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Global optimization of atomistic structure rely on the generation of new candidate structures in order to drive the exploration of the potential energy surface (PES) in search for the global minimum energy (GM) structure. In this work, we discuss a type of structure generation, which locally optimizes structures in complementary energy (CE) landscapes. These landscapes are formulated temporarily during the searches as machine learned potentials (MLPs) using local atomistic environments sampled from collected data. The CE landscapes are deliberately incomplete MLPs that rather than mimicking every aspect of the true PES are sought to become much smoother, having only few local minima. This means that local optimization in the CE landscapes may facilitate identification of new funnels in the true PES. We discuss how to construct the CE landscapes and we test their influence on global optimization of a reduced rutile SnO2(110)-(4x1) surface, and an olivine (Mg2SiO4)4 cluster for which we report a new global minimum energy structure., Comment: 13 pages, 9 figures

The use of deep learning models in computational biology has increased massively in recent years, and it is expected to continue with the current advances in the fields such as Natural Language Processing. These models, although able to draw complex relations between input and target, are also inclined to learn noisy deviations from the pool of data used during their development. In order to assess their performance on unseen data (their capacity to generalize), it is common to split the available data randomly into development (train/validation) and test sets. This procedure, although standard, has been shown to produce dubious assessments of generalization due to the existing similarity between samples in the databases used. In this work, we present SpanSeq, a database partition method for machine learning that can scale to most biological sequences (genes, proteins and genomes) in order to avoid data leakage between sets. We also explore the effect of not restraining similarity between sets by reproducing the development of two state-of-the-art models on bioinformatics, not only confirming the consequences of randomly splitting databases on the model assessment, but expanding those repercussions to the model development. SpanSeq is available at https://github.com/genomicepidemiology/SpanSeq.

This paper introduces a new tool, OccCANINE, to automatically transform occupational descriptions into the HISCO classification system. The manual work involved in processing and classifying occupational descriptions is error-prone, tedious, and time-consuming. We finetune a preexisting language model (CANINE) to do this automatically, thereby performing in seconds and minutes what previously took days and weeks. The model is trained on 14 million pairs of occupational descriptions and HISCO codes in 13 different languages contributed by 22 different sources. Our approach is shown to have accuracy, recall, and precision above 90 percent. Our tool breaks the metaphorical HISCO barrier and makes this data readily available for analysis of occupational structures with broad applicability in economics, economic history, and various related disciplines., Comment: All code and guides on how to use OccCANINE is available on GitHub https://github.com/christianvedels/OccCANINE

We propose to estimate the weight matrix used for forecast reconciliation as parameters in a general linear model in order to quantify its uncertainty. This implies that forecast reconciliation can be formulated as an orthogonal projection from the space of base-forecast errors into a coherent linear subspace. We use variance decomposition together with the Wishart distribution to derive the central estimator for the forecast-error covariance matrix. In addition, we prove that distance-reducing properties apply to the reconciled forecasts at all levels of the hierarchy as well as to the forecast-error covariance. A covariance matrix for the reconciliation weight matrix is derived, which leads to improved estimates of the forecast-error covariance matrix. We show how shrinkage can be introduced in the formulated model by imposing specific priors on the weight matrix and the forecast-error covariance matrix. The method is illustrated in a simulation study that shows consistent improvements in the log-score. Finally, standard errors for the weight matrix and the variance-separation formula are illustrated using a case study of forecasting electricity load in Sweden., Comment: 51 pages

Our work introduces a module for assessing the trajectory safety of autonomous vehicles in dynamic environments marked by high uncertainty. We focus on occluded areas and occluded traffic participants with limited information about surrounding obstacles. To address this problem, we propose a software module that handles blind spots (BS) created by static and dynamic obstacles in urban environments. We identify potential occluded traffic participants, predict their movement, and assess the ego vehicle's trajectory using various criticality metrics. The method offers a straightforward and modular integration into motion planner algorithms. We present critical real-world scenarios to evaluate our module and apply our approach to a publicly available trajectory planning algorithm. Our results demonstrate that safe yet efficient driving with occluded road users can be achieved by incorporating safety assessments into the planning process. The code used in this research is publicly available as open-source software and can be accessed at the following link: https://github.com/TUM-AVS/Frenetix-Occlusion., Comment: 8 Pages. Submitted to IEEE IV Conference, Korea

Our research introduces a modular motion planning framework for autonomous vehicles using a sampling-based trajectory planning algorithm. This approach effectively tackles the challenges of solution space construction and optimization in path planning. The algorithm is applicable to both real vehicles and simulations, offering a robust solution for complex autonomous navigation. Our method employs a multi-objective optimization strategy for efficient navigation in static and highly dynamic environments, focusing on optimizing trajectory comfort, safety, and path precision. The algorithm is used to analyze the algorithm performance and success rate in 1750 virtual complex urban and highway scenarios. Our results demonstrate fast calculation times (8ms for 800 trajectories), a high success rate in complex scenarios (88%), and easy adaptability with different modules presented. The most noticeable difference exhibited was the fast trajectory sampling, feasibility check, and cost evaluation step across various trajectory counts. We demonstrate the integration and execution of the framework on real vehicles by evaluating deviations from the controller using a test track. This evaluation highlights the algorithm's robustness and reliability, ensuring it meets the stringent requirements of real-world autonomous driving scenarios. The code and the additional modules used in this research are publicly available as open-source software and can be accessed at the following link: https://github.com/TUM-AVS/Frenetix-Motion-Planner., Comment: Submitted to IEEE ACCESS

Let $F:(\mathbb{C}^2,0)\to (\mathbb{C}^n,0)$ be the germ of a finite map and $(X,0)$ be its image. We will in this article using the topology of the link show that $(X,0)$ has to be a quotient singularity if it is normal and describe the possible topological types. Including a discussion of the groups and examples of how to construct a map to a given topology. We will also discuss the case when the image is not normal., Comment: 17 Pages

It is known from Earth that ionizing high-energy radiation can lead to ion-induced nucleation of cloud condensation nuclei in the atmosphere. Since the amount of high-energy radiation can vary greatly based on the radiative environment of a host star, understanding the effect of high-energy radiation on cloud particles is critical to understand exoplanet atmospheres. This study aims to explore how high-energy radiation affects the aggregation and charging of mineral cloud particles. We present experiments conducted in an atmosphere chamber on mineral SiO$_2$ particles with diameters of 50 nm. The particles were exposed to gamma radiation in either low-humidity (RH $\approx$ 20%) or high-humidity (RH $>$ 50%) environments. The aggregation and charging state of the particles were studied with a Scanning Mobility Particle Sizer. We find that the single SiO$_2$ particles (N1) cluster to form larger aggregates (N2 - N4), and that this aggregation is inhibited by gamma radiation. We find that gamma radiation shifts the charging of the particles to become more negative, by increasing the charging state of negatively charged particles. Through an independent T-test we find that this increase is statistically significant within a 5% significance level for all aggregates in the high-humidity environment, and for all except the N1 particles in the low-humidity environment. For the positively charged particles the changes in charging state are not within the 5% significance level. We suggest that the overall effect of gamma radiation could favor the formation of a high number of small particles over a lower number of larger particles., Comment: Accepted to ApJ, 19 pages, 9 figures

(Abridged). We explore the fraction of radio loud quasars in the eHAQ+GAIA23 sample, which contains quasars from the High A(V) Quasar (HAQ) Survey, the Extended High A(V) Quasar (eHAQ) Survey, and the Gaia quasar survey. All quasars in this sample have been found using a near-infrared color selection of target candidates that have otherwise been missed by the Sloan Digital Sky Survey (SDSS). We implemented a redshift-dependent color cut in g-i to select red quasars in the sample and divided them into redshift bins, while using a nearest-neighbors algorithm to control for luminosity and redshift differences between our red quasar sample and a selected blue sample from the SDSS. Within each bin, we cross-matched the quasars to the Faint Images of the Radio Sky at Twenty centimeters (FIRST) survey and determined the radio-detection fraction. We find similar radio-detection fractions for red and blue quasars within 1 sigma, independent of redshift. This disagrees with what has been found in the literature for red quasars in SDSS. It should be noted that the fraction of broad absorption line (BAL) quasars in red SDSS quasars is about five times lower than in our sample. BAL quasars have been observed to be more frequently radio quiet than other quasars, therefore the difference in BAL fractions could explain the difference in radio-detection fraction. The observed higher proportion of BAL quasars in our dataset relative to the SDSS sample, along with the higher rate of radio detections, indicates an association of the redness of quasars and the inherent BAL fraction within the overall quasar population. This finding highlights the need to explore the underlying factors contributing to both the redness and the frequency of BAL quasars, as they appear to be interconnected phenomena., Comment: 10 pages, 4 figures. Accepted for publication in A&A

The increasing capability of Large Language Models to act as human-like social agents raises two important questions in the area of opinion dynamics. First, whether these agents can generate effective arguments that could be injected into the online discourse to steer the public opinion. Second, whether artificial agents can interact with each other to reproduce dynamics of persuasion typical of human social systems, opening up opportunities for studying synthetic social systems as faithful proxies for opinion dynamics in human populations. To address these questions, we designed a synthetic persuasion dialogue scenario on the topic of climate change, where a 'convincer' agent generates a persuasive argument for a 'skeptic' agent, who subsequently assesses whether the argument changed its internal opinion state. Different types of arguments were generated to incorporate different linguistic dimensions underpinning psycho-linguistic theories of opinion change. We then asked human judges to evaluate the persuasiveness of machine-generated arguments. Arguments that included factual knowledge, markers of trust, expressions of support, and conveyed status were deemed most effective according to both humans and agents, with humans reporting a marked preference for knowledge-based arguments. Our experimental framework lays the groundwork for future in-silico studies of opinion dynamics, and our findings suggest that artificial agents have the potential of playing an important role in collective processes of opinion formation in online social media., Comment: 9 pages, 6 figures, 3 tables, 1 page in appendix

In the hydrodynamics of integrable models, diffusion is a subleading correction to ballistic propagation. Here we quantify the diffusive contribution for one-dimensional Bose gases and find it most influential in the crossover between the main thermodynamic regimes of the gas. Analysing the experimentally measured dynamics of a single density mode, we find diffusion to be relevant only for high wavelength excitations. Instead, the observed relaxation is solely caused by a ballistically driven dephasing process, whose time scale is related to the phonon lifetime of the system and is thus useful to evaluate the applicability of the phonon bases typically used in quantum field simulators., Comment: 16 pages, 7 figures. Submission to SciPost Physics. Comments are welcome

We consider a class of toy models describing a fermion field coupled with a boson field. The model can be viewed as a Yukawa model but with scalar fermions. As in our first paper, the interaction kernels are assumed bounded in the fermionic momentum variable and decaying like $|q|^{-p}$ for large boson momenta $q$. With no restrictions on the coupling strength, we prove norm resolvent convergence to an ultraviolet renormalized Hamiltonian, when the ultraviolet cutoff is removed. We do this by subtracting a sufficiently large, but finite, number of recursively defined self-energy counter-terms, which may be interpreted as arising from a perturbation expansion of the ground state energy. The renormalization procedure requires a spatial cutoff and works in three dimensions provided $p>\frac12$, which is as close as one may expect to the physically natural exponent $p = \frac12$.

Let $X=\sum_{k=1}^\infty X_k \beta^{-k}$ be the base-$\beta$ expansion of a continuous random variable $X$ on the unit interval where $\beta$ is the golden ratio. We study the asymptotic distribution and convergence rate of the scaled remainder $\sum_{k=n+1}^\infty X_k \beta^{n-k}$ when $n$ tends to infinity., Comment: 7 pages

In this study, the performance of the doubles-corrected higher random-phase approximation (HRPA(D)) has been investigated in calculations of NMR spin-spin coupling constants (SSCCs) for 58 molecules with the experimental values used as the reference values. HRPA(D) is an approximation to the second-order polarization propagator approximation (SOPPA), and is therefore computationally less expensive than SOPPA. HRPA(D) performs comparable and sometimes even better than SOPPA, and therefore when calculating SSCCs it should be considered as an alternative to SOPPA. Furthermore, it was investigated whether a CCSD(T) or MP2 geometry optimization was optimal for a SOPPA and a HRPA(D) SSCCs calculation for 8 smaller molecules. CCSD(T) is the optimal geometry optimization for the SOPPA calculation, and MP2 was optimal for the HRPA(D) SSCC calculations.

In October 2011, Denmark introduced the world's first and, to date, only tax targeting saturated fat. However, this tax was subsequently abolished in January 2013. Leveraging exogenous variation from untaxed Northern-German consumers, we employ a difference-in-differences approach to estimate the causal effects of both the implementation and repeal of the tax on consumption and expenditure behavior across eight product categories targeted by the tax. Our findings reveal significant heterogeneity in the tax's impact across these products. During the taxed period, there was a notable decline in consumption of bacon, liver sausage, and cheese, particularly among low-income households. In contrast, expenditure on butter, cream, and margarine increased as prices rose. Interestingly, we do not observe any difference in expenditure increases between high and low-income households, suggesting that the latter were disproportionately affected by the tax. After the repeal of the tax, we do not observe any significant decline in consumption. On the contrary, there was an overall increase in consumption for certain products, prompting concerns about unintended consequences resulting from the brief implementation of the tax. Finally, we find strong evidence on an overall increase purchases of butter abroad for households living less than 50 km from the German boarder but we do not find strong evidence of spatial heterogeneous effects of the tax.

One-dimensional Mott insulators can be described using the sine-Gordon model, an integrable quantum field theory that provides the low-energy effective description of several one-dimensional gapped condensed matter systems, including recent realizations with trapped ultra-cold atoms. Employing the theory of Generalized Hydrodynamics, we demonstrate that this model exhibits separation of the transport of topological charge vs. energy. Analysis of the quasiparticle dynamics reveals that the mechanism behind the separation is the reflective scattering between topologically charged kinks/antikinks. The effect of these scattering events is most pronounced at strong coupling and low temperatures, where the distribution of quasiparticles is narrow compared to the reflective scattering amplitude. This effect results in a distinctively shaped "arrowhead" light cone for the topological charge., Comment: 19 pages, 16 figures

Based on previous work, we assess the use of NIR-HSI images for calibrating models on two datasets, focusing on protein content regression and grain variety classification. Limited reference data for protein content is expanded by subsampling and associating it with the bulk sample. However, this method introduces significant biases due to skewed leptokurtic prediction distributions, affecting both PLS-R and deep CNN models. We propose adjustments to mitigate these biases, improving mean protein reference predictions. Additionally, we investigate the impact of grain-to-background ratios on both tasks. Higher ratios yield more accurate predictions, but including lower-ratio images in calibration enhances model robustness for such scenarios., Comment: 19 pages, 17 figures

Conventional hydrodynamics describes systems with few long-lived excitations. In one dimension, however, many experimentally relevant systems feature a large number of long-lived excitations even at high temperature, because they are proximate to integrable limits. Such models cannot be treated using conventional hydrodynamics. The framework of generalized hydrodynamics (GHD) was recently developed to treat the dynamics of one-dimensional models: it combines ideas from integrability, hydrodynamics, and kinetic theory to come up with a quantitative theory of transport. GHD has successfully settled several longstanding questions about one-dimensional transport; it has also been leveraged to study dynamical questions beyond the transport of conserved quantities, and to systems that are not integrable. In this article we introduce the main ideas and predictions of GHD, survey some of the most recent theoretical extensions and experimental tests of the GHD framework, and discuss some open questions in transport that the GHD perspective might elucidate., Comment: 16 pages, 5 figures, perspective article, feedback is welcome

This paper explores the potential application of quantum and hybrid quantum-classical neural networks in power flow analysis. Experiments are conducted using two datasets based on 4-bus and 33-bus test systems. A systematic performance comparison is also conducted among quantum, hybrid quantum-classical, and classical neural networks. The comparison is based on (i) generalization ability, (ii) robustness, (iii) training dataset size needed, (iv) training error, and (v) training process stability. The results show that the developed hybrid quantum-classical neural network outperforms both quantum and classical neural networks, and hence can improve deep learning-based power flow analysis in the noisy-intermediate-scale quantum (NISQ) and fault-tolerant quantum (FTQ) era., Comment: 8 pages, 13 figures

We present a campaign designed to train the GRANDMA network and its infrastructure to follow up on transient alerts and detect their early afterglows. In preparation for O4 II campaign, we focused on GRB alerts as they are expected to be an electromagnetic counterpart of gravitational-wave events. Our goal was to improve our response to the alerts and start prompt observations as soon as possible to better prepare the GRANDMA network for the fourth observational run of LIGO-Virgo-Kagra (which started at the end of May 2023), and future missions such as SM. To receive, manage and send out observational plans to our partner telescopes we set up dedicated infrastructure and a rota of follow-up adcates were organized to guarantee round-the-clock assistance to our telescope teams. To ensure a great number of observations, we focused on Swift GRBs whose localization errors were generally smaller than the GRANDMA telescopes' field of view. This allowed us to bypass the transient identification process and focus on the reaction time and efficiency of the network. During 'Ready for O4 II', 11 Swift/INTEGRAL GRB triggers were selected, nine fields had been observed, and three afterglows were detected (GRB 220403B, GRB 220427A, GRB 220514A), with 17 GRANDMA telescopes and 17 amateur astronomers from the citizen science project Kilonova-Catcher. Here we highlight the GRB 220427A analysis where our long-term follow-up of the host galaxy allowed us to obtain a photometric redshift of $z=0.82\pm0.09$, its lightcurve elution, fit the decay slope of the afterglows, and study the properties of the host galaxy.

The ability to manipulate the space-time structure of light waves diversifies light-matter interaction and light-driven applications. Conventionally, metasurfaces are employed to locally control the amplitude and phase of light fields by the material response and structure of small meta-atoms. However, the fixed spatial structures of metasurfaces offer limited opportunities. Here, using quantum control we introduce a new approach that enables the amplitude, sign, and even configuration of the generated light fields to be manipulated in an all-optical manner. Following this approach, we demonstrate the generation of flying doughnut terahertz (THz) pulses. We show that the single-cycle THz pulse radiated from the dynamic semiconductor ring current has an electric field structure that is azimuthally polarized and that the space- and time-resolved magnetic field has a strong, isolated longitudinal component. As a first application, we detect absorption features from ambient water vapor on the spatiotemporal structure of the measured electric fields and the calculated magnetic fields. Quantum control is a powerful and flexible route to generating any structured light pulse in the THz range, while pulse compression of cylindrical vector beams is available for very high-power magnetic-pulse generation from the mid-infrared to near UV spectral region. Pulses such as these will serve as unique probes for spectroscopy, imaging, telecommunications, and magnetic materials.

Purpose: Field-to-susceptibility inversion in quantitative susceptibility mapping (QSM) is ill-posed and needs numerical stabilization through either regularization or oversampling by acquiring data at three or more object orientations. Calculation Of Susceptibility through Multiple Orientations Sampling (COSMOS) is an established oversampling approach and regarded as QSM gold standard. It achieves a well-conditioned inverse problem, requiring rotations by 0{\deg}, 60{\deg} and 120{\deg} in the yz-plane. However, this is impractical in vivo, where head rotations are typically restricted to a range of +-25{\deg}. Non-ideal sampling degrades the conditioning with residual streaking artifacts whose mitigation needs further regularization. Moreover, susceptibility anisotropy in white matter is not considered in the COSMOS model, which may introduce additional bias. The current work presents a thorough investigation of these effects in primate brain. Methods: Gradient-recalled echo (GRE) data of an entire fixed chimpanzee brain were acquired at 7 T (350 microns resolution, 10 orientations) including ideal COSMOS sampling and realistic rotations in vivo. Comparisons of the results included ideal COSMOS, in-vivo feasible acquisitions with 3-8 orientations and single-orientation iLSQR QSM. Results: In-vivo feasible and optimal COSMOS yielded high-quality susceptibility maps with increased SNR resulting from averaging multiple acquisitions. COSMOS reconstructions from non-ideal rotations about a single axis required additional L2-regularization to mitigate residual streaking artifacts. Conclusion: In view of unconsidered anisotropy effects, added complexity of the reconstruction, and the general challenge of multi-orientation acquisitions, advantages of sub-optimal COSMOS schemes over regularized single-orientation QSM appear limited in in-vivo settings., Comment: Text: 2593 words (without legends, references and statements) Abstract: 239 words References: 33 Figures: 4 Tables: 1

A polarization-sensitive optical coherence tomography (PS-OCT) system is able to not only show the structure of samples through the analysis of backscattered light, but is also capable of determining their polarimetric properties. This is an extra functionality to OCT which allows the retardance and axis orientation of a bulk sample to be determined. Here, we describe the temperature instabilities of a depth-encoded, multiple input state PS-OCT system, where two waves corresponding to two orthogonal states in the interrogating beam are delayed using a 5-meter long polarization-maintaning (PM) fiber. It is shown that the temperature not only affects the delay between the two relatively delayed waves, but also the amount of mismatched dispersion in the interferometer, which ultimately affects the achievable axial resolution in the system. To this end, the technique of complex master/slave interferometry (CMSI) can be used as an option to mitigate this effect., Comment: The first two authors contributed equally to this work

The chemical enrichment of dust and metals in the interstellar medium (ISM) of galaxies throughout cosmic time is one of the key driving processes of galaxy evolution. Here we study the evolution of the gas-phase metallicities, dust-to-gas (DTG), and dust-to-metal (DTM) ratios of 36 star-forming galaxies at $1.7 < z < 6.3$ probed by gamma-ray bursts (GRBs). We compile all GRB-selected galaxies with intermediate (R=7000) to high (R>40,000) resolution spectroscopic data for which at least one refractory (e.g. Fe) and one volatile (e.g. S or Zn) element have been detected at S/N>3. This is to ensure that accurate abundances and dust depletion patterns can be obtained. We first derive the redshift evolution of the dust-corrected, absorption-line based gas-phase metallicity [M/H]$_{\rm tot}$ in these galaxies, for which we determine a linear relation with redshift ${\rm [M/H]_{tot}}(z) = (-0.21\pm 0.04)z -(0.47\pm 0.14)$. We then examine the DTG and DTM ratios as a function of redshift and through three orders of magnitude in metallicity, quantifying the relative dust abundance both through the direct line-of-sight visual extinction $A_V$ and the derived depletion level. We use a novel method to derive the DTG and DTM mass ratios for each GRB sightline, summing up the mass of all the depleted elements in the dust-phase. We find that the DTG and DTM mass ratios are both strongly correlated with the gas-phase metallicity and show a mild evolution with redshift as well. While these results are subject to a variety of caveats related to the physical environments and the narrow pencil-beam sightlines through the ISM probed by the GRBs, they provide strong implications for studies of dust masses to infer the gas and metal content of high-redshift galaxies, and particularly demonstrate the large offset from the average Galactic value in the low-metallicity, high-redshift regime., Comment: Accepted in A&A

We present an analysis of microlensing event OGLE-2019-BLG-0825. This event was identified as a planetary candidate by preliminary modeling. We find that significant residuals from the best-fit static binary-lens model exist and a xallarap effect can fit the residuals very well and significantly improves $\chi^2$ values. On the other hand, by including the xallarap effect in our models, we find that binary-lens parameters like mass-ratio, $q$, and separation, $s$, cannot be constrained well. However, we also find that the parameters for the source system like the orbital period and semi major axis are consistent between all the models we analyzed. We therefore constrain the properties of the source system better than the properties of the lens system. The source system comprises a G-type main-sequence star orbited by a brown dwarf with a period of $P\sim5$ days. This analysis is the first to demonstrate that the xallarap effect does affect binary-lens parameters in planetary events. It would not be common for the presence or absence of the xallarap effect to affect lens parameters in events with long orbital periods of the source system or events with transits to caustics, but in other cases, such as this event, the xallarap effect can affect binary-lens parameters., Comment: 19 pages, 7 figures, 6 tables. Accepted by AJ

Container shipping drives the global economy and is an eco-friendly mode of transportation. A key objective is to maximize the utilization of vessels, which is challenging due to the NP-hardness of stowage planning. This article surveys the literature on the Container Stowage Planning Problem (CSPP). We introduce a classification scheme to analyze single-port and multi-port CSPPs, as well as the hierarchical decomposition of CSPPs into the master and slot planning problem. Our survey shows that the area has a relatively small number of publications and that it is hard to evaluate the industrial applicability of many of the proposed solution methods due to the oversimplification of problem formulations. To address this issue, we propose a research agenda with directions for future work, including establishing a representative problem definition and providing new benchmark instances where needed.

Let $X_1,X_2,...$ be the digits in the base-$q$ expansion of a random variable $X$ defined on $[0,1)$ where $q\ge2$ is an integer. For $n=1,2,...$, we study the probability distribution $P_n$ of the (scaled) remainder $T^n(X)=\sum_{k=n+1}^\infty X_k q^{n-k}$: If $X$ has an absolutely continuous CDF then $P_n$ converges in the total variation metric to the Lebesgue measure $\mu$ on the unit interval. Under weak smoothness conditions we establish first a coupling between $X$ and a non-negative integer valued random variable $N$ so that $T^N(X)$ follows $\mu$ and is independent of $(X_1,...,X_N)$, and second exponentially fast convergence of $P_n$ and its PDF $f_n$. We discuss how many digits are needed and show examples of our results. The convergence results are extended to the case of a multivariate random variable defined on a unit cube., Comment: 22 pages, 3 figures