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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.
The Levenberg–Marquardt regularization for the backward heat equation with fractional derivative
(2022)
The backward heat problem with time-fractional derivative in Caputo's sense is studied. The inverse problem is severely ill-posed in the case when the fractional order is close to unity. A Levenberg-Marquardt method with a new a posteriori stopping rule is investigated. We show that optimal order can be obtained for the proposed method under a Hölder-type source condition. Numerical examples for one and two dimensions are provided.
Congenital adrenal hyperplasia (CAH) is the most common form of adrenal insufficiency in childhood; it requires cortisol replacement therapy with hydrocortisone (HC, synthetic cortisol) from birth and therapy monitoring for successful treatment. In children, the less invasive dried blood spot (DBS) sampling with whole blood including red blood cells (RBCs) provides an advantageous alternative to plasma sampling.
Potential differences in binding/association processes between plasma and DBS however need to be considered to correctly interpret DBS measurements for therapy monitoring. While capillary DBS samples would be used in clinical practice, venous cortisol DBS samples from children with adrenal insufficiency were analyzed due to data availability and to directly compare and thus understand potential differences between venous DBS and plasma. A previously published HC plasma pharmacokinetic (PK) model was extended by leveraging these DBS concentrations.
In addition to previously characterized binding of cortisol to albumin (linear process) and corticosteroid-binding globulin (CBG; saturable process), DBS data enabled the characterization of a linear cortisol association with RBCs, and thereby providing a quantitative link between DBS and plasma cortisol concentrations. The ratio between the observed cortisol plasma and DBS concentrations varies highly from 2 to 8. Deterministic simulations of the different cortisol binding/association fractions demonstrated that with higher blood cortisol concentrations, saturation of cortisol binding to CBG was observed, leading to an increase in all other cortisol binding fractions.
In conclusion, a mathematical PK model was developed which links DBS measurements to plasma exposure and thus allows for quantitative interpretation of measurements of DBS samples.
In this article we prove upper bounds for the Laplace eigenvalues lambda(k) below the essential spectrum for strictly negatively curved Cartan-Hadamard manifolds. Our bound is given in terms of k(2) and specific geometric data of the manifold. This applies also to the particular case of non-compact manifolds whose sectional curvature tends to -infinity, where no essential spectrum is present due to a theorem of Donnelly/Li. The result stands in clear contrast to Laplacians on graphs where such a bound fails to be true in general.
Diffusion maps is a manifold learning algorithm widely used for dimensionality reduction. Using a sample from a distribution, it approximates the eigenvalues and eigenfunctions of associated Laplace-Beltrami operators. Theoretical bounds on the approximation error are, however, generally much weaker than the rates that are seen in practice. This paper uses new approaches to improve the error bounds in the model case where the distribution is supported on a hypertorus. For the data sampling (variance) component of the error we make spatially localized compact embedding estimates on certain Hardy spaces; we study the deterministic (bias) component as a perturbation of the Laplace-Beltrami operator's associated PDE and apply relevant spectral stability results. Using these approaches, we match long-standing pointwise error bounds for both the spectral data and the norm convergence of the operator discretization. We also introduce an alternative normalization for diffusion maps based on Sinkhorn weights. This normalization approximates a Langevin diffusion on the sample and yields a symmetric operator approximation. We prove that it has better convergence compared with the standard normalization on flat domains, and we present a highly efficient rigorous algorithm to compute the Sinkhorn weights.
Our input is a complete graph G on n vertices where each vertex has a strict ranking of all other vertices in G. The goal is to construct a matching in G that is popular. A matching M is popular if M does not lose a head-to-head election against any matching M ': here each vertex casts a vote for the matching in {M,M '} in which it gets a better assignment. Popular matchings need not exist in the given instance G and the popular matching problem is to decide whether one exists or not. The popular matching problem in G is easy to solve for odd n. Surprisingly, the problem becomes NP-complete for even n, as we show here. This is one of the few graph theoretic problems efficiently solvable when n has one parity and NP-complete when n has the other parity.
We establish a new approach of treating elliptic boundary value problems (BVPs) on manifolds with boundary and regular corners, up to singularity order 2. Ellipticity and parametrices are obtained in terms of symbols taking values in algebras of BVPs on manifolds of corresponding lower singularity orders. Those refer to Boutet de Monvel's calculus of operators with the transmission property, see Boutet de Monvel (Acta Math 126:11-51, 1971) for the case of smooth boundary. On corner configuration operators act in spaces with multiple weights. We mainly study the case of upper left entries in the respective 2 x 2 operator block-matrices of such a calculus. Green operators in the sense of Boutet de Monvel (Acta Math 126:11-51, 1971) analogously appear in singular cases, and they are complemented by contributions of Mellin type. We formulate a result on ellipticity and the Fredholm property in weighted corner spaces, with parametrices of analogous kind.
The spatio-temporal epidemic type aftershock sequence (ETAS) model is widely used to describe the self-exciting nature of earthquake occurrences. While traditional inference methods provide only point estimates of the model parameters, we aim at a fully Bayesian treatment of model inference, allowing naturally to incorporate prior knowledge and uncertainty quantification of the resulting estimates. Therefore, we introduce a highly flexible, non-parametric representation for the spatially varying ETAS background intensity through a Gaussian process (GP) prior. Combined with classical triggering functions this results in a new model formulation, namely the GP-ETAS model. We enable tractable and efficient Gibbs sampling by deriving an augmented form of the GP-ETAS inference problem. This novel sampling approach allows us to assess the posterior model variables conditioned on observed earthquake catalogues, i.e., the spatial background intensity and the parameters of the triggering function. Empirical results on two synthetic data sets indicate that GP-ETAS outperforms standard models and thus demonstrate the predictive power for observed earthquake catalogues including uncertainty quantification for the estimated parameters. Finally, a case study for the l'Aquila region, Italy, with the devastating event on 6 April 2009, is presented.
The Arnoldi process can be applied to inexpensively approximate matrix functions of the form f (A)v and matrix functionals of the form v*(f (A))*g(A)v, where A is a large square non-Hermitian matrix, v is a vector, and the superscript * denotes transposition and complex conjugation. Here f and g are analytic functions that are defined in suitable regions in the complex plane. This paper reviews available approximation methods and describes new ones that provide higher accuracy for essentially the same computational effort by exploiting available, but generally not used, moment information. Numerical experiments show that in some cases the modifications of the Arnoldi decompositions proposed can improve the accuracy of v*(f (A))*g(A)v about as much as performing an additional step of the Arnoldi process.
Hidden semi-Markov models generalise hidden Markov models by explicitly modelling the time spent in a given state, the so-called dwell time, using some distribution defined on the natural numbers. While the (shifted) Poisson and negative binomial distribution provide natural choices for such distributions, in practice, parametric distributions can lack the flexibility to adequately model the dwell times. To overcome this problem, a penalised maximum likelihood approach is proposed that allows for a flexible and data-driven estimation of the dwell-time distributions without the need to make any distributional assumption. This approach is suitable for direct modelling purposes or as an exploratory tool to investigate the latent state dynamics. The feasibility and potential of the suggested approach is illustrated in a simulation study and by modelling muskox movements in northeast Greenland using GPS tracking data. The proposed method is implemented in the R-package PHSMM which is available on CRAN.
In this paper, we develop the mathematical tools needed to explore isotopy classes of tilings on hyperbolic surfaces of finite genus, possibly nonorientable, with boundary, and punctured. More specifically, we generalize results on Delaney-Dress combinatorial tiling theory using an extension of mapping class groups to orbifolds, in turn using this to study tilings of covering spaces of orbifolds. Moreover, we study finite subgroups of these mapping class groups. Our results can be used to extend the Delaney-Dress combinatorial encoding of a tiling to yield a finite symbol encoding the complexity of an isotopy class of tilings. The results of this paper provide the basis for a complete and unambiguous enumeration of isotopically distinct tilings of hyperbolic surfaces.
The application of the fractional calculus in the mathematical modelling of relaxation processes in complex heterogeneous media has attracted a considerable amount of interest lately.
The reason for this is the successful implementation of fractional stochastic and kinetic equations in the studies of non-Debye relaxation.
In this work, we consider the rotational diffusion equation with a generalised memory kernel in the context of dielectric relaxation processes in a medium composed of polar molecules. We give an overview of existing models on non-exponential relaxation and introduce an exponential resetting dynamic in the corresponding process.
The autocorrelation function and complex susceptibility are analysed in detail.
We show that stochastic resetting leads to a saturation of the autocorrelation function to a constant value, in contrast to the case without resetting, for which it decays to zero. The behaviour of the autocorrelation function, as well as the complex susceptibility in the presence of resetting, confirms that the dielectric relaxation dynamics can be tuned by an appropriate choice of the resetting rate.
The presented results are general and flexible, and they will be of interest for the theoretical description of non-trivial relaxation dynamics in heterogeneous systems composed of polar molecules.
We present a Bayesian inference scheme for scaled Brownian motion, and investigate its performance on synthetic data for parameter estimation and model selection in a combined inference with fractional Brownian motion. We include the possibility of measurement noise in both models. We find that for trajectories of a few hundred time points the procedure is able to resolve well the true model and parameters. Using the prior of the synthetic data generation process also for the inference, the approach is optimal based on decision theory. We include a comparison with inference using a prior different from the data generating one.
We show how to deduce Rellich inequalities from Hardy inequalities on infinite graphs. Specifically, the obtained Rellich inequality gives an upper bound on a function by the Laplacian of the function in terms of weighted norms. These weights involve the Hardy weight and a function which satisfies an eikonal inequality. The results are proven first for Laplacians and are extended to Schrodinger operators afterwards.
In this article, we propose an all-in-one statement which includes existence, uniqueness, regularity, and numerical approximations of mild solutions for a class of stochastic partial differential equations (SPDEs) with non-globally monotone nonlinearities. The proof of this result exploits the properties of an existing fully explicit space-time discrete approximation scheme, in particular the fact that it satisfies suitable a priori estimates. We also obtain almost sure and strong convergence of the approximation scheme to the mild solutions of the considered SPDEs. We conclude by applying the main result of the article to the stochastic Burgers equations with additive space-time white noise.
We introduce and study a family of lattice equations which may be viewed either as a strongly nonlinear discrete extension of the Gardner equation, or a non-convex variant of the Lotka-Volterra chain. Their deceptively simple form supports a very rich family of complex solitary patterns. Some of these patterns are also found in the quasi-continuum rendition, but the more intriguing ones, like interlaced pairs of solitary waves, or waves which may reverse their direction either spontaneously or due a collision, are an intrinsic feature of the discrete realm.
Transition path theory (TPT) for diffusion processes is a framework for analyzing the transitions of multiscale ergodic diffusion processes between disjoint metastable subsets of state space. Most methods for applying TPT involve the construction of a Markov state model on a discretization of state space that approximates the underlying diffusion process. However, the assumption of Markovianity is difficult to verify in practice, and there are to date no known error bounds or convergence results for these methods. We propose a Monte Carlo method for approximating the forward committor, probability current, and streamlines from TPT for diffusion processes. Our method uses only sample trajectory data and partitions of state space based on Voronoi tessellations. It does not require the construction of a Markovian approximating process. We rigorously prove error bounds for the approximate TPT objects and use these bounds to show convergence to their exact counterparts in the limit of arbitrarily fine discretization. We illustrate some features of our method by application to a process that solves the Smoluchowski equation on a triple-well potential.
A theory for diffusivity estimation for spatially extended activator-inhibitor dynamics modeling the evolution of intracellular signaling networks is developed in the mathematical framework of stochastic reaction-diffusion systems. In order to account for model uncertainties, we extend the results for parameter estimation for semilinear stochastic partial differential equations, as developed in Pasemann and Stannat (Electron J Stat 14(1):547-579, 2020), to the problem of joint estimation of diffusivity and parametrized reaction terms. Our theoretical findings are applied to the estimation of effective diffusivity of signaling components contributing to intracellular dynamics of the actin cytoskeleton in the model organism Dictyostelium discoideum.
In this work, we present Raman lidar data (from a Nd:YAG operating at 355 nm, 532 nm and 1064 nm) from the international research village Ny-Alesund for the time period of January to April 2020 during the Arctic haze season of the MOSAiC winter. We present values of the aerosol backscatter, the lidar ratio and the backscatter Angstrom exponent, though the latter depends on wavelength. The aerosol polarization was generally below 2%, indicating mostly spherical particles. We observed that events with high backscatter and high lidar ratio did not coincide. In fact, the highest lidar ratios (LR > 75 sr at 532 nm) were already found by January and may have been caused by hygroscopic growth, rather than by advection of more continental aerosol. Further, we performed an inversion of the lidar data to retrieve a refractive index and a size distribution of the aerosol. Our results suggest that in the free troposphere (above approximate to 2500 m) the aerosol size distribution is quite constant in time, with dominance of small particles with a modal radius well below 100 nm. On the contrary, below approximate to 2000 m in altitude, we frequently found gradients in aerosol backscatter and even size distribution, sometimes in accordance with gradients of wind speed, humidity or elevated temperature inversions, as if the aerosol was strongly modified by vertical displacement in what we call the "mechanical boundary layer". Finally, we present an indication that additional meteorological soundings during MOSAiC campaign did not necessarily improve the fidelity of air backtrajectories.
We prove a homology vanishing theorem for graphs with positive Bakry-' Emery curvature, analogous to a classic result of Bochner on manifolds [3]. Specifically, we prove that if a graph has positive curvature at every vertex, then its first homology group is trivial, where the notion of homology that we use for graphs is the path homology developed by Grigor'yan, Lin, Muranov, and Yau [11]. We moreover prove that the fundamental group is finite for graphs with positive Bakry-' Emery curvature, analogous to a classic result of Myers on manifolds [22]. The proofs draw on several separate areas of graph theory, including graph coverings, gain graphs, and cycle spaces, in addition to the Bakry-Emery curvature, path homology, and graph homotopy. The main results follow as a consequence of several different relationships developed among these different areas. Specifically, we show that a graph with positive curvature cannot have a non-trivial infinite cover preserving 3-cycles and 4-cycles, and give a combinatorial interpretation of the first path homology in terms of the cycle space of a graph. Furthermore, we relate gain graphs to graph homotopy and the fundamental group developed by Grigor'yan, Lin, Muranov, and Yau [12], and obtain an alternative proof of their result that the abelianization of the fundamental group of a graph is isomorphic to the first path homology over the integers.
Variational bayesian inference for nonlinear hawkes process with gaussian process self-effects
(2022)
Traditionally, Hawkes processes are used to model time-continuous point processes with history dependence. Here, we propose an extended model where the self-effects are of both excitatory and inhibitory types and follow a Gaussian Process. Whereas previous work either relies on a less flexible parameterization of the model, or requires a large amount of data, our formulation allows for both a flexible model and learning when data are scarce. We continue the line of work of Bayesian inference for Hawkes processes, and derive an inference algorithm by performing inference on an aggregated sum of Gaussian Processes. Approximate Bayesian inference is achieved via data augmentation, and we describe a mean-field variational inference approach to learn the model parameters. To demonstrate the flexibility of the model we apply our methodology on data from different domains and compare it to previously reported results.
We derive Onsager-Machlup functionals for countable product measures on weighted l(p) subspaces of the sequence space R-N. Each measure in the product is a shifted and scaled copy of a reference probability measure on R that admits a sufficiently regular Lebesgue density. We study the equicoercivity and Gamma-convergence of sequences of Onsager-Machlup functionals associated to convergent sequences of measures within this class. We use these results to establish analogous results for probability measures on separable Banach or Hilbert spaces, including Gaussian, Cauchy, and Besov measures with summability parameter 1 <= p <= 2. Together with part I of this paper, this provides a basis for analysis of the convergence of maximum a posteriori estimators in Bayesian inverse problems and most likely paths in transition path theory.
We introduce the class of "smooth rough paths" and study their main properties. Working in a smooth setting allows us to discard sewing arguments and focus on algebraic and geometric aspects. Specifically, a Maurer-Cartan perspective is the key to a purely algebraic form of Lyons' extension theorem, the renormalization of rough paths following up on [Bruned et al.: A rough path perspective on renormalization, J. Funct. Anal. 277(11), 2019], as well as a related notion of "sum of rough paths". We first develop our ideas in a geometric rough path setting, as this best resonates with recent works on signature varieties, as well as with the renormalization of geometric rough paths. We then explore extensions to the quasi-geometric and the more general Hopf algebraic setting.
The evaluation of process-oriented cognitive theories through time-ordered observations is crucial for the advancement of cognitive science. The findings presented herein integrate insights from research on eye-movement control and sentence comprehension during reading, addressing challenges in modeling time-ordered data, statistical inference, and interindividual variability. Using kernel density estimation and a pseudo-marginal likelihood for fixation durations and locations, a likelihood implementation of the SWIFT model of eye-movement control during reading (Engbert et al., Psychological Review, 112, 2005, pp. 777–813) is proposed. Within the broader framework of data assimilation, Bayesian parameter inference with adaptive Markov Chain Monte Carlo techniques is facilitated for reliable model fitting. Across the different studies, this framework has shown to enable reliable parameter recovery from simulated data and prediction of experimental summary statistics. Despite its complexity, SWIFT can be fitted within a principled Bayesian workflow, capturing interindividual differences and modeling experimental effects on reading across different geometrical alterations of text. Based on these advancements, the integrated dynamical model SEAM is proposed, which combines eye-movement control, a traditionally psychological research area, and post-lexical language processing in the form of cue-based memory retrieval (Lewis & Vasishth, Cognitive Science, 29, 2005, pp. 375–419), typically the purview of psycholinguistics. This proof-of-concept integration marks a significant step forward in natural language comprehension during reading and suggests that the presented methodology can be useful to develop complex cognitive dynamical models that integrate processes at levels of perception, higher cognition, and (oculo-)motor control. These findings collectively advance process-oriented cognitive modeling and highlight the importance of Bayesian inference, individual differences, and interdisciplinary integration for a holistic understanding of reading processes. Implications for theory and methodology, including proposals for model comparison and hierarchical parameter inference, are briefly discussed.
The Bayesian solution to a statistical inverse problem can be summarised by a mode of the posterior distribution, i.e. a maximum a posteriori (MAP) estimator. The MAP estimator essentially coincides with the (regularised) variational solution to the inverse problem, seen as minimisation of the Onsager-Machlup (OM) functional of the posterior measure. An open problem in the stability analysis of inverse problems is to establish a relationship between the convergence properties of solutions obtained by the variational approach and by the Bayesian approach. To address this problem, we propose a general convergence theory for modes that is based on the Gamma-convergence of OM functionals, and apply this theory to Bayesian inverse problems with Gaussian and edge-preserving Besov priors. Part II of this paper considers more general prior distributions.
Let X be an infinite linearly ordered set and let Y be a nonempty subset of X. We calculate the relative rank of the semigroup OP(X,Y) of all orientation-preserving transformations on X with restricted range Y modulo the semigroup O(X,Y) of all order-preserving transformations on X with restricted range Y. For Y = X, we characterize the relative generating sets of minimal size.
In this paper we introduce a fractional variant of the characteristic function of a random variable. It exists on the whole real line, and is uniformly continuous. We show that fractional moments can be expressed in terms of Riemann-Liouville integrals and derivatives of the fractional characteristic function. The fractional moments are of interest in particular for distributions whose integer moments do not exist. Some illustrative examples for particular distributions are also presented.
Forecast verification
(2021)
The philosophy of forecast verification is rather different between deterministic and probabilistic verification metrics: generally speaking, deterministic metrics measure differences, whereas probabilistic metrics assess reliability and sharpness of predictive distributions. This article considers the root-mean-square error (RMSE), which can be seen as a deterministic metric, and the probabilistic metric Continuous Ranked Probability Score (CRPS), and demonstrates that under certain conditions, the CRPS can be mathematically expressed in terms of the RMSE when these metrics are aggregated. One of the required conditions is the normality of distributions. The other condition is that, while the forecast ensemble need not be calibrated, any bias or over/underdispersion cannot depend on the forecast distribution itself. Under these conditions, the CRPS is a fraction of the RMSE, and this fraction depends only on the heteroscedasticity of the ensemble spread and the measures of calibration. The derived CRPS-RMSE relationship for the case of perfect ensemble reliability is tested on simulations of idealised two-dimensional barotropic turbulence. Results suggest that the relationship holds approximately despite the normality condition not being met.
We study the diffusive motion of a particle in a subharmonic potential of the form U(x) = |x|( c ) (0 < c < 2) driven by long-range correlated, stationary fractional Gaussian noise xi ( alpha )(t) with 0 < alpha <= 2. In the absence of the potential the particle exhibits free fractional Brownian motion with anomalous diffusion exponent alpha. While for an harmonic external potential the dynamics converges to a Gaussian stationary state, from extensive numerical analysis we here demonstrate that stationary states for shallower than harmonic potentials exist only as long as the relation c > 2(1 - 1/alpha) holds. We analyse the motion in terms of the mean squared displacement and (when it exists) the stationary probability density function. Moreover we discuss analogies of non-stationarity of Levy flights in shallow external potentials.