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 $1\<p\<\infty$. 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 $\varphi$ be a substitution of Pisot type on the alphabet $𝒜=\{1,2,...,d\}$; $\varphi$ satisfies theif for every $i,j\in 𝒜$, there are integers $k,n$ such that ${\varphi }^n\left(i\right)$ and ${\varphi }^n\left(j\right)$ have the same $k$-th letter, and the prefixes of length $k-1$ of ${\varphi }^n\left(i\right)$ and ${\varphi }^n\left(j\right)$ have the same image under the abelianization map. We prove that the strong coincidence condition is satisfied if $d=2$ and provide a partial result for $d\ge 2$.

Let $\mu$ be a probability measure on $[0,1]$ which is invariant and ergodic for $T_a\left(x\right)=ax\mathrm{𝚖𝚘𝚍}1$, and $0<\mathrm{𝚍𝚒𝚖}\mu <1$. Let $f$ be a local diffeomorphism on some open set. We show that if $E\subseteq ℝ$ and $\left(f\mu \right){\left|{}_E\sim \mu \right|}_E$, then $f^{\text{'}}\left(x\right)\in \left\{\pm a^r:r\in \mathbb{Q}\right\}$ at $\mu$-a.e. point $x\in f^{-1}E$. In particular, if $g$ is a piecewise-analytic map preserving $\mu$ then there is an open $g$-invariant set $U$ containing supp $\mu$ such that $g\left|{}_U\right.$ is piecewise-linear with slopes which are rational powers of $a$. In a similar vein, for $\mu$ as above, if $b$ is another integer and $a,b$ are not powers of a common integer, and if $\nu$ is...

In this paper, we consider the linear and circular consecutive $k$-out-of-$n$ systems consisting of arbitrarily dependent components. Under the condition that at least $n-r+1$ components ($r\le n$) of the system are working at time $t$, we study the reliability properties of the residual lifetime of such systems. Also, we present some stochastic ordering properties of residual lifetime of consecutive $k$-out-of-$n$ systems. In the following, we investigate the inactivity time of the component with lifetime...

The aim of this paper is to establish an existence and uniqueness result for a class of the set functional differential equations of neutral type$$\left\{\begin{array}{c}{D}_{H}X\left(t\right)=F(t,X_t,D_H{X}_t),\hfill \\ {X|}_{[-r,0]}=\Psi ,\hfill \end{array}\right.$$ where $F:[0,b]\times {𝒞}_0\times {𝔏}_0^1\to K_c\left(E\right)$ is a given function, $K_c\left(E\right)$ is the family of all nonempty compact and convex subsets of a separable Banach space $E$, ${𝒞}_0$ denotes the space of all continuous set-valued functions $X$ from $[-r,0]$ into $K_c\left(E\right)$, ${𝔏}_0^1$ is the space of all integrally bounded set-valued functions $X:[-r,0]\to K_c\left(E\right)$, $\Psi \in {𝒞}_0$ and $D_H$ is the Hukuhara derivative. The continuous dependence of solutions on initial...

For a real number $q>1$ and a positive integer $m$, let $Y_m\left(q\right):=\left\{{\sum }_{i=0}^n{ϵ}_i{q}^i:{ϵ}_i\in \left\{0,\pm 1,...,\pm m\right\},n=0,1,...\right\}$. In this paper, we show that $Y_m\left(q\right)$ is dense in $ℝ$ if and only if $q<m+1$ and $q$ is not a Pisot number. This completes several previous results and answers an open question raised by Erdös, Joó and Komornik [8].

Let $\Omega \in L^s\left({\mathrm{S}}^{n-1}\right)$ for $s\ge 1$ be a homogeneous function of degree zero and $b$ a BMO function. The commutator generated by the Marcinkiewicz integral ${\mu }_{\Omega }$ and $b$ is defined by $${\displaystyle [b,{\mu }_{\Omega }]\left(f\right)\left(x\right)=({\int }_0^{\infty }{\left|{\int }_{|x-y|\le t}\frac{\Omega (x-y)}{{|x-y|}^{n-1}}[b\left(x\right)-b\left(y\right)]f\left(y\right)\mathrm{d}y{|}^2\frac{\mathrm{d}t}{t^3}\right)}^{1/2}.}$$ In this paper, the author proves the $(L^{p(·)}\left({ℝ}^n\right),L^{p(·)}\left({ℝ}^n\right))$-boundedness of the Marcinkiewicz integral operator ${\mu }_{\Omega }$ and its commutator $[b,{\mu }_{\Omega }]$ when $p(·)$ satisfies some conditions. Moreover, the author obtains the corresponding result about ${\mu }_{\Omega }$ and $[b,{\mu }_{\Omega }]$ on Herz spaces with variable exponent.

Let $S_n^{\left(2\right)}$ denote the iterated partial sums. That is, $S_n^{\left(2\right)}=S_1+S_2+\cdots +S_n$, where $S_i=X_1+X_2+\cdots +X_i$. Assuming $X_1,X_2,...,X_n$ are integrable, zero-mean, i.i.d. random variables, we show that the persistence probabilities $$p_n^{\left(2\right)}:=\mathbb{P}\left(\underset{1\le i\le n}{max}{S}_i^{\left(2\right)}lt;0\right)\le c\sqrt{\frac{𝔼|S_{n+1}|}{(n+1)𝔼|X_1|}},$$ with $c\le 6\sqrt{30}$ (and $c=2$ whenever $X_1$ is symmetric). The converse inequality holds whenever the non-zero $min(-X_1,0)$ is bounded or when it has only finite third moment and in addition $X_1$ is squared integrable. Furthermore, $p_n^{\left(2\right)}\asymp n^{-1/4}$ for any non-degenerate squared integrable, i.i.d., zero-mean $X_i$. In contrast, we show that for any $0lt;\gamma lt;1/4$ there exist integrable,...

We consider “nonconventional” averaging setup in the form $\frac{\mathrm{d}{X}^{\epsilon }\left(t\right)}{\mathrm{d}t}=\epsilon B(X^{\epsilon }\left(t\right)$, $𝛯\left(q_1\left(t\right)\right),𝛯\left(q_2\left(t\right)\right),...,𝛯\left(q_{\ell }\left(t\right)\right))$ where $𝛯\left(t\right)$, $t\ge 0$ is either a stochastic process or a dynamical system with sufficiently fast mixing while $q_j\left(t\right)={\alpha }_{j}t$, ${\alpha }_{1}lt;{\alpha }_{2}lt;\cdots lt;{\alpha }_k$ and $q_j$, $j=k+1,...,\ell$ grow faster than linearly. We show that the properly normalized error term in the “nonconventional” averaging principle is asymptotically Gaussian.

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...

The seminal theorem of Cobham has given rise during the last 40 years to a lot of work about non-standard numeration systems and has been extended to many contexts. In this paper, as a result of fifteen years of improvements, we obtain a complete and general version for the so-called substitutive sequences. Let $\alpha$ and $\beta$ be two multiplicatively independent Perron numbers. Then a sequence $x\in A^{\mathbb{N}}$, where $A$ is a finite alphabet, is both $\alpha$-substitutive and $\beta$-substitutive if and only if $x$ is ultimately...