Some additive results on Drazin inverse

In this paper, we investigate additive results of the Drazin inverse of elements in a ring R. Under the condition ab = ba, we show that a + b is Drazin invertible if and only if aa^D(a+b) is Drazin invertible, where the superscript D means the Drazin inverse. Furthermore we find an expression of (a + b)^D. As an application we give some new representations for the Drazin inverse of a 2 × 2 block matrix.Supported by the National Natural Science Foundation of China (11361009), the Guangxi Provincial Natural Science Foundation of China (2013GXNSFAA019008), and Science Research Project 2013 of the China-ASEAN Study Center (Guangxi Science Experiment Center) of Guangxi University for Nationalities.Liu, X.; Qin, X.; Benítez López, J. (2015). Some additive results on Drazin inverse. Applied Mathematics - A Journal of Chinese Universities. 30(4):479-490. https://doi.org/10.1007/s11766-015-3333-4S479490304A Ben-Israel, T N E Greville. Generalized Inverses, Theory and Applications, 2nd edition, Springer-Verlag, 2003.S L Campbell, C D Meyer. Generalized Inverses of Linear Transformations, Pitman (Advanced Publishing Program), Boston, MA, 1979.N Castro-González, J J Koliha. Additive perturbation results for the Drazin inverse, Linear Algebra Appl, 2005, 397: 279–297.N Castro-González, E Dopazo, M F Martínez-Serrano. On the Drazin inverse of the sum of two operators and its application to operator matrices, J Math Anal Appl, 2008, 350: 207–215.N Castro-González, M F Martínez-Serrano. Expressions for the g-Drazin inverse of additive perturbed elements in a Banach algebra, Linear Algebra Appl, 2010, 432: 1885–1895.N Castro-González, J J Koliha. New additive results for the Drazin inverse, Proc Roy Soc Edinburgh Sect A, 2004, 134: 1085–1097.M Catral, D D Olesky, P van den Driessche. Block representations of the Drazin inverse of a bipartite matrix, Electron J Linear Algebra, 2009, 18: 98–107.J L Chen, G F Zhuang, Y Wei. The Drazin inverse of a sum of morphisms, Acta Math Sci Ser A Chin Ed, 2009, 29(3): 538–552.D S Cvetković-Ilić, D S Djordjević, Y Wei. Additive results for the generalized Drazin inverse in a Banach algebra, Linear Algebra Appl, 2006, 418, 53–61.D S Cvetković-Ilić. A note on the representation for the Drazin inverse of 2 × 2 block matrices, Linear Algebra Appl, 2008, 429: 242–248.C Deng. The Drazin inverses of sum and difference of idempotents, Linear Algebra Appl, 2009, 430: 1282–1291.C Deng, Y Wei. Characterizations and representations of the Drazin inverse of idempotents, Linear Algebra Appl, 2009, 431: 1526–1538.C Deng, Y Wei. New additive results for the generalized Drazin inverse, J Math Anal Appl, 2010, 370: 313–321.D S Djordjević, P S Stanimirović. On the generalized Drazin inverse and generalized resolvent, Czechoslovak Math J, 2001, 51(126): 617–634.D S Djordjević, Y Wei. Additive results for the generalized Drazin inverse, J Aust Math Soc, 2002, 73: 115–125.D S Djordjević, V Rakočević. Lectures on Generalized inverses, University of Niš, 2008.E Dopazo, M F Martínez-Serrano. Further results on the representation of the Drazin inverse of a 2 × 2 block matrices, Linear Algebra Appl, 2010, 432: 1896–1904.M P Drazin. Pseudo-inverses in associative rings and semiproup, Amer Math Monthly, 1958, 65: 506–514.R E Hartwig, G R Wang, Y Wei. Some additive results on Drazin inverse, Linear Algebra Appl, 2001, 322: 207–217.R E Hartwig, X Li, Y Wei. Representations for the Drazin inverse of a 2×2 block matrix, SIAM J Matrix Anal Appl, 2006, 27: 757–771.Y Liu, C G Cao. Drazin inverse for some partitioned matrices over skew fields, J Nat Sci Heilongjiang Univ, 2004, 24: 112–114.J Ljubisavljević, D S Cvetković-Ilić. Additive results for the Drazin inverse of block matrices and applications, J Comput Appl Math, 2011, 235: 3683–3690.C D Meyer ffixJr, N J Rose. The index and the Drazin inverse of block triangular matrices, SIAM J Appl Math, 1977, 33(1): 1–7.L Wang, H H Zhu, X Zhu, J L Chen. Additive property of Drazin invertibility of elements, arXiv: 1307.1816v1 [math.RA], 2013.H Yang, X Liu. The Drazin inverse of the sum of two matrices and its applications, J Comput Appl Math, 2011, 235: 1412–1417

Application of hybrid boundary element method: Example of semishperical ground inhomogeneity

One new, so-called hybrid boundary element method (HBEM) is presented in this paper. It is a recently proposed numerical method for stationary and quasi-stationary EM field analysis. The method application is illustrated on the example of solving the problem of modelling hemispherical ground inhomogeneity influence on grounding system. The applied procedure also includes using of quasi-stationary image-theory. The obtained results are compared with those ones based on using the Green’s function for the point source inside semi-spherical inhomogeneities as well as with the results obtained by applying COMSOL program package. [TR 33008

Axillobifemoral bypass grafting

INTRODUCTION Axillo-femoral bypass (AxF) means connecting the axillar and femoral artery with the graft that is placed subcutaneously [1]. Usually, this graft is connected with contralateral femoral artery via one accessory subcutaneous graft, and this connection is known as axillobifemoral bypass (AxFF). This extra-anatomic procedure is an alternative method to the standard reconstruction of aortoiliac region when there are contraindications for general or local reasons. OBJECTIVE The objective of this paper is to show early and late results of AxFF bypass grafting as well as to show the indications for AxFF bypass. METHODS The sample consisted of 37 patients. The procedure was performed in 28 patients who suffered from aortoiliac occlusive disease and who were at high risk due to the comorbidity- in one patient with the rupture of juxtarenal aneurysm of abdominal aorta; in five patients with aortoenteric fistula, in two patients with iatrogenic lesion of abdominal aorta and in one female patient with anus preternaturalis definitivus who was treated for rectovaginal fistula. Donor's right axillary artery was used in 26 cases (70.3%), and donor's left axillary artery was used in 9 cases (29.7%). Dacron graft was used in 34 patients and Polytetrafluo-roethlylene graft was used in three patients. Simultaneously, profundo-plastic was done in four patients and femoro-popliteal bypass was performed in three patients. In five patients who suffered from aortoenteric fistula, simultaneous intervention of gastrointerstinal system has been done, x2 test was used for statistical evaluation and life table method was used for verification of late graft patency. RESULTS The rate of early postoperative mortality was 13.5%. The causes of death were: sepsis -1, MOFS - 3, and infarct myocardium -1. The mean follow up period was 40.1 months, ranging from six months to 17 years. During the follow up period, an early graft thrombosis was identified in two and late graft occlusion was reported in four patients. As the cause of occlusion, the progression of occlusive disease of receptive artery was identified in three patients, while anastomotic neointimae hyperplasia of recipient artery was identified in one patient. Three patients died during the follow up period. As the cause of death, CVI was reported in two patients and malignancy of the urinary tract was fpund in one patient. The other complications were - artery angulation on the level of proximal anastomosis in one patient (Figure 1), false aneurysm in one patient, perigraft seroma in one patient and graft infection in three patients. Life table method has shown that cumulative rate of late graft patency is 80.39% after five years (Graph 1). DISCUSSION Our results were analyzed and compared with the results of the study on 283 patients who had undergone aortobifemoral bypass (AFF) operation due to the aortoiliac occlusive disease. This study was completed in 1995 (18). The results showed that there was no statistically significant differences between AxFF and AFF group (p>0.05), considering early mortality rate and late graft patency (Graph 2). The review of mortality and late patency rate after AxFF bypass grafting in a world well known studies has shown the similar results (Table 1). CONCLUSION The authors suggest that axilobifemoral bypass is indicated when there are contraindications or difficulties to perform anatomic reconstruction due to the abdomen condition (infection, adhesion, comorbidity) as well as in high risk patients with low life expectancy