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.

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.

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.

We study the stability of the inverse transmission eigenvalue problem for the Schrödinger operator with the Neumann boundary condition. When $\int_0^1q(t)dt=0$ and $q(1)\neq 0$, there are infinitely many real eigenvalues. In case, by using the theory of transformation operators and the properties of Riesz basis, we give the estimates for the difference of two potentials in the sense of the weak form and $W_2^1$-norm, according to the difference of two corresponding spectral data, which imply the stability of the inverse spectral problem.

Let $Y$ be a real Banach space, $\varepsilon\geq0$, and let $f:c_0\rightarrow Y$ be a standard $\varepsilon$-norm-additive mapping, i.e., $f(0)=0$ and $ \big|\|f(x)+f(y)\|-\|x+y\|\big|\leq\varepsilon,\;\forall\, x,y\in c_0. $ In this paper, we show that if $f$ is $\delta$-surjective for some $\delta>0$, then there exists a linear surjective isometry $U:c_0\rightarrow Y$ such that $ \|f(x)-U(x)\|\leq\frac{3}{2}\varepsilon,\;\forall\, x\in c_0. $ The constant $\frac{3}{2}$ is optimal.

A Hom-structure on a Lie algebra $(L, [\cdot])$ is a linear map $\varphi: L\to L$ which satisfies the Hom-Jacobi identity $[[x,y],\varphi(z)]+[[z,x], \varphi(y)] +[[y,z], \varphi(x)]=0$ for any $x,y,z\in L.$ A Hom-structure is called regular (respectively, a derivation double Lie algebra) if $\varphi$ is also a Lie algebra isomorphism (respectively, derivation). The $n$-th Schrödinger algebra is the semi-direct product of the simple Lie algebra $\mathfrak{s l}_{2}$ with the $n$-th Heisenberg Lie algebra $\mathfrak{h}_{n}$. In this paper, we prove that any Hom- Lie algebra structure is a sum of a scalar multiplication and a central Hom-structure. Furthermore, any regular Hom-structure is an identity mapping, and any derivation double Lie algebra is a zero mapping.

We present a generalization of a technical lemma due to Boccardo and Orsina, and then give an application to regularity of minima for integral functionals noncoercive in the energy space.

We study the split feasibility problem with multiple output sets in uniformly convex Banach spaces. By transforming it into a fixed point problem, we construct parallel and cyclic iterative methods to solve the problem respectively, and prove their strong convergence under weak geometric conditions.

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.

Let $\alpha_{2}(D) = \max\{|X|: X \subseteq V(D)$ and $D[X]$ has no 2-cycle$\}$ be the $\alpha_{2}(D)$-stable number of a digraph $D$. In [$Proc. London Math. Soc.$, 42 (1981) 231-251], Thomassen constructed non-hamiltonian digraphs $D$ with $\kappa(D) = \alpha(D)$ to show that the well-known Chvátal-Erdös theorem does not have obvious extension to digraphs. Bang-Jensen and Thomassé conjectured that every digraph with arc strong-connectivity at least its stable number must have a spanning closed trail. The problem also remains unanswered whether a digraph with its arc strong-connectivity at least its $\alpha_{2}(D)$-stable number has spanning trails or not. A digraph $D$ is weakly trail-connected if for any two vertices $x$ and $y$ of $D$, $D$ admits a spanning $(x,y)$-trail or a spanning $(y,x)$-trail, and is strongly trail-connected if for any two vertices $x$ and $y$ of $D$, $D$ contains both a spanning $(x,y)$-trail and a spanning $(y,x)$-trail. We determine two well-characterized families of strong digraphs $ \mathcal{M}$ and $\mathcal{H}$, and prove each of the following for any strong digraph $D$ with $\alpha_{2}(D) =2$: (i) $D$ is hamiltonian if and only if $D\not\in \mathcal{M}$. (ii) $D$ is weakly trail-connected. (iii) $D$ is strongly trail-connected if and only if $D\not\in\mathcal{H}$. In particular, every strong digraph $D$ with $\alpha_{2}(D) =2$ has a hamiltonian path and every 2-strong digraph $D$ with $\alpha_{2}(D) =2$ is strongly trail-connected.

The theory of tensor products of Hilbert spaces is the mathematical framework of multipartite quantum systems. The lossless decoupling of quantum entanglement in multipartite systems is one of the hot topics of current quantum information and computing theory. One always wants to know under what conditions multipartite entanglement can be losslessly decoupled. Existed known lossless decoupling conditions are obtained in the case of three-qubit systems. In the paper, we devote to extending the above entanglement lossless decoupling conclusions into more general tripartite systems. We prove that for some $2\otimes2\otimes3$ and $2\otimes2\otimes4$ pure states, the lossless decoupling can be realized.

In this paper, we study the generalized integral composition operators between different Fock spaces $F^p$ and $F^q$ with $0<p, q\leq \infty$. We first characterize bounded and compact differences of two generalized integral composition operators, and then estimate the essential norm of the difference. In addition, we study the path connected subsets of the space of the generalized integral composition operators.

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.

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.

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.

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 $m$.

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.

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.

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 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 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.

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 Demazure product gives a natural monoid structure on any Coxeter group. Such structure occurs naturally in many different areas in Lie Theory. This paper studies the Demazure product of an extended affine Weyl group. The main discovery is a close connection between the Demazure product of an extended affine Weyl group and the quantum Bruhat graph of the finite Weyl group. As applications, we obtain explicit formulas on the generic Newton points and the Demazure products of elements in the lowest two-sided cell, and obtain an explicit formula on the LusztigVogan map from the coweight lattice to the set of dominant coweights.

We survey some recent developments of the restriction problems for real untiary group. In particular we briefly explain a proof of the local Gan-Gross-Prasad conjecture for real unitary groups.

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.

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.

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.

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 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.

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 $T_1\leq\gamma\leq T_2$.

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.

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.

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.

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.

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.

This paper establishes the existence and uniqueness results for solutions to multidimensional backward stochastic differential equation ($G$-BSDE) driven by $G$-Brownian motion, in which the generators $f$ and $g$ of $G$-BSDE satisfy the $\beta$-order Mao's condition in $y$ and the Lipschitz conditon in $z$.

In the paper, Viterbi algorithms for hidden Markov models are studied. When partial states, initial probability distributions, transition probability matrices and observation probability matrices are given, the optimal state sequences are estimated by the Viterbi algorithms. Compared with existing algorithms, the algorithms presented in the paper have not only considered the influence of partially visible states on the initial conditions and recursion formulas, but also ensured that the predicted state sequences are overall optimal. Finally, fault recognition is investigated to verify the feasibility of the algorithms.

We introduce a modified iterative algorithm for solving the split common fixed point problem for multi-valued demicontractive mapping in real Hilbert spaces. Under very mild conditions, we prove a strong convergence theorem for our algorithm by using inertial and hybrid projection technique. Further, we give the application and example of the algorithm to illustrate the effectiveness of the algorithm. Our result improves and extends the corresponding ones announced by some others in the earlier and recent literature.