In algebraic topology, a k-chainis a formal linear combination of the k-cells in a cell complex. In simplicial complexes (respectively, cubical complexes), k-chains are combinations of k-simplices (respectively, k-cubes).[1][2][3] Chains are used in homology; the elements of a homology group are equivalence classes of chains.

Integration on chains[edit]

Integration is defined on chains by taking the linear combination of integrals over the simplices in the chain with coefficients (which are typically integers).The set of all k-chains forms a group and the sequence of these groups is called a chain complex.

Boundary operator on chains[edit]

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The boundary of a polygonal curve is a linear combination of its nodes; in this case, some linear combination of A1 through A6. Assuming the segments all are oriented left-to-right (in increasing order from Ak to Ak+1), the boundary is A6 − A1.

A chain is a serial assembly of connected pieces, called links, typically made of metal, with an overall character similar to that of a rope in that it is flexible and curved in compression but linear, rigid, and load-bearing in tension. A group of atoms, often of the same element, bound together in a line, branched line, or ring to form a molecule.♦ In a straight chain, each of the constituent atoms is attached to other single atoms, not to groups of atoms.♦ In a branched chain, side groups are attached to the chain.♦ In a closed chain, the atoms are arranged in the shape of a ring.

A closed polygonal curve, assuming consistent orientation, has null boundary.

The boundary of a chain is the linear combination of boundaries of the simplices in the chain. The boundary of a k-chain is a (k−1)-chain. Note that the boundary of a simplex is not a simplex, but a chain with coefficients 1 or −1 – thus chains are the closure of simplices under the boundary operator.

Example 1: The boundary of a path is the formal difference of its endpoints: it is a telescoping sum. To illustrate, if the 1-chain c=t1+t2+t3{displaystyle c=t_{1}+t_{2}+t_{3},} is a path from point v1{displaystyle v_{1},} to point v4{displaystyle v_{4},}, where t1=[v1,v2]{displaystyle t_{1}=[v_{1},v_{2}],},t2=[v2,v3]{displaystyle t_{2}=[v_{2},v_{3}],} andt3=[v3,v4]{displaystyle t_{3}=[v_{3},v_{4}],} are its constituent 1-simplices, then

1c=1(t1+t2+t3)=1(t1)+1(t2)+1(t3)=1([v1,v2])+1([v2,v3])+1([v3,v4])=([v2][v1])+([v3][v2])+([v4][v3])=[v4][v1].{displaystyle {begin{aligned}partial _{1}c&=partial _{1}(t_{1}+t_{2}+t_{3})&=partial _{1}(t_{1})+partial _{1}(t_{2})+partial _{1}(t_{3})&=partial _{1}([v_{1},v_{2}])+partial _{1}([v_{2},v_{3}])+partial _{1}([v_{3},v_{4}])&=([v_{2}]-[v_{1}])+([v_{3}]-[v_{2}])+([v_{4}]-[v_{3}])&=[v_{4}]-[v_{1}].end{aligned}}}

Example 2: The boundary of the triangle is a formal sum of its edges with signs arranged to make the traversal of the boundary counterclockwise.

A chain is called a cycle when its boundary is zero. A chain that is the boundary of another chain is called a boundary. Boundaries are cycles,so chains form a chain complex, whose homology groups (cycles modulo boundaries) are called simplicial homology groups.

Example 3: A 0-cycle is a linear combination of points such that the sum of all the coefficients is 0. Public beta mac os. Thus, the 0-homology group measures the number of path connected components of the space.

Example 4: The plane punctured at the origin has nontrivial 1-homology group since the unit circle is a cycle, but not a boundary.

In differential geometry, the duality between the boundary operator on chains and the exterior derivative is expressed by the general Stokes' theorem.


  1. ^Hatcher, Allen (2002). Algebraic Topology. Cambridge University Press. ISBN0-521-79540-0.
  2. ^1950-, Lee, John M. (2011). Introduction to topological manifolds (2nd ed.). New York: Springer. ISBN978-1441979391. OCLC697506452.CS1 maint: numeric names: authors list (link)
  3. ^Tomasz, Kaczynski (2004). Computational homology. Mischaikow, Konstantin Michael, Mrozek, Marian. New York: Springer. ISBN9780387215976. OCLC55897585.
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Helping Dogs Become a Part of the Family

Your dogs wants to be with you.

Food Chain

You, the human family, have become your dog’s pack. When not with you, the dog suffers mentally and physically, shut out of the pack and not knowing why.

Chain link fence

If dogs don’t learn how to live with humans, they are more likely to bite.

Your pet is 2.8 times more likely to bite when chained or penned especially is an un-neutered male or a mother with pups. There were 304 children in the US killed or seriously injured by dogs living chained or penned between October 2003 and April 2010.

Dogs are not meant to fight.

Chain Reaction

If you got your dogs for fighting, the animal is feeling abused. The dog does not want to breed more dogs to die on chains, in shelters, or in a dog fight.

They don’t want to live chained, penned, shut away in a basement or garage. They want to be a part of your family.

Change a dog’s life!

Bring the pet into the home and family. Fence the yard. Take him for walks, it will be a great exercise for both of you. Play with him, take him to training classes, make the dog a part of the pack!

Riding in a car; Scared

Chain Minecraft

UnchainOK is here to help. Do you need assistance with food, shelter, or fencing? Our volunteers will work to assist you in providing a happier home for you pet. Do you know of a chained dog that needs assistance? Please contact us.

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Unchain OK mission: Unchain OK is dedicated to helping dogs on chains through education and providing animals in need with food, water, and shelter from the elements. We are working towards legislation to prevent chaining in the state of Oklahoma.


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