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Homological Coordinatization [article]

Andrew Tausz, Gunnar Carlsson
2011 arXiv   pre-print
Coordinatization on a Manifold. In this section we investigate how the method previously described can be used to find manifold-valued coordinates for a given simplicial complex.  ...  In this case, since the tree is contractible, the direct use of homology will not be useful, since homology vanishes on contractible spaces.  ... 
arXiv:1107.0511v2 fatcat:zymzs4zo4rewrlkg44tkdyk4ba

Page 1381 of Mathematical Reviews Vol. 41, Issue 5 [page]

1971 Mathematical Reviews  
Artmann (Giessen) Ostrom, Theodore G. 7525 Translation planes admitting homologies of large order. Math. Z. 114 (1970), 79-92.  ...  Coordinatization of H-planes by H-modules. Math. Z. 115 (1970), 79-103.  ... 

Page 6429 of Mathematical Reviews Vol. , Issue 88m [page]

1988 Mathematical Reviews  
E*(C(-)) is isomorphic to Kasparov’s homological K-functor KK(-;C).  ...  to the usual K-functor and proves that it is really a K-homology in Schochet’s sense.  ... 

Generalized conference matrices and projective planes

Yury J. Ionin
2007 European journal of combinatorics (Print)  
existence of an (a, A)-transitive finite projective plane with a nonincident point-line pair (a, A) is equivalent to the existence of a generalized conference matrix of index 1 over the group of all (a, A)-homologies  ...  an application, I will obtain a short proof of the following result: if a finite projective plane of even order is (a, A)-and (b, B) -transitive with B = ab and b = AB and if the group of all (a, A)-homologies  ...  Let H be the group of all (c, A)-homologies of Π and let C be the core of a GC(H ; 1) corresponding to an H -coordinatization of Π .  ... 
doi:10.1016/j.ejc.2006.08.006 fatcat:vdzqkej7cvhpvp3k22obl22gty

Page 3066 of Mathematical Reviews Vol. , Issue 86g [page]

1986 Mathematical Reviews  
J. 21(96) (1971), 157- 164; MR 43 #3898], is a configuration of 4 triangles, no two with a common vertex, each two of them in 6-fold homology such that the homology centers of any two triangles are the  ...  Remarks on planar ternary rings coordinatizing a projective plane. Arch. Math. (Basel) 42 (1984), no. 6, 573-576.  ... 

Page 7243 of Mathematical Reviews Vol. , Issue 94m [page]

1994 Mathematical Reviews  
They classify those planes which admit nontrivial homology groups with respect to at least 2” + 2 axes [resp. coaxes].  ...  With this coordinatization the authors are able to give an explicit construction of the Ree unital on the quadric Q(6, q), q = 374+! h EN.  ... 

Transitivities in projective planes

T. G. Ostrom
1957 Canadian Journal of Mathematics - Journal Canadien de Mathematiques  
Baer (4) has considered the coordinatization of planes which admit certain groups of perspectivities.  ...  In Part 2, we consider the coordinatization of planes admitting two p -L transitive groups. (See definition in Part 1.)  ...  Let p and a be two homologies with the same center p, and let the axes of p and a intersect in some point q ^ p.  ... 
doi:10.4153/cjm-1957-046-8 fatcat:pvu6d3w2gzdrzjwavpnjlhgsue

Transitive parabolic unitals in translation planes of odd order

V. Abatangelo, G. Korchmáros, B. Larato
2001 Discrete Mathematics  
Correspondingly, the a ne plane coordinatized by Q(+; ?)  ...  Assume that the collineation group G of ÿxing U contains an a ne homology.  ... 
doi:10.1016/s0012-365x(00)00301-0 fatcat:rk2o3f74d5ddpo2lsx6jzahcnq

On symmetric incidence matrices of projective planes

P. Dey, J. L. Hayden
1995 Designs, Codes and Cryptography  
Evans [4] has shown that a plane of prime order p which admits a homology of order (p -1)/2 and an elation with the same axis must be Desarguesian.  ...  Pott [8] has studied planes which admit a collineation group F = E x H where E is a Cartesian group of order n and H is a group of n -1 homologies, i.e., hFI = n 2 -n.  ... 
doi:10.1007/bf01388473 fatcat:n2dtxyhp2fgipcpra4uqzgxsom

Some translation planes constructed by multiple derivation

N. L. Johnson
1981 Bulletin of the Australian Mathematical Society  
In [4], Rao and Rao construct a class of translation planes in the following manner: 2 Let Z denote the Desarguesian affine plane of order q coordinatized by GF[q ) .  ...  It not only has the eight homology centers in an orbit but also has 18 homologies of order 2 whose centers are in a single orbit.  ...  In [4] Johnson and Os+rom considered translation planes admitting several homology centers and constructed a class of planes admitting homologies of order 3 with exactly eight centers all of which are  ... 
doi:10.1017/s0004972700007164 fatcat:f7awtjlzpjf7zepuzv4t7abk5i

Page 961 of Mathematical Reviews Vol. 32, Issue 5 [page]

1966 Mathematical Reviews  
. > is HOMOLOGICAL ALGEBRA If the e,’s in the expression 1 = > e, are pairwise connect- ed, then the indicator ¢ is unique and independent of i and j.  ...  By combining the results for the special cases ¢=}, 0, a coordinatization theorem for non- commutative Jordan algebras is obtained, namely, if A has n 23 connected orthogonal indempotents, then either  ... 

Finite atomic lattices and resolutions of monomial ideals [article]

Sonja Mapes
2010 arXiv   pre-print
In particular, we formally define the notion of coordinatizing a finite atomic lattice P to produce a monomial ideal whose LCM lattice is P, and we give a complete characterization of all such coordinatizations  ...  Thus it has the same reduced homology. Clearly no other elements of Q have nonzero Betti numbers since Q is assumed to be in the same strata of total Betti numbers as P.  ...  To show that such graded lattices have strongly generic coordinatizations, coordinatize P using the ECCV coordinatization.  ... 
arXiv:1009.1430v1 fatcat:xxnei44jfzamvbdqmikwry67uq

A note on Z-planes

Michael J Kallaher
1974 Journal of Algebra  
Let T be a translation plane of order p' coordinatized 63 the quas$eld (Q, +, .), let G be the Q-autotopism group of T, 1st H, alzd HV be the groups of homologies of G with axis y = 0 and x = 0, respectively  ...  The group K of (0, &,)-homologies of 7~ is called the kernel of n-.  ... 
doi:10.1016/0021-8693(74)90042-8 fatcat:imu7j2iyozdxxm3eeapg773wna

Page 2723 of Mathematical Reviews Vol. 58, Issue 4 [page]

1979 Mathematical Reviews  
A Desarguesian K-plane can be coordinatized by a local ring R.  ...  Then there exists a projective plane 7 [7’] which contains a point R and a line r with R Er [a point R’ and a line 7’ with R’ ¢r’} such that the group of all (R, r)-elations of m [(R’, r’)-homologies of  ... 

Generalized Hall planes of even order

Alan Rahilly
1974 Pacific Journal of Mathematics  
Kirkpatrick who has obtained a characterization of the odd order Hall planes, within the class of odd order generalized Hall planes, in terms of their homologies with affine axis. N. L.  ...  characterization of Hall planes of odd order which may be stated as follows: A generalized Hall plane of odd order is a Hall plane if and only if each point of M is the 543 centre of a nontrivial involutory homology  ...  Let us coordinatize π over the quadrangle 0, /, X, Y in π 0 (where XY = IJ) by a generalized Hall system F with subfield JP 0 coordinatizing π 0 and defining functions /, g, h, and k.  ... 
doi:10.2140/pjm.1974.55.543 fatcat:yntxv3gwx5a6zclhn56eujto24
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