I read this in a paper:
$ \lambda_m = \mbox{const} \quad \mbox{for all} \quad m \in \left\{1,2,\cdots,M\right\} $
Does this mean that all $\lambda_m$ are the same, or that they're all constant functions, but with different constants?
The context:
I read this in a paper:
$ \lambda_m = \mbox{const} \quad \mbox{for all} \quad m \in \left\{1,2,\cdots,M\right\} $
Does this mean that all $\lambda_m$ are the same, or that they're all constant functions, but with different constants?
The context:
I agree with Yuval Filmus's comment both in the abstract and in the specific. That is, generally we would write something like $ \lambda_n = \lambda_m \qquad \forall n,m\in\mathbb{N} $ or $ \lambda_1 = \lambda_2 = \cdots = \lambda_n = \cdots $ or $ \lambda_n = C \qquad \forall n \in\mathbb{N}$ if we want it to mean that all of the $\lambda_n$ are equal.
This is also the case, I believe, in the context. Note that the author refers to functions $e_m(k) = e(k - \lambda_m)$ and states that
i.e., all $e_m(k)$ have identical shape and can differ only by a time-shift (latency) $\lambda_m$.
If all $\lambda_m$ were to be the same constant, there's hardly any point in defining $e_m$'s as different functions! Hence it is more natural to interpret the statement as requiring that $\lambda_m$ being independent of $k$, with the possibility that $\lambda_n\neq \lambda_m$.