Lecture 2: Addition (and free abelian groups)

Transcription

1 Lectur: Addition (and free abelian groups) of a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305) Topics in Topology: Scientific and Engineering Applications of Algebraic Topology Target Audience: Anyone interested in topological data analysis including graduate students, faculty, industrial researchers in bioinformatics, biology, computer science, cosmology, engineering, imaging, mathematics, neurology, physics, statistics, etc. Isabel K. Darcy Mathematics Department/Applied Mathematical & Computational Sciences University of Iowa

2 A free abelian group generated by the elements x 1, x 2,, x k consists of all elements of the form n 1 x 1 + n 2 x n k x k where n i are integers. Z = The set of integers = {, -2, -1, 0, 1, 2, } = the set of all whole numbers (positive, negative, 0) Addition: (n 1 x 1 + n 2 x n k x k ) + (m 1 x 1 + m 2 x m k x k ) = (n 1 + m 1 ) x 1 + (n 2 + m 2 )x (n k + m k )x k

3 Will add video clips when video becomes available. Formal sum: 4 cone flower + 2 rose + 3 cone flower + 1 rose = 7 cone flower + 3 rose

4 A free abelian group generated by the elements x 1, x 2,, x k consists of all elements of the form n 1 x 1 + n 2 x n k x k where n i are integers. Z = The set of integers = {, -2, -1, 0, 1, 2, } = the set of all whole numbers (positive, negative, 0) Addition: (n 1 x 1 + n 2 x n k x k ) + (m 1 x 1 + m 2 x m k x k ) = (n 1 + m 1 ) x 1 + (n 2 + m 2 )x (n k + m k )x k

5 A free abelian group generated by the elements x 1, x 2,, x k consists of all elements of the form n 1 x 1 + n 2 x n k x k where n i are integers. Example: Z[x 1, x 2 ] 4x 1 + 2x 2 x 1-2x 2-3x 1 kx 1 + nx 2 Z = The set of integers = {, -2, -1, 0, 1, 2, }

6 A free abelian group generated by the elements x 1, x 2,, x k consists of all elements of the form n 1 x 1 + n 2 x n k x k where n i are integers. Example: Z[x, x ] 4 ix + 2 I k x 2 i -3 x + n iii Z = The set of integers = {, -2, -1, 0, 1, 2, }

7 A free abelian group generated by the elements x 1, x 2,, x k consists of all elements of the form n 1 x 1 + n 2 x n k x k where n i are integers. Example: Z[x 1, x 2 ] 4x 1 + 2x 2 x 1-2x 2-3x 1 kx 1 + nx 2 Z = The set of integers = {, -2, -1, 0, 1, 2, }

8 Addition: (n 1 x 1 + n 2 x n k x k ) + (m 1 x 1 + m 2 x m k x k ) = (n 1 + m 1 ) x 1 + (n 2 + m 2 )x (n k + m k )x k Example: Z[x 1, x 2 ] (4x 1 + 2x 2 ) + (3x 1 + x 2 ) = 7x 1 + 3x 2 (4x 1 + 2x 2 ) + (x 1-2x 2 ) = 5x 1

9 Addition: (n 1 x 1 + n 2 x n k x k ) + (m 1 x 1 + m 2 x m k x k ) = (n 1 + m 1 ) x 1 + (n 2 + m 2 )x (n k + m k )x k Example: Z[x, x 2 ] (4x + 2x 2 ) + (3 x + x 2 ) = 7 x + 3x 2 (4x 1 + 2x 2 ) + (x 1-2x 2 ) = 5x 1

10 Addition: (n 1 x 1 + n 2 x n k x k ) + (m 1 x 1 + m 2 x m k x k ) = (n 1 + m 1 ) x 1 + (n 2 + m 2 )x (n k + m k )x k Example: Z[x 1, x 2 ] (4x 1 + 2x 2 ) + (3x 1 + x 2 ) = 7x 1 + 3x 2 (4x 1 + 2x 2 ) + (x 1-2x 2 ) = 5x 1

11 Example: 4 vertices + 5 edges + 1 faces 4v + 5e + f. v = vertex e = edge f = face

12 Exampl: 4 vertices + 5 edges 4v + 5e v = vertex e = edge

13

14 + + + in Z[,,,, ] in Z[,,,,, ]

15 Note that + + is a cycle. Technical note: In graph theory, the cycle also includes vertices. I.e, this cycle in graph theory is the path,,,,,,,. Since we are interested in simplicial complexes (see later lecture), we only need the edges, so + + is a cycle.

16 Note that + + is a cycle. Note that + + is a cycle. Technical note: In graph theory, the cycle also includes vertices. I.e, the cycle in graph theory is the path,,,,,,,. Since we are interested in simplicial complexes (see later lecture), we only need the edges, so + + is a cycle.

17 Note that + + is a cycle. Note that + + is a cycle.

18 Note that + + is a cycle. Note that + + is a cycle. Objects: oriented edges e i

19 Note that + + is a cycle. Note that + + is a cycle. Objects: oriented edges in Z[,,,, ] e i

20 Note that + + is a cycle. Note that + + is a cycle. Objects: oriented edges in Z[,,,, ] e i e i

21 ( + + ) + ( + + ) = + + +

22 = =

23 The boundary of =

24 The boundary of = The boundary of = The boundary of = The boundary of + + = + + = 0

25 Add a face

26 Add an oriented face

27 Add an oriented face

28 Add an oriented face Note that the boundary of this face is the cycle + +

29 Simplicial complex

30 0-simplex = vertex = v 1-simplex = oriented edge = (v j, v k ) Note that the boundary v j v k e of this edge is v k v j 2-simplex = oriented face = (v i, v j, v k ) v 3 Note that the boundary of this face is the cycle + +

31 3-simplex = (,,, ) = tetrahedron 4-simplex = (,,,, v 5 )