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A groupoid rack and spatial surfaces

Katsunori Arai

A spatial surface is a compact surface embedded in the $3$-sphere.
We assume that each connected component has non-empty boundary.
Spatial surfaces are
represented by diagrams of spatial trivalent graphs.
In this paper,
we introduce the notion of a groupoid rack, which is an algebraic structure
that can be used for colorings of diagrams of oriented spatial surfaces.
Furthermore, we show that the groupoid rack has a universal property
on colorings for diagrams of spatial surfaces.

§ INTRODUCTION

A quandle <cit.> is an algebraic structure
with axioms corresponding to the three Reidemeitser moves in knot theory.
The number of colorings of a knot diagram using a fixed finite quandle is an invariant of a knot.
A rack <cit.> is a generalization of a quandle, whose axioms
correspond to two of the three Reidemeister moves.

A handlebody-knot <cit.> is an embedding of a handlebody into the $3$-sphere $S^{3} = \mathbb{R}^{3} \sqcup \left\{\infty\right\}$.

Isotopy classes of knots in $S^{3}$ correspond to isotopy classes of handlebody-knots by considering regular neighborhoods.
In this sense, handlebody-knot theory is a generalization of knot theory.
Handlebody-knots can be represented by spatial trivalent graphs and their diagrams <cit.>.

A multiple conjugation quandle <cit.> is an algebra with certain axioms
derived from Reidemeister moves for diagrams of handlebody-knots.
It has a universal property on colorings for diagrams of handlebody-knots, i.e.,
any algebraic structure used to define colorings for handlebody-knot diagrams must have a structure of a multiple conjugation quandle <cit.>.
When we fix a multiple conjugation quandle,
the number of multiple conjugation quandle colorings is an invariant of handlebody-knots.

A spatial surface is a compact surface embedded in the $3$-sphere $S^{3}$.
Throughout this paper we assume that each connected component has non-empty boundary.
Then for spatial surfaces,
their diagrams and Reidemeister moves are defined in <cit.>.

A multiple group rack <cit.> is an algebraic structure
derived from Reidemeister moves for diagrams of spatial surfaces.
It is a generalization of a multiple conjugation quandle.
In <cit.>, another algebraic structure, we call a heap rack, is defined,
which is also derived from Reidemeister moves for spatial surfaces.

In this paper, we introduce an algebraic structure, a groupoid rack,
that can be used for colorings of spatial surface diagrams.
We show that for a given groupoid rack, the number of colorings is an invariant of a spatial surface (Theorem <ref>).
Furthermore, a groupoid rack has a universal property on colorings of spatial surface diagrams (Theorem <ref>).
Multiple group racks <cit.> and heap racks <cit.> are examples of groupoid racks.

§ A GROUPOID RACK

In this section we introduce the notion of a groupoid rack and give some examples.

A rack (cf.<cit.>) is a nonempty set $R$ with a binary operation $\ast : R \times R \to R$ defined by $(x, y) \mapsto x \ast y$ satisfying the following conditions.

(i) For any $y \in R$, the map $S_{y} : R \to R$ defined by $x \mapsto x \ast y$ is bijective.

(ii) For any $x, y, z \in R$, $(x \ast y) \ast z = (x \ast z) \ast (y \ast z)$.

For any $x, y \in R$ and $n \in \mathbb{Z}$, we denote $S_{y}^{n}(x)$ by $x \ast^{n} y$.
For simplicity, the rack $(R, \ast)$ is sometimes written as $R$.

A good involution (cf. <cit.>) of a rack $R$ is a map $\rho : R \to R$ satisfying the following conditions.

(i) $\rho$ is involutive.

(ii) For any $x, y \in R$, $\rho(x \ast y) = \rho(x) \ast y$.

(iii) For any $x, y \in R$, $x \ast \rho(y) = x \ast^{-1} y$.

We call a pair $R = (R, \rho)$ a symmetric rack.

A groupoid rack $X$ is the set of all morphisms of a groupoid $\mathcal{C}$ equipped with a binary operation
$\ast : X \times X \to X$ satisfying the following conditions.

(i) For any $x \in X$ and $f, g \in X$ with $\textrm{cod}(f) = \textrm{dom}(g)$,
we have $x \ast (fg) = (x \ast f) \ast g$ and  for any object $\lambda$ of $\mathcal{C}$, $x \ast \textrm{id}_{\lambda} = x$,
where $\textrm{id}_{\lambda}$ is the identity morphism of the object $\lambda$.

(ii) For any $x, y, z \in X$, $(x \ast y) \ast z = (x \ast z) \ast (y \ast z)$.

(iii) For any $x \in X$ and $f, g \in X$ with $\textrm{cod}(f) = \textrm{dom}(g)$,
we have
$(fg) \ast x = (f \ast x) (g \ast x)$.

A groupoid rack $X$ is regarded as a symmetric rack with the good involution $\rho : X \to X$ defined by $\rho(x) = x^{-1}$.

We show some examples of a groupoid rack.
The first two examples below show that a multiple group rack introduced in <cit.> and a heap rack introduced in <cit.> are groupoid racks in our sense.

Let $\left\{G_{\lambda}\right\}_{\lambda \in \Lambda}$ be a family of groups and let $e_{\lambda}$ be the identity element of $G_{\lambda}$.
A multiple group rack $X$ is the disjoint union of groups $G_{\lambda}$
with a binary operation $\ast : X \times X \to X$ satisfying following conditions.

(i) For any $x \in X$ and $y_{1}, y_{2} \in G_{\lambda}$,
$x \ast (y_{1} y_{2}) = (x \ast y_{1}) \ast y_{2}$ and $x \ast e_{\lambda} = x$.

(ii) For any $x, y, z \in X$, $(x \ast y) \ast z = (x \ast z) \ast (y \ast z)$.

(iii) For any $x \in X$ and $\lambda \in \Lambda$,
there exists $\mu \in \Lambda$ such that
for any $y_{1}, y_{2} \in G_{\lambda}$,
$(y_{1} \ast x), (y_{2} \ast x) \in G_{\mu}$ and $(y_{1} y_{2}) \ast x = (y_{1} \ast x)(y_{2} \ast x)$.

Then $X$ is a groupoid rack with a groupoid $\mathcal{C}$ defined by the following.

* $\textrm{Ob}(\mathcal{C}) = \Lambda$.

* For any $\lambda \in \Lambda$, $\textrm{Hom}(\lambda, \lambda) = G_{\lambda}$ and
for any $\lambda \neq \mu \in \Lambda$, $\textrm{Hom}(\lambda, \mu) = \varnothing$.

* The composition $G_{\lambda} \times G_{\lambda} \to G_{\lambda}$ is defined by $(x, y) \mapsto xy$.

* The identity morphism of $\lambda \in \Lambda$ is the identity element $e_{\lambda}$.

* The inverse morphism of $x \in G_{\lambda}$ is the inverse element $x^{-1}$.

In particular, when a multiple group rack $X$ satisfies the follwing condition (iv),
$X$ is a multiple conjugation quandle <cit.>.

(iv) For any $\lambda \in \Lambda$ and $x, y \in G_{\lambda}$, $x \ast y = y^{-1}xy$.

Let $G$ be a group.
A binary operation $\ast : G^{2} \times G^{2} \to G^{2}$ defined by $(x, y) \ast (z, w) = (xz^{-1}w, yz^{-1}w)$ is a rack operation on $G^{2}$.
In this paper, we call the rack $G^{2}$ with the partial product defined by $(x, y)(y, z) = (x, z),\  x,y,z \in G, $ a heap rack.

Then a heap rack is a groupoid rack with the groupoid $\mathcal{C}$ defined by the following.

* $\textrm{Ob}(\mathcal{C}) = G$.

* For any $x, y \in G$, $\textrm{Hom}(x, y) = \left\{(x, y)\right\}$.

* The composition $\mathrm{Hom}(x, y) \times \mathrm{Hom}(y, z) \to \mathrm{Hom}(x, z)$ defined by $((x, y), (y, z)) \mapsto (x, z)$.

* The identity morphism of $x \in G$ is  $(x, x)$.

* The inverse morphism of $(x, y) \in \textrm{Hom} (x, y)$ is $(y, x)$.

The next example implies that an analogous result to Example <ref> holds for racks.

Let $R = (R, \ast)$ be a rack.
A binary operation $\triangleright : R^{2} \times R^{2} \to R^{2}$ defined by $(x, y) \triangleright (z, w) = ((x \ast^{-1} z) \ast w, (x \ast^{-1} z) \ast w)$ is a rack operation on $R^{2}$.

Then the rack $R^{2} =(R^{2}, \triangleright)$ with the partial product $(x,y)(y, z) = (x, z),\ x,y,z \in R, $ is a groupoid rack with the groupoid $\mathcal{C}$ defined by the following.

* $\textrm{Ob}(\mathcal{C}) = R$.

* For any $x, y \in R$, $\textrm{Hom}(x, y) = \left\{(x, y)\right\}$.

* The composition $\textrm{Hom}(x, y) \times \textrm{Hom}(y, z) \to \mathrm{Hom}(x, z)$ is defined by $((x, y), (y, z)) \mapsto (x, z)$.

* The identity morphism of $x \in R$ is  $(x, x)$.

* The inverse morphism of $(x, y) \in \textrm{Hom}(x, y)$ is $(y, x)$.

We can construct a groupoid rack from an augmented rack.

Let $R = (R, \ast)$ be a rack equipped with a right group action by a group $G$ and let $\partial : R \to G$ be a map.
The triple $(R, G, \partial)$ is an augmented rack if the map $\partial : R \to G$ satisfying the following conditions.

(i) For any $x, y \in R$, $x \ast y = x \cdot \partial (y)$.

(ii) For any $x \in R$ and $g \in G$, $\partial (x \cdot g) = g^{-1} \partial (x) g$.

Let $(R, G, \partial)$ be an augmented rack.

Define a groupoid $\mathcal{C}$ by the following.

* $\textrm{Ob}(\mathcal{C}) = R$.

* For any $x, y \in R$,
$\textrm{Hom}(x, y) = \left\{(x, y, g) \in R \times R \times G \mid y = x \cdot g\right\}$.

* The composition $\textrm{Hom}(x, y) \times \textrm{Hom}(y, z) \to \textrm{Hom}(x, z)$ is defined by $((x,y,g), (y,z,h)) \mapsto (x, z, gh)$.

* The identity morphism of $x \in R$ is $(x, x, e)$, where $e$ is the identity element of $G$.

* The inverse morphism of $(x, y, g)$ is $(y, x, g^{-1})$.

Fix an integer $n$ and $\delta \in \left\{0,1\right\}$. Then the set of all morphisms $X$ of $\mathcal{C}$ is a groupoid rack with a binary operation $\ast : X \times X \to X$ defined by

$$(x, y, g) \ast (z, w, h) = (x \cdot {\partial (z)}^{n} h^{\delta} {\partial (w)}^{-n}, y \cdot {\partial (z)}^{n} h^{\delta} {\partial (w)}^{-n}, {\partial(w)}^{n} h^{-\delta}{\partial (z)}^{-n} g {\partial (z)}^{n} h^{\delta} {\partial (w)}^{-n}).$$

When a rack $(R, \ast)$ is given,
we may regard it as an augmented rack $(R, G, \partial)$ by considering $G$ as the free group on $R$ and $\partial$ is the right action as in <cit.>.
Examples <ref> and <ref> show methods of constructing groupoid racks from a given rack.

§ COLORINGS FOR SPATIAL SURFACES

A spatial surface is a compact surface embedded in $S^{3} = \mathbb{R}^{3} \sqcup \left\{\infty\right\}$.
In this paper, we assume the following.

* A spatial surface is oriented.

* Each component of a spatial surface has a non-empty boundary.

* A spatial surface has no disk components.

Two spatial surfaces are equivalent if there exists an ambient isotopy of $S^{3}$
which sends one to the other.

A spatial trivalent graph is a finite trivalent graph embedded in $S^{3}$.
In this paper, we assume that a trivalent graph may have loops, multi-edges and
a spatial trivalent graph may have some $S^{1}$-components, i.e., circles embedded in $S^{3}$.
A spatial trivalent graph can be presented by a diagram in $S^{2} = \mathbb{R}^{2} \sqcup \left\{\infty\right\}$ as usual in knot theory.

We obtain a spatial surface $F$ from a spatial trivalent graph diagram $D$ by
taking a regular neighborhood $N(D)$ of $D$ in $\mathbb{R}^{2}$ and
giving the subset of $N(D)$ corresponding to the crossings of $D$ with over or under information induced from the information of the crossings of $D$.

We give $F$ an orientation induced from the orientation of $\mathbb{R}^{2}$ (See Fig. <ref>).

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The construction of a spatial surface from a spatial trivalent graph diagram

Then we say that $D$ represents the spatial surface $F$.
Any spatial surface is equivalent to a spatial surface obtained by this way.
A diagram of a spatial surface $F$ means a spatial trivalent graph diagram which represents a spatial surface equivalent to $F$.

Two spatial surfaces are equivalent if and only if
their diagrams are related by a finite sequence of R$2$, R$3$, R$5$ and R$6$ moves on $S^{2}$ depicted in Fig. <ref>.

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Reidemeister moves for spatial surface diagrams

A Y-orientation of a spatial trivalent graph is an assignment of orientations to edges and $S^{1}$-components such that there exist no sinks and no sources (Fig. <ref>).

A Y-oriented spatial trivalent graph is
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Any spatial trivalent graph admits at least one Y-orientation.
A Y-oriented diagram of a spatial surface is a diagram of the spatial surface with a Y-orientation.

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See Fig.<ref> for Y-oriented R$5$ and R$6$ moves.

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Y-oriented R$5$ and R$6$ moves

In this paper, we call an operation which reverses an orientation of one $S^{1}$-component an inverse move.

Matsuzaki <cit.> showed the following theorem.

Two spatial surfaces are equivalent if and only if
their Y-oriented diagrams are related by a finite sequence of Y-oriented Reidemeister moves and inverse moves.

Let $D$ be a Y-oriented diagram of a spatial surface $F$.
An arc of $D$ means a simple arc or a loop in $S^{2}$ which is obtained from $D$ by cutting the diagram at under crossings and vertices.
We denote the set of all arcs by $\mathcal{A}(D)$.
Let $X$ be a groupoid rack.
An $X$-coloring or a coloring by $X$ is a map $C : \mathcal{A}(D) \to X$ satisfying the conditions indicated in Fig. <ref>.
We denote the set of all $X$-colorings by $\textrm{Col}_{X}(D)$.

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$X$-coloring conditions ($x, y, f, g \in X,\ \text{cod}(f) = \text{dom}(g)$)

Let $X$ be a groupoid rack and let $D$ and $D'$ be Y-oriented diagrams which represent equivalent spatial surfaces.
Then there is a bijection between $\textrm{Col}_{X}(D)$ and $\textrm{Col}_{X}(D')$.
In particular,
the cardinality of $\textrm{Col}_{X}(D)$ is an invariant of a spatial surface.

Since $X$ is a symmetric rack,
an $X$-coloring is compatible with inverse moves,
if $D'$ is obtained from $D$ by reversing orientations of some of $S^{1}$-components then there exists a bijection between $\mathrm{Col}_{X}(D)$ and $\mathrm{Col}_{X}(D')$, <cit.>.

An $X$-coloring is compatible with Y-oriented Reidemeister moves,

$$x \ast (fg) = (x \ast f) \ast g,\ x \ast \textrm{id}_{\lambda} = x,\ %\textrm{cod}(f \ast x) = \textrm{dom}(g \ast x) \
\mbox{and} \  (fg) \ast x = (f \ast x)(g \ast x)$$

hold for R$2$ and R$5$ moves, and

$$(x \ast y) \ast z = (x \ast z) \ast (y \ast z)$$

holds for R$3$ moves,
and the associativity of the composition of morphisms holds for the R$6$ move.

As seen in Examples <ref> and <ref>,
multiple group racks and heap racks are regarded as groupoid racks.
colorings using multiple group racks defined in <cit.> and
the colorings using heap racks defined in <cit.>
are regarded as colorings using groupoid racks in our sense.

§ THE UNIVERSALITY OF GROUPOID RACKS ON COLORINGS FOR SPATIAL SURFACE DIAGRAMS

Theorem <ref> implies that a groupoid rack has a universal property on colorings for spatial surface diagrams, i.e.
any algebraic structure defining colorings for spatial surface diagrams
which is compatible with Y-oriented Redemeister moves and inverse moves must have a structure of a groupoid rack.

Let $R = (R, \ast, \rho)$ be a symmetric rack.
Assume that a subset $P \subset R \times R$ and a map $\mu : P \to R$
satisfy the following conditions (1) – (5), where we denote $\mu(a, b)$ by $ab$.

(1) For any $a, b, c \in R$, the following two statements are equivalent.

$$\left[(a, b) \in P \wedge (ab, c) \in P\right] \ \mbox{and} \  \left[(b, c) \in P \wedge (a, bc) \in P\right].$$

(2) For any $(a, b), (ab, c) \in P$, we have

$$(ab)c = a(bc).$$

(3) For any $a, b, x \in R$, the following three statements are equivalent.

$$(a, b) \in P,\  (\rho(b), \rho(a)) \in P \  \mbox{and} \  (a \ast x, b \ast x) \in P.$$

(4) For any $(a, b) \in P$, we have

$$(b, \rho(ab)) \in P \  \mbox{and} \  (\rho(ab), a) \in P.$$

(5) For any $(a, b) \in P$ and $x \in R$, we have

$$\rho(b) \rho(a) = \rho(ab), \  (ab)\rho(b) = a, \  (ab) \ast x = (a \ast x)(b \ast x) \  \mbox{and} \  x \ast (ab) = (x \ast a) \ast b.$$

Put $R' = \bigcup_{(a, b) \in P}\left\{a, b\right\}$.

$(\mathrm{i})$ $R'$ is a subrack of $(R, \ast)$ and the restriction of $\rho$ to $R'$ is a good involution
on $R'$.

$(\mathrm{ii})$ $(R', \ast)$ is a groupoid rack.

Before giving the proof,
we show how the conditions (1) – (5) of the theorem are obtained.

Let $R = (R, \ast, \rho)$ be a triple of a set $R$, a binary operation $\ast$ on $R$ and a map $\rho : R \to R$.
For the first, we assume that the condition on colorings shown in Fig. <ref> is satisfied at each crossing.
For compatibility with R2 and R3 moves, we see that $R = (R, \ast)$ is a rack, <cit.>.

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Coloring condition around a crossing ($x, y \in R$)

For the second, we assume that when we reverse an orientation of an arc of a diagram,
the color $x$ on the arc is switched with $\rho(x)$ (Fig. <ref>).
Then we see that $\rho$ is a good involution, <cit.>.
Thus $(R, \ast, \rho)$ is a symmetric rack.

A basic inversion is reversing the orientation of an arc and changing the color $x$ on the arc with $\rho(x)$.

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A basic inversion ($x \in X$)

For the third,
we assume the conditions shown in Fig. <ref> are satisfied
at vertices, where we assume $(a, b) \in P$.

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Coloring conditions around vertices ($(a, b) \in P$)

For well-definedness of a coloring at a vertex under basic inversions,
we have the following conditions.

* For any $a, b, x \in R$, the following two statements are equivalent.

$$(a, b) \in P \ \mbox{and} \  (\rho(b), \rho(a)) \in P.$$

* For any $(a, b) \in P$, we have

$$(b, \rho(ab)) ,\  (\rho(ab), a),\  (ab, \rho(b)) \  \mbox{and} \  (\rho(a), ab) \  \mbox{belong to} \  P.$$

* For any $(a, b) \in P$ and for any $x \in R$, we have

$$b \rho(ab) = \rho(a),\  \rho(ab) a = \rho(b),\
\rho(b) \rho(a) = \rho(ab),\ % \newline
(ab) \rho(b) = a,\ \mbox{and} \  \rho(a) (ab) = b.

The following two conditions can be reduced to the conditions of the Theorem <ref>.

* For any $(a, b) \in P$, we have

$$%(b, \rho(ab)) \in P, \quad (\rho(ab), a) \in P,
(ab, \rho(b)) \in P
\ \mbox{and} \ (\rho(a), ab) \in P.$$

* For any $(a, b) \in P$, we have

$$b \rho(ab) = \rho(a), \ \rho(ab) a = \rho(b),% \quad \rho(b) \rho(a) = \rho(ab), \quad (ab) \rho(b) = a,
\ \mbox{and} \ \rho(a) (ab) = b.$$

The following two coditions are required for Y-oriented R$5$ moves (Fig. <ref>).

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Y-oriented R$5$ moves

* For any $a, b, x \in R$, the following two statements are equivalent.

$$(a, b) \in P \  \mbox{and} \  (a \ast x, b \ast x) \in P.$$

* For any $(a, b) \in P$ and $x \in R$, we have

$$(ab) \ast x = (a \ast x)(b \ast x) \  \mbox{and} \  x \ast (ab) = (x \ast a) \ast b.$$

The following two conditions are required for Y-oriented R$6$ moves (Fig. <ref>).

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(6,1.5) – (7,0.5);
(5,1.5) circle(0.15);
(6,1.5) circle(0.15);
(4.5,2) node-45$\blacktriangleright$;
(4.5,1) node45$\blacktriangleleft$;
(5.5,1.5) node$\blacktriangleright$;
(6.5,2) node45$\blacktriangleright$;
(6.5,1) node-45$\blacktriangleright$;
(4,2.5) node[left]$a(bc)$;
(7,2.5) node[right]$c$;
(4,0.5) node[left]$a$;
(5.5,1) node$bc$;
(7,0.5) node[right]$b$;

[line width=1.6pt]
(-2,2.5) – (-3,1.5) – (-4,1.5) – (-5,2.5);
(-5,0.5) – (-4,1.5);
(-2,0.5) – (-3,1.5);
(-3,1.5) circle(0.15);
(-4,1.5) circle(0.15);
(-4.5,2) node-45$\blacktriangleright$;
(-4.5,1) node45$\blacktriangleleft$;
(-3.5,1.5) node$\blacktriangleleft$;
(-2.5,2) node45$\blacktriangleleft$;
(-2.5,1) node-45$\blacktriangleright$;
(-5,2.5) node[left]$b$;
(-2,2.5) node[right]$c$;
(-5,0.5) node[left]$a$;
(-3.5,1) node$\rho(b) a$;
(-2,0.5) node[below]$\rho(\rho(b) a) c$;

[<->] (-0.5,1.5) – (-1.5,1.5);
(-1,1) nodeR$6$;

(0,3) – (1,2) – (2,3);
(1,2) – (1,1);
(0,0) – (1,1) – (2,0);
(0.5,2.5) node-45$\blacktriangleright$;
(1.5,2.5) node45$\blacktriangleleft$;
(1,1.5) node$\blacktriangledown$;
(0.5,0.5) node45$\blacktriangleleft$;
(1.5,0.5) node-45$\blacktriangleright$;
(1,2) circle(0.15);
(1,1) circle(0.15);
(0,3) node[left]$b$;
(2,3) node[right]$c$;
(0,0) node[left]$a$;
(1,1.5) node[left]$bc$;
(2,0) node[right]$\rho(a) (bc)$;

[<->] (2.5,1.5) – (3.5,1.5);
(3,1) nodeR$6$;

(4,2.5) – (5,1.5) – (6,1.5) – (7,2.5);
(4,0.5) – (5,1.5);
(6,1.5) – (7,0.5);
(5,1.5) circle(0.15);
(6,1.5) circle(0.15);
(4.5,2) node-45$\blacktriangleright$;
(4.5,1) node45$\blacktriangleleft$;
(5.5,1.5) node$\blacktriangleright$;
(6.5,2) node45$\blacktriangleleft$;
(6.5,1) node-45$\blacktriangleright$;
(4,2.5) node[left]$b$;
(7,2.5) node[right]$c$;
(4,0.5) node[left]$a$;
(5.5,1) node$\rho(a) b$;
(7,0.5) node[right]$(\rho(a) b) c$;

[line width=1.6pt]
(-2,2.5) – (-3,1.5) – (-4,1.5) – (-5,2.5);
(-5,0.5) – (-4,1.5);
(-2,0.5) – (-3,1.5);
(-3,1.5) circle(0.15);
(-4,1.5) circle(0.15);
(-4.5,2) node-45$\blacktriangleright$;
(-4.5,1) node45$\blacktriangleleft$;
(-3.5,1.5) node$\blacktriangleleft$;
(-2.5,2) node45$\blacktriangleright$;
(-2.5,1) node-45$\blacktriangleleft$;
(-5,2.5) node[left]$b$;
(-2,2.5) node[right]$c$;
(-5,0.5) node[left]$a$;
(-3.5,1) node$\rho(b) a$;
(-2,0.5) node[below]$c (\rho(b) a)$;

[<->] (-0.5,1.5) – (-1.5,1.5);
(-1,1) nodeR$6$;

(0,3) – (1,2) – (2,3);
(1,2) – (1,1);
(0,0) – (1,1) – (2,0);
(0.5,2.5) node-45$\blacktriangleright$;
(1.5,2.5) node45$\blacktriangleright$;
(1,1.5) node$\blacktriangledown$;
(0.5,0.5) node45$\blacktriangleleft$;
(1.5,0.5) node-45$\blacktriangleleft$;
(1,2) circle(0.15);
(1,1) circle(0.15);
(0,3) node[left]$b$;
(2,3) node[right]$c$;
(0,0) node[left]$a$;
(1,1.5) node[left]$b\rho(c)$;
(2,0) node[right]$\rho(b \rho(c)) a$;

[<->] (2.5,1.5) – (3.5,1.5);
(3,1) nodeR$6$;

(4,2.5) – (5,1.5) – (6,1.5) – (7,2.5);
(4,0.5) – (5,1.5);
(6,1.5) – (7,0.5);
(5,1.5) circle(0.15);
(6,1.5) circle(0.15);
(4.5,2) node-45$\blacktriangleright$;
(4.5,1) node45$\blacktriangleleft$;
(5.5,1.5) node$\blacktriangleright$;
(6.5,2) node45$\blacktriangleright$;
(6.5,1) node-45$\blacktriangleleft$;
(4,2.5) node[left]$b$;
(7,2.5) node[right]$c$;
(4,0.5) node[left]$a$;
(5.5,1) node$\rho(a) b$;
(7,0.5) node[right]$c \rho(\rho(a) b)$;

Y-oriented R$6$ moves

* For any $a, b, c \in R$, the following are equivalent.

$$\left[(a, b) \in P \wedge (ab, c) \in P\right] \  \mbox{and} \  \left[(b, c) \in P \wedge (a, bc) \in P\right].$$

* For any $(a, b), (ab, c) \in P$, we have

$$(ab)c = a(bc).$$

From the above,
we obtain the conditions (1) – (5) of Theorem <ref>.

Therefore, whenever we consider colorings assuming the conditions shown in Figs. <ref>, <ref> and <ref>, the condition (1) – (5) of Theorem <ref> must be satisfied.
Suppose that $R = (R, \ast, \rho)$ and $\mu : P \to R$ satisfy the conditions (1) – (5) and let $R' = \bigcup_{(a,b) \in P}\left\{a,b\right\}$.

For any $a \in R'$,
$(\rho(a), a) \in P$ and $(a, \rho(a)) \in P$.

From the definition of $R'$,
there exists an element $b \in R$ such that $(a, b) \in P$ or $(b, a) \in P$.

If $(a, b) \in P$, we have $(\rho(ab), a) \in P$ by ($4$).
From ($3$), $(\rho(a), ab) \in P$.
Using ($1$), we have $(\rho(a), a) \in P$.
Furthermore, using ($4$) and ($6$), we have $(a, \rho(\rho(a)a)) = (a, \rho(a) a) \in P$.
Therefore, from ($1$), we have $(a, \rho(a)) \in P$.

If $(b, a) \in P$, we have $(a, \rho(ba)) \in P$ by ($4$).
From ($3$), $(ba, \rho(a)) \in P$.
Using ($1$), we have $(a, \rho(a)) \in P$.
Furthermore, using ($4$) and ($6$), we have $(\rho(a), \rho(a \rho(a))) = (\rho(a), a \rho(a)) \in P$.
Therefore, from ($1$), we have $(\rho(a), a) \in P$.

For any $a \in R'$, the following (i) and (ii) hold.

(i) For any $(x, a \rho(a)) \in P$, $x (a\rho(a)) = x$.

(ii) For any $(a \rho(a), y) \in P$, $(a \rho(a)) y = y$.

(i) :
($1$) and Lemma <ref> imply $(x, a) \in P$.
From ($2$) and ($5$), we have
$x (a \rho(a)) = (xa) \rho(a) = x$.

(ii) :
($1$) and Lemma <ref> imply $(\rho(a), y) \in P$.
Using ($2$), ($3$) and ($5$), we have
$(a \rho(a))y = a (\rho(a) y)
= \rho^{2}(a(\rho(a)y))
= \rho((\rho(y)a)\rho(a))
= \rho^{2}(y)
= y$.

$\rho(R') = R'$.

By Lemma <ref>,
we see that
for any $a \in R'$, $\rho(a) \in R'$.
Then we have $\rho(R') \subset R'$.
And for any $a \in R'$, $a = \rho^{2} (a) \in \rho(R')$ implies $R' \subset \rho(R')$.

For any $x \in R$,
$S_{x} (R') = R'$.

By ($3$),
we have $a \ast x \in R'$ for any $a \in R'$.
Therefore $S_{x} (R') \subset R'$.
Furthermore $R' = S_{x} \circ S_{\rho(x)} (R') \subset S_{x} (R')$.

(i): From Lemmas <ref> and <ref>,
$R'$ is closed under $\ast$, $\ast^{-1}$ and $\rho$.
Therefore $R'$ is a subrack of $R$ and the restriction of $\rho$ to $R'$ is a good involution on $R'$.

(ii): For each element $x$ of $R$ with $(x, y) \in P$ or $(y, x) \in P$ for some $y \in R$, let us introduce two symbols $s_{x}$ and $t_{x}$.
Let $X$ be the set $\bigcup_{(a, b) \in P} \left\{s_{a}, s_{b}, t_{a}, t_{b}\right\}$
and $\sim$ be the equivalence relation on $X$ generated by
$\left\{(t_{a}, s_{b}) \mid (a, b) \in P\right\} \subset X \times X$.
We consider the groupoid $\mathcal{C}$ defined by the following.

* $\textrm{Ob}(\mathcal{C}) = X/\sim$.

* For any $x, y \in \textrm{Ob}(\mathcal{C})$, $\textrm{Hom}(x, y) = \left\{a \in R \mid s_{a} \in x, t_{a} \in y\right\}$.

* The composition $\mathrm{Hom}(x, y) \times \mathrm{Hom}(y, z) \to \mathrm{Hom}(x, z)$ is defined by $(a, b) \mapsto ab$.

* Let $x \in \mathrm{Ob}(\mathcal{C})$.
If $x = \left[s_{a}\right]$ then
$a \rho(a)$ is the identity for $x \in \textrm{Ob}(\mathcal{C})$.
If $x = \left[t_{a}\right]$ then $\rho(a) a$ is the identity for $x \in \textrm{Ob}(\mathcal{C})$.

* For any $a \in \textrm{Hom}(x, y)$, $\rho(a)$ is the inverse of $a$.

The identity morphisms are well-defined by Lemmas <ref> and <ref>.
If $\left[s_{a}\right] = \left[s_{b}\right]$ then
we have $a \rho(a) = b \rho(b)$.
If $\left[t_{a}\right] = \left[t_{b}\right]$ then
we have $\rho(a) a = \rho(b) b$.
If $\left[s_{a}\right] = \left[t_{b}\right]$ then
we have $a \rho(a) = \rho(b) b$.

Then $R' = \textrm{Hom}(\mathcal{C})$ as a set,
$\mu$ is equal to the composition of morphisms as a map and
$\ast$ satisfies the conditions of a groupoid rack.
Therefore $(R', \ast)$ is a groupoid rack.

§ ACKNOWLEDGEMENT

The author would like to thank Seiichi Kamada, Atsushi Ishii and Yuta Taniguchi for helpful advice and discussion on this research.

(K. Arai) Department of Mathematics, Graduate School of Science, Osaka University, 1-1, Machikaneyama, Toyonaka, Osaka, 560-0043, Japan


# Relating exponents of truncated power laws for self-similar signals

Michael Heisel<EMAIL_ADDRESS>Department of Atmospheric and Oceanic
Sciences, University of California in Los Angeles, Los Angeles, USA

###### Abstract

Stochastic simulations are conducted to evaluate relations between statistics
for processes governed by a power law probability distribution with exponent
$\alpha$. The power law exponent is determined for statistics of simulated
signals, namely the box-counting fractal dimension $D$, energy spectrum
exponent $\beta$, and an intermittency exponent $\mu$. For a binary signal
with no variability in amplitude, the parameters are related linearly as
$D=2-\beta=1-\mu$. The relations are unchanged if the sampled power law
distribution is truncated to a finite range of values, e.g. for a distribution
exhibiting a cutoff. However, truncating the distribution yields statistics
that are not truly scale-invariant, and distorts the connection between the
statistic exponents and $\alpha$. The behavior is due to the survival
function, or the complementary cumulative distribution, which for a finite-
sized power law is only approximately self-similar and has an effective
exponent differing from $\alpha$. An expression for the effective exponent is
presented. The results are discussed in the context of turbulent flows, but
are generally applicable to any statistically self-similar signal.

##  I Introduction

Attributing a self-similar (scale-invariant) power law to real-world data can
be controversial [1, 2, 3], yet power laws have been applied to various topics
including allometry [4], anomalous transport [5], geology [6], and hydrology
[7], among many others. Mechanistic concepts underlying these power laws
include random walks [8, 9], fractal geometries [10, 11], and self-organized
criticality [12, 13, 14].

In addition to the examples above, turbulent fluid dynamics is one of
relatively few fields where the existence of self-similarity is supported by
both theory [15] and extensive observation [16]. The most well-known power law
in turbulence occurs within the energy spectrum: for intermediate scales the
fluctuating energy decays as $E\sim f^{-\beta}$, where $f$ is the frequency
(or wavenumber) and $\beta$ is the spectral exponent. Attempts to relate self-
similarity in turbulence to concepts such as fractal geometries have been less
definitive [17, 18].

Relating $\beta$ in turbulence to other power law statistics such as the
fractal dimension is inhibited by two challenges. First, a limited span of
$\beta$ values is observed from turbulent measurements, which precludes
empirical fits across the parameter space of $\beta$. Second, the power law
behavior is restricted to a narrow range of scales based on the Reynolds
number $Re$. For instance, laboratory and numerical experiments typically have
Taylor miscroscale Reynolds number $Re_{\lambda}\sim O(10^{2})$, corresponding
to one decade (i.e. order of magnitude) of approximately self-similar behavior
[16].

In terms of probability distributions, a power law may be truncated to a
finite range of values due to undersampling [19], computational modeling
limitations [20], or the natural bounds of the system [21] as in the
turbulence example. In these cases, there are finite-size effects and
cumulative probabilities deviate from a true power law [19, 21]. While the
effect on estimated probabilities is well documented, less attention has been
paid to the statistics of processes governed by truncated power law
distributions. Specifically, it is not clear how deviations in cumulative
probabilities translate to various statistics like the energy spectrum, and
whether any correction is required for existing relations derived assuming an
unbounded power law.

To investigate these effects, stochastic simulations are used here to compute
power law statistics across a range of exponent values. The truncation of the
underlying power law distribution is also varied. The statistics evaluated
here are the fractal box-counting dimension, the energy spectrum, and an
intermittency parameter. The simulations are analogous to a Monte Carlo
analysis, except the goal is to identify the ensemble average of statistics
rather than their uncertainty. While turbulent flows motivate the study, the
simulations are purely stochastic and do not directly model any governing
physics. This stochastic approach is used to address two general questions
applicable to any truncated self-similar process: (i) for a signal governed by
a power law probability distribution, how are the ensuing statistics related?
(ii) how do these relations change when the range of self-similarity is
increasingly confined?

The study is organized into the following sections: Sec. II describes the
stochastic simulations, Sec. III presents results of the simulations, Sec. IV
introduces a correction for truncated distributions, and Sec. V summarizes the
findings. Finally, the appendix provides equations for the simulations.

##  II Stochastic simulations

The premise of the stochastic simulations is to create a synthetic signal
$s(t)$ defined by a sequence of “events”, where the interval $\tau$ between
events is governed by a power law probability density function (PDF). Here,
$t$ and $s$ can represent any independent and dependent variable,
respectively. The power law PDF is defined as

$PDF(\tau)=\frac{1}{1-\left(\tau_{2}/\tau_{1}\right)^{1-\alpha}}\frac{\alpha-1}{\tau_{1}}\left(\frac{\tau}{\tau_{1}}\right)^{-\alpha},$
(1)

where $\tau_{1}$ and $\tau_{2}$ are the minimum and maximum values of the
distribution, respectively, and $\alpha$ is the distribution exponent. The
power law in Eq. (1) is equivalent to a Pareto distribution with exponent
$\alpha-1$. The integral of the PDF is equal to unity – as required by the PDF
definition – only for $\alpha>$ 1 and if a minimum value is imposed. The
integral is infinite and the PDF is not well-defined for $\alpha\leq$ 1\. The
minimum $\tau_{1}=$ 1 is used here for simplicity. While a maximum value
$\tau_{2}$ is typically $\infty$ for an unbounded power law,
$\tau_{2}=10^{300}$ is employed to avoid infinite values in the simulations.
This value is close to the largest definable number in double-precision
floating-point format, and yields 300 decades of power law. For the present
goals, this case is considered equivalent to an unbounded power law. An
example PDF is shown in Fig. 1(a).

Figure 1: Example signal constructed from a power law probability distribution
$PDF\sim\tau^{-\alpha}$ spanning 300 orders of magnitude (i.e. decades). (a)
Probability distribution for the interval $\tau$ between events in a signal
following Eq. (1). (b) Determination of a single $\tau$ value using inverse
transform sampling of the cumulative distribution $CDF$ and Eq. (3). (c)
Position $t$ of events based on ten $\tau$ values. (d) Signal $s(t)$ whose
value 0 or 1 changes at each event.

Inverse transform sampling is used to select values of $\tau$ from the
distribution. In this approach, the cumulative distribution function

$CDF(\tau)=\frac{1}{1-\left(\tau_{2}/\tau_{1}\right)^{1-\alpha}}\left[1-\left(\frac{\tau}{\tau_{1}}\right)^{1-\alpha}\right]$
(2)

is inverted to define $\tau$ as a function of $CDF$:

$\tau(CDF)=\tau_{1}\left[1-\left(1-\left(\frac{\tau_{2}}{\tau_{1}}\right)^{1-\alpha}\right)CDF\right]^{\frac{1}{1-\alpha}}.$
(3)

The $CDF$ value is simulated by selecting a random value between 0 and 1, and
the corresponding interval $\tau$ is determined from Eq. (3) as shown in Fig.
1(b). A small sample of events is shown in Fig. 1(c), where each event is
separated by simulated intervals $\tau$.

The position $t$ of each event is used to build a synthetic signal $s(t)$ on a
discrete domain between 0 and $t_{max}=10^{6}$. The resolution between points
on the domain is $\tau_{1}=$ 1, thus allowing for six decades of statistics in
the signal. The choice for $t_{max}$ is limited here by the computational
resources required to define $s(t)$. As seen in Fig. 1(d), the signal $s(t)$
is designed to alternate its value between 0 and 1 at each event.

Later results present simulations for one hundred $\alpha$ values between 1.02
and 3. The value $\alpha=1$ is excluded because the PDF is not well-defined as
discussed above. For a given $\alpha$, intervals $\tau$ are simulated until
$t_{max}$ is exceeded to ensure the signal is fully populated. Statistics are
thereafter calculated using $s(t)$, and the process is repeated until the
ensemble average statistics are converged. The number of signals contributing
to each statistic varies between $10^{2}$ for large $\alpha$ and $10^{4}$ for
small $\alpha$. The latter returns sparse signals, which require a larger
number of realizations to converge statistics.

With respect to turbulent flows, $s(t)$ is analogous to the so-called
telegraph approximation (TA) [22, 23] based on zero crossings of velocity
fluctuations [24, 25, 26]. The TA signal is 1 when the fluctuating velocity is
positive, and is 0 when the velocity is negative. The “event” in this case
occurs when the fluctuating velocity crosses zero, and the interval $\tau$
between crossings is known as the interpulse period [25, 23] or persistence
[27, 28, 29]. The interpulse PDF is well approximated by a lognormal
distribution for small $\tau$ [30, 31], a power law for intermediate $\tau$
[22, 23], and an exponential cutoff for the largest $\tau$ [25, 23, 28].

The same PDF shape – a blend of lognormal, power law, and exponential
distributions – is used here to simulate the effect of a truncated power law
region. The case represents a weak power law because a power law expression
does not describe the full range of values in the distribution [32]. Same as
for the unbounded power law, inverse transform sampling is used to simulate
$s(t)$ based on a random selection of values for $\tau$. The construction of
the distribution and definitions for $PDF(\tau)$ and $\tau(CDF)$ are provided
in the appendix. Figure 2 shows a synthetic truncated power law. An example
turbulent zero-crossing (interpulse) PDF is included for reference. The
interpulse is estimated from hotwire anemometry measurements of boundary layer
turbulence [33], and exhibits a similar shape to the simulated PDF with one
decade of power law. Results for distributions with one, three, and five
decades of power law are presented herein.

Figure 2: Power law distribution whose extent is truncated by lognormal and
exponential curves. The example exhibits one decade of power law. (a)
Probability distribution for $\tau$ following Eq. (12). The gray line is an
example zero-crossing distribution from boundary layer turbulence
measurements, shifted for visual comparison. (b) Inverse transform sampling of
$CDF$ following Eq. (15).

##  III Results

###  III.1 Fractal Dimension

The fractal dimension of the one-dimensional signal $s(t)$ is estimated here
using the box counting method [see, e.g., 34, 11, 35]. In this approach,
$s(t)$ is discretized into segments of size $\Delta t$ and the number of
segments $N$ containing at least one event is counted. If the resulting
dependency $N(\Delta t)$ follows a power law

$N(\Delta t)\sim\Delta t^{-D},$ (4)

the signal is considered statistically self-similar. Whether the signal is
also considered to be a fractal object depends on the definition, as
fractality is sometimes reserved for geometric shapes. The fractal dimension
$D$ in Eq. (4) is bounded between 0 and 1. These limits correspond to a signal
with $\tau>\Delta t$ (for $D=0$) or $\tau<\Delta t$ (for $D=1$) for all
intervals $\tau$ across the tested range of $\Delta t$.

Example box-counting results are shown in Fig. 3(a,b) for a range of $\alpha$
and power law PDF extent. The expected power law in Eq. (4) is observed for
the infinite distribution. However, $N(\Delta t)$ becomes increasingly
dissimilar from a power law as the self-similar region in $PDF(\tau)$ is
increasingly truncated. This trend may help to explain conflicting findings on
the fractal dimension of isosurfaces in turbulence, where several studies have
observed $D$ to depend on $\Delta t$ [e.g., 36, 37]. The observed scale
dependence may be due to an insufficient Reynolds number (and thus a limited
power law extent) for an approximately self-similar region in $N(\Delta t)$ to
develop. The reason for the absence of a power law in Fig. 3(b) is discussed
later in Sec. IV.

Figure 3: Statistics for the fractal dimension $D$ estimated via box counting.
(a) Box count $N\sim\Delta t^{-D}$ for the unbounded case and $\alpha=1.1$
(dotted line), $\alpha=1.5$ (solid), and $\alpha=2$ (dashed). (b) Box count
for $\alpha=1.5$ and a varying power law extent, where the vertical lines
delineate the power law region for the one decade case. (c) Fractal dimension
$D$ as a function of $\alpha$ with $D=\alpha-1$ (line) for reference. In this
and later figures, the legend indicates the number of power law decades in
$PDF(\tau)$.

Despite the departure from a power law, the dimension $D$ is estimated by
fitting Eq. (4) to the Fig. 3(a,b) curves assuming $D$ is constant. The fit is
performed within the range of $\Delta t$ corresponding to the self-similar
region in $PDF(\tau)$. The fitted values for $D$ across the tested range of
$\alpha$ are shown in Fig. 3(c). The error bars correspond to the change in
$D$ when the region where the power law is fitted is shifted by a factor of
two in either direction. The error bars therefore increase as the dependence
$D(\Delta t)$ increases. The large error bars corresponding to small $\alpha$
in Fig. 3(c) reflect the dissimilarity from a true power law observed in Fig.
3(b). The same method is used to calculate the error bars in later figures.

For the unbounded PDF case in Fig. 3(c), $D(\alpha)$ follows a linear trend
$D=\alpha-1$ up to approximately $\alpha\approx$ 1.5, and thereafter $D$
asymptotically approaches 1. The same asymptotic behavior is observed for the
truncated distributions. However, the results for small $\alpha$ depart from
the linear relation as the power law extent decreases. The reason for the
departure is related to the trends in Fig. 3(b) and is discussed in Sec. IV.

###  III.2 Energy spectrum

The energy spectrum – more accurately the energy spectral density – is defined
as

$E(f)\sim\lvert\hat{s}(f)\rvert^{2},$ (5)

where $\hat{s}(f)$ is the Fourier transform of $s(t)$ in frequency or
wavenumber space. Prior to computing the transform, the signal is filtered
using a Hamming window and zero-padded to prevent aliasing in $\hat{s}(f)$.

Using the same format as Fig. 3, the energy spectra are shown in Fig. 4.
Spectra for the truncated distributions in Fig. 4(b) exhibit stronger self-
similarity than the box counts in Fig. 3(b). The Fourier transform efficiently
isolates local (in scale) contributions to the variance. In contrast, the box-
counting effectively measures a cumulative effect capturing all intervals
smaller than the given $\Delta t$. For the cumulative statistics, the non-
power-law behavior is spread across scales to the expected self-similar
region.

Figure 4: Statistics for the energy spectrum power law exponent $\beta$. (a)
Spectrum $E$ for the unbounded case and $\alpha=1.1$ (dotted line),
$\alpha=1.5$ (solid), and $\alpha=2$ (dashed). (b) Spectrum for $\alpha=1.5$
and a varying power law extent, where the vertical lines delineate the power
law region for the one decade case. (c) Exponent $\beta$ as a function of
$\alpha$ with $\beta=3-\alpha$ (line) for reference.

The spectral exponent $\beta$ is estimated by fitting the power law $E\sim
f^{-\beta}$ to each individual spectrum. The dependency of $\beta$ on $\alpha$
is shown in Fig. 4(c). The apparent “roughness” of the curves is attributed to
the shape of $s(t)$. Artificial oscillations appear in the energy spectrum
when the sinusoidal basis functions of the Fourier transform are used to
decompose the discontinuous signal. These oscillations may propagate to
$\beta$ as the fitted power law region varies with $\alpha$.

As before, the error bars are largest for the one decade case, due to the
curvature of $E(f)$ immediately adjacent to the expected self-similar region.
The trend for small $\alpha$ observed in Fig. 3(c) is similarly present for
the spectrum exponent. The unbounded PDF case follows a linear relation
$\beta=3-\alpha$ in Fig. 4(c) up to $\alpha\approx$ 2\. This relation is
applicable to a superposition of Poisson processes [14], and appears similarly
applicable to the self-similar process simulated here. For $\alpha>$ 2, the
results slowly deviate from the linear relation and there is agreement across
cases. It is assumed that $\beta$ asymptotically approaches 0 as $\alpha$
increases, but this trend cannot be confirmed due to the limited tested range
of $\alpha$.

###  III.3 Intermittency

The intermittency is another quantitative measure of variability in the
distribution of events. Intermittency can be parameterized using Obukhov’s
local moving average [38]

$\chi(t,\Delta t)=\frac{1}{\Delta t}\int_{t}^{t+\Delta
t}\bigg{|}\frac{ds^{2}}{dt}\bigg{|}dt.$ (6)

Given the amplitude of $s(t)$ is invariable, the integral corresponds to the
number of events occurring within “windows” of size $\Delta t$. The
intermittency is quantified using the scaling [39, 40, 22]

$\frac{\langle\chi^{2}\rangle}{\langle\chi\rangle^{2}}\sim\Delta t^{-\mu},$
(7)

Figure 5: Statistics for the intermittency exponent $\mu$. (a) Intermittency
parameter $\langle\chi^{2}\rangle/\langle\chi\rangle^{2}$ for the unbounded
case and $\alpha=1.1$ (dotted line), $\alpha=1.5$ (solid), and $\alpha=2$
(dashed). (b) Intermittency for $\alpha=1.5$ and a varying power law extent,
where the vertical lines delineate the power law region for the one decade
case. (c) Exponent $\mu$ as a function of $\alpha$ with $\mu=2-\alpha$ (line)
for reference.

where angled brackets $\langle\cdot\rangle$ indicate an ensemble average
across $t$. Equation (7) is defined here using the second-order moment, but
the same principle can be applied to higher-order moments. The exponent $\mu$
represents how the variability in number of events across windows changes as
the window size is increased.

The intermittency parameter statistics are shown in Fig. 5. Results for the
truncated distributions in Fig. 5(b) exhibit a lack of self-similarity. As for
the box counting methodology, Eq. (6) accounts for all intervals smaller than
$\Delta t$, resulting in a cumulative metric.

Values for $\mu$, fitted using Eq. (7), are plotted as a function of $\alpha$
in Fig. 5(c). The curves follow the same trends as $D$ and $\beta$. A linear
relation $\mu=2-\alpha$ is observed for the unbounded distribution and small
$\alpha$ values. Truncating the power law distribution leads to a departure
from the linear relation. For larger $\alpha$, all cases deviate from
$\mu=2-\alpha$ as $\mu$ asymptotically approaches zero.

###  III.4 Exponent relations

The relations between power law exponents $D$, $\beta$ and $\mu$ are plotted
in Fig. 6 for the tested range of $\alpha$. The lines correspond to the linear
trends in panel (c) of Figs. 3, 4, and 5. For visual clarity, the error bars
from previous figures are only reproduced in Fig. 6 for the case with one
decade of self-similarity.

Figure 6: Relations between power law exponents of the presented statistics.
(a) Fractal dimension $D$ and intermittency exponent $\mu$, compared to
$\mu=D-1$ (line). (b) Dimension $D$ and spectrum exponent $\beta$, compared to
$\beta=2-D$ (line). (c) Exponents $\mu$ and $\beta$, compared to $\beta=\mu+1$
(line).

A robust inverse relation between $\mu$ and $D$ is observed in Fig. 6(a). The
results are in close agreement with the prediction $\mu=1-D$ [41, 40], which
can be derived via the correlation dimension [42]. The linear trend is
invariant to truncation of the power law PDF, and the primary difference
across cases is the observed range in $\mu$ and $D$ values.

Linear trends $\beta=2-D$ and $\beta=\mu+1$ also exist for the spectral
exponent in Fig. 6(b,c). However, the linearity is limited to $\beta\gtrapprox
1.2$. The behavior for smaller $\beta$ is attributed to the slower rate at
which $\beta$ asymptotically approaches zero, compared to the corresponding
rates for $D$ and $\mu$.

The results for $PDF(\tau)$ with one decade of power law are visibly offset
from the other cases in Fig. 6(b,c). The difference in $\beta$ is
approximately 0.2, which is within the extent of the error bars. The
difference may therefore be due to the lack of self-similarity in the
statistics and the precise range chosen to fit the power law exponents. The
result highlight the challenge in recovering the expected relations when the
power law behavior is truncated to a narrow range of probability and the
statistics lose the signature of self-similarity.

Aside from the offset, the relations in Fig. 6 do not depend on the bounds of
the power law PDF. Direct linear relations can be expected between power law
statistics, even if the governing process is self-similar across a finite
range of values. The effect of a truncated power law is therefore limited to
the altered connection between the statistics and the probability exponent
$\alpha$.

##  IV Effect of a truncated power law

The probability distribution definitions in Eqs. (1) and (2) impose a finite
maximum value $\tau_{2}$. As a result, intervals above $\tau_{2}$ are under-
sampled relative to an infinite power law distribution [19, 21]. Values within
the power law region may be under- or over-sampled, depending on the shape of
the cutoff regions bounding the power law.

The consequence of the sampling discrepancy is apparent in the survival
function $1-CDF(\tau)$, also known as the complementary cumulative
distribution. For the power law in Eq. (2), the survival function can be
expressed as

$1-CDF(\tau)=\frac{\left(\tau/\tau_{1}\right)^{1-\alpha}-\left(\tau_{2}/\tau_{1}\right)^{1-\alpha}}{1-\left(\tau_{2}/\tau_{1}\right)^{1-\alpha}}.$
(8)

Equation (8) is only a power law for $\tau_{2}/\tau_{1}\rightarrow\infty$, and
otherwise has a finite additive constant distorting the probability that the
next interval exceeds a given value of $\tau$.

The survival functions for three values of $\alpha$ are shown in Fig.
7(a,b,c). The trends are consistent with the previously observed dependencies
on $\alpha$: the survival function for the truncated distributions only
approximate a power law, and the departure from the unbounded case is largest
for small $\alpha$ values.

Figure 7: Effective exponent $\alpha_{e}$ resulting from the distortion of the
survival function by the truncated power law. Rows correspond the the survival
function $1-CDF$ (a,b,c) and associated exponent $\alpha_{e}$ following Eq.
(10) (d,e,f). Columns correspond to values of $\alpha$: 1.1 (a,d); 1.5 (b,e);
and 2 (c,f).

To quantify the deviation from the unbounded case, the effective exponent
$\alpha_{e}$ can be calculated from the slope of the curves in Fig. 7(a,b,c).
Mathematically, the exponent is

$1-\alpha_{e}=\frac{d}{d\log(\tau)}\left[\log\left(1-CDF\right)\right].$ (9)

The chain rule can be used to simplify the derivative operation as
$d/d\log(\tau)=\tau d/d\tau$. Using Eq. (8) to compute the derivative of
$\log(1-CDF)$, the effective exponent can be expressed as

$\alpha_{e}(\tau)=1+\frac{\tau PDF(\tau)}{1-CDF(\tau)}.$ (10)

Importantly, the exponent changes as a function of $\tau$, reflecting the fact
that the survival function is not a true power law for a truncated
distribution. The effective exponent is shown in Fig. 7(d,e,f) for each
simulated case. Within the range of $\tau$ where a power law is expected,
$\alpha_{e}$ becomes increasingly larger than $\alpha$ as $\alpha$ decreases
and as the extent of the power law shortens. The decrease in $\alpha$ and the
power law extent both represent an increase in the portion of the $CDF$ that
is “missing” due to the cutoff. For a power law defined only in the range
between $\tau_{1}$ and $\tau_{2}$, $\alpha_{e}$ is

$\alpha_{e}(\tau)=1+\left(\alpha-1\right)\frac{\left(\tau/\tau_{1}\right)^{1-\alpha}}{\left(\tau/\tau_{1}\right)^{1-\alpha}+\left(\tau_{2}/\tau_{1}\right)^{1-\alpha}}.$
(11)

The result $\alpha_{e}=\alpha$ is recovered when
$\tau_{2}/\tau_{1}\rightarrow\infty$. The equation for the truncated
distribution in Fig. 2 follows a similar form to Eq. 11, except the right-side
term in the denominator depends on the prescribed cutoff behavior.

In practice, $\alpha_{e}$ in Eq. 10 can be estimated from discrete histograms
approximating $PDF(\tau)$. A single representative value of $\alpha_{e}$
should be taken as the average of $\alpha_{e}(\tau)$ within the range of
$\tau$ where power law statistics are fitted.

Fig. 8(a) compares the spectrum exponent as a function of $\alpha$ and
$\alpha_{e}$. Substituting for the effective exponent $\alpha_{e}$ accounts
for the trends observed in Fig. 4(c), and the results follow
$\beta=3-\alpha_{e}$. The remaining deviation in the one decade case is
attributed to the strong departure from self-similar statistics previously
discussed.

Figure 8: Spectrum exponent $\beta$ as a function of $\alpha$ (transparent
markers) and the effective exponent $\alpha_{e}$ (opaque). (a) Simulated
results. (b) Boundary layer turbulence measurements of the longitudinal
($\bullet$) and wall-normal ($\times$) velocity components.

As a practical example, Fig. 8(b) shows the exponent correction for turbulent
TA signals. The signals represent a range of positions within a wind tunnel
boundary layer above both smooth and rough surfaces [33]. The results for
$\beta(\alpha_{e})$ align with the expected linear relation and exhibit
reduced scatter compared to $\beta(\alpha)$. The observed range in $\beta$ is
due to the Reynolds number of the experiment, which is insufficient for a
distinct self-similar region to emerge in the spectrum, particularly for the
wall-normal velocity component. The effective exponent $\alpha_{e}$ in Fig.
8(b) successfully accounts for this limitation.

The discrepancy between $\beta(\alpha)$ in Fig. 8(b) and $\beta=3-\alpha$ has
been previously explained as an effect of intermittency. Specifically, the
correction $\beta=3-\mu/2-\alpha$ was proposed [22]. The present simulations
demonstrate that $\beta=3-\alpha$ is applicable to intermittent, self-similar
processes governed by an unbounded power law. Adjusting the unbounded power
law in Fig. 4 for intermittency would lead to incorrect results. Rather, the
relations between the power law statistics and $PDF(\tau)$ must consider the
effective exponent $\alpha_{e}$ in Eq. (10) resulting from a truncated
distribution. The success of the correction in Fig. 8 demonstrates that the
distortion of the survival function propagates to statistics based on the
simulated signals, and that the resulting parameters $D$, $\beta$, and $\mu$
depend on the survival function and its effective exponent $1-\alpha_{e}$.

##  V Summary

For a stochastic process described by an unbounded power law probability
distribution, linear equations exist to relate the power law exponent of
various statistics. The fractal dimension $D$, energy spectrum exponent
$\beta$, and intermittency exponent $\mu$ are related as $D=2-\beta=1-\mu$.
The relation between $D$ and $\mu$ matches the predicted analytical solution,
and the relation between $D$ and $\beta$ is a new empirical finding for binary
signals. The statistics are also linearly related to the probability exponent
$\alpha$, e.g. as $\beta=3-\alpha$, for small values of $\alpha$. For larger
values $\alpha\gtrapprox 2$, the statistics asymptotically approach their
limiting values. While a small selection of statistics are evaluated here,
similar relations and trends are expected for other parameters like the
correlation integral [43].

The same connection between $D$, $\beta$, and $\mu$ is retained if the power
law probability distribution is truncated to a finite range of values.
However, the finite limits change the effective exponent $\alpha_{e}$ of the
survival function, which propagates to the ensuing statistics. The original
relation $\beta=3-\alpha_{e}$ is recovered by replacing $\alpha$ with the
effective exponent in Eq. (10). Yet, the statistics are only approximately
self-similar, as $\alpha_{e}(\tau)$ is scale-dependent when the distribution
is finite. The departure from self-similarity may be even greater in practice,
as the effect of an exponential cutoff on the scale invariance in the
probability distribution is not considered here [44].

The effect of a truncated distribution is consistent with observations in
turbulence, where the self-similar range of scales is limited in extent by the
flow Reynolds number. Previous findings on the scale dependence of the fractal
dimension [36, 37] can be explained as a statistical consequence of the
truncated distribution. Additionally, deviations from $\beta=3-\alpha$ in
turbulent TA signals (Fig. 8) are well-described by the proposed correction
derived from the survival function.

Based on the design of the simulations, the linear trends and effective
exponent discussed above are purely statistical properties of truncated power
laws. The findings are independent of any governing physics or underlying
mechanisms such as self-organized criticality. In this regard, the conclusions
apply to any binary process defined by a truncated power law probability
distribution. Importantly, the linear expressions cannot be directly applied
to processes exhibiting amplitude variability with $s(t)$ values beyond 0 or
1. In this case, certain relations also depend on the phase of the signal
[45].

###### Acknowledgements.

The author is financially supported by the US National Science Foundation
(NSF-AGS-2031312). The author is also grateful to Professors M. Chamecki and
G. G. Katul for fruitful discussions that helped to shape this work.

*

##  Appendix A Equations for the truncated power law distribution

The truncated power law blended with a lognormal distribution and exponential
cutoff is introduced in Sec. II and shown in Fig. 2(a). The equations required
to simulate the bounded power law are presented here.

For simplicity, the lognormal portion of the distribution is defined using the
parameters $\mu_{*}=$ 1 and $\sigma_{*}^{2}=$ 1, which respectively correspond
to the mean and variance of log($\tau$). From these parameters, the mode of
the lognormal curve is fixed at $\tau_{1}=$ 1\. The resulting PDF for $\tau$
is given by the piecewise function

$PDF(\tau)=\begin{cases}\frac{C_{1}}{\tau}e^{\frac{-\left(\log(\tau)-1\right)^{2}}{2}}&\tau\leq
e^{\alpha}\\\ C_{2}\tau^{-\alpha}&e^{\alpha}<\tau\leq b\\\
C_{3}e^{-\lambda\tau}&b<\tau\leq\tau_{2}\\\ \end{cases}$ (12)

The transition from the lognormal to the power law curve occurs at
$e^{\alpha}$. This point corresponds to $d\log(PDF)/d\log(\tau)=\alpha$ along
the lognormal curve, ensuring a smooth transition to the power law. The
transition to the exponential cutoff is imposed at $b=10^{x}e^{\alpha}$, where
$x$ is the desired number of power law decades. The exponential parameter
$\lambda=\alpha/b$ enforces a smooth transition to the exponential cutoff,
i.e. $d\log(PDF)/d\log(\tau)=\alpha$ at $\tau=b$. The factors are defined as

$\displaystyle C_{1}$ $\displaystyle=$
$\displaystyle\left[C_{4}+\frac{e^{\frac{\alpha^{2}-1}{\alpha}}}{1-\alpha}C_{5}-\frac{e^{\frac{1}{2}\left(\alpha^{2}-1\right)+\alpha}}{\alpha
b^{\alpha-1}}\left(e^{-\lambda\tau_{2}}-e^{-\alpha}\right)\right]^{-1}$
$\displaystyle C_{2}$ $\displaystyle=$ $\displaystyle
C_{1}e^{\frac{\alpha^{2}-1}{2}}$ $\displaystyle C_{3}$ $\displaystyle=$
$\displaystyle C_{2}\left(\frac{e}{b}\right)^{\alpha}$ $\displaystyle C_{4}$
$\displaystyle=$
$\displaystyle\sqrt{\frac{\pi}{2}}\left(\mathrm{erf}\left(\frac{\alpha-1}{\sqrt{2}}\right)+1\right)$
$\displaystyle C_{5}$ $\displaystyle=$ $\displaystyle
b^{1-\alpha}-e^{\alpha\left(1-\alpha\right)}.$

The constants $C_{2}$ and $C_{3}$ are defined relative to $C_{1}$ to ensure
the amplitude of $PDF(\tau)$ is matched at the transition points, and $C_{1}$
is defined to achieve $\int_{0}^{\tau_{2}}PDF(\tau)=$ 1\. The cumulative
distribution corresponding to Eq. (12) is

$CDF(\tau)=\begin{cases}C_{1}\sqrt{\frac{\pi}{2}}\left(\mathrm{erf}\left(\frac{\log(\tau)-1}{\sqrt{2}}\right)+1\right)&\tau\leq
e^{\alpha}\\\
C_{1}C_{4}+\frac{C_{2}}{1-\alpha}\left(\tau^{1-\alpha}-e^{\alpha(1-\alpha)}\right)&e^{\alpha}<\tau\leq
b\\\
C_{1}C_{4}+\frac{C_{2}C_{5}}{1-\alpha}+\frac{C_{3}}{\lambda}\left(e^{-\lambda\tau}-e^{-\alpha}\right)&b<\tau\leq\tau_{2}\\\
\end{cases}$ (14)

Finally, the inversion of Eq. (14) yields the transform equation used to
simulate the finite power law:

$\tau=\begin{cases}\exp\left[\sqrt{2}\,\mathrm{erfinv}\left(\sqrt{\frac{\pi}{2}}\frac{CDF}{C_{1}}-1\right)+1\right]&CDF\leq
C_{1}C_{4}\\\ \left[\frac{1-\alpha}{C_{2}}\left(CDF-
C_{1}C_{4}\right)+e^{\alpha\left(1-\alpha\right)}\right]^{\frac{1}{1-\alpha}}&C_{1}C_{4}<CDF\leq
C_{1}C_{4}+\frac{C_{2}C_{5}}{1-\alpha}\\\
-\frac{1}{\lambda}\log\left[\frac{\lambda}{C_{3}}\left(C_{1}C_{4}+\frac{C_{2}C_{5}}{1-\alpha}-CDF\right)+e^{-\alpha}\right]&CDF>C_{1}C_{4}+\frac{C_{2}C_{5}}{1-\alpha}.\\\
\end{cases}$ (15)

In Eqs. (14) and (15), erf($x$) and erfinv($x$) refer to the error function
and its inverse, respectively, and the two notations for the exponential
function $e^{x}$ and $\exp(x)$ are used interchangeably for readability.

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