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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:

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Without context, this is impossible to answer. – Phira May 30 '12 at 15:03
It's from Some Improvements in the Measurement of Variable Latency Acoustically Evoked Potentials in Human EEG. I'll supply some more context. – Johan May 30 '12 at 15:10
I'm not allowed to post images, so could someone edit this into the question? – Johan May 30 '12 at 15:14
Usually this notation means that each $\lambda_m$ is constant, but the constants can be different. Otherwise the constant should be given a name, e.g. $\lambda_m = C$ where $C$ is constant, or they could just write $\lambda_1 = \cdots = \lambda_m = \text{const}$. – Yuval Filmus May 30 '12 at 15:22
@Mark: even so it is worth saving a click – Henry May 30 '12 at 17:35
up vote 5 down vote accepted

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 $\lambda_n\neq \lambda_m$.

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Congratulations on hitting 20k. Finally you can be trusted. – Asaf Karagila Jun 1 '12 at 7:24

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