Download Combinatorial geometry with application to field theory by Mao L. PDF

By Mao L.

ISBN-10: 1599731002

ISBN-13: 9781599731001

This monograph is inspired with surveying arithmetic and physics through CC conjecture, i.e., a mathematical technology should be reconstructed from or made by way of combinatorialization. issues lined during this booklet contain primary of mathematical combinatorics, differential Smarandache n-manifolds, combinatorial or differentiable manifolds and submanifolds, Lie multi-groups, combinatorial vital 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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Validated and invalided, or only invalided but in multiple distinct ways. A Smarandache system (Σ; R) is a mathematical system which has at least one Smarandachely denied rule in R. 2 For an integer m ≥ 2, let (Σ1 ; R1 ), (Σ2 ; R2 ), · · ·, (Σm ; Rm ) be m mathematical systems different two by two. A Smarandache multi-space is a pair (Σ; R) with m Σ= m and R = Σi , i=1 Ri . i=1 Certainly, we can construct Smarandache systems by applying Smarandache multi-spaces, particularly, Smarandache geometries appeared in the next chapter.

It is still a poset with |X \ {a1 , a2 , · · · , at−1 , at }| ≤ k. By the induction assumption, it is a multi-chain. Whence, (X, P ) = (X \ {a1 , a2 , · · · , at−1 , at }, P ) Lt is also a multi-chain. In conclusion, we get that (X, P ) is a multi-chain in the case of |X| = k + 1. By the induction principle, we get that (X, P ) is a multi-chain for any X with |X| ≥ 1. , when is a multi-poset a poset? We find conditions in the following result. 3 An s-poset (X, P ) = s (Xi , Pi ) is a poset if and only if for any i=1 integer i, j, 1 ≤ i, j ≤ s, (x, y) ∈ Pi and (y, z) ∈ Pj imply that (x, z) ∈ P .

1 A graph G is a tree if and only if G is connected and E(G) = |V (G)| − 1. 1 Combinatorics with Graphs C2. Hamiltonian graph. A graph G is hamiltonian if it has a circuit, called a hamiltonian circuit containing all vertices of G. Similarly, a path containing all vertices of a graph G is called a hamiltonian path. A graph Bn = (Vb , Eb ; Ib ) with Vb = { O }, Eb = C3. Bouquet and dipole. {e1 , e2 , · · · , en } and Ib (ei ) = (O, O) for any integer i, 1 ≤ i ≤ n is called a bouquet of n edges. t = (Vd , Ed ; Id ) is called a dipole if Vd = {O1, O2 }, Ed = {e1 , e2 , · · · , es , es+1 , · · · , es+l , es+l+1 , · · · , es+l+t} and ⎧ ⎪ ⎪ ⎨ (O1 , O1), Id (ei ) = (O1 , O2), ⎪ ⎪ ⎩ (O , O ), 2 2 if 1 ≤ i ≤ s, if s + 1 ≤ i ≤ s + l, if s + l + 1 ≤ i ≤ s + l + t.

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