Let $l⩾2$ be an integer. Recently, Hu and Lü offered the asymptotic formula for the sum of the higher divisor function $$\sum _{1⩽n_1,n_2,...,n_l⩽x^{1/2}}{\tau }_k(n_1^2+n_2^2+\cdots +n_l^2),$$ where ${\tau }_k\left(n\right)$ represents the $k$th divisor function. We give the Goldbach-type analogy of their result. That is to say, we investigate the asymptotic behavior of the sum $$\sum _{1⩽p_1,p_2,...,p_l⩽x}{\tau }_k(p_1+p_2+\cdots +p_l),$$ where $p_1,p_2,\cdots ,p_l$ are prime variables.

A new derivation of the classic asymptotic expansion of the $n$-th prime is presented. A fast algorithm for the computation of its terms is also given, which will be an improvement of that by Salvy (1994). Realistic bounds for the error with ${li}^{-1}\left(n\right)$, after having retained the first $m$ terms, for $1\le m\le 11$, are given. Finally, assuming the Riemann Hypothesis, we give estimations of the best possible $r_3$ such that, for $n\ge r_3$, we have $p_n\>s_3\left(n\right)$ where $s_3\left(n\right)$ is the sum of the first four terms of the asymptotic...

We consider certain class of second order nonlinear nonautonomous delay differential equations of the form $$a\left(t\right)x^{\text{'}\text{'}}+b\left(t\right)g(x,x^{\text{'}})+c\left(t\right)h\left(x(t-r)\right)m\left(x^{\text{'}}\right)=p(t,x,x^{\text{'}})$$ and $${\left(a\left(t\right)x^{\text{'}}\right)}^{\text{'}}+b\left(t\right)g(x,x^{\text{'}})+c\left(t\right)h\left(x(t-r)\right)m\left(x^{\text{'}}\right)=p(t,x,x^{\text{'}}),$$ where $a$, $b$, $c$, $g$, $h$, $m$ and $p$ are real valued functions which depend at most on the arguments displayed explicitly and $r$ is a positive constant. Different forms of the integral inequality method were used to investigate the boundedness of all solutions and their derivatives. Here, we do not require construction of the Lyapunov-Krasovski functional to establish our results....

Let $K_{s,t}$ be the complete bipartite graph with partite sets $\{x_1,...,x_s\}$ and $\{y_1,...,y_t\}$. A split bipartite-graph on $(s+s^{\text{'}})+(t+t^{\text{'}})$ vertices, denoted by ${\mathrm{SB}}_{s+s^{\text{'}},t+t^{\text{'}}}$, is the graph obtained from $K_{s,t}$ by adding $s^{\text{'}}+t^{\text{'}}$ new vertices $x_{s+1},...,x_{s+s^{\text{'}}}$, $y_{t+1},...,y_{t+t^{\text{'}}}$ such that each of $x_{s+1},...,x_{s+s^{\text{'}}}$ is adjacent to each of $y_1,...,y_t$ and each of $y_{t+1},...,y_{t+t^{\text{'}}}$ is adjacent to each of $x_1,...,x_s$. Let $A$ and $B$ be nonincreasing lists of nonnegative integers, having lengths $m$ and $n$, respectively. The pair $(A;B)$ is potentially ${\mathrm{SB}}_{s+s^{\text{'}},t+t^{\text{'}}}$-bigraphic if there is a simple bipartite graph containing ${\mathrm{SB}}_{s+s^{\text{'}},t+t^{\text{'}}}$ (with $s+s^{\text{'}}$ vertices $x_1,...,x_{s+s^{\text{'}}}$ in the part of size $m$...

In this paper, we study a model for the magnetization in thin ferromagnetic films. It comes as a variational problem for $S^1$-valued maps $m^{\text{'}}$ (the magnetization) of two variables $x^{\text{'}}$: $E_{\epsilon }\left(m^{\text{'}}\right)=\epsilon \int {|{\nabla }^{\text{'}}·m^{\text{'}}|}^{2}d{x}^{\text{'}}+\frac{1}{2}\int {\left||{\nabla }^{\text{'}}{|}^{-1/2}{\nabla }^{\text{'}}·m^{\text{'}}\right|}^{2}d{x}^{\text{'}}$. We are interested in the behavior of minimizers as $\epsilon \to 0$. They are expected to be $S^1$-valued maps $m^{\text{'}}$ of vanishing distributional divergence ${\nabla }^{\text{'}}·m^{\text{'}}=0$, so that appropriate boundary conditions enforce line discontinuities. For finite $\epsilon >0$, these line discontinuities are approximated by smooth transition layers, the so-called Néel...

The half-linear differential equation $$\left(\right|u^{\text{'}}{|}^{\alpha }\mathrm{sgn}{u}^{\text{'}}{)}^{\text{'}}=\alpha ({\lambda }^{\alpha +1}+b\left(t\right)){\left|u\right|}^{\alpha }\mathrm{sgn}u,t\ge t_0,$$ is considered, where $\alpha$ and $\lambda$ are positive constants and $b\left(t\right)$ is a real-valued continuous function on $[t_0,\infty )$. It is proved that, under a mild integral smallness condition of $b\left(t\right)$ which is weaker than the absolutely integrable condition of $b\left(t\right)$, the above equation has a nonoscillatory solution $u_0\left(t\right)$ such that $u_0\left(t\right)\sim {\mathrm{e}}^{-\lambda t}$ and $u_0^{\text{'}}\left(t\right)\sim -\lambda {\mathrm{e}}^{-\lambda t}$ ($t\to \infty$), and a nonoscillatory solution $u_1\left(t\right)$ such that $u_1\left(t\right)\sim {\mathrm{e}}^{\lambda t}$ and $u_1^{\text{'}}\left(t\right)\sim \lambda {\mathrm{e}}^{\lambda t}$ ($t\to \infty$).

Let d be a positive integer and μ a generalized Cantor measure satisfying $\mu ={\sum }_{j=1}^m{a}_j\mu \circ S_j^{-1}$, where $0<a_j<1$, ${\sum }_{j=1}^m{a}_j=1$, $S_j=\rho R+b_j$ with 0 < ρ < 1 and R an orthogonal transformation of ${ℝ}^d$. Then ⎧1 < p ≤ 2 ⇒ ⎨$su{p}_{r>0}{r}^{d(1/{\alpha }^{\text{'}}-1/p^{\text{'}})}({\int }_{J_x^r}{\left|\mu ̂\left(y\right)\right|}^{p^{\text{'}}}{dy)}^{1/p^{\text{'}}}\le D₁{\rho }^{-d/{\alpha }^{\text{'}}}$, $x\in {ℝ}^d$, ⎩ p = 2 ⇒ infr≥1 rd(1/α’-1/2) (∫J₀r|μ̂(y)|² dy)1/2 ≥ D₂ρd/α’$,$where $J_x^r={\prod }_{i=1}^d(x_i-r/2,x_i+r/2)$, α’ is defined by ${\rho }^{d/{\alpha }^{\text{'}}}={\left({\sum }_{j=1}^m{a}_j^p\right)}^{1/p}$ and the constants D₁ and D₂ depend only on d and p.

We consider elementary operators $x\mapsto {\sum }_{j=1}^n{a}_{j}x{b}_j$, acting on a unital Banach algebra, where $a_j$ and $b_j$ are separately commuting families of generalized scalar elements. We give an ascent estimate and a lower bound estimate for such an operator. Additionally, we give a weak variant of the Fuglede-Putnam theorem for an elementary operator with strongly commuting families $a_j$ and $b_j$, i.e. $a_j=a_j^{\text{'}}+i{a}_j^{\text{'}\text{'}}$ ($b_j=b_j^{\text{'}}+i{b}_j^{\text{'}\text{'}}$), where all $a_j^{\text{'}}$ and $a_j^{\text{'}\text{'}}$ ($b_j^{\text{'}}$ and $b_j^{\text{'}\text{'}}$) commute. The main tool is an L¹ estimate of the Fourier transform of a certain class...

We determine the duals of the homogeneous matrix-weighted Besov spaces ${Ḃ}_p^{\alpha q}\left(W\right)$ and ${ḃ}_p^{\alpha q}\left(W\right)$ which were previously defined in [5]. If W is a matrix $A_p$ weight, then the dual of ${Ḃ}_p^{\alpha q}\left(W\right)$ can be identified with ${Ḃ}_{p^{\text{'}}}^{-\alpha q^{\text{'}}}\left(W^{-p^{\text{'}}/p}\right)$ and, similarly, $\left[{ḃ}_p^{\alpha q}\left(W\right)\right]*\approx {ḃ}_{p^{\text{'}}}^{-\alpha q^{\text{'}}}\left(W^{-p^{\text{'}}/p}\right)$. Moreover, for certain W which may not be in the $A_p$ class, the duals of ${Ḃ}_p^{\alpha q}\left(W\right)$ and ${ḃ}_p^{\alpha q}\left(W\right)$ are determined and expressed in terms of the Besov spaces ${Ḃ}_{p^{\text{'}}}^{-\alpha q^{\text{'}}}\left(A_Q^{-1}\right)$ and ${ḃ}_{p^{\text{'}}}^{-\alpha q^{\text{'}}}\left(A_Q^{-1}\right)$, which we define in terms of reducing operators ${A_Q}_Q$ associated with W. We also develop the basic theory of these reducing operator Besov spaces....

For a double complex $(A,d^{\text{'}},d^{\text{'}\text{'}})$, we show that if it satisfies the $d^{\text{'}}{d}^{\text{'}\text{'}}$-lemma and the spectral sequence $\left\{E_r^{p,q}\right\}$ induced by $A$ does not degenerate at $E_0$, then it degenerates at $E_1$. We apply this result to prove the degeneration at $E_1$ of a Hodge-de Rham spectral sequence on compact bi-generalized Hermitian manifolds that satisfy a version of $d^{\text{'}}{d}^{\text{'}\text{'}}$-lemma.

Criteria for oscillatory behavior of solutions of fourth order half-linear differential equations of the form $$\left(\right|y^{\text{'}\text{'}}{|}^{\alpha }\mathrm{sgn}{y}^{\text{'}\text{'}}{)}^{\text{'}\text{'}}+q\left(t\right){\left|y\right|}^{\alpha }\mathrm{sgn}y=0,t\ge a>0,A$$ where $\alpha >0$ is a constant and $q\left(t\right)$ is positive continuous function on $[a,\infty )$, are given in terms of an increasing continuously differentiable function $\omega \left(t\right)$ from $[a,\infty )$ to $(0,\infty )$ which satisfies ${\int }_a^{\infty }1/\left(t\omega \left(t\right)\right)dt<\infty$.

A couple ($\sigma ,\tau$) of lower and upper slopes for the resonant second order boundary value problem $$x^{\text{'}\text{'}}=f(t,x,x^{\text{'}}),x\left(0\right)=0,x^{\text{'}}\left(1\right)={\int }_0^1{x}^{\text{'}}\left(s\right)\mathrm{d}g\left(s\right),$$ with $g$ increasing on $[0,1]$ such that ${\int }_0^{1}dg=1$, is a couple of functions $\sigma ,\tau \in C^1\left([0,1]\right)$ such that $\sigma \left(t\right)\le \tau \left(t\right)$ for all $t\in [0,1]$, $$\begin{array}{c}{\sigma }^{\text{'}}\left(t\right)\ge f(t,x,\sigma \left(t\right)),\sigma \left(1\right)\le {\int }_0^1\sigma \left(s\right)\mathrm{d}g\left(s\right),\\ {\tau }^{\text{'}}\left(t\right)\le f(t,x,\tau \left(t\right)),\tau \left(1\right)\ge {\int }_0^1\tau \left(s\right)\mathrm{d}g\left(s\right),\end{array}$$ in the stripe ${\int }_0^t\sigma \left(s\right)\mathrm{d}s\le x\le {\int }_0^t\tau \left(s\right)\mathrm{d}s$ and $t\in [0,1]$. It is proved that the existence of such a couple $(\sigma ,\tau )$ implies the existence and localization of a solution to the boundary value problem. Multiplicity results are also obtained.

Let $p_1,p_2,\cdots$ be the sequence of all primes in ascending order. Using explicit estimates from the prime number theory, we show that if $k\ge 5$, then $$(p_{k+1}-1)!\mid \left(\frac{1}{2}(p_{k+1}-1)\right)!p_k!,$$ which improves a previous result of the second author.

We study the existence of solutions of the system $$\left\{\begin{array}{c}{\left({\phi }_1\left(u_1^{\text{'}}\left(t\right)\right)\right)}^{\text{'}}=f_1(t,u_1\left(t\right),u_2\left(t\right),u_1^{\text{'}}\left(t\right),u_2^{\text{'}}\left(t\right)),\text{a.e.}t\in [0,T],{\left({\phi }_2\left(u_2^{\text{'}}\left(t\right)\right)\right)}^{\text{'}}=f_2(t,u_1\left(t\right),u_2\left(t\right),u_1^{\text{'}}\left(t\right),u_2^{\text{'}}\left(t\right)),\text{a.e.}t\in [0,T],\hfill \end{array}\right.$$ submitted to nonlinear coupled boundary conditions on $[0,T]$ where ${\phi }_1,{\phi }_2:(-a,a)\to ℝ$, with $0<a<+\infty$, are two increasing homeomorphisms such that ${\phi }_1\left(0\right)={\phi }_2\left(0\right)=0$, and $f_i:[0,T]\times {ℝ}^4\to ℝ$, $i\in \{1,2\}$ are two $L^1$-Carathéodory functions. Using some new conditions and Schauder fixed point Theorem, we obtain solvability result.

Let A denote the space of all analytic functions in the unit disc Δ with the normalization f(0) = f’(0) − 1 = 0. For β < 1, let $P{⁰}_{\beta }=f\in A:Re{f}^{\text{'}}\left(z\right)>\beta ,z\in \Delta$. For λ > 0, suppose that denotes any one of the following classes of functions: $M_{1,\lambda }^{\left(1\right)}=f\in :Rez{\left(z{f}^{\text{'}}\left(z\right)\right)}^{\text{'}\text{'}}>-\lambda ,z\in \Delta$, $M_{1,\lambda }^{\left(2\right)}=f\in :Rez{\left(z²f^{\text{'}\text{'}}\left(z\right)\right)}^{\text{'}\text{'}}>-\lambda ,z\in \Delta$, $M_{1,\lambda }^{\left(3\right)}=f\in :Re1/2{\left(z{\left(z²f^{\text{'}}\left(z\right)\right)}^{\text{'}\text{'}}\right)}^{\text{'}}-1>-\lambda ,z\in \Delta$. The main purpose of this paper is to find conditions on λ and γ so that each f ∈ is in ${}_{\gamma }$ or ${}_{\gamma }$, γ ∈ [0,1/2]. Here ${}_{\gamma }$ and ${}_{\gamma }$ respectively denote the class of all starlike functions of order γ and the class of all convex functions of order γ. As a consequence, we obtain...