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

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

Personality disorders and their relation to treatment outcomes in cognitive behavioural therapy for depression: a systematic review and meta-analysis

Background Previous reviews indicate that depressed patients with a comorbid personality disorder (PD) tend to benefit less from psychotherapies for depression and thus personality pathology needs to be the primary focus of treatment. This review specifically focused on studies of Cognitive Behavioural Therapy (CBT) for depression examining the influence of comorbid PD on post-treatment depression outcomes. Methods This was a systematic review and meta-analysis of studies identified through PubMed, PsychINFO, Web of Science, and Scopus. A review protocol was pre-registered in the PROSPERO database (CRD42019128590). Results Eleven eligible studies (N = 769) were included in a narrative synthesis, and ten (N = 690) provided sufficient data for inclusion in random effects meta-analysis. All studies were rated as having “low” or “moderate” risk of bias and there was no significant evidence of publication bias. A small pooled effect size indicated that patients with PD had marginally higher depression severity after CBT compared to patients without PD (g = 0.26, [95% CI: 0.10, 0.43], p = .002), but the effect was not significant in controlled trials (p = .075), studies with low risk of bias (p = .107) and studies that adjusted for intake severity (p = .827). Furthermore, PD cases showed symptomatic improvements across studies, particularly those with longer treatment durations (16–20 sessions). Conclusions The apparent effect of PD on depression outcomes is likely explained by higher intake severity rather than treatment resistance. Excluding these patients from evidence-based care for depression is unjustified, and adequately lengthy CBT should be routinely offered

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

Fundamental scaling laws of on-off intermittency in a stochastically driven dissipative pattern forming system

Noise driven electroconvection in sandwich cells of nematic liquid crystals exhibits on-off intermittent behaviour at the onset of the instability. We study laser scattering of convection rolls to characterize the wavelengths and the trajectories of the stochastic amplitudes of the intermittent structures. The pattern wavelengths and the statistics of these trajectories are in quantitative agreement with simulations of the linearized electrohydrodynamic equations. The fundamental $\tau^{-3/2}$ distribution law for the durations $\tau$ of laminar phases as well as the power law of the amplitude distribution of intermittent bursts are confirmed in the experiments. Power spectral densities of the experimental and numerically simulated trajectories are discussed.Comment: 20 pages and 17 figure