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.

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.

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.

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.

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 $m$.

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.

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.

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.

We study some properties of Toeplitz operators with positive operator-valued function symbols on the vector-valued exponential weighted Bergman spaces $A^p_{\varphi}(\mathcal{H})\ (1 ＜ p ＜ \infty)$. Firstly, we discuss when the Bergman projection from $L^p_{\varphi}(\mathcal{H})$ onto $A^p_{\varphi}(\mathcal{H})$ is bounded and get the dual of the vector-valued exponential weighted Bergman spaces. Secondly, we obtain several equivalent descriptions of Carleson condition to characterize the boundedness and compactness of Toeplitz operators on $A^p_{\varphi}(\mathcal{H})$. Finally, we consider the Schatten-$p$ class membership of Toeplitz operators acting on $A^2_{\varphi}(\mathcal{H})$.

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.

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.

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.

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.

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.

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.

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.

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.

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.

The main purpose of this paper is using the elementary methods, the number of the solutions of some congruence equations and the properties of the classical Gauss sums to study the calculating problem of the fifth power mean of one kind two-term exponential sums, and give the exact calculating formula for it.

In this paper, we investigate the reciprocal sums of the cubes of odd and even terms in the Fibonacci sequence and we obtain two interesting identities for the Fibonacci numbers.

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.

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.

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.

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.

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 d ≤ n - 1 with small function coefficients, p_j (j = 1, 2, 3) are non-zero constants, α_j (j = 1, 2, 3) are three distinct non-zero constants. Moreover, some examples are given to illustrate our results.

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.

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 $GL_2$.

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.

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 $T_1\leq\gamma\leq T_2$.

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.

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.

A hole is an induced cycle of length at least 4, a hole of odd length (resp. even length) is called an odd hole (resp. even hole). An HVN is a graph composed by a vertex adjacent to both ends of an edge in $K_4$. Let $H$ be the complement of a cycle on 7 vertices. Chudnovsky et al. in [J. Combin. Theory B, 2010, 100: 313—331] proved that every (odd hole, $K_4$)-free graph is 4-colorable and is 3-colorable if it does not contain $H$ as an induced subgraph. In this paper, we use the idea and proving technique of Chudnovsky et al. to generalize this conclusion to $($odd hole, HVN$)$-free graphs. Let $G$ be an $($odd hole, HVN$)$-free graph. We prove that if $G$ contains $H$ as an induced subgraph, then it either has a special cutset or is in two classes of pre-defined graphs. As its corollary, we show that $\chi(G)\le \omega(G)+1$, and the equality holds if and only if $\omega(G)=3$ and $G$ has $H$ as an induced subgraph.

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.

We study a new algorithm to solve a common solution of the split feasibility problem and the fixed point problem involving quasi-nonexpansive mappings in Hilbert spaces. Based on the common solutions of these two classes of problems, we solve the variational inequality problem. Compared with the predecessors, the self-adaptive technique and the inertial iteration method are added, which can speed up the convergence rate of the iterative sequence generated by our algorithms. At the same time, we extend the involving previous nonexpansive mappings to extensive quasi-nonexpansive mappings. In addition, a strong positive bounded operator is added to the algorithm, which extends the original viscous iterative algorithm to a more general viscous iterative algorithm. The effectiveness of the algorithm is verified by numerical examples.

Let $\rho$ be an orthogonal representation on a Euclidean space $V$, and $SV$ be the unit sphere of $V$. Let $\bar{d}_{\mathcal{H}}$ and $d_{\mathcal{H}}$ be the horizontal metrics on $V$ and $SV$ induced by $\rho$, respectively. Our main result is to show that the following conditions are equivalent: (1) The representation $\rho$ is polar. (2) $(V, \bar{d}_{\mathcal{H}})$ is a CAT$(0)$ space. (3) $(SV, d_{\mathcal{H}})$ is a CAT$(1)$ space.

Based on the complete classification of the torsion subgroup by Mazur, and results of the related diophantine equation, we determine all elliptic curves defined over $\mathbb{Q}$ with a rational point of the order $n\ (n \geq 6, n\neq 11)$ and the conductor $p^{a}q^{b}r^{c}$, where $p, q, r$ are distinct primes, and $a, b, c$ are positive integers. In particular, an upper bound of the minimal discriminant for these elliptic curves are given.

We investigate the asymptotic approximation of solutions of initial and boundary value problems for set-valued differential equations with maxima. By introducing the concepts of Hausdorff measure and semideviation measure, the approximate relationship of the solutions between the original equations and the average equations is discussed via the averaging method when the average limit of the right-hand function exists or does not exist.

We extend the definition of central strong approximation with Brauer- Manin obstruction which is valid for all singular varieties. We show that a variety defined by a polynomial represented by an isotropic binary quadratic form satisfies central strong approximation with Brauer-Manin obstruction by explicit blowing-up. This is the last case of the whole generalization of Watson’s results about Diophantine equations reducible to quadratics.

This paper is concerned with the existence and time-asymptotic nonlinear stability of traveling wave solutions to the Cauchy problem of the one-dimensional compressible planar magnetohydrodynamics system, which governs the motions of a conducting fluid in an electro-magnetic field. Motivated by the relationship between planar magnetohydrodynamics system and Navier—Stokes system, we can prove that the solutions to the compressible planar magnetohydrodynamics system tend time-asymptotically to the traveling wave, provided that the initial disturbance is small and of integral zero.