The discussion revolves around the meaning and implications of the colon notation, :, particularly in the context of integrals and weak solutions in mathematics. Participants explore its application in various mathematical constructs, including inner products and dyadics.
Participants express differing views on the interpretation of the colon notation and its application, indicating that multiple competing perspectives remain unresolved.
The discussion highlights ambiguities in definitions and the context of the notation, particularly regarding the nature of the variables involved and the operations being performed.
Can you be a bit more precise and descriptive?Integrals said:What does the colon notation, :, mean when it is inside an integral for weak solutions?
Apologies for the confusion, is it correct to say that $$\mathbf{x}\otimes\mathbf{x}:y$$ is equivalent notation for $$\langle\mathbf{x}\otimes\mathbf{x}, y\rangle$$ where the latter denotes the inner product. Perhaps, this question was more elementary than I had meant...fresh_42 said:Can you be a bit more precise and descriptive?
Locally integrable functions - where $$\mathbf{x}=(x_1({\cdot}), x_2({\cdot}), x_3(\cdot))$$fresh_42 said:I haven't seen the first one, and the second one calls for the question, what are you multiplying there?
If I build the ##\mathbf{x} \otimes \mathbf{x}## from this, I get a rank one ##3\times 3## matrix. How can this be multiplied by an ##y## via an inner product, wherever this is from?Integrals said:Locally integrable functions - where $$\mathbf{x}=(x_1({\cdot}), x_2({\cdot}), x_3(\cdot))$$