In this paper, the chaotic behavior of the generalized elliptic Sitnikov (N+1)-body problems is analytically studied. First, based on the perturbation method of integrable Hamiltonian systems, the generalized elliptic Sitnikov (N+1)-body problem is regarded as the perturbation of the generalized circular Sitnikov (N+1)-body problem. Then, we prove that the Melnikov integral function has a simple zero, arriving at the existence of transversal homoclinic orbits. Moreover, since the equilibrium point is a degenerate hyperbolic saddle, the standard Smale- Birkhoff theorem cannot be used directly to prove the existence of Smale horseshoes. We alternatively construct an invertible map f and check that f satisfies the Conley-Moser condition, which shows that the generalized elliptic Sitnikov (N+1)-body problem possess chaotic behaviour of Smale horseshoe type.

We determine the genus distributions for circular ladder-like sequences of graphs and Möbius ladder-like sequences of graphs. First, we obtain the production matrix for genus distributions of ladder-like sequences of graphs. Then, any circular (Möbius) ladder-like graph is obtained by adding edges to a ladder-like graph by using edge addition rule, and we obtain the genus polynomials for circular ladder-like sequences of graphs and Möbius ladder-like sequences of graphs. In addition, we also verify the asymptotic normality of the classical ladder graphs.

We establish a Freidlin- Wentzell type large deviation principle for twodimensional incompressible Magneto-hydrodynamics equations driven by multiplicative noises when the noises converge to zero that are uniform with respect to initial conditions in bounded subsets of the infinite dimensional Banach space. The proof is based on the weak convergence approach.

This paper is based on the Hayman Conjecture for the zeros problem of complex differential polynomials. Using an important estimate on the zeros of higher derivative of meromorphic functions given by Yamanoi, we obtain the improved results on the zeros of delay-differential polynomials. For instance, we have that if f is a transcendental meromorphic function with hyper-order less than one and q ≥ p+s+t+1, then [Q(f)P(f(z +c))]^(k) -a has infinitely many zeros, where a is a non-zero constant, P(z) is a polynomial of degree p with t different zeros and Q(z) is a polynomial of degree q with s different zeros. Our results improve the former results which obtained mainly by the second main theorem of Nevanlinna.

Based on the Peaceman-Rachford splitting method, combined with the linear approximate technique and Bregman distance, in this paper, we present a linear approximation Bregman-type Peaceman-Rachford splitting method for solving the nonconvex nonseparable optimization problem with linear constraints. Under the conventional assumptions, we get the global convergence of the proposed algorithm. On the premise that the merit function satisfies the Kurdyka-Ƚojasiewicz property, the strong convergence of the proposed algorithm is proved. When the associated Kurdyka-Ƚojasiewicz property function has a special structure, the convergence rate results of the proposed algorithm are analyzed and obtained. Finally, some preliminary numerical results show that the proposed algorithm has numerical validity.

In this paper, some new classes of n-cycle permutations of the form x^rh(x^s) over finite fields are presented, which are the further study on a recent work of Chen, Wang and Zhu. In addition, based on some detailed discussions, four interesting problems are proposed.

We obtain the existence and multiplicity of solutions for the fractional Schrödinger equation with Hardy-Sobolev critical exponent in R^N by Ekeland’s variational principle and Nehari decomposition.

This paper discusses the computations of Hausdorff centered measure and packing measure for self-similar sets. Let E be a self-similar sets with the strong separation condition, s be the Hausdorff dimension of E, μ be the self-similar measure defined on E, In this paper, we obtained the following results: (1) Lettirg x₀ ∈ E, if $\overline{D}^s(\mu,x_0)=\overline{d}$, then $\overline{D}^s(\mu,x) \ge \overline{d}$; for μ-almost all x ∈ E; (2) Lettirg y₀ ∈ E, if $\underline{D}^s(\mu,y_0)=\underline{d}$, then $\underline{D}^s(\mu,y) \le \underline{d}$ for μ-almost all y ∈ E. Some computational problems of measures of the self-similar sets are discussed by the use of those results.

Let m, n, L be positive integer. The following conclusion are proved: If m<n≤m+Lm^ε, ε∈(0,1), and m>(123789L√L)1/1-ε, or j>10.25×10¹²log⁴(2(L+1)(123789L√L)1/1-ε, then positive integer solutions of simultaneous Pell equations $x^{2}-(m^{2}-1)y^{2}=z^{2}-(n^{2}-1)y^{2}=1$ satisfy $1≤k\leq\delta L^{2}$, where $\delta\in[\frac{1}{2}(123787L\sqrt{L})^{\frac{1}{\varepsilon-1}},1]$,$ and $$y=\frac{(m+\sqrt{m^{2}{-}1})^{j}{-}(m{-}\sqrt{m^{2}{-}1})^{j}}{2\sqrt{m^{2}{-}1}}=\frac{(n{+}\sqrt{n^{2}{-}1})^{k}{-}(n{-}\sqrt{n^{2}{-}1})^{k}}{2\sqrt{n^{2}{-}1}},$ and j$j=k=1$ or $k+2\leq j<\frac{1}{3}(5-2\varepsilon)k$,$2\,|\,(j+k)$, $k>\frac{3}{1-\varepsilon}$. It improves the previous work of [Proc. Amer. Math. Soc., 2015, 143(11): 4685-4693].

In the paper, we give the definition of the Markov chains indexed by a tree taking value on R. Then, we prove the equivalent properties of it.

In this paper, a subclass of g-starlike mappings of complex order λ on the unit ball $\mathscr {B}^{n}\subset \mathbb{C}^{n}$ is introduced, which unifies almost starlike mappings of complex order λ and g-starlike mappings. The growth theorem for g-starlike mappings of complex order λ is established by using the parameter method of Loewner chains. By giving scalar conditions of homogenous polynormials of degree k, we prove that the modified Roper-Suffridge extension operator on the unit ball $\mathscr {B}^{n}$ given by Φ_Pk(f)(z) = (f(z₁) + P_k(z₀)f' (z₁), [f' (z₁)] 1/k z₀)^T preserves the property of g-starlike mappings of complex order λ. Our results not only generalize some well-known growth theorems of different classes of starlike mappings on $\mathscr {B}^{n}$, but also give more concise geometric characteristics of polynomial P_k.

In this note, we prove that the conformal dimension of the Sierpinski carpet S_p satisfies $1{+}\frac{\log (p{-}1)}{logp}\!\leq\dim_{C}S_{p}\leq \frac{\log((p^{2}-1)^{4}-8)}{4\log p}$ , where $p\geq3$ is odd. This result implies that S_p is not a quasisymmetrically minimal set.

In 2021, Zhang and Liao gave a necessary and sufficient condition for that the composite polynomial of two involution polynomials over finite fields is also an involution polynomial. Since the involution polynomial is a special class of permutation polynomials, based on the relationship between the sets of fixed points and non-fixed points of polynomials, we obtain a necessary and sufficient condition for the composition of an involution polynomial and a permutation polynomial to be involuted.

Let L² be the set consisting of all the Lebesgue integrable functions on the unit circle T. Define the Hardy space to be the closed subspace spanned by the analytic polynomials in L². For any point z in the unit disk D of the complex plane, K_z(w) = 1/1-zw ˉ is the reproducing kernel function in H². It is well known that T_fK_z = f(z)K_z, that is, K_z is the eigenvector of T_f corresponding to the eigenvalue f(z). Conversely, if there exists some z ∈ D (or, for any z ∈ D), K_z is an eigenvector of T_f, whether there must be $f\in \overline{H^{\infty}}$? For the above questions, in this paper, we give a complete characterization of the Toeplitz operators as well as the bounded linear operators which take the reproducing kernels K_z as their eigenvectors. Moreover, we partially describe the Toeplitz operators with f(z) (z ∈ D) as their eigenvalues.

We construct two 3-generated metabelian groups from the upper triangular matrices of order 2 over two kinds of domains. Their structures are clear and their residual finiteness is studied. One of the groups with infinite rank is a residually finite p-group, where p is a prime. And the other group with finite rank does not have this neat residual finiteness property, although its structure is simpler.

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.

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.

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.

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 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.

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, 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 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.

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.

This paper studies the $M/G/1$ queueing system with startup time, bi-level threshold $(m,N)$-policy and single server vacation without interruption. In this system, when the server is transferred on vacation, the server starts the system immediately if the number of waiting customers is no less than a given positive integer threshold $m\ (m\ge 1)$, and when the system startup is complete, the server begins service immediately if the number of waiting customers is no less than another given positive integer threshold $N\ (N\ge m)$. Assume that the server's vacation time and the startup time of the system follow general distributions, both the transient queue-length distribution and the steady-state queue-length distribution of the system are discussed by using the renewal process theory, the total probability decomposition technique and Laplace transform tool. The expressions of the Laplace transformation of the transient queue-length distribution with respect to time $t$ are obtained. Furthermore, the recursive expressions of the steady-state queue-length distribution are derived by a direct calculation. Meanwhile, the stochastic decomposition structure of the steady-state queue-length and the explicit expression of the additional queue-length distribution are presented. Finally, the explicit expression of the long-run expected cost per unit time is derived under a given cost model. And the numerical example is given to determine the optimal control policy $({{m}^{*}},{{N}^{*}})$ for minimizing the long-run expected cost per unit time.

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 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.

We first give definitions of derivations, generalized derivations and quasi derivations of multiplicative Hom-Lie color triple systems, and then provide some properties. In particular, we prove that ${\rm QDer}(T)+{\rm QC}(T)= {\rm GDer}(T)$, and both the central derivation algebras and the centroids are Hom-ideals of ${\rm GDer}(T)$. Also, we show that ${\rm ZDer}(T)={\rm C}(T)\cap {\rm Der}(T)$ when the characteristic of the field $\mathbb{F}$ is not equal to 2.

The first eigenvalue of discrete weighted p-Laplacian on (maybe infinite) the tree with the unique root as Dirichlet boundary is studied in this paper, which is equivalent to the optimal constant of a class of weighted Hardy inequality. By investigating the properties of the eigenfunction corresponding to the eigenvalue, three kinds of variational formulas for the first eigenvalue with mixed boundary are presented, which are meaningful supplement to the classical variational formula. As applications, a basic estimation for the first eigenvalue of p-Laplacian is obtained and two examples are presented to illustrate the results.

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.

Let $G$ be a finite group. The commuting graph of $G$ is a graph whose vertex set is the set of all non-central elements, and two distinct vertices $x$ and $y$ are adjacent if and only if $xy=yx$. The graph perfect code problem is as follows: determine whether a graph admits perfect codes; if a graph admits a perfect code, then how do we find the perfect codes. In this paper, we solve the perfect code problem of the commuting graphs of symmetric groups and alternating groups.

This paper studies the global well-posedness issue on the $d$-dimensional $(d\geq 2)$ tropical climate model without thermal diffusion for small initial data. The model considered here has fractional dissipations $\Lambda^{2\alpha}u$ and $\Lambda^{2\beta}v$ on the equations of the barotropic mode and the first baroclinic mode of velocity, respectively. We establish the global existence and uniqueness of the solutions for the case when $0\leq \alpha<1$, $\frac{1}{2}\leq \beta\leq 1$ and $\alpha+\beta\geq 1$ under small initial data.

We investigate Toeplitz operators with harmonic symbols acting on the harmonic Bergman space for the punctured domain $D\setminus\{0\}$, where $D$ is the unit disc in the complex plane. First, we investigate the construction of $b^p(\Omega)$ and derive the image of every element of $b^p(\Omega)$ under the harmonic Bergman projection. Second, we prove that the special Toeplitz operator $T_{\log|w|}: b^2(D\setminus\{0\})\rightarrow b^2(D\setminus\{0\})$ is a bounded linear operator and obtain some sufficient and necessary conditions that a Toeplitz operator with harmonic or holomorphic symbol can commute with $T_{\log|w|}$. Third, We obtain a sufficient and necessary condition for two Toeplitz operators with holomorphic symbols commuting with each other. Fourth, we give a characterization of normal Toeplitz operators with holomorphic symbols. Finally, we derive a necessary condition for Toeplitz operators with harmonic symbols commuting with each other.

The concept of Whittaker modules over finite dimensional simple Lie algebras was introduced by Kostant [Invent. Math., 1978, 48: 101-184], where he described the structure of such modules in full detail with respect to nonsingular Whittaker functions. In this paper, we study the simple Whittaker modules for the Lie algebra of type B₂ with singular Whittaker functions.

We study the existence of traveling wave fronts and the minimal speed selection mechanism of a competitive-cooperative system with three species. Some sufficient conditions on the existence of traveling wave fronts have been given by constructing some suitable super-sub solutions and strict analysis. Based on the existence results, some sufficient conditions on the minimal speed, which is linearly determined, have been proved. These conclusions enrich the content of the research on such system.

We consider the compressed data separation problem under $\ell_{p}$ bounded noise, that is, to reconstruct different sparse subcomponents of signals from compressed measurements. In order to reconstruct different subcomponents that are (approximate) sparse in terms of different frames ${{D}_1}\in{\mathbb{R}^{n\times{d_1}}}$ and ${{D}_2}\in{\mathbb{R}^{n\times{d_2}}}$, we first propose the $\ell_{1}- \alpha\ell_{2}$ split analysis algorithm, which can deal with the problem of signal separation under the corruption of different kinds of noises, when the measurement matrix meets a certain restricted isometry property and the dictionaries meet a certain mutual coherence condition. In addition, based on the classical Dantzig Selector, we also introduce $ \ell_{1}- \alpha\ell_{2}$ split analysis Dantzig Selector. It can also stably separate compressed data under appropriate conditions. Numerical experiments are carried out to show that $ \ell_{1}- \alpha\ell_{2}$ minimization are robust and stable for the separation of compressed data with redundant tight frames.