Partitions of Unity: Exploring Their Meaning

[Terilien]

what are they exactly?

 

[mathwonk]

smooth functions which are each zero off a small interval, but which add up to the constant function 1 on the whole line or whole mANIFOLD. hence they "partition" the constant function 1, i.e. unity.

they aRE USED TO PATCH TOGETHER THIngS WHICH ARE only CONSTRUCTED LOCALLY.

i.e. given another function f which we want to integrate over a whole manifold M, WE COVER M by small nbhds and take a pof1 subordinate to tht cover. then multiplying f by one of our pof1 functions makes the product non zero only in SMLL NBHD AND WE INTEGRATE THERE USING LOCAL COORDINATES.

doing this over all nbhds we then add the results.in affine algebraic geometry, a similar technique is uised when we haVE GENERATORS f1,...fm FOR the unit IDEAL R. i.e. this emans there exist multipliers g1,...,gm such that the sum of the products figi equals 1.

then we can make a local construction using the gi, and modify it with the fi to get a global construction.

this technique for example can be used to prove the first cohomology of the structure sheaf O on an affine variety is zero.

 

[tacman]

Partitions of Unity refer to a mathematical concept that is commonly used in the fields of analysis and topology. They were first introduced by the mathematician Augustin-Louis Cauchy in the 19th century.

In simple terms, partitions of unity are a way of breaking down a larger mathematical object into smaller, more manageable pieces. These smaller pieces, or "subsets", are then combined in a way that allows us to reconstruct the original object.

More specifically, a partition of unity is a collection of functions that satisfy certain properties. These functions are typically referred to as "bump functions" because they are non-zero only on a specific subset of the original object.

The key idea behind partitions of unity is that they allow us to "patch together" different functions or sets in a smooth and continuous manner. This is particularly useful in situations where we need to work with functions or sets that are defined on different domains.

For example, in topology, partitions of unity can be used to construct a global function from local functions defined on different open sets. This allows us to study the properties of a function on a larger scale, without having to work with complicated functions defined on the entire space.

In summary, partitions of unity are a powerful tool in mathematics that allow us to break down and reconstruct complex objects in a smooth and continuous manner. They have numerous applications in various fields of mathematics and are an important concept to understand for anyone studying advanced mathematics.