Dr. rer. nat. Fabian Joachim Erich Telschow

Profil

• EXC 2046/1: MATH+ Junior Research Group on Statistical Inversion and Quantification of Uncertainties

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  Förderer: DFG Exzellenzstrategie Cluster Zeitraum: 09/2020 - 12/2025 Projektleitung: Dr. rer. nat. Fabian Joachim Erich Telschow

• International Centre for Higher Education Research Kassel

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• Center for International Forestry Research

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• WWF-Indonesia

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• Science & Startups Berlin

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  • Zuwendung im Rahmen des Programms „exist – Existenzgründungen aus der Wissenschaft“ aus dem Bundeshaushalt, Einzelplan 09, Kapitel 02, Titel 68607, Haushaltsjahr 2026, sowie aus Mitteln des Europäischen Strukturfonds (hier Euro-päischer Sozialfonds Plus – ESF Plus) Förderperiode 2021-2027 – Kofinanzierung für das Vorhaben: „exist Women“
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• Isogen GmbH & Co KG

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• Fortiss GmbH

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• Österreichische Studiengesellschaft für Kybernetik

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• Stichting Centrum voor Wiskunde en Informatica

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• Barcelona Supercomputing Center - Centro Nacional de Supercomputation

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  • The Novel Materials Discovery Laboratory (NoMaD)
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    49.8%

• Bloomfield Science Museum Jerusalem

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• Confidence Sets for Cohen’s d effect size images

  2020

  NeuroImage · 55 Zitationen · DOI

  Current statistical inference methods for task-fMRI suffer from two fundamental limitations. First, the focus is solely on detection of non-zero signal or signal change, a problem that is exacerbated for large scale studies (e.g. UK Biobank, N=40,000+) where the 'null hypothesis fallacy' causes even trivial effects to be determined as significant. Second, for any sample size, widely used cluster inference methods only indicate regions where a null hypothesis can be rejected, without providing any notion of spatial uncertainty about the activation. In this work, we address these issues by developing spatial Confidence Sets (CSs) on clusters found in thresholded Cohen's d effect size images. We produce an upper and lower CS to make confidence statements about brain regions where Cohen's d effect sizes have exceeded and fallen short of a non-zero threshold, respectively. The CSs convey information about the magnitude and reliability of effect sizes that is usually given separately in a t-statistic and effect estimate map. We expand the theory developed in our previous work on CSs for %BOLD change effect maps (Bowring et al., 2019) using recent results from the bootstrapping literature. By assessing the empirical coverage with 2D and 3D Monte Carlo simulations resembling fMRI data, we find our method is accurate in sample sizes as low as N=60. We compute Cohen's d CSs for the Human Connectome Project working memory task-fMRI data, illustrating the brain regions with a reliable Cohen's d response for a given threshold. By comparing the CSs with results obtained from a traditional statistical voxelwise inference, we highlight the improvement in activation localization that can be gained with the Confidence Sets.

• Spatial confidence sets for raw effect size images

  2019

  NeuroImage · 26 Zitationen · DOI

  The mass-univariate approach for functional magnetic resonance imaging (fMRI) analysis remains a widely used statistical tool within neuroimaging. However, this method suffers from at least two fundamental limitations: First, with sufficient sample sizes there is high enough statistical power to reject the null hypothesis everywhere, making it difficult if not impossible to localize effects of interest. Second, with any sample size, when cluster-size inference is used a significant p-value only indicates that a cluster is larger than chance. Therefore, no notion of confidence is available to express the size or location of a cluster that could be expected with repeated sampling from the population. In this work, we address these issues by extending on a method proposed by Sommerfeld et al. (2018) (SSS) to develop spatial Confidence Sets (CSs) on clusters found in thresholded raw effect size maps. While hypothesis testing indicates where the null, i.e. a raw effect size of zero, can be rejected, the CSs give statements on the locations where raw effect sizes exceed, and fall short of, a non-zero threshold, providing both an upper and lower CS. While the method can be applied to any mass-univariate general linear model, we motivate the method in the context of blood-oxygen-level-dependent (BOLD) fMRI contrast maps for inference on percentage BOLD change raw effects. We propose several theoretical and practical implementation advancements to the original method formulated in SSS, delivering a procedure with superior performance in sample sizes as low as N=60. We validate the method with 3D Monte Carlo simulations that resemble fMRI data. Finally, we compute CSs for the Human Connectome Project working memory task contrast images, illustrating the brain regions that show a reliable %BOLD change for a given %BOLD threshold.

• Simultaneous Confidence Bands for Functional Data Using the Gaussian Kinematic Formula

  2019

  arXiv (Cornell University) · 23 Zitationen · DOI

  This article constructs simultaneous confidence bands (SCBs) for functional parameters using the Gaussian Kinematic formula of $t$-processes (tGKF). Although the tGKF relies on Gaussianity, we show that a central limit theorem (CLT) for the parameter of interest is enough to obtain asymptotically precise covering rates even for non-Gaussian processes. As a proof of concept we study the functional signal-plus-noise model and derive a CLT for an estimator of the Lipschitz-Killing curvatures, the only data dependent quantities in the tGKF SCBs. Extensions to discrete sampling with additive observation noise are discussed using scale space ideas from regression analysis. Here we provide sufficient conditions on the processes and kernels to obtain convergence of the functional scale space surface. The theoretical work is accompanied by a simulation study comparing different methods to construct SCBs for the population mean. We show that the tGKF works well even for small sample sizes and only a Rademacher multiplier-$t$ bootstrap performs similarily well. For larger sample sizes the tGKF often outperforms the bootstrap methods and is computational faster. We apply the method to diffusion tensor imaging (DTI) fibers using a scale space approach for the difference of population means. R code is available in our Rpackage SCBfda.

• Peak p-values and false discovery rate inference in neuroimaging

  2019

  NeuroImage · 10 Zitationen · DOI

• Confidence tubes for curves on SO(3) and identification of subject-specific gait change after kneeling

  2023

  Journal of the Royal Statistical Society Series C (Applied Statistics) · 8 Zitationen · DOI

  Abstract In order to identify changes of gait patterns, e.g. due to prolonged occupational kneeling, which might be a major risk factor for the development of knee osteoarthritis, we develop confidence tubes for curves following a perturbation model on SO(3) using the Gaussian kinematic formula which are equivariant under gait similarities and have precise coverage even for small sample sizes. Applying them to gait curves from eight volunteers undergoing kneeling tasks and adjusting for different walking speeds and marker replacement at different visits, allows us to identify at which phases of the gait cycle the gait pattern changed due to kneeling.

• Functional Inference on Rotational Curves and Identification of Human Gait at the Knee Joint

  2016

  arXiv (Cornell University) · 7 Zitationen · DOI

  We extend Gaussian perturbation models in classical functional data analysis to the three-dimensional rotational group where a zero-mean Gaussian process in the Lie algebra under the Lie exponential spreads multiplicatively around a central curve. As an estimator, we introduce point-wise extrinsic mean curves which feature strong perturbation consistency, and which are asymptotically a.s. unique and differentiable, if the model is so. Further, we consider the group action of time warping and that of spatial isometries that are connected to the identity. The latter can be asymptotically consistently estimated if lifted to the unit quaternions. Introducing a generic loss for Lie groups, the former can be estimated, and based on curve length, due to asymptotic differentiability, we propose two-sample permutation tests involving various combinations of the group actions. This methodology allows inference on gait patterns due to the rotational motion of the lower leg with respect to the upper leg. This was previously not possible because, among others, the usual analysis of separate Euler angles is not independent of marker placement, even if performed by trained specialists.

• Estimation of expected Euler characteristic curves of nonstationary smooth random fields

  2023

  The Annals of Statistics · 6 Zitationen · DOI

  The expected Euler characteristic (EEC) of excursion sets of a smooth Gaussian-related random field over a compact manifold approximates the distribution of its supremum for high thresholds. Viewed as a function of the excursion threshold, the EEC of a Gaussian-related field is expressed by the Gaussian kinematic formula (GKF) as a finite sum of known functions multiplied by the Lipschitz–Killing curvatures (LKCs) of the generating Gaussian field. This paper proposes consistent estimators of the LKCs as linear projections of “pinned” Euler characteristic (EC) curves obtained from realizations of zero-mean, unit variance Gaussian processes. As observed, data seldom is Gaussian and the exact mean and variance is unknown, yet the statistic of interest often satisfies a CLT with a Gaussian limit process; we adapt our LKC estimators to this scenario using a Gaussian multiplier bootstrap approach. This yields consistent estimates of the LKCs of the possibly nonstationary Gaussian limiting field that have low variance and are computationally efficient for complex underlying manifolds. For the EEC of the limiting field, a parametric plug-in estimator is presented, which is more efficient than the nonparametric average of EC curves. The proposed methods are evaluated using simulations of 2D fields, and illustrated on cosmological observations and simulations on the 2-sphere and 3D fMRI volumes.

• Simultaneous confidence bands for functional data using the Gaussian Kinematic formula

  2021

  Journal of Statistical Planning and Inference · 6 Zitationen · DOI

• Functional delta residuals and applications to simultaneous confidence bands of moment based statistics

  2022

  Journal of Multivariate Analysis · 5 Zitationen · DOI

• Estimation of Expected Euler Characteristic Curves of Nonstationary Smooth Gaussian Random Fields

  2019

  arXiv (Cornell University) · 4 Zitationen · DOI

  The expected Euler characteristic (EEC) curve of excursion sets of a Gaussian random field is used to approximate the distribution of its supremum for high thresholds. Viewed as a function of the excursion threshold, the EEC is expressed by the Gaussian kinematic formula (GKF) as a linear function of the Lipschitz-Killing curvatures (LKCs) of the field, which solely depend on the domain and covariance function of the field. So far its use for non-stationary Gaussian fields over non-trivial domains has been limited because in this case the LKCs are difficult to estimate. In this paper, consistent estimators of the LKCs are proposed as linear projections of "pinned" observed Euler characteristic curves and a linear parametric estimator of the EEC curve is obtained, which is more efficient than its nonparametric counterpart for repeated observations. A multiplier bootstrap modification reduces the variance of the estimator, and allows estimation of LKCs and EEC of the limiting field of non-Gaussian fields satisfying a functional CLT. The proposed methods are evaluated using simulations of 2D fields and illustrated in thresholding of 3D fMRI brain activation maps and cosmological simulations on the 2-sphere.

• Estimation of Expected Euler Characteristic Curves of Nonstationary Smooth Gaussian Random Fields

  2019

  arXiv (Cornell University) · 4 Zitationen

  The expected Euler characteristic (EEC) curve of excursion sets of a Gaussian random field is used to approximate the distribution of its supremum for high thresholds. Viewed as a function of the excursion threshold, the EEC is expressed by the Gaussian kinematic formula (GKF) as a linear function of the Lipschitz-Killing curvatures (LKCs) of the field, which solely depend on the domain and covariance function of the field. So far its use for non-stationary Gaussian fields over non-trivial domains has been limited because in this case the LKCs are difficult to estimate. In this paper, consistent estimators of the LKCs are proposed as linear projections of pinned observed Euler characteristic curves and a linear parametric estimator of the EEC curve is obtained, which is more efficient than its nonparametric counterpart for repeated observations. A multiplier bootstrap modification reduces the variance of the estimator, and allows estimation of LKCs and EEC of the limiting field of non-Gaussian fields satisfying a functional CLT. The proposed methods are evaluated using simulations of 2D fields and illustrated in thresholding of 3D fMRI brain activation maps and cosmological simulations on the 2-sphere.

• Inverse set estimation and inversion of simultaneous confidence intervals

  2024

  Journal of the Royal Statistical Society Series C (Applied Statistics) · 3 Zitationen · DOI

  Motivated by the questions of risk assessment in climatology (temperature change in North America) and medicine (impact of statin usage and coronavirus disease 2019 on hospitalized patients), we address the problem of estimating the set in the domain of a function whose image equals a predefined subset of the real line. Existing methods require strict assumptions. We generalize the estimation of such sets to dense and nondense domains with protection against inflated Type I error in exploratory data analysis. This is achieved by proving that confidence sets of multiple upper, lower, or interval sets can be simultaneously constructed with the desired confidence nonasymptotically through inverting simultaneous confidence intervals. Nonparametric bootstrap algorithm and code are provided.

• Functional inference on rotational curves under sample‐specific group actions and identification of human gait

  2020

  Scandinavian Journal of Statistics · 3 Zitationen · DOI

  Abstract Inspired by the problem of gait reproducibility (reidentifying individuals across doctor's visits) we develop two‐sample permutation tests under a sample‐specific group action on Lie groups with a bi‐invariant Riemannian metric. These tests rely on consistent estimators and pairwise curve alignment. To this end, we propose Gaussian perturbation models and for the special case of curves on the group of 3D rotations we provide asymptotic consistency and, employing a quaternion point of view, fast spatial alignment of pointwise extrinsic mean curves. In our application to rotations of the tibia versus the femur at the knee joint under the spatial action of marker placement and the temporal action of different walking speeds, obtained from an experiment, we solve the problem of gait reproducibility.

• Representation by Integrating Reproducing Kernels

  2012

  arXiv (Cornell University) · 3 Zitationen · DOI

  Based on direct integrals, a framework allowing to integrate a parametrised family of reproducing kernels with respect to some measure on the parameter space is developed. By pointwise integration, one obtains again a reproducing kernel whose corresponding Hilbert space is given as the image of the direct integral of the individual Hilbert spaces under the summation operator. This generalises the well-known results for finite sums of reproducing kernels; however, many more special cases are subsumed under this approach: so-called Mercer kernels obtained through series expansions; kernels generated by integral transforms; mixtures of positive definite functions; and in particular scale-mixtures of radial basis functions. This opens new vistas into known results, e.g. generalising the Kramer sampling theorem; it also offers interesting connections between measurements and integral transforms, e.g. allowing to apply the representer theorem in certain inverse problems, or bounding the pointwise error in the image domain when observing the pre-image under an integral transform.

• Robust FWER control in Neuroimaging using Random Field Theory: Riding the SuRF to Continuous Land Part 2

  2023

  arXiv (Cornell University) · 2 Zitationen · DOI

  Historically, applications of RFT in fMRI have relied on assumptions of smoothness, stationarity and Gaussianity. The first two assumptions have been addressed in Part 1 of this article series. Here we address the severe non-Gaussianity of (real) fMRI data to greatly improve the performance of voxelwise RFT in fMRI group analysis. In particular, we introduce a transformation which accelerates the convergence of the Central Limit Theorem allowing us to rely on limiting Gaussianity of the test-statistic. We shall show that, when the GKF is combined with the Gaussianization transformation, we are able to accurately estimate the EEC of the excursion set of the transformed test-statistic even when the data is non-Gaussian. This allows us to drop the key assumptions of RFT inference and enables us to provide a fast approach which correctly controls the voxelwise false positive rate in fMRI. We employ a big data \cite{Eklund2016} style validation in which we process resting state data from 7000 subjects from the UK BioBank with fake task designs. We resample from this data to create realistic noise and use this to demonstrate that the error rate is correctly controlled.

• Functional delta residuals and applications to functional effect sizes

  2020

  arXiv (Cornell University) · 2 Zitationen

  Given a functional central limit (fCLT) and a parameter transformation, we use the functional delta method to construct random processes, called functional delta residuals, which asymptotically have the same covariance structure as the transformed limit process. Moreover, we prove a multiplier bootstrap fCLT theorem for these transformed residuals and show how this can be used to construct simultaneous confidence bands for transformed functional parameters. As motivation for this methodology, we provide the formal application of these residuals to a functional version of the effect size parameter Cohen's $d$, a problem appearing in current brain imaging applications. The performance and necessity of such residuals is illustrated in a simulation experiment for the covering rate of simultaneous confidence bands for the functional Cohen's $d$ parameter.

• Asymptotics for Object Descriptors

  2014

  Biometrical Journal · 2 Zitationen · DOI

  This is a discussion of the following paper: “Overview of object oriented data analysis” by J. Steve Marron and Andrés M. Alonso.

• SCoRE Sets: A Versatile Framework for Simultaneous Inference

  2023

  arXiv (Cornell University) · 1 Zitationen · DOI

  We study asymptotic statistical inference in the space of bounded functions endowed with the supremums norm over an arbitrary metric space $S$ using a novel concept: Simultaneous COnfidence Region of Excursion (SCoRE) Sets. They simultaneously quantify the uncertainty of several lower and upper excursion sets of a target function. We investigate their connection to multiple hypothesis tests controlling the familywise error rate in the strong sense and show that they grant a unifying perspective on several statistical inference tools such as simultaneous confidence bands, quantification of uncertainties in level set estimation, for example, CoPE sets, and multiple hypothesis testing over $S$, for example, finding relevant differences or regions of equivalence within $S$. In particular, our abstract setting allows us to refine and reduce the assumptions in recent articles on CoPE sets and relevance and equivalence testing using the supremums norm.

• Inverse set estimation and inversion of simultaneous confidence intervals

  2022

  arXiv (Cornell University) · 1 Zitationen · DOI

  Motivated by the questions of risk assessment in climatology (temperature change in North America) and medicine (impact of statin usage and COVID-19 on hospitalized patients), we address the problem of estimating the set in the domain of a function whose image equals a predefined subset. Existing methods that construct confidence sets require strict assumptions. We generalize the estimation of such sets to dense and non-dense domains with protection against "data peeking" by proving that confidence sets of multiple levels can be simultaneously constructed with the desired confidence non-asymptotically through inverting simultaneous confidence bands. A non-parametric bootstrap algorithm and code are provided.

• Confidence Tubes for Curves on SO(3) and Identification of Subject-Specific Gait Change after Kneeling

  2019

  arXiv (Cornell University) · 1 Zitationen · DOI

  In order to identify changes of gait patterns, e.g. due to prolonged occupational kneeling, which is believed to be major risk factor, among others, for the development of knee osteoarthritis, we develop confidence tubes for curves following a Gaussian perturbation model on SO(3). These are based on an application of the Gaussian kinematic formula to a process of Hotelling statistics and we approximate them by a computible version, for which we show convergence. Simulations endorse our method, which in application to gait curves from eight volunteers undergoing kneeling tasks, identifies phases of the gait cycle that have changed due to kneeling tasks. We find that after kneeling, deviation from normal gait is stronger, in particular for older aged male volunteers. Notably our method adjusts for different walking speeds and marker replacement at different visits.

• Spatial Confidence Sets for Raw Effect Size Images

  2019

  bioRxiv (Cold Spring Harbor Laboratory) · 1 Zitationen · DOI

  Abstract The mass-univariate approach for functional magnetic resonance imagery (fMRI) analysis remains a widely used and fundamental statistical tool within neuroimaging. However, this method suffers from at least two fundamental limitations: First, with sample sizes growing to 4, 5 or even 6 digits, the entire approach is undermined by the null hypothesis fallacy, i.e. with sufficient sample size, there is high enough statistical power to reject the null hypothesis everywhere, making it difficult if not impossible to localize effects of interest. Second, with any sample size, when cluster-size inference is used a significant p -value only indicates that a cluster is larger than chance, and no notion of spatial uncertainty is provided. Therefore, no perception of confidence is available to express the size or location of a cluster that could be expected with repeated sampling from the population. In this work, we address these issues by extending on a method proposed by Sommerfeld, Sain, and Schwartzman (2018) to develop spatial Confidence Sets (CSs) on clusters found in thresholded raw effect size maps. While hypothesis testing indicates where the null, i.e. a raw effect size of zero, can be rejected, the CSs give statements on the locations where raw effect sizes exceed, and fall short of, a non-zero threshold, providing both an upper and lower CS. While the method can be applied to any parameter in a mass-univariate General Linear Model, we motivate the method in the context of BOLD fMRI contrast maps for inference on percentage BOLD change raw effects. We propose several theoretical and practical implementation advancements to the original method in order to deliver an improved performance in small-sample settings. We validate the method with 3D Monte Carlo simulations that resemble fMRI data. Finally, we compute CSs for the Human Connectome Project working memory task contrast images, illustrating the brain regions that show a reliable %BOLD change for a given %BOLD threshold.

• Peak p-values and false discovery rate inference in neuroimaging

  2018

  bioRxiv (Cold Spring Harbor Laboratory) · 1 Zitationen · DOI

  Abstract Peaks are a mainstay of neuroimage analysis for reporting localization results. The current peak detection procedure in SPM12 requires a pre-threshold for approximating p-values and a false discovery rate (FDR) nominal level for inference. However, the pre-threshold is an undesirable feature, while the FDR level is meaningless if the signal is assumed to be nonzero everywhere. This article provides: 1) a peak height distribution for smooth Gaussian error fields that does not require a screening pre-threshold; 2) a signal-plus-noise model where FDR of peaks can be controlled and properly interpreted. Matlab code for calculation of p-values using the exact peak height distribution is available as an SPM extension.

• Spatial Confidence Regions for Piecewise Continuous Processes

  2025

  ArXiv.org · DOI

  In scientific disciplines such as neuroimaging, climatology, and cosmology it is useful to study the uncertainty of excursion sets of imaging data. While the case of imaging data obtained from a single study condition has already been intensively studied, confidence statements about the intersection, or union, of the excursion sets derived from different subject conditions have only been introduced recently. Such methods aim to model the images from different study conditions as asymptotically Gaussian random processes with differentiable sample paths. In this work, we remove the restricting condition of differentiability and only require continuity of the sample paths. This allows for a wider range of applications including many settings which cannot be treated with the existing theory. To achieve this, we introduce a novel notion of convergence on piecewise continuous functions over finite partitions. This notion is of interest in its own right, as it implies convergence results for maxima of sequences of piecewise continuous functions over sequences of sets. Generalizing well-known results such as the extended continuous mapping theorem, this novel convergence notion also allows us to construct for the first time confidence regions for mathematically challenging examples such as symmetric differences of excursion sets.

• Applicability of CSD-based resilience analyses to remotely sensed Vegetation Indices in the Tropics

  2024

  DOI

  Tropical forests are vital for climate change mitigation as carbon sinks. Yet, research suggests that climate change, deforestation and other human influences threaten these systems, potentially pushing them across a tipping point where the tropical vegetation might collapse into a low-treecover state. Signs for this trend are reductions of resilience defined as the system's capability to recover from perturbations. When resilience decreases, according to dynamic system theory, a critical slowing down (CSD) induces changes in statistical measures such as the variance and the autocorrelation. This allows to indirectly examine resilience changes in the absence of observations of strong perturbations. Yet, deriving estimates of the statistical measures indicating resilience changes based on CSD impose several assumptions on the system under observation. For tropical vegetation, it is not obvious that these assumptions are fulfilled.Additionally, the conditions of tropical rainforests pose difficulties on the observation of the vegetation. Among other factors, cloud cover, aerosols, and the dense vegetation hinder the reliable retrieval of Vegetation Indices (Vis), especially from data gathered in the optical spectrum. Thus, such data might not be suitable for resilience analyses based on CSD, even if the theory is applicable in principle.We investigate the different assumptions of CSD and test them on a diverse set of remotely sensed VIs. Hereby, we establish a framework that allows to decide whether a specific dataset is appropriate for resilience analyses based on CSD.

• On the finiteness of the second moment of the number of critical points of Gaussian random fields

  2022

  arXiv (Cornell University) · DOI

  We prove that the second moment of the number of critical points of any sufficiently regular random field, for example with almost surely $ C^3 $ sample paths, defined over a compact Whitney stratified manifold is finite. Our results hold without the assumption of stationarity - which has traditionally been assumed in other work. Under stationarity we demonstrate that our imposed conditions imply the generalized Geman condition of Estrade 2016.

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Name

Dr. rer. nat. Fabian Joachim Erich Telschow

Titel

Dr. rer. nat.

Fakultät

Mathematisch-Naturwissenschaftliche Fakultät

Institut

Institut für Mathematik

Arbeitsgruppe

Angewandte Mathematik

Telefon

+49 30 2093-45449

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