We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
In this paper, we establish a translation theorem for the generalised analytic Feynman integral of functionals that belong to the Banach algebra 
${\mathcal{F}}(C_{a,b}[0,T])$.
This article explores the properties of fractal interpolation functions with variable scaling parameters, in the context of smooth fractal functions. The first part extends the Barnsley–Harrington theorem for differentiability of fractal functions and the fractal analogue of Hermite interpolation to the present setting. The general result is applied on a special class of iterated function systems in order to develop differentiability of the so-called 
$\boldsymbol{{\it\alpha}}$-fractal functions. This leads to a bounded linear map on the space 
${\mathcal{C}}^{k}(I)$ which is exploited to prove the existence of a Schauder basis for 
${\mathcal{C}}^{k}(I)$ consisting of smooth fractal functions.
Some well-known results about the 2-density topology on 
${\mathcal R}$ (in particular in the context of the Lusin–Menchoff property) are extended to τbm , i.e. the m-density topology on ${\mathcal R}$ n with m ∈ (n,+∞). Every set of finite perimeter in ${\mathcal R}$ n is equivalent (in measure) to a set in τb m 0 , where m 0=n+1+ 
${1\over n-1}$ . There exists a set of finite perimeter in ${\mathcal R}$ n which is not equivalent (in measure) to any member in the a.e.-modification of τbm , whatever m ∈ [n,+∞).
We consider classes 
of subsets of [0, 1], originally introduced by Falconer, that are closed under countable intersections, and such that every set in the class has Hausdorff dimension at least s. We provide a Frostman-type lemma to determine if a limsup set is in such a class. Suppose that E = lim sup En ⊂ [0, 1], and that μn are probability measures with support in En. If there exists a constant C such that
for all n, then, under suitable conditions on the limit measure of the sequence (μn), we prove that the set E is in the class .
As an application we prove that, for α > 1 and almost all λ ∈ (½, 1), the set
where 
and ak ∈ {0, 1}}, belongs to the class 
. This improves one of our previously published results.
Let 
$Q$ be an infinite subset of 
$\mathbb{N}$. For any 
${\it\tau}>2$, denote 
$W_{{\it\tau}}(Q)$ (respectively 
$W_{{\it\tau}}$) to be the set of 
${\it\tau}$ well-approximable points by rationals with denominators in 
$Q$ (respectively in 
$\mathbb{N}$). We consider the Hausdorff dimension of the liminf set 
$W_{{\it\tau}}\setminus W_{{\it\tau}}(Q)$ after Adiceam. By using the tools of continued fractions, it is shown that if 
$Q$ is a so-called 
$\mathbb{N}\setminus Q$-free set, the Hausdorff dimension of 
$W_{{\it\tau}}\setminus W_{{\it\tau}}(Q)$ is the same as that of 
$W_{{\it\tau}}$, i.e. 
$2/{\it\tau}$.
We study the distribution of the orbits of real numbers under the beta-transformation 
$T_{{\it\beta}}$ for any 
${\it\beta}>1$. More precisely, for any real number 
${\it\beta}>1$ and a positive function 
${\it\varphi}:\mathbb{N}\rightarrow \mathbb{R}^{+}$, we determine the Lebesgue measure and the Hausdorff dimension of the following set: 
$$\begin{eqnarray}E(T_{{\it\beta}},{\it\varphi})=\{(x,y)\in [0,1]\times [0,1]:|T_{{\it\beta}}^{n}x-y|<{\it\varphi}(n)\text{ for infinitely many }n\in \mathbb{N}\}.\end{eqnarray}$$
Let 
$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathbb{F}_q$ be the finite field of 
$q$ elements. An analogue of the regular continued fraction expansion for an element 
$\alpha $ in the field of formal Laurent series over 
$\mathbb{F}_q$ is given uniquely by where 
$$\begin{equation*} \alpha = A_0(\alpha )+\cfrac {1}{A_1(\alpha )+\cfrac {1}{A_2(\alpha )+\ddots }}, \end{equation*}$$
$(A_n(\alpha ))_{n=0}^\infty $ is a sequence of polynomials with coefficients in 
$\mathbb{F}_q$ such that 
$\deg (A_n(\alpha ))\ge 1$ for all 
$n\ge 1.$ We first prove the exactness of the continued fraction map in positive characteristic. This fact implies a number of strictly weaker properties. Particularly, we then use the weak-mixing property and ergodicity to establish various metrical results regarding the averages of partial quotients of continued fraction expansions. A sample result that we prove is that if 
$(p_n)_{n=1}^\infty $ denotes the sequence of prime numbers, we have for almost every 
$$\begin{equation*} \lim _{n\to \infty }\frac {1}{n}\sum _{j=1}^n \deg (A_{p_j}(\alpha )) = \frac {q}{q-1} \end{equation*}$$
$\alpha $ with respect to Haar measure. In the case where the sequence 
$(p_n)_{n=1}^\infty $ is replaced by 
$(n)_{n=1}^\infty ,$ this result is due to V. Houndonougbo, V. Berthé and H. Nakada. Our proofs rely on pointwise subsequence and moving average ergodic theorems.
The Kohlrausch functions 
$\exp (- {t}^{\beta } )$, with 
$\beta \in (0, 1)$, which are important in a wide range of physical, chemical and biological applications, correspond to specific realizations of completely monotone functions. In this paper, using nonuniform grids and midpoint estimates, constructive procedures are formulated and analysed for the Kohlrausch functions. Sharper estimates are discussed to improve the approximation results. Numerical results and representative approximations are presented to illustrate the effectiveness of the proposed method.
