The Owen’s T function is presented in four new ways, one of them as a series similar to the Euler’s arctangent series divided by 2π, which is its majorant series. All possibilities enable numerically stable ...The Owen’s T function is presented in four new ways, one of them as a series similar to the Euler’s arctangent series divided by 2π, which is its majorant series. All possibilities enable numerically stable and fast convergent computation of the bivariate normal integral with simple recursion. When tested computation on a random sample of one million parameter triplets with uniformly distributed components and using double precision arithmetic, the maximum absolute error was 3.45 × 10<sup>-</sup><sup>16</sup>. In additional testing, focusing on cases with correlation coefficients close to one in absolute value, when the computation may be very sensitive to small rounding errors, the accuracy was retained. In rare potentially critical cases, a simple adjustment to the computation procedure was performed—one potentially critical computation was replaced with two equivalent non-critical ones. All new series are suitable for vector and high-precision computation, assuming they are supplemented with appropriate efficient and accurate computation of the arctangent and standard normal cumulative distribution functions. They are implemented by the R package Phi2rho, available on CRAN. Its functions allow vector arguments and are ready to work with the Rmpfr package, which enables the use of arbitrary precision instead of double precision numbers. A special test with up to 1024-bit precision computation is also presented.展开更多
Gel-incorporated single-crystals provide unique combinational properties of long-range order and composite structures, which is desired for semiconducting and conducting materials. However, the reported gel-incorporat...Gel-incorporated single-crystals provide unique combinational properties of long-range order and composite structures, which is desired for semiconducting and conducting materials. However, the reported gel-incorporated single-crystals are limited to insulating crystals. Here, we examine crystals of two typical semiconductors, lead sulfide （PbS） and lead iodide （Pbl2）, grown from both silica gels and agarose gels. In all the four crystal-gel pairs, single-crystals of the cubic phase of PbS and the hexagonal phase of Phi2 were obtained according to the X-ray diffraction analysis. Dissolution of the gel-grown crystals exposed insoluble materials with the shape similar to the original crystals, indicative of gelincorporation inside the crystals. As such, this work creates a facile strategy to construct 3D heterostructures inside serniconducting single-crystals without destroying their long-range order.展开更多
文摘The Owen’s T function is presented in four new ways, one of them as a series similar to the Euler’s arctangent series divided by 2π, which is its majorant series. All possibilities enable numerically stable and fast convergent computation of the bivariate normal integral with simple recursion. When tested computation on a random sample of one million parameter triplets with uniformly distributed components and using double precision arithmetic, the maximum absolute error was 3.45 × 10<sup>-</sup><sup>16</sup>. In additional testing, focusing on cases with correlation coefficients close to one in absolute value, when the computation may be very sensitive to small rounding errors, the accuracy was retained. In rare potentially critical cases, a simple adjustment to the computation procedure was performed—one potentially critical computation was replaced with two equivalent non-critical ones. All new series are suitable for vector and high-precision computation, assuming they are supplemented with appropriate efficient and accurate computation of the arctangent and standard normal cumulative distribution functions. They are implemented by the R package Phi2rho, available on CRAN. Its functions allow vector arguments and are ready to work with the Rmpfr package, which enables the use of arbitrary precision instead of double precision numbers. A special test with up to 1024-bit precision computation is also presented.
基金supported by Zhejiang Province Natural Science Foundation(No.LZ13E030002)the 973 Program(No. 2014CB643503)+1 种基金the National Natural Science Foundation of China (Nos.51222302,51373150,51461165301)Fundamental Research Funds for the Central Universities
文摘Gel-incorporated single-crystals provide unique combinational properties of long-range order and composite structures, which is desired for semiconducting and conducting materials. However, the reported gel-incorporated single-crystals are limited to insulating crystals. Here, we examine crystals of two typical semiconductors, lead sulfide （PbS） and lead iodide （Pbl2）, grown from both silica gels and agarose gels. In all the four crystal-gel pairs, single-crystals of the cubic phase of PbS and the hexagonal phase of Phi2 were obtained according to the X-ray diffraction analysis. Dissolution of the gel-grown crystals exposed insoluble materials with the shape similar to the original crystals, indicative of gelincorporation inside the crystals. As such, this work creates a facile strategy to construct 3D heterostructures inside serniconducting single-crystals without destroying their long-range order.
