Defining Direct Products in Exponents

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In summary, the conversation discusses the definition of the direct product when it is in the exponent of some variable. One speaker suggests it is defined via a Taylor expansion or an eigenfunction expansion, while the other speaker asks for clarification on the notation, specifically referencing G^{\otimes} as a possible notation. The conversation also mentions the notation \overset{k}{\otimes}V and Equation 14 in relation to the tensor product of N copies of rho.
  • #1
Nusc
760
2
When the direct product is in the exponent of some variable, how is it defined?
 
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  • #2
I would say that it's defined via the Taylor expansion. Can you give the explicit expression?
 
  • #3
Like O(3) ^ direct product blah
 
  • #4
You mean something like

[itex]

G^{\otimes}

[/itex]
? I never saw such a thing, but I would then guess it's a notation for

[itex]
G \otimes G \otimes G \otimes \ldots \otimes G
[/itex]

Does that make sense in your context? Otherwise you should give the exact expression in LaTeX :)
 
  • #5
t is defined either by a Taylor expansion or by an eigenfunction expansion.
 
  • #6
Nusc said:
When the direct product is in the exponent of some variable, how is it defined?

There seems to be confusion in this thread (at least for me).

Please write down clearly, completely, and precisely what you mean, or give a reference to a text or paper which uses the notation that you want want clarified.
 
  • #7
haushofer said:
You mean something like

[itex]

G^{\otimes}

[/itex]

Something like that. What does it mean?
 
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  • #8
Nusc said:
Something like that. What does it mean?

Do you mean "something like" or "exactly like"? You have to be precise.

Do you mean

[tex]\overset{k}{\otimes}V?[/tex]

This is standard notation for

[tex]V \otimes V \otimes \ldots \otimes V[/tex]

with [itex]V[/itex] repeated [itex]k[/itex] times.
 
  • #10
It's the tensor product of N copies of rho.
 
  • #11
thanks
 

1. What is a direct product in exponents?

A direct product in exponents is a mathematical operation that combines two or more numbers or variables using multiplication. It is represented by the symbol "x" and is read as "times" or "multiplied by". For example, 2x3 is a direct product of 2 and 3.

2. How do you define a direct product in exponents?

A direct product in exponents is defined as the product of multiplying the base of the exponents together and adding the exponents. For example, (2^3)(2^4) can be written as (2x2x2)(2x2x2x2) and simplified to 2^(3+4) = 2^7. This is also known as the "Product Rule" for exponents.

3. What is the difference between a direct product and an exponentiation?

A direct product is the result of multiplying two or more numbers or variables together, while exponentiation is the result of raising a number to a power. In other words, a direct product is a way of writing repeated multiplication, while exponentiation is a way of writing repeated multiplication by the same number.

4. How is a direct product used in real-life applications?

Direct products in exponents are commonly used in scientific and mathematical calculations, such as calculating compound interest, growth rates, and population growth. They are also used in physics and engineering to represent quantities with both size and direction, such as velocity and force.

5. What are some common mistakes when using direct products in exponents?

One common mistake is confusing the order of operations in a direct product, as multiplication is typically done before addition. Another mistake is not simplifying the direct product and leaving it in its expanded form, which can make calculations more complicated than necessary. It is also important to pay attention to negative exponents, as they can change the outcome of a direct product.

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