We consider a class of nonlocal N-Kirchhoff type problems involving a singular exponential critical growth nonlinearity:m(||u||^N)(-Δ_Nu + V (x)|u|^N-2u)=f(x,u)/|x|^β + ϵh(x), in R^N, where N ≥ 2,||u||^N=∫_R^N (|∇u|^N + V (x)|u|^N)dx, Δ_N^u=div(|∇u|^N-2∇u) is the N-Laplacian, m:R⁺ → R⁺ is a Kirchhoff function, V:R^N → R is a continuous potential, f:R^N×R → R is a continuous function, and behaves like e^α|s|N/N-1 when|s|→ ∞ for some α > 0, 0 ≤ β < N, h(x) ∈ (W₀^1,N(R^N))^*, h(x) ≥ 0 and h(x) ≢ 0, ϵ is a small positive parameter. Applying variational methods together with singular Trudinger-Moser inequality in the whole R^N, when is small enough, we obtain the existence and multiplicity of solutions.

We define unstable local entropies for arbitrary Borel probability measures in partially hyperbolic systems. In order to characterize the multifractal spectrum of unstable local entropies, we introduce the concept of unstable (q, μ)-entropy, provide some basic properties of (q, μ)-entropy and establish a relation formula between the Bowen unstable entropy of the multifractal spectrum and the (q, μ)-entropy.

In this paper, the dominating set of the Bergman space in the unit ball are characterized in terms of the pseudohyperbolic metric ball. Our method is to generalize Luecking's three key lemmas on the unit disc to the unit ball. We then apply those three lemmas to give a complete description of the dominating set of the Bergman space on the unit ball.

Under the natural assumption of saturation effect, this paper proves the global existence and uniform boundedness of the classical solutions to the 3D initial boundary value problem for a double chemotaxis-Stokes system. Due to the strong nonlinearity in the system, the method developed in this paper can be applied to the related models for the coral spawning, which have attracted much attention recently.

The present paper is devoted to the study of the so-called bivariate partial theta function which is first introduced by the author and contains the classical partial theta function as a special case. We focus on its possible product formula, recurrence relation, and series expansion and so on. As main results, we establish a product formula of any two bivariate partial theta functions. It is a generalization of Andrews-Warnaar's product formula for the classical partial theta functions. At the same time, we obtain a second order recurrence relation satisfied by this bivariate partial theta function. Finally, we present two series expansions of the bivariate partial theta function θ(q, x; ab) with respect to {θ(q, axqⁿ; b)|n ≥ 0} and {θ(q, xqⁿ; b)|n ≥ 0}, respectively. As further applications of these results, we also find a product formula of two ₃φ₂ series and a ternary representation of the bivariate partial theta function.

Let H be a complex infinite dimensional Hilbert space. B(H) denotes the algebra of all bounded linear operators on H. In this paper, we characterize the operators in B(H) for which f(T) satisfies Weyl's theorem, where f denotes the analytic function on some neighbourhood of the spectrum of T. Also, the relationships between Weyl's theorem for functions of operators and the stability of Weyl's theorem are explored.

We introduce a new algorithm for solving pseudomonotone variational inequality problems and fixed point problems by using the subgradient extragradient method. A weak convergence theorem of proposed algorithm is obtained under some suitable assumptions imposed on the parameters. The results obtained in this paper extend and improve many recent ones in the literature.

Let Z and N be the set of all integers and positive integers, respectively. M_m (Z) be the set of m×m matrix over Z where m ∈ N. In this paper, by using the result of Fermat's Last Theorem, we show that the following second-order matrix equation has only trivial solutions:Xⁿ + Yⁿ=λⁿI (λ ∈ Z, λ ≠ 0, X, Y ∈ M₂(Z)), where X has an eigenvalue that is a rational number and n ∈ N, n ≥ 3; By using the result of primitive divisors, we show that the second-order matrix equation Xⁿ +Yⁿ=(±1)ⁿI (n ∈ N, n ≥ 3, X, Y ∈ M₂(Z)) has nontrivial solutions if and only if n=4 or gcd(n,6)=1 and all nontrivial solutions are given; By constructing integer matrix, we show that the following matrix equation has an infinite number of nontrivial solutions:∀_n ∈ N, Xⁿ + Yⁿ=λⁿI (λ ∈ Z, λ ≠ 0, X, Y ∈ M_n(Z)); X³ + Y³=λ³I (λ ∈ Z, λ ≠ 0, m ∈ N, m ≥ 2, X, Y ∈ M_m(Z)).

We study the following polyharmonic Dirichlet problems in a punctured unit ball

where B is the unit ball in R^N, ν is the unit outward normal vector of ∂B, N > 2k, k ≥ 2. Under certain assumptions on f, we use the moving plane method to show radial symmetry of any singular positive solution provided that 0 is a nonremovable singularity point. As an application, we can obtain nonexistence of positive solutions for a critical Dirichlet biharmonic problem.

It was proved that a function with exact one discontinuity may have a continuous iterate of second order. It actually shows that its discontinuity may be repaired to be a continuous one by its own pair of functions under iteration. If a function has at least two discontinuities, then each of its discontinuities may be repaired to be a continuous one by either its own pair of functions or the other's pair of functions under iteration. In this paper we investigate those functions having more than one but finitely many discontinuities of the same type and give necessary and sufficient conditions for those functions whose second order iterates are continuous.

In this paper, the quadratic numerical radius inequalities of off-diagonal block operator matrixwhose entries are bounded operators on the Hilbert space is studied. According to the classical convexity inequalities of non-negative real numbers, the quadratic numerical radius inequalities of A is generalized.

For a graph G=(V (G), E(G)), if a mapping ?:E(G) → {1, 2,..., k} such that ?(e₁) =?(e₂) for any adjacent edges e₁, e₂, and there are no bicolored cycles in G, then ? is called an acyclic edge coloring of G. For a list assignment L={L(e)|e ∈ V (E)}, if there exists an acyclic coloring ? such that ?(e) ∈ L(e) for each e ∈ E(G), then ? is called an acyclic L-list coloring of G. If for any L with|L(e)| ≥ k for each e ∈ E(G), there exists an acyclic L-list coloring of G, then we say G is acyclically k-edge choosable. The minimum integer k making G is acyclically k-edge choosable is called the acyclic list chromatic index of G, denoted by a_l'(G). In this paper, it is proved that for a connected graph G with maximum degree Δ ≤ 4 and|E(G)| ≤ 2|V(G)|-1, it follows that a_l'(G) ≤ 6, which extends the result of Basavaraju and Chandran[J. Graph Theory, 2009, 61(3):192-209].

The iterated function system with two-element digit set is the simplest case and the most important case in the study of spectrality or non-spectrality of self- affine measures. The one-dimensional case corresponds to the Bernoulli convolution whose spectral property is understandable. However, the higher dimensional analogue, especially the two-dimensional case has not been solved completely. Also, there is a conjecture to illustrate that in the plane, the remaining cases correspond to nonspectrality of self-affine measures. Motivated by this problem, we provide in this paper some non-spectral conditions for the planar self-affine measures with two-element digit set. Under one of the conditions, we determine the maximal cardinality of orthogonal exponentials. An application of this result and the validity of the conditions are also presented.

In this note, it is proved that the Coleman outer automorphism group of a generalized quaternion group is either 1 or an elementary abelian 2-group by using the projection limit property of the group.

Periodicity is one of the most common factor in time series analysis. In the time series analysis of discrete-valued response variables, we use maximum likelihood estimation with penalty to establish a consistent estimator of the unknown period. Given the estimator of the period, we take B-spline to approximate the trend term and the additive function, and at the same time obtain the √n-consistent estimator of the periodic term and the initial estimators of the trend term and the additive function. Then based on the idea of back-fitting, we establish the improved estimators of the trend term and additive function, and the asymptotic normality and efficiency of them are also demonstrated. Simulation experiments and empirical analysis confirm that our proposed method performs well for the finite sample.

This paper is concerned with a third order in time dissipative Moore-Gibson-Thompson equation with infinite degenerate memory
$\tau u_{ttt}+\alpha(x)u_{tt}-c^{2}\Delta u - b\Delta u_{t}+\displaystyle\int_{0}^{\infty}g(s)\text{div}[a(x)\nabla u(t-s)]{d}s= 0$,
in which the function $a(x)\geq0 $ and $\alpha(x)\geq0$ can be degenerate but satisfy $a(x)+\alpha(x)\geq \delta >0$. This equation arises as a linearization of a model for wave propagation in viscous thermally relaxing fluids. By using the Faedo-Galerkin approximations together with some energy estimates, we prove that the above system is well-posedness. Besides, under appropriate assumptions, we establish the exponential or general decay results of energy via constructing suitable Lyapunov functionals.

We show that if the following delay differential equation
$ \left[w(z+1)w(z)-1\right]\left[w(z)w(z-1)-1\right]+a(z)\dfrac{w'(z)}{w(z)}=\dfrac{\sum_{i=0}^pa_i(z)w^i}{\sum_{j=0}^qb_j(z)w^j}$
with rational coefficients $a(z), a_i(z), b_j(z)$, admits a transcendental meromorphic solution $w$ of finite many poles with hyper-order less than one, then it reduces into a more simple delay differential equation, which improves some known theorems obtained most recently by Liu and Song. Moreover, we also study the delay differential equations of Tumura-Clunie type and obtain some quantitative properties of transcendental meromorphic solutions.

Let $(\Sigma_M,\sigma)$ be the shift of finite type where $M=(m_{ij})$ is a $b\times b$ matrix with $m_{ij}\in\{0,1\}$. In this paper, we consider the first return rate of the system $(\Sigma_M,\sigma)$. Let $\tau_k(x)$ be the first return time of $x\in \Sigma_M$ to the $k$-th cylinder containing $x$. Denote
$E_{\alpha,\beta}=\Big\{x\in\Sigma_M: \liminf_{k\rightarrow\infty}\frac{\log \tau_k(x)}{k}=\alpha,\, \limsup_{k\rightarrow\infty}\frac{\log \tau_k(x)}{k}=\beta\Big\}.$
We prove that if $M$ is irreducible, then $E_{\alpha,\beta}$ has full Hausdorff dimension for any $0\leq\alpha\leq\beta\leq+\infty$ and has Markov measure either 0 or 1.

We consider the constrained group sparse regression problem, where the loss function is convex. In order to obtain the exact continuous relaxation of the problem, we use the group Capped-L₁ regularization to relax the group sparse regularization. First, we introduce three types of stationary points for the relaxed problem, that is, D(irectional)-stationary points, C(ritical)-stationary points, and L(ifted)-stationary points. We provide some equivalent descriptions for the three types of stationary points, based on which, we investigate the relationship among the three types of stationary points and obtain some properties of them. Furthermore, we analyze the necessary and sufficient optimality conditions for the original group sparse problem and the relaxed problem. At last, we investigate the relationship of the solutions between the original group sparse problem and the relaxed problem, not only from the point of view of global minimizers but also from the point of view of local minimizers. The results reveal the equivalence of the original problem and the relaxed problem.

The bifurcation phenomenon of the operator equation $\lambda (f_1'(x)+f_2'(x))=g_1'(x)+g_2'(x)$ is studied in this paper. Suppose $f_2'\equiv0$, $f_1$ and $g_1$ are $a$-homogeneous, and some other suitable conditions hold, Fučík et al. obtained that each normalized LS-eigenvalue of $\lambda f_1'(x)= g_1'(x)$ is a bifurcation point of the operator equation above. This paper studies the inhomogeneous case of $f_1+f_2$. We establish the same results as theirs when $f_1$, $f_2$, $g_1$ and $g_2$ satisfy some suitable conditions. A Lyusternik-Shnirel'man theorem is obtained as a preliminary result. And for the application of our abstract theorems, the bifurcation phenomenon from arbitrary LS-eigenvalues is studied for a nonlocal elliptic problem.

Let f(z) be a holomorphic Hecke eigenform of even weight k for the full modular group. L(s, f) is the automorphic L-function associated of f. By the smooth shifted second moment of L(s, f), it is proved that there exist infinitely many consecutive zeros of L(s, f) on the critical line whose gaps are greater than 1.88 times of the averaging spacing.

This paper is concerned with the existence of periodic solutions for the fully second order ordinary differential equation
$ -u''(t)=f(t,\,u(t),\,u'(t)),\ \ t\in\mathbb{R} $
where the nonlinearity $f:\mathbb{R}^{3}\to\mathbb{R}$ is continuous and $f(t, x, y)$ is $2\pi$-periodic in $t$. Some existence results of odd $2\pi$-periodic solutions are obtained under that $f$ satisfies some precise inequality conditions. These inequality conditions allow that $f(t, x, y)$ may be superlinear or sublinear growth on $(x, y)$ as $|(x, y)|\to 0$ and $|(x, y)|\to \infty$.

Let $\mathscr{M}$ be a separable type II$_1$ von Neumann algebra. We prove that, if $\mathscr{M}$ has Property $\Gamma$, $G$ is a countable amenable group and $\alpha$ is a trace preserving, properly outer action of $G$ on $\mathscr M$, then the crossed product $\mathscr{M} \rtimes_{\alpha} G$ is a type II$_1$ von Neumann algebra with Property $\Gamma$.

In this paper, we establish two endpoint estimates for the commutator, $[b,T]$, of a class of pseudodifferential operators $T$ with symbols in Hörmander class $S^{m}_{\rho,\delta}(\mathbb R^{n})$. The first one is that the commutator $[b,T]$ is bounded from Hardy space $H^{1}(\mathbb R^{n})$ into weak $L^{1}(\mathbb R^{n})$ space. The second one is an estimate in the Hardy type spaces associated with $b$, where $b\in {\rm BMO}(\mathbb R^{n})$.

In this paper, we first investigate the correspondence between order boundedness and Hilbert-Schmidt of weighted composition operators $W_{\phi,\varphi}(f):=\phi f\circ\varphi$. Then, by resorting to the estimates of the norms of point evaluation functionals $\delta_z$ and derivative point evaluation functionals $\delta'_z$ on weighted Dirichlet spaces $D_{\beta}^q (0<q<\infty,\, -1<\beta<\infty)$ and derivative Hardy spaces $S^p (0<p<\infty)$, the order boundedness of weighted composition operators $W_{\phi,\varphi}$ between weighted Dirichlet spaces $D_{\beta}^q$ and derivative Hardy spaces $S^p$ are completely characterized.

An adjacent vertex distinguishing edge-colorings of a graph $G$ is a proper edge coloring of $G$ such that any pair of adjacent vertices have distinct sets of colors. The minimum number of color required for an adjacent vertex distinguishing edge-coloring of $G$ is denoted by $\chi_{a}'(G)$. In this paper, we prove that if $G$ is a planar graph with girth at least 5 and without isolated edges, then $\chi_a'(G)\leq$ max$\{8,\Delta(G)+1\}$.

We study the geometry and classification problems of biharmonic hypersurfaces in a pseudo-Euclidean 5-space $\mathbb{E}^5_s \ (s=1,2,3,4)$. We prove that if $M^4_{r}$ is a nondegenerate hypersurface in $\mathbb{E}^5_s$ with diagonal shape operator, then $M^4_{r}$ is minimal. Furthermore, based on the results due to Turgay et al. we show that any Lorentz biharmonic hypersurfaces in $\mathbb{E}^5_1$ is minimal. This result gives supporting answers to the biharmonic conjecture for hypersurfaces in 5-dimensional pseudo-Euclidean space.

Let q be a power of the prime, m ≥ 2 be an integer and p₁, p₂ be two distinct odd primes with gcd(q, p₁p₂) = 1 and m | gcd(p₁ - 1, p₂ - 1). Based on the idea of m-th residues, the present paper gives two constructions for the m-th residue code with length n = p₁p₂ over finite fields. For each construction, a necessary and sufficient condition for the q-ary m-th residue code and the corresponding counting formula are given. Furthermore, a criterion for that these codes are self-orthogonal or complementary dual is obtained, respectively. In some cases, a lower bound of the minimal distance for these codes is obtained.

We study transcendental meromorphic solutions of a certain type of nonlinear complex differential equations $f^{4}+a(z)ff^{(k)}=p_{1}(z){\rm e}^{\alpha_{1}(z)}+p_{2}(z){\rm e}^{\alpha_{2}(z)}$, where $a$, $p_{1}$, $p_{2}$ are non-zero rational functions and $\alpha_{1}$, $\alpha_{2}$ are nonconstant polynomials. Further, we can derive the conditions concerning the terms $\alpha_{1}$, $\alpha_{2}$, $p_{1}$ and $p_{2}$ that are necessary for the existence and the form of a transcendental meromorphic solution of the equation above. In addition, we analyze the existence of meromorphic solutions of another type of nonlinear complex differential equations $f^{3}+a(z)f'=p_{1}(z){\rm e}^{\nu(z)}+p_{2}(z){\rm e}^{-\nu(z)}$, where $a$, $p_{1}$, $p_{2}$ are non-zero rational functions and $\nu$ is a nonconstant polynomial. These results extend some known results obtained most recently.

This paper mainly studies the Hermite ring conjecture on valuation rings. According to the properties of the univariate polynomial ring $V[x]$ on the valuation ring $V$, we investigate and obtain a series of equivalent properties for the unimodular row vector $(a_1(x),a_2(x),\ldots,a_n(x))$ on $V[x]$. And then we prove that the Hermite ring conjecture on the valuation ring holds, that is, for an arbitrary valuation ring $V$, $V[x]$ is a Hermite ring.

We consider stochastic differential equations (SDEs) driven by mixed fractional Brownian motions under non-Lipschitz conditions. The mixed fractional Brownian motion is a linear combination of Brownian motion and fractional Brownian moiton. We give the p-th moment estimates and the continuity for solutions of considered SDEs by divergence-type Itô formula and Malliavin calculus for mixed fractional Brownianmotion.

In this paper, we investigate the normalized torsion units of the integral group ring of the direct product of the symmetric group $S_5$ and the cyclic group $C_3$. As a consequence, we confirm the Zassenhaus's conjecture about this group.

In the theory of formations of finite soluble groups, Bryce and Cossey proved an important theorem: A soluble local formation $\frak{F}$ is a Fitting class if and only if every value of the canonical formation function $F$ of $\frak{F}$ is a Fitting class. In this paper, basing on the theory of $\sigma$-groups, we generalized the results of Bryce and Cossey. We proved that A $\sigma$-local formation $\frak{F}$ is a Fitting class if and only if every value of the canonical $\sigma$-local definition $F$ of $\frak{F}$ is a Fitting class.

Let $\mathscr{H}$ be complex separable infinite-dimensional Hilbert spaces. Given the operators $A\in\mathscr{B}(\mathscr{H})$ and $B\in\mathscr{B}(\mathscr{H}),$ we define $M_{X}:= \begin{bmatrix}\begin{smallmatrix} A& X\\ 0& B \end{smallmatrix}\end{bmatrix}$ where $X\in \mathscr{S}(\mathscr{H})$ is a self-adjoint operator. In this paper, a necessary and sufficient condition is given for $M_{X}$ to be a left (right) Fredholm operator for some $X\in\mathscr{S}(\mathscr{H})$. Moreover, it is shown that \[\begin{array}{l} \bigcap\limits_{X\in\mathscr{S}(\mathscr{H})} \sigma_{*}(M_{X}) =\bigcap\limits_{X\in\mathscr{B}(\mathscr{H})} \sigma_{*}(M_{X})\cup\Delta, \end{array} \] where $\sigma_{*}$ is the left (right) essential spectrum. Finally, we further characterize the perturbation of the left (right) essential spectrum for Hamiltonian operators.

We consider the growth of solutions of differential-difference equation
$f^{n}(z)+q(z){\rm e}^{Q(z)}f^{(k)}(z+c)=p_{1}{\rm e}^{\alpha_{1}z}+p_{2}{\rm e}^{\alpha_{2}z}$
and
$f^{n}(z)+q(z){\rm e}^{Q(z)}\Delta_{c}f=p_{1}{\rm e}^{\lambda{z}}+p_{2}{\rm e}^{-\lambda{z}}$,
where $n\geq{1}$ and $k\geq{1}$ are two integers, $q(z)$ is a non-zero polynomial and $Q(z)$ is a non-constant polynomial. $c,\lambda,\alpha_{1},\alpha_{2},p_{1}$ and $p_{2}$ are non-zero constants, $\alpha_{1}\neq{\alpha_{2}}$. In particular, we show that exponential polynomial solutions satisfying certain conditions must reduce to rather specific forms, which is an improvement of previous results.

We investigate and obtain a necessary and sufficient condition for three Hankel operators on the Hardy spaces to be commuting, that is, suppose $f, g$ and $u$ are nonconstant functions in $\bigcap_{q>1}H^{q}$, then $H_{\bar{f}},H_{\bar{g}}$ and $H_{\bar{u}}$ commute if and only if a nontrivial linear combination of any two functions of $f, g, u$ is constant.

For $\alpha>-1$, let $A_\alpha^2(\mathbb{B}_N)$ be the weighted Bergman space on the unit ball $\mathbb{B}_N$ in $\mathbb{C}^N$. We prove that the multiplication operator $M_{z^n}$ is quasi-similar to $\bigoplus_1^{\prod_{i=1}^N n_i}M_z$ on $A_\alpha^2(\mathbb{B}_N)$ for the multi-index $n=(n_1,n_2,\ldots,n_N)$.

In this paper, we obtain gradient estimates for positive solutions of two nonlinear parabolic equations as follows $\frac{\partial u}{\partial t}=\Delta_V u+au\log u$ and $(\Delta_V-\frac{\partial}{\partial t})u(x,t)+p(x,t)u^\beta(x,t)+q(x,t)u(x,t)=0$ where $\alpha,\, \beta\in\mathbb{R}$, $\Delta_V(\cdot):=\Delta+\langle V,\nabla(\cdot)\rangle$ on the complete Riemannian manifold with Bakry--Emery Ricci curvature bounded below. Due to the introduction of $\Delta_V$, the Laplacian comparison theorem is replaced by the $V$-Laplacian comparison theorem in the process of proving the gradient estimates. Applications of these estimates yield Harnack inequalities and Liouville type theorem.

We investigate the differentiability and compactness of the $C_0$-semigroup generated by the $k/G:N$ redundant system with finite repair time. We show that the $C_0$-semigroup is eventually differentiable and eventually compact when the repair time is finite. However, this is not true for the case when the repair time is infinite.

We study the quasi-minimizers to nonhomogeneous energy functional on metric measure spaces. Assuming that the metric spaces satisfy doubling condition and Poincar\'{e} inequality, local boundedness for quasi-minimizers is obtained by establishing Caccioppoli inequality and De Giorgi iteration.