We consider the one dimensional Schrödinger operator with potential 1/x 4 on the half line. It is known that a generalized Titchmarsh-Weyl function can be associated to it. For other strongly singular potentials in some previous works it was possible to give an operator theoretic interpretation of this fact. However, for the present potential we show that such an interpretation does not exist.
Background: Processing whole-slide images (WSI) to train neural networks can be intricate and laborious. We developed an open-source library covering recurrent tasks in processing of WSI and in evaluating the performance of the trained networks for classification tasks. Methods: Two histopathology use-cases were selected. First we aimed to train a CNN to distinguish H&E-stained slides obtained from neuropathologically classified low-grade epilepsy-associated dysembryoplastic neuroepithelial tumor (DNET) and ganglioglioma (GG). The second project we trained a convolutional neural network (CNN) to predict the hormone expression of pituitary adenoms only from hematoxylin and eosin (H&E) stained slides. In the same approach, we addressed the issue to also predict clinically silent corticotroph adenoma. We included four clinico-pathological disease conditions in a multilabel approach. Results: Our best performing CNN achieved an area under the curve (AUC) of 0.97 for the receiver operating characteristic (ROC) for corticotroph adenoma, 0.86 for silent corticotroph adenoma and 0.98 for gonadotroph adenoma. Our DNET-GG classifier achieved an AUC of 1.00 for the ROC curve. All scores were calculated with the help of our library on predictions on a case basis. Conclusions: Our comprehensive library is most helpful to standardize the work-flow and minimize the work-burden in training CNN. It is also compatible with fastai. Indeed, our new CNNs reliably extracted neuropathologically relevant information from the H&E staining only. This approach will supplement the clinico-pathological diagnosis of brain tumors, which is currently based on cost-intense microscopic examination and variable panels of immunohistochemical stainings.
In this paper self-adjoint realizations of the formal expression Aα:=A+α⟨ϕ,⋅⟩ϕ are described, where α∈R∪{∞}, the operator A is self-adjoint in a Hilbert space H and ϕ is a supersingular element from the scale space H−n−2(A)∖H−n−1(A) for n⩾1. The crucial point is that the spectrum of A may consist of the whole real line. We construct two models to describe the family (Aα). It can be interpreted in a Hilbert space with a twisted version of Krein's formula, or with a more classical version of Krein's formula but in a Pontryagin space. Finally, we compare the two approaches in terms of the respective Q-functions.
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