We improve the bound on Kühnel's problem to determine the smallest such that the -skeleton of an -simplex does not embed into a compact PL -manifold by showing that if embeds into , then . As a consequence we obtain improved Radon and He…
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RED detects sleep EEG events using deep neural networks, outperforming previous methods.
problem Manual detection of sleep EEG events is time-consuming and variable.
method Deep Recurrent Neural Networks (RNNs) with convolutional and recurrent components.
result RED outperforms state-of-the-art methods in sleep spindle and K-complex detection.
The paper proves embedding conditions for complexes in manifolds using matrix rank criteria.
problem Embedding -dimensional simplicial complexes into -connected PL manifolds.
method Proves embedding conditions using a skew-symmetric matrix with low rank over .
result Embedding conditions for -complexes in -manifolds are equivalent to low-rank matrix conditions.
Background: Electroencephalography (EEG) monitors brain activity during sleep and is used to identify sleep disorders. In sleep medicine, clinicians interpret raw EEG signals in so-called sleep stages, which are assigned by experts to every 30s window of signal. For diagnosis, they also rely on shorter prototypical mic…
Electroencephalography (EEG) during sleep is used by clinicians to evaluate various neurological disorders. In sleep medicine, it is relevant to detect macro-events (> 10s) such as sleep stages, and micro-events (<2s) such as spindles and K-complexes. Annotations of such events require a trained sleep expert, a time co…
A simplified proof for embedding higher-dimensional complexes into manifolds.
problem Embedding higher-dimensional complexes into manifolds with constraints.
method A short and accessible proof for the Patak-Tancer theorem.
result A simplified proof for the Heawood inequality in higher dimensions.
Regularisation method studies Lie algebroids via foliated structures.
problem Geometric structures on Lie algebroids.
method Regularisation procedure applied to various Lie algebroids.
result Proof of Weinstein conjecture for specific Lie algebroids.
In this article, we derive many properties of étale stacks in various contexts, and prove that étale stacks may be characterized categorically as those stacks that arise as prolongations of stacks on a site of spaces and local homeomorphisms. Moreover, we show that the bicategory of étale differentiable stacks and loca…
A Theory of Complex Adaptive Learning and a Non-Localized Wave Equation in Quantum Mechanicsq-fin.GN
The abstract explores a new wave equation linking quantum mechanics and complex adaptive systems.
problem Understanding the underlying mechanism of distribution formation in complex quantum entanglement.
method Exploring the logical relationship between Schrödinger's wave equation and Shi's trading volume-price wave equation in finance.
result A non-localized wave equation in quantum mechanics reveals the invariance of interaction as a universal law.