The main purpose of this paper is to establish the elliptic gradient estimate for the heat equation on compact Riemannian manifold with control on integral Ricci curvature. We also derived the volume comparison theorem under the new integral Ricci curvature condition which extended Petersen-Wei's volume comparison theorem.

A subgroup $H$ of a finite group $G$ is a BNA-subgroup of $G$ if either $H^{x}=H$ or $x\in \langle H, H^{x}\rangle $ for all $x\in G$. A finite group $G$ is called a CBNA-group if its all cyclic subgroups of order prime or $4$ are BNA-subgroups of $G$. The main aim of this paper is to investigate the structure of CBNA-groups, and the groups whose all proper subgroups are CBNA-groups are classified completely.

In general, the properties of modules over a triangular matrix ring $T=\left(\begin{smallmatrix}A & U \\0 & B\end{smallmatrix}\right)$ are studied via modules over diagonal "small rings" $A$ and $B$. However, we use model structures on the category of $T$-modules to characterize the stable categories $\underline{\mathcal{GP}}(A)$, $\underline{\mathcal{GP}}(B)$ of Gorenstein projective modules over $A$ and $B$. To this end, we introduce two subcategories of Gorenstein $T$-modules, and obtain two corresponding complete cotorsion pairs. Moreover, cotorsion pairs of modules are lifted to $T$-complexes, and the equivalences and recollements of homotopy categories of complexes are studied.

In this paper, a new conception called perfect permutation will be introduced. We focus on its algebraic properties and construction methods. The main result is that there exists a perfect permutation of order n when 2n + 1 is a prime. Furthermore, we use perfect permutations to construct cyclic spatially balanced Latin squares and symmetric spatially balanced Latin squares both of which are widely used in experimental designs.

Let $(Z_n)$ be a supercritical branching process with immigration in an independent and identically distributed random environment. Based on the structure of $Z_n$, using related results on random walks and technique of measure change, we establish a Cramér's large deviation expansion for $\log Z_n$.

In this paper, we study $(C,\varepsilon)$-super subdifferentials of set-valued maps. First, we introduce a notion of $(C,\varepsilon)$-super efficient point of a set. Some properties and equivalent characterizations of the $(C,\varepsilon)$-super efficient points are presented. Scalarization theorems of the set-valued optimization problem are obtained in the sense of $(C,\varepsilon)$-super efficiency. Second, we define $(C,\varepsilon)$-subdifferentials of set-valued maps and research the existence conditions of $(C,\varepsilon)$-subdifferentials. Moreau-Rockafellar type theorems characterized by $(C,\varepsilon)$-subdifferentials are also established. Finally, as the applications, we establish some optimality conditions of the set-valued optimization problem involving the $(C,\varepsilon)$-super subdifferentials. The results obtained in this paper unify and generalize some results characterized by the super subdifferentials or $\varepsilon$-super subdifferentials of the set-valued maps in the literature.

We introduce a new algorithm for solving pseudomonotone variational inequality problems in Banach spaces. We prove that the sequence generated by this algorithm converges strongly an element of solutions for variational inequality problems under some suitable conditions imposed on the parameters. The results obtained in this paper extend and improve many recent ones in the literature.

Combing Wintner, Delange, Ushiroya and Tóth's works from 1976 to 2017, we have that the multi-variable arithmetic functions defined on integer ring can be expanded through the Ramanujan sums. This is an analogue of the Fourier expansion for periodic functions in the classical analysis. In this paper we further investigate the properties of Ramanujan sums in the polynomial ring $\mathbb{F}_{q}{[T]}$, and show that the multi-variable arithmetic functions defined on $\mathbb{F}_{q}{[T]}$ can also be expanded through the polynomial Ramanujan sums and the unitary polynomial Ramanujan sums.

In this paper, a class of infinite dimensional complete Lie algebras is obtained by extending the one dimensional homogeneous derivations of Loop algebras over simple Lie algebras. It is proved that every 2-local homogeneous derivation of this kind of infinite dimensional complete Lie algebra is a derivation.

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 ${\mathbb L}$ be a super Heisenberg-Virasoro algebra with the ${\mathbb C}$-basis {$L_{n},I_{n},G_{n}|$ $ n\in {\mathbb Z}$}, which satisfies the relations $[L_{m},L_{n}]=(m-n)L_{m+n}$, $[L_{m},I_{n}]=\!-nI_{m+n}$, $[L_{m},G_{n}]=-nG_{m+n}$ and $ [G_{m},G_{n}]=I_{m+n}.$ In this paper, we prove that all super-skewsymmetric super-biderivations of ${\mathbb L}$ are inner. Furthermore, we prove that every linear super-commuting map on ${\mathbb L}$ has the form $\Psi(x)=f(x)I_{0}$ for all $x\in{\mathbb L}$, where $f(x)$ is a linear map from ${\mathbb L}$ to ${\mathbb C}$.

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.

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, 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.

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.

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.

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.

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.

Quantum states are positive operators with unit traces on a Hilbert space. The set of all quantum states is convex. In the paper, we give a characterization of maps leave the maximal eigenvalue of convex combinations of quantum states.

We define the T-DMP inverse and T-CMP inverse of third-order F-square tensors by using the T-Moore-Penrose inverse, T-Drazin inverse and T-core-nilpotent decomposition theorem of tensors via the T-product. Then, we present some characterizations and properties. Finally, the Cayley-Hamilton theorem of third-order tensors is extended to T-Drazin inverses and T-DMP inverses. Examples are also given to illustrate these theorems.

In this article, the refined estimates of all homogeneous expansions for a subclass of biholomorphic spirallike mappings which have a concrete parametric representation on the unit ball in complex Banach spaces and the unit polydisk in $\mathbb{C}^n$ are mainly established. In particular, the result is sharp for the ($k+1$)-th homogeneous expansion. Meanwhile the estimates of all homogeneous expansions for a subclass of $k$-fold symmetric biholomorphic spirallike mappings which have parametric representation on the unit ball in complex Banach spaces and the unit polydisk in $\mathbb{C}^n$ are also given, and the result is sharp for the ($k+1$)-th homogeneous expansion as well. Our obtained results include many known results in some prior literatures.

We get an explicit and recursive representation for high order moments of additive functionals for discrete-time single death chains. Meanwhile, the polynomial convergence and a central limit theorem are investigated.

For $n=2$, Bian and Liu in[Algebra Colloquium, 2017, 24(2):297-308] provided a presentation for little $q$-Schur algebra in terms of generators and relations. It is more difficult for the case $n>2$. In this paper, we give a presentation for little $q$-Schur algebra $u_k(3,3)$ at odd roots.

In this paper, conformal biderivations and automorphism groups of two classes of Schrödinger-Virasoro type Lie conformal algebras ${\rm TSV}(a,b)$ and ${\rm TSV}(c)$ are completely determined, respectively. The results for the Lie conformal algebras $W(a,b)$ are also obtained as a corollary.

In Ringel-Hall algebra of Dynkin type, the set $S$ of isomorphism classes of indecomposable modules forms a minimal Gröbner-Shirshov basis of the ideal $\hbox{Id}(S)$ generated by the set $S$, and the corresponding irreducble elements forms a PBW basis of the corresponding Ringel-Hall algbera. Our aim is to generalize this result to the derived Hall algebra of type $G_2$. First, we compute the skew commutator relations between the isomorphism classes of indecomposable objects in the bounded derived category of type $G_2$ by using the Auslander-Reiten quiver of the bounded derived category of type $G_2$. Then, we prove that the compositions between these skew commutator relations are trivial. Finally, we construct a PBW basis of the derived Hall algebra of type $G_2.$

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.

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 $\mathbb F_p$.

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.

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.

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, 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 $A$.

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 $G$.

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.

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.

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.

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.