Convolution of multifractals and the local magnetization in a random field Ising chain

The local magnetization in the one-dimensional random-field Ising model is essentially the sum of two effective fields with multifractal probability measure. The probability measure of the local magnetization is thus the convolution of two multifractals. In this paper we prove relations between the multifractal properties of two measures and the multifractal properties of their convolution. The pointwise dimension at the boundary of the support of the convolution is the sum of the pointwise dimensions at the boundary of the support of the convoluted measures and the generalized box dimensions of the convolution are bounded from above by the sum of the generalized box dimensions of the convoluted measures. The generalized box dimensions of the convolution of Cantor sets with weights can be calculated analytically for certain parameter ranges and illustrate effects we also encounter in the case of the measure of the local magnetization. Returning to the study of this measure we apply the general inequalities and present numerical approximations of the D_q-spectrum. For the first time we are able to obtain results on multifractal properties of a physical quantity in the one-dimensional random-field Ising model which in principle could be measured experimentally. The numerically generated probability densities for the local magnetization show impressively the gradual transition from a monomodal to a bimodal distribution for growing random field strength h.Comment: An error in figure 1 was corrected, small additions were made to the introduction and the conclusions, some typos were corrected, 24 pages, LaTeX2e, 9 figure

Orbits and phase transitions in the multifractal spectrum

We consider the one dimensional classical Ising model in a symmetric dichotomous random field. The problem is reduced to a random iterated function system for an effective field. The D_q-spectrum of the invariant measure of this effective field exhibits a sharp drop of all D_q with q < 0 at some critical strength of the random field. We introduce the concept of orbits which naturally group the points of the support of the invariant measure. We then show that the pointwise dimension at all points of an orbit has the same value and calculate it for a class of periodic orbits and their so-called offshoots as well as for generic orbits in the non-overlapping case. The sharp drop in the D_q-spectrum is analytically explained by a drastic change of the scaling properties of the measure near the points of a certain periodic orbit at a critical strength of the random field which is explicitly given. A similar drastic change near the points of a special family of periodic orbits explains a second, hitherto unnoticed transition in the D_q-spectrum. As it turns out, a decisive role in this mechanism is played by a specific offshoot. We furthermore give rigorous upper and/or lower bounds on all D_q in a wide parameter range. In most cases the numerically obtained D_q coincide with either the upper or the lower bound. The results in this paper are relevant for the understanding of random iterated function systems in the case of moderate overlap in which periodic orbits with weak singularity can play a decisive role.Comment: The article has been completely rewritten; the title has changed; a section about the typical pointwise dimension as well as several references and remarks about more general systems have been added; to appear in J. Phys. A; 25 pages, 11 figures, LaTeX2

The randomly driven Ising ferromagnet, Part I: General formalism and mean field theory

We consider the behavior of an Ising ferromagnet obeying the Glauber dynamics under the influence of a fast switching, random external field. After introducing a general formalism for describing such systems, we consider here the mean-field theory. A novel type of first order phase transition related to spontaneous symmetry breaking and dynamic freezing is found. The non-equilibrium stationary state has a complex structure, which changes as a function of parameters from a singular-continuous distribution with Euclidean or fractal support to an absolutely continuous one.Comment: 12 pages REVTeX/LaTeX format, 12 eps/ps figures. Submitted to Journal of Physics

Rascacielos, en Hamburgo

This project consists of 21 buildings and 1061 flats, and is divided into two units. One of them contains 716 flats, constructed by the «Barets» prefabrication method, and the other 345 flats, for which the «Camus» system has been adopted. Both techniques make use of solid panels of reinforced concrete for partition walls and floorings, and sandwich type panels for the outer walls. The construction of the balconies is noteworthy, which consist of elements hanging from the outer walls by means of overhangs and cables. Another outstanding feature is the variety of plan and elevation designs, which give a dynamic quality to the total aspect of the buildings.<br><br>El conjunto erigido —21 casas con un total de 1.061 viviendas— consta de dos unidades: una de ellas con 716 viviendas, construidas con el sistema de prefabricación «Barets»; y la otra con 345 viviendas, construidas con el sistema «Camus»; ambos sistemas a base de paneles macizos —realizados en hormigón armado— en muros interiores y forjados; y de paneles tipo «Sandwich» en cerramientos exteriores. Digna de mención es la construcción de los balcones, concebidos y realizados como elementos sustentados en el espacio, y colgados de las fachadas, por medio de ménsulas y elementos metálicos. Igualmente cabe destacar la variedad de plantas y alzados y el extraordinario dinamismo y movimiento del conjunto

Rascacielos, en Hamburgo

This project consists of 21 buildings and 1061 flats, and is divided into two units. One of them contains 716 flats, constructed by the «Barets» prefabrication method, and the other 345 flats, for which the «Camus» system has been adopted. Both techniques make use of solid panels of reinforced concrete for partition walls and floorings, and sandwich type panels for the outer walls. The construction of the balconies is noteworthy, which consist of elements hanging from the outer walls by means of overhangs and cables. Another outstanding feature is the variety of plan and elevation designs, which give a dynamic quality to the total aspect of the buildings.El conjunto erigido —21 casas con un total de 1.061 viviendas— consta de dos unidades: una de ellas con 716 viviendas, construidas con el sistema de prefabricación «Barets»; y la otra con 345 viviendas, construidas con el sistema «Camus»; ambos sistemas a base de paneles macizos —realizados en hormigón armado— en muros interiores y forjados; y de paneles tipo «Sandwich» en cerramientos exteriores. Digna de mención es la construcción de los balcones, concebidos y realizados como elementos sustentados en el espacio, y colgados de las fachadas, por medio de ménsulas y elementos metálicos. Igualmente cabe destacar la variedad de plantas y alzados y el extraordinario dinamismo y movimiento del conjunto

Randomly Evolving Idiotypic Networks: Structural Properties and Architecture

We consider a minimalistic dynamic model of the idiotypic network of B-lymphocytes. A network node represents a population of B-lymphocytes of the same specificity (idiotype), which is encoded by a bitstring. The links of the network connect nodes with complementary and nearly complementary bitstrings, allowing for a few mismatches. A node is occupied if a lymphocyte clone of the corresponding idiotype exists, otherwise it is empty. There is a continuous influx of new B-lymphocytes of random idiotype from the bone marrow. B-lymphocytes are stimulated by cross-linking their receptors with complementary structures. If there are too many complementary structures, steric hindrance prevents cross-linking. Stimulated cells proliferate and secrete antibodies of the same idiotype as their receptors, unstimulated lymphocytes die. Depending on few parameters, the autonomous system evolves randomly towards patterns of highly organized architecture, where the nodes can be classified into groups according to their statistical properties. We observe and describe analytically the building principles of these patterns, which allow to calculate number and size of the node groups and the number of links between them. The architecture of all patterns observed so far in simulations can be explained this way. A tool for real-time pattern identification is proposed.Comment: 19 pages, 15 figures, 4 table

Randomly Evolving Idiotypic Networks: Modular Mean Field Theory

We develop a modular mean field theory for a minimalistic model of the idiotypic network. The model comprises the random influx of new idiotypes and a deterministic selection. It describes the evolution of the idiotypic network towards complex modular architectures, the building principles of which are known. The nodes of the network can be classified into groups of nodes, the modules, which share statistical properties. Each node experiences only the mean influence of the groups to which it is linked. Given the size of the groups and linking between them the statistical properties such as mean occupation, mean life time, and mean number of occupied neighbors are calculated for a variety of patterns and compared with simulations. For a pattern which consists of pairs of occupied nodes correlations are taken into account.Comment: 14 pages, 8 figures, 4 table

Evaluating Functional Diversity as Potential Early-Warning Indicator of Rangeland Degradation

Droughts and overgrazing play a crucial role in the degradation of semi-arid rangelands. This is evident in the loss of palatable long-lived grass species and bush encroachment. Early warning indicators are needed to mitigate long-term degradation and decline in essential forage provision. Functional diversity provides valuable information on ecosystem health. However, functional diversity indices have not yet been tested regarding their applicability as early warning indicators, revealing non-linear threshold behaviour. We therefore examined the following questions: (1) How do functional diversity indices respond to grazing pressure? (2) Does land tenure affect the relationship between functional diversity and grazing pressure? (3) Are functional diversity indices suitable early-warning indicators? To answer these questions, we conducted a space-for-time substitution of land use intensity of semi-arid rangelands in Namibia. Some 16 grazing gradients were selected, each starting at a cattle watering point where grazing pressure was highest. Gradients were located in four communal and four freehold farms. Communal farms were characterised by continuous grazing, while freehold farms by rotational grazing. In each transect we recorded plant species composition of the grass layer in 9 plots of 10 × 10 m each (N = 162 plots). Within each transect, these plots were logarithmically distributed. Various plant functional traits—all relating to plant life history or resource acquisition strategy—were measured for 142 dominant species, accounting for more than 80 % of the biomass, and indices of functional diversity were calculated. We found potential threshold behaviour in functional richness on freehold farms. Certain functional diversity indices revealed non-linear patterns in rangelands but are currently not a user-friendly early-warning indicator. To harness functional diversity, we need a more standardized method of calculation, and more functional trait databases for sub-Saharan species