A Mathematical Gift I: The Interplay Between Topology, - download pdf or read online

By Kenji Ueno, Koji Shiga, Shigeyuki Morita

This ebook will convey the wonder and enjoyable of arithmetic to the study room. It bargains severe arithmetic in a full of life, reader-friendly kind. integrated are routines and plenty of figures illustrating the most techniques.
The first bankruptcy provides the geometry and topology of surfaces. between different subject matters, the authors talk about the Poincaré-Hopf theorem on serious issues of vector fields on surfaces and the Gauss-Bonnet theorem at the relation among curvature and topology (the Euler characteristic). the second one bankruptcy addresses quite a few elements of the concept that of size, together with the Peano curve and the Poincaré technique. additionally addressed is the constitution of third-dimensional manifolds. specifically, it's proved that the three-d sphere is the union of 2 doughnuts.
This is the 1st of 3 volumes originating from a sequence of lectures given by means of the authors at Kyoto college (Japan).

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Additional info for A Mathematical Gift I: The Interplay Between Topology, Functions, Geometry, and Algebra (Mathematical World, Volume 19)

Example text

By monotonicity, this ends the proof. 41 Suppose H(P|B) < ∞. Then H(P|Q ∨ B) = lim ↓ H(P|Q(m) ∨ B). 10. 1) 1 − sup i=1 where P = {Ai , i ∈ N}, Q = {Bi , i ∈ N}, denotes the symmetric difference and the infimum (and supremum) runs through all permutations π of the natural numbers. This pseudometric becomes a metric once factored to classes of partitions modulo measure zero. It is elementary to see that d1 (P1 ∨ P2 , Q1 ∨ Q2 ) ≤ d1 (P1 , Q1 ) + d1 (P2 , Q2 ). 2) One of the important features of this metric is the possibility of approximating a partition measurable with respect to k Qk by Qk -measurable partitions.

The elements Ai of a partition P will be referred to as cells. A partition P is finer than (or is a refinement of) Q, which we write as P Q, when each cell of P is (up to measure) contained in a cell of Q. By disjointness, each cell of Q is then a union of some cells of P; we will also say that Q is Pmeasurable. 2 Partitions and sigma-algebras 31 P ∨ Q = {A ∩ B : A ∈ P, B ∈ Q}. 1) It is easy to see that P Q ⇐⇒ P ∨ Q = P. We will also consider sub-sigma-algebras B of A, and call them simply sigma-algebras, always assuming that B are completed.

N}, equipped with the product sigma-algebra A = A×{all subsets} and the product measure μ = μ × Prob, where Prob is the normalized counting measure on {1, . . , n}. Now, (X , A , μ ) is again a standard probability space. , n} has measure (n − 1)/n. Let Q = {A, B} be the associated partition of X . Now define partitions Pi (i = 1, . . , they all contain the large set B in one piece. We fix a nonempty set F ⊂ {1, . . , k} and we calculate the Shannon entropy of the join PF . Because Q is refined by each Pi , it is also refined by PF , thus we have H(PF ) = H(PF ∨ Q) = H(PF |Q) + H(Q) = μ (B)HB (PF ) + μ (A)HA (PF ) + H(Q) = n−1 n · 0 + n1 Hμ (PF ) + H(Q) = 1 n Hμ (PF ) + H(Q).

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