By Mao L.
This monograph is encouraged with surveying arithmetic and physics by means of CC conjecture, i.e., a mathematical technology might be reconstructed from or made by way of combinatorialization. themes lined during this ebook contain basic of mathematical combinatorics, differential Smarandache n-manifolds, combinatorial or differentiable manifolds and submanifolds, Lie multi-groups, combinatorial central fiber bundles, gravitational box, quantum fields with their combinatorial generalization, additionally with discussions on primary questions in epistemology. All of those are precious for researchers in combinatorics, topology, differential geometry, gravitational or quantum fields.
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Additional info for Combinatorial geometry with application to field theory
1 Let (A ; ◦) be an associative system. Then for a1 , a2 , · · · , an ∈ A , the product a1 ◦a2 ◦· · ·◦an is uniquely determined and independent on the calculating order. Proof The proof is by induction. 2 Algebraic Systems (· · · ((a1 ◦ a2 ) ◦ a3 ) ◦ · · ·) ◦ an . If n = 3, the claim is true by deﬁnition. Assume the claim is true for any integers n ≤ k. We consider the case of n = k + 1. , ◦ = 1 . 2 Apply the inductive assumption, we can assume that = (· · · ((a1 ◦ a2 ) ◦ a3 ) ◦ · · ·) ◦ al 1 and = (· · · ((al+1 ◦ al+2 ) ◦ al+3 ) ◦ · · ·) ◦ ak+1 .
5. 5 For a given proposition, what can we say it is true or false? A proposition and its non-proposition jointly exist in the world. Its truth or false can be only decided by logic inference, independent on one knowing it or not. A norm inference is called implication. , if p then q, is a proposition that is false when p is true but q false and true otherwise. There are three propositions related with p → q, namely, q → p, ¬q → ¬p and ¬p → ¬q, called the converse, contrapositive and inverse of p → q.
Embedded graph. A graph G is embeddable into a topological space R if there is a one-to-one continuous mapping f : G → R in such a way that edges are disjoint except possibly on endpoints. A embedded graph on a topological space R is a graph embeddable on this space. , topological graph theory and combinatorial map theory. Readers can ﬁnd more information in references [GrT1], [Liu1]-[Liu3], [Mao1], [MoT1], [Tut1] and [Whi1]. 5 Operations on Graphs. A union G1 G2 of graphs G1 with G2 is deﬁned by V (G1 G2 ) = V 1 V2 , E(G1 G2 ) = E1 E2 , I(E1 E2 ) = I1 (E1 ) I2 (E2 ).
Combinatorial geometry with application to field theory by Mao L.