arXiv subjects
Publications and source records attributed to Marcus Weber.
"After reading two different crime stories, an artificial intelligence concludes that in both stories the police has found the murderer just by random." -- To what extend and under which assumptions this is a description of a realistic scenario?
Bismuth compounds are of growing interest with regard to potential applications in catalysis, medicine and electronics, for which their environmentally benign nature is one of the key factors. The most common starting material is bismuth nitrate, which easily hydrolyses to give a large number of condensation products. The so-called bismuth subnitrates are composed of bismuth oxido clusters of varying composition and nuclearity. One reason that hampers the further development of bismuth oxido-based materials, however is the low solubility of the subnitrates, which makes targeted immobilisation on substrates challenging. We present an approach towards solubilisation of bismuth oxido clusters by introducing an amino carboxylate as functional group and a study of the growth mode of these atom-precise nanoclusters on gold surfaces. For this purpose the bismuth oxido cluster [Bi38O45(NO3)20(dmso)28](NO3)4*4dmso (dmso=dimethyl sulfoxide) was reacted with the sodium salt of tert-butyloxycabonyl(Boc)-protected phenylalanine (Phe) to give the soluble and chiral nanocluster [Bi38O45(Boc-Phe)24(dmso)9]. The hydrodynamic diameter of the cluster was estimated with (1.4-1.6) nm (in CH3CN) and (2.2 nm-2.9) nm (in Ethanol) based on dynamic light scattering (DLS). The full exchange of the nitrates by the amino carboxylates was proven by NMR and FTIR as well as elemental analysis (EA) and XPS. The solubility of the bismuth oxido cluster in a protic as well as an aprotic polar organic solvent and the growth mode of the clusters on Au upon spin-, dip-, and drop-coating on gold surfaces were studied. Successful deposition of bismuth oxido cluster was proven by powder XRD, FTIR, and XPS while the microstructure of the resulting films was investigated as a function of the deposition method and the solvent used by SEM, AFM, and optical microscopy.
The real numbers, it is taught at universities, correspond to our idea of a continuum, although the hyperreal numbers are located ``in between'' the real numbers. The number $x + dx$, where $dx$ should be an infinitesimal number and $x$ real, is infinitesimally close to $x$ but ``infinitely'' far away from all other real numbers. Analogously: If $f'(x_0)$ and $f(x_0)$ are given for a differentiable function $f:\mathbb{R}\rightarrow\mathbb{R}$ at $x_0\in\mathbb{R}$, we can not determine $f(x)$ at {\em any} point $x\in \mathbb{R}$ different from $x_0$. These points seem to be ``infinitely'' far away. That is one conceptual problem of solving differential equations in numerical mathematics. In this article, we will present a numerical algorithm to solve very simple initial value problems. However, the change of paradigm is, that we will not ``leave'' the point $x_0$. Solving ordinary differential equations is like searching for ``recipes'' $f$. Instead of trying to find these recipes for values $x\in\mathbb{R}$, we will learn them from special relations in the ``monad'' of $x_0$.
There is a heated debate about how far computers can map the complexity of text analysis compared to the abilities of the whole team of human researchers. A "deep" analysis of a given text is still beyond the possibilities of modern computers. In the heart of the existing computational text analysis algorithms there are operations with real numbers, such as additions and multiplications according to the rules of algebraic fields. However, the process of "comparing" has a very precise mathematical structure, which is different from the structure of an algebraic field. The mathematical structure of "comparing" can be expressed by using Boolean rings. We build on this structure and define the corresponding algebraic equations lifting algorithms of comparative text analysis onto the "correct" algebraic basis. From this point of view, we can investigate the question of {\em computational} complexity of comparative text analysis.
The aim of this paper is to investigate the rebinding effect, a phenomenon describing a "short-time memory" which can occur when projecting a Markov process onto a smaller state space. For guaranteeing a correct mapping by the Markov State Model, we assume a fuzzy clustering in terms of membership functions, assigning degrees of membership to each state. The macro states are represented by the membership functions and may be overlapping. The magnitude of this overlap is a measure for the strength of the rebinding effect, caused by the projection and stabilizing the system. A minimal bound for the rebinding effect included in a given system is computed as the solution of an optimization problem. Based on membership functions chosen as a linear combination of Schur vectors, this generalized approach includes reversible as well as non-reversible processes.
Molecular dynamics are extremely complex, yet understanding the slow components of their dynamics is essential to understanding their macroscopic properties. To achieve this, one models the molecular dynamics as a stochastic process and analyses the dominant eigenfunctions of the associated Fokker-Planck operator, or of closely related transfer operators. So far, the calculation of the discretized operators requires extensive molecular dynamics simulations. The Square-root approximation of the Fokker-Planck equation is a method to calculate transition rates as a ratio of the Boltzmann densities of neighboring grid cells times a flux, and can in principle be calculated without a simulation. In a previous work we still used molecular dynamics simulations to determine the flux. Here, we propose several methods to calculate the exact or approximate flux for various grid types, and thus estimate the rate matrix without a simulation. Using model potentials we test computational efficiency of the methods, and the accuracy with which they reproduce the dominant eigenfunctions and eigenvalues. For these model potentials, rate matrices with up to $\mathcal{O}(10^6)$ states can be obtained within seconds on a single high-performance compute server if regular grids are used.
Modern methods of simulating molecular systems are based on the mathematical theory of Markov operators with a focus on autonomous equilibrated systems. However, non-autonomous physical systems or non-autonomous simulation processes are becoming more and more important. We present a representation of non-autonomous Markov jump processes as autonomous Markov chains on space-time. Augmenting the spatial information of the embedded Markov chain by the temporal information of the associated jump times, we derive the so-called augmented jump chain. The augmented jump chain inherits the sparseness of the infinitesimal generator of the original process and therefore provides a useful tool for studying time-dependent dynamics even in high dimensions. We furthermore discuss possible generalizations and applications to the computation of committor functions and coherent sets in the non-autonomous setting. After deriving the theoretical foundations we illustrate the concepts with a proof-of-concept Galerkin discretization of the transfer operator of the augmented jump chain applied to simple examples.
Raman spectroscopy is a well established tool for the analysis of vibration spectra, which then allow for the determination of individual substances in a chemical sample, or for their phase transitions. In the Time-Resolved-Raman-Sprectroscopy the vibration spectra of a chemical sample are recorded sequentially over a time interval, such that conclusions for intermediate products (transients) can be drawn within a chemical process. The observed data-matrix $M$ from a Raman spectroscopy can be regarded as a matrix product of two unknown matrices $W$ and $H$, where the first is representing the contribution of the spectra and the latter represents the chemical spectra. One approach for obtaining $W$ and $H$ is the non-negative matrix factorization. We propose a novel approach, which does not need the commonly used separability assumption. The performance of this approach is shown on a real world chemical example.
For the analysis of molecular processes, the estimation of time-scales, i.e., transition rates, is very important. Estimating the transition rates between molecular conformations is -- from a mathematical point of view -- an invariant subspace projection problem. A certain infinitesimal generator acting on function space is projected to a low-dimensional rate matrix. This projection can be performed in two steps. First, the infinitesimal generator is discretized, then the invariant subspace is approxi-mated and used for the subspace projection. In our approach, the discretization will be based on a Voronoi tessellation of the conformational space. We will show that the discretized infinitesimal generator can simply be approximated by the geometric average of the Boltzmann weights of the Voronoi cells. Thus, there is a direct correla-tion between the potential energy surface of molecular structures and the transition rates of conformational changes. We present results for a 2d-diffusion process and Alanine dipeptide.
This article is about molecular simulation. However, the theoretical results apply for general overdamped Langevin dynamics simulations. Molecular simulation is often used for determining the stability of a complex (e.g., ligand-receptor). The stability can be measured by computing the expected holding time of the complex before its dissociation. This dissociation can be seen as an exit event from a certain part S of the conformational state space. Determining exit rates (i.e, for SDE-based simulations exiting from a given starting set S) for a stochastic process in which the exit event occurs very rarely is obviously hard to solve by straight forward simulation methods. Finding a low variance procedure for computing rare event statistics is still an open problem. Imagine now, e.g., a simulation of a diffusion process. As long as the time-dependent state trajectory is inside the starting set S, no information is gained about the rare event statistics. Only at that point of time, when the process leaves the starting set, a piece of information about the exit rate is collected. If S, however, is a fuzzy set given by a membership function, then there might be additional information of the kind "the process is about to leave the set". However, how to define an exit rate from a fuzzy set?
In this paper, we present a new, optimization-based method to exhibit cyclic behavior in non-reversible stochastic processes. While our method is general, it is strongly motivated by discrete simulations of ordinary differential equations representing non-reversible biological processes, in particular molecular simulations. Here, the discrete time steps of the simulation are often very small compared to the time scale of interest, i.e., of the whole process. In this setting, the detection of a global cyclic behavior of the process becomes difficult because transitions between individual states may appear almost reversible on the small time scale of the simulation. We address this difficulty using a mixed-integer programming model that allows us to compute a cycle of clusters with maximum net flow, i.e., large forward and small backward probability. For a synthetic genetic regulatory network consisting of a ring-oscillator with three genes, we show that this approach can detect the most productive overall cycle, outperforming classical spectral analysis methods. Our method applies to general non-equilibrium steady state systems such as catalytic reactions, for which the objective value computes the effectiveness of the catalyst.
Importance sampling is a widely used technique to reduce the variance of a Monte Carlo estimator by an appropriate change of measure. In this work, we study importance sam- pling in the framework of diffusion process and consider the change of measure which is realized by adding a control force to the original dynamics. For certain exponential type expectation, the corresponding control force of the optimal change of measure leads to a zero-variance estimator and is related to the solution of a Hamilton-Jacobi-Bellmann equation. We focus on certain diffu- sions with both slow and fast variables, and the main result is that we obtain an upper bound of the relative error for the importance sampling estimators with control obtained from the limiting dynamics. We demonstrate our approximation strategy with an illustrative numerical example.
The gluon-gluon induced terms for Higgs production through weak boson fusion (WBF) are computed. Formally, these are of NNLO in the strong coupling constant. This is the lowest order at which non-zero color exchange occurs between the scattering quarks, leading to a color field and thus additional hadronic activity between the outgoing jets. Using a minimal set of cuts, the numerical impact of these terms is at the percent level with respect to the NLO rate for weak boson fusion. Applying the so-called WBF cuts leads to an even stronger suppression, so that we do not expect a significant deterioration of the WFB signal by these color exchange effects.