Discrete Steps Research Articles

Hematopoietic differentiation is characterized by a transient peak of entropy at a single-cell level

BackgroundMature blood cells arise from hematopoietic stem cells in the bone marrow by a process of differentiation along one of several different lineage trajectories. This is often represented as a series of discrete steps of increasing progenitor cell commitment to a given lineage, but as for differentiation in general, whether the process is instructive or stochastic remains controversial. Here, we examine this question by analyzing single-cell transcriptomic data from human bone marrow cells, assessing cell-to-cell variability along the trajectories of hematopoietic differentiation into four different types of mature blood cells. The instructive model predicts that cells will be following the same sequence of instructions and that there will be minimal variability of gene expression between them throughout the process, while the stochastic model predicts a role for cell-to-cell variability when lineage commitments are being made.ResultsApplying Shannon entropy to measure cell-to-cell variability among human hematopoietic bone marrow cells at the same stage of differentiation, we observed a transient peak of gene expression variability occurring at characteristic points in all hematopoietic differentiation pathways. Strikingly, the genes whose cell-to-cell variation of expression fluctuated the most over the course of a given differentiation trajectory are pathway-specific genes, whereas genes which showed the greatest variation of mean expression are common to all pathways. Finally, we showed that the level of cell-to-cell variation is increased in the most immature compartment of hematopoiesis in myelodysplastic syndromes.ConclusionsThese data suggest that human hematopoietic differentiation could be better conceptualized as a dynamical stochastic process with a transient stage of cellular indetermination, and strongly support the stochastic view of differentiation. They also highlight the need to consider the role of stochastic gene expression in complex physiological processes and pathologies such as cancers, paving the way for possible noise-based therapies through epigenetic regulation.

Stability Analysis of Continuous Stochastic Linear Model

Many scholars have conducted research on the traffic oscillations and reproduced the growth pattern by establishing stochastic models and simulations. However, the growth pattern of oscillations caused by uncertainty have not been thoroughly studied. Recently, a frequency domain stability analysis method was proposed to analyze the discrete stochastic model. This paper extends this analysis to a continuous situation based on frequency domain tools (e.g., Laplace transform) by introducing a continuous bandlimited white noise. The analytical expression for the evolution of speed standard deviation has been derived. Our study of a homogeneous case reveals an interesting phenomenon: when |G(ω)|∞<1, the speed variance will converge to a constant value, which only depends on the self-disturbance of vehicles. The simulation results verified that the continuous models and corresponding discrete model tend to be consistent when the discrete time step tends to the infinitesimal. Overall, this paper makes up for the deficiency of previous studies on continuous oscillations in car-following theory and can potentially be used to develop new control strategies to help dampen traffic oscillations.

An Efficient Method of Classification the Gestational Diabetes Using ID3 Classifier

Artificial intelligence algorithms have an important and effective role in the medical field, especially in the field of diagnosing diseases. This research focuses on predicting the diagnosis of gestational diabetes by using the Iterative Dichotomiser3 (ID3) classifier algorithm, which is utilized to identify gestational diabetes; it was one of the most significant algorithms employed in this study. Training and testing are two critical phases of the research study. This study employed the Pima Indians Diabetes dataset, which comprised 768 women aged 21 and abovewith the eight reported traits. A feature selection stage, a discretization step, and using the classifier model for producing decision rules are all part of the Pima Indians diabetes data gathering process (Diabetes Dataset). In this study, the decision tree is employed to develop the classifier model, which is based on Diabetes training. The Iterative Dichotomiser3 (ID3) technique may be used to run the decision tree classification process. Diabetes is tested using decision rules, and the classifier implementation confusion matrix was retrieved from the testing portion. The system delivered high-quality results with a 94 percent accuracy rate.

Compressed sensing based on QUBO formulation

Ising machines efficiently solve the combinatorial optimization problems described by the Ising model or the quadratic unconstrained binary optimization (QUBO) formulation. A hybrid method based on the QUBO formulation for compressed sensing is proposed. The proposed method comprises alternative steps of discrete and continuous optimization. In the discrete optimization step, the objective function is described by the QUBO formulation. Successful examples obtained via the proposed method are demonstrated. The performance of the proposed method depends highly on the initial conditions.

Tunable x-ray free electron laser multi-pulses with nanosecond separation

X-ray Free Electron Lasers provide femtosecond x-ray pulses with narrow bandwidth and unprecedented peak brightness. Special modes of operation have been developed to deliver double pulses for x-ray pump, x-ray probe experiments. However, the longest delay between the two pulses achieved with existing single bucket methods is less than 1 picosecond, thus preventing the exploration of longer time-scale dynamics. We present a novel two-bucket scheme covering delays from 350 picoseconds to hundreds of nanoseconds in discrete steps of 350 picoseconds. Performance for each pulse can be similar to the one in a single pulse operation. The method has been experimentally tested with the Linac Coherent Light Source (LCLS-I) and the copper linac with LCLS-II hard x-ray undulators.

Implementation of apertures in a proton pencil-beam dose algorithm

The use of field-specific apertures, routine in scattered or uniform-scanned proton fields, are still a necessity in pencil-beam scanned (PBS) fields to sharpen the penumbral edge at low energies and in high fraction dose application beyond that achievable with small spot size. We describe a model implemented in our clinical pencil-beam algorithm that models the insertion of a shaped aperture, including shapes adapted per energy layer such as may be achieved with a multi-leaf collimator. The model decomposes the spot transport into discrete steps. The first step transport a uniform intensity field of high-resolution sub-pencil-beams at the layer energy through the medium. This transport only considers primary scattering in both the patient and an optional range-shifter. The second step models the aperture areas and edge penumbral transition as a modulation of the uniform intensity. The third step convolves individual steps over the uniform-transported field including the aperture-modified intensities. We also introduce an efficient model based on a Clarkson sector integration for nuclear scattered halo protons. This avoids the explicit modeling of long range halo protons to the detriment of computational efficiency in calculation and optimization. We demonstrate that the aperture effect is primarily due to in-patient and shifter scattering with a small contribution from the apparent beam source position. The model provides insight into the primary physics contributions to the penumbra and the nuclear halo. The model allowed us to fully deploy our PBS capacity at our two-gantry center without which PBS treatments would have been inferior compared to scattered fields with apertures. Finally, Monte Carlo calculations have (nearly) replaced phenomenological pencil-beam models for collimated fields. Phenomenological models do, however, allow exposition of underlying clinical phenomena and closer connection to representative clinical observables.

Insights into polyamine metabolism: homospermidine is double-oxidized in two discrete steps by a single copper-containing amine oxidase in pyrrolizidine alkaloid biosynthesis.

Polyamines are important metabolites in plant development and abiotic and biotic stress responses. Copper-containing amine oxidases (CuAOs) are involved in the regulation of polyamine levels in the cell. CuAOs oxidize primary amines to their respective aldehydes and hydrogen peroxide. In plants, aldehydes are intermediates in various biosynthetic pathways of alkaloids. CuAOs are thought to oxidize polyamines at only one of the primary amino groups, a process frequently resulting in monocyclic structures. These oxidases have been postulated to be involved in pyrrolizidine alkaloid (PA) biosynthesis. Here, we describe the identification and characterization of homospermidine oxidase (HSO), a CuAO of Heliotropium indicum (Indian heliotrope), involved in PA biosynthesis. Virus-induced gene silencing of HSO in H. indicum leads to significantly reduced PA levels. By in vitro enzyme assays after transient in planta expression, we show that this enzyme prefers Hspd over other amines. Nuclear magnetic resonance spectroscopy and mass spectrometry analyses of the reaction products demonstrate that HSO oxidizes both primary amino groups of homospermidine (Hspd) to form a bicyclic structure, 1-formylpyrrolizidine. Using tracer feeding, we have further revealed that 1-formylpyrrolizidine is an intermediate in the biosynthesis of PAs. Our study therefore establishes that HSO, a canonical CuAO, catalyzes the second step of PA biosynthesis and provides evidence for an undescribed and unusual mechanism involving two discrete steps of oxidation that might also be involved in the biosynthesis of complex structures in other alkaloidal pathways.

Extraction of fucoxanthin from Chaetoceros calcitrans by electropermeabilization-assisted liquid biphasic flotation system

Electropermeabilization-assisted liquid biphasic flotation (LBF) system is an emerging extraction system facilitated by the adsorptive bubble separation and the electroporation in an aqueous two-phase medium. This integrative extraction system is well suited for the direct recovery of intracellular biocompounds from cells without the needs of discrete steps of mid- or down-stream bioprocessing. The potential of electropermeabilization-assisted LBF system in bioseparation was demonstrated in this study by using the diatom Chaetoceros calcitrans (C. calcitrans) as the source of fucoxanthin, which is a promising antioxidant highly demanded by food and pharmaceutical industries. The extraction performances of LBF and electropermeabilization-assisted LBF system were comprehensively evaluated under the optimal operating conditions. Comparatively, the optimized LBF and electropermeabilization-assisted LBF systems achieved the satisfactory yields of fucoxanthin, i.e., 14.78 mg/g and 16.09 mg/g, respectively. The good recovery of fucoxanthin using electropermeabilization-assisted LBF system could be attributed to the higher release of fucoxanthin from the electrotreated C. calcitrans, allowing a higher partition of fucoxanthin to the top phase of LBF system (236.72 as compared to 152.15 from LBF system). In addition, the fucoxanthin extracted suing both methods exhibit satisfactory antioxidant activities. The application of electropermeabilization-assisted LBF system in the extraction of fucoxanthin is a greener and highly efficient bioprocessing route which can be prospectively extended to other biocompound extraction from microalgal sources.

Fast evaluation of pressure and saturation predictions with a deep learning surrogate flow model

Numerical models for flow through porous media are essential for forecasting subsurface fluid flow response to support optimal decision-making to develop subsurface resources such as groundwater, geothermal, and oil and gas. For reservoir engineering, numerical flow simulation modeling is applied to support and maximize well ultimate recovery and recoverable reserves to maximize project economics and safety while minimizing environmental impacts. Subsurface models explore subsurface uncertainty based on integrating geological, geophysical, petrophysical, and reservoir engineering data and expert interpretations.To evaluate uncertainty, we rely on multiple geostatistical subsurface heterogeneity realizations paired with flow simulation forecasts to test the sensitivity of various reservoir development parameters and build an uncertainty model of reservoir performance. However, with large reservoir models, numerical flow simulation time increases, leading to a significant amount of professional time and computational effort that increases project costs, and increases cycle times resulting in delay or diminished decision quality. This issue motivates the utilization of surrogate, computationally efficient approximative flow models. Current methods focus on prediction accuracy and minimizing prediction error. When uncertainty is significant, prediction accuracy is insufficient, and we must consider the entire uncertainty distribution.We propose a new and general workflow to generate accurate and precise machine learning-based surrogate flow models to predict the relationship between the reservoir rock and fluids, development parameters, and the reservoir flow simulation responses. We train a deep convolutional neural network using the results from a three-dimensional two-phase flow simulator. Next, we use the trained model to generate ensemble predictions from the flow surrogate to evaluate the uncertainty based on the development parameters and geological information.The machine learning-based surrogate model uses exhaustive subsurface predictor features, porosity, permeability, well position, and an engineered feature representing non-dimensional time as input to predict exhaustive subsurface response features, pressure, and saturation distribution over discrete time steps as the output. The proposed workflow integrates the spatiotemporal aspect of subsurface flow modeling. It allows the practitioner engineer to explore subsurface uncertainty without the entire computational cost of numerical flow simulation to support optimum, timely development decision-making.

Stock-recruitment models from the viewpoint of density-dependent survival and the onset of strong density-dependence when a carrying capacity limit is reached

When viewed on a discrete time step (e.g., one year) rather than instantaneously, the Ricker and Beverton-Holt stock recruitment relationships describe a strong influence of changes in adult abundance on juvenile survival even at low biomass levels. In fact, they describe a stronger influence at lower adult abundance than at high adult abundance. This is not consistent with a simplistic interpretation of carrying capacity, where survival reduces more rapidly as the juvenile abundance approaches its carrying capacity due to resource limitation. It implies that other processes (e.g., changes in predation risk with increased abundance as described by arena theory) control density dependence. A simple habitat limitation model suggests that a similar relationship can occur if the number of juveniles produced by the adults is always more than the juvenile carrying capacity. However, the resulting stock-recruitment relationship is constant recruitment, while that of the Beverton-Holt is increasing asymptotically and the Ricker is dome shaped. Fluctuations around an average recruitment level for a wide range of adult abundance levels is common for highly fecund pelagic spawners (e.g., tuna), suggesting that the survival and recruitment relationships implied by the simple habitat limitation model might be appropriate for these species. Underpinning this argument is the concept that the adult population will saturate the juvenile carrying capacity unless the adult population is fished to extremely low levels. However, it is reasonable to expect that survival should be density independent, or possibly even decompensatory, at some low level of adult abundance. Ignoring depensation, the survival relationship would therefore be well described by a general logistic equation where the decline in survival initially accelerates as abundance increases and then deaccelerates. The resulting stock-recruitment relationship would be a hockey stick shape with the possibility of reduced recruitment at large stock sizes. The model could also be used for low fecund species (e.g., sharks) for which the juvenile carrying capacity is not saturated until the adult abundance is large (i.e., survival is density-independent for a wide range of adult population sizes). In our model, particularly for high fecund species, temporal variation in recruitment would be controlled by changes in the juvenile carrying capacity. We provide equations for the generalized logistic survival stock-recruitment model and investigate the yield consequences. We suggest that an important management consideration is the level of adult abundance at which the juvenile carrying capacity becomes saturated (i.e., when does density dependence become influential on juvenile survival).

Delineating the role of the k-loop in the KIF1A chemomechanical cycle

The kinesin-3 KIF1A has three prominent biophysical characteristics: fast velocity, superprocessivity, and low resistance to load. Using a combination of single molecule motility assay and stopped-flow spectroscopy, we measured the discrete steps of the chemomechanical cycle. The fast rear-head detachment suggests that the fast velocity originates from a reduced waiting time in the 2-head-bound state, resulting in KIF1A predominantly residing in the vulnerable 1-head-bound state during its stepping cycle and therefore being susceptible to load; while the superprocessivity arises from the relatively high microtubule affinity in the post-hydrolysis state.

Coevolutionary theory of hosts and parasites.

Host and parasite evolution are closely intertwined, with selection for adaptations and counter‐adaptations forming a coevolutionary feedback loop. Coevolutionary dynamics are often difficult to intuit due to these feedbacks and are hard to demonstrate empirically in most systems. Theoretical models have therefore played a crucial role in shaping our understanding of host–parasite coevolution. Theoretical models vary widely in their assumptions, approaches and aims, and such variety makes it difficult, especially for non‐theoreticians and those new to the field, to: (1) understand how model approaches relate to one another; (2) identify key modelling assumptions; (3) determine how model assumptions relate to biological systems; and (4) reconcile the results of different models with contrasting assumptions. In this review, we identify important model features, highlight key results and predictions and describe how these pertain to model assumptions. We carry out a literature survey of theoretical studies published since the 1950s (n = 219 papers) to support our analysis. We identify two particularly important features of models that tend to have a significant qualitative impact on the outcome of host–parasite coevolution: population dynamics and the genetic basis of infection. We also highlight the importance of other modelling features, such as stochasticity and whether time proceeds continuously or in discrete steps, that have received less attention but can drastically alter coevolutionary dynamics. We finish by summarizing recent developments in the field, specifically the trend towards greater model complexity, and discuss likely future directions for research.

Single-Cell Transcriptomics Reveals Discrete Steps in Regulatory T Cell Development in the Human Thymus.

CD4+CD25+FOXP3+ regulatory T (Treg) cells control immunological tolerance. Treg cells are generated in the thymus (tTreg) or in the periphery. Their superior lineage fidelity makes tTregs the preferred cell type for adoptive cell therapy (ACT). How human tTreg cells develop is incompletely understood. By combining single-cell transcriptomics and flow cytometry, we in this study delineated three major Treg developmental stages in the human thymus. At the first stage, which we propose to name pre-Treg I, cells still express lineage-inappropriate genes and exhibit signs of TCR signaling, presumably reflecting recognition of self-antigen. The subsequent pre-Treg II stage is marked by the sharp appearance of transcription factor FOXO1 and features induction of KLF2 and CCR7, in apparent preparation for thymic exit. The pre-Treg II stage can further be refined based on the sequential acquisition of surface markers CD31 and GPA33. The expression of CD45RA, finally, completes the phenotype also found on mature recent thymic emigrant Treg cells. Remarkably, the thymus contains a substantial fraction of recirculating mature effector Treg cells, distinguishable by expression of inflammatory chemokine receptors and absence of CCR7. The developmental origin of these cells is unclear and warrants caution when using thymic tissue as a source of stable cells for ACT. We show that cells in the major developmental stages can be distinguished using the surface markers CD1a, CD27, CCR7, and CD39, allowing for their viable isolation. These insights help identify fully mature tTreg cells for ACT and can serve as a basis for further mechanistic studies into tTreg development.

Distribution Agnostic Symbolic Representations for Time Series Dimensionality Reduction and Online Anomaly Detection

Due to the importance of the lower bounding distances and the attractiveness of symbolic representations, the family of symbolic aggregate approximations (SAX) has been used extensively for encoding time series data. However, typical SAX-based methods rely on two restrictive assumptions; the Gaussian distribution and equiprobable symbols. This paper proposes two novel data-driven SAX-based symbolic representations, distinguished by their discretization steps. The first representation, oriented for general data compaction and indexing scenarios, is based on the combination of kernel density estimation and Lloyd-Max quantization to minimize the information loss and mean squared error in the discretization step. The second method, oriented for high-level mining tasks, employs the Mean-Shift clustering method and is shown to enhance anomaly detection in the lower-dimensional space. Besides, we verify on a theoretical basis a previously observed phenomenon of the intrinsic process that results in a lower than the expected variance of the intermediate piecewise aggregate approximation. This phenomenon causes an additional information loss but can be avoided with a simple modification. The proposed representations possess all the attractive properties of the conventional SAX method. Furthermore, experimental evaluation on real-world datasets demonstrates their superiority compared to the traditional SAX and an alternative data-driven SAX variant.

Weak Approximations of the Wright–Fisher Process

In this paper, we construct first- and second-order weak split-step approximations for the solutions of the Wright–Fisher equation. The discretization schemes use the generation of, respectively, two- and three-valued random variables at each discretization step. The accuracy of constructed approximations is illustrated by several simulation examples.

Сходимость приближенных решений ядром теплопроводности для уравнения переноса-диффузии в полуплоскости

Рассматривается уравнение параболического типа (уравнение переноса-диффузии), которое используется, например, в задачах распространения загрязняющих веществ в атмосферном воздухе или водной среде. Имеется нескольких классических методов решения этого параболического уравнения, но также хорошо известно и вероятностное представление решения уравнения переноса-диффузии. Поскольку плотность распределения винеровского процесса соответствует ядру уравнения теплопроводности, с использованием ядра теплопроводности и оператора переноса на каждом шаге дискретизации времени можно построить приближенные решения уравнения переноса-диффузии в $ \mathbb{R}^d $. В предыдущих работах была доказана равномерная сходимость этих приближенных решений к функции, удовлетворяющей уравнению переноса-диффузии и начальному условию. Так как эти приближенные решения определяются только интегральным оператором и оператором переноса, доказательство их сходимости проводится без использования вероятностных понятий. В данной работе рассматривается уравнение переноса-диффузии в полуплоскости $ \mathbb{R}^2_+ $ с граничным условием на $ \{ x_2 = 0 \}$ и доказывается сходимость приближенных решений, построенных ядром теплопроводности и оператором переноса, к функции, удовлетворяющей уравнению переноса-диффузии в $ \mathbb{R}^2_{+} $, начальному и граничному условиям. Для получения этого результата используется метод нечетного продолжения заданных в $ \mathbb{R}^2_{+} $ функций на $ \mathbb{R}^2 $, поэтому применяются технические приемы предыдущих работ для задач в $ \mathbb{R}^d $. Тем не менее из-за наличия граничного условия остается проблема гладкости приближенных решений, для разрешения которой получаются оценки для производных приближенных решений, на которые влияют особенные данные на $ \{ x_2 = 0 \}$.

From LSAT: The Progress and Challenges of Complex Reasoning

Complex reasoning aims to draw a correct inference based on complex rules. As a hallmark of human intelligence, it involves a degree of explicit reading comprehension, interpretation of logical knowledge and complex rule application. In this paper, we take a step forward in complex reasoning by systematically studying the three challenging and domain-general tasks of the Law School Admission Test (LSAT), including analytical reasoning, logical reasoning and reading comprehension. We propose a hybrid reasoning system to integrate these three tasks and achieve impressive overall performance on the LSAT tests. The experimental results demonstrate that our system endows itself a certain complex reasoning ability, especially the fundamental reading comprehension and challenging logical reasoning capacities. Further analysis also shows the effectiveness of combining the pre-trained models with the task-specific reasoning module, and integrating symbolic knowledge into discrete interpretable reasoning steps in complex reasoning. We further shed a light on the potential future directions, like unsupervised symbolic knowledge extraction, model interpretability, few-shot learning and comprehensive benchmark for complex reasoning.

Computing Maximum Matchings in Temporal Graphs

Temporal graphs are graphs whose topology is subject to discrete changes over time. Given a static underlying graph $G$, a temporal graph is represented by assigning a set of integer time-labels to every edge $e$ of $G$, indicating the discrete time steps at which $e$ is active. We introduce and study the complexity of a natural temporal extension of the classical graph problem Maximum Matching, taking into account the dynamic nature of temporal graphs. In our problem, Maximum Temporal Matching, we are looking for the largest possible number of time-labeled edges (simply time-edges) $(e,t)$ such that no vertex is matched more than once within any time window of $\Delta$ consecutive time slots, where $\Delta \in \mathbb{N}$ is given. The requirement that a vertex cannot be matched twice in any $\Delta$-window models some necessary recovery period that needs to pass for an entity (vertex) after being paired up for some activity with another entity. We prove strong computational hardness results for Maximum Temporal Matching, even for elementary cases. To cope with this computational hardness, we mainly focus on fixed-parameter algorithms with respect to natural parameters, as well as on polynomial-time approximation algorithms.

Tensor-Based Online Network Anomaly Detection and Diagnosis

This paper presents an online anomaly detection system capable of handling operational network traffic of large networks (such as an ISP). We also aim for an effective and practical diagnosis of anomalies diagnosis to produce actionable intelligence that enables automated response. To achieve these objectives, we use the following approaches. (1) We model the status of the network by a stream of tensors where each tensor cell contains a time series. (2) We detect anomalous tensors at discrete time steps using an unsupervised tensor representation learning model. (3) We produce actionable intelligence through the diagnosis of anomaly detection results and by identifying the abnormal time series that are the most likely causes of each anomaly in the tensor. (4) We further analyze the traffic corresponding to each anomalous time series by an innovative method that extracts and isolates the attack traffic. (5) We provide solutions for challenges in streaming data anomaly detection such as large volume, high velocity, seasonality, and concept drift. We apply our approach to the complete test set of UGR data to show its practicality and effectiveness. Not only can we detect and isolate most of the labelled attack traffic, but we also identify many organic attack activities in the UGR data. Our results on the complete UGR dataset show high detection and isolation rate for the labelled attacks in the dataset. We also report on additional organic attacks we detected that were originally labelled as background in the dataset. Our analysis shows that the isolated background traffic represents interesting and potentially malicious behavior and can provide invaluable insight for cyber-threat researchers.

Asymptotics of Yule’s nonsense correlation for Ornstein-Uhlenbeck paths: A Wiener chaos approach

In this paper, we study the distribution of the so-called Yule's nonsense correlation on a time interval $[0,T]$ for a time horizon $T>0$ , when $T$ is large, for a pair $(X_{1},X_{2})$ of independent Ornstein-Uhlenbeck processes. This statistic is by definition equal to : \begin{equation*} \rho (T):=\frac{Y_{12}(T)}{\sqrt{Y_{11}(T)}\sqrt{Y_{22}(T)}}, \end{equation*} where the random variables $Y_{ij}(T)$, $i,j=1,2$ are defined as \begin{equation*} Y_{ij}(T):=\int_{0}^{T}X_{i}(u)X_{j}(u)du-T\bar{X}_{i}\bar{X_{j}}, \bar{X}_{i}:=\frac{1}{T}\int_{0}^{T}X_{i}(u)du. \end{equation*} We assume $X_{1}$ and $X_{2}$ have the same drift parameter $\theta >0$. We also study the asymptotic law of a discrete-type version of $\rho (T)$, where $Y_{ij}(T)$ above are replaced by their Riemann-sum discretizations. In this case, conditions are provided for how the discretization (in-fill) step relates to the long horizon $T$. We establish identical normal asymptotics for standardized $\rho (T)$ and its discrete-data version. The asymptotic variance of $\rho (T)T^{1/2}$ is $\theta ^{-1}$. We also establish speeds of convergence in the Kolmogorov distance, which are of Berry-Esseen-type (constant*$T^{-1/2}$) except for a $\ln T$ factor. Our method is to use the properties of Wiener-chaos variables, since $\rho (T)$ and its discrete version are comprised of ratios involving three such variables in the 2nd Wiener chaos. This methodology accesses the Kolmogorov distance thanks to a relation which stems from the connection between the Malliavin calculus and Stein's method on Wiener space.