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Issue No. 06 - June (1972 vol. 21)
ISSN: 0018-9340
pp: 606-610
H. R. Hwa , Basser Computing Department, University of Sydney, Sydney, New South Wales, Australia.
This note attempts to show that, in a vertex weight method [1], every contradiction equation bears a one-to-one correspondence with the summability pair C<inf>1S</inf>, C<inf>2S</inf>, where C<inf>1S</inf> = {X<inf>11</inf>, X<inf>12</inf>, ..., X<inf>1k</inf>}¿ C<inf>1</inf> C<inf>2S</inf> = {X<inf>21</inf>, X<inf>22</inf>,..., X<inf>2k</inf>} ¿ C<inf>2</inf> and vector sums of the vertices plz check [Eqa] The vertices, X<inf>ki</inf>'s, K = 1, or 2, are not necessarily distinct, and C<inf>1</inf>, C<inf>2</inf> are two disjoint sets of vertices in E<sup>n</sup> space. As a consequence, the contradiction equation is a necessary and sufficient condition that the homogeneous system, solved for a threshold function of order r, has no solution. This tells that the threshold function is of order greater than r.

H. R. Hwa, "Contradiction Equations in a B Matrix of Vertex Weight Method and Their Correspondence with the k-Summability Property of Vertices," in IEEE Transactions on Computers, vol. 21, no. , pp. 606-610, 1972.
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