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We consider the case of scattering by several obstacles in Rd for d ≥ 2.
In this setting, the absolutely continuous part of the Laplace operator Δ with Dirichlet boundary conditions and the free Laplace operator Δ0 are unitarily equivalent.
For suitable functions that decay sufficiently fast, we have that the difference g(Δ) - g(Δ0) is a trace-class operator and its trace is described by the Krein spectral shift function.
In this article, we study the contribution to the trace (and hence the Krein spectral shift function) that arises from assembling several obstacles relative to a setting where the obstacles are completely separated. In the case of two obstacles, we consider the Laplace operators Δ1 and Δ2 obtained by imposing Dirichlet boundary conditions only on one of the objects.
Our main result in this case states that then g(Δ) - g(Δ1) - g(Δ2) C g(Δ0) is a trace-class operator for a much larger class of functions (including functions of polynomial growth) and that this trace may still be computed by a modification of the Birman–Krein formula. In case g(x) D x 2 , 1 the relative trace has a physical meaning as the vacuum energy of the massless scalar field and is expressible as an integral involving boundary layer operators.
Such integrals have been derived in the physics literature using nonrigorous path integral derivations and our formula provides both a rigorous justification as well as a generalization.
We present the extension of the Kalmag model, proposed as a candidate for IGRF-13, to the twentieth century.
The dataset serving its derivation has been complemented by new measurements coming from satellites, ground-based observatories and land, marine and airborne surveys.
As its predecessor, this version is derived from a combination of a Kalman filter and a smoothing algorithm, providing mean models and associated uncertainties. These quantities permit a precise estimation of locations where mean solutions can be considered as reliable or not.
The temporal resolution of the core field and the secular variation was set to 0.1 year over the 122 years the model is spanning.
Nevertheless, it can be shown through ensembles a posteriori sampled, that this resolution can be effectively achieved only by a limited amount of spatial scales and during certain time periods.
Unsurprisingly, highest accuracy in both space and time of the core field and the secular variation is achieved during the CHAMP and Swarm era. In this version of Kalmag, a particular effort was made for resolving the small-scale lithospheric field.
Under specific statistical assumptions, the latter was modeled up to spherical harmonic degree and order 1000, and signal from both satellite and survey measurements contributed to its development.
External and induced fields were jointly estimated with the rest of the model. We show that their large scales could be accurately extracted from direct measurements whenever the latter exhibit a sufficiently high temporal coverage.
Temporally resolving these fields down to 3 hours during the CHAMP and Swarm missions, gave us access to the link between induced and magnetospheric fields. In particular, the period dependence of the driving signal on the induced one could be directly observed.
The model is available through various physical and statistical quantities on a dedicated website at https://ionocovar.agnld.uni-potsdam.de/Kalmag/.
The diffusion process of water in swelling (expansive) soil often deviates from normal Fick diffusion and belongs to anomalous diffusion.
The process of water adsorption by swelling soil often changes with time, in which the microstructure evolves with time and the absorption rate changes along a fractal dimension gradient function.
Thus, based on the material coordinate theory, this paper proposes a variable order derivative fractal model to describe the cumulative adsorption of water in the expansive soil, and the variable order is time dependent linearly.
The cumulative adsorption is a power law function of the anomalous sorptivity, and patterns of the variable order.
The variable-order fractal derivative model is tested to describe the cumulative adsorption in chernozemic surface soil, Wunnamurra clay and sandy loam.
The results show that the fractal derivative model with linearly time dependent variable-order has much better accuracy than the fractal derivative model with a constant derivative order and the integer order model in the application cases.
The derivative order can be used to distinguish the evolution of the anomalous adsorption process. The variable-order fractal derivative model can serve as an alternative approach to describe water anomalous adsorption in swelling soil.
DrDimont: explainable drug response prediction from differential analysis of multi-omics networks
(2022)
Motivation:
While it has been well established that drugs affect and help patients differently, personalized drug response predictions remain challenging.
Solutions based on single omics measurements have been proposed, and networks provide means to incorporate molecular interactions into reasoning.
However, how to integrate the wealth of information contained in multiple omics layers still poses a complex problem.
Results:
We present DrDimont, Drug response prediction from Differential analysis of multi-omics networks.
It allows for comparative conclusions between two conditions and translates them into differential drug response predictions.
DrDimont focuses on molecular interactions.
It establishes condition-specific networks from correlation within an omics layer that are then reduced and combined into heterogeneous, multi-omics molecular networks. A novel semi-local, path-based integration step ensures integrative conclusions. Differential predictions are derived from comparing the condition-specific integrated networks.
DrDimont's predictions are explainable, i.e. molecular differences that are the source of high differential drug scores can be retrieved. We predict differential drug response in breast cancer using transcriptomics, proteomics, phosphosite and metabolomics measurements and contrast estrogen receptor positive and receptor negative patients. DrDimont performs better than drug prediction based on differential protein expression or PageRank when evaluating it on ground truth data from cancer cell lines. We find proteomic and phosphosite layers to carry most information for distinguishing drug response.
We consider the initial value problem for the Navier-Stokes equations over R-3 x [0, T] with time T > 0 in the spatially periodic setting.
We prove that it induces open injective mappings A(s): B-1(s) -> B-2(s-1) where B-1(s), B-2(s-1) are elements from scales of specially constructed function spaces of Bochner-Sobolev typeparametrized with the smoothness index s is an element of N.
Finally, we prove that a map Asis surjective if and only if the inverse image A(s)(- 1) (K) of any pre compact set K from the range of the map Asis bounded in the Bochner space L-s([0, T], L-r(T-3))with the Ladyzhenskaya-Prodi-Serrin numbers s, r.
We consider Bayesian inference for large-scale inverse problems, where computational challenges arise from the need for repeated evaluations of an expensive forward model.
This renders most Markov chain Monte Carlo approaches infeasible, since they typically require O(10(4)) model runs, or more.
Moreover, the forward model is often given as a black box or is impractical to differentiate.
Therefore derivative-free algorithms are highly desirable. We propose a framework, which is built on Kalman methodology, to efficiently perform Bayesian inference in such inverse problems.
The basic method is based on an approximation of the filtering distribution of a novel mean-field dynamical system, into which the inverse problem is embedded as an observation operator.
Theoretical properties are established for linear inverse problems, demonstrating that the desired Bayesian posterior is given by the steady state of the law of the filtering distribution of the mean-field dynamical system, and proving exponential convergence to it.
This suggests that, for nonlinear problems which are close to Gaussian, sequentially computing this law provides the basis for efficient iterative methods to approximate the Bayesian posterior.
Ensemble methods are applied to obtain interacting particle system approximations of the filtering distribution of the mean-field model; and practical strategies to further reduce the computational and memory cost of the methodology are presented, including low-rank approximation and a bi-fidelity approach.
The effectiveness of the framework is demonstrated in several numerical experiments, including proof-of-concept linear/nonlinear examples and two large-scale applications: learning of permeability parameters in subsurface flow; and learning subgrid-scale parameters in a global climate model.
Moreover, the stochastic ensemble Kalman filter and various ensemble square-root Kalman filters are all employed and are compared numerically.
The results demonstrate that the proposed method, based on exponential convergence to the filtering distribution of a mean-field dynamical system, is competitive with pre-existing Kalman-based methods for inverse problems.
In this paper, we define a variant of Roe algebras for spaces with cylindrical ends and use this to study questions regarding existence and classification of metrics of positive scalar curvature on such manifolds which are collared on the cylindrical end.
We discuss how our constructions are related to relative higher index theory as developed by Chang, Weinberger, and Yu and use this relationship to define higher rho-invariants for positive scalar curvature metrics on manifolds with boundary.
This paves the way for the classification of these metrics.
Finally, we use the machinery developed here to give a concise proof of a result of Schick and the author, which relates the relative higher index with indices defined in the presence of positive scalar curvature on the boundary.
In this paper we consider surfaces which are critical points of the Willmore functional subject to constrained area.
In the case of small area we calculate the corrections to the intrinsic geometry induced by the ambient curvature.
These estimates together with the choice of an adapted geometric center of mass lead to refined position estimates in relation to the scalar curvature of the ambient manifold.
As the loop space of a Riemannian manifold is infinite-dimensional, it is a non-trivial problem to make sense of the "top degree component " of a differential form on it.
In this paper, we show that a formula from finite dimensions generalizes to assign a sensible "top degree component " to certain composite forms, obtained by wedging with the exponential (in the exterior algebra) of the canonical presymplectic 2-form on the loop space.
This construction is a crucial ingredient for the definition of the supersymmetric path integral on the loop space.
State space models enjoy wide popularity in mathematical and statistical modelling across disciplines and research fields. Frequent solutions to problems of estimation and forecasting of a latent signal such as the celebrated Kalman filter hereby rely on a set of strong assumptions such as linearity of system dynamics and Gaussianity of noise terms.
We investigate fallacy in mis-specification of the noise terms, that is signal noise
and observation noise, regarding heavy tailedness in that the true dynamic frequently produces observation outliers or abrupt jumps of the signal state due to realizations of these heavy tails not considered by the model. We propose a formalisation of observation noise mis-specification in terms of Huber’s ε-contamination as well as a computationally cheap solution via generalised Bayesian posteriors with a diffusion Stein divergence loss resulting in the diffusion score matching Kalman filter - a modified algorithm akin in complexity to the regular Kalman filter. For this new filter interpretations of novel terms, stability and an ensemble variant are discussed. Regarding signal noise mis-specification, we propose a formalisation in the frame work of change point detection and join ideas from the popular CUSUM algo-
rithm with ideas from Bayesian online change point detection to combine frequent reliability constraints and online inference resulting in a Gaussian mixture model variant of multiple Kalman filters. We hereby exploit open-end sequential probability ratio tests on the evidence of Kalman filters on observation sub-sequences for aggregated inference under notions of plausibility.
Both proposed methods are combined to investigate the double mis-specification problem and discussed regarding their capabilities in reliable and well-tuned uncertainty quantification. Each section provides an introduction to required terminology and tools as well as simulation experiments on the popular target tracking task and the non-linear, chaotic Lorenz-63 system to showcase practical performance of theoretical considerations.
Hardy inequalities on graphs
(2024)
The dissertation deals with a central inequality of non-linear potential theory, the Hardy inequality. It states that the non-linear energy functional can be estimated from below by a pth power of a weighted p-norm, p>1. The energy functional consists of a divergence part and an arbitrary potential part. Locally summable infinite graphs were chosen as the underlying space. Previous publications on Hardy inequalities on graphs have mainly considered the special case p=2, or locally finite graphs without a potential part.
Two fundamental questions now arise quite naturally: For which graphs is there a Hardy inequality at all? And, if it exists, is there a way to obtain an optimal weight? Answers to these questions are given in Theorem 10.1 and Theorem 12.1. Theorem 10.1 gives a number of characterizations; among others, there is a Hardy inequality on a graph if and only if there is a Green's function. Theorem 12.1 gives an explicit formula to compute optimal Hardy weights for locally finite graphs under some additional technical assumptions. Examples show that Green's functions are good candidates to be used in the formula.
Emphasis is also placed on illustrating the theory with examples. The focus is on natural numbers, Euclidean lattices, trees and star graphs. Finally, a non-linear version of the Heisenberg uncertainty principle and a Rellich inequality are derived from the Hardy inequality.
We present general existence and uniqueness results for marked models with pair interactions, exemplified through Gibbs point processes on path space.
More precisely, we study a class of infinite-dimensional diffusions under Gibbsian interactions, in the context of marked point configurations: the starting points belong to R-d, and the marks are the paths of Langevin diffusions.
We use the entropy method to prove existence of an infinite-volume Gibbs point process and use cluster expansion tools to provide an explicit activity domain in which uniqueness holds.
Mathematical modelling and statistical inference provide a framework to evaluate different non-pharmaceutical and pharmaceutical interventions for the control of epidemics that has been widely used during the COVID-19 pandemic. In this paper, lessons learned from this and previous epidemics are used to highlight the challenges for future pandemic control. We consider the availability and use of data, as well as the need for correct parameterisation and calibration for different model frameworks. We discuss challenges that arise in describing and distinguishing between different interventions, within different modelling structures, and allowing both within and between host dynamics. We also highlight challenges in modelling the health economic and political aspects of interventions. Given the diversity of these challenges, a broad variety of interdisciplinary expertise is needed to address them, combining mathematical knowledge with biological and social insights, and including health economics and communication skills. Addressing these challenges for the future requires strong cross disciplinary collaboration together with close communication between scientists and policy makers.
This paper deals with the long-term behavior of positive operator semigroups on spaces of bounded functions and of signed measures, which have applications to parabolic equations with unbounded coefficients and to stochas-tic analysis. The main results are a Tauberian type theorem characterizing the convergence to equilibrium of strongly Feller semigroups and a generalization of a classical convergence theorem of Doob. None of these results requires any kind of time regularity of the semigroup.
We present a Reduced Order Model (ROM) which exploits recent developments in Physics Informed Neural Networks (PINNs) for solving inverse problems for the Navier-Stokes equations (NSE). In the proposed approach, the presence of simulated data for the fluid dynamics fields is assumed. A POD-Galerkin ROM is then constructed by applying POD on the snapshots matrices of the fluid fields and performing a Galerkin projection of the NSE (or the modified equations in case of turbulence modeling) onto the POD reduced basis. A POD-Galerkin PINN ROM is then derived by introducing deep neural networks which approximate the reduced outputs with the input being time and/or parameters of the model. The neural networks incorporate the physical equations (the POD-Galerkin reduced equations) into their structure as part of the loss function. Using this approach, the reduced model is able to approximate unknown parameters such as physical constants or the boundary conditions. A demonstration of the applicability of the proposed ROM is illustrated by three cases which are the steady flow around a backward step, the flow around a circular cylinder and the unsteady turbulent flow around a surface mounted cubic obstacle.
The past three decades of policy process studies have seen the emergence of a clear intellectual lineage with regard to complexity. Implicitly or explicitly, scholars have employed complexity theory to examine the intricate dynamics of collective action in political contexts. However, the methodological counterparts to complexity theory, such as computational methods, are rarely used and, even if they are, they are often detached from established policy process theory. Building on a critical review of the application of complexity theory to policy process studies, we present and implement a baseline model of policy processes using the logic of coevolving networks. Our model suggests that an actor's influence depends on their environment and on exogenous events facilitating dialogue and consensus-building. Our results validate previous opinion dynamics models and generate novel patterns. Our discussion provides ground for further research and outlines the path for the field to achieve a computational turn.
Instruments for measuring the absorbed dose and dose rate under radiation exposure, known as radiation dosimeters, are indispensable in space missions. They are composed of radiation sensors that generate current or voltage response when exposed to ionizing radiation, and processing electronics for computing the absorbed dose and dose rate. Among a wide range of existing radiation sensors, the Radiation Sensitive Field Effect Transistors (RADFETs) have unique advantages for absorbed dose measurement, and a proven record of successful exploitation in space missions. It has been shown that the RADFETs may be also used for the dose rate monitoring. In that regard, we propose a unique design concept that supports the simultaneous operation of a single RADFET as absorbed dose and dose rate monitor. This enables to reduce the cost of implementation, since the need for other types of radiation sensors can be minimized or eliminated. For processing the RADFET's response we propose a readout system composed of analog signal conditioner (ASC) and a self-adaptive multiprocessing system-on-chip (MPSoC). The soft error rate of MPSoC is monitored in real time with embedded sensors, allowing the autonomous switching between three operating modes (high-performance, de-stress and fault-tolerant), according to the application requirements and radiation conditions.
In this work we consider the first encounter problems between a fixed and/or mobile target A and a moving trap B on Bethe lattices and Cayley trees. The survival probabilities (SPs) of the target A on the both kinds of structures are considered analytically and compared. On Bethe lattices, the results show that the fixed target will still prolong its survival time, whereas, on Cayley trees, there are some initial positions where the target should move to prolong its survival time. The mean first encounter time (MFET) for mobile target A is evaluated numerically and compared with the mean first passage time (MFPT) for the fixed target A. Different initial settings are addressed and clear boundaries are obtained. These findings are helpful for optimizing the strategy to prolong the survival time of the target or to speed up the search process on Cayley trees, in relation to the target's movement and the initial position configuration of the two walkers. We also present a new method, which uses a small amount of memory, for simulating random walks on Cayley trees. (C) 2020 Elsevier B.V. All rights reserved.
An instance of the marriage problem is given by a graph G = (A boolean OR B, E), together with, for each vertex of G, a strict preference order over its neighbors. A matching M of G is popular in the marriage instance if M does not lose a head-to-head election against any matching where vertices are voters. Every stable matching is a min-size popular matching; another subclass of popular matchings that always exists and can be easily computed is the set of dominant matchings. A popular matching M is dominant if M wins the head-to-head election against any larger matching. Thus, every dominant matching is a max-size popular matching, and it is known that the set of dominant matchings is the linear image of the set of stable matchings in an auxiliary graph. Results from the literature seem to suggest that stable and dominant matchings behave, from a complexity theory point of view, in a very similar manner within the class of popular matchings. The goal of this paper is to show that there are instead differences in the tractability of stable and dominant matchings and to investigate further their importance for popular matchings. First, we show that it is easy to check if all popular matchings are also stable; however, it is co-NP hard to check if all popular matchings are also dominant. Second, we show how some new and recent hardness results on popular matching problems can be deduced from the NP-hardness of certain problems on stable matchings, also studied in this paper, thus showing that stable matchings can be employed to show not only positive results on popular matchings (as is known) but also most negative ones. Problems for which we show new hardness results include finding a min-size (resp., max-size) popular matching that is not stable (resp., dominant). A known result for which we give a new and simple proof is the NP-hardness of finding a popular matching when G is nonbipartite.