Concept Of Ultrametricity Research Articles

Formula omitted]-Adic mathematics and theoretical biology

The principal objective of this article is a brief overview of the main parts of p-adic mathematics, which have already had valuable applications and may have a significant impact in the near future on the further development of some fields of theoretical and mathematical biology. In particular, we present the basics of ultrametrics, p-adic numbers and p-adic analysis, as well as insight into their applications for modeling some cognitive processes, genetic code and protein dynamics. We also argue that ultrametric concepts and p-adic mathematics are natural tools for the viable description of biological systems and phenomena with a hierarchical structure.

Hierarchical structures of correlations networks among Turkey’s exports and imports by currencies

We have examined the hierarchical structures of correlations networks among Turkey’s exports and imports by currencies for the 1996–2010 periods, using the concept of a minimal spanning tree (MST) and hierarchical tree (HT) which depend on the concept of ultrametricity. These trees are useful tools for understanding and detecting the global structure, taxonomy and hierarchy in financial markets. We derived a hierarchical organization and build the MSTs and HTs during the 1996–2001 and 2002–2010 periods. The reason for studying two different sub-periods, namely 1996–2001 and 2002–2010, is that the Euro (EUR) came into use in 2001, and some countries have made their exports and imports with Turkey via the EUR since 2002, and in order to test various time-windows and observe temporal evolution. We have carried out bootstrap analysis to associate a value of the statistical reliability to the links of the MSTs and HTs. We have also used the average linkage cluster analysis (ALCA) to observe the cluster structure more clearly. Moreover, we have obtained the bidimensional minimal spanning tree (BMST) due to economic trade being a bidimensional problem. From the structural topologies of these trees, we have identified different clusters of currencies according to their proximity and economic ties. Our results show that some currencies are more important within the network, due to a tighter connection with other currencies. We have also found that the obtained currencies play a key role for Turkey’s exports and imports and have important implications for the design of portfolio and investment strategies.

Topological properties of stock market networks: The case of Brazil

This paper investigates the topological properties of the Brazilian stock market networks. We build the minimum spanning tree, which is based on the concept of ultrametricity, using the correlation matrix for a variety of stocks of different sectors. Our results suggest that stocks tend to cluster by sector. We employ a dynamic approach using complex network measures and find that the relative importance of different sectors within the network varies. The financial, energy and material sectors are the most important within the network.

TOPOLOGICAL PROPERTIES OF BANK NETWORKS: THE CASE OF BRAZIL

This paper investigates the topological properties of bank networks. We build the minimum spanning tree, which is based on the concept of ultrametricity, using the correlation matrix for a variety of banking variables. Empirical results suggest that the private and foreign banks tend to form clusters within the network. Furthermore, banks with different sizes are also strongly connected and tend to form clusters. These results are robust to the use of different variables to build the network, such as bank profitability, assets, equity, revenue and loans.

The expectation hypothesis of interest rates and network theory: The case of Brazil

This paper investigates the topological properties of the Brazilian term structure of interest rates network. We build the minimum spanning tree (MST), which is based on the concept of ultrametricity, using the correlation matrix for interest rates of different maturities. We show that the short-term interest rate is the most important within the interest rates network, which is in line with the Expectation Hypothesis of interest rates. Furthermore, we find that the Brazilian interest rates network forms clusters by maturity.

Dynamic asset trees and Black Monday

The minimum spanning tree, based on the concept of ultrametricity, is constructed from the correlation matrix of stock returns. The dynamics of this asset tree can be characterised by its normalised length and the mean occupation layer, as measured from an appropriately chosen centre called the ‘central node’. We show how the tree length shrinks during a stock market crisis, Black Monday in this case, and how a strong reconfiguration takes place, resulting in topological shrinking of the tree.

Dynamic asset trees and portfolio analysis

The minimum spanning tree, based on the concept of ultrametricity, is constructed from the correlation matrix of stock returns and provides a meaningful economic taxonomy of the stock market. In order to study the dynamics of this asset tree we characterise it by its normalised length and by the mean occupation layer, as measured from an appropriately chosen centre called the `central node'. We show how the tree evolves over time, and how it shrinks strongly, in particular, during a stock market crisis. We then demonstrate that the assets of the optimal Markowitz portfolio lie practically at all times on the outskirts of the tree. We also show that the normalised tree length and the investment diversification potential are very strongly correlated.

Spin-glass dynamics

Abstract The configuration space of spin-glasses is characterized by the existence of quasi-degenerate equilibrium states separated by barriers, Δ( T ). A temperature cycling protocol enables us to extract the temperature dependence of a specific barrier . We determine ϖΔ( T )/ϖ T vs T , and ϖΔ( T )/ϖ T vs Δ for barriers between 25 T g and 35 T g at temperatures 8, 9, 9.5 and 10 K ( T g = 10.4 K) for Ag:Mn (2.6 at%). We find, in the range of temperature investigated and within experimental accuracy, that ϖΔ( T )/ϖ T depends only on the particular value of Δ( T ) and not on the temperature. dΔ( T )/d T r can be fitted using all the data points to either a power law ( a Δ 6 ) or an exponential [α exp(βΔ)], where r = T / T g . Integration leads to a divergence of Δ( T ) at temperature T ∗ (integration constant) as the temperature decreases, characteristic of the particular barrier considered. The existence of aging effects at all temperatures below T g in the whole range of accessible times impliesa continuous distribution of barrier heights at any temperature. Hence, at any temperature below T g there are diverging barriers, associated with a distribution of T ∗ with an upper cutoff value of T g . Said another way, as the temperature is lowered to T g , the first divergence of a barrier occurs, separating phase space into two (or more) regimes. States separated by divergent barriers are the “pure states”. As the temperature is lowered, a continuous series of barrier divergences breaks up phase space into ever smaller regions, creating an ever increasing number of pure states. Between these divergent barriers, barriers of finite height exist. Those between 25 T g and 35 T g are responsible for the dynamics accessible to us in our experiments. We interpret the time decay of the thermoremanent magnetization, M TRM ( t ), at times t much larger than the waiting time, t w , as a measure of the correlation of the spin-glass state at t with the field cooled state. Using the concept of ultrametricity, this enables us to extract the branching ratio of the “tree” into which the spin-glass states are organized from a plot of log M TRM ( t ) vs log t , for t a t w . Experiements times t up to to 5540 min (4 days!) exhibits a linear waiting time of 10.2 min and measurement times t up to 5 540 min (4 days!) exhibit a linear relationship between log M TRM ( t ) and log t . Within the approximations made, this implies a constant branching ratio for the ultrametric tree, at least within the (small) range of overlaps q accessible to us because of our experimental time window. These conclusions suggest that our dynamical measurements can be interpreted within the mean-field picture for spin-glasses.