Related papers: Probability and Fourier duality for affine iterate…
We compute the full probability distribution of the spectral form factor in the self-dual kicked Ising model by providing an exact lower bound for each moment and verifying numerically that the latter is saturated. We show that at large…
Consider the following probabilistic contracting on average iterated function system $$\Phi = \left\{f_i (x) = \lambda_i x + d_i,\;i=1,2 ;\;\; p = \left(\frac{1}{2} , \frac{1}{2}\right) \right\},$$ where the contraction ratios $\lambda_1 ,…
We exhibit a large class of symbols $m$ on $\R^d$, $d\geq 2$, for which the corresponding Fourier multipliers $T_m$ satisfy the following inequality. If $D$, $E$ are measurable subsets of $\R^d$ with $E\subseteq D$ and $|D|<\infty$, then $$…
We consider the class $PW(\mathbb R^n)$ of functions in $L^2(\mathbb R^n)$, whose Fourier transform has bounded support. We obtain a description of continuous maps $\varphi : \mathbb R^m\rightarrow\mathbb R^n$ such that $f\circ\varphi\in…
Fourier series multiscale method, a concise and efficient analytical approach for multiscale computation, will be developed out of this series of papers. The second paper is concerned with simultaneous approximation to functions and their…
Finite dimensional irreducible modules of the two-parameter quantum enveloping algebra $U_{r,s}(\mathfrak{sl}_n)$ are explicitly constructed using the fusion procedure when $rs^{-1}$ is generic. This provides an alternative and…
The eigenvalue problem for a linear function L centers on solving the eigen-equation Lx = rx. This paper generalizes the eigenvalue problem from a single linear function to an iterated function system F consisting of possibly an infinite…
Multifractal formalism is designed to describe the distribution at small scales of the elements of $\mathcal M^+_c(\R^d)$, the set of positive, finite and compactly supported Borel measures on $\R^d$. It is valid for such a measure $\mu$…
Knowledge of exact analytical functional forms for the pair correlation function $g_2(r)$ and its corresponding structure factor $S(k)$ of disordered many-particle systems is limited. For fundamental and practical reasons, it is highly…
Certain Bernoulli convolution measures (\mu) are known to be spectral. Recently, much work has concentrated on determining conditions under which orthonormal Fourier bases (i.e. spectral bases) exist. For a fixed measure known to be…
Assume $n=2m\geq 2$ and let $F(x)=x^{d_1}+x^{d_2}$ be a binomial vectorial function over $\F_{2^n}$ possessing the maximal number (i.e. $2^n-2^m$) of bent components. Suppose the $2$-adic Hamming weights $\wt_2(d_1)$ and $\wt_2(d_2)$ are…
We introduce the novel concept of hypercomplex iterated function system (IFS) on the complete metric space $(\mathbb{A}_{n+1}^k,d)$ and define its hypercomplex attractor. Systems of hypercomplex function systems arising from hypercomplex…
Let $\{S_i\}_{i=1}^\ell$ be an iterated function system (IFS) on $\R^d$ with attractor $K$. Let $(\Sigma,\sigma)$ denote the one-sided full shift over the alphabet $\{1,..., \ell\}$. We define the projection entropy function $h_\pi$ on the…
This paper gives the pointwise H\"older (or multifractal) spectrum of continuous functions on the interval $[0,1]$ whose graph is the attractor of an iterated function system consisting of $r\geq 2$ affine maps on $\mathbb{R}^2$. These…
Iterated functions system (IFS) is defined by specifying a set of functions in a classical phase space, which act randomly on an initial point. In an analogous way, we define a quantum iterated functions system (QIFS), where functions act…
Let $ \mu $ be the self-similar measure associated with a homogeneous iterated function system $ \Phi = \{ \lambda x + t_j \}_{j=1}^m $ on ${\Bbb R}$ and a probability vector $ (p_{j})_{j=1}^m$, where $0\neq \lambda\in (-1,1)$ and $t_j\in…
One can realize higher laminations as positive configurations of points in the affine building. The duality pairings of Fock and Goncharov give pairings between higher laminations for two Langlands dual groups $G$ and $G^{\vee}$. These…
We provide an overview of iterated function systems (IFS), where randomly chosen state-to-state maps are applied iteratively to a state. We aim to summarize the state of art and, where possible, identify fundamental challenges and…
We study invariant sets and measures generated by iterated function systems defined on countable discrete spaces that are uniform grids of a finite dimension. The discrete spaces of this type can be considered as models of spaces in which…
In this paper we consider equimeasurable symmetric(rearrangement invariant) spaces $\mathbf{E}_1 = \mathbf{E}_1(\Omega_1,\mathcal{F}_1,\mu_1)$ and $\mathbf{E}_2 = \mathbf{E}_2(\Omega_2,\mathcal{F}_2,\mu_2)$ on a measure spaces $(\Omega_1,…