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I have the expression

$y_i = \displaystyle\sum_j x_j \!\cdot\!a_{ij} \!\cdot\! \exp \Big(\sum_k b_{ijk} \!\cdot\! x_k \Big)$

which I want to shorten without introducing more notation than necessary. For example, $\displaystyle\,y_i = \sum_j a_{ij}\! \cdot\! x_j\,$ can be written as $\,\vec{y} = A \,\vec{x}\,$ without any further explanation.

So what about this expression for the first sum above,

$\vec{y} = \Big(A \!\circ\! \exp\big(\, \underline{B} \!\cdot \!\vec{x} \,\big) \Big) \!\cdot\! \vec{x}$

where $A$ is the matrix with entries $a_{ij}$ and $B$ is the 3rd order tensor-like thing with entries $b_{ijk}$? Of course, $\exp$ is taken element-wise.

I am saying tensor-like because I am not sure which notation I would have to use when I were to use tensor index notation – for example, $b^{ij}_k$ or $b^i_{jk}$. All beginner articles about tensor notation seem to use the concept of a changing basis to determine which index goes super- or subscript; however, I only have one single basis, so proceeding like this does not really help me.

Update: I think my main question is this: If $\,\underline{B}\,$ is a 3rd order tensor and $\,x\,$ a vector, is it obvious (or usually assumed) that $\,\displaystyle\underline{B} \!\cdot\!\vec{x} = \sum_k b_{ijk}\!\cdot\!x_k $? Or could this be understood as $\,\sum_j b_{ijk}\!\cdot\!x_j\,$ or even $\,\displaystyle\sum_i b_{ijk}\!\cdot\!x_i$?

Another update: Similar to the above: What about $\,\vec{x}^{\,T} \!\!\!\cdot\! \underline{B}$? I understand that as $\,\displaystyle\sum_i x_i \!\cdot\!b_{ijk}$, but is this obvious?

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  • $\begingroup$ I am curious where your desired expression comes from. Tensor notation is not really intended to handle element-wise operations (aside from addition and scalar multiplication, of course) and usually the exponential of a second-order tensor is defined like the matrix exponential. $\endgroup$
    – Matt Dickau
    Oct 7, 2015 at 20:22
  • $\begingroup$ The exponential comes from the Beer-Lambert law; the expression describes a physical model of a photon-propagation process. $\endgroup$
    – bers
    Oct 7, 2015 at 20:23
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    $\begingroup$ In a slightly different direction, the Einstein summation convention would be helpful here. As @MattDickau said, it's really only useful for 'flat' expressions, but it's helpful (and generally cleaner than the vector-and-dot-product approach). $\endgroup$
    – anomaly
    Jan 27, 2016 at 15:55
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    $\begingroup$ The problem with that notation is, how to you pronounce the difference between $\exp(\sum_j a_{ij} x_j)$ and $\sum_j \exp(a_{ij} x_j)$? Both will be $\exp(a_{ij} x_j)$, I guess. $\endgroup$
    – bers
    Jan 27, 2016 at 15:59
  • $\begingroup$ This is one of the reasons why Einstein summation convention exists, and why past 2-D matrices, you usually write all the sub/superscripts. There are just too many different ways to multiply tensors together to have a different symbol for each case. With Einstein summation, you could write this as $x_i \cdot a_{ij} \cdot exp(b_i^{jk} \cdot x_k)$, which somewhat works. $\endgroup$
    – weux082690
    Feb 3, 2016 at 4:39

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I don't think there is any standard tensor notation convention for expressions like yours. Both your "tensor-like" proposal and the version suggested by weux082690 have their merits. In my opinion, which version to prefer depends on how often you are going to use expressions of this kind in your paper (or other document), and so how much effort you are prepared to invest in explaning your notation.

If you only use the expression a few times, you may want a short and concise explanation. Then you can, like in weux082690's suggestion, apply Einstein's summation convention, using upper and lower indices to specify which repeated indices are to be summed over (only those which appear as both upper and lower ones are), and with the additional rule that summation is carried out as soon as bothof these indices have been encountered in the evaluation order of the arithmetic operations.

On the other hand, your own "tensor-like" notation looks very clean and intuitive, but it requires a bit more explanation in your text. This task may be facilitated by the notation being "Matlab style", in the sense that functions on arrays are, by default, evaluated element-wise and that there is an element-wise multiplication of arrays, indicated by a dot "." in Matlab and in your notation by "$\circ$". So it will appear familiar to most readers.

Another aspect to take into consideration is whether you want to perfom calculations in the compact notation, or just state results. If you want to do calculations in it that will be intelligible to the reader, you should insure that the notation obeys simple and clear transformation rules between equivalent expressions.

Note added: Concerning your updates, I guess you just have to tell the reader in the text how you want him or her to understand it!

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