Homomorphism of an elementwise sum and dot product

In summary, a homomorphism is a function that preserves the operations and properties of two algebraic systems it maps between. An elementwise sum is a mathematical operation that adds corresponding elements of two vectors or matrices to create a new one. A dot product is a mathematical operation that calculates the sum of products of corresponding elements in two vectors. These two operations are related through a homomorphism, meaning that the sum of two operations is equal to the operation on the sum. Homomorphism is significant in this context as it simplifies calculations and helps us understand the relationship between different mathematical operations.
  • #1
NotASmurf
150
2
∑ab is needed but is impractical to implement.

Specifically ∑i ai.10i-|i| in any form where I can work with ∑i ai = α and ∑i 10i-|i| separately.

Is there a homomorphic function I can run it through such that ∑ab can be expressed as ∑a∑b? Note: for current problem i cannot simply set it up such that ∑a∑b - ∑ab polynomial expansion saves the day, any help appreciated.
 
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  • #2
No. A simple counterexample: Let all ai=1 and all bi=1 Then Σab = Σa = Σb.
 

Related to Homomorphism of an elementwise sum and dot product

1. What is a homomorphism?

A homomorphism is a mathematical concept that describes a structure-preserving map between two algebraic systems. In other words, it is a function that preserves the operations and properties of the systems it maps between.

2. What is an elementwise sum?

An elementwise sum is a mathematical operation that takes two vectors or matrices of the same size and adds the corresponding elements together to create a new vector or matrix of the same size.

3. What is a dot product?

A dot product is a mathematical operation that takes two vectors of the same size and calculates the sum of the products of their corresponding elements. It is also known as an inner product.

4. How are the elementwise sum and dot product related?

The elementwise sum and dot product are related through a homomorphism. This means that if we apply a dot product to the elementwise sum of two vectors, it is equivalent to the elementwise sum of the dot products of those vectors. In other words, the homomorphism states that the sum of two operations is equal to the operation on the sum.

5. What is the significance of homomorphism in this context?

Homomorphism is significant in this context because it allows us to simplify calculations by breaking them down into smaller, more manageable operations. It also helps us understand the relationship between different mathematical operations and how they can be applied to different types of data.

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