A space $X$ is said to have the Rothberger property (or simply $X$ is Rothberger) if for every sequence $\langle {𝒰}_n:n\in \omega \rangle$ of open covers of $X$, there exists $U_n\in {𝒰}_n$ for each $n\in \omega$ such that $X={\bigcup }_{n\in \omega }{U}_n$. For any $n\in \omega$, necessary and sufficient conditions are obtained for $C_p{(\Psi \left(𝒜\right),2)}^n$ to have the Rothberger property when $𝒜$ is a Mrówka mad family and, assuming CH (the Continuum Hypothesis), we prove the existence of a maximal almost disjoint family $𝒜$ for which the space $C_p{(\Psi \left(𝒜\right),2)}^n$ is Rothberger for all $n\in \omega$.

We consider the equation $$-{\left(r\left(x\right)y^{\text{'}}\left(x\right)\right)}^{\text{'}}+q\left(x\right)y\left(x\right)=f\left(x\right),x\in ℝ,$$ where $f\in L_p\left(ℝ\right)$, $p\in (1,\infty )$ and $$r>0,\frac{1}{r}\in L_1^{\mathrm{loc}}\left(ℝ\right),q\in L_1^{\mathrm{loc}}\left(ℝ\right).$$ For particular equations of this form, we suggest some methods for the study of the question on requirements to the functions $r$ and $q$ under which the above equation is correctly solvable in the space $L_p\left(ℝ\right),$ $p\in (1,\infty ).$

We give existence theorems for weak and strong solutions with trichotomy of the nonlinear differential equation $$\dot{x}\left(t\right)=ℒ\left(t\right)x\left(t\right)+f(t,x\left(t\right)),t\in ℝ\left(\mathrm{P}\right)$$ where $\left\{ℒ\right(t):t\in ℝ\}$ is a family of linear operators from a Banach space $E$ into itself and $f:ℝ\times E\to E$. By $L\left(E\right)$ we denote the space of linear operators from $E$ into itself. Furthermore, for $a<b$ and $d>0$, we let $C\left(\right[-d,0],E)$ be the Banach space of continuous functions from $[-d,0]$ into $E$ and $f^d:[a,b]\times C([-d,0],E)\to E$. Let $\widehat{ℒ}:[a,b]\to L\left(E\right)$ be a strongly measurable and Bochner integrable operator on $[a,b]$ and for $t\in [a,b]$ define ${\tau }_{t}x\left(s\right)=x(t+s)$ for each $s\in [-d,0]$. We prove that, under certain...

Let $\alpha \&gt;1$ be irrational. Several authors studied the numbers $${\ell }^m\left(\alpha \right)=inf\left\{\right|y|:y\in {\Lambda }_m,y\ne 0\},$$ where $m$ is a positive integer and ${\Lambda }_m$ denotes the set of all real numbers of the form $y={ϵ}_0{\alpha }^n+{ϵ}_1{\alpha }^{n-1}+\cdots +{ϵ}_{n-1}\alpha +{ϵ}_n$ with restricted integer coefficients $|{ϵ}_i|\le m$. The value of ${\ell }^1\left(\alpha \right)$ was determined for many particular Pisot numbers and ${\ell }^m\left(\alpha \right)$ for the golden number. In this paper the value of ${\ell }^m\left(\alpha \right)$ is determined for irrational numbers $\alpha$, satisfying ${\alpha }^2=a\alpha \pm 1$ with a positive integer $a$.

We give new results on square functions $${\parallel x\parallel }_F=\parallel {\left({\int }_0^{\infty }{\left|F\left(tA\right)x\right|}^2\frac{\mathrm{d}t}{t}\right)}^{1/2}{\parallel }_p$$ associated to a sectorial operator $A$ on $L^p$ for $1\&lt;p\&lt;\infty$. Under the assumption that $A$ is actually $R$-sectorial, we prove equivalences of the form $K^{-1}{\parallel x\parallel }_G\le {\parallel x\parallel }_F\le K{\parallel x\parallel }_G$ for suitable functions $F,G$. We also show that $A$ has a bounded $H^{\infty }$ functional calculus with respect to ${\parallel .\parallel }_F$. Then we apply our results to the study of conditions under which we have an estimate $\parallel {({\int }_0^{\infty }|C{\mathrm{e}}^{-tA}{\left(x\right)|}^2\mathrm{d}t{)}^{1/2}\parallel }_q\le M{\parallel x\parallel }_p$, when $-A$ generates a bounded semigroup ${\mathrm{e}}^{-tA}$ on $L^p$ and $C:D\left(A\right)\to L^q$ is a linear mapping.

Let $q\ge 2$ and denote by $s_q$ the sum-of-digits function in base $q$. For $j=0,1,\cdots ,q-1$ consider $$\#\{0\le n\&lt;N:s_q\left(2n\right)\equiv j(modq)\}.$$ In 1983, F. M. Dekking conjectured that this quantity is greater than $N/q$ and, respectively, less than $N/q$ for infinitely many $N$, thereby claiming an absence of a drift (or Newman) phenomenon. In this paper we prove his conjecture.

Let $\Omega$ be a bounded domain in ${ℝ}^n$ with smooth boundary $\partial \Omega$. We consider the equation $d^2\Delta u-u+u^{\frac{n-k+2}{n-k-2}}=0\text{in}\Omega$, under zero Neumann boundary conditions, where $\Omega$ is open, smooth and bounded and $d$ is a small positive parameter. We assume that there is a $k$-dimensional closed, embedded minimal submanifold $K$ of $\partial \Omega$, which is non-degenerate, and certain weighted average of sectional curvatures of $\partial \Omega$ is positive along $K$. Then we prove the existence of a sequence $d=d_j\to 0$ and a positive solution $u_d$ such that $d^2{\left|\nabla u_d\right|}^2\rightharpoonup S{\delta }_K\mathrm{as}d\to 0$ in the sense of measures,...

In this paper, we introduce an explicit formula and discuss the global behavior of solutions of the difference equation $$x_{n+1}=\frac{a{x}_{n-3}}{b+c{x}_{n-1}{x}_{n-3}},n=0,1,\cdots$$ where $a,b,c$ are positive real numbers and the initial conditions $x_{-3}$, $x_{-2}$, $x_{-1}$, $x_0$ are real numbers.

Let $D$ be a ${𝒞}^d$ $q$-convex intersection, $d\ge 2$, $0\le q\le n-1$, in a complex manifold $X$ of complex dimension $n$, $n\ge 2$, and let $E$ be a holomorphic vector bundle of rank $N$ over $X$. In this paper, ${𝒞}^k$-estimates, $k=2,3,\cdots ,\infty$, for solutions to the $\overline{\partial }$-equation with small loss of smoothness are obtained for $E$-valued $(0,s)$-forms on $D$ when $n-q\le s\le n$. In addition, we solve the $\overline{\partial }$-equation with a support condition in ${𝒞}^k$-spaces. More precisely, we prove that for a $\overline{\partial }$-closed form $f$ in ${𝒞}_{0,q}^k(X\setminus D,E)$, $1\le q\le n-2$, $n\ge 3$, with compact support and for $\epsilon$ with $0<\epsilon <1$ there...

We will consider the following problem $$-\Delta u-\lambda \frac{u}{{\left|x\right|}^2}={\left|\nabla u\right|}^p+cf,u\&gt;0\text{in}\Omega ,$$ where $\Omega \subset {ℝ}^N$ is a domain such that $0\in \Omega$, $N\ge 3$, $c\&gt;0$ and $\lambda \&gt;0$. The main objective of this note is to study the precise threshold $p_{+}=p_{+}\left(\lambda \right)$ for which there is novery weak supersolutionif $p\ge p_{+}\left(\lambda \right)$. The optimality of $p_{+}\left(\lambda \right)$ is also proved by showing the solvability of the Dirichlet problem when $1\le p\&lt;p_{+}\left(\lambda \right)$, for $c\&gt;0$ small enough and $f\ge 0$ under some hypotheses that we will prescribe.

In this paper we prove the following theorems in incidence geometry. 1. There is $\delta >0$ such that for any $P_1,\cdots ,P_4$, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le n^{(1+\delta )/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then $P_1,\cdots ,P_4$ are collinear. If the number of the distinct lines is $<c{n}^{1/2}$ then the cross ratio of the four points is algebraic. 2. Given $c>0$, there is $\delta >0$ such that for any $P_1,P_2,P_3\in {ℂ}^2$ noncollinear, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le c{n}^{1/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then for any $P\in {ℂ}^2\setminus \{P_1,P_2,P_3\}$, we have $\delta n$ distinct lines between $P$ and $Q_j$. 3. Given...