha, \beta)\in \mathbb{C}^2$ are investigated. Using Halmos decomposition theory for orthogonal projections we give some equivalent conditions for which $T$ is the generalized pencil and study the spectrum properties of this generalized pencil $T$. We prove that the generalized pencil $T$ is similar to a diagonal operator under some conditions. The spectrum relations among the generalized pencil $T$ and projections $P$, $Q$ are established. Further, we give the necessary and sufficient conditions under which the generalized pencil $T$ is a Fredholm operator, a compact operator or a selfadjoint operator, respectively. Finally, the generalized pencils of pairs of idempotents are studied.
• Yun GAO, Fang Wei FU
Acta Mathematica Sinica, Chinese Series. 2024, 67(3): 413-427. https://doi.org/10.12386/A20220016
Let $\mathbb{F}_q$ be a finite field of cardinality $q$, where $q$ is a power of a prime $p$, $t \ge 2$ is an even number satisfying $t\not \equiv 1\ (\bmod \,p)$ and $\mathbb{F}_{{q^t}}$ is an extension field of $\mathbb{F}_q$ with degree $t$. Firstly, a trace bilinear form on $\mathbb{F}_{{q^t}}^n$ which is called $\Delta$-bilinear form is given, where $n$ is a positive integer coprime to $q$. Then according to this trace bilinear form, bases and enumeration of cyclic $\Delta$-self-orthogonal and cyclic $\Delta$-self-dual $\mathbb{F}_q$-linear $\mathbb{F}_{{q^t}}$-codes are investigated when $t=2$. Furthermore, some good $\mathbb{F}_q$-linear $\mathbb{F}_{{q^2}}$-codes are obtained.
• Acta Mathematica Sinica, Chinese Series. 2024, 67(2): 209-210. https://doi.org/10.12386/A20240400
• Liu Yan LI, Jun Ping LI
Acta Mathematica Sinica, Chinese Series. 2023, 66(5): 815-826. https://doi.org/10.12386/b20210647
This paper considers a supercritical Galton-Watson process with immigration $\{X_n\}_{n\geq0}$. It is well-known that there is a sequence of constants $\{c_n\}_{n\geq0}$ such that $X_n/c_n\to V$ almost surely as $n\rightarrow\infty$. Using Cramér transforms, we obtain lower deviations for the process $\{X_n\}_{n\geq0}$, which refer to the asymptotic properties of $P(X_n=k)$ for sufficiently large $k$ satisfying $k_n\leq k\leq c_n$ and $k_n\rightarrow \infty$.
• Da Qing WAN, Jun ZHANG
Acta Mathematica Sinica, Chinese Series. 2024, 67(2): 211-219. https://doi.org/10.12386/A20220143
Counting zeros of polynomials over finite fields is one of the most important topics in arithmetic algebraic geometry. In this paper, we consider the problem for complete symmetric polynomials. The homogeneous complete symmetric polynomial of degree $m$ in the $k$-variables $\{x_1,x_2,\ldots,x_k\}$ is defined to be $h_m(x_1,x_2,\ldots$, $x_k):=\sum_{1\leq i_1\leq i_2\leq \cdots \leq i_m\leq k}x_{i_1}x_{i_2}\cdots x_{i_m}.$ A complete symmetric polynomial of degree $m$ over $\mathbb{F}$q in the $k$-variables $\{x_1,x_2,\ldots,x_k\}$ is defined to be $h(x_1,\ldots$, $x_k):=\sum_{e=0}^m a_eh_e(x_1,x_2,\ldots$, $x_k),$ where $a_e\in$ $\mathbb{F}$q and $a_m\not=0$. Let $N_q(h):= \#\{(x_1,\ldots, x_k)\in$ $\mathbb{F}$q |$h(x_1,\ldots, x_k)=0\}$ denote the number of $\mathbb{F}$q-rational points on the affine hypersurface defined by $h(x_1,\ldots, x_k)=0.$ In this paper, we improve the bounds given in [J. Zhang and D. Wan, "Rational points on complete symmetric hypersurfaces over finite fields", Discrete Mathematics, 343(11): 112072, 2020] and [D. Wan and J. Zhang, "Complete symmetric polynomials over finite fields have many rational zeros" Scientia Sinica Mathematica, 51(10): 1677-1684, 2021]. Explicitly, we obtain the following new bounds:
(1) Let $h(x_1,\ldots, x_k)$ be a complete symmetric polynomial in $k\geq 3$ variables over $\mathbb{F}$q of degree $m$ with $1\leq m\leq q-2$. If $q$ is odd, then $N_q(h)\geq\!\frac{\lceil \frac{q-1}{m+1}\rceil}{q-\lceil \frac{q-1}{m+1}\rceil}(q-m-1)q^{k-2}.$
(2) Let $h(x_1,\ldots, x_k)$ be a complete symmetric polynomial in $k\geq 4$ variables over $\mathbb{F}$q of degree $m$ with $1\leq m\leq q-2$. If $q$ is even, then $N_q(h)\geq\!\frac{\lceil \frac{q-1}{m+1}\rceil}{q-\lceil \frac{q-1}{m+1}\rceil}(q-\frac{m+1}{2})(q-1)q^{k-3}.$\newline Note that our new bounds roughly improve the bounds mentioned in the above two papers by the factor $\frac{q^2}{6m}$ for small degree $m$.
• Yan Xun CHANG, Shuang Fei TAN, Jun Ling ZHOU
Acta Mathematica Sinica, Chinese Series. 2023, 66(6): 1019-1030. https://doi.org/10.12386/A20220122
The Chinese mathematician Jiaxi Lu introduced the concepts of LD design and LD* design in the process of resolving the existence problem of large sets of Steiner triple systems (LSTSs). Lu also presented several recursive constructions and direct constructions for the two types of designs, which played vital roles in producing LSTSs. In order to deal with the remaining six possible exceptions for the existence of LSTSs, Teirlinck still employed LD designs in the recursive constructions using pairwise balanced designs and then he resolved the problem of LSTSs completely. This paper proves that the necessary conditions are all sufficient for the existence of LD designs and there leaves only four possible exceptions for LD* designs.
• Zhi-Wei SUN
Acta Mathematica Sinica, Chinese Series. 2024, 67(2): 286-295. https://doi.org/10.12386/A20220195
In this paper we study some determinants and permanents. In particular, we investigate the new-type determinants $$\det [(i^2+cij+dj^2)^{p-2}]_{0≤ i,j≤ p-1}{and}det [(i^2+cij+dj^2)^{p-2}]_{1≤ i,j≤ p-1}$$ modulo an odd prime $p$, where $c$ and $d$ are integers. We also pose some conjectures for further research.
• Wan Xia MA, Mi Xia WU, Guo Wang LUO
Acta Mathematica Sinica, Chinese Series. 2023, 66(6): 1031-1044. https://doi.org/10.12386/A20220063
In this paper, we study the estimation of the partial linear spatial autoregressive model with the response variable missing at random. Based on the nonparametric B-spline method, the marginal maximum Likelihood estimation and the EM algorithms for the maximum Likelihood estimation and the pseudo-restricted maximum Likelihood estimation are proposed, respectively. Numerical simulations are carried out under different sample sizes, missing rates and spatial weight matrix settings to compare the performances of the three methods. Finally, the effectiveness of the three estimation methods are verified by a real data analysis.
• Yi XUAN
Acta Mathematica Sinica, Chinese Series. 2024, 67(3): 428-443. https://doi.org/10.12386/B20220154
We study weighted fractional Sobolev-Poincaré inequalities in irregular domains. The weights considered here are distances to the boundary to certain powers, and the domains are the so-called $s$-John domains and $beta$-Hölder domains. Our main results extend that of Hajlasz-Koskela [J. Lond. Math. Soc., 1998, 58(2): 425-450] from the classical weighted Sobolev-Poincaré inequality to its fractional counter-part and Guo [Chin. Ann. Math., 2017, 38B(3): 839-856] from the fractional Sobolev-Poincaré inequality to its weighted case.
• Wei CAO, Wei Hua LI, Bi Yun XU
Acta Mathematica Sinica, Chinese Series. 2024, 67(4): 624-633. https://doi.org/10.12386/A20220014
Let $\mathbb{F}_{q}$ be the finite field of $q$ elements, and $\mathbb{F}_{q^{n}}$ be its extension of degree $n$. An element $\alpha\in \mathbb{F}_{q^{n}}$ is called a normal element of $\mathbb{F}_{q^{n}}/\mathbb{F}_{q}$ if $\{\alpha,\alpha^{q},\ldots, \alpha^{q^{n-1}}\}$ constitutes a basis of $\mathbb{F}_{q^{n}}/\mathbb{F}_{q}$. Normal elements over finite fields have proved very useful for fast arithmetic computations with potential applications to coding theory and to cryptography. The minimal polynomial of a normal element is certainly an irreducible polynomial with nonzero trace, while the converse does not hold in general. Using linearized polynomials, we give some necessary and sufficient conditions for this problem, which extend the known results.
• Ya Ling WANG, Xu DONG, Chun Na ZENG, Xing Xing WANG
Acta Mathematica Sinica, Chinese Series. 2024, 67(1): 127-136. https://doi.org/10.12386/A20220102
The curvature integral inequalities play an important role in geometric inequalities. In this paper, we first obtain an integral inequality about periodic functions by using the Fourier analysis method. Furthermore, we obtain the strengthened form of the famous Ros inequality on the plane. On the other hand, by applying the obtained lemma, we combine Green-Osher inequality with Steiner polynomial, then the curvature integral inequalities of higher power of planar convex curve are obtained. These inequalities are generalizations and improvements of known Green-Osher inequalities on the Euclidean plane.
• Xin Yi YUAN
Acta Mathematica Sinica, Chinese Series. 2024, 67(2): 227-249. https://doi.org/10.12386/A20220154
In this paper, we explicitly compute the Kodaira-Spencer map over a quaternionic Shimura curve over the field of rational numbers, and also compute its effect on the metrics of the Hodge bundle. The former is based on moduli interpretation and deformation theory, and the latter is based on the theory of complex abelian varieties.
• Jian YANG, Sheng Fan ZHOU
Acta Mathematica Sinica, Chinese Series. 2024, 67(1): 21-44. https://doi.org/10.12386/A20220079
We mainly consider the existence of random uniform exponential attractors in the weighted space of infinite sequences for second order lattice systems with quasi-periodic forces and multiplicative white noise. We first present some sufficient conditions for the existence of a random uniform exponential attractor for a jointly continuous random dynamical system defined on a product space of weighted space of infinite sequences. Secondly, by using Ornstein-Uhlenbeck process, a reversible variable substitution is constructed to transform the stochastic second-order lattice system (SDE) with white noise into a random system (RDE) without white noise, whose solutions generate a jointly continuous random dynamical system. Then we verify the Lipschitz continuity of the jointly continuous random dynamical system and decompose the difference between the two solutions of system into a sum of the two parts, and estimate the expectations of some random variables. Finally, we obtain the existence of random uniform exponential attractors for the considered system.
• Zi Ling HENG, De Xiang LI, Xiao WANG
Acta Mathematica Sinica, Chinese Series. 2024, 67(1): 195-208. https://doi.org/10.12386/A20210122
Projective codes over finite fields have important applications in combinatorial designs and strongly regular graphs. In this paper, we first construct a family of linear codes and then study their parameters and weight distributions in four cases. It turns out that the proposed linear codes are projective and are optimal in two cases. The duals of these codes are either optimal or almost optimal according to the sphere-packing bound. As applications, these codes are used to construct $t$-designs and strongly regular graphs.
• Ya Wen LI, Jin Hua QIAN
Acta Mathematica Sinica, Chinese Series. 2023, 66(6): 1045-1056. https://doi.org/10.12386/A20220058
In this article, the concept of lightlike growth surface is proposed by evolving a lightlike curve as dictated direction and growth velocity in Minkowski 3-space. The geometric structure of the lightlike growth surfaces are investigated by the aid of the structure function of its generating lightlike curve. Meanwhile, the expression forms of the lightlike growth surfaces initiated by the lightlike helices are explored accompanied with several typical examples to characterize the growth process of such kind of surfaces explicitly.
• Yong Gao CHEN, Rui Jing WANG
Acta Mathematica Sinica, Chinese Series. 2024, 67(2): 259-272. https://doi.org/10.12386/A20220173
We prove that there is a positive proportion of positive integers which can be uniquely represented as the sum of a Fibonacci number and a prime. We also study the integers of the form $p+a_k$, where $p$ is a prime and $\{ a_k\}$ is an exponential type sequence of integers.
• Yi Feng LIU
Acta Mathematica Sinica, Chinese Series. 2024, 67(2): 273-285. https://doi.org/10.12386/A20220177
In this note, we confirm a conjecture on the existence of test functions for trilinear zeta integrals with regular support, for representations with maximal exponent strictly less than 1/22.
• Yan Hui ZHANG, Tao QIAN
Acta Mathematica Sinica, Chinese Series. 2023, 66(5): 835-844. https://doi.org/10.12386/b20220026
In this paper, using complex analysis methods, we will show that an $h^p$ harmonic function $u$ can be decomposed into the sum of one singular function and one absolutely continuously function on unit ball $B$ of $\mathbb{R}^n$ for $p\geq 1.$ Then we will obtain the corresponding results of functions in $h^p$ space of the upper half space of $\mathbb{R}^n$ by the Kelvin transform.
• He Ying WANG, Rui LIU, Qi Yao BAO
Acta Mathematica Sinica, Chinese Series. 2023, 66(5): 827-834. https://doi.org/10.12386/A20220008
Quantum detection investigates the injectivity of quantum measurements on quantum states. Since every measurement of quantum system can be characterized by a positive operator-valued measure (POVM), and every Parseval frame corresponds to a rank-one POVM. In this paper, we mainly consider the quantum injectivity problem of Gabor frames, and give a sufficient condition for the quantum injectivity of a Gabor frame $\left\{\pi (m,n) \varphi \right\}_{(m,n) \in \Lambda}$, namely it is a full Gabor frame and satisfies $\langle \pi (m,n) \varphi,\varphi \rangle \ne 0$ for $m=0,\ 1 \le n \le \frac{N}{2}$, $1 \le m \le \frac{N-1}{2},\ 0 \le n \le N-1$ and $\frac{N-1}{2} < m \le \frac{N}{2},\ 0 \le n \le \frac{N}{2}$. We also give its stability with a quantitative error estimate and its applications for low dimensional cases.
• Xiao Fei SUN, Kang Ning WANG, Lu LIN
Acta Mathematica Sinica, Chinese Series. 2023, 66(5): 899-916. https://doi.org/10.12386/A20220037
Composite quantile regression has good properties in robustness and estimation efficiency. For the longitudinal data single-index models, we propose profile composite quantile regression based estimating equations and smooth-threshold variable selection methods. The new methods can incorporate the intra-subject correlation by using copula functions, and inherit the advantages of composite quantile regression. Under some mild conditions, the asymptotical properties are established. Simulation studies and real data analysis are included to illustrate the finite sample performance.
• Yun Chuan YIN, Xiao Dan CAO
Acta Mathematica Sinica, Chinese Series. 2024, 67(1): 173-186. https://doi.org/10.12386/A20210180
We further develop the theory of $W$-graph ideals in a Coxeter system $(W,S)$. We mainly study the structural coefficients of the corresponding modules, the direct and iterative algorithms for the canonical basis elements. Compared with standard recursive algorithms, this algorithm has the advantage of fast computation and memory saving when computing specific canonical basis elements. Due to the generality of the concept of $W$-graph ideal, our results are also the generalizations of those in some classical cases.
• Zhao Han LIU, Li Ming TANG
Acta Mathematica Sinica, Chinese Series. 2023, 66(6): 1111-1120. https://doi.org/10.12386/A20220053
In this paper, we first introduce the concept of primitive Lie superalgebras and investigation three types of primitive Lie superalgebra and some related structural properties. Then, the concept of chief factors of Lie superalgebra is introduced. According to the properties of the third type of primitive Lie superalgebra, the L connection relation between the chief factors of Lie superalgebra is given. Finally, we introduce the CAP-subalgebras of Lie superalgebra L, and prove that L is solvable if all maximal subalgebras of Lie superalgebra are CAP-subalgebras.
• Min Feng CHEN, Zong Xuan CHEN
Acta Mathematica Sinica, Chinese Series. 2023, 66(6): 1205-1220. https://doi.org/10.12386/A20220093
The main purpose of this paper is to give the expressions of meromorphic solutions of the following non-linear differential equation \begin{equation*}f^{n}(z)+P_{d}(z,f)=p_{1}{\rm e}^{\alpha_{1}z}+p_{2}{e}^{\alpha_{2}z}+p_{3}{\rm e}^{\alpha_{3}z}\end{equation*} under certain conditions, where n ≥ 3 is an integer, $P_{d}(z,f)\not\equiv0$ is a differential polynomial in f of degree dn - 1 with small function coefficients, pj (j = 1, 2, 3) are non-zero constants, αj (j = 1, 2, 3) are three distinct non-zero constants. Moreover, some examples are given to illustrate our results.
• Jian Ya LIU, Ting Ting WEN, Jie WU
Acta Mathematica Sinica, Chinese Series. 2024, 67(2): 347-356. https://doi.org/10.12386/A20230032
Manin's conjecture predicts the quantitative behaviour of rational points on algebraic varieties. For a primitive positive definite quadratic form $Q$ with integer coefficients, the equation $x^3=Q(\boldsymbol{y})z$ represents a class of singular cubic hypersurfaces. In this paper, we introduce Manin's conjecture for these hypersurfaces, and describe the ideas, methods, and related results. Generalizations are treated in the last section.
• Ran Ran ZHANG, Zhi Bo HUANG, Chuang Xin CHEN
Acta Mathematica Sinica, Chinese Series. 2023, 66(5): 855-866. https://doi.org/10.12386/A20220020
We consider the uniqueness of the meromorphic solution $f(z)$ of the second order linear difference equation $p_2(z)y(z+2)+p_1(z)y(z+1)+p_0(z)y(z)=0,$ where $p_2(z), p_1(z), p_0(z)$ are nonzero polynomials with $p_2(z)+p_1(z)+p_0(z)\not\equiv0$. We give the forms of $f(z)$ if $f(z)$ shares $0, 1, \infty$ CM with any meromorphic function $g(z)$. Furthermore, if $g(z)$ is also a solution of the above equation, we obtain the exact forms of this equation. As a corollary, we see that if a meromorphic function $g(z)$ shares $0, 1, \infty$ CM with the gamma function $\Gamma(z)$, then $g(z)\equiv \Gamma (z)$.
• Yong Xin BAI, Man Ling QIAN, Mao Zai TIAN
Acta Mathematica Sinica, Chinese Series. 2024, 67(3): 444-467. https://doi.org/10.12386/A20220026
We propose an effective iterative screening method for the ultra-high dimensional additive quantile regression with missing data. Specifically, the canonical correlation analysis is introduced into the maximum correlation coefficient based on the optimal transformation, and the marginal contribution of important variables is sorted by the maximum correlation coefficient after the optimal transformation of covariates and model residuals. On the basis of variable screening, the sparse smooth penalty is used to make further variable selection. The proposed variable selection method has three advantages: (1) The maximum correlation based on optimal transformation can reflect the nonlinear dependent structure of response variable to covariable more comprehensively; (2) In the iteration process, the residual can be used to obtain the relevant information of the model so as to improve the accuracy of variable screening; (3) The variable screening process can be separated from model estimation to avoid regression of redundant covariables. Under appropriate conditions, the sure independent screening property of the variable screening method and the sparsity and consistency of the estimator under the sparse-smooth penalty are proved. Finally, the performance of the proposed method is given by Monte Carlo simulation and the rat genome data is used to illustrate the effectiveness of the proposed method.
• Xin GUAN, Jin Hong YOU, Yong ZHOU, Guo Ying XU
Acta Mathematica Sinica, Chinese Series. 2024, 67(1): 45-71. https://doi.org/10.12386/A20220066
This paper studies a novel dynamic single index varying coefficient quantile regression model, which reflects the dynamic interaction between explanatory variables and the response variable, and covers many important models as special cases. In order to improve the interpretability and estimation accuracy, this paper further discusses the semi-varying structure of the model. Firstly, we use the B-spline method to obtain the estimators of the varying coefficient function and the index function. Secondly, the semi-varying model is identified based on the penalty function method. We also propose an estimation procedure for this semi-parametric model. In addition, We establish the consistency and asymptotic normality of each estimator, and both parametric and non-parametric estimators can achieve the optimal convergence rate. Numerical simulations show that the proposed models and estimation methods enjoy excellent properties. Finally, we analyze a NO$_2$ data set to demonstrate the performance of the proposed method in practical applications.
• Hou Rong QIN
Acta Mathematica Sinica, Chinese Series. 2024, 67(2): 341-346. https://doi.org/10.12386/A20230028
We give an introduction to the Vandiver conjecture and some related research in the literature. We show that $A_0=A_2=\cdots=A_{32}=0$, where $A$ is the $p$-Sylow subgroup of the ideal class group of $\mathbb{Q}(\zeta_{p})$. Finally, we propose a new conjecture on the distribution of irregular primes with numerical verifications.
• Yong Quan HU
Acta Mathematica Sinica, Chinese Series. 2024, 67(2): 377-392. https://doi.org/10.12386/A20230173
This paper is a survey on mod $p$ Langlands program, with a focus on the history of development and some recent progress in the case of $GL_2$.
• Yi Chao TIAN
Acta Mathematica Sinica, Chinese Series. 2024, 67(2): 357-376. https://doi.org/10.12386/A20230162
This article is a survey on some recent developement of the prismatic cohomology theory. We will start with some motivation from classical p-adic Hodge theory, and discuss the origine of the prismatic cohomolgy theory and its basic results. We will then put emphasis on the notion of prismatic crystals, their cohomological properties, and the relationship with the cohomology of classical crystalline crystals.
• Yue Lu ZHANG, Gang CAI
Acta Mathematica Sinica, Chinese Series. 2024, 67(3): 599-610. https://doi.org/10.12386/A20230043
This paper introduces a Bregman extragradient method and applies it to solve pseudo-monotone variational inequality problems in Hilbert spaces. Under some reasonable assumptions imposed on the parameters, a weak convergence theorem for the suggested method is achieved. The results obtained in this paper generalize and improve many recent ones in the literature.
• Shao Tao HU, Yuan Heng WANG, Gang CAI
Acta Mathematica Sinica, Chinese Series. 2023, 66(5): 845-854. https://doi.org/10.12386/A20220013
We introduce a new Tseng’s extragradient algorithm for solving pseudomonotone variational inequality problems in Hilbert spaces. We prove that the sequence generated by our proposed algorithm converges strongly to an element of solution set for variational inequality problems. The results obtained in this paper extend and improve many recent ones in the literature.
• Li Yan XI, Quan Wu MU
Acta Mathematica Sinica, Chinese Series. 2024, 67(1): 187-194. https://doi.org/10.12386/A20210134
Let $k\in \{5, 6\}$ and $\eta$ be any given real number. Suppose that $\lambda_1, \lambda_2, \ldots, \lambda_7$ are nonzero real numbers, not all of the same sign and $\lambda_1/\lambda_2$ is irrational. It is proved that the inequality $|\lambda_1x_1^2+\lambda_2x_2^3+\lambda_3x_3^3+\lambda_4x_4^3+\lambda_5x_5^3+\lambda_6x_6^4+\lambda_7x_7^k+\eta|<(\max_{1\leq j\leq 7} x_j)^{-\sigma}$ has infinitely many solutions in positive integers $x_1, x_2, \ldots, x_7$ for $0<\sigma<\frac{1}{12(k-3)}$. This result constitutes an improvement upon that of Li and Gong.
• Hai Wei SUN, Yang Bo YE
Acta Mathematica Sinica, Chinese Series. 2024, 67(2): 406-412. https://doi.org/10.12386/A20230025
In this paper, aggregate zero density bounds for a family of automorphic $\mathrm{L}$-functions are deduced from bounds for a sum of integral power moments of such $\mathrm{L}$-functions. More precisely, let $I$ be a set of certain automorphic representations $\pi$, and let $c(\pi)$ be a non-negative coefficient for each $\pi\in I$ such that $\sum_{\pi\in I}c(\pi)$ converges. Assume that \begin{equation*} \sum_{\pi\in I} c(\pi) \int_T^{T+T^\alpha} \bigg| \mathrm{L}\bigg(\frac12+{\rm i}t,\pi\bigg) \bigg|^{2\ell} dt \ll_\varepsilon T^{\theta+\varepsilon} \sum_{\pi\in I} c(\pi) \end{equation*} for certain $\ell\geq1$, $0<\alpha\leq1$ and $\theta\geq\alpha$. Upper bounds for the following aggregate zero density \begin{equation*} \sum_{\pi\in I} c(\pi) N_\pi(\sigma,T,T+T^\alpha) \end{equation*} will be proved, where $N_\pi(\sigma,T_1,T_2)$ is the number of zeros $\rho=\beta+{\rm i}\gamma$ of $\mathrm{L}(s,\pi)$ in $\sigma<\beta<1$ and $T_1\leq\gamma\leq T_2$.
• Da Xin XU
Acta Mathematica Sinica, Chinese Series. 2024, 67(2): 250-258. https://doi.org/10.12386/A20230001
Faltings proposed a $p$-adic analogue of Simpson's correspondence between Higgs bundles on projective complex manifolds and finite dimensional $\mathbb{C}$-representation of the fundamental group. In this paper, we will give an overview of this work and recent progress on finite dimensional $p$-adic representations of the fundamental group of a $p$-adic curve. In the last section, we will briefly discuss some related works.
Quantum toroidal algebras or double affine quantum algebras are special cases ($N=2$) of the recently defined quantum $N$-toroidal algebras, which generalize the toroidal Lie algebras and $N$-toroidal Lie algebras. In this paper, we will construct a level one representation of the quantum $N$-toroidal algebra for the exceptional type $G_2$, which can be regarded as a generalization of the basic representation of the quantum affine algebra in type $G_2$.