Operators

Part 2 of 8 · Previous: Basics · Index · Next: Functions

7. Sums and products

Summation

$$\sum _{i=1}^k{x}_i$$

means:

$$x_1+x_2+\cdots +x_k$$

The bottom gives the initial index:

$$i=1$$

The top gives the final index:

$$k$$

Example:

$$\sum _{i=1}^3{x}_i=x_1+x_2+x_3$$

Constraint example:

$$\sum _{i=1}^k{x}_i=N$$

Read:

The ants assigned to all $k$ paths add up to $N$.

Product

$$\prod _{i=1}^k{x}_i$$

means:

$$x_1{x}_2\cdots x_k$$

Example:

$$\prod _{i=1}^{4}i=1\cdot 2\cdot 3\cdot 4=24$$

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8. Minimum and maximum

Minimum

$$\min (a,b,c)$$

returns the smallest value.

$$\min (4,9,2)=2$$

Maximum

$$\max (a,b,c)$$

returns the largest value.

$$\max (4,9,2)=9$$

Minimum over a domain

$$\underset{x\in S}{\min }f(x)$$

Read:

The smallest value of $f(x)$ among all valid $x\in S$.

Example:

$$\underset{x\in \mathbb{N},x\ge 5}{\min }x=5$$

Constraints under an operator

$$\underset{\begin{array}{c}{x}_i\in {\mathbb{N}}_0\\ {\sum }_i{x}_i=N\end{array}}{\min }F(x_1,\dots ,x_k)$$

Read:

Minimize $F$, considering only nonnegative integer values $x_i$ whose sum is $N$.

Important

The material below $\min$ is not being minimized. It defines the allowed choices.

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9. min versus argmin

These are different.

Minimum value

$$\underset{x\in S}{\min }f(x)$$

returns the smallest value of $f$.

Argument attaining the minimum

$${\mathrm{argmin}}_{x\in S}f(x)$$

returns the value of $x$ that produces the minimum.

Example:

$$f(x)=(x-3{)}^2$$

Then:

$$\underset{x\in \mathbb{R}}{\min }f(x)=0$$

while:

$${\mathrm{argmin}}_{x\in \mathbb{R}}f(x)=3$$

So:

• min answers: What is the best score?
• argmin answers: Which solution gives the best score?

Example:

$$T^{\ast }=\underset{\mathcal{P}}{\min }T(\mathcal{P})$$

is the optimal number of turns.

$${\mathcal{P}}^{\ast }={\mathrm{argmin}}_{\mathcal{P}}T(\mathcal{P})$$

is the best path family.

The star often means optimal.

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10. Ceiling and floor

Symbol	Name	Meaning
$\lfloor x\rfloor$	Floor	Greatest integer not larger than $x$
$\lceil x\rceil$	Ceiling	Smallest integer not smaller than $x$

Examples:

$$\lfloor 4.8\rfloor =4$$ $$\lceil 4.2\rceil =5$$

In scheduling, ceiling appears when a fractional result must become a whole number of turns:

$$T=\lceil \frac{N+D_k-k}{k}\rceil$$

If the fraction is $5.2$, the solution needs 6 turns:

$$\lceil 5.2\rceil =6$$

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11. Absolute value, cardinality, and norm

The same vertical bars can mean different things.

Absolute value

For a number:

$$|x|$$

means its distance from zero.

$$|-5|=5$$

Set cardinality

For a set:

$$|S|$$

means the number of elements in $S$.

If:

$$S=\{a,b,c\}$$

then:

$$|S|=3$$

Vector norm

For a vector:

$$\Vert v\Vert$$

means its length or magnitude.

For a two-dimensional Euclidean vector:

$$v=(x,y)$$ $$\Vert v{\Vert }_2=\sqrt{x^2+y^2}$$

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12. Parentheses and brackets

Notation	Common purpose
$(a+b)$	Grouping
$f(x)$	Function application
$[a,b]$	Closed interval
$(a,b)$	Open interval
$[a,b)$	Includes $a$, excludes $b$
$\{a,b\}$	Set
$⟨u,v⟩$	Tuple or inner product

Intervals

$$x\in [2,5]$$

means:

$$2\le x\le 5$$ $$x\in (2,5)$$

means:

$$2<x<5$$ $$x\in [2,5)$$

means:

$$2\le x<5$$

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