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...

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].

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 $F(x_1,x_2)$.

It is proved that real functions on $ℝ$ which can be represented as the difference of two semiconvex functions with a general modulus (or of two lower $C^1$-functions, or of two strongly paraconvex functions) coincide with semismooth functions on $ℝ$ (i.e. those locally Lipschitz functions on $ℝ$ for which $f_{+}^{\text{'}}\left(x\right)={lim}_{t\to x+}{f}_{+}^{\text{'}}\left(t\right)$ and $f_{-}^{\text{'}}\left(x\right)={lim}_{t\to x-}{f}_{-}^{\text{'}}\left(t\right)$ for each $x$). Further, for each modulus $\omega$, we characterize the class $DS{C}_{\omega }$ of functions on $ℝ$ which can be written as $f=g-h$, where $g$ and $h$ are semiconvex with modulus $C\omega$ (for some $C>0$) using a new...

Let $G$ be a group and $\omega \left(G\right)$ be the set of element orders of $G$. Let $k\in \omega \left(G\right)$ and $m_k\left(G\right)$ be the number of elements of order $k$ in $G$. Let nse$\left(G\right)=\{m_k\left(G\right):k\in \omega \left(G\right)\}$. Assume $r$ is a prime number and let $G$ be a group such that nse$\left(G\right)=$ nse$\left(S_r\right)$, where $S_r$ is the symmetric group of degree $r$. In this paper we prove that $G\cong S_r$, if $r$ divides the order of $G$ and $r^2$ does not divide it. To get the conclusion we make use of some well-known results on the prime graphs of finite simple groups and their components.

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....

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...

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...

Let $\Omega$ be a bounded domain of class $C^2$ in $ℝ$N and let $K$ be a compact subset of $\partial \Omega$. Assume that $q\ge (N+1)/\left(N-1\right)$ and denote by $U_K$ the maximal solution of $-\Delta u+u^q=0$ in $\Omega$ which vanishes on $\partial \Omega \setminus K$. We obtain sharp upper and lower estimates for $U_K$ in terms of the Bessel capacity $C_{2/q,q^{\text{'}}}$ and prove that $U_K$ is $\sigma$-moderate. In addition we describe the precise asymptotic behavior of $U_K$ at points $\sigma \in K$, which depends on the “density” of $K$ at $\sigma$, measured in terms of the capacity $C_{2/q,q^{\text{'}}}$.

Let $R$ be a prime ring of characteristic different from 2, $Q_r$ its right Martindale quotient ring and $C$ its extended centroid. Suppose that $F$, $G$ are generalized skew derivations of $R$ with the same associated automorphism $\alpha$, and $p(x_1,...,x_n)$ is a non-central polynomial over $C$ such that $$\left[F\right(x),\alpha (y\left)\right]=G\left(\right[x,y\left]\right)$$ for all $x,y\in \{p(r_1,...,r_n):r_1,...,r_n\in R\}$. Then there exists $\lambda \in C$ such that $F\left(x\right)=G\left(x\right)=\lambda \alpha \left(x\right)$ for all $x\in R$.

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 $p$.

Let α,β,γ,δ ≥ 0 and ϱ:= γβ + αγ + αδ > 0. Let ψ(t) = β + αt, ϕ(t) = γ + δ - γt, t ∈ [0,1]. We study the existence of positive solutions for the m-point boundary value problem ⎧u” + h(t)f(u) = 0, 0 < t < 1, ⎨$\alpha u\left(0\right)-\beta u^{\text{'}}\left(0\right)={\sum }_{i=1}^{m-2}{a}_{i}u\left({\xi }_i\right)$ ⎩$\gamma u\left(1\right)+\delta u^{\text{'}}\left(1\right)={\sum }_{i=1}^{m-2}{b}_{i}u\left({\xi }_i\right)$, where ${\xi }_i\in (0,1)$, $a_i,b_i\in (0,\infty )$ (for i ∈ 1,…,m-2) are given constants satisfying $\varrho -{\sum }_{i=1}^{m-2}{a}_i\varphi \left({\xi }_i\right)>0$, $\varrho -{\sum }_{i=1}^{m-2}{b}_i\psi \left({\xi }_i\right)>0$ and $\Delta :=\left|\begin{array}{cc}-{\sum }_{i=1}^{m-2}{a}_i\psi \left({\xi }_i\right)& \varrho -{\sum }_{i=1}^{m-2}{a}_i\varphi \left({\xi }_i\right)\\ \varrho -{\sum }_{i=1}^{m-2}{b}_i\psi \left({\xi }_i\right)& -{\sum }_{i=1}^{m-2}{b}_i\varphi \left({\xi }_i\right)\end{array}\right|<0$. We show the existence of positive solutions if f is either superlinear or sublinear by a simple application of a fixed point theorem in cones. Our result extends a result established by Erbe and...

In analogy to the classical isomorphism between ((ℝⁿ), ${}^{\text{'}}\left({ℝ}^m\right))$ and ${}^{\text{'}}\left({ℝ}^{m+n}\right)$ (resp. $\left(\left(ℝⁿ\right){,}^{\text{'}}\left({ℝ}^m\right)\right)$ and ${}^{\text{'}}\left({ℝ}^{m+n}\right)$), we show that a large class of moderate linear mappings acting between the space ${}_C\left(ℝⁿ\right)$ of compactly supported generalized functions and (ℝⁿ) of generalized functions (resp. the space ${}_{}\left(ℝⁿ\right)$ of Colombeau rapidly decreasing generalized functions and the space ${}_{\tau }\left(ℝⁿ\right)$ of temperate ones) admits generalized integral representations, with kernels belonging to specific regular subspaces of $\left({ℝ}^{m+n}\right)$ (resp. ${}_{\tau }\left({ℝ}^{m+n}\right)$). The main novelty is to use...

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 $G$ be a finite group and let $\pi \left(G\right)=\{p_1,p_2,...,p_k\}$ be the set of prime divisors of $\left|G\right|$ for which $p_1<p_2<\cdots <p_k$. The Gruenberg-Kegel graph of $G$, denoted $GK\left(G\right)$, is defined as follows: its vertex set is $\pi \left(G\right)$ and two different vertices $p_i$ and $p_j$ are adjacent by an edge if and only if $G$ contains an element of order $p_i{p}_j$. The degree of a vertex $p_i$ in $\mathrm{GK}\left(G\right)$ is denoted by $d_G\left(p_i\right)$ and the $k$-tuple $D\left(G\right)=(d_G\left(p_1\right),d_G\left(p_2\right),...,d_G\left(p_k\right))$ is said to be the degree pattern of $G$. Moreover, if $\omega \subseteq \pi \left(G\right)$ is the vertex set of a connected component of $GK\left(G\right)$, then the largest $\omega$-number which divides $\left|G\right|$, is...

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 $p_n\&gt;s_3\left(n\right)$ where $s_3\left(n\right)$ is the sum of the first four terms of the asymptotic...