Questions tagged [index-notation]
For questions about index notations, e.g. abstract index notation, Einstein summation convention, topics in introductive tensor calculus, Levi-Civita Symbol, Kronecker Delta symbol, proofs of vector calculus identities or fluid dynamics formulae using index notation.
482
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Summation with inner products: properties and rearrangement
OPTION 1.
I have this expression,
$$\sum \limits_{l=1}^{L}a_l^2 + \sum \limits_{l<j}^{L}a_l^2 \frac{a_j}{a_l}$$
and I would like to take out $$\sum \limits_{l=1}^{L}a_l^2$$, as to obtain something ...
- 131
0
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0
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41
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Tensor Index Manipulation
I am trying to study General Relativity and I thought about starting with some index gymnastics. I found a worksheet online and I am stuck with a simple problem.
I have to prove that $\partial_{\mu} g^...
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0
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24
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Checking an induction proof for a summation.
In my textbook practice problem, I want to prove the following by induction:
$\sum_{i=0}^{n}\sum_{k=0}^{n-i}a_{i, k} = \sum_{m=0}^{n}\sum_{k=0}^{m}a_{m-k, k}$
For my "n+1 implies n" ...
- 161
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1
answer
23
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Evaluation of an indexed sum. [closed]
I am going through a textbook solution to an induction problem and I'm not sure why it says this summation holds.
$\sum_{m=0}^0 \sum_{k=0}^m a_{m - k, k} = a_{0, 0}$
This is very simple I know, but ...
- 161
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1
answer
47
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What formula can I use to translate $m$ indices into $n$ indices?
NOTE: The tensors are considered row-major. The sequence of elements is preserved.
Let $A$ be a $m$-dimensional tensor of size $m_1 \times m_2 \times m_3 \cdots$ and $B$ be a $n$-dimensional tensor of ...
- 1,573
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1
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69
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Derivative of block matrix using einstein notation
Let $Y = [A \quad XB \quad C]$, where $A,B,C,X$ are all matrices with appropriate size. What is the derivative of $Y$ w.r.t. $X$?
The part that confuses me is that $\frac{\partial A}{\partial X}$ ...
- 1
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1
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14
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Indexing and $∪_{p∈P}C_p = (∪_{i∈I} A_i)\times (∪_{i∈I} B_i)$
The following is a problem question in How to Prove It (4.1.14b):
For each $(i, j) ∈ I × I$ let $C(i,,j) = A_i × B_j$, and let $P = I × I$.
Prove that $∪_{p∈P}C_p = (∪_{i∈I} A_i)\times (∪_{i∈I} B_i)$
...
- 53
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Confusion about Levi-Civita determinant identity
From the wiki page on the Levi-Civita symbol, they write the following equation
$$
\sum_{i_1,i_2,...}\epsilon_{i_1...i_n}a_{i_1j_1}...a_{i_n j_n}=\det(\mathbf{A})\epsilon_{j_1...j_n},
$$
which I ...
- 15
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1
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139
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What do the symbols $\delta_{ij}$ and $\epsilon_{ijk}$ mean?
I am reading David Tong's notes on vector calculus (which are amazing), and the symbols $\delta_{ij}$ and $\epsilon_{ijk}$ keep coming up in the notes. I do not know what these actually mean. I think ...
- 33
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1
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Question regarding the index notation: Interpretation of $\partial_n r_k$
I have recently been studying and using the index notation in physics, but I have a specific question, which is not very clear to me.
Say we have the radial vector $\textbf{r}$ and the usual Del-...
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1
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94
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Proof of this identity without using index notation
I want to see the following identity that is used in the proof of the first version of Bernoulli's theorem without using index notation
$(\mathbf{u} \cdot \mathbf{\nabla})\mathbf{u} = (\nabla \times \...
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0
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Notation ${}_{\mathcal B}[h]_{\mathcal A}$ or $[h]_{\mathcal A}^{\mathcal B}$ for the matrix of a linear map $h$
I've seen a few times the notation ${}_{\mathcal B}[h]_{\mathcal A}$ for the matrix of a linear map $h\colon E\to F$ with respect to a base $\mathcal A$ of $E$ and a base $\mathcal B$ of $F$.
I ...
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How to prove $\det{AB}=\det{A}\det{B}$ with Leibniz formula in terms of Levi-Civita symbol and Einstein summation notation? [closed]
Prove that $\det{AB}=\det{A}\det{B}$ with Leibniz formula in terms of Levi-Civita symbol and Einstein summation notation
Here is a similar question asked 6 years ago. The OP answered in the question ...
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Compact notation for multiple sums
Consider a multiple series of the form:
$$
\sum_{k_1=0}^\infty\cdots\sum_{k_n=0}^\infty f(k_1,\dots,k_n).
$$
I have a more complicated multiple series where the indicies are $k_{jn}$ with $j=1,\dots,J$...
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1
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38
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Possibility (generality) of symmetrization wrt certain indices
Consider the following contraction between two vector fields
$$
A_{k,i}B_{k,j}
$$
Summation over $k$ is implied. I want to decompose this into parts that are symmetric/antisymmetric w.r.t the ...
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1
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109
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Deriving the coordinate formula for the Gauss curvature under conformal map using the moving frames method
As an exercise, I wanted to derive this formula for the Gaussian curvature (with n=2) under a conformal map:
$$\tilde{K}=e^{-2\rho}K-e^{-2\rho}\Delta\rho$$
with the method of moving frames.
Given two ...
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73
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Equivalent formulas for Laplace-Beltrami operator
I am trying to derive the following equivalence:
$$
g^{jk}\frac{\partial^2 f}{\partial x^j \partial x^k} +
\frac{\partial g^{jk}}{\partial x^j}\frac{\partial f}{\partial x^k}+
\frac{1}{2}g^{jk}g^{il}\...
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0
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19
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Difference and meaning in the position of index of a quantity
I'm a beginner in this type of math, we are just starting to study it, but I need some clarifications about the meaning and the difference of when we write
$$\partial_i \qquad \text{and}\qquad \...
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2
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133
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Index Notation - An Inconsistency with a deeper meaning?
Background
Suppose we have a linear transformation $T : V \to W$ where $V$ and $W$ are finite dimensional vector spaces with bases $(e_i)$ and $(f_i)$ respectively. Using index notation, it is ...
- 5,567
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1
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135
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How to write expressions in tensor notation?
I’m a self-taught math student and i want to learn how to work and calculate with tensors.
I’m pretty familiar with the theory behind them, but something about the the different notations just does ...
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1
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34
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Index Notation Simplification Help
Given a real matrix $F$, I'm trying to simplify the expression
$$ C_{ijkl} = \frac{\partial (F_{ip}F_{jp})}{\partial F_{km}} F_{lm}$$
One reference I saw online says the right-hand side simplifies to:
...
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32
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Why Does the Trace of the following Double Derivative Expression Vanish?
I need some help completing the last part of this problem.
Problem:
For the position vector $r$ such that $r = x^2 + y^2 + z^2 = \sum_{i=1}x_i^2$, show that the trace of the following expression
$$\...
- 21
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vote
1
answer
41
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Identity tensor outer product representation
There is an equation (from Chapman-Enskog theory) that uses index notation:
$$\sum_i w_ie_{i\alpha}e_{i\beta}e_{i\gamma}e_{i\mu}=
c_s^4(\delta_{\alpha\beta}\delta_{\gamma\mu}+
\delta_{\alpha\gamma}\...
2
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0
answers
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Usage of dummy indices in a sum
A current textbook I'm reading has the following example of how to take derivatives of summations
$$\frac{d}{dt}\sqrt{\sum_jA_jB_j} = \frac{\sum_k\dot{A}_kB_k + A_k\dot{B}_k}{2\sqrt{\sum_jA_jB_j}}$$
...
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Showing that a function is a Fourier Multiplier (Umarov)
Hello. In example 1.8.2, it is shown that $|t^mp^{m}(t)|\leq C$ for all $t>0$ and finally, the function $m(\xi)$ satisfies the condition 1.68.
In my case, I am playing with the function $\rho(t)=t/(...
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Problem with Einsteinian notation
I'm studying the Einsteinian notation at the beginning of McConnell's tensor analysis. I'm puzzled by the following exercise:
Here $r, s$, are superscripts that, at the left side of the equation, ...
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In a rigorous setting, how do I do this derivative using Ricci Calculus?
Let $R = \sqrt{x_k x_k}$. I want to calculate ${\partial R \over \partial x_i}$ and ${\partial^2 R \over \partial x_i \partial x_j}$. The solution was given as such:
$$ {\partial R \over \partial x_j} ...
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2
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How to describe a multidimensional array with one index?
I apologize in advance for the lack of rigor, as I am not a professional mathematician.
Consider the following array of real numbers:
\begin{equation}
A_{i_{1},i_{2},\ldots,i_{k}}\in\mathbb{R}^{n_{1}}\...
1
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2
answers
82
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Algebraic proof for Vandermonde's identity + book recommendations for practising summation notation.
I am trying to digest the proof for Vandermonde's identity, but due to my lack of experience with manipulating sigma notation, I am unable to understand how they got the RHS in the first step (...
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50
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Líenard-Weichert potential satisfies the Lorenz gauge
It can be shown in 3-vector notation that the Líenard-Weichert potential satisfies the Lorenz gauge condition $\frac{\text{d}\phi}{\text{d}t}+\vec{\nabla}\vec{A}=0$; I am interested in showing this in ...
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Does $\langle x_i\rangle_{i\in I}$ refer to a function $x$ or to the image of $x$ ($x[I]$)?
I am confused with the notation and jargon for families.
Wikipedia (https://en.wikipedia.org/wiki/Indexed_family) says "indexed families and mathematical functions are technically identical ...&...
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0
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Conversion of fourth-order tensor multiplication from indices notation to matrix form
Considering a 2D fourth-order tensor $C_{IJKL}$ which can be represented in Voigt notation as:
$$
C_{IJKL} = \begin{bmatrix}C_{1111}&C_{1122}&C_{1112}\\ C_{2211}&C_{2222}&C_{2212}\\ C_{...
0
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0
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35
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notation for index of integers in some range
If I have an expression like
$$
\mu^i(x,t) = 0 \qquad, \qquad \qquad i=1,\dots, K
$$
I want a more compact way of specifying that the index $i$ takes integer values from $1$ to $K$. I'm looking for ...
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Simplifying the product of $(x-x_j)$'s when $x_k$ has multiplicty $\geq 2$
Let $w(x)$ be defined as such: $w(x) = \prod_{j=0}^{n}(x-x_j)$, where $x_j$'s are distinct real numbers, $j=0,1,...,n$.
Suppose there exists exactly one point $x_k$, $k \in {0,1,...,n}$ such that $x_k$...
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Difference between letters used as indices of arbitrary elements in a matrix/vector and letters used as indices of the last element in a matrix/vector
Let's say I want to define a vector $A$.
When I intend to use a particular letter (eg. letter $j$) to index an arbitrary element of vector $A$, can I use the same letter to index the last element of ...
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Simplifying a subindex equation
Consider the following equation
$$
\frac{e^{\sum_i \alpha_i x_i}}{\sum_i x_i}=\sum_k\frac{e^{\sum_i \alpha_i y_{i, k}}}{\sum_i y_{i, k}}
$$
Is it possible to write $x_i$ as a function of the terms $y_{...
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Simplification of multi-index notation of the dot product of a non-constant vector and the nabla operator $(a(\vec{x})\nabla )^\alpha f(\vec{x})=?$
Is there a known simplification for the multi-index notation of
$$
(\vec{v}\cdot \vec{A})^kf(\vec{x}) =\sum_{|\alpha|=k} \frac{k!}{\alpha!} (\vec{v} \circ \vec{A})^\alpha f(\vec{x}) = \sum_{|\alpha|=...
- 81
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2
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Interchanging summations with complicated, nested indices
I have a question regarding interchanging the order of three nested summations. My expression looks like
\begin{align}
\sum_{n=0}^\infty \sum_{k=0}^n \sum_{\nu=0}^{4n-2k}\frac{C_{nk\nu}}{k!(n-k)!}\...
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Product of a specific $(0,2)$ and $(2,0)$ tensor (Minkowski Metric tensor) [closed]
How to calculate
$$\eta_{\mu\nu}\eta^{\mu\nu}$$
Where
$$\eta=\begin{bmatrix} -1 \\ &1 \\&&1\\&&&1\end{bmatrix}$$
All other entries are $0$.
1
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1
answer
31
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Help with understanding indices for Navier Stokes Equation
I am using Kundu and Cohen's textbook on Fluid Mechanics. I am not a mechanical engineering major, but I am trying to understand the indices for a project that I am doing. This is what I have so far:
...
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0
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22
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Vector product notation for a prior density formula
Given a binary vector $\boldsymbol{\gamma}$ which is a $p$-dimensional vector and a matrix $\mathbf{G}$, which is a $p$x$p$ matrix, suppose we have probability distribution defined as follows :
$$p(\...
- 115
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1
answer
158
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Matrix manipulations with Levi-Civita symbol
My question relates to this reply on math.stackexchange.
More precisely, I am wondering about the following sequence of expressions involving elements of an invertible square matrix $M$ and a pair of ...
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1
vote
0
answers
43
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Understanding the relationship between the covariant and contravarient basis in Euclidean space using traditional matricies
For a contravariant basis vector in ${\Bbb R}^n$ that is defined using a covariant basis vector ${\bf Z}_j$ in ${\Bbb R}^n$ in terms of the contravarient metric tensor as:
$${\bf Z}^i = Z^{ij}{\bf Z}...
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1
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41
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How to prove this equality using Einstein summation (index notation)
I have to prove this equality using index notation (Einstein summation), but I don't know how to proceed from here:
$\nabla(A \cdot B)=A \times (\nabla \times B)+(A \cdot \nabla)B+B \times (\nabla \...
-1
votes
1
answer
57
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Could you help me with this problem on natural numbers?
Pick two natural numbers $q,n \in \mathbb{N}$ such that $1<q<n$ and for $0 \leq j\leq q-1$ define $a(j) = \lfloor \frac{n-j}{q}\rfloor$.
The claim is that the elements of the set
$\{rq+j \mid 0\...
2
votes
1
answer
64
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Is $\int_{-\infty}^\infty \sum_{g(x)=a}\frac{f(a)}{|g’(x)|} da=\left(\frac1{|g’(a_1)|}+\dots+\frac1{|g’(a_j)|}\right)\int_{-\infty}^\infty f(a)da$?
In
A strange integral: $\int_{-\infty}^{+\infty} {dx \over 1 + \left(x + \tan x\right)^2} = \pi.$
@robjohn posts that:
$$\int_{—\infty}^\infty f(g(x))dx=\int_{-\infty}^\infty \sum_{g(x)=a}\frac{f(a)...
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1
answer
76
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How would one differentiate Einstein summation and matrix entry notation?
In this video on Tensor calculus(timestamped), the professor explains how the matrix entry is written in a tensor-like notation:
$$\text{(i,j)}^{th}\text{ entry of matrix A}=A_{ij}=A_j^i$$
Later in ...
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1
vote
1
answer
55
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Question on matrix over matrix derivation
Consider recursive relations
$$\mathbf{H}_k=\sigma(\mathbf{Z}_k),\ \mathbf{Z}_k=\mathbf{A}\mathbf{H}_{k-1}\mathbf{W}_k$$
where $\mathbf{Z}_k\in\mathbb{R}^{m\times n_k}$, $\mathbf{A}\in\mathbb{R}^{m\...
- 461
1
vote
1
answer
41
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Notation for multi-sigma sum where index-values are never equal
I'm wondering if the following:
$$\sum_{i_1=1}^n \biggr(\sum_{i_2=1}^{i_1-1} f(i_2) + \sum_{i_2=i_1+1}^n f(i_2)\biggr)$$
Can be abbreviated to this:
$$\sum_{i_1=1}^n \sum_{i_2=1, i_2\ne i_1}^n f(i_2) \...
- 1,013
1
vote
1
answer
36
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Declaring a variable to be any element of a given set
Let $A \in \mathbb{R}^{m\times n}$. The element in the i:th row, j:th column is denoted as $A(i,j)$. I am writing an algorithm/a pseudocode where I want to declare a variable pair $(i, j)$ which ...
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