Set Of Power Laws Research Articles

The low-mass content of the massive young star cluster RCW 38

RCW 38 is a deeply embedded young (~1 Myr), massive star cluster located at a distance of 1.7 kpc. Twice as dense as the Orion Nebula Cluster, orders of magnitude denser than other nearby star forming regions, and rich in massive stars, RCW 38 is an ideal place to look for potential differences in brown dwarf formation efficiency as a function of environment. We present deep, high resolution adaptive optics data of the central ~0.5x0.5 pc^2 obtained with NACO at the Very Large Telescope. Through comparison with evolutionary models we determine masses and extinction for ~480 candidate members, and derive the first Initial Mass Function (IMF) of the cluster extending into the substellar regime. Representing the IMF as a set of power laws in the form dN/dM~M^(-alpha), we derive the slope alpha = 1.60+-0.13 for the mass range 0.5 - 20 MSun which is shallower than the Salpeter slope, but in agreement with results in several other young massive clusters. At the low-mass side, we find alpha = 0.71+-0.11 for masses between 0.02 and 0.5 MSun, or alpha = 0.81+-0.08 for masses between 0.02 and 1 MSun. Our result is in agreement with the values found in other young star-forming regions, revealing no evidence that a combination of high stellar densities and the presence of numerous massive stars affect the formation efficiency of brown dwarfs and very-low mass stars. We estimate that the Milky Way galaxy contains between 25 and 100 billion brown dwarfs (with masses > 0.03 MSun).

Fractals and Fractal Dimension of Systems of Blood Vessels: An Analogy between Artery Trees, River Networks, and Urban Hierarchies

An analogy between the fractal nature of networks of arteries and that of systems of rivers has been drawn in the previous works. However, the deep structure of the hierarchy of blood vessels has not yet been revealed. This paper is devoted to researching the fractals, allometric scaling, and hierarchy of blood vessels. By analogy with Horton-Strahler's laws of river composition, three exponential laws have been put forward. These exponential laws can be reconstructed and transformed into three linear scaling laws, which can be named composition laws of blood vessels network. From these linear scaling laws it follows a set of power laws, including the three-parameter Zipf's law on the rank-size distribution of blood vessel length and the allometric scaling law on the length-diameter relationship of blood vessels in different orders. The models are applied to the observed data on human beings and animals early given by other researchers, and an interesting finding is that human bodies more conform to natural rules than dog's bodies. An analogy between the hierarchy of blood vessels, river networks, and urban systems are further drawn, and interdisciplinary studies of hierarchies will probably provide new revealing examples for the science of complexity.

Order of Magnitude Scaling: A Systematic Approach to Approximation and Asymptotic Scaling of Equations in Engineering

This work introduces the “order of magnitude scaling” (OMS) technique, which permits for the first time a simple computer implementation of the scaling (or “ordering”) procedure extensively used in engineering. The methodology presented aims at overcoming the limitations of the current scaling approach, in which dominant terms are manually selected and tested for consistency. The manual approach cannot explore all combinations of potential dominant terms in problems represented by many coupled differential equations, thus requiring much judgment and experience and occasionally being unreliable. The research presented here introduces a linear algebra approach that enables unassisted exhaustive searches for scaling laws and checks for their self-consistency. The approach introduced is valid even if the governing equations are nonlinear, and is applicable to continuum mechanics problems in areas such as transport phenomena, dynamics, and solid mechanics. The outcome of OMS is a set of power laws that estimates the characteristic values of the unknowns in a problem (e.g., maximum velocity or maximum temperature variation). The significance of this contribution is that it extends the range of applicability of scaling techniques to large systems of coupled equations and brings objectivity to the selection of small terms, leading to simplifications. The methodology proposed is demonstrated using a linear oscillator and thermocapillary flows in welding.

Electron–acoustic-phonon energy-loss rate in multicomponent electron systems with symmetric and asymmetric coupling constants

We consider electron-phonon (\textit{e-ph}) energy loss rate in 3D and 2D multi-component electron systems in semiconductors. We allow general asymmetry in the \textit{e-ph} coupling constants (matrix elements), i.e., we allow that the coupling depends on the electron sub-system index. We derive a multi-component \textit{e-ph}power loss formula, which takes into account the asymmetric coupling and links the total \textit{e-ph} energy loss rate to the density response matrix of the total electron system. We write the density response matrix within mean field approximation, which leads to coexistence of\ symmetric energy loss rate $F_{S}(T)$ and asymmetric energy loss rate $F_{A}(T)$ with total energy loss rate $ F(T)=F_{S}(T)+F_{A}(T)$ at temperature $T$. The symmetric component F_{S}(T) $ is equivalent to the conventional single-sub-system energy loss rate in the literature, and in the Bloch-Gr\"{u}neisen limit we reproduce a set of well-known power laws $F_{S}(T)\propto T^{n_{S}}$, where the prefactor and power $n_{S}$ depend on electron system dimensionality and electron mean free path. For $F_{A}(T)$ we produce a new set of power laws F_{A}(T)\propto T^{n_{A}}$. Screening strongly reduces the symmetric coupling, but the asymmetric coupling is unscreened, provided that the inter-sub-system Coulomb interactions are strong. The lack of screening enhances $F_{A}(T)$ and the total energy loss rate $F(T)$. Especially, in the strong screening limit we find $F_{A}(T)\gg F_{S}(T)$. A canonical example of strongly asymmetric \textit{e-ph} matrix elements is the deformation potential coupling in many-valley semiconductors.

Statistics of the two self-avoiding random walks on the three-dimensional fractal lattices

Abstract We present results of the effects of interpenetration of two interacting self-avoiding walks that take place in a member of a three-dimensional Sierpinski Gasket (SG) fractal family. We focus our attention on finding number of point contacts between the two SAW paths, which turns out to be a set of power laws whose characteristics depend predominantly on the given interactions between SAW steps. To establish statistics of the defining model, we apply an exact Renormalization Group Method for the few members ( b = 2 , 3 and 4 ) of the SG fractal family, as well as a Monte Carlo RG method for 2 ⩽ b ⩽ 25 . The phase diagrams have been established and relevant values of the contact critical exponents, associated with the two-path mutual contacts, are determined.

A spatial network explanation for a hierarchy of urban power laws

Power laws in socioeconomic systems are generally explained as being generated by multiplicative growth of aggregate objects. In this paper we formulate a model of geographic activity distribution with spatial correlations on the level of land lots where multiplicative growth is assumed to be dominant but not exclusive. The purpose is to retain the explanatory power of earlier models due to Simon, Gibrat and others while attaining some additional properties that are attractive for both empirical and modelling purposes. In this sense, the model presented here is a combination of the two factors that have been identified as central to urban evolution but rarely appear unified in the same model: transportation costs and multiplicative growth. The model is an elaboration of a previously reported complex network model of geographical land value evolution. We reproduce statistical properties of an empirical geographical distribution of land values on multiple hierarchical levels: land value per unit area, cluster areas, aggregated land value per cluster and cluster area/perimeter ratios. It is found that transportation effects are not strong enough to disturb the power law distribution of land values per unit area but strong enough to sort nodes to generate a new set of power laws on a higher level of aggregation. The main hypothesis is that all these relations can be understood as consequences of an underlying growing scale-free network of geographic economic interdependencies.

Characterizing Indian Ocean manganese nodule-bearing seafloor using multi-beam angular backscatter

Application of quantitative angular backscatter modelling to manganese nodule-bearing areas of the Central Indian Ocean Basin (CIOB) has been initiated at NIO during the year 1998. Studies were aimed to establish the suitability of seafloor backscattering in delineating seafloor parameters characteristic of nodule-rich sediments. In this paper, processed Hydrosweep multi-beam backscatter data from 45 spot locations in the CIOB (where nodule samples are available) were analysed to estimate seafloor and sediment volume roughness parameters. The application of a composite roughness model to a nodule-bearing region (6,600 km2) of the CIOB, to determine seafloor interface roughness parameters from a multi-beam backscatter dataset, shows only four power law sets. The results attest 80% of the nodule-bearing seafloor to be smooth in terms of interface roughness parameters at micro-topographic level. The sediment volume roughness parameters are dominant only in 29% of the smooth interface roughness sites. This indicates that 51% of the seafloor area possesses negligible (interface and volume) roughness. A critical analysis using pseudo-side-scan records from 12 selected locations in the study area affirms the combined importance of the seafloor interface and sediment volume roughness parameters for precise determination of manganese nodule abundance.

A long-time tail for random walk in random scenery

Consider a simple random walk on ℤ d whose sites are colored black or white independently with probabilityq, resp. 1−q. Walk and coloring are independent. Letn k be the number of steps by the walk between itskth and (k+1) th visits to a black site (i.e., the length of itskth white run), and letΔ k =E(n k )−q −1. Our main result is a proof that (*) lim k → ∞ k d/2 Δ k = (1 −q)q d/2 − 2(d/2π) d/2. Since it is known thatq − 1 Δ k =E(n 1 n k + 1 ∣B) −E(n 1 ∣B)E(n k + 1 ∣B), withB the event that the origin is black, (*) exhibits a long-time tail in the run length autocorrelation function. Numerical calculations ofΔ k (1⩽k⩽100) ind=1, 2, and 3 show that there is an oscillatory behavior ofΔ k for smallk. This damps exponentially fast, following which the power law sets in fairly rapidly. We prove that if the coloring is not independent, but is convex in the sense of FKG, then the decay ofΔ k cannot be faster than (*).