Divisible Probability Research Articles

Completely monotonic functions involving the gamma and $q$-gamma functions

We give an infinite family of functions involving the gamma function whose logarithmic derivatives are completely monotonic. Each such function gives rise to an infinitely divisible probability distribution. Other similar results are also obtained for specific combinations of the gamma and $q$-gamma functions.

Triage Decision for ICU Admission and Patients' Outcome

Background: We constructed a prospective study to evaluate the outcome of patients referred to an intensivist for ICU admission as well as the factors associated with ICU admission refusal. Methods: Patients referred for ICU admission to medico-surgical ICU in our hospital for 16 weeks were included in this study. We classified the reasons for refusal into three categories: inappropriate referral; triage; futility. Also we classified admitted patients into two categories: immediate admission; delayed admission. After initial evaluation of a patient by an intensivist, we checked the patient's outcome for following 28 days. Results: 632 patients had been referred for ICU admission during study period. Among them, 445 (70%) patients were admitted and 187 (30%) patients were refused ICU admission. 116 patients were refused because of inappropriate referral, 52 for triage and 19 for futility. 394 patients were admitted immediately and 51 were refused initially but were later admitted. When 116 inappropriate referral patients were excluded, the mortality rates for immediate admission, delayed admission and triage/futility were 14.4%, 39.2% and 56.3%, respectively (P ＜ 0.05). Standardized mortality ratio was 0.70 for immediate admission, 1.20 for delayed admission, 1.28 for triage and 1.30 for futility (P ＜ 0.05). The factors associated with refusal for ICU admission were age, medical division, diagnostic group and high Mortality Probability Model (MPM ) grades. Conclusions: Because the triage decision for ICU admission influences the patients' outcome, an intensivist must carefully come to a decision when admitting patients to ICU by considering the appropriate guidelines for ICU admission and triage.

A CONNECTION BETWEEN FREE AND CLASSICAL INFINITE DIVISIBILITY

In this paper we continue our studies, initiated in Refs. 2–4, of the connections between the classes of infinitely divisible probability measures in classical and in free probability. We show that the free cumulant transform of any freely infinitely divisible probability measure equals the classical cumulant transform of a certain classically infinitely divisible probability measure, and we give several characterizations of the latter measure, including an interpretation in terms of stochastic integration. We find, furthermore, an alternative definition of the Bercovici–Pata bijection, which passes directly from the classical to the free cumulant transform, without passing through the Lévy–Khintchine representations (classical and free, respectively).

Models for the dynamics and order of immunoglobulin isotype switching

Experiments show that class switch recombination (CSR) depends on the number of divisions that the cell has performed rather than on the time since stimulation. Using computer simulations of CSR dynamics in B cell populations, we addressed the following questions. How does the probability of CSR depend on the number of divisions that a cell has performed? How does the cell decide which isotype to switch to? Does this decision depend on the distance between the genes of the pre-switch and the post-switch isotype? Our results indicate that post-switch isotype choice may be determined indirectly by the probabilities of division (which is fixed) and of switching per division (which increases as a function of the number of divisions that a cell performs), and more directly by a bias in the choice of the post-switch C gene segment towards those proximal to the pre-switch C gene.

Asymmetric division and lineage commitment at the level of hematopoietic stem cells: inference from differentiation in daughter cell and granddaughter cell pairs.

How hematopoietic stem cells (HSCs) commit to a particular lineage is unclear. A high degree of HSC purification enabled us to address this issue at the clonal level. Single-cell transplantation studies revealed that 40% of the CD34−/low, c-Kit+, Sca-1+, and lineage marker− (CD34−KSL) cells in adult mouse bone marrow were able, as individual cells, to reconstitute myeloid and B- and T-lymphoid lineages over the long-term. Single-cell culture showed that >40% of CD34−KSL cells could form neutrophil (n)/macrophage (m)/erythroblast (E)/megakaryocyte (M) (nmEM) colonies. Assuming that a substantial portion of long-term repopulating cells can be detected as nmEM cells within this population, we compared differentiation potentials between individual pairs of daughter and granddaughter cells derived in vitro from single nmEM cells. One of the two daughter or granddaughter cells remained an nmEM cell. The other showed a variety of combinations of differentiation potential. In particular, an nmEM cell directly gave rise, after one cell division, to progenitor cells committed to nm, EM, or M lineages. The probability of asymmetric division of nmEM cells depended on the cytokines used. These data strongly suggest that lineage commitment takes place asymmetrically at the level of HSCs under the influence of external factors.

The rescaling method for quantifying the turnover of cell populations

The dynamic nature of immune responses requires the development of appropriate experimental and theoretical tools to quantitatively estimate the division and death rates which determine the turnover of immune cells. A number of papers have used experimental data from BrdU and D-glucose labels together with a simple random birth–death model to quantify the turnover of immune cells focusing on HIV/SIV infections [Mohri et al. 279 (1998) 1223–1227, Hellerstein et al. 5 (1999) 83–89, Bonhoeffer et al. 164 (2000) 5049–5054, Mohri et al. 87 (2001) 1277–1287]. We show how uncertainties in the assumptions of the random birth–death model may lead to substantial errors in the parameters estimated. We then show how more accurate estimates can be obtained from the more recent CFSE data which allow to track the number of divisions each cell has undergone. Specifically, we: (i) describe a general stage-structured model of cell division where the probabilities of division and death are functions of time since the previous division; (ii) develop a rescaling method to identify invariant parameters (i.e. the ones that are independent of the specific functions describing division and death); (iii) show how these invariant parameters can be estimated, and (iv) illustrate this technique by applying it to CFSE data taken from the literature.

Lévy laws in free probability.

This article and its sequel outline recent developments in the theory of infinite divisibility and Lévy processes in free probability, a subject area belonging to noncommutative (or quantum) probability. The present paper discusses the classes of infinitely divisible probability measures in classical and free probability, respectively, via a study of the Bercovici-Pata bijection between these classes.

Lévy processes in free probability.

This is the continuation of a previous article that studied the relationship between the classes of infinitely divisible probability measures in classical and free probability, respectively, via the Bercovici-Pata bijection. Drawing on the results of the preceding article, the present paper outlines recent developments in the theory of Lévy processes in free probability.

Affinely infinitely divisible distributions and the embedding problem

Let A be a locally compact abelian group and let μ be a probability measure on A. A probability measure λ on A is an affine k-th root of μ if there exists a continuous automorphism ρ of A such that ρ k =I (the identity transformation) and λ∗ρ(λ)∗ρ 2 ( λ)∗… ∗ρ k-1 (λ ) = μ, and μ is affinely infinitely divisible if it has affine k-th roots for all orders. Clearly every infinitely divisible probability measure is affinely infinitely divisible. In this paper we prove the converse for connected abelian Lie groups: Every affinely infinitely divisible probability measure on a connected abelian Lie group A is infinitely divisible. If G is a locally compact group, A a closed abelian subgroup of G, and μ a probability measure on G which is supported on A and infinitely divisible on G, we give sufficient conditions which ensure that μ is infinitely divisible on A.

Type G Distributions on \({\mathbb{R}}\) d

This paper presents a systematic study of the class of multivariate distributions obtained by a Gaussian randomization of jumps of a Levy process. This class, called the class of type G distributions, constitutes a closed convolution semigroup of the family of symmetric infinitely divisible probability measures. Spectral form of Levy measures of type G distributions is obtained and it is shown that type G property can not be determined by one dimensional projections. Conditionally Gaussian structure of type G random vectors is exhibited via series representations.

Radiofrequency electromagnetic fields do not alter the cell cycle progression of C3H 10T and U87MG cells.

The effects of exposure to radiofrequency electromagnetic fields (RF EMFs) on cell cycle progression of mouse fibroblasts C3H 10T(1/2) and human glioma U87MG cells were determined by the flow cytometric bromodeoxyuridine pulse-chase method. Cells were exposed to a frequency-modulated continuous wave at 835.62 MHz or a code division multiple access RF EMF centered on 847.74 MHz at an average specific absorption rate of 0.6 W/kg. Five cell cycle parameters, including the transit of cells through G(1), G(2) and S phase and the probability of cell division, were examined immediately after the cells were placed in the fields or after they had been kept in the fields for up to 100 h. The only significant change observed in the study was that associated with C3H 10T(1/2) cell cultures moving into plateau phase toward the later times in the long-exposure experiment. No changes in the cell cycle parameters were observed in cells exposed to either mode of RF EMFs when compared to sham-exposed cells in either of the cell lines studied during the entire experimental period. The results show that exposure to RF EMFs, at the frequencies and power tested, does not have any effect on cell progression in vitro.

Completely monotonic functions

In this expository article we survey some properties of completely monotonic functions and give various examples, including some famous special functions. Such function are useful, for example, in probability theory. It is known, [1, p.450], for example, that a function w is the Laplace transform of an infinitely divisible probability distribution on (0,∞), if and only if w = e-h , where the derivative of h is completely monotonic and h(0+) = 0.

Age-dependent Cell Cycle Models

This paper presents age-dependent cell cycle models i.e., models where cell generation time is a random variable given by some distribution function, and the probability of cell division per unit time is a function only of cell age (not, for example, of cell mass). It is shown that there does not exist a stable mass distribution if the cells grow exponentially. In the case of linear growth, conditions for stability of the mass distribution are derived. To show these, the methods different from those considered up till now in the literature, are used. It is also shown that one can consider the cell mass growth as a linear dynamical system with a stochastic perturbation. The sister cell model as an improvement of the Transition Probability Model is derived. Statistical data are obtained for that model, and comparisons are made with some experimental data. As a verification tool, alpha and beta curves, are used.

Partly Divisible Probability Measures on Locally Compact Abelian Groups

A notion of admissible probability measures $\mu$ on a locally compact Abelian group (LCA-group) $G$ with connected dual group $\hat G=\R^d\times \T^n$ is defined. To such a measure $\mu$, a closed semigroup $\Lambda(\mu)\subseteq (0,\infty)$ can be associated, such that, for $t\in \Lambda(\mu)$, the Fourier transform to the power $t$, $(\hat \mu)^t$, is a characteristic function. We prove that the existence of roots for non admissible probability measures underlies some restrictions, which do not hold in the admissible case. As we show for the example $\Z_2$, in the case of LCA-groups with non connected dual group, there is no canonical definition of the set $\Lambda(\mu)$.

Immunoglobulin class switch recombination machinery: progress and challenges.

Experiments show that class switch recombination (CSR) depends on the number of divisions that the cell has performed rather than on the time since stimulation. Using computer simulations of CSR dynamics in B cell populations, we addressed the following questions. How does the probability of CSR depend on the number of divisions that a cell has performed? How does the cell decide which isotype to switch to? Does this decision depend on the distance between the genes of the pre-switch and the post-switch isotype? Our results indicate that post-switch isotype choice may be determined indirectly by the probabilities of division (which is fixed) and of switching per division (which increases as a function of the number of divisions that a cell performs), and more directly by a bias in the choice of the post-switch C gene segment towards those proximal to the pre-switch C gene.

Time-lapse video reveals immediate heterogeneity and heritable damage among human ileocecal carcinoma HCT-8 cells treated with raltitrexed (ZD1694)

Cellular heterogeneity in drug response has important clinical implications, and is believed to develop over many generations during clonal evolution in human tumors. The purpose of this study was to determine the level of heterogeneity exhibited by sister cells soon after their birth. Human ileocecal carcinoma cells (HCT-8) were followed up to 11 days in vitro after a 2-h exposure to 1 microM raltitrexed (IC(95)) in a time-lapse video system. Over five experiments, 414 cells were followed after exposure to raltitrexed. Immediate sterility occurred in 74% of treated cells. Only 6% of cells could produce more than two generations of offspring, and heterogeneity in drug response was seen. Comparing sister cells < 24 h old, the more proliferative sibling produced up to 73 times more offspring, with a median ratio of 9.0 (control median = 1.19). Offspring of prolific drug-treated cells had a decreased probability of division (68% compared with 92%) and an increased average interdivision time (19.0 h compared with 15.1 h). Short-term exposure to raltitrexed resulted in increased interdivision times and production of sterile offspring extending seven generations. Cellular heterogeneity (difference in proliferation potential comparing drug-treated sister cells) was evident without a period of clonal evolution.

MODEL OF THE DYNAMICS OF A BRANCHING SYSTEM OF THE GLIAL CELL LINEAGESIN VITRO

A genealogical model describing the dynamics of a binary branching system of astrocytes and microglia which takes into account a developmental hierarchy, is proposed. The model consists of a scheme of developmental pathways interconnecting the elements at various stages of development from a common progenitor to a nonproliferating end stage. To the elements at each stage are attributed probabilities of division, differentiation and quiescence. The pathway of any particular element at the end of each cycle is determined by a random-number generator according to the predetermined probabilities. The model is applied to colony formation in vitro. The development of each colony is followed for several cycles of division and theoretical results are compared to experimental values. Comparison of values obtained from several variants of the theoretical model with experimental data is then used to derive the most plausible scheme of branching pathways under given experimental conditions. The model is defined as follows: a common unlabeled progenitor with a high self-renewal potential differentiates into unlabeled monopotential precursors which further develop into astrocytes and microglia, identified experimentally as GFAP-positive cells and CR3-positive cells, respectively. Both the monopotential unlabeled cells and the identifiable progeny also have the capability of self-renewal.

Infinitely divisible probabilities on linearp-adic groups

In this paper we extend the work of Shah, on the structure of infinitely divisible probabilities onp-adic linear groups, to give a classification for all such probabilities.

ENTRANCE LAWS FOR DAWSON-WATANABE SUPERPROCESSES WITH NONLOCAL BRANCHING

This paper proves a 1-1 correspondence between minimal probability entrance laws for the superprocess and entrance laws for its underlying process. From this the author deduces that an infinitely divisible probability entrance law for the superprocess is uniquely determined by an infinitely divisible probability measure on the space of the underlying entrance laws. Under an additional condition, a characterization is given for all entrance laws for the superprocess, generalizing the results of Dynkin (1989). An application to immigration processes is also discussed.

Partly divisible probability distributions

AbstractGiven a probability distribution µ a set Λ(µ) of positive real numbersis introduced, so that Λ(µ) measures the ”divisibility” of µ. The basicproperties of Λ(µ) are described and examples of probability distributionsare given, which exhibit the existence of a continuum of situations inter-polating the extreme cases of infinitely and minimally divisible probabilitydistributions. Mathematics Subject Classification (1991): Primary 42A85;Secondary 60B15, 60E10Key Words and Phrases: admissible probability distributions, convolutionsemigroups, divisibility, Fourier transforms. Introduction In probability theory it is often interesting to know whether a random variableXcan be decomposed into a sum of nindependent identical distributed randomvariablesX= X 1 +··· +X n .In terms of the probability distribution µof Xthe above equation meansthatµ= µ 1n ∗···∗µ 1 | {z n } n times , (1)where µ 1n is the probability distribution of X 1 .The notion of infinitely divisible probability distribution characterizes a spe-cial case of such phenomena, where the process of dividing a probability distri-bution µinto n-fold convolutions of probability distributions µ