The random division of faces in a planar graph

1996 ◽  
Vol 28 (2) ◽  
pp. 377-383 ◽  
Author(s):  
Richard Cowan ◽  
Simone Chen

A planar graph contains faces which can be classified into types depending on the number of edges on the face boundaries. Under various natural rules for randomly dividing faces by the addition of new edges, we investigate the limiting distribution of face type as the number of divisions increases.

1996 ◽  
Vol 28 (02) ◽  
pp. 377-383 ◽  
Author(s):  
Richard Cowan ◽  
Simone Chen

A planar graph contains faces which can be classified into types depending on the number of edges on the face boundaries. Under various natural rules for randomly dividing faces by the addition of new edges, we investigate the limiting distribution of face type as the number of divisions increases.


1996 ◽  
Vol 28 (2) ◽  
pp. 331-331
Author(s):  
Richard Cowan ◽  
Simone Chen

Consider a connected planar graph. A bounded face is said to be of type k, or is called a k-face, if the boundary of that face contains k edges. Under various natural rules for randomly dividing bounded faces by the addition of new edges, we investigate the limiting distribution of face type as the number of divisions increases.


1996 ◽  
Vol 28 (02) ◽  
pp. 331
Author(s):  
Richard Cowan ◽  
Simone Chen

Consider a connected planar graph. A bounded face is said to be of type k, or is called a k-face, if the boundary of that face contains k edges. Under various natural rules for randomly dividing bounded faces by the addition of new edges, we investigate the limiting distribution of face type as the number of divisions increases.


2004 ◽  
Vol 19 (supp02) ◽  
pp. 478-509
Author(s):  
Y. YAMADA

We present face-type elliptic solutions to the Yang-Baxter equation. They have 2N-2 real parameters. When specializing them to definite values, we recover the various models so far known. The intertwining relation between the face models above and the ZN-symmetric vertex model of Belavin is also given.


2019 ◽  
Vol 8 (2) ◽  
pp. 1552-1557
Author(s):  
D.R. Omotoso ◽  
A.J. Olanrewaju ◽  
U.C. Okwuonu ◽  
O Adagboyin ◽  
E.O. Bienonwu

This article has been retracted by the Editor.Cephalometry is an important branch of anthropometry which involves the morphological study of structures present in the human head or scientific measurement of the dimensions of the head. Some of the most important cephalometric parameters include the length/height and breadth/width of the head, the face and the nose as well as their respective indices. These cephalometric parameters are vital in the description of variation which is a common phenomenon that characterizes human physiognomy. They are also useful in the description of human inter-racial and intra-racial similarities both within and across gender. This study involved 450 Bini children (235 males and 215 females) between ages 5-12 years. The length and width of the head and face of each subject was measured between the appropriate anatomical landmarks using spreading and sliding calipers. The measurements were used to calculate the cephalic and facial indices for each subject. The result showed sexual variation in both cephalic and facial indices among the Bini children with the males having higher values than the females. Also, the result of this study showed that prevalence of brachycephalic head type among both male (51.1%) and female (49.8%) Bini children. The mesoproscopic face type was the most prevalent face type among both male (62.6%) and female (47.4%) Bini children. The cephalo-facial indices are vital in demonstrating similarity and variation in physical morphologies of individuals or group of people of different ethnicity, races, gender and geographical locations.Keywords: Cephalometry, Cephalic index, facial index, Bini children, Nigeria


2019 ◽  
Vol 8 (2) ◽  
pp. 1580-1585
Author(s):  
D.R. Omotoso ◽  
A.J. Olanrewaju ◽  
U.C. Okwuonu ◽  
O. Adagboyin ◽  
E.O. Bienonwu

Cephalometry is an important branch of anthropometry which involves the morphological study of structures present in the human head or scientific measurement of the dimensions of the head. Some of the most important cephalometric parameters include the length/height and breadth/width of the head, the face and the nose as well as their respective indices. These cephalometric parameters are vital in the description of variation which is a common phenomenon that characterizes human physiognomy. They are also useful in the description of human inter-racial and intra-racial  similarities both within and across gender. This study involved 450 Bini children (235 males and 215 females) between ages 5-12 years. The length and width of the head and face of each subject was measured between the appropriate anatomical landmarks using spreading and sliding calipers. The measurements were used to calculate the cephalic and facial indices for each subject. The result showed sexual variation in both cephalic and facial indices among the Bini children with the males having higher values than the females. Also, the result of this study showed that prevalence of brachycephalic head type among both male (51.1%) and female (49.8%) Bini children. The mesoproscopic face type was the most prevalent face type among both male (62.6%) and female (47.4%) Bini children. The cephalo-facial indices are vital in demonstrating similarity and variation in physical morphologies of individuals or group of people of different ethnicity, races, gender and geographical locations. Keywords: Cephalometry, Cephalic index, facial index, Bini children, Nigeria


2021 ◽  
Vol 27 (2) ◽  
pp. 39-46
Author(s):  
O.S. Prokopenko

Considering the differences in the values of teleroentgenograhic (TRG) indicators in different racial, gender, ethnic, age, population, geographical population groups and numerical methods of TRG analysis, determination of normative values of cephalometric, gnatometric TRG-indicators, their interdependencies in the population of different countries is extremely important both for human anatomy and for the practice of dentists. This fully applies to residents of Ukraine of different age groups. The aim of the work is to develop and analyze regression models of teleroentgenograhic indicators of tooth position and facial soft tissue profile according to Schwarz A.M. in Ukrainian young men and young women with different facial types. Lateral teleroentgenograms of 49 young men (aged 17 to 21 years) and 76 young women (aged 16 to 20 years) with physiological occlusion as close as possible to orthognathic were analyzed. In the license package “Statistica 6.0” regression models of indicators of position of teeth and profile of soft tissues of the face according to the method of Schwarz A.M. depending on basic cephalometric and gnatometric indicators separately for boys and for girls with different types of the face are constructed. In young men, 16 of the 24 possible reliable regression models were constructed, in which the coefficient of determination is greater than 0.6 (R2=from 0.609 to 0.998). For young men with 1st (back face type) and 3rd (front face type) face types, the following models were created with respect to 5 indicators from 8 possible (respectively, angles Max1-SpP S-arz, distances Sn-Pn and Pog’-Por, angles Gl’LsPog’ and SnPog’-Pn; R2 = from 0.609 to 0.998 and angles Max1-SpP S-arz and Mand1-MP Schwars, distances Sn-Pn and Pog’-Por, angle Gl’LsPog’; R2 = from 0.609 to 0.946), and for young men with 2nd face types (average face type according to Schwarz AM) – for 6 indicators (angles Max1-SpP S-arz and Mand1-MP Schwars, distances Sn-Pn and Pog’-Por, angles Gl’LsPog’and SnPog’-Pn; R2 = from 0.690 to 0.990). In young women, 17 of the 24 possible reliable models with R2 greater than 0.6 (R2 = 0.628 to 0.958) were constructed. For young women with 1st type of face 6 indicators are modeled – angles Max1-SpP S-arz, distances Sn-Pn and Pog’-Por, angles Gl’LsPog’and SnPog’-Pn, the distance of Li-SnPog’(R2= from 0.663 to 0.958). For young women with 2nd face type, the following models were created for 7 indicators – angles Max1-SpP S-arz and Mand1-MP Schwars, distances Sn-Pn and Pog’-Por, angles Gl’LsPog’ and SnPog’-Pn, distances Li-SnPog’(R2= from 0.628 to 0.891). For young women with 3rd face type the smallest number of models with R2 larger than 0.6 was built, compared to all groups of persons with different face types – only 4 models for Max1-SpP S-arz angle, Pog’-Por distance, angles Gl’LsPog’ and SnPog’-Pn (R2 from 0.718 to 0.847).


1997 ◽  
Vol 488 ◽  
Author(s):  
Hideki Yamochi ◽  
Kiyohiko Tsutsumi ◽  
Takeshi Kawasaki ◽  
Gunzi Saito

AbstractIn the course of the preparation of conductive molecular crystals using ‘percyano anions’, the molecular structure of HCDAH2- was revealed to be a planar geometry having a long diagonal line (≈ 10.4 Å). In (TTF)2(HCDAH)(CH3CN) [block-shaped TTF complex] (triclinic, P1, a = 8.258(5), b = 9.572(6), c= 10.070(4) Å, α = 89.89(5), β = 70.76(4), γ = 80.76(5)°, V = 740.7(7) Å3, Z = 2, final R-value = 0.074), the face-to-face type overlap between the side-byside arranged HCDAH dianions was observed. The donor packing pattern in (BEDOTTF) 6(HCDAH) (monoclinic, P21/a, a = 20.243(3), b = 4.086(1), c = 16.820(4) Å, b = 99.45(1)°, V = 1372.4(5) Å3, Z = 2/3, final R-value = 0.048) was determined to be HCP-type which is the second example in BEDO-TTF complexes.


2018 ◽  
Vol 41 ◽  
Author(s):  
Samuel G. B. Johnson

AbstractZero-sum thinking and aversion to trade pervade our society, yet fly in the face of everyday experience and the consensus of economists. Boyer & Petersen's (B&P's) evolutionary model invokes coalitional psychology to explain these puzzling intuitions. I raise several empirical challenges to this explanation, proposing two alternative mechanisms – intuitive mercantilism (assigning value to money rather than goods) and errors in perspective-taking.


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