We consider the final-data problem for systems of nonlinear Schrödinger equations (NLS) with L 2 subcritical nonlinearity. An asymptotically free solution is uniquely obtained for almost every randomized asymptotic profile in L 2 (R d ), extending the result of J. Murphy [29] to powers equal to or lower than the Strauss exponent. In particular, systems with quadratic nonlinearity can be treated in three space dimensions, and by the same argument, the Gross-Pitaevskii equation in the energy space. The extension is by use of the Strichartz estimate with a time weight.
We go along a knot diagram, and get a sequence of over-and under-crossing points. We will study which kinds of sequences are realized by diagrams of the trefoil knot. As an application, we will characterize the Shimizu warping polynomials for diagrams of the trefoil knot.
Association projections from cortical pyramidal neurons connect disparate intrahemispheric cortical areas, which are implicated in higher cortical functions. The underlying developmental processes of these association projections, especially the initial phase before reaching the target areas, remain unknown. To visualize developing axons of individual neurons with association projections in the mouse neocortex, we devised a sparse labeling method that combined in utero electroporation and confocal imaging of flattened and optically cleared cortices. Using the promoter of an established callosal neuron marker gene that was expressed in over 80% of L2/3 neurons in the primary somatosensory cortex (S1) that project to the primary motor cortex (M1), we found that an association projection of a single neuron was the longest among the interstitial collaterals that branched out in L5 from the earlier-extended callosal projection. Collaterals to M1 elongated primarily within the cortical gray matter with little branching before reaching the target. Our results suggest that dual-projection neurons in S1 make a significant fraction of the association projections to M1, supporting the directed guidance mechanism in long-range corticocortical circuit formation over random projections followed by specific pruning.
retracted] We found out that the difference was dependent on the Chainer library, and does not replicate with another library (PyTorch) which indicates that the results are probably due to a bug in Chainer, rather than being hardware-dependent.-old abstract Deep neural networks often present uncertainties such as hardwareand software-derived noise and randomness. We studied the effects of such uncertainty on learning outcomes, with a particular focus on the function of graphics processing units (GPUs), and found that GPU-induced uncertainty increased learning accuracy of a certain deep neural network. When training a predictive deep neural network using only the CPU without the GPU, the learning error is higher than when training the same number of epochs using the GPU, suggesting that the GPU plays a different role in the learning process than just increasing the computational speed. Because this effect cannot be observed in learning by a simple autoencoder, it could be a phenomenon specific to certain types of neural networks. GPU-specific computational processing is more indeterminate than that by CPUs, and hardware-derived uncertainties, which are often considered obstacles that need to be eliminated, might, in some cases, be successfully incorporated into the training of deep neural networks. Moreover, such uncertainties might be interesting phenomena to consider in brain-related computational processing, which comprises a large mass of uncertain signals.
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