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 $u_1^{\text{'}}\left(t\right)\sim \lambda {\mathrm{e}}^{\lambda t}$ ($t\to \infty$).

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...

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.

Let $r(G_1,G_2)$ be the Ramsey number of the two graphs $G_1$ and $G_2$. For $n_1\ge n_2\ge 1$ let $S(n_1,n_2)$ be the double star given by $V\left(S(n_1,n_2)\right)=\{v_0,v_1,...,v_{n_1},w_0$, $w_1,...,w_{n_2}\}$ and $E\left(S(n_1,n_2)\right)=\{v_0{v}_1,...,v_0{v}_{n_1},v_0{w}_0,w_0{w}_1,...,w_0{w}_{n_2}\}$. We determine $r(K_{1,m-1},$ $S(n_1,n_2))$ under certain conditions. For $n\ge 6$ let $T_n^3=S(n-5,3)$, $T_n^{\text{'}\text{'}}=(V,E_2)$ and $T_n^{\text{'}\text{'}\text{'}}=(V,E_3)$, where $V=\{v_0,v_1,...,v_{n-1}\}$, $E_2=\{v_0{v}_1,...,v_0{v}_{n-4},v_1{v}_{n-3}$, $v_1{v}_{n-2},v_2{v}_{n-1}\}$ and $E_3=\{v_0{v}_1,...,v_0{v}_{n-4},v_1{v}_{n-3},$ $v_2{v}_{n-2},v_3{v}_{n-1}\}$. We also obtain explicit formulas for $r(K_{1,m-1},T_n)$, $r(T_m^{\text{'}},T_n)$ $(n\ge m+3)$, $r(T_n,T_n)$, $r(T_n^{\text{'}},T_n)$ and $r(P_n,T_n)$, where $T_n\in \{T_n^{\text{'}\text{'}},T_n^{\text{'}\text{'}\text{'}},T_n^3\}$, $P_n$ is the path on $n$ vertices and $T_n^{\text{'}}$ is the unique tree with $n$ vertices and maximal degree $n-2$.

Let $G$ be a finite group and write $\mathrm{cd}\left(G\right)$ for the degree set of the complex irreducible characters of $G$. The group $G$ is said to satisfy the two-prime hypothesis if for any distinct degrees $a,b\in \mathrm{cd}\left(G\right)$, the total number of (not necessarily different) primes of the greatest common divisor $gcd(a,b)$ is at most $2$. We prove an upper bound on the number of irreducible character degrees of a nonsolvable group that has a composition factor isomorphic to PSL${}_2\left(q\right)$ for $q\ge 7$.

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 $m$...

Let $\mathbb{Z}$ be the set of integers, ${\mathbb{N}}_0$ the set of nonnegative integers and $F(x_1,x_2)=u_1{x}_1+u_2{x}_2$ be a binary linear form whose coefficients $u_1$, $u_2$ are nonzero, relatively prime integers such that $u_1{u}_2\ne \pm 1$ and $u_1{u}_2\ne -2$. Let $f:\mathbb{Z}\to {\mathbb{N}}_0\cup \left\{\infty \right\}$ be any function such that the set $f^{-1}\left(0\right)$ has asymptotic density zero. In 2007, M. B. Nathanson (2007) proved that there exists a set $A$ of integers such that $r_{A,F}\left(n\right)=f\left(n\right)$ for all integers $n$, where $r_{A,F}\left(n\right)=\left|\{(a,a^{\text{'}}):n=u_{1}a+u_2{a}^{\text{'}}:a,a^{\text{'}}\in A\}\right|$. We add the structure of difference for the binary linear form $F(x_1,x_2)$.

We discuss the validity of the Helmholtz decomposition of the Muckenhoupt $A_p$-weighted $L^p$-space ${\left(L_w^p\left(\Omega \right)\right)}^n$ for any domain $\Omega$ in ${ℝ}^n$, $n\in \mathbb{Z}$, $n\ge 2$, $1<p<\infty$ and Muckenhoupt $A_p$-weight $w\in A_p$. Set $p^{\text{'}}:=p/(p-1)$ and $w^{\text{'}}:=w^{-1/(p-1)}$. Then the Helmholtz decomposition of ${\left(L_w^p\left(\Omega \right)\right)}^n$ and ${\left(L_{w^{\text{'}}}^{p^{\text{'}}}\left(\Omega \right)\right)}^n$ and the variational estimate of $L_{w,\pi }^p\left(\Omega \right)$ and $L_{w^{\text{'}},\pi }^{p^{\text{'}}}\left(\Omega \right)$ are equivalent. Furthermore, we can replace $L_{w,\pi }^p\left(\Omega \right)$ and $L_{w^{\text{'}},\pi }^{p^{\text{'}}}\left(\Omega \right)$ by $L_{w,\sigma }^p\left(\Omega \right)$ and $L_{w^{\text{'}},\sigma }^{p^{\text{'}}}\left(\Omega \right)$, respectively. The proof is based on the reflexivity and orthogonality of $L_{w,\pi }^p\left(\Omega \right)$ and $L_{w,\sigma }^p\left(\Omega \right)$ and the Hahn-Banach theorem. As a corollary of our main result, we obtain the extrapolation...

Given an integer $n\ge 3$, let $u_1,...,u_n$ be pairwise coprime integers $\ge 2$, $𝒟$ a family of nonempty proper subsets of $\{1,...,n\}$ with “enough” elements, and $\epsilon$ a function $𝒟\to \{\pm 1\}$. Does there exist at least one prime $q$ such that $q$ divides ${\prod }_{i\in I}{u}_i-\epsilon \left(I\right)$ for some $I\in 𝒟$, but it does not divide $u_1\cdots u_n$? We answer this question in the positive when the $u_i$ are prime powers and $\epsilon$ and $𝒟$ are subjected to certain restrictions. We use the result to prove that, if ${\epsilon }_0\in \{\pm 1\}$ and $A$ is a set of three or more primes that contains all prime divisors of any...

Let $K$ be a number field defined by an irreducible polynomial $F\left(X\right)\in \mathbb{Z}\left[X\right]$ and ${\mathbb{Z}}_K$ its ring of integers. For every prime integer $p$, we give sufficient and necessary conditions on $F\left(X\right)$ that guarantee the existence of exactly $r$ prime ideals of ${\mathbb{Z}}_K$ lying above $p$, where $\overline{F}\left(X\right)$ factors into powers of $r$ monic irreducible polynomials in ${𝔽}_p\left[X\right]$. The given result presents a weaker condition than that given by S. K. Khanduja and M. Kumar (2010), which guarantees the existence of exactly $r$ prime ideals of ${\mathbb{Z}}_K$ lying above $p$....

Let ${𝒫}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. Suppose that $a$ and $q$ are positive integers satisfying $(a,q)=1$. Denote by ${𝒫}_2(a,q)$ the least almost-prime ${𝒫}_2$ which satisfies ${𝒫}_2\equiv a(modq)$. It is proved that for sufficiently large $q$, there holds $${𝒫}_2(a,q)\ll q^{1.8345}.$$ This result constitutes an improvement upon that of Iwaniec (1982), who obtained the same conclusion, but for the range $1.845$ in place of $1.8345$.

In this paper, we examine a natural question concerning the divisors of the polynomial $x^n-1$: “How often does $x^n-1$ have a divisor of every degree between $1$ and $n$?” In a previous paper, we considered the situation when $x^n-1$ is factored in $\mathbb{Z}\left[x\right]$. In this paper, we replace $\mathbb{Z}\left[x\right]$ with ${𝔽}_p\left[x\right]$, where $p$ is an arbitrary-but-fixed prime. We also consider those $n$ where this condition holds for all $p$.

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...

For the dynamics $x^{\text{'}\text{'}}=-{\nabla }_{x}V\left(x\right)$, an equilibrium point $\underline{x}$ are always unstable when on a neighborhood of $\underline{x}$ the non constant $V$ satisfies $P(x,\partial )V=0$ for a real second order elliptic $P$. The proof uses a result of Kozlov [6].