Structural transformation and dynamical heterogeneity in Germania melt under compression: molecular dynamic simulation

The structural transformation and dynamical heterogeneity in Germania (GeO2) are investigated via molecular dynamics (MD) simulation. The MD model with 5499 atoms was constructed under pressure up to 150 GPa and at a temperature of 3500 K. The structural transformation mechanism has been studied by observing domain structures and boundary oxygen atoms. The simulation result reveals that GeO2 consists of separate domains and boundaries in its melt structure. Under compression, the structure of GeO2 changes gradually and represents many types of structures. The melt structure exhibits many structural domains Dx, and polymorphism appears at pressures of 12 and 20 GPa. The change of tetrahedral structure to octahedral structure in germanium coordination occurred in parallel with the process of merging and splitting of domain structure. Moreover, the existence of high- and low-density phases in GeO2 melt is indicated. The high-density phase is D6 domain and boundary oxygen while the low-density phase is D4 and D5 domain. The compression mechanism in GeO2 melt mainly is a reduction of average Voronoi volume of oxygen and Voronoi volume of D6, boundary atoms oxygen. Furthermore, we find the dynamical heterogeneity at ambient pressure. The separate “fast” regions and “slow” regions in GeO2 are detected via link-cluster function.

Notes on cluster algebras and some all-loop Feynman integrals

We study cluster algebras for some all-loop Feynman integrals, including box-ladder, penta-box-ladder, and double-penta-ladder integrals. In addition to the well-known box ladder whose symbol alphabet is $$ {D}_2\simeq {A}_1^2 $$ D 2 ≃ A 1 2 , we show that penta-box ladder has an alphabet of D3 ≃ A3 and provide strong evidence that the alphabet of seven-point double-penta ladders can be identified with a D4 cluster algebra. We relate the symbol letters to the u variables of cluster configuration space, which provide a gauge-invariant description of the cluster algebra, and we find various sub-algebras associated with limits of the integrals. We comment on constraints similar to extended-Steinmann relations or cluster adjacency conditions on cluster function spaces. Our study of the symbol and alphabet is based on the recently proposed Wilson-loop d log representation, which allows us to predict higher-loop alphabet recursively; by applying it to certain eight-point and nine-point double-penta ladders, we also find D5 and D6 cluster functions respectively.

Distribution of sodium and dynamical heterogeneity in sodium silicate liquid

The structural and dynamical properties in sodium silicate liquid were investigated by molecular dynamics method. To clarify the distribution of sodium atoms in model, characteristics of simplex have been investigated. The simulation results reveal that Na2O⋅4SiO2 (NS4) liquid has a lot of simplexes with four sodium atoms inside but about half of simplexes do not have sodium. The spatial distribution of sodium is nonuniform, sodium tends to be in the nonbridging oxygen-simplexes and in larger-radius simplex. Moreover, the sodium density for nonbridging oxygen region is significantly higher than the one for Si-region. Namely, link-cluster function F[Formula: see text](r, t) has been used to clarify dynamical heterogeneity in NS4 liquid. The F[Formula: see text](r, t) for sets of random, immobile and mobile network atoms is quite different, which indicates that the dynamics of network atoms is heterogeneous. The Si–O network has the structure with two separated domains (immobile and mobile domains). These types of domain are significantly different in local microstructure, mobility of atoms and chemical composition.

Hox cluster genes and collinearities throughout the tree of animal life

The discovery of Hox gene clusters, first in Drosophila (a protostome) and then as homologues in vertebrates (deuterostomes), was a major step in our understanding of both developmental and evolutionary biology. Hox genes in both species perform the same overall function: that is, organization of the body along its head-tail axis. The conclusion is that the protostome-deuterostome ancestor, founder of 99% of all described animal species, must already have had this same basic Hox cluster, and that it probably used it in the same way to establish its body plan. A striking feature of Hox genes is the spatial collinearity rule: that order of the genes along the chromosome corresponds with the order of their expression domains along the embryo. For vertebrates, though not Drosophila, there is also the temporal collinearity rule: that order of genes along the chromosome corresponds with timing of Hox expressions in the embryo. Although Hox genes are clearly recognized in pre-bilaterians (Cnidaria), it is only in bilaterians that the characteristic clustered Hox arrangement and function is commonly found. Spatial collinearity in expression is conserved widely throughout Bilateria but temporal collinearity is so far limited to vertebrates, cephalochordates, and some arthropods and annelids. In addition to conserved use of Hox genes to pattern the head-tail axis, some animal groups, particularly lophotrochozoans, have extensively co-opted Hox genes, outside collinearity rules, to regulate development of novel structures. Satisfactory understanding of Hox cluster function requires better understanding of the bilaterian last common ancestor (Urbilateria). Xenacoelomorpha may provide useful living models of the ancestral bilaterian condition.

The set of filter cluster functions

In this work, we are concerned with the concepts of F-?-convergence, F-pointwise convergence and F-uniform convergence for sequences of functions on metric spaces, where F is a filter on N. We define the concepts of F-limit function, F-cluster function and limit function respectively for each of these three types of convergence, and obtain some results about the sets of F-cluster and F-limit functions for sequences of functions on metric spaces. We use the concept of F-exhaustiveness to characterize the relations between these points.