# Index: Symbolic Notation

In a collaborative project, a unified notation is important for both the authors and readers. In the below list, you will find the symbolic notation we choose for mathematical concepts. Because we occasionally use the same symbols across different branches of mathematics to express different things, the symbols are attached to different branches.

### Algebra

Notation Description Comment
$x^{-1}$ inverse element of $x$
$G=\langle g \rangle$ cyclic group
$\operatorname{ord}(a)$ order of the element $a$
$\operatorname{char}( F )$ characteristic (here of a field)
$\operatorname{char}( F )$ characteristic (here of a ring)
$(R,+,\cdot)$ ring, unit ring
$a\sim_G b$ conjugate elements $a,b$ in the group $G$
$a\mid b$ divisor (generalization)
$a\sim b$ Associate
$\operatorname{conv}(X)$ convex hull of $X$
$[x_1,x_2,\ldots]$ continued fraction
$A\simeq B$ isomorphic sets $A$ and $B$
$aH$ left coset of the subgroup $H\subseteq G$ with respect to $a\in G$
$Ha$ right coset of the subgroup $H\subseteq G$ with respect to $a\in G$
$\langle v,w\rangle$ dot product of the vectors $v$ and $w$
$\langle v,w\rangle$ inner product of the vectors $v$ and $w$
$\langle v,w\rangle$ scalar product of the vectors $v$ and $w$
$\operatorname {dim}(V)$ dimension of a vector space $V$
$x^n$ $n$-th power $x$
$\bigwedge ^{n}V$ exterior algebra over the vector space $V$
$L/F$ field extension $L$ over the field $F$
$(F,+,\cdot)$ field with the addition $+$ and multiplication $\cdot$
$\gcd(M)$ greatest common divisor of the set $M$
$\langle A \rangle$ group generated by the set $A$
$\langle A\rangle_R$ ideal generated by the set $A$
$\left\lvert G \right\rvert$ order of the group $G$
$(G,\ast)$ group
$\left\lvert G \right\rvert$ order of the group $G$
$I\lhd R$ ideal of the ring $R$
$I$ identity matrix
$A^{-1}$ inverse matrix
$\ker(f)$ kernel of a group homomorphism $f$
$\operatorname{im}(f)$ image of a group homomorphism $f$
$Span(A)$ linear span of a set of a set $A$ of vectors
$Span(S)$ linear hull of a set of a set $A$ of vectors
$(X,\ast)$ magma
$M_{m\times n}(F)$ matrices (set of)d over a field $F$ with $m$ rows and $n$ columns
$_RM$ left module of a ring $R$
$M_R$ right module of a ring $R$
$N\unlhd G$ normal subgroup
$\deg(p)$ degree of a polynomial $p$
$R[X]$ polynomial ring (over the commutative ring $R$).
$(a)$ principal ideal generated by $a$
$(S,\ast)$ semigroup
$\operatorname{Spec}(R)$ spectrum of a commutative ring
$A^T$ transposed matrix
$x,y,a,b,\ldots$ vectors (Latin letters)
$\alpha,\beta,\gamma,\ldots$ scalars (Greek letters)
$v=\pmatrix{ \alpha_{1}\cr \alpha_{2}\cr \vdots \cr \alpha_{n} \cr }$ vector (column notation)
$O$ zero matrix

### Analysis

Notation Description Comment
$\left(D_{v}f\right)\left(x\right)$ directional derivative of $f$ along vector $v$ at point $x$
$\Delta f(b,a)$ difference quotient of function $f$
$f'(x)$ derivative of $f$ at $x$
$\frac {df(x)}{dx}$ derivative of $f$ at $x$
$\mathbb R^n$ Euclidean vector space of dimension $n$
$a_{-k}a_{-k+1}\cdots a_{-1}a_{0}.a_{1}a_{2}a_{3}\cdots$ $b$-adic fraction
$(a_n)\sim(b_n)$ asymptotically equivalent sequences
$\lim_{n\to\infty} x_n$ limit of convergent series
$\sum_{k=0}^\infty x_k$ series
$\sum_{k=0}^\infty x_k$ convergent series (real case)
$\overline{\mathbb R}$ extended real numbers
$\int_a^bf(x)dx=F(x)\;\Rule{1px}{4ex}{2ex}^{b}_{a}$ Riemann integral calculation of $f$ using its antiderivative $F$
$\int_a^bf(x)dx=F(x)\;\Rule{1px}{4ex}{2ex}^{b}_{a}$ Riemann integral calculation of $f$ using its antiderivative $F$
$\Gamma(x)$ Gamma function of $x$
$D_{v_{n}}(...D_{v_{2}}(D_{v_{1}}f(x)))$ higher order directional derivative of $f$ along the vectors $v_{1},\ldots ,v_{n}$ at the point $x$
$f^{(k)}$ derivative of order $k$ of $f$
$\frac{d^kf(x)}{dx^k}$ derivative of order $k$ of $f$
$\left(\frac d{dx}\right)^kf(x)$ derivative of order $k$ of $f$
$\cosh$ hyperbolic cosine
$\sinh$ hyperbolic sine
$\inf(D)$ infimum of a subset of real numbers $D\subseteq\mathbb R$
$\arccos(x)$ inverse cosine
$\operatorname{arcosh}(x)$ inverse hyperbolic cosine
$\operatorname{arsinh}(x)$ inverse hyperbolic sine
$\arcsin(x)$ inverse sine of $x$
$\arctan(x)$ inverse tangent of $x$
$\varliminf$ limit inferior
$\varlimsup$ limit superior
$\lim_{\xi\to x} f(\xi)$ limit of $f$ at $x$
$\lim_{\xi\searrow x} f(\xi)$ limit (from above) of $f$ at $x$
$\lim_{\xi\nearrow x} f(\xi)$ limit (from below) of $f$ at $x$
$\lim_{\xi\to +\infty} f(\xi)$ limit (at infinity) of $f$
$\lim_{\xi\to +\infty} f(\xi)$ limit (at minus infinity) of $f$
$\log_a(x)$ logarithm of $x$ to the base $a$
$\max(D)$ maximum of the real set $D$ (if exists)
$(M,{\mathcal {A}})$ measureable set with a $\sigma$-algebra $\mathcal {A}$
$(M,\mathcal{A},\mu)$ measurable space, consisting of a set $M$, a $\sigma$-algebra $\mathcal{A}$ and a measure $\mu$
$\min(D)$ minimum of the real subset $D\subset R$ (if it exists)
$\ln(x)$ natural logarithm
$x^n$ $n$-th power of $x$
$\sqrt[n]{x}$ $n$-th root of $x$
$f'_+(x)$ right-derivative of $f$ at $x$
$f'_-(x)$ left-derivative of $f$ at $x$
$[a,b]$ closed real interval
$(a,b)$ open real interval
$(a,b]$ left-open, right-closed real interval
$[a,b)$ right-open, left-closed real interval
$(- \infty,b)$ left-unbounded, right-open real interval
$(- \infty,b]$ left-unbounded, right-closed real interval
$(a,+ \infty)$ right-unbounded, left-open real interval
$[a,+ \infty)$ right-unbounded, left-closed real interval
$\sup(D)$ supremum of a subset of real numbers $D\subseteq\mathbb R$
$\tan(x)$ tangent of $x$
$(D\varphi)_x$ total derivative of $\varphi$ at the point $x$

### Combinatorics

Notation Description Comment
$\binom nk$ binomial coefficient
$n!$ factorial
$n^{\underline{k}}$ falling factorial power
$n^{\overline{k}}$ rising factorial power
$(\begin{matrix}\pi(1) & \pi(2) & \pi(3) & \cdots & \pi(n-1) & \pi(n)\end{matrix})$ permutation of $n$ elements
$n!$ factorial

### Geometry / Euclidean Geometry

Notation Description Comment
${A}\cong{B}$ congruence of the objects $A$ and $B$
${A}\sim{B}$ similarity of the geometrical figures $A$ and $B$

### Geometry / Euclidean Geometry / Elements Euclid / Book 1 Plane Geometry

Notation Description Comment
$\angle{PQR}$ angle between $\overline{PQ}$ and $\overline{QR}$ (called legs of the angle) being rotated counter-clockwise around $Q$ (called the vertex of the angle)
$\angle{PQR}$ angle between $\overline{PQ}$ and $\overline{QR}$ (called legs of the angle) being rotated counter-clockwise around $Q$ (called the vertex of the angle)
$\angle{PQR}$ angle between $\overline{PQ}$ and $\overline{QR}$ (called legs of the angle) being rotated counter-clockwise around $Q$ (called the vertex of the angle)
$A,B,C,\ldots$ point
$\overline EF$ ray beginning at point $E$ and going through point $F$
$\overline{AC}\perp\overline{DB}$ right angle between $\overline{AC}$ and $\overline{DB}$ being perpendicular
$\overline{AC}\perp\overline{DB}$ right angle between $\overline{AC}$ and $\overline{DB}$ being perpendicular
$\overline{CD}$ segment between the endpoints $C$ and $D$
$AB$ straight line connecting the points $A$ and $B$
$\odot AB$ circle with center $A$ and radius $\overline{AB}$]
$\odot AB$ circle with center $A$ and radius $\overline{AB}$]
$\triangle ABC$ triangle between the points $A,B,C$]
$\boxdot{ABCD}$ square with the edges $A,B,C,D$]
$\boxdot{ABCD}$ parallelogram with the edges $A,B,C,D$]
$\boxdot{ABCD}$ rhombus with the edges $A,B,C,D$]
$\boxdot{ABCD}$ rectangle with the edges $A,B,C,D$]
$AB\parallel CD$ parallel lines
$AB\not\parallel CD$ non-parallel lines

### Geometry / Euclidean Geometry / Elements Euclid / Book 7 Elementary Number Theory

Notation Description Comment
$\sim$ proportional
$d\not\mid n$ $d$ does not divide $n$
$\mathbb P$ prime numbers (set of)

### Geometry / Projective Geometry

Notation Description Comment
$P=m\cdot l$ concurrent straight lines $m$ and $l$ meeting at the point $P$
$l=PQ$ collinear points $P$ and $Q$ on a line $l$ joining them
$\alpha=PQ$ coplanar points $P$ and $Q$ on a plane $\alpha$ joining them
$\alpha=Pl$ coplanar point $P$ and line $l$ on a plane $\alpha$ joining them
$p\overline{\overline\wedge}q$ perspectivity of two straight lines
$P\overline{\overline\wedge}Q$ perspectivity of two points
$P\overline\wedge p$ projectivity of a point $P$ and a straight line $p$

### Graph Theory

Notation Description Comment
$G^\ast_{\mathcal D}$ dual graph of a planar graph $G$ in its planar drawing $\mathcal D$
$D=(V,E,\alpha,\omega)$ Digraph
$\alpha(e)$ initial vertex assigned to the edge $e$
$\omega(e)$ terminal vertex assigned to the edge $e$
$C_n$ cycle graph (with $n$ vertices)
$\overline G$ complement graph
$K_{m,n}$ complete bipartite graph
$K_n$ complete graph (with $n$ vertices)
$N_G(v)$ neighbors of $v$ in the undirected graph $G$, (short form $N(v)$)
$\delta_G(v)$ edges incident to $v$ in the undirected graph $G$, (short form $\delta(v)$)
$N_D^-(v)$ predecessor vertices of $v$ in the digraph $D$, (short form $(N^-(v)$)
$N_D^+(v)$ successor vertices of $v$ in the digraph $D$, (short form $N^+(v)$)
$\delta_D^-(v)$ edges incoming to $v$ in the digraph $D$, (short form $\delta^-(v)$)
$\delta_D^+(v)$ edges outgoing from $v$ in the digraph $D$, (short form $\delta^+(v)$)
$D_1\equiv D_2$ isomorphic digraphs $D_1$ and $D_2$
$G_1\equiv G_2$ isomorphic undirected graphs $G_1$ and $G_2$
$\tau(G)$ minimal tree decomposability of the graph $G$
$N_n$ null graph (with $n$ vertices)
$\left\lvert G \right\rvert$ order of the graph $G$
$\mathcal D$ planar drawing
$S\subseteq D$ subdigraph $S$ of a digraph $D$
$D[V]$ induced subdigraph in $D$ spanned by vertices in $V$
$D[v_1,\ldots,v_k]$ induced subdigraph in $D$ spanned by the vertices $v_1,\ldots,v_k$
$S\subseteq G$ subgraph $S$ of a graph $G$
$G[V]$ induced subgraph in $G$ spanned by vertices in $V$
$G[v_1,\ldots,v_k]$ induced subgraph in $G$ spanned by the vertices $v_1,\ldots,v_k$
$\tilde G$ suppressed multigraph
$G=(V,E,\gamma)$ undirected graph
$G=(V,E)$ simple graph
$\left\lvert G \right\rvert$ order of a (directed or undirected) $G$
$d_D^+(v)$ outer degree of $v$ in the digraph $D$, ($d^+(v)$ short form)
$d_D^-(v)$ inner degree of $v$ in the digraph $D$, ($d^-(v)$ short form)
$d_D(v)$ degree of $v$ in the digraph $D$, ($d(v)$ short form)
$\deg_G(v)$ degree of $v$ in the undirected graph $G$, ($\deg(v)$ short form)
$P^k$ path of length $k$
$T^k$ trail of length $k$
$W^k$ walk of length $k$

### Knot Theory

Notation Description Comment
$U$ unknot

### Logic

Notation Description Comment
$(\mathbb B,\sqcap,\sqcup,1,0)$ Boolean algebra
$\operatorname{cnf}(\phi)$ canonical normal form of $\phi$
$x \Leftrightarrow y$ equivalence "$x$ if and only if $y$"
$x \vee y$ disjunction of the propositions $x$ and $y$ ("$x$ or $y$")
$\vdash_L\phi$ derivable formula $\phi$ in logical calculus $L$
$x \wedge y$ conjunction of two propositions ("$x$ and $y$")
$\operatorname{ccnf}(\phi)$ conjunctive canonical normal form of $\phi$
$\operatorname{dcnf}(\phi)$ disjunctive canonical normal form of $\phi$
$x \Rightarrow y$ conclusion ("if $x$ then $y$")
$\neg x$ negation of the proposition $x$ ("not $x$")
$\operatorname{PL1}$ predicate logic of first order
$\exists$ existential quantifier "there exists"
$\forall$ universal quantifier "for all"
$\vdash$ derivable relation
$\models$ model relation
$\models \phi$ valid expression $\phi$
$\models \phi$ tautology $\phi$
$\not{\models} \phi$ invalid expression $\phi$
$\not{\models} \phi$ contradiction $\phi$
$\mathbb B:=\{0,1\}$ binary logical values (set of) (contains true and false)
$PL0$ propositional logic
$\Sigma$ alphabet
$\Sigma^*$ strings (set of) over an alphabet $\Sigma$

### Number Systems Arithmetics

Notation Description Comment
$e$ Euler's constant
$\left\lvert z \right\rvert$ absolute value (complex case) $z$
$\left\lvert x\right\rvert$ absolute value (real case)
$\left\lvert x-y\right\rvert$ distance (here real case) of $x$ and $y$
$\mathbb C$ complex numbers (set of)
$\Re(z)$ real part of the complex number $z$
$\Im(z)$ imaginary part of the complex number $z$
$\mathbb Z$ integers (set of)
$\mathbb Q$ rational numbers (set of)
$\mathbb R$ real numbers (set of)
$+$ addition (here natural numbers)
$(\mathbb C, +, \cdot)$ field of complex numbers
$(\mathbb C, +)$ commutative group of complex numbers with respect to addition
$(\mathbb Z, + )$ group of integers with respect to addition
$(\mathbb C^*, \cdot)$ commutative group of complex numbers with respect to multiplication
$(\mathbb Q^*, \cdot)$ commutative group of rational numbers with respect to multiplication
$(\mathbb R^*, \cdot)$ commutative group of real numbers with respect to multiplication
$(\mathbb Q,+,\cdot))$ field of rational numbers
$(\mathbb Q,+)$ commutative group of rational numbers with respect to addition
$(\mathbb R,+,\cdot))$ field of real numbers
$(\mathbb R, +)$ commutative group of real numbers with respect to addition
$z^*$ complex conjugate of the complex number $z$
$i$ imaginary unit
$\cdot$ multiplication sign
$\mathbb N$ natural numbers (set of) (including 0)
$x-y$ subtraction (here integers)
$a_1+a_2+\cdots+a_n$ sum (three dots-notation)
$\sum_{k=1}^n a_k$ sum (delimited notation)
$\sum_{ \substack{ 1\text{st condition on }k \\ 2\text{nd condition on }k\\ \vdots }}a_k$ sum (general notation)
$\Delta_n$ triangle number ($n$'th)

### Number Theory

Notation Description Comment
$p$ prime number
$p_i$ prime number (i-th)
$d\mid n$ $d$ divides $n$
$a\equiv b\mod m$ congruent modulo $m$
$a\equiv b(m)$ residue modulo $m$
$a\perp b$ co-prime (also relatively prime numbers) numbers $a$ and $b$
$a\perp b$ co-prime (also relatively prime numbers) numbers $a$ and $b$
$\mathbb Z_m$ quotient set of congruence class modulo $m$
$\mathbb Z_m$ quotient set of congruence class modulo $m$
$a(m)$ representative $a$ of the congruence class modulo $m$
$a(m)$ representative $a$ of the congruence class modulo $m$
$\lfloor x \rfloor$ floor of the real number $x$
$\lceil x \$ ceiling of the real number $x$
$\gcd(a,b)$ greatest common divisor
$\operatorname{lcm}(a,b)$ least common multiple
$\left(\frac np\right)$ Legendre symbol modulo $p$
$(\mathbb Z_m^*,\cdot)$ multiplicative group modulo $m$
$\mathbb N_d$ natural numbers relatively prime to a given number (set of) $d$
$\mathbb P_d$ prime numbers not dividing the natural number $d$
$\Lambda(n)$ von Mangoldt function of $n$

### Probability Theory And Statistics

Notation Description Comment
$\Omega$ certain event
$\emptyset$ impossible event
$p(X\le x)$ probability of the event that the random variable $X$ has a realization less or equal $x$
$p(\overline A)$ probability of the complement event of $A$
$\Omega$ probability space
$(A,B,C,\ldots$ random events
$X,Y$ random variables (denoted by roman upper-case letters)
$f_n(A)$ absolute frequency of the event $A$
$F_n(A)$ relative frequency of the event $A$

### Set Theory

Notation Description Comment
$A=B$ equal sets $A$ and $B$
$A\neq B$ unequal sets $A$ and $B$
$[a]_R$ equivalence class with the representative element $a$ under the relation $R$
$[a]$ equivalence class with the representative element $a$ (short form, if relation $R$ is clear from the context)
$a\sim_R b$ equivalence relation $R$ of the elements $a$ and $b$
$a\sim b$ equivalence relation (short notation), if $R$ is clear from the context of the elements $a$ and $b$
$\left\lvert A \right\rvert$ cardinality of the set $A$ (also cardinal number)
$\left\lvert A \right\rvert$ cardinality of the set $A$ (also cardinal number)
$A\times B$ Cartesian product of the sets $A$ and $B$
$A_1\times A_2\times\cdots\times A_n$ Cartesian product of the sets $A_1\ldots A_n$
$A^n$ Cartesian product of the set $A$ with itself ($n$ times)
$a\prec b$ $a$ is greater than $b$
$a\succ b$ $a$ is smaller than $b$
$a\preceq b$ $a$ is greater than or equal to $b$
$a\succeq b$ $a$ is smaller than or equal to $b$
$a=b$ $a$ is equal to $b$
$(R_2\circ R_1)(x)$ composition of the relations $R_2$ and $R_1$
$R_2(R_1(x))$ composition of the relations $R_2$ and $R_1$
$(g\circ f)(x)$ composition of the functions $f$ and $g$
$g(f(x))$ composition of the functions $f$ and $g$
$\Gamma_f$ graph $\Gamma$ of a function $f$
$f^{-1}$ inverse function
$\omega$ minimal inductive set
$(x_1,x_2)$ ordered pair
$(x_1,\ldots,x_n)$ ordered n-tuple of set elements
$\alpha, \beta,\gamma, \ldots$ ordinal numbers
$f(a)$ image of element $a$ under the function $f$
$f[A]$ image of a set $A$ under the function $f$
$f^{-1}[B]$ inverse image of a set $B$ under the function $f$
$f^{-1}(b)$ fiber of the element $b$ under the function $f$
$f:A\mapsto B$ map
$\mathcal P(X)$ power set of $X$
$(V,\preccurlyeq)$ ordered set
$\preccurlyeq$ order
$\preccurlyeq$ preorder
$V/_{R}$ quotient set of the set $V$ under the relation $R$ (also called "$V$ modulo $R$")
${f\right\rvert}_C : A \to$ restriction of the function $f$ on the subset $C\subseteq A$
$A^C$ complement of the set $A\subseteq X$ (in a given set $X$)
$A\setminus B$ set difference
$A\cap B$ intersection of the sets $A$ and $B$
$A\cup B$ union of the sets $A$ and $B$
$x\in X$ [$x$ is an element of the set $X$
$x\notin X$ [$x$ is not an element of the set $X$
$\emptyset$ empty set
$\{X\}$ singleton of the set $X$
$\inf(S)$ infimum of $S$
$\sup(S)$ supremum of $S$
$\max(S)$ maximum of $S$
$\min(S)$ minimum of $S$
$A\subseteq X$ $A$ is a subset of $X$
$A\subset X$ $A$ is a proper subset of $X$
$X\supseteq A$ $X$ is a superset of $A$
$X\supset A$ $X$ is a proper superset of $A$
$\mathcal P(X)$ power set of $X$
$(V,\preccurlyeq)$ chain $V$ with a total order $\preccurlyeq$

### Theoretical Computer Science / Complexity Theory

Notation Description Comment
$\mathcal O$ big-O or asymptotic notation
$\mathcal O$ big-O or asymptotic notation

### Theoretical Computer Science / Computability

Notation Description Comment
$G O T O$ GOTO-computable functions (set of)
$G O T O^{part}$ partially GOTO-computable function (set of)
$L O O P$ LOOP-computable functions (set of)
$R A M$ Unit-Cost Random Access Machine
$W H I L E$ WHILE-computable functions (set of)
$W H I L E^{part}$ partially WHILE-computable functions (set of)

### Theoretical Computer Science / Data Structures

Notation Description Comment
$(n_1,n_2,...,n_k)$ linked list (data structure) of $k$ nodes

### Theoretical Physics / Classical Physics

Notation Description Comment
$\bar{v}$ average velocity (in physics)
$\dot x(t)$ velocity (instantaneous)

### Theoretical Physics / Special Relativity

Notation Description Comment
$m$ meter
$s$ second (SI unit)
$\gamma$ Lorentz factor (special relativity)

### Topology

Notation Description Comment
$\lim_{n\rightarrow\infty} a_n=a$ limit of a convergent sequence
$\operatorname{diam} (A)$ diameter of set $A$
$\delta X$ boundary of $X$
$X^e$ exterior of $X$
$X^-$ closure of $X$
$X^\circ$ interior of $X$
$\operatorname {lim} _{x\rightarrow a}\,f(x)=b$ limit of a function
$(X,d)$ metric space of the set $X$ with a metric $d$
$\omega_f$ modulus of continuity of $f$
$\left\lVert \right\rVert$ norm
$(V,\left\lvert \right\rvert)$ normed vector space $V$
$B(c,r)$ open ball with center $c$ and radius $r$
$\left\lVert x\right\rVert_p$ $p$-norm
$(a_n)_{n\in\mathbb N}$ sequence
$(a_{n_k})_{k\in\mathbb N}$ subsequence of the sequence $(a_n)_{n\in\mathbb N}$
$X,\mathcal {T}$ topological space consisting of a set $X$ and its topology $\mathcal {T}$

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