Flexible toggles and symmetric invertible asynchronous elementary cellular automata

A sequential dynamical system (SDS) consists of a graph $G$ with vertices $v_1,\dots ,v_n$, a state set $A$, a collection of “vertex functions” ${\left\{f_{v_i}\right\}}_{i=1}^n$, and a permutation $\pi \in S_n$ that specifies how to compose these functions to yield the SDS map $[G,{\left\{f_{v_i}\right\}}_{i=1}^n,\pi ]:A^n\to A^n$. In this paper, we study symmetric invertible SDS defined over the cycle graph $C_n$ using the set of states ${\mathbb{F}}_2$. These are, in other words, asynchronous elementary cellular automata (ECA) defined using ECA rules 150 and 105. Each of these SDS defines a group action on the set ${\mathbb{F}}_2^n$ of $n$-bit binary vectors. Because the SDS maps are products of involutions, this relates to generalized toggle groups, which Striker recently introduced. In this paper, we further generalize the notion of a generalized toggle group to that of a flexible toggle group; the SDS maps we consider are examples of Coxeter elements of flexible toggle groups.Our main result is the complete classification of the dynamics of symmetric invertible SDS defined over cycle graphs using the set of states ${\mathbb{F}}_2$ and the identity update order $\pi =123?n$. More precisely, if $T$ denotes the SDS map of such an SDS, then we obtain an explicit formula for $\left|{\text{Per}}_r\left(T\right)\right|$, the number of periodic points of $T$ of period $r$, for every positive integer $r$. It turns out that if we fix $r$ and vary $n$ and $T$, then $\left|{\text{Per}}_r\left(T\right)\right|$ only takes at most three nonzero values.     

Partitioning edge-coloured complete symmetric digraphs into monochromatic complete subgraphs

Let ${\overrightarrow{K}}_{\mathbb{N}}$ be the complete symmetric digraph on the positive integers. Answering a question of DeBiasio and McKenney, we construct a 2-colouring of the edges of ${\overrightarrow{K}}_{\mathbb{N}}$ in which every monochromatic path has density 0.However, if we restrict the length of monochromatic paths in one colour, then no example as above can exist: We show that every $(r+1)$-edge-coloured complete symmetric digraph (of arbitrary infinite cardinality) containing no directed paths of edge-length ${?}_i$ for any colour $i\le r$ can be covered by ${\prod }_{i\in \left[r\right]}{?}_i$ pairwise disjoint monochromatic complete symmetric digraphs in colour $r+1$.Furthermore, we present a stability version for the countable case of the latter result: We prove that the edge-colouring is uniquely determined on a large subgraph, as soon as the upper density of monochromatic paths in colour $r+1$ is bounded by ${\prod }_{i\in \left[r\right]}\frac{1}{{?}_i}$.     

Optimizing and improving of the C-to-R method for solving complex symmetric linear systems

For complex symmetric linear systems, Axelsson et al. (2014) proposed the C-to-R method. In this paper, by further studying the C-to-R method with $W$ and $T$ being symmetric positive semidefinite, the optimal iteration parameter for the C-to-R method ${\alpha }_{opt}=\frac{\sqrt{2\sqrt{2}}}{2}$ is obtained and the C-to-R method is optimized. Furthermore, based on the optimized C-to-R method, we further propose an optimized preconditioner. Eigenvalue properties of the optimized preconditioned matrix are analyzed, which show that all the eigenvalues of the preconditioned matrix are located in tighter interval. Numerical results are presented, not only confirm the validity of the theoretical analysis, but also demonstrate the feasibility and effectiveness of the proposed optimized C-to-R method.     

A remark on the orbit structure of the complexification of a semisimple symmetric space

We consider the action of a real semisimple Lie group G on the complexification $G_{\mathbf{C}}/H_{\mathbf{C}}$ of a semisimple symmetric space $G/H$ and we present a refinement of Matsuki?s results (Matsuki, 1997 [1]) in this case. We exhibit a finite set of points in $G_{\mathbf{C}}/H_{\mathbf{C}}$, sitting on closed G-orbits of locally minimal dimension, whose slice representation determines the G-orbit structure of $G_{\mathbf{C}}/H_{\mathbf{C}}$. Every such point $\overline{p}$ lies on a compact torus and occurs at specific values of the restricted roots of the symmetric pair $(\mathfrak{g},\mathfrak{h})$. The slice representation at $\overline{p}$ is equivalent to the isotropy representation of a real reductive symmetric space, namely $Z_G(p^4)/G_{\overline{p}}$. In principle, this gives the possibility to explicitly parametrize all G-orbits in $G_{\mathbf{C}}/H_{\mathbf{C}}$.     

Denseness of volatile and nonvolatile sequences of functions

In a recent paper by Jonasson and Steif, definitions to describe the volatility of sequences of Boolean functions, $f_n:{\{?1,1\}}^n\to \{?1,1\}$ were introduced. We continue their study of how these definitions relate to noise stability and noise sensitivity. Our main results are that the set of volatile sequences of Boolean functions is a natural way “dense” in the set of all sequences of Boolean functions, and that the set of non-volatile Boolean sequences is not “dense” in the set of noise stable sequences of Boolean functions.