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.

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.

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 $A^2_{\varphi}(\mathcal{H})$.

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 investigate a class of domains $\Omega^{n+1}_k =\{(z,w)\in \mathbb{C}^n\times \mathbb{C}: |z|^k < |w| < 1\}$ for $k \in \mathbb{Z}^+$ which generalizes the Hartogs triangle. We first obtain the new explicit formulas for the Bergman kernel function on these domains and further give a range of $p$ values for which the $L^p$ boundedness of the Bergman projection holds. This range of $p$ is shown to be sharp.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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 present a method to study isochronous centers in 3-dimensional polynomial differential systems. Firstly, the isochronous constants of the three dimensional system are defined and recursive formulas to obtain them are given. The conditions for the isochronicity of a center are determined by the computation of isochronous constants for which there is no need to compute center manifolds of the three dimensional systems. Then the isochronous center conditions of two specific systems are discussed as an application of our method. Our method is a generalization of the formal series method proposed by Yirong Liu for determining the order of a fine focus of planar differential systems. This method with the recursive formulas can be easily implemented on a computer using a computer algebra system.

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 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 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 Lorenz-type maps are piecewise expanding maps with discontinuous points, the discontinuity comes from the singularities of Lorenz equations showing butterfly effect, and the observable statistical properties of such maps are given by the absolutely continuous invariant measures. In this paper, we consider the perturbation $f_t=f+tX\circ f$ of an improved Lorenz-type map $f $, and denote by $\mu_t$ the perturbation of the corresponding absolutely continuous measure $\mu$. We prove that if $X $ takes zero on all image sets of the discontinuous point of $f $, then its sensitivity formula $$\Psi(\lambda)=\sum\limits_{n=0}^\infty \lambda^n \int \mu(dx)X(x)\dfrac{\partial(\varphi(f^nx))}{\partial x},\quad\varphi\in C^1, $$ converges at $ \lambda = 1 $, thus the linear response formula $\frac{d}{dt}|_{t=0}\mu_t(\varphi)=\Psi(1)$ is established.

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.

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.

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.

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.

In this paper, we study a generalized comparison question about singular metrics of twisted pluricanonical bundles over complex varieties with canonical singularities. We prove that for an algebraic fibration over the unit disc, the restriction of the twisted relative $m$-Bergman metric on a singular fiber with canonical singularities at worst coincides with the intrinsic twisted $m$-Bergman metric on itself, provided that the singular metric on the twisted line bundle has slope zero at each point belonging to the singular fiber.

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.

In this paper, we consider the persistence of invariant tori in the following system \begin{equation*} \begin{array}{ll} \left\{\begin{array}{ll} \dot{x}=\omega+y+f(x,y),\\[0.1cm] \dot{y}=g(x,y), \end{array}\right. \end{array} \end{equation*} where $x\in \mathbb{T}^\Lambda$, $y\in\mathbb{R}^\Lambda$, $\Lambda$ is a countable subset of $\mathbb{Z}$, ${\omega}=(\ldots,{\omega}_\lambda,\ldots)_{\lambda\in \Lambda}\in\mathbb{R}^\Lambda$ is the frequency vector, ${\omega}=(\ldots,{\omega}_\lambda,\ldots)_{\lambda\in \Lambda}$ is a bilateral infinite sequence of rationally independent frequency, in other words, any finite segments of ${\omega}=(\ldots,{\omega}_\lambda,\ldots)_{\lambda\in \Lambda}$ are rationally independent, and the perturbations $f, g$ are real analytic functions. We also assume that the above system is reversible with respect to the involution $\mathcal{M }: (x,y) \mapsto (-x,y)$. By the KAM method, we prove the persistence of invariant tori for the above reversible system.

In this paper, we study the computational problems of one kind congruent equation modulo $p$, and give some exact computational formulae for them.

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.

Linear regression models are often used to study the relationship between variables in various fields of scientific research, such as medicine, genetics, economics. However, main effects may not be sufficient to characterize the relationship between the response and predictors in complex situations, the interaction effects between variables will also have an important influence on the response variable in many practical problems. Interaction model that considers both the main effect and the interaction effect can describe the relationship between variables more comprehensively. For high-dimensional data, the number of variables $p$ is relatively large, and the number of second-order interaction terms $\frac{p(p+1)}{2}$ is much larger, the statistical analysis of the interaction model faces many difficulties and challenges. How to select the important interaction effects that have a significant impact on the event of interest from huge number of interaction effects is a very important problem. The existing research on this problem mainly focuses on the complete data under the framework of the linear model. In this paper, we will consider this problem for ultrahigh-dimensional right-censored survival data. Based on distance correlation and the two-step analysis method, we propose a model-free screening method for interaction effects which does not depend on any model assumptions. This method can select the important main effects and important interaction effects at the same time, and can handle ultrahigh-dimensional data with large $p$. Extensive simulation studies are carried out to evaluate the finite sample performance of the proposed procedure, and the results show that this method can effectively select the important interaction effects for ultrahigh-dimensional right-censored survival data. As an illustration, we apply the proposed method to analyze the diffuse large-B-cell lymphoma (DLBCL) data.

In this paper, we investigate the $\ell$-convex Legendre curves on the plane, which are the natural generalization of strictly convex curves. On the one hand, by using the method of Green and Osher, we obtain the necessary and sufficient condition for the Green-Osher inequality's equality of $\ell$-convex Legendre curves, that is, when $F(x)$ is a strictly convex function in $\mathbb{R}$, the equality holds if and only if the curve $\gamma$ is a circle. On the other hand, we obtain a series of curvature integral inequalities of $\ell$-convex Legendre curves. In particular, when $\gamma$ is a strictly convex curve, the corollaries obtained are the improved forms of Ros inequality, Green-Osher inequality and Gage isoperimetric inequality.

In this paper, a nonnegative projection correlation coefficient (NPCC) is proposed to measure the dependence between two random vectors, where the projection direction comes from the standard multivariate normal distribution. The NPCC is nonnegative and is zero if and only if the two random vectors are independent. Also, its estimation is free of tuning parameters and does not require any moment conditions on the random vectors. Based on the NPCC, we further propose a novel feature screening procedure for ultrahigh dimensional data, which is robust, model-free and enjoys both sure screening and rank consistency properties under weak assumptions. Monte Carlo simulation studies indicate that the NPCC-based screening procedure have strong competitive advantages over the existing methods. Lastly, we also use a real data example to illustrate the application of the proposed procedure.

We demonstrate the importance of duality and traces in symmetric monoidal categories through a series of concrete examples. In particular, we give an introduction to a new application in ′etale cohomology: the characterization of universal local acyclicity and the relative Lefschetz-Verdier trace formula.

The $q$-deformed $W(2,2)$ algebra is a Hom-Lie algebra, denoted by $\ W ^ q$. In this paper, we compute its second cohomology with values in the adjoint module by elementary and direct calculations, and obtain that the second cohomology group $H^2(\ W ^q,\ W ^ q)$ is two-dimensional.

Bayesian networks utilize directed acyclic graphs (DAGs) to constrain conditional independencies in multivariate joint probability distribution, so as to realize its modular decomposition in uncertainty reasoning and reduce the computational complexity of probabilistic reasoning. They are widely used in probabilistic reasoning, machine learning and causal inference. In practice, if structure learning or statistical inference was performed by adopting the idea of dividing and conquering or model collapsing, we have to establish the marginal models by finding their minimal Markov subgraphs (or minimal independence maps). Therefore, this paper details minimal Markov subgraphs for marginal models of Bayesian networks, and provides the refined characterization on them from the perspectives of statistics and graph theory. For the collapsibility of DAG, this paper gives more intuitive equivalent conditions based on the properties of directed inducing paths, and also proposes some sufficient conditions, which provides more theoretical tools for judging whether the considered models can be collapsible onto local sub-models.

We study Toeplitz operators with positive symbols on $A_2$ weighted harmonic Bergman spaces in bounded smooth domain of $n$-dimensional real space. We characterize sufficient and necessary conditions for bounded or compact Toeplitz operators via average function and Berezin transform. The Schatten class Toeplitz operators are obtained as well.