Area-weighted thickness distributions of fundamental illite particles for samples of illite and illite-smectite from seven locations (including bentonites and hydrothermally altered pyroclastics) were measured by Pt-shadowing technique, by transmission electron microscopy. Most thickness distributions are described by lognormal distributions, which suggest a unique crystallization process. The shapes of lognormal distributions of fundamental illite particles can be calculated from the distribution mean because the shape parameters α and β2 are interrelated: β2= 0.107α − 0.03. This growth process was simulated by the mathematical Law of Proportionate Effect that generates lognormal distributions. Simulations indicated that illite particles grow from 2-nm thick illite nuclei by surface-controlled growth, i.e., the rate of growth is restricted by how rapid crystallization proceeds given a near infinite supply of reactants, and not by the rate of supply of reactants to the crystal surface. Initially formed, 2-nm thick crystals may nucleate and grow within smectite interlayers from material produced by dissolution of single smectite 2:1 layers, thereby transforming the clay from randomly interstratified (Reichweite, R = 0) to ordered (R = 1) illite-smectite after the smectite single layers dissolve. In this initial period of illite nucleation and growth, during which expandable layers range from 100 to 20%, illite crystals grow parallel to [001]* direction, and the dimensions of the (001) plane are confined to the size of the original smectite 2:1 layers. After nucleation ceases, illite crystals may continue to grow by surface-controlled growth, and the expandable-layer content ranges from 20 to 0%. This latter period of illitization is characterized by three-dimensional growth. Other crystal-growth mechanisms, such as Ostwald ripening, supply-controlled growth, and the coalescence of smectite layers, do not produce the observed evolution of α and β2and the observed shapes of crystal thickness distributions.

The crystal growth of NH4-illite (NH4-I) from the hydrothermal system of Harghita Bãi (Eastern Carpathians) was deduced from the shapes of crystal thickness distributions (CTDs). The <2 mm and the <2-0.2 mm fractions of clay samples collected from the argillized andesite rocks consist of NH4-illite-smectite (I-S) interstratified structures (R1, R2, and R3-type ordering) with a variable smectite-layer content. The NH4-I-S (40-5% S) structures were identified underground in a hydrothermal breccia structure, whereas the K-I/NH4-I mixtures were found at the deepest level sampled (-110 m). The percentage of smectite interlayers generally decreases with increasing depth in the deposit. This decrease in smectite content is related to the increase in degree of fracturing in the breccia structure and corresponds to a general increase in mean illite crystal thickness. In order to determine the thickness distributions of NH4-I crystals (fundamental illite particles) which make up the NH4-I-S interstratified structures and the NH4-I/K-I mixtures, 27 samples were saturated with Li and aqueous solutions of PVP-10 to remove swelling and then were analyzed by X-ray diffraction. The profiles for the mean crystallite thickness (Tmean) and crystallite thickness distribution (CTD) of NH4-I crystallites were determined by the Bertaut-Warren-Averbach method using the MudMaster computer code. The Tmean of NH4-I from NH4-I-S samples ranges from 3.4 to 7.8 nm. The Tmean measured for the NH4-I/K-I mixture phase ranges from 7.8 nm to 11.7 nm (NH4-I) and from 12.1 to 24.7 nm (K-I).

The CTD shapes of NH4-I fundamental particles are asymptotic and lognormal, whereas illites from NH4-I/K-I mixtures have bimodal shapes related to the presence of two lognormal-like CTDs corresponding to NH4-I and K-I.

The crystal-growth mechanism for NH4-I samples was simulated using the Galoper code. Reaction pathways for NH4-I crystal nucleation and growth could be determined for each sample by plotting their CTD parameters on an α-ß2 diagram constructed using Galoper. This analysis shows that NH4-I crystals underwent simultaneous nucleation and growth, followed by surface-controlled growth without simultaneous nucleation.

Two independent methods of crystal-size distribution analysis were compared: the Bertaut-Warren-Averbach XRD technique (MudMaster computer program) and high-resolution transmission electron microscopy (HRTEM). These techniques were used to measure thickness distributions of illite crystals (fundamental particles) from sets of illite-smectites from shales and bentonites that had expandabilities ranging from 86%S to 6%S. The illite-smectites were treated with a polymer (polyvinylopyrolidone, PVP) to separate them into fundamental particles for XRD and HRTEM investigations.

A systematic difference between XRD and HRTEM results was observed: XRD (area-weighted distributions) detected a larger fraction of thick (>4 nm) and a smaller fraction of thin crystals as compared to HRTEM (number-weighted distributions). As a result, XRD-determined distributions have larger mean thickness values and larger distribution parameters (α and β2). The measurements performed by the two techniques were verified by modeling XRD patterns of the PVP-illites, using the measured distributions as inputs. The modeling indicated that the XRD-determined distributions are very accurate. Selecting broader thickness distributions in MudMaster further improved the modeling results. The HRTEM measurements underestimate the proportion of coarse particles, in particular in shale samples, and this inaccuracy is attributed to the effect of using number-weighted (rather than area-weighted) distributions and to inaccurate counting statistics for thick crystals.

The distributions of illite crystal (‘fundamental particle’) thickness in <0.2 µm fractions of 13 shale samples (from the Carpathian Foredeep, Poland), obtained using the Bertaut-Warren-Averbach X-ray diffraction method (MudMaster computer program), were analyzed and interpreted in terms of the mechanism of smectite illitization. All illite crystal thickness distributions in the analyzed shales are characterized by an ‘asymptotic’ shape. The origin of the asymptotic-type distributions is explained by the heterogeneity of illite crystals in shales, i.e. superposition of different populations of crystals, those included in illite-smectite (I-S) interstratifications, and those which occur as discrete illite. The analysis of the distributions in shales shows that the most frequent thickness class of illite crystals forming I-S is 2 nm. Discrete illite is composed of thicker crystals; though crystals as thin as 2 nm can also contribute to this population. The modeling of asymptotic distributions in shales by using a number of theoretical lognormal distributions shows that with progressive diagenesis, the mean thickness of illite crystals forming the I-S component increases gradually, whereas the discrete illite does not show a clear trend.

The diverse origins of illite crystals in shales do not permit determination of the mechanism of smectite illitization from the evolution of the shape of the overall crystal-thickness distribution during diagenesis. Therefore, in order to understand the mechanism of smectite illitization in shales, an attempt was made to trace the relative gains and losses of crystals of different thicknesses during illitization. This approach indicates the following mechanism of smectite illitization in the investigated shales: dissolution of smectite monolayers accompanied by growth of 2 nm crystals which are largely of detrital origin. Nucleation of 2 nm illites seems to be very limited.

The authors use the mathematical tool of Maccone's lognormal distribution to further factor the Drake equation, which calculates the number of advanced civilizations in the galaxy, from the seven original levels of the Drake equation to 49 levels of overall analysis. The Maccone approach, in fact, supported by the central limit theorem, becomes more reliable the more levels are introduced. The resulting study necessarily draws upon an array of disciplines ranging from astronomy, chemistry and geology to biology, palaeontology and futurology. The final result calculates the number of planetary systems suitable for life in its various stages of development: those which have probably hosted life in the past and those which still host it at its various evolutionary levels. The final evolutionary level is the so-called galactic civilization (often called ETC, or extraterrestrial civilizations). The number of resulting galactic civilizations is divided between static civilizations, which do not move around the galaxy and whose Kardašëv rating is still low (<1.4), of which we find three examples (we ourselves plus, perhaps, two others), and potentially dynamic civilizations, which move around the galaxy and have a sufficiently high Kardašëv rating (≥1.4), of which we find 2000.

Let (X1,...,Xn) be multivariate normal, with mean vector 𝛍 and covariance matrix 𝚺, and let Sn=eX1+⋯+eXn. The Laplace transform ℒ(θ)=𝔼e-θSn∝∫exp{-hθ(𝒙)}d𝒙 is represented as ℒ̃(θ)I(θ), where ℒ̃(θ) is given in closed form and I(θ) is the error factor (≈1). We obtain ℒ̃(θ) by replacing hθ(𝒙) with a second-order Taylor expansion around its minimiser 𝒙*. An algorithm for calculating the asymptotic expansion of 𝒙* is presented, and it is shown that I(θ)→ 1 as θ→∞. A variety of numerical methods for evaluating I(θ) is discussed, including Monte Carlo with importance sampling and quasi-Monte Carlo. Numerical examples (including Laplace-transform inversion for the density of Sn) are also given.

Quantum Monte Carlo data are often afflicted with distributions that resemble lognormal probability distributions and consequently their statistical analysis cannot be based on simple Gaussian assumptions. To this extent a method is introduced to estimate these distributions and thus give better estimates to errors associated with them. This method entails reconstructing the probability distribution of a set of data, with given mean and variance, that has been assumed to be lognormal prior to undergoing a blocking or renormalization transformation. In doing so, we perform a numerical evaluation of the renormalized sum of lognormal random variables. This technique is applied to a simple quantum model utilizing the single-thread Monte Carlo algorithm to estimate the ground state energy or dominant eigenvalue of a Hamiltonian matrix.

Approximate aggregation techniques consist of introducing certain approximations thatallow one to reduce a complex system involving many coupled variables obtaining a simplerʽʽaggregated systemʼʼ governed by a few variables. Moreover, they give results that allowone to extract information about the complex original system in terms of the behavior ofthe reduced one. Often, the feature that allows one to carry out such a reduction is thepresence of different time scales in the system under consideration. In this work we dealwith aggregation techniques in stochastic discrete time models and their application tothe study of multiregional models, i.e., of models for an age structured populationdistributed amongst different spatial patches and in which migration between the patchesis usually fast with respect to the demography (reproduction-survival) in each patch.Stochasticity in population models can be of two kinds: environmental and demographic. Wereview the formulation and the main properties of the dynamics of the different models forpopulations evolving in discrete time and subjected to the effects of environmental anddemographic stochasticity. Then we present different stochastic multiregional models withtwo time scales in which migration is fast with respect to demography and we review themain relationships between the dynamics of the original complex system and the aggregatedsimpler one. Finally, and within the context of models with environmental stochasticity inwhich the environmental variation is Markovian, we make use these techniques to analyzequalitatively the behavior of two multiregional models in which the original complexsystem is intractable. In particular we study conditions under which the population goesextinct or grows exponentially.

This paper considers the application of loglinear models to claims run-off triangles which contain negative incremental claims. Maximum likelihood estimation is applied using the three parameter lognormal distribution. The method can be used in conjunction with any model which can be expressed in lognormal form. In particular the chain ladder technique is considered. An example is given and the results compared with the basic actuarial method.

The model introduced may be treated as a mixed two-way analysis of variance with fixed company effects and random time effects. Further, the risk volumes are integrated into the model in such a way that the unexplained variance is inversely proportional to the risk volume of each company. The proposed model is used to analyze loss ratio data from the general insurance market in Kuwait. The maximum likelihood estimates of the structural parameters are obtained. These estimates are then used to compute the loss ratios and solvency margins for the four domestic insurance companies.

The run off-pattern of long-term reinsurance treaties is described by means and standard deviations of logarithmic increments of premiums and loss ratios in a normal distribution. From this description forecasts of ultimate claims and current IBNR-reserves are derived, with associated distributions and confidence limits. Aggregation from individual treaties to portfolio level is proposed by normal approximation. Security loading of IBNR-reserves is proposed by a contingency reserve at portfolio level.

The sequential broken stick model has appeared in numerous contexts, including biology, physics, engineering and geology. Kolmogorov showed that under appropriate conditions, sequential breakage processes often yield a lognormal distribution of particle sizes. Of particular interest to ecologists is the observed variance of the logarithms of the sizes, which characterizes the evenness of an assemblage of species. We derive the first two moments for the logarithms of the sizes in terms of the underlying distribution used to determine the successive breakages. In particular, for a process yielding n pieces, the expected sample variance behaves asymptotically as log(n). These results also yield a new identity for moments of path lengths in random binary trees.

Since the pioneering work of Siegert (1951), the problem of determining the first passage time distribution for a preassigned continuous and time homogeneous Markov process described by a diffusion equation has been deeply analyzed and satisfactorily solved. Here we discuss the “inverse problem” — of applicative interest — consisting in deciding whether a given function can be considered as the first passage time probability density function for some continuous and homogeneous Markov diffusion process. A constructive criterion is proposed, and some examples are provided. One of these leads to a singular diffusion equation representing a dynamical model for the genesis of the lognormal distribution.