We first study the basic coupling and obtain an equation between total variation norm and the basic coupling. Then by use this equation we investigate the ergodicity property of continuous time Markov processes in general state space. For an ergodic continuous-time Markov processes, adding condition $\pi(f)<\infty$, by using the coupling method, there exists the full absorption set, such that the continuous time Markov processes are $f$-ergodic on it.

The relationship between a fractional diffusion process and its integration by parts formula is studied. By constructing a pull back formula, the integration by parts formula for fractional diffusion process is established. Conversely, a fractional diffusion process can be characterized through its integration by parts formula.

In this paper, using large deviations for a Brownian sheet and increments of a Brownian sheet, we obtain local functional law of the iterated logarithm for a Brownian sheet and increments of a Brownian sheet.

Spectral collocation method is investigated for the nonlinear Caputo fractional multi-point value problems. The main idea of the presented method is to solve the corresponding nonlinear weakly singular Volterra-Fredholm integral equations obtained from the nonlinear Caputo fractional multi-point value problems. In order to carry out convergence analysis for the presented method, we investigate the Gronwall type inequality with Volterra-Fredholm integral terms. The provided convergence analysis shows that the presented method has spectral convergence, which is confirmed by the provided numerical experiments. At present, numerical methods for fractional multi-point value problems are rarely studied. The method and convergence analysis in this paper are useful references for the researches of related subjects.

Let $\mathbb{C}_v$ be an algebraically closed non-archimedean field, complete with respect to a valuation $v$. Let $\varphi:\mathbb{P}^N\rightarrow \mathbb{P}^N$ be a morphism of degree greater than one defined over $\mathbb{C}_v$, $\Phi$ a lift of $\varphi$. Let $\mathcal{G}_\Phi$ be the Green function of $\Phi$ and $\rho$ the chordal metric on $\mathbb{P}^N(\mathbb{C}_v)$. In this paper, we first study the properties of reduction of points in high dimensional projective space and reduction of automorphisms of $\mathbb{P}^N$ with degree one. With the help of Green function $\mathcal{G}_\Phi$ of $\Phi$, we introduce the arithmetic distance of morphisms and investigate its property. The necessary and sufficient condition which $\Phi$ has good reduction is obtained in this paper. We also describe explicitly the Filled Julia set of $\Phi$ by its Green function.

In this paper, we study the transmission eigenvalue problem with the Robin boundary condition. Applying the related properties of entire function of exponential type, we show the relationship between the density of eigenvalues and the length of the support interval of the potential function. Meanwhile, we prove that the transmission eigenvalue problem is equivalent to a kind of Sturm-Liouville problem with spectral parameter in the boundary condition.

In this paper, I consider that the actuarial model is affected by the environmental process $\Theta$, premium income counting process $\eta$, claim counting process I and the claim process B, and establish a compound binomial risk model with random income in Markov chain environment, which is called MRICM, for short. The characteristic five-tuple set is given. It is proved that there exists a probabilistic space $(\Omega,\mathscr{F},P)$, and MRICM$(\Theta,\eta,I,B)$ defined on it, and its characteristic five-tuple set coincides with the given one. The recursive equations of conditional ruin probability for finite time and infinite time are obtained.

The advantage of time series with matrix cross-section data is that multiple attributes of multiple objects can be characterized simultaneously. This paper focuses on autoregression model of time series with matrix cross-section data and presents the methods of parameter estimation, model identification and white noise test. Finally, the daily yield series and daily volume change rate series of two bank stocks are analyzed by this model.

For any real number $x\in(0,1)$, there exists a unique Engel continued fractions of $x$. In this paper, we mainly discuss the exceptional set which the logarithms of the partial quotients grow with non-linear rate. We completely characterize the Hausdorff dimension of the relevant exceptional set.

In differential geometry, there is a classical result, named Schur's Theorem, which is about the comparison of chords of two curves in $\mathbb E^3$. Inspired by it, this paper presents Schur-type theorems about the comparison of chord tangent angles of two curves, and the comparison of heights of two curves relative to their chords.

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.

We study perturbation bounds and convergence rates for uniformly ergodic Markov chains on general state spaces in terms of uniform moments of the first hitting times on some set. For reversible and non-negative definite Markov chains, we first investigate the geometrically ergodic convergence rates by the spectral theory. Based on the estimates, together with a first passage formula, we then get the convergence rates and perturbation bounds of uniform ergodicity. If the Markov chain is only reversible, we transfer to study the skeleton chain with transition kernel P². Finally, we investigate perturbation bounds for general Markov chains.

This paper firstly gives the definition of closed strongly irreducible operators on Banach spaces and gives an example of unbounded strongly irreducible operator. It obtains some properties of closed strongly irreducible operators. In particular, it obtains some equivalent descriptions of closed strongly irreducible operators. It also demonstrates some sufficient conditions for the strongly irreducibility of closed operators which have the forms of upper triangular operator matrices.

Let $\overline{W}_{F(t)}$, corresponding to a rotation invariant function $F(t(\nu))$ with a convexity condition on the upper hyperboloid $\mathbb{H}_+^n$, be a compact space-like Wulff shape bounded by a light-like $(n-1)$-round sphere. By applying perturbation metric and some integral formulae, we show that the only spacelike hypersurface with constant $r$th $F$-mean curvature in $\mathbb{L}^{n+1}$, which is tangent to $\overline{W}_{F(t)}$ on the boundary, is the Wulff shape.

A simplified dynamical systems method for solving the nonlinear equation $F(u)=f$ is studied in this paper. Under certain conditions of the operator $F$ and the exact solution $y$, the error estimate of the solution of the dynamical systems equation is given, and the discrepancy principle of the posterior selection of regularized parameter is proposed to ensure the optimal rate of convergence of the solution of the dynamical systems equation. Compared with the traditional dynamical systems method, the simplified dynamical systems method reduces the computation amount of derivatives.

Consider any prime number $p$. In this paper, we use analytic methods, properties of Legendre's symbol modulo $p$, and estimation for character sums to study distribution properties of triples of consecutive quadratic residues (named 3-CQR) and consecutive quadratic non-residues (3-CQN) modulo $p$. We provide exact formulas for the numbers $S_1(p)$ and $S_2(p)$ of 3-CQRs and 3-CQNs when $p\equiv 3$ or $7\pmod{8}$. Asymptotic formulas are given for $p\equiv 1$ or $5\pmod{8}$. Similarly, triples of quadratic residues with equal distance 2 are investigated and corresponding enumeration formulas are given. As an application, we further apply 3-CQRs to construct magic squares of squares of full degree over $\mathbb F_p$.

Let $A$ be a free abelian group of rank $n$. It is well known that the automorphism group $\operatorname{Aut}(A)$ of $A$ is $\operatorname{GL}(n,\mathbb{Z})$. Let $f(\lambda)=\lambda^{n}+a_{n-1}\lambda^{n-1}+\cdots+a_{1}\lambda+a_{0}$ be an irreducible polynomial in $\mathbb{Z}[\lambda]$, where $a_{0}=\pm1$. Let $T=\langle\alpha\rangle$ be an infinite cyclic group. Let $\alpha$ act on $A$ via the automorphism of $A$ induced by the Frobenius companion matrix of the monic polynomial $f(\lambda)$. Assume that $G=A\rtimes T$. Let $p$ be a prime. We prove that $G$ is a residually-finite $p$-group if and only if $p$ divides $f(1)$.

We give the definition of geominimal L_p (p ≥ 1) integral curvature. We mainly prove the existence and uniqueness of the L_p entropy Petty body for the convex body containing the origin in its interiors. Moreover, we also study the continuity of the geominimal L_p integral curvature and the L_p entropy Petty body.

The paper proposes a hypothesis testing of spatial autoregressive and parametric component in partial functional linear spatial autoregressive model. The functional principle component analysis is employed to approximate the slope function. And generalized method of moments (GMM) is used to estimate parameters. Basis on consistent estimators, we construct a test statistic of the residual sums of squares under null and alternative hypothesis. In addition, we establish the asymptotic properties of the proposed test. Simulation studies show the proposed test has good size and power with finite sample size. Finally, a real data analysis of growth data is conducted to investigate the significance of spatial autoregressive and parametric coefficients with partial functional linear spatial autoregressive model.

Constructing examples of groups is an important aspect in the theory of groups. We will study the residual finiteness of two concrete matrix groups. Let $p$ be a prime, let $C=\langle c\rangle$ be an infinite cyclic group, let $R=\mathbb{Z}C$ be the integral group ring over $C$, and let $U(n,R)$ be the upper unitriangular group over $R$ of order $n$, where $n\geq 2$, which is a nilpotent group of infinite rank of class $n-1$. Firstly, we prove that $U(n,R)$ is a residually finite $p$-group. Secondly, let $ G=\langle\alpha\rangle\ltimes U(3,R)$, where $\alpha={\rm diag}(c,1,c)$ is a diagonal matrix of order 3. We will study the structure of $G$ and prove that $G$ is a residually finite $p$-group, $G$ is a 3-generated soluble group of derived length 3. Moreover, we will construct two quotient groups of $G$, neither of which is residually finite. These two quotient groups seem to be more elementary and concrete than the classical examples discovered by Hall.

We present an inexact Newton-Krylov subspace method with incomplete line search technique for solving symmetric nonlinear equations, in which the Krylov subspace method uses the Lanczos-type decomposition technique. The iterative direction is obtained by approximately solving the Newton’s equations of the nonlinear equations using the Lanczos method. The global convergence and local superlinear convergence rate of the proposed algorithm are established under some reasonable conditions. Finally, the numerical results show the effectiveness of the proposed algorithm.

We study Leibniz algebras equipped with higher derivations. We call such a tuple of a Leibniz algebra and a higher derivation by LeibHDer pair. First, we define representations of LeibHDer pairs and construct the semi-direct product. Finally, we define a suitable cohomology for a LeibHDer pair with coefficients in a representation, and study central extensions and deformations of LeibHDer pairs.

In this paper, we discuss the Gröbner--Shirshov bases and a linear bases of the cyclotomic Hecke algebra of type $A$. First, by computing the compositions, we construct a Gröbner--Shirshov bases of the cyclotomic Hecke algebra of type $A$. Then using this Gröbner--Shirshov bases and the composition-diamond lemma we give a linear bases of the cyclotomic Hecke algebra of type $A$.

We systematically study the recurrent set and non recurrent set of the continuous time Markov process in the general state space, and focus on the determination methods of the recurrent set and non recurrent set of the Markov process, which provides a strong support for the study of the recurrence of the continuous time Markov process in the general state space.

We mainly use the nonnegative continuous function $\{\zeta_n(r)\}_{n\ge 0}$ to redefine the Bohr radius for the class of analytic functions satisfying ${\rm Re} f(z)<1$ in the unit disk $|z|<1$ and redefine the Bohr radius of the alternating series $A_f(r)$ with analytic functions $f$ of the form $f(z)=\sum_{n=0}^{\infty}a_{pn+m}z^{pn+m}$ in $|z|<1$. In the latter case, one can also get information about Bohr radius for even and odd analytic functions. Moreover, the relationships between the majorant series $M_f(r)$ and the odd and the even bits of $f(z)$ are also established. We will prove that most of results are sharp.

The main purpose of this paper is using the elementary methods and the properties of the solutions of some congruence equations to study the calculating problem of a certain fourth power means of the generalized exponential sums, and give three interesting identities for them.

We focus on the distributed statistical inference for linear models with multi-source massive heterogeneous data. First, a communication-efficient distributed aggregation method is proposed to estimate the unknown parameter vector, and the derived estimator is proved to be best linear unbiased and asymptotically normal under some regularity conditions. Then, a distributed test method is proposed to test the heterogeneity among a large number of data sources. Finally, the simulations are conducted to illustrate the effectiveness of the proposed method.

In this paper, by establishing new monotone formulas along the shortening flows for plane curves, we present new proofs for three geometric inequalities. In particular, a new proof of the Ros theorem on $\mathbb R^2$ is given by the classical curve shortening flow. And new proofs of the Ros theorem on $\mathbb R^2$ and its refined form and an entropy inequality for plane curve are given by an area-preserving curve shortening flow.

INGARCH models are often constructed based on Poisson distribution, negative binomial distribution and so on. Beta negative binomial (BNB) distribution is a flexible distribution. Recently, the related BNB-INGARCH model was proposed, whose conditional mean is linear, the parameters are restricted to non-negative and negative autocorrelation cannot be modeled. In this paper, we first propose the log-linear BNB-INGARCH model to solve the above problems, but the simple form of linear mean and ARMA-like structures are lost. So we further construct softplus BNB-INGARCH$(p,q)$ model by using the softplus function, which is the main research object. When $p$ and $q$ are equal to 1, the stationarity and ergodicity of the model are proved and the conditions for the existence of the second moment are given. In addition, the strong consistency and the asymptotic normality of the maximum likehood estimator are shown. Finally, the analysis of real-data examples show the usefulness of the proposed model.

An edge-partition of a graph $G$ is a decomposition of $G$ into subgraphs $G_1, G_2,\ldots,G_m$ such that $E(G)=E(G_1)\cup\cdots\cup E(G_m)$ and $E(G_i)\cap E(G_j)=\emptyset$ for any $i\neq j$. A linear forest is forest in which each connected component is a path. The linear arboricity ${\rm la}(G)$ is the least integer $m$ such that $G$ can be edge-partitioned into $m$ linear forests. In this paper, we use the discharging method to study the linear arboricity ${\rm la}(G)$ of 1-planar graphs, and prove that ${\rm la}(G)=\lceil\frac{\Delta(G)}{2}\rceil$ for each 1-planar graph $G$ with $\Delta(G)\ge25$, where $\Delta(G)$ is the maximum degree of $G$.

We study the geometric characteristics of C-Bézier curves that possess the Pythagorean Hodograph (PH) property. Based on the algebraic necessary and sufficient conditions for PH C-curves, we prove that a C-Bézier curve is a PH C-curve if and only if the interior angles of its control polygon are equal, and the second leg length of the control polygon is the geometric mean of the first and the last ones. Our main idea is to represent a planar parametric curve in complex form. We claim that the geometric characteristics of PH C-curves are quite similar to polynomial PH curves, which can be used to identify PH C-curves and their constructions. As an application, we give some examples of $G^1$ Hermite interpolation using PH C-curves. We point out that there are no more than two PH C-curves for any given $G^1$ Hermite conditions.

We study the variable selection problem of the spatial autoregressive quantile model with fixed effects. By penalizing the relevant parameters, we can identify the spatial effects, estimate the unknown parameters and select the explanatory variables simultaneously. In addition, we give an algorithm of variable selection and prove the large sample property of penalty estimator. Numerical simulation and real data analysis show the excellent performance of the proposed method.

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.

In this paper, the two dimensional Riemann problem of the Euler system for Chaplygin gas with three pieces of constant states is studied. The three states are divided by the $x$-axis and the positive semi-axis of the $y$-axis. Based on the assumption that each jump in initial data outside of the origin projects exactly one planar wave of shocks, centered rarefaction waves, or slip planes, and by using of the method of generalized characteristic analysis, we give the structures of the solution in detail. In fact, we divide the analysis into ten cases and among them, only four subcases are reasonable.

In this paper, by means of the continuation theorem of the Mawhin coincidence degree theory and the existence of periodic solutions of generalized ordinary differential equations, under the condition of the existence of equivalent relation between impulsive retarded functional differential equations and generalized ordinary differential equations, the existence theorem of periodic solutions for impulsive retarded functional differential equations is established.

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 $k_n\rightarrow \infty$.

Let $\Omega\subset \mathbb{R}^n$ be a Gromov hyperbolic domain which satisfies a quasihyperbolic boundary condition. In this paper we prove that there is a bi-Hölder identification between the internal boundary of $\Omega$ and the Gromov boundary endowed with a visual metric by using a diameter type Gehring—Hayman inequality and also the uniformization of Bonk—Heinonen—Koskela. As an application, we establish the internal boundary continuity not only for quasiconformal homeomorphisms, but also for rough quasi-isometries between the domains with respect to the quasihyperbolic metrics.

This article studies global well-posedness of generalized Hartree equation with a nonlinear damping \begin{equation*} \left\{ \begin{array}{ll} i\partial_tu+\Delta u+ i a|u|^{q-2}u=\pm\left(I_\alpha\ast |u|^{p}\right)|u|^{p-2}u,\quad (t,x)\in \mathbb R^+ \times \mathbb R^d, \\ u(0)=u_0,\quad x\in \mathbb R^d, \end{array} \right. \end{equation*} where $d\geq3$, $a > 0$, $0< \alpha < d$. When $2\leq p<\frac{d+\alpha}{d-2}$ and $2\leq q<\frac{2d}{d-2}$, we get the local well-posedness for this equation; When $p = 1 + \frac{2+\alpha}{d}$, $2\leq q<\frac{2d}{d-2}$ and $\|u_0\|_{L^2} \leq \|Q\|_{L^2}$, we prove the solution is global; When $p = 1 + \frac{2+\alpha}{d}$ and $q= 2+\frac{4}{d}$, we find that the damping prevents blow-up, and further study the scattering problem.

We study a new iterative algorithm for solving pseudomonotone variational inequality problems in real Hilbert spaces. The proposed algorithm combines the subgradient extragradient method, the inertial method and the viscosity method. Under appropriate conditions imposed on the parameters, we accelerate and improve the convergence of the algorithm by introducing different parameters. Finally, some numerical experiments are proposed to show the efficiency of our algorithm through comparison with related algorithms.

Schwarz lemma plays significant roles on function theory of holomorphic mappings or harmonic mappings. In this paper, we establish the Schwarz lemma at the boundary for self-mappings solutions satisfying the Poisson's equation of the unit ball in $\mathbb{R}^n$. As an application, the boundary Schwarz lemma for harmonic self-mappings on the unit ball is obtained which extends the result of pluriharmonic mappings to harmonic mappings in higher dimensions.