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We introduce the concept of TRAP (Traces and Permutations), which can roughly be viewed as a wheeled PROP (Products and Permutations) without unit. TRAPs are equipped with a horizontal concatenation and partial trace maps.
Continuous morphisms on an infinite-dimensional topological space and smooth kernels (respectively, smoothing operators) on a closed manifold form a TRAP but not a wheeled PROP.
We build the free objects in the category of TRAPs as TRAPs of graphs and show that a TRAP can be completed to a unitary TRAP (or wheeled PROP).
We further show that it can be equipped with a vertical concatenation, which on the TRAP of linear homomorphisms of a vector space, amounts to the usual composition. The vertical concatenation in the TRAP of smooth kernels gives rise to generalised convolutions.
Graphs whose vertices are decorated by smooth kernels (respectively, smoothing operators) on a closed manifold form a TRAP. From their universal properties we build smooth amplitudes associated with the graph.
We prove that optimal lower eigenvalue estimates of Zhong-Yang type as well as a Cheng-type upper bound for the first eigenvalue hold on closed manifolds assuming only a Kato condition on the negative part of the Ricci curvature.
This generalizes all earlier results on Lp-curvature assumptions.
Moreover, we introduce the Kato condition on compact manifolds with boundary with respect to the Neumann Laplacian, leading to Harnack estimates for the Neumann heat kernel and lower bounds for all Neumann eigenvalues, which provides a first insight in handling variable Ricci curvature assumptions in this case.
Each completely regular semigroup is a semilattice of completely simple semigroups. The more specific concept of a strong semilattice provides the concrete product between two arbitrary elements.
We characterize strong semilattices of rectangular groups by so-called disjunctions of identities. Disjunctions of identities generalize the classical concept of an identity and of a variety, respectively.
The rectangular groups will be on the one hand left zero semigroups and right zero semigroups and on the other hand groups of exponent p is an element of P, where P is any set of pairwise coprime natural numbers.
Devising optimal interventions for constraining stochastic systems is a challenging endeavor that has to confront the interplay between randomness and dynamical nonlinearity.
Existing intervention methods that employ stochastic path sampling scale poorly with increasing system dimension and are slow to converge.
Here we propose a generally applicable and practically feasible methodology that computes the optimal interventions in a noniterative scheme.
We formulate the optimal dynamical adjustments in terms of deterministically sampled probability flows approximated by an interacting particle system.
Applied to several biologically inspired models, we demonstrate that our method provides the necessary optimal controls in settings with terminal, transient, or generalized collective state constraints and arbitrary system dynamics.
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.