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Does the use of auto-differentiation yield reasonable updates to deep neural networks that represent neural ODEs? Through mathematical analysis and numerical evidence, we find that when the neural network employs high-order forms to approximate the underlying ODE flows (such as the Linear Multistep Method (LMM)), brute-force computation using auto-differentiation often produces non-converging artificial oscillations. In the case of Leapfrog, we propose a straightforward post-processing technique that effectively eliminates these oscillations, rectifies the gradient computation and thus respects the updates of the underlying flow.

We prove inheritance of measure zero property of the set of singular vectors for affine subspaces and submanifolds inside those affine subspaces. We define a notion of $n$-singularity for matrices, which is closely related to the uniform exponent of irrationality. For certain affine subspaces, we show that the set of singular vectors has measure zero if and only if the parametrizing matrix is not $n$-singular. In particular, we show for affine hyperplanes the set of singular vectors has measure zero if and only if the parametrizing matrix is not rational., 11 pages. Theorem 1.2 is improved by fixing an error in a lemma. arXiv admin note: substantial text overlap with arXiv:2203.09716

We show that the only $\psi$-Dirichlet numbers in a function field over a finite field are rational functions, unlike $\psi$-Dirichlet numbers in $\mathbb{R}$. We also prove that there are uncountably many totally irrational singular vectors with large uniform exponent in quadratic surfaces over a positive characteristic field., Comment: We have split version 1 into two parts. The present paper addresses totally irrational singular vectors and $psi$-Dirichlet numbers in function field

We show that the only $ψ$-Dirichlet numbers in a function field over a finite field are rational functions, unlike $ψ$-Dirichlet numbers in $\mathbb{R}$. We also prove that there are uncountably many totally irrational singular vectors with large uniform exponent in quadratic surfaces over a positive characteristic field., We have split version 1 into two parts. The present paper addresses totally irrational singular vectors and $psi$-Dirichlet numbers in function field

Six new bis(benzimidazolyl) monomers containing ether, sulfonyl and ketone groups were synthesized efficiently. A series of poly(N-arylenebenzimidazole)s were prepared using an aromatic nucleophilic displacement reaction that removed the N–H sites from the novel bis(benzimidazolyl) derivatives with activated aromatic difluorides in sulfolane. The reaction was carried out at 210 °C in the presence of anhydrous potassium carbonate. All resulting polymers showed essentially amorphous patterns. This was consistent with the calculated results. Differential scanning calorimetry and thermogravimetric measurements showed that the polymers had high glass transition temperatures (>240 °C), good thermostability and high decomposition temperatures (>400 °C). These novel polymers also showed easy solubility. © 2013 Society of Chemical Industry

A series of novel poly(ether ether ketone)s containing benzimidazolone groups (PNBEEKs) with precise structures in high yields were synthesized from various stoichiometric ratio mixtures of benzimidazolone, 4,4′-dihydroxybenzophenone and 4,4′-difluorobnzophenone via a C–N/C–O coupling reaction process using sulfolane as a solvent. The reaction was carried out at 210 °C in the presence of anhydrous potassium carbonate. The structures of the resulted polymers were characterized by means of FT-IR, 1H NMR spectroscopy, 13C NMR spectroscopy, and elemental analysis, and the results were largely consistent with the proposed structure. XRD studies revealed that the incorporation of benzimidazolone groups increased the crystallinity of the resulted polymers. At the same time, as the benzimidazolone unit content in the copolymer increased, the solvent resistance properties and thermal properties of the prepared polymers improved. The polymers showed high glass transition temperatures (Tg = 126–221 °C) and high thermal stability (Td5% = 497–593 °C in nitrogen, 466–588 °C in air). Moreover, the resulted polymers showed good fluorescence properties and the fluorescence emission peak was 435 nm.

Poly(vincylcyclohexane) (PVCH) was prepared with polystyrene(PS) saturated using heterogeneous hydrogenation techniques. The structure of the synthesized polymers was characterized by means of FT-IR, 1H NMR spectroscopy and GPC, and the results show a good agreement with the proposed structure. The optimum condition for PVCH synthesis was obtained via many experiments. DSC and TG measurements show that PVCH possesses good thermal stability (Tg=146 ℃, TD>345 ℃) compared with PS. The solubility of PVCH is improved, i.e. it can completely dissolve in petroleum ether and cyclohexane.