I"m spring at some formulas including matrices (in the context of machine learning, but I"m not certain it"s relevant) and also I came throughout \$odot\$. What might this mean? The context is \$M odot N\$, whereby \$M\$ is a matrix and also \$N\$ can be a vector, or a matrix, or a scalar, it"s a bit thick so it"s difficult to tell. I have reason to believe it may be the Hadamard product, is there anything rather it might mean? In the LSTM equations, the circled dot operator is typically used to stand for element-wise multiplication.

You are watching: Circle with dot in middle math When i researched around the prize ⊙. I acquired two things:

\$1\$. In the publication Meaning, Logic and Ludics-By Alain Lecomte, The writer says that: let \$U\$ and \$B\$ be two confident designs we signify tensor product that \$U\$ and \$B\$ by \$U\$⊙\$B\$.

\$2\$. In the book Dag Prawitz top top Proofs and Meaning: The writer says that \$alpha⊙eta\$ means that one of two people \$eta \$ is derivable indigenous \$alpha\$ or is \$alpha\$ derivable from \$eta \$.

I think that second on e renders sense, as for tensor product ns have constantly seen the prize ⊗ gift used. I provided you the info which ns had, expect it helps.

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answered Dec 2 "16 in ~ 17:01 Vidyanshu MishraVidyanshu Mishra
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\$egingroup\$ This is the correct answer! \$endgroup\$
–user575518
Nov 7 "19 in ~ 10:43
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We use the circle with a dot notation in nonlinear continuum mechanics. The is identified in table of contents notation aseginalignleft(oldsymbolM ⊙ oldsymbolN ight)_ABCD= dfrac12left(M_ACN_BD+M_ADN_BC ight)endalignwhere \$oldsymbolM\$ and \$oldsymbolN\$ are second order tensors. The result of this procedure is a 4th order tensor. You can think of the as concerned the outer product of 2 matrices or \$oldsymbolMotimes oldsymbolN\$ in the it take away two 2nd order tensors and generates a fourth order tensor. Remember the external product is identified for two 2nd order tensors as

eginalignleft(oldsymbolM otimes oldsymbolN ight)_ABCD= M_ABN_CDendalign

The circle with a dot procedure \$⊙\$ occurs once calculating the elastic modulus \$lungemine.combfunderlineC\$ (a fourth order tensor) from constitutive laws. Examples encompass the Mooney-Rivlin or Neohookean product models. To acquire the elastic modulus you need to take derivatives of tensors v respect to various other tensors. Because that instance,

eginalignunderlinelungemine.combfC = 2dfracdoldsymbolSdoldsymbolCendalignor in index notation aseginalign extC_ABCD = 2dfracpartial S_ABpartial C_CDendalignwhere \$oldsymbolC\$ is the left Cauchy-Green deformation tensor and \$oldsymbolS\$ is the 2nd Piola-Kirchhoff anxiety tensor. Based on the constitutive models, i m sorry relate stress in a product to the strain, it turns out \$oldsymbolS\$ is a duty of the train station of \$oldsymbolC\$. If you occupational it the end you will discover that girlfriend will need to compute \$fracpartial oldsymbolC^-1partial oldsymbolC\$. The proof is a headache yet the result comes the end toeginaligndfracpartial oldsymbolC^-1partial oldsymbolC = -oldsymbolC^-1 ⊙ oldsymbolC^-1endalignIn index notationeginaligndfracpartial C^-1_ABpartial C_CD = -dfrac12left(C^-1_ACC^-1_BD+C^-1_ADC^-1_BC ight)=-left(oldsymbolC^-1 ⊙ oldsymbolC^-1 ight)_ABCDendalignThe circle with a dot operation only arises due to the fact that \$oldsymbolC\$ is a symmetric matrix, i.e., \$oldsymbolC=oldsymbolC^T\$ and \$oldsymbolC_sym=dfrac12left(oldsymbolC+oldsymbolC^T ight) = oldsymbolC\$. Keep in mind that if acquisition the derivative that an inverse of a nonsymmetric tensor with respect to itself yieldseginaligndfracpartial A_AB^-1partial A_CD=-A^-1_ACA^-1_DBendalign and also this is no the external product. This operation has not however been provided a symbol.

Note:

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

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

In physics, I have actually seen it typical a point resource such together a suggest charge or gravity resource like a planet.In physics, I have actually seen it average the vector point out out that the page \$⊙\$. And also \$otimes\$ method the direction the the vector is into the page. I have actually seen this in E&M because that B-fields and E-fields and mechanics for torques.In lungemine.com it might mean a role composition operator, i m sorry maps functions to functions, e.g., \$,f⊙g\$.

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This is what ns think it is provided for in that long Short-Term storage neural networking great https://www.youtube.com/watch?v=iX5V1WpxxkY&feature=youtu.be at 46:00.

In lungemine.comematics, practical compositions room usually denoted by a small circle \$circ\$. Because that example, Eulerian \$f(oldsymbolx,t)\$ and Lagrangian \$F(oldsymbolX,t)\$ explanation are regarded each various other by a duty composition:eginalignF(oldsymbolX,t)=f(oldsymbolPhi(oldsymbolX,t),t) extrm or F = f circ oldsymbolPhiendalign