On Sets Closed Under the Cross Product and a Property of the Octahedron

2020 ◽  
Vol 127 (2) ◽  
pp. 151-158
Author(s):  
Mohammad Javaheri
Keyword(s):  
2021 ◽  
Vol 42 (2) ◽  
pp. 435-450
Author(s):  
Chanakan Sungboonchoo ◽  
Thuntida Ngamkham ◽  
Wararit Panichkitkosolkul ◽  
Andrei Volodin

2003 ◽  
Vol 2003 (38) ◽  
pp. 2425-2445 ◽  
Author(s):  
Heath Emerson

For every hyperbolic groupΓwith Gromov boundary∂Γ, one can form the cross productC∗-algebraC(∂Γ)⋊Γ. For each such algebra, we construct a canonicalK-homology class. This class induces a Poincaré duality mapK∗(C(∂Γ)⋊Γ)→K∗+1(C(∂Γ)⋊Γ). We show that this map is an isomorphism in the case ofΓ=𝔽2, the free group on two generators. We point out a direct connection between our constructions and the Baum-Connes conjecture and eventually use the latter to deduce our result.


2017 ◽  
Vol 9 (5) ◽  
pp. 71
Author(s):  
Yevhen Mykolayovych Kharchenko

The theory of angular vectors, which allows modelling of the properties of angular physical quantities, is considered. The meaning of the cross product of vectors was radically revised and changed. Formulas for finding torque and angular velocity in a coordinate-vector form with a correct mapping of their directions were deduced. Described definition of the inverse vector and its properties. The inversed vector allows us to perform vector division operations.


2000 ◽  
Vol 18 (10) ◽  
pp. 2179-2184 ◽  
Author(s):  
Lawrence H. Schwartz ◽  
Michelle S. Ginsberg ◽  
Douglas DeCorato ◽  
Lawrence N. Rothenberg ◽  
Steven Einstein ◽  
...  

PURPOSE: To evaluate the variability in bidimensional computed tomography (CT) measurements obtained of actual tumors and of tumor phantoms by use of three measurement techniques: hand-held calipers on film, electronic calipers on a workstation, and an autocontour technique on a workstation. MATERIALS AND METHODS: Three radiologists measured 45 actual tumors (in the lung, liver, and lymph nodes) on CT images, using each of the three techniques. Bidimensional measurements were recorded, and their cross-products calculated. The coefficient of variation was calculated to assess interobserver variability. CT images of 48 phantoms were measured by three radiologists with each of the techniques. In addition to the coefficient of variation, the differences between the cross-product measurements of tumor phantoms themselves and the measurements obtained with each of the techniques were calculated. RESULTS: The differences between the coefficients of variation were statistically significantly different for the autocontour technique, compared with the other techniques, both for actual tumors and for tumor phantoms. There was no statistically significant difference in the coefficient of variation between measurements obtained with hand-held calipers and electronic calipers. The cross-products for tumor phantoms were 12% less than the actual cross-product when calipers on film were used, 11% less using electronic calipers, and 1% greater using the autocontour technique. CONCLUSION: Tumor size is obtained more accurately and consistently between readers using an automated autocontour technique than between those using hand-held or electronic calipers. This finding has substantial implications for monitoring tumor therapy in an individual patient, as well as for evaluating the effectiveness of new therapies under development.


2019 ◽  
Vol 50 (4) ◽  
pp. 298-299
Author(s):  
Suk-Geun Hwang ◽  
Ik-Pyo Kim
Keyword(s):  

2009 ◽  
Vol 30 (10) ◽  
pp. 1027-1037 ◽  
Author(s):  
Pei-Kuang Chao ◽  
Hsiao-Lung Chan ◽  
Fuk-Tan Tang ◽  
Yu-Chuan Chen ◽  
May-Kuen Wong

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