What does a dot in a circle mean?

Solution 1:

In the LSTM equations, the circled dot operator is typically used to represent element-wise multiplication.

Solution 2:

When I researched about the symbol ⊙. I got two things:

$1$. In the book Meaning, Logic and Ludics-By Alain Lecomte, The writer says that: Let $U$ and $B$ be two positive designs we denote tensor product of $U$ and $B$ by $U$⊙$B$.

$2$. In the book Dag Prawitz on Proofs and Meaning: The writer says that $\alpha⊙\beta$ means that either $\beta $ is derivable from $\alpha$ or is $\alpha$ derivable from $\beta $.

I think that second on e makes sense, as for tensor product I have always seen the symbol ⊗ being used. I provided you the info which I had, hope it helps.

Solution 3:

We use the circle with a dot notation in nonlinear continuum mechanics. It is defined in index notation as \begin{align} \left(\boldsymbol{M} ⊙ \boldsymbol{N}\right)_{ABCD}= \dfrac{1}{2}\left(M_{AC}N_{BD}+M_{AD}N_{BC}\right) \end{align} where $\boldsymbol{M}$ and $\boldsymbol{N}$ are second order tensors. The result of this operation is a 4th order tensor. You can think of it as related to the outer product of two matrices or $\boldsymbol{M}\otimes \boldsymbol{N}$ in that it takes two 2nd order tensors and generates a 4th order tensor. Remember the outer product is defined for two second order tensors as

\begin{align} \left(\boldsymbol{M} \otimes \boldsymbol{N}\right)_{ABCD}= M_{AB}N_{CD} \end{align}

The circle with a dot operation $⊙$ occurs when calculating the elastic modulus $\mathbf{\underline{C}}$ (a 4th order tensor) from constitutive laws. Examples include the Mooney-Rivlin or Neohookean material models. To get the elastic modulus you must take derivatives of tensors with respect to other tensors. For instance,

\begin{align} \underline{\mathbf{C}} = 2\dfrac{d\boldsymbol{S}}{d\boldsymbol{C}} \end{align} or in index notation as \begin{align} \text{C}_{ABCD} = 2\dfrac{\partial S_{AB}}{\partial C_{CD}} \end{align} where $\boldsymbol{C}$ is the left Cauchy-Green deformation tensor and $\boldsymbol{S}$ is the 2nd Piola-Kirchhoff stress tensor. Based on the constitutive models, which relate stress in a material to the strain, it turns out $\boldsymbol{S}$ is a function of the inverse of $\boldsymbol{C}$. If you work it out you will find that you will have to compute $\frac{\partial \boldsymbol{C}^{-1}}{\partial \boldsymbol{C}}$. The proof is a headache but the result comes out to \begin{align} \dfrac{\partial \boldsymbol{C}^{-1}}{\partial \boldsymbol{C}} = -\boldsymbol{C}^{-1} ⊙ \boldsymbol{C}^{-1} \end{align} In index notation \begin{align} \dfrac{\partial C^{-1}_{AB}}{\partial C_{CD}} = -\dfrac{1}{2}\left(C^{-1}_{AC}C^{-1}_{BD}+C^{-1}_{AD}C^{-1}_{BC}\right)=-\left(\boldsymbol{C}^{-1} ⊙ \boldsymbol{C}^{-1}\right)_{ABCD} \end{align} The circle with a dot operation only arises because $\boldsymbol{C}$ is a symmetric matrix, i.e., $\boldsymbol{C}=\boldsymbol{C}^{T}$ and $\boldsymbol{C}_{sym}=\dfrac{1}{2}\left(\boldsymbol{C}+\boldsymbol{C}^{T}\right) = \boldsymbol{C}$. Note that if taking the derivative of an inverse of a nonsymmetric tensor with respect to itself yields \begin{align} \dfrac{\partial A_{AB}^{-1}}{\partial A_{CD}}=-A^{-1}_{AC}A^{-1}_{DB} \end{align} and this is not the outer product. This operation has not yet been given a symbol.

Note:

• the outer product $\otimes$ is also called the tensor product.
• The indices are capital letters ABCD since in continuum mechanics capital letters denote Lagrangian/material (or reference configuration) coordinates. Lower case indices ijkl denote spatial/Eulerian coordinates.

Addendum: Other uses I have seen for $⊙$ include

1. In physics, I have seen it mean a point source such as a point charge or gravity source like a planet.
2. In physics, I have seen it mean the vector points out of the page $⊙$. And $\otimes$ means the direction of the vector is into the page. I have seen this in E&M for B-fields and E-fields and mechanics for torques.
3. In mathematics it could mean a function composition operator, which maps functions to functions, e.g., $\,f⊙g$.

This is what I think it is used for in that Long Short-Term Memory neural networking lecture https://www.youtube.com/watch?v=iX5V1WpxxkY&feature=youtu.be at 46:00.

In mathematics, functional compositions are usually denoted by a small circle $\circ$. For example, Eulerian $f(\boldsymbol{x},t)$ and Lagrangian $F(\boldsymbol{X},t)$ descriptions are related to each other by a function composition: \begin{align} F(\boldsymbol{X},t)=f(\boldsymbol{\Phi}(\boldsymbol{X},t),t) \textrm{ or } F = f \circ \boldsymbol{\Phi} \end{align}