# Symbols:S

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### Second

$\mathrm s$

The symbol for the second is $\mathrm s$.

Its $\LaTeX$ code is \mathrm s .

### Set

$S$

Used to denote a general set.

The $\LaTeX$ code for $S$ is S .

### Algebraic Structure

$S$

Used to denote a general algebraic structure, in particular a semigroup.

In this context, frequently seen in the compound symbol $\struct {S, \circ}$ where $\circ$ represents an arbitrary binary operation.

The $\LaTeX$ code for $\struct {S, \circ}$ is \struct {S, \circ} .

### South

$\mathrm S$

South is the direction on (or near) Earth's surface along the meridian directly towards the South Pole.

The $\LaTeX$ code for $\mathrm S$ is \mathrm S .

### Southeast

$\mathrm {SE}$

Southeast is the direction on (or near) Earth's surface halfway between south and east.

The $\LaTeX$ code for $\mathrm {SE}$ is \mathrm {SE} .

### Southwest

$\mathrm {SW}$

Southwest is the direction on (or near) Earth's surface halfway between south and west.

The $\LaTeX$ code for $\mathrm {SW}$ is \mathrm {SW} .

### Set of Permutations

$S_n$

The set of permutations of $\N_n$ is denoted $S_n$.

The $\LaTeX$ code for $S_n$ is S_n .

### Symmetric Group

$\struct {S_n, \circ}$

Let $S_n$ denote the set of permutations on $n$ letters.

Let $\struct {S_n, \circ}$ denote the symmetric group on $S_n$.

Then $\struct {S_n, \circ}$ is referred to as the symmetric group on $n$ letters.

The $\LaTeX$ code for $\struct {S_n, \circ}$ is \struct {S_n, \circ} .

### Signum Function

$\map \sgn x$

Let $X \subseteq \R$ be a subset of the real numbers.

The signum function $\sgn: X \to \set {-1, 0, 1}$ is defined as:

$\forall x \in X: \map \sgn x := \sqbrk {x > 0} - \sqbrk {x < 0}$

where $\sqbrk {x > 0}$ etc. denotes Iverson's convention.

That is:

$\forall x \in X: \map \sgn x := \begin{cases} -1 & : x < 0 \\ 0 & : x = 0 \\ 1 & : x > 0 \end{cases}$

The $\LaTeX$ code for $\map \sgn x$ is \map \sgn x .

### Filtering Function

$\map {\operatorname {sinc} } x$

The filtering function is the real function $\operatorname {sinc}: \R \to \R$ defined as:

$\forall x \in \R: \map {\operatorname {sinc} } x := \dfrac {\sin \pi x} {\pi x}$

where $\sin$ denotes the (real) sine function.

The $\LaTeX$ code for $\map {\operatorname {sinc} } x$ is \map {\operatorname {sinc} } x .

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