Per l'equazione differenziale ordinaria non lineare del 3° ordine indicata nel titolo, studiata da numerosi autori sotto l'ipotesi $h(x)\text{sgn}x\ge 0$ $for|x|>R$, si dimostra l'esistenza di almeno una soluzione limitata sopprimendo l'ipotesi suddetta.

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.

Per l'equazione differenziale ordinaria non lineare del 3° ordine indicata nel titolo, studiata da numerosi autori sotto l'ipotesi $h(x)\text{sgn}x\ge 0$ $for|x|>R$, si dimostra l'esistenza di almeno una soluzione limitata sopprimendo l'ipotesi suddetta.

We obtain estimates for the average value of the largest prime factor P(n) in short intervals [x,x+y] and of h(P(n)+1), where h is a complex-valued additive function or multiplicative function satisfying certain conditions. Letting $s_q\left(n\right)$ stand for the sum of the digits of n in base q ≥ 2, we show that if α is an irrational number, then the sequence ${\left(\alpha s_q\left(P\left(n\right)\right)\right)}_{n\in \mathbb{N}}$ is uniformly distributed modulo 1.

Improving on a theorem of Heath-Brown, we show that if X is sufficiently large then a positive proportion of the values n³ + 2: n ∈ (X,2X] have a prime factor larger than $X^{1+{10}^{-52}}$.

Viene dimostrata l'esistenza di soluzioni del problema di Darboux per l'equazione iperbolica $z_{xy}^{\mathrm{\prime \prime }}=f(x,y,z,Z_x^{\prime },z_y)$ sul planiquarto $x\ge 0$, $y\ge 0$. Qui, $f$ è una funzione continua, con valori in uno spazio Banach che soddisfano alcune condizioni di regolarità espresse in termini della misura di non-compattezza $\alpha$.

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 μ ∈ ℂ such that Re μ > 0 let ${ℱ}_{\mu }$ denote the class of all non-vanishing analytic functions f in the unit disk with f(0) = 1 and $Re(2\pi /\mu z{f}^{\text{'}}\left(z\right)/f\left(z\right)+(1+z)/(1-z))>0$ in . For any fixed z₀ in the unit disk, a ∈ ℂ with |a| ≤ 1 and λ ∈ ̅, we shall determine the region of variability V(z₀,λ) for log f(z₀) when f ranges over the class ${ℱ}_{\mu }\left(\lambda \right)=f\in {ℱ}_{\mu }:f^{\text{'}}\left(0\right)=(\mu /\pi )(\lambda -1)and{f}^{\text{'}\text{'}}\left(0\right)=(\mu /\pi )\left(a\right(1-\left|\lambda \right|²)+(\mu /\pi )(\lambda -1)²-(1-\lambda ²))$. In the final section we graphically illustrate the region of variability for several sets of parameters.

Let $G$ be a finite group and ${\pi }_e\left(G\right)$ be the set of element orders of $G$. Let $k\in {\pi }_e\left(G\right)$ and $m_k$ be the number of elements of order $k$ in $G$. Set $\mathrm{nse}\left(G\right):=\{m_k:k\in {\pi }_e\left(G\right)\}$. In fact $\mathrm{nse}\left(G\right)$ is the set of sizes of elements with the same order in $G$. In this paper, by $\mathrm{nse}\left(G\right)$ and order, we give a new characterization of finite projective special linear groups $L_2\left(p\right)$ over a field with $p$ elements, where $p$ is prime. We prove the following theorem: If $G$ is a group such that $\left|G\right|=|L_2\left(p\right)|$ and $\mathrm{nse}\left(G\right)$ consists of $1$, $p^2-1$, $p(p+ϵ)/2$ and some numbers divisible by $2p$, where $p$ is a prime...

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...

This paper concerns the following system of nonlinear third-order boundary value problem: $$u_i^{\text{'}\text{'}\text{'}}\left(t\right)+f_i(t,u_1\left(t\right),\cdots ,u_n\left(t\right),u_1^{\text{'}}\left(t\right),\cdots ,u_n^{\text{'}}\left(t\right))=0,0<t<1,i\in \{1,\cdots ,n\}$$ with the following multi-point and integral boundary conditions: $$\left\{\begin{array}{c}{u}_i\left(0\right)=0u_i^{\text{'}}\left(0\right)=0u_i^{\text{'}}\left(1\right)=\sum _{j=1}^p{\beta }_{j,i}{u}_i^{\text{'}}\left({\eta }_{j,i}\right)+{\int }_0^1{h}_i(u_1\left(s\right),\cdots ,u_n\left(s\right))ds\hfill \end{array}\right.$$ where ${\beta }_{j,i}>0$, $0<{\eta }_{1,i}<\cdots <{\eta }_{p,i}<\frac{1}{2}$, $f_i:[0,1]\times {ℝ}^n\times {ℝ}^n\to ℝ$ and $h_i:[0,1]\times {ℝ}^n\to ℝ$ are continuous functions for all $i\in \{1,\cdots ,n\}$ and $j\in \{1,\cdots ,p\}$. Using Guo-Krasnosel’skii fixed point theorem in cone, we discuss the existence of positive solutions of this problem. We also prove nonexistence of positive solutions and we give some examples to illustrate our results.

Some sufficient conditions on the existence and multiplicity of solutions for the damped vibration problems with impulsive effects ⎧ u”(t) + g(t)u’(t) + f(t,u(t)) = 0, a.e. t ∈ [0,T ⎨ u(0) = u(T) = 0 ⎩ $\Delta u^{\text{'}}\left(t_j\right)=u^{\text{'}}(t{⁺}_j-u^{\text{'}}\left(t{¯}_j\right)=I_j\left(u\left(t_j\right)\right)$, j = 1,...,p, are established, where $t₀=0<t₁<\cdots <t_p<t_{p+1}=T$, g ∈ L¹(0,T;ℝ), f: [0,T] × ℝ → ℝ is continuous, and $I_j:ℝ\to ℝ$, j = 1,...,p, are continuous. The solutions are sought by means of the Lax-Milgram theorem and some critical point theorems. Finally, two examples are presented to illustrate the effectiveness...

Let G be a finite group, and $M=D_p\left(3\right)$ (p ≥ 3). It is proved that G ≅ M if G and M have the same order components.

Let M ∈ Mₙ(ℤ) be expanding such that |det(M)| = p is a prime and pℤⁿ ⊈ M²(ℤⁿ). Let D ⊂ ℤⁿ be a finite set with |D| = |det(M)|. Suppose the attractor T(M,D) of the iterated function system ${{\varphi }_d\left(x\right)=M^{-1}(x+d)}_{d\in D}$ has positive Lebesgue measure. We prove that (i) if D ⊈ M(ℤⁿ), then D is a complete set of coset representatives of ℤⁿ/M(ℤⁿ); (ii) if D ⊆ M(ℤⁿ), then there exists a positive integer γ such that $D=M^{\gamma }D₀$, where D₀ is a complete set of coset representatives of ℤⁿ/M(ℤⁿ). This improves the corresponding results...