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We calculate the gravitational waves (GWs) induced by the density fluctuations due to the inhomogeneous distribution of primordial black holes (PBHs) in the case where PBHs eventually dominate and reheat the universe by Hawking evaporation. The initial PBH density fluctuations are isocurvature in nature. We find that most of the induced GWs are generated right after evaporation, when the universe transits from the PBH dominated era to the radiation dominated era and the curvature perturbation starts to oscillate wildly. The strongest constraint on the amount of the produced GWs comes from the big bang nucleosynthesis (BBN). We improve previous constraints on the PBH fraction and find that it cannot exceed 10-4. Furthermore, this maximum fraction decreases as the mass increases and reaches 10-12 for MPBH∼ 5×108 g, which is the largest mass allowed by the BBN constraint on the reheating temperature. Considering that PBH may cluster above a given clustering scale, we also derive a lower bound on the scale of clustering. Interestingly, the GW spectrum for MPBH∼ 104 -108 g enters the observational window of LIGO and DECIGO and could be tested in the future. Although we focus on the PBH dominated early universe in this paper, our methodology is applicable to any model with early isocurvature perturbation.

As the current research of face gear drive cannot realize fluctuating gear ratio, a design method of orthogonal fluctuating gear ratio face gear drive is proposed. The mathematical model of orthogonal fluctuating gear ratio face gear drive is found based on the space engagement theory. The equation of the pitch curve, addendum curve and dedendum curve of face gear are derived. The design method of tooth surface of the face gear is available based on the envelope method. The conversion relationship of enveloping coordinate systems is obtained during the enveloping process of orthogonal fluctuating gear ratio face gear drive after the establishment of enveloping coordinate systems. Then combining with equation of generating surface, the tooth surface equation of the face gear is obtained. The three-dimensional model of orthogonal fluctuating gear ratio face gear is acquired on the basis of a modeling program, which is developed under the environment of VB and Solidworks (API). Furthermore, localization of the bearing contact is achieved by the manufacturing method and it is justified by the finite element method analysis result. Finally, the kinematics of the orthogonal fluctuating gear ratio face gear drive is analyzed, and the change laws of transmission ratio, angular displacement and angular acceleration of the face gear are acquired.

Reaction of O-ethyl thiocarbamate (4) with (chlorocarbonyl)sulfenyl chloride (5) gives 3-ethoxy-1,2,4-dithiazolin-5-one (2) and 3,5-diethoxy-1,2,4-thiadiazole (3), with the relative amounts of 2 and 3 formed depending very much on the solvent (e.g., diethyl ether favors 2; chloroform favors 3). The effects of added base, order of addition, concentration, and temperature were also studied. Mechanisms for the observed transformations have been proposed and are supported by the characterization of relatively unstable acyclic intermediates, e.g., formimidoyl(chlorocarbonyl)disulfane 8, symmetrical bis(formimidoyl)disulfane 10, and ethoxythiocarbonyl imidate 11, which are obtained under alternative conditions. Compound 2 is converted with concentrated aqueous hydrochloric acid upon short reflux to 1,2,4-dithiazolidine-3,5-dione (1), rearranges upon prolonged melting to give principally N-ethyl-1,2,4-dithiazolidine-3,5-dione (13), and is desulfurized with various trivalent phosphorus compounds to yield O-ethyl cyanate (15) plus carbonyl sulfide. X-ray crystallographic structures of 1 and 2 have been solved; the planarity and aromatic character of these molecules help to explain some of their reactions.

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