By Mao L.

ISBN-10: 1599731002

ISBN-13: 9781599731001

This monograph is stimulated with surveying arithmetic and physics by way of CC conjecture, i.e., a mathematical technological know-how might be reconstructed from or made by way of combinatorialization. issues coated during this booklet comprise basic of mathematical combinatorics, differential Smarandache n-manifolds, combinatorial or differentiable manifolds and submanifolds, Lie multi-groups, combinatorial relevant fiber bundles, gravitational box, quantum fields with their combinatorial generalization, additionally with discussions on basic questions in epistemology. All of those are precious for researchers in combinatorics, topology, differential geometry, gravitational or quantum fields.

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**Extra resources for Combinatorial geometry with application to field theory**

**Example text**

Then |Y | |Sur(Y X )| = (−1)|Y | (−1)i i=0 |Y | i i|X| . Proof For any sets X = {x1 , x2 , · · · , xn } and Y , by the product principle we know that |Y X | = |Y {x1 } × Y {x2 } × · · · × Y {xn } | = |Y {x1 } ||Y {x2 } | · · · |Y {xn } | = |Y ||X| . Now let Φ : Y X → P(Y ) be a mapping deﬁned by Φ(f ) = Y f (X) − Y f (X). Notice that f ∈ Sur(Y ) is a surjection if and only if Φ(f ) = ∅. For any subset X S ⊆ Y , let XS = {f ∈ Y X |S ⊆ Φ(f )}. Then calculation shows that |XS | = |{f ∈ Y X |S ⊆ Φ(f )}| = |{f ∈ Y X |f (X) ⊆ Y = |Y S−Y S−Y S}| S||X| = (|Y | − |S|)|X|.

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 . 2 Therefore, we get that ◦ = 1 2 = (· · · (a1 ◦ a2 ) ◦ · · ·) ◦ al ◦ (· · · (al+1 ◦ al+2 ) ◦ · · ·) ◦ ak+1 = (· · · (a1 ◦ a2 ) ◦ · · ·) ◦ al ◦ ((· · · (al+1 ◦ al+2 ) ◦ · · · ◦ ak ) ◦ ak+1 ) = ((· · · (a1 ◦ a2 ) ◦ · · ·) ◦ al ◦ (· · · (al+1 ◦ al+2 ) ◦ · · · ◦ ak )) ◦ ak+1 = (· · · ((a1 ◦ a2 ) ◦ a3 ) ◦ · · ·) ◦ ak+1 by the inductive assumption.

For any subset X S ⊆ Y , let XS = {f ∈ Y X |S ⊆ Φ(f )}. Then calculation shows that |XS | = |{f ∈ Y X |S ⊆ Φ(f )}| = |{f ∈ Y X |f (X) ⊆ Y = |Y S−Y S−Y S}| S||X| = (|Y | − |S|)|X|. 5 Enumeration Applying the inclusion-exclusion principle, we ﬁnd that |Sur(Y X )| = |Y X \ XS | ∅=S⊆Y |Y | (−1)|S| (|Y | − |S|)|X| = |Y X | − i=1 |Y | (|Y | − i)|X| (−1)i = |S|=i i=0 |Y | |Y | (−1)i = i i=0 |Y | = (−1)|Y | (−1)i i=0 The last equality applies the fact |Y | i = (|Y | − i)|X| |Y | i i|X| . |Y | |Y | − i on binomial coeﬃ- cients.

### Combinatorial geometry with application to field theory by Mao L.

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