Particle size distribution of coarse aggregates through mechanical sieving gives results in terms of cumu- lative mass percent. But digital image processing generated size distribution of particles, while being fast a...Particle size distribution of coarse aggregates through mechanical sieving gives results in terms of cumu- lative mass percent. But digital image processing generated size distribution of particles, while being fast and accurate, is often expressed in terms of area function or number of particles. In this paper, a mass model is developed which converts the image obtained size distribution to mass-wise distribution, mak- ing it readily comparable to mechanical sieving data. The concept of weight/particle ratio is introduced for mass reconstruction from 2D images of particle aggregates. Using this mass model, the effects of several particle shape parameters (such as major axis, minor axis, and equivalent diameter) on sieve-size of the particles is studied. It is shown that the sieve-size of a particle strongly depend upon the shape param- eters, 91% of its variation being explained by major axis, minor axis, bounding box length and equivalent diameter. Furthermore, minor axis gives an overall accurate estimate of particle sieve-size, error in mean size (D-50) being just 0.4%. However, sieve-size of smaller particles (〈20 ram) strongly depends upon the length of the smaller arm of the bounding box enclosing them and sieve-sizes of larger particles (〉20 mm) are highly correlated to their equivalent diameters. Multiple linear regression analysis has been used to generate overall mass-wise particle size distribution, considering the influences of all these shape parameters on particle sieve-size. Multiple linear regression generated overall mass-wise particle size distribution shows a strong correlation with sieve generated data. The adjusted R-square value of the regression analysis is found to be 99 percent (w.r,t cumulative frequency). The method proposed in this paper provides a time-efficient way of producing accurate (up to 99%) mass-wise PSD using digital image processing and it can be used effectively to renlace the mechanical sieving.展开更多
The k-tuple conjecture of Hardy and Little wood predicts that there are infinitely many primes p such that p+2 and p+6 are primes simultaneously.In this paper,we prove that there are infinitely many primes p such that...The k-tuple conjecture of Hardy and Little wood predicts that there are infinitely many primes p such that p+2 and p+6 are primes simultaneously.In this paper,we prove that there are infinitely many primes p such that Ω(p+2)≤3 and Ω(p+6)≤6,where Ω(n) denotes the total number of prime divisors of an integer n.We also prove a better conditional result,with the above Ω(p+6)≤6 replaced by Ω(p+6)≤3,under the Elliott-Halberstam conjecture.展开更多
基金Indian Institute of Technology,Kharagpur in India for supporting this work
文摘Particle size distribution of coarse aggregates through mechanical sieving gives results in terms of cumu- lative mass percent. But digital image processing generated size distribution of particles, while being fast and accurate, is often expressed in terms of area function or number of particles. In this paper, a mass model is developed which converts the image obtained size distribution to mass-wise distribution, mak- ing it readily comparable to mechanical sieving data. The concept of weight/particle ratio is introduced for mass reconstruction from 2D images of particle aggregates. Using this mass model, the effects of several particle shape parameters (such as major axis, minor axis, and equivalent diameter) on sieve-size of the particles is studied. It is shown that the sieve-size of a particle strongly depend upon the shape param- eters, 91% of its variation being explained by major axis, minor axis, bounding box length and equivalent diameter. Furthermore, minor axis gives an overall accurate estimate of particle sieve-size, error in mean size (D-50) being just 0.4%. However, sieve-size of smaller particles (〈20 ram) strongly depends upon the length of the smaller arm of the bounding box enclosing them and sieve-sizes of larger particles (〉20 mm) are highly correlated to their equivalent diameters. Multiple linear regression analysis has been used to generate overall mass-wise particle size distribution, considering the influences of all these shape parameters on particle sieve-size. Multiple linear regression generated overall mass-wise particle size distribution shows a strong correlation with sieve generated data. The adjusted R-square value of the regression analysis is found to be 99 percent (w.r,t cumulative frequency). The method proposed in this paper provides a time-efficient way of producing accurate (up to 99%) mass-wise PSD using digital image processing and it can be used effectively to renlace the mechanical sieving.
基金supported by the National Key Research and Development Program of China (Grant No. 2021YFA1000700)National Natural Science Foundation of China (Grant No. 12031008)。
文摘The k-tuple conjecture of Hardy and Little wood predicts that there are infinitely many primes p such that p+2 and p+6 are primes simultaneously.In this paper,we prove that there are infinitely many primes p such that Ω(p+2)≤3 and Ω(p+6)≤6,where Ω(n) denotes the total number of prime divisors of an integer n.We also prove a better conditional result,with the above Ω(p+6)≤6 replaced by Ω(p+6)≤3,under the Elliott-Halberstam conjecture.