This elucidation investigates the Hausdorff dimension of the output space of multi-layer neural networks. When the factor map from the covering space of the output space to the output space has a synchronizing word, the Hausdorff dimension of the output space relates to its topological entropy. This clarifies the geometrical structure of the output space in more details.

Abstract:
This elucidation studies ergodicity and equilibrium measures for additive cellular automata with prime states. Additive cellular automata are ergodic with respect to Bernoulli measure unless it is either an identity map or constant. The formulae of measure-theoretic and topological entropies can be expressed in closed forms and the topological pressure is demonstrated explicitly for potential functions that depend on finitely many coordinates. According to these results, Parry measure is inferred to be an equilibrium measure.

Abstract:
The use of recombinant BoNT domains has been proposed as a means to develop strategies to treat and prevent botulism. Here, details on the molecular cloning, protein expression, purification, and immunoreactivity of BoNT/F domains from Clostridium botulinum are presented. Initially, full-length synthetic genes encoding recombinant BoNT/F domains (catalytic, translocation, and receptor binding) were designed and cloned into Escherichia coli for expression. Recombinant proteins were then purified through GST affinity chromatography preceding elution of GST-free recombinant domains by thrombin protease. Soluble recombinant proteins encoding catalytic light chain and translocation N-terminal heavy chain were subsequently used to perform in vivo immunization. Polyclonal mouse antibodies specific to these domains were raised, confirmed by Western blot analysis and elevated immunoreactivity was identified through indirect ELISA. In conclusion, availability of the recombinant protein provides an effective system to study the immunological aspects of BoNT/F and corresponding applications in pathogen detection and vaccine candidacy.

Abstract:
Topological behavior, such as chaos, irreducibility, and mixing of a one-sided shift of finite type, is well elucidated. Meanwhile, the investigation of multidimensional shifts, for instance, textile systems is difficult and only a few results have been obtained so far. This paper studies shifts defined on infinite trees, which are called tree-shifts. Infinite trees have a natural structure of one-sided symbolic dynamical systems equipped with multiple shift maps and constitute an intermediate class in between one-sided shifts and multidimensional shifts. We have shown not only an irreducible tree-shift of finite type, but also a mixing tree-shift that are chaotic in the sense of Devaney. Furthermore, the graph and labeled graph representations of tree-shifts are revealed so that the verification of irreducibility and mixing of a tree-shift is equivalent to determining the irreducibility and mixing of matrices, respectively. This extends the classical results of one-sided symbolic dynamics. A necessary and sufficient condition for the irreducibility and mixing of tree-shifts of finite type is demonstrated. Most important of all, the examination can be done in finite steps with an upper bound.

Abstract:
Let $\mathbf{Y}$ be the solution space of an $n$-layer cellular neural network, and let $\mathbf{Y}^{(i)}$ and $\mathbf{Y}^{(j)}$ be the hidden spaces, where $1 \leq i, j \leq n$. ($\mathbf{Y}^{(n)}$ is called the output space.) The classification and the existence of factor maps between two hidden spaces, that reaches the same topological entropies, are investigated in [Ban et al., J.~Differential Equations \textbf{252}, 4563-4597, 2012]. This paper elucidates the existence of factor maps between those hidden spaces carrying distinct topological entropies. For either case, the Hausdorff dimension $\dim \mathbf{Y}^{(i)}$ and $\dim \mathbf{Y}^{(j)}$ can be calculated. Furthermore, the dimension of $\mathbf{Y}^{(i)}$ and $\mathbf{Y}^{(j)}$ are related upon the factor map between them.

Abstract:
Let $1<\beta \leq 2$. It is well-known that the set of points in $% [0,1/(\beta -1)]$ having unique $\beta $-expansion, in other words, those points whose orbits under greedy $\beta $-transformation escape a hole depending on $\beta $, is of zero Lebesgue measure. The corresponding escape rate is investigated in this paper. A formula which links the Hausdorff dimension of univoque set and escape rate is established in this study. Then we also proved that such rate forms a devil's staircase function with respect to $\beta $.

Abstract:
This paper studies the entropy of tree-shifts of finite type with and without boundary conditions. We demonstrate that computing the entropy of a tree-shift of finite type is equivalent to solving a system of nonlinear recurrence equations. Furthermore, the entropy of the binary Markov tree-shifts over two symbols is either $0$ or $\ln 2$. Meanwhile, the realization of a class of reals including multinacci numbers is elaborated, which indicates that tree-shifts are capable of rich phenomena. By considering the influence of three different types of boundary conditions, say, the periodic, Dirichlet, and Neumann boundary conditions, the necessary and sufficient conditions for the coincidence of entropy with and without boundary conditions are addressed.

Abstract:
Autoimmune thyroid diseases (AITDs), including Graves’ disease (GD) and Hashimoto’s thyroiditis (HT), are among the commonest autoimmune disorders, affecting approximately 2% - 5% of the population. Epidemiological data support strong genetic influences on the development of AITD. The identification of genes placing individuals at an increased risk for the development of AITD has been a slow process. However, over the last 20 years or so real progress has been made with the mapping of novel loci, via a number of different approaches. The first AITD gene discovered, Human Leucocyte Antigen (HLA)/Major Histocompatibility Complex (MHC), is associated with both GD and HT. Non-MHC genes that confer susceptibility to AITD can be classified into two groups: (1) immune-regulatory genes (e.g., CD40, CTLA-4, and PTPN22); (2) thyroid-specific genes—thyroglobulin and TSH receptor genes. These genes interact with environmental factors, such as infection, likely through epigenetic mechanisms to trigger disease. In this review, we will summarize the latest findings on AITD susceptibility genes in non-Caucasians.

A weak value of an observable is studied for a quantum system which
is placed under the influence of an environment, where a quantum system
irreversibly evolves from a pre-selected state to a post-selected state. A
general expression for a weak value influenced by an environment is provided.
For a Markovian environment, the weak value is calculated in terms of the
predictive and retrodictive density matrices, or by means of the quantum
regression theorem. For a non-Markovian environment, a weak value is examined
by making use of exactly solvable models. It is found that although the
anomalous property is significantly suppressed by a Markovian
environment, it can survivea non-Markovian environment.

Abstract:
Insomnia is reported to chronically affect 10~15% of the adult population. However, very little is known about the genetics and metabolism of insomnia. Here we surveyed 10,038 Korean subjects whose genotypes have been previously profiled on a genome-wide scale. About 16.5% reported insomnia and displayed distinct metabolic changes reflecting an increase in insulin secretion, a higher risk of diabetes, and disrupted calcium signaling. Insomnia-associated genotypic differences were highly concentrated within genes involved in neural function. The most significant SNPs resided in ROR1 and PLCB1, genes known to be involved in bipolar disorder and schizophrenia, respectively. Putative enhancers, as indicated by the histone mark H3K4me1, were discovered within both genes near the significant SNPs. In neuronal cells, the enhancers were bound by PAX6, a neural transcription factor that is essential for central nervous system development. Open chromatin signatures were found on the enhancers in human pancreas, a tissue where PAX6 is known to play a role in insulin secretion. In PLCB1, CTCF was found to bind downstream of the enhancer and interact with PAX6, suggesting that it can probably inhibit gene activation by PAX6. PLCB4, a circadian gene that is closely located downstream of PLCB1, was identified as a candidate target gene. Hence, dysregulation of ROR1, PLCB1, or PLCB4 by PAX6 and CTCF may be one mechanism that links neural and pancreatic dysfunction not only in insomnia but also in the relevant psychiatric disorders that are accompanied with circadian rhythm disruption and metabolic syndrome.