E that ( (sl ), (sl ))l , L H = 0. l ln,q In specific, (sl ) = 0, hence sl Hl ( L H ). Certainly, s = sl (s l ) and ls l L(2)n,q-(Y, L H ). Therefore,s Hn,q ( L H, L H )by definition. The injectivity is now obvious. Proof of Corollary 1. Look at the quick precise sequence 0 KX L I ( L ) KX L I ( L ) KX L I ( L )/I ( L ) 0. The related cohomology lengthy exact sequence implies that the surjectivity of H 0 ( X, KX L I ( L )) H 0 ( X, KX L I ( L )/I ( L )) is equivalent to the injectivity of H 1 ( X, KX L I ( L )) H 1 ( X, KX L I ( L )). Applying Proposition five, it reduces to prove that (6)Hn,1 ( L, L ) Hn,1 ( L, L )is well-defined. n,1 In actual fact, let l Hl ( L, L ) and l Ln,0 (Y, L) L such that (two) = l l .Symmetry 2021, 13,14 ofn,1 Clearly, l Ln,0 (Y, L) L . Here, Hl ( L, L ), Ln,0 (Y, L) L and Ln,0 (Y, L) L are (two) (2) (2) understood in an apparent way. Additionally, applying Proposition 2 with L , we acquire that L l = 0 and ([i L, L , ]l , l )l , L = 0.Applying Proposition 2 one particular a lot more time with L , then 0 ( L l , L l )l , L=( L l , L l )l , L ([i L, L , ]l , l )l , L .Considering the fact that L = (-1)n(n2)1 = L , we also have L l = 0. Alternatively,([i L, L , ]l , l )l , L ([i L, L , ]l , l )l , L 1 (1 )([i L, L , ]l , l )l , L =0,The last inequality comes from the assumption that for (0, ), i L, L (1 )i 0.n,1 In summary, L l = 0. As a result, l Hl ( L, L ). Hn,1 ( L, L ) by definition. The proof is complete.6. Conclusions We establish an injectivity theorem on a weakly pseudoconvex K ler manifold X with damaging sectional curvature. In particular, X isn’t necessary to be compact. As an application, we get an L2 -extension theorem regarding the subvariety that is not necessary to be lowered. Such variety of extension theorem is of PHA-543613 Autophagy critical value in complex geometry.Funding: This analysis was funded by China Postdoctoral Science Foundation Grant No. 2019M661328. Acknowledgments: The author thanks the referees for detailed and constructive criticism of your original manuscript. Conflicts of Interest: The author declares no conflict of interest.
technologiesArticleElectrospun PVP/TiO2 Nanofibers for Filtration and Doable Protection from Many Viruses like COVID-Ankush Sharma 1 , Dinesh Pathak 1, , Deepak S. Patil 2 , Naresh Dhiman 3 , Viplove Bhullar four and Aman Mahajan3School of Physics and Components Science, Shoolini University of Biotechnology and Management Sciences, Solan 173212, India; [email protected] Department of Civil and Betamethasone disodium manufacturer Environmental Engineering, University of California Los Angeles, Los Angeles, CA 90095, USA; [email protected] Government Degree College Sujanpur Tihra, Sujanpur Tira 176110, India; [email protected] Division of Physics, Guru Nanak Dev University, Amritsar 143005, India; [email protected] (V.B.); [email protected] (A.M.) Correspondence: [email protected]: Sharma, A.; Pathak, D.; Patil, D.S.; Dhiman, N.; Bhullar, V.; Mahajan, A. Electrospun PVP/TiO2 Nanofibers for Filtration and Probable Protection from Various Viruses like COVID-19. Technologies 2021, 9, 89. https://doi.org/10.3390/ technologies9040089 Academic Editor: Manoj Gupta Received: 18 October 2021 Accepted: 15 November 2021 Published: 19 NovemberAbstract: Within this study, TiO2 nanofibers had been ready with Polyvinylpyrrolidone (PVP) polymer working with sol-gel strategy by means of electrospinning technique. Owing to the benefits of smal.