On Functional Decomposition of Multivariate Polynomials with Differentiation and Homogenization

In this paper, we give a theoretical analysis for the algorithms to compute functional decomposition for multivariate polynomials based on differentiation and homogenization which are proposed by Ye, Dai, Lam (1999) and Faug$\mu$ere, Perret (2006, 2008, 2009). We show that a degree proper functional decomposition for a set of randomly decomposable quartic homogenous polynomials can be computed using the algorithm with high probability. This solves a conjecture proposed by Ye, Dai, and Lam (1999). We also propose a conjecture such that the decomposition for a set of polynomials can be computed from that of its homogenization with high probability. Finally, we prove that the right decomposition factors for a set of polynomials can be computed from its right decomposition factor space. Combining these results together, we prove that the algorithm can compute a degree proper decomposition for a set of randomly decomposable quartic polynomials with probability one when the base field is of characteristic zero, and with probability close to one when the base field is a finite field with sufficiently large number under the assumption that the conjeture is correct

Rational general solutions of algebraic ordinary differential equations

We give a necessary and sufficient condition for an alge-braic ODE to have a rational type general solution. For an autonomous first order ODE, we give an algorithm to com-pute a rational general solution if it exists. The algorithm is based on the relation between rational solutions of the first order ODE and rational parametrizations of the plane algebraic curve defined by the first order ODE and Padé approximants

Cooling mechanical resonators to quantum ground state from room temperature

Ground-state cooling of mesoscopic mechanical resonators is a fundamental requirement for test of quantum theory and for implementation of quantum information. We analyze the cavity optomechanical cooling limits in the intermediate coupling regime, where the light-enhanced optomechanical coupling strength is comparable with the cavity decay rate. It is found that in this regime the cooling breaks through the limits in both the strong and weak coupling regimes. The lowest cooling limit is derived analytically at the optimal conditions of cavity decay rate and coupling strength. In essence, cooling to the quantum ground state requires $Q_{\mathrm{m}}>2.4n_{\mathrm{th}}$, with $Q_{\mathrm{m}}$ being the mechanical quality factor and $n_{\mathrm{th}}$ being the thermal phonon number. Remarkably, ground-state cooling is achievable starting from room temperature, when mechanical $Q$-frequency product $Q_{\mathrm{m}}{\nu>1.5}\times10^{13}$, and both of the cavity decay rate and the coupling strength exceed the thermal decoherence rate. Our study provides a general framework for optimizing the backaction cooling of mesoscopic mechanical resonators

Effects of vitamins and trace-elements supplementation on milk production in dairy cows: A review

During the past decade, significant advances were made in understanding the effects of vitamins and trace-elements supplement on milk production of dairy cows. This work discussed the effects of vitamins and trace-elements supplementation on milk production of dairy cows. Studies have indicated that vitamin A (VA) and β-carotene (BC) supplementation have some effects on udder health and milk yield in dairy cows whose intake is below 110 IU/kg BW/day. If low quality forage is fed, supplementation of VA should be considered. Supplementation of B-vitamin has important effects on milk production and could increase milk yield and milk component production. The effect of vitamin E (VE) and selenium (Se) supplementation on the milk yield and milk components are not unified due to the optimum dose, route and timing of VE administration in lactational dairy cows. Zinc (Zn) supplementation increases lactation performance and reduces milk somatic cell count (SCC) in most studies. Limited research has indicated that copper (Cu) supplementation could reduce milk SCC. Before deciding to supplement vitamins and trace-minerals indicated earlier for the improvement of milk production of lactational cows, farmers should have their animals fed with tested and evaluated rations to be sure of the levels of supplementation which may be warranted.Keywords: Vitamin, trace-element, dairy cow, milk productio

Memory augment is All You Need for image restoration

Image restoration is a low-level vision task, most CNN methods are designed as a black box, lacking transparency and internal aesthetics. Although some methods combining traditional optimization algorithms with DNNs have been proposed, they all have some limitations. In this paper, we propose a three-granularity memory layer and contrast learning named MemoryNet, specifically, dividing the samples into positive, negative, and actual three samples for contrastive learning, where the memory layer is able to preserve the deep features of the image and the contrastive learning converges the learned features to balance. Experiments on Derain/Deshadow/Deblur task demonstrate that these methods are effective in improving restoration performance. In addition, this paper's model obtains significant PSNR, SSIM gain on three datasets with different degradation types, which is a strong proof that the recovered images are perceptually realistic. The source code of MemoryNet can be obtained from https://github.com/zhangbaijin/MemoryNe

(E)-N′-[(2-Hy­droxy­naphthalen-1-yl)methyl­idene]-4-methyl­benzohydrazide

In the title compound, C19H16N2O2, the benzene ring and the naphthyl ring system form a dihedral angle of 8.7 (3)° and an intra­molecular O—H⋯N hydrogen bond generates an S(6) ring. In the crystal, mol­ecules are linked by N—H⋯O hydrogen bonds, forming C(4) chains propagating in [001]