Self-avoiding random walks and Olbers' paradox

In this paper, we prove that a self-avoiding walk of infinite length provides a structure that would resolve Olbers' paradox. That is, if the stars of a universe were distributed like the vertices of an infinite random walk with each segment length of about a parsec, then the night sky could be as dark as actually observed on the Earth. Self-avoiding random walk structure can therefore resolve the Olbers' paradox even in a static universe.

The odd couple: contrasting phylogeographic patterns in two sympatric sibling species of woodlouse-­‐hunter spiders in the Canary Islands

Comparative phylogeography seeks for commonalities in the spatial demographic history of sympatric organisms to characterize the mechanisms that shaped such patterns. The unveiling of incongruent phylogeographic patterns in co-occurring species, on the other hand, may hint to overlooked differences in their life histories or microhabitat preferences. The woodlouse-hunter spiders of the genus Dysdera have undergone a major diversi cation on the Canary Islands. The species pair Dysdera alegranzaensis and Dysdera nesiotes are endemic to the island of Lanzarote and nearby islets, where they co-occur at most of their known localities. The two species stand in sharp contrast to other sympatric endemic Dysdera in showing no evidence of somatic (non-genitalic) differentiation. Phylogenetic and population genetic analyses of mitochondrial cox1 sequences from an exhaustive sample of D. alegranzaensis and D. nesiotes specimens, and additional mitochondrial (16S, L1, nad1) and nuclear genes (28S, H3) were analysed to reveal their phylogeographic patterns and clarify their phylogenetic relationships. Relaxed molecular clock models using ve calibration points were further used to estimate divergence times between species and populations. Striking differences in phylogeography and population structure between the two species were observed. Dysdera nesiotes displayed a metapopulation-like structure, while D. alegranzaensis was characterized by a weaker geographical structure but greater genetic divergences among its main haplotype lineages, suggesting more complex population dynamics. Our study con rms that co-distributed sibling species may exhibit contrasting phylogeographic patterns in the absence of somatic differentiation. Further ecological studies, however, will be necessary to clarify whether the contrasting phylogeographies may hint at an overlooked niche partitioning between the two species. In addition, further comparisons with available phylogeographic data of other eastern Canarian Dysdera endemics con rm the key role of lava ows in structuring local populations in oceanic islands and identify localities that acted as refugia during volcanic eruptions

Treatment patterns and health care resource utilization in a 1-year observational cohort study of outpatients with schizophrenia at risk of nonadherence treated with long-acting injectable antipsychotics

Purpose: To describe (1) the clinical profiles and the patterns of use of long-acting injectable (LAI) antipsychotics in patients with schizophrenia at risk of nonadherence with oral antipsychotics, and in those who started treatment with LAI antipsychotics, (2) health care resource utilization and associated costs. Patients and methods: A total of 597 outpatients with schizophrenia at risk of nonadherence, according to the psychiatrist's clinical judgment, were recruited at 59 centers in a noninterventional prospective observational study of 1-year follow-up when their treatment was modified. In a post hoc analysis, the profiles of patients starting LAI or continuing with oral antipsychotics were described, and descriptive analyses of treatments, health resource utilization, and direct costs were performed in those who started an LAI antipsychotic. Results: Therapy modifications involved the antipsychotic medications in 84.8% of patients, mostly because of insufficient efficacy of prior regimen. Ninety-two (15.4%) patients started an LAI antipsychotic at recruitment. Of these, only 13 (14.1%) were prescribed with first-generation antipsychotics. During 1 year, 16.3% of patients who started and 14.9% of patients who did not start an LAI antipsychotic at recruitment relapsed, contrasting with the 20.9% who had been hospitalized only within the prior 6 months. After 1 year, 74.3% of patients who started an LAI antipsychotic continued concomitant treatment with oral antipsychotics. The mean (median) total direct health care cost per patient per month during the study year among the patients starting any LAI antipsychotic at baseline was 1,407 ( 897.7). Medication costs (including oral and LAI antipsychotics and concomitant medication) represented almost 44%, whereas nonmedication costs accounted for more than 55% of the mean total direct health care costs. Conclusion: LAI antipsychotics were infrequently prescribed in spite of a psychiatrist-perceived risk of nonadherence to oral antipsychotics. Mean medication costs were lower than nonmedication costs.

Comparisons of intensity-duration patterns of physical activity in the US, Jamaica and 3 African countries.

BACKGROUND: This difference in how populations living in low-, middle or upper-income countries accumulate daily PA, i.e. patterns and intensity, is an important part in addressing the global PA movement. We sought to characterize objective PA in 2,500 participants spanning the epidemiologic transition. The Modeling the Epidemiologic Transition Study (METS) is a longitudinal study, in 5 countries. METS seeks to define the association between physical activity (PA), obesity and CVD risk in populations of African origin: Ghana (GH), South Africa (SA), Seychelles (SEY), Jamaica (JA) and the US (suburban Chicago). METHODS: Baseline measurements of objective PA, SES, anthropometrics and body composition, were completed on 2,500 men and women, aged 25-45 years. Moderate and vigorous PA (MVPA, min/d) on week and weekend days was explored ecologically, by adiposity status and manual labor. RESULTS: Among the men, obesity prevalence reflected the level of economic transition and was lowest in GH (1.7%) and SA (4.8%) and highest in the US (41%). SA (55%) and US (65%) women had the highest levels of obesity, compared to only 16% in GH. More men and women in developing countries engaged in manual labor and this was reflected by an almost doubling of measured MPVA among the men in GH (45 min/d) and SA (47 min/d) compared to only 28 min/d in the US. Women in GH (25 min/d), SA (21 min/d), JA (20 min/d) and SEY (20 min/d) accumulated significantly more MPVA than women in the US (14 min/d), yet this difference was not reflected by differences in BMI between SA, JA, SEY and US. Moderate PA constituted the bulk of the PA, with no study populations except SA men accumulating > 5 min/d of vigorous PA. Among the women, no sites accumulated >2 min/d of vigorous PA. Overweight/obese men were 22% less likely to engage in manual occupations. CONCLUSION: While there is some association for PA with obesity, this relationship is inconsistent across the epidemiologic transition and suggests that PA policy recommendations should be tailored for each environment.

Linking genetic and species diversities in a community of freshwater gastropods

Summary Biodiversity is usually studied through species or genetic diversities. To date, these two levels of diversity have remained the independent .fields of investigations of community ecologists and population geneticists. However, recent joint analyses of species and genetic diversities have suggested that common processes may underlie the two levels. Positive correlations between species diversity and genetic diversity may arise when the effects of drift and migration overwhelm selective effects. The first goal of this thesis was to make a joint investigation of the patterns of species and genetic diversity in a community of freshwater gastropods living in a floodplain habitat. The second goal was to determine, as far as possible, the relative influences of the processes underlying the patterns observed at each level. In chapter 2 we investigate the relative influences of different evolutionary forces in shaping the genetic structure of Radix balthica populations. Results revealed that the structure inferred using quantitative traits was lower or equal to the one inferred using neutral molecular markers. Consequently, the pattern of structure observed could be only due to random drift, possibly to uniform selection, but definitely not to selection for local optima. In chapter 3, we analyze the temporal variation of species and genetic diversities in five localities. An extended period of drought occurred at the end of the study period leading to decay of both species and genetic diversities. This parallel loss of diversity following a natural perturbation highlighted the role sometimes predominant of random drift over selection on patterns of biodiversity in a floodplain habitat. In chapter 4, we compare the spatial genetic structures of two sympatric species: Radix balthica and Planorbis carinatus. We found that R. balthica populations are weakly structured and have moderate to high values of gene diversity. In contrast, P. carinatus populations are highly structured and poorly diverse. Then we measured correlations between various indices of species and genetic diversity using genetic data .from the two species. We found only one significant correlation: between species richness and gene diversity of P. carinatus. This result highlights the .need to use genetic date from more than one species to infer correlations between species and genetic diversities. Overall, this thesis provided new insights into the common processes underlying patterns of species and genetic diversity. Résumé La biodiversité est généralement étudiée au niveau de la diversité génétique ou spécifique. Ces deux niveaux sont restés jusqu'à maintenant les domaines d'investigation séparés des généticiens des populations et des écologistes des communautés. Cependant, des analyses conjointes des diversités génétique et spécifique ont récemment suggéré que des processus similaires pouvaient influencer ces deux niveaux. Des corrélations positives entre les diversités génétique et spécifique pourraient être dues aux effets de migration et de dérive qui dominent les effets sélectifs. Le premier but de cette thèse était de faire une étude conjointe des diversités génétique et spécifique dans une communauté de gastéropodes d'eau douce. Le second objectif était de déterminer les influences relatives des différents processus liés à chaque niveau de diversité. Dans le chapitre 2 nous cherchons à déterminer quelles forces évolutives influencent la structure génétique de quatre populations de Radix balthica. La structure mesurée sur des traits quantitatifs s'est révélée être plus faible ou égale à celle mesurée avec des marqueurs moléculaires neutres. La structure observée pourrait ainsi être due uniquement à la dérive génétique, potentiellement à la sélection uniforme, mais en aucun cas à la sélection locale pour différents optima. Dans le chapitre 3 nous analysons la variation temporelle des diversités génétique et spécifique dans cinq localités. Une récente période de sécheresse a causé une diminution parallèle des deux niveaux de diversité. Cette perturbation à mis en évidence le rôle parfois prépondérant de la dérive par rapport à celui de la sélection dans le déterminisme de la biodiversité dans un écosytème alluvial. Dans le chapitre 4, nous comparons la structure génétique spatiale de deux espèces vivant en sympatrie : Radix balthica et Planorbis carinatus. Les populations de R. balthica sont peu structurées et présentent un niveau de diversité relativement élevé alors que celles de P. carinatus sont fortement structurées et peu diversifiées. Nous avons ensuite mesuré différentes corrélations entre les diversités génétique et spécifique, mais la seule relation significative a été trouvée entre la richesse spécifique et la diversité génétique de P. carinatus. Ainsi, cette thèse a permis de découvrir de nouveaux aspects des processus qui influencent en parallèle la diversité aux niveaux génétique et spécifique.

Patterns of neurovascular compression in patients with classic trigeminal neuralgia: A high-resolution MRI-based study.

PURPOSE: To describe the anatomical characteristics and patterns of neurovascular compression in patients suffering classic trigeminal neuralgia (CTN), using high-resolution magnetic resonance imaging (MRI). MATERIALS AND METHODS: The analysis of the anatomy of the trigeminal nerve, brain stem and the vascular structures related to this nerve was made in 100 consecutive patients treated with a Gamma Knife radiosurgery for CTN between December 1999 and September 2004. MRI studies (T1, T1 enhanced and T2-SPIR) with axial, coronal and sagital simultaneous visualization were dynamically assessed using the software GammaPlan?. Three-dimensional reconstructions were also developed in some representative cases. RESULTS: In 93 patients (93%), there were one or several vascular structures in contact, either, with the trigeminal nerve, or close to its origin in the pons. The superior cerebellar artery was involved in 71 cases (76%). Other vessels identified were the antero-inferior cerebellar artery, the basilar artery, the vertebral artery, and some venous structures. Vascular compression was found anywhere along the trigeminal nerve. The mean distance between the nerve compression and the origin of the nerve in the brainstem was 3.76±2.9mm (range 0-9.8mm). In 39 patients (42%), the vascular compression was located proximally and in 42 (45%) the compression was located distally. Nerve dislocation or distortion by the vessel was observed in 30 cases (32%). CONCLUSIONS: The findings of this study are similar to those reported in surgical and autopsy series. This non-invasive MRI-based approach could be useful for diagnostic and therapeutic decisions in CTN, and it could help to understand its pathogenesis.

Trends in risk factors, patterns and causes in hospitalized strokes over 25 years: The Lausanne Stroke Registry.

BACKGROUND AND OBJECTIVE: The Lausanne Stroke Registry includes, from 1979, all patients admitted to the department of Neurology of the Lausanne University Hospital with the diagnosis of first clinical stroke. Using the Lausanne Stroke Registry, we aimed to determine trends in risk factors, causes, localization and inhospital mortality over 25 years in hospitalized stroke patients. METHODS: We assessed temporal trends in stroke patients characteristics through the following consecutive periods: 1979-1987, 1988-1995 and 1996-2003. Age-adjusted cardiovascular risk factors, etiologies, stroke localizations and mortality were compared between the three periods. RESULTS: Overall, 5,759 patients were included. Age was significantly different among the analyzed periods (p < 0.001), showing an increment in older patients throughout time. After adjustment for age, hypercholesterolemia increased (p < 0.001), as opposed to cigarette smoking (p < 0.001), hypertension (p < 0.001) and diabetes and hyperglycemia (p < 0.001). In patients with ischemic strokes, there were significant changes in the distribution of causes with an increase in cardioembolic strokes (p < 0.001), and in the localization of strokes with an increase in entire middle cerebral artery (MCA) and posterior circulation strokes together with a decrease in superficial middle cerebral artery stroke (p < 0.001). In patients with hemorrhagic strokes, the thalamic localizations increased, whereas the proportion of striatocapsular hemorrhage decreased (p = 0.022). Except in the older patient group, the mortality rate decreased. CONCLUSIONS: This study shows major trends in the characteristics of stroke patients admitted to a department of neurology over a 25-year time span, which may result from referral biases, development of acute stroke management and possibly from the evolution of cerebrovascular risk factors.

Self-Organising Maps and Correlation Analysis as a Tool to Explore Patterns in Excitation-Emission Matrix Data Sets and to Discriminate Dissolved Organic Matter Fluorescence Components

Dissolved organic matter (DOM) is a complex mixture of organic compounds, ubiquitous in marine and freshwater systems. Fluorescence spectroscopy, by means of Excitation-Emission Matrices (EEM), has become an indispensable tool to study DOM sources, transport and fate in aquatic ecosystems. However the statistical treatment of large and heterogeneous EEM data sets still represents an important challenge for biogeochemists. Recently, Self-Organising Maps (SOM) has been proposed as a tool to explore patterns in large EEM data sets. SOM is a pattern recognition method which clusterizes and reduces the dimensionality of input EEMs without relying on any assumption about the data structure. In this paper, we show how SOM, coupled with a correlation analysis of the component planes, can be used both to explore patterns among samples, as well as to identify individual fluorescence components. We analysed a large and heterogeneous EEM data set, including samples from a river catchment collected under a range of hydrological conditions, along a 60-km downstream gradient, and under the influence of different degrees of anthropogenic impact. According to our results, chemical industry effluents appeared to have unique and distinctive spectral characteristics. On the other hand, river samples collected under flash flood conditions showed homogeneous EEM shapes. The correlation analysis of the component planes suggested the presence of four fluorescence components, consistent with DOM components previously described in the literature. A remarkable strength of this methodology was that outlier samples appeared naturally integrated in the analysis. We conclude that SOM coupled with a correlation analysis procedure is a promising tool for studying large and heterogeneous EEM data sets.

Emergence of oriented circuits driven by synaptic pruning associated with spike-timing-dependent plasticity (STDP)

ABSTRACT: Massive synaptic pruning following over-growth is a general feature of mammalian brain maturation. Pruning starts near time of birth and is completed by time of sexual maturation. Trigger signals able to induce synaptic pruning could be related to dynamic functions that depend on the timing of action potentials. Spike-timing-dependent synaptic plasticity (STDP) is a change in the synaptic strength based on the ordering of pre- and postsynaptic spikes. The relation between synaptic efficacy and synaptic pruning suggests that the weak synapses may be modified and removed through competitive "learning" rules. This plasticity rule might produce the strengthening of the connections among neurons that belong to cell assemblies characterized by recurrent patterns of firing. Conversely, the connections that are not recurrently activated might decrease in efficiency and eventually be eliminated. The main goal of our study is to determine whether or not, and under which conditions, such cell assemblies may emerge out of a locally connected random network of integrate-and-fire units distributed on a 2D lattice receiving background noise and content-related input organized in both temporal and spatial dimensions. The originality of our study stands on the relatively large size of the network, 10,000 units, the duration of the experiment, 10E6 time units (one time unit corresponding to the duration of a spike), and the application of an original bio-inspired STDP modification rule compatible with hardware implementation. A first batch of experiments was performed to test that the randomly generated connectivity and the STDP-driven pruning did not show any spurious bias in absence of stimulation. Among other things, a scale factor was approximated to compensate for the network size on the ac¬tivity. Networks were then stimulated with the spatiotemporal patterns. The analysis of the connections remaining at the end of the simulations, as well as the analysis of the time series resulting from the interconnected units activity, suggest that feed-forward circuits emerge from the initially randomly connected networks by pruning. RESUME: L'élagage massif des synapses après une croissance excessive est une phase normale de la ma¬turation du cerveau des mammifères. L'élagage commence peu avant la naissance et est complété avant l'âge de la maturité sexuelle. Les facteurs déclenchants capables d'induire l'élagage des synapses pourraient être liés à des processus dynamiques qui dépendent de la temporalité rela¬tive des potentiels d'actions. La plasticité synaptique à modulation temporelle relative (STDP) correspond à un changement de la force synaptique basé sur l'ordre des décharges pré- et post- synaptiques. La relation entre l'efficacité synaptique et l'élagage des synapses suggère que les synapses les plus faibles pourraient être modifiées et retirées au moyen d'une règle "d'appren¬tissage" faisant intervenir une compétition. Cette règle de plasticité pourrait produire le ren¬forcement des connexions parmi les neurones qui appartiennent à une assemblée de cellules caractérisée par des motifs de décharge récurrents. A l'inverse, les connexions qui ne sont pas activées de façon récurrente pourraient voir leur efficacité diminuée et être finalement éliminées. Le but principal de notre travail est de déterminer s'il serait possible, et dans quelles conditions, que de telles assemblées de cellules émergent d'un réseau d'unités integrate-and¬-fire connectées aléatoirement et distribuées à la surface d'une grille bidimensionnelle recevant à la fois du bruit et des entrées organisées dans les dimensions temporelle et spatiale. L'originalité de notre étude tient dans la taille relativement grande du réseau, 10'000 unités, dans la durée des simulations, 1 million d'unités de temps (une unité de temps correspondant à une milliseconde), et dans l'utilisation d'une règle STDP originale compatible avec une implémentation matérielle. Une première série d'expériences a été effectuée pour tester que la connectivité produite aléatoirement et que l'élagage dirigé par STDP ne produisaient pas de biais en absence de stimu¬lation extérieure. Entre autres choses, un facteur d'échelle a pu être approximé pour compenser l'effet de la variation de la taille du réseau sur son activité. Les réseaux ont ensuite été stimulés avec des motifs spatiotemporels. L'analyse des connexions se maintenant à la fin des simulations, ainsi que l'analyse des séries temporelles résultantes de l'activité des neurones, suggèrent que des circuits feed-forward émergent par l'élagage des réseaux initialement connectés au hasard.

Three essays on the determinants of output, inflation and interest rates

Résumé: Output, inflation and interest rates are key macroeconomic variables, in particular for monetary policy. In modern macroeconomic models they are driven by random shocks which feed through the economy in various ways. Models differ in the nature of shocks and their transmission mechanisms. This is the common theme underlying the three essays of this thesis. Each essay takes a different perspective on the subject: First, the thesis shows empirically how different shocks lead to different behavior of interest rates over the business cycle. For commonly analyzed shocks (technology and monetary policy errors), the patterns square with standard models. The big unknown are sources of inflation persistence. Then the thesis presents a theory of monetary policy, when the central bank can better observe structural shocks than the public. The public will then seek to infer the bank's extra knowledge from its policy actions and expectation management becomes a key factor of optimal policy. In a simple New Keynesian model, monetary policy becomes more concerned with inflation persistence than otherwise. Finally, the thesis points to the huge uncertainties involved in estimating the responses to structural shocks with permanent effects.

Measuring irregularities and surface defects from printed patterns

Quality inspection and assurance is a veryimportant step when today's products are sold to markets. As products are produced in vast quantities, the interest to automate quality inspection tasks has increased correspondingly. Quality inspection tasks usuallyrequire the detection of deficiencies, defined as irregularities in this thesis. Objects containing regular patterns appear quite frequently on certain industries and science, e.g. half-tone raster patterns in the printing industry, crystal lattice structures in solid state physics and solder joints and components in the electronics industry. In this thesis, the problem of regular patterns and irregularities is described in analytical form and three different detection methods are proposed. All the methods are based on characteristics of Fourier transform to represent regular information compactly. Fourier transform enables the separation of regular and irregular parts of an image but the three methods presented are shown to differ in generality and computational complexity. Need to detect fine and sparse details is common in quality inspection tasks, e.g., locating smallfractures in components in the electronics industry or detecting tearing from paper samples in the printing industry. In this thesis, a general definition of such details is given by defining sufficient statistical properties in the histogram domain. The analytical definition allowsa quantitative comparison of methods designed for detail detection. Based on the definition, the utilisation of existing thresholding methodsis shown to be well motivated. Comparison of thresholding methods shows that minimum error thresholding outperforms other standard methods. The results are successfully applied to a paper printability and runnability inspection setup. Missing dots from a repeating raster pattern are detected from Heliotest strips and small surface defects from IGT picking papers.

Identifying quantitative operation principles in metabolic pathways: a systematic method for searching feasible enzyme activity patterns leading to cellular adaptive responses

Background: Optimization methods allow designing changes in a system so that specific goals are attained. These techniques are fundamental for metabolic engineering. However, they are not directly applicable for investigating the evolution of metabolic adaptation to environmental changes. Although biological systems have evolved by natural selection and result in well-adapted systems, we can hardly expect that actual metabolic processes are at the theoretical optimum that could result from an optimization analysis. More likely, natural systems are to be found in a feasible region compatible with global physiological requirements. Results: We first present a new method for globally optimizing nonlinear models of metabolic pathways that are based on the Generalized Mass Action (GMA) representation. The optimization task is posed as a nonconvex nonlinear programming (NLP) problem that is solved by an outer- approximation algorithm. This method relies on solving iteratively reduced NLP slave subproblems and mixed-integer linear programming (MILP) master problems that provide valid upper and lower bounds, respectively, on the global solution to the original NLP. The capabilities of this method are illustrated through its application to the anaerobic fermentation pathway in Saccharomyces cerevisiae. We next introduce a method to identify the feasibility parametric regions that allow a system to meet a set of physiological constraints that can be represented in mathematical terms through algebraic equations. This technique is based on applying the outer-approximation based algorithm iteratively over a reduced search space in order to identify regions that contain feasible solutions to the problem and discard others in which no feasible solution exists. As an example, we characterize the feasible enzyme activity changes that are compatible with an appropriate adaptive response of yeast Saccharomyces cerevisiae to heat shock Conclusion: Our results show the utility of the suggested approach for investigating the evolution of adaptive responses to environmental changes. The proposed method can be used in other important applications such as the evaluation of parameter changes that are compatible with health and disease states.

Dissemination of periodic mammography and patterns of use, by birth cohort, in Catalonia (Spain)

Background: In Catalonia (Spain) breast cancer mortality has declined since the beginning of the 1990s. The dissemination of early detection by mammography and the introduction of adjuvant treatments are among the possible causes of this decrease, and both were almost coincident in time. Thus, understanding how these procedures were incorporated into use in the general population and in women diagnosed with breast cancer is very important for assessing their contribution to the reduction in breast cancer mortality. In this work we have modeled the dissemination of periodic mammography and described repeat mammography behavior in Catalonia from 1975 to 2006. Methods: Cross-sectional data from three Catalan Health Surveys for the calendar years 1994, 2002 and 2006 was used. The dissemination of mammography by birth cohort was modeled using a mixed effects model and repeat mammography behavior was described by age and survey year. Results: For women born from 1938 to 1952, mammography clearly had a period effect, meaning that they started to have periodic mammograms at the same calendar years but at different ages. The age at which approximately 50% of the women were receiving periodic mammograms went from 57.8 years of age for women born in 1938–1942 to 37.3 years of age for women born in 1963–1967. Women in all age groups experienced an increase in periodic mammography use over time, although women in the 50–69 age group have experienced the highest increase. Currently, the target population of the Catalan Breast Cancer Screening Program, 50–69 years of age, is the group that self-reports the highest utilization of periodic mammograms, followed by the 40–49 age group. A higher proportion of women of all age groups have annual mammograms rather than biennial or irregular ones. Conclusion: Mammography in Catalonia became more widely implemented during the 1990s. We estimated when cohorts initiated periodic mammograms and how frequently women are receiving them. These two pieces of information will be entered into a cost-effectiveness model of early detection in Catalonia.

Scaling behavior of random knots.

Using numerical simulations we investigate how overall dimensions of random knots scale with their length. We demonstrate that when closed non-self-avoiding random trajectories are divided into groups consisting of individual knot types, then each such group shows the scaling exponent of approximately 0.588 that is typical for self-avoiding walks. However, when all generated knots are grouped together, their scaling exponent becomes equal to 0.5 (as in non-self-avoiding random walks). We explain here this apparent paradox. We introduce the notion of the equilibrium length of individual types of knots and show its correlation with the length of ideal geometric representations of knots. We also demonstrate that overall dimensions of random knots with a given chain length follow the same order as dimensions of ideal geometric representations of knots.