#  Quantum Theory in Design

Zhengtang Tan, Shouchuan Zhang
Department of Mathematics, Hunan University
Changsha 410082, P.R. China, Emails<EMAIL_ADDRESS>
###### Abstract

In this paper, we use braiding diagrams to present rules of shapes and
designs. That is, we design colour, design size, design brightness, design
codes by means of braiding.

2000 Mathematics Subject Classification: 16B50.

Keywords: code, braiding, design.

##  0 Introduction

The Yang-Baxter equation first came up in a paper by Yang as factorization
condition of the scattering S-matrix in the many-body problem in one dimension
and in the work of Baxter on exactly solvable models in statistical mechanics.
It has been playing an important role in mathematics and physics ( see [BD82]
, [YG89] ). Attempts to find solutions of The Yang-Baxter equation in a
systematic way have led to the theory of quantum groups [Ka95].

Shape grammars have been used as a computational design tool for over two
decades. Shape grammars are a production system created by taking a sample of
the whole for which one is trying to write a language [St06]. From this sample
a vocabulary of shapes can be written that represent all the basic forms of
that sample. By defining the spatial relationships between those forms and how
the forms are related to each other, shape rules can be written. A shape rule
consists of a left and right side. If the shape in the left side matches a
shape in a drawing then the rule can be applied, and the matching shape
changes to match the right side of the rule. The shape rules allow the
addition and subtraction of shapes, which in the end are perceived as shape
modifications. These shape rules, combined with an initial shape, produce a
shape grammar that represents the language of the design [St06]. Shapes
themselves can exist as points, lines, planes, volumes, or any combination
thereof [St06]. All shape generation must start with an initial shape: a
point, a coordinate axis, or some foundation from which to start the shape
grammar. If the grammar is going to end, it can end with a terminal rule,
which prevents any other rules from being applied after it. This forces there
to be closure in the rule sequence. Alternatively, a design sequence can
continue indefinitely and designs could be chosen at any point in the design
process. The method discussed here fundamentally changes the method of
developing the shape grammar. Mackenzie [Ma69] demonstrates in a simplified
case that if the fundamental shapes in the language are defined, the
relationship between the shapes can be inferred through examples. These
inferred relationships can then be mapped to trees which in turn can be used
to automatically create shape grammar rules for that language. While this was
shown effective in a particular use, it is more common in practice that the
vocabulary of the design to be used as the foundation of a shape grammar is
determined by the creator of the grammar. The creator of the grammar looks at
the sample and subjectively derive the vocabulary. From that vocabulary, the
rules are formed based upon the creator s experience and intention. It is
quite possible that two different persons looking at the same sample of shapes
would create two very different shape grammars.

In [OCB08], the results from the principal component analysis are used to
create a new coupe shape grammar based upon these discovered shape
relationships. The shape grammar is then used to create new coupe vehicles.
Although the focus of this paper is on vehicle design, the methods developed
here are applicable to any class of physical products based on a consistent
form language. In [Ho10], visual 3D spatial grammars are studied.

In this paper we use braiding diagrams to represent rule of shapes and
designs. This simplifies the expression of designs. The shape rules allow
braiding of two shapes, which in the end are perceived as shape modifications.
We represent new design by means of restricted PBW of Nichols algebras. That
is, we designs in braided tensor categories. We design colour, design size,
design brightness, design codes by means of braiding. This completely is a new
method and will be applied more and more in various designs.

##  1 Preliminaries

We begin with the tensor category (see [Ma98], [Ka95] and [Zh99]). We define
the product ${\cal C}\times{\cal D}$ of two category ${\cal C}$ and ${\cal D}$
whose objects are pairs of objects $(U,V)\in(ob{\cal C},ob{\cal D})$ and whose
morphisms are given by

$Hom_{{\cal C}\times{\cal D}}((V,W)(V^{\prime},W^{\prime}))=Hom_{\cal
C}(V,V^{\prime})\times Hom_{\cal D}(W,W^{\prime}).$

Let ${\cal C}$ be a category and $\otimes$ be a functor from ${\cal
C}\times{\cal C}$ to ${\cal C}$. This means

(i) we have object $V\otimes W$ for any $V,W\in ob{\cal C};$

(ii) we have morphism $f\otimes g$ from $U\otimes V$ to $X\otimes Y$ for any
morphisms $f$ and $g$ from $U$ to $X$ and from $V$ to $Y$;

(iii) we have

$(f\otimes g)(f^{\prime}\otimes g^{\prime})=(ff^{\prime})\otimes(gg^{\prime})$

for any morphisms $f:U\rightarrow X,g:V\rightarrow
Y,f^{\prime}:U^{\prime}\rightarrow V$ and $g^{\prime}:V^{\prime}\rightarrow
V;$

(iv) $id_{U\otimes V}=id_{U}\otimes id_{V}.$

Let $\otimes\tau$ denote the functor from ${\cal C}\times{\cal C}$ to ${\cal
C}$ such that $(\otimes\tau)(U,V)=(V\otimes U)$ and
$(\otimes\tau)(f,g)=g\otimes f$, for any objects $U,V,X,Y$ in ${\cal C},$ and
for any morphisms $f:U\rightarrow X$ and $g:V\rightarrow Y.$

An associativity constraint $a$ for tensor $\otimes$ is a natural isomorphism

$a:\otimes(\otimes\times id)\rightarrow\otimes(id\times\otimes).$

This means that, for any triple $(U,V,W)$ of objects of ${\cal C}$, there is a
morphism $a_{U,V,W}:(U\otimes V)\otimes W\rightarrow U\otimes(V\otimes W)$
such that

$a_{U^{\prime},V^{\prime},W^{\prime}}((f\otimes g)\otimes
h)=(f\otimes(g\otimes h))a_{U,V,W}$

for any morphisms $f,g$ and $h$ from $U$ to $U^{\prime}$, from $V$ to
$V^{\prime}$ and from $W$ to $W^{\prime}$ respectively.

Let $I$ be an object of ${\cal C}$. If there exist natural isomorphisms

$l:\otimes(I\times id)\rightarrow id\hbox{ \ \ \ and \ \ \ }r:\otimes(id\times
I)\rightarrow id\ ,$

then $I$ is called the unit object of ${\cal C}$ with left unit constraint $l$
and right unit constraint $r$.

######  Definition 1.1.

$({\cal C},\otimes,I,a,l,r)$ is called a tensor category if ${\cal C}$ is
equipped with a tensor product $\otimes$, with a unit object $I$, an
associativity constraint $a$ , a left unit constraint $l$ and a right unit
constraint $r$ such that the Pentagon Axiom and the Triangle Axiom are
satisfied, i.e.

$(id_{U}\otimes a_{V,W,X})a_{U,V\otimes W,X}(a_{U,V,W}\otimes
id_{X})=a_{U,V,W\otimes X}a_{U\otimes V,W,X}$

and

$(id_{V}\otimes l_{W})a_{V,I,W}=r_{V}\otimes id_{W}$

for any $U,V,W,X\in ob{\cal C}.$

Furthermore, if there exists a natural isomorphism

$C:\otimes\rightarrow\otimes\tau$

such that the Hexagon Axiom holds, i.e.

$a_{V,W,U}C_{U,V\otimes W}a_{U,V,W}=(id_{V}\otimes
C_{U,W})a_{V,U,W}(C_{U,V}\otimes id_{W})$

and

$a^{-1}_{W,U,V}C_{U\otimes V,W}a^{-1}_{U,V,W}=(C_{U,W}\otimes
id_{V})a^{-1}_{U,W,V}(id_{U}\otimes C_{V,W}),$

for any $U,V,W\in ob{\cal C},$ then $({\cal C},\otimes,I,a,l,r,C)$ is called a
braided tensor category. In this case, $C$ is called a braiding of ${\cal C}.$
If $C_{U,V}=C_{V,U}^{-1}$ for any $U,V\in ob{\cal C},$ then $({\cal C},C)$ is
called a symmetric braided tensor category or a symmetric tensor category.
Here functor $\tau:{\cal C}\times{\cal C}\rightarrow{\cal C}\times{\cal C}$ is
the flip functor defined by $\tau(U\times V)=V\times U$ and $\tau(f\times
g)=g\times f,$ for any $U,V\in{\cal C},$ and morphisms $f$ and $g.$ Note that
we denote the braiding $C$ in braided tensor category $({\cal
C},\otimes,I,a,l,r,C)$ by ${}^{\cal C}C$ sometimes.

######  Example 1.2.

(The tensor category of vector spaces ) The most fundamental example of a
tensor category is given by the category ${\cal C}={\cal V}ect(k)$ of vector
spaces over field $k$. ${\cal V}ect(k)$ is equipped with tensor structure for
which $\otimes$ is the tensor product of the vector spaces over $k$, the unit
object $I$ is the ground field $k$ itself, and the associativity constraint
and unit constraint are the natural isomorphisms

$a_{U,V,W}((u\otimes v)\otimes w):=u\otimes(v\otimes w)\hbox{ \ \ and \ \
}l_{V}(1\otimes v):=v:=r_{V}(v\otimes 1)$

for any vector space $U,V,W$ and $u\in U,v\in V,w\in W.$

Furthermore, the most fundamental example of a braided tensor category is
given by the tensor category ${\cal V}ect(k)$, whose braiding is usual twist
map from $U\otimes V$ to $V\otimes U$ defined by sending $a\otimes b$ to
$b\otimes a$ for any $a\in U,b\in V$.

Now we define some notations. If $f$ is a morphism from $U$ to $V$ and $g$ is
a morphism from $V$ to $W$, we denote the composition $gf$ by

$\begin{array}[c]{l}{\hbox{\kern-3.95901pt${U}$}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){1000.0}}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){200.0}}\put(0.0,800.0){\line(0,1){200.0}}\put(0.0,500.0){\oval(1200.0,600.0)}\put(0.0,500.0){\makebox(0.0,0.0)[cc]{\scriptsize$gf$}}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){1000.0}}}\@arraycr{\hbox{\kern-5.41667pt${W}$}}\@arraycr\end{array}\kern
9.9945pt=\kern
9.9945pt\begin{array}[c]{l}{\hbox{\kern-3.95901pt${U}$}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){400.0}}\put(0.0,1600.0){\line(0,1){400.0}}\put(0.0,1000.0){\oval(1200.0,1200.0)}\put(0.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$f$}}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){400.0}}\put(0.0,1600.0){\line(0,1){400.0}}\put(0.0,1000.0){\oval(1200.0,1200.0)}\put(0.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$g$}}}\@arraycr{\hbox{\kern-5.41667pt${W}$}}\@arraycr\end{array}\
\ \ .$

We usually omit $I$ and the morphism $id_{I}$ in any diagrams. In particular,
If $f$ is a morphism from $I$ to $V$, $g$ is a morphism from $V$ to $I$,we
denote $f$ and $g$ by

$\begin{array}[c]{l}\hbox{\put(0.0,0.0){\line(0,1){400.0}}\put(0.0,1000.0){\oval(1200.0,1200.0)}\put(0.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$f$}}}\@arraycr{\hbox{\kern-4.02777pt${V}$}}\@arraycr\end{array}\kern
9.9945pt\hbox{ , }\kern
9.9945pt\begin{array}[c]{l}{\hbox{\kern-4.02777pt${V}$}}\@arraycr\hbox{\put(0.0,1600.0){\line(0,1){400.0}}\put(0.0,1000.0){\oval(1200.0,1200.0)}\put(0.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$g$}}}\@arraycr\end{array}\
\ \ \ .$

If $f$ is a morphism from $U\otimes V$ to $P$, $g$ is a morphism from
$U\otimes V$ to $I$ and $\zeta$ is a morphism from $U\otimes V$ to $V\otimes
U$, we denote $f$, $g$ and $\zeta$ by

$\begin{array}[c]{l}{\hbox{\kern-3.95901pt${U}$}}\kern
19.98901pt{\hbox{\kern-4.02777pt${V}$}}\@arraycr\hbox{\put(400.0,1500.0){\oval(800.0,800.0)[lb]}\put(1600.0,1500.0){\oval(800.0,800.0)[rb]}\put(0.0,1500.0){\line(0,1){500.0}}\put(2000.0,1500.0){\line(0,1){500.0}}\put(1000.0,0.0){\line(0,1){500.0}}\put(1000.0,1100.0){\oval(1200.0,1200.0)}\put(1000.0,1100.0){\makebox(0.0,0.0)[cc]{\scriptsize$f$}}}\@arraycr\kern
9.9945pt{\hbox{\kern-3.90451pt${P}$}}\@arraycr\end{array}\kern 9.9945pt,\kern
9.9945pt\begin{array}[c]{l}{\hbox{\kern-3.95901pt${U}$}}\kern
19.98901pt{\hbox{\kern-4.02777pt${V}$}}\@arraycr\hbox{\put(0.0,1600.0){\line(0,1){400.0}}\put(2000.0,1600.0){\line(0,1){400.0}}\put(400.0,1600.0){\oval(800.0,800.0)[lb]}\put(1600.0,1600.0){\oval(800.0,800.0)[rb]}\put(1000.0,1200.0){\oval(1200.0,1200.0)}\put(1000.0,1200.0){\makebox(0.0,0.0)[cc]{\scriptsize$g$}}}\@arraycr\end{array}\kern
9.9945pt,\kern 9.9945pt\begin{array}[c]{l}{\hbox{\kern-3.95901pt${U}$}}\kern
19.98901pt{\hbox{\kern-4.02777pt${V}$}}\@arraycr\hbox{\put(1000.0,1000.0){\oval(1400.0,1400.0)}\put(1000.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$\zeta$}}
<EMAIL_ADDRESS>19.98901pt{\hbox{\kern-3.95901pt${U}$}}\@arraycr\end{array}\kern 9.9945pt.$

In particular, we denote the braiding $C_{U,V}$ and its inverse $C_{U,V}^{-1}$
by

$\begin{array}[c]{l}{\hbox{\kern-3.95901pt${U}$}}\kern
19.98901pt{\hbox{\kern-4.02777pt${V}$}}\<EMAIL_ADDRESS>19.98901pt{\hbox{\kern-3.95901pt${U}$}}\@arraycr\end{array}\kern
9.9945pt,\kern 9.9945pt\begin{array}[c]{l}{\hbox{\kern-4.02777pt${V}$}}\kern
19.98901pt{\hbox{\kern-3.95901pt${U}$}}\<EMAIL_ADDRESS>19.98901pt{\hbox{\kern-4.02777pt${V}$}}\@arraycr\end{array}\kern 9.9945pt.\\\
$

$\xi$ is called an $R$-matrix of ${\cal C}$ if $\xi$ is a natural isomorphism
from $\otimes$ to $\otimes\tau$ and for any $U,V,W\in ob{\cal C}$, the Yang-
Baxter equation of ${\cal C}$:

(YBE):

$\begin{array}[c]{l}{\hbox{\kern-3.95901pt${U}$}}\kern
19.98901pt{\hbox{\kern-4.02777pt${V}$}}\kern
19.98901pt{\hbox{\kern-5.41667pt${W}$}}\@arraycr\hbox{\put(1000.0,1000.0){\oval(1400.0,1400.0)}\put(1000.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$\xi$}}
}\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\hbox{\put(1000.0,1000.0){\oval(1400.0,1400.0)}\put(1000.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$\xi$}}
}\@arraycr\hbox{\put(1000.0,1000.0){\oval(1400.0,1400.0)}\put(1000.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$\xi$}}
}\kern
<EMAIL_ADDRESS>19.98901pt{\hbox{\kern-4.02777pt${V}$}}\kern
19.98901pt{\hbox{\kern-3.95901pt${U}$}}\@arraycr\end{array}\ \ =\
\begin{array}[c]{l}{\hbox{\kern-3.95901pt${U}$}}\kern
19.98901pt{\hbox{\kern-4.02777pt${V}$}}\kern
19.98901pt{\hbox{\kern-5.41667pt${W}$}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\hbox{\put(1000.0,1000.0){\oval(1400.0,1400.0)}\put(1000.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$\xi$}}
}\@arraycr\hbox{\put(1000.0,1000.0){\oval(1400.0,1400.0)}\put(1000.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$\xi$}}
}\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\hbox{\put(1000.0,1000.0){\oval(1400.0,1400.0)}\put(1000.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$\xi$}}
<EMAIL_ADDRESS>19.98901pt{\hbox{\kern-4.02777pt${V}$}}\kern
19.98901pt{\hbox{\kern-3.95901pt${U}$}}\@arraycr\end{array}\ $

holds. In particular, the above equation is called the Yang-Baxter equation on
$V$ when $U=V=W.$

######  Lemma 1.3.

(i) The braiding $C$ of braided tensor category $({\cal C},C)$ is an
$R$-matrix of ${\cal C}$, i.e.

$\begin{array}[c]{l}{\hbox{\kern-3.95901pt${U}$}}\kern
19.98901pt{\hbox{\kern-4.02777pt${V}$}}\kern
19.98901pt{\hbox{\kern-5.41667pt${W}$}}\@arraycr\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\hbox{}\@arraycr\kern
<EMAIL_ADDRESS>19.98901pt{\hbox{\kern-4.02777pt${V}$}}\kern
19.98901pt{\hbox{\kern-3.95901pt${U}$}}\@arraycr\end{array}\ \ =\
\begin{array}[c]{l}{\hbox{\kern-3.95901pt${U}$}}\kern
19.98901pt{\hbox{\kern-4.02777pt${V}$}}\kern
19.98901pt{\hbox{\kern-5.41667pt${W}$}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\hbox{}\@arraycr\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
<EMAIL_ADDRESS>19.98901pt{\hbox{\kern-4.02777pt${V}$}}\kern
19.98901pt{\hbox{\kern-3.95901pt${U}$}}\@arraycr\end{array}\ $

holds.

(ii)

$\begin{array}[c]{l}{\hbox{\kern-3.95901pt${U}$}}\kern
19.98901pt{\hbox{\kern-4.02777pt${V}$}}\kern
19.98901pt{\hbox{\kern-5.41667pt${W}$}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\hbox{}\@arraycr\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
<EMAIL_ADDRESS>29.98352pt{\hbox{\kern-3.90451pt${P}$}}\@arraycr\end{array}\kern
9.9945pt=\kern 9.9945pt\begin{array}[c]{l}{\hbox{\kern-3.95901pt${U}$}}\kern
19.98901pt{\hbox{\kern-4.02777pt${V}$}}\kern
9.9945pt{\hbox{\kern-5.41667pt${W}$}}\@arraycr\hbox{\put(400.0,1500.0){\oval(800.0,800.0)[lb]}\put(1600.0,1500.0){\oval(800.0,800.0)[rb]}\put(0.0,1500.0){\line(0,1){500.0}}\put(2000.0,1500.0){\line(0,1){500.0}}\put(1000.0,0.0){\line(0,1){500.0}}\put(1000.0,1100.0){\oval(1200.0,1200.0)}\put(1000.0,1100.0){\makebox(0.0,0.0)[cc]{\scriptsize$f$}}}\kern
9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\kern
9.9945pt\hbox{}\@arraycr\kern 9.9945pt{\hbox{\kern-5.41667pt${W}$}}\kern
19.98901pt{\hbox{\kern-3.90451pt${P}$}}\@arraycr\end{array}\kern
9.9945pt,\kern 9.9945pt\begin{array}[c]{l}{\hbox{\kern-3.95901pt${U}$}}\kern
19.98901pt{\hbox{\kern-5.41667pt${W}$}}\kern
19.98901pt{\hbox{\kern-4.02777pt${V}$}}\@arraycr\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
<EMAIL_ADDRESS>29.98352pt{\hbox{\kern-3.90451pt${P}$}}\@arraycr\end{array}\kern
9.9945pt=\kern 9.9945pt\begin{array}[c]{l}{\hbox{\kern-3.95901pt${U}$}}\kern
19.98901pt{\hbox{\kern-5.41667pt${W}$}}\kern
9.9945pt{\hbox{\kern-4.02777pt${V}$}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\hbox{}\@arraycr\hbox{\put(400.0,1500.0){\oval(800.0,800.0)[lb]}\put(1600.0,1500.0){\oval(800.0,800.0)[rb]}\put(0.0,1500.0){\line(0,1){500.0}}\put(2000.0,1500.0){\line(0,1){500.0}}\put(1000.0,0.0){\line(0,1){500.0}}\put(1000.0,1100.0){\oval(1200.0,1200.0)}\put(1000.0,1100.0){\makebox(0.0,0.0)[cc]{\scriptsize$f$}}}\kern
9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\kern
9.9945pt\hbox{}\@arraycr\kern 9.9945pt{\hbox{\kern-5.41667pt${W}$}}\kern
19.98901pt{\hbox{\kern-3.90451pt${P}$}}\@arraycr\end{array}\ \ .$

Proof. (i)

$\begin{array}[c]{l}{\hbox{\kern-3.95901pt${U}$}}\kern
19.98901pt{\hbox{\kern-4.02777pt${V}$}}\kern
19.98901pt{\hbox{\kern-5.41667pt${W}$}}\@arraycr\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\hbox{}\@arraycr\kern
<EMAIL_ADDRESS>19.98901pt{\hbox{\kern-4.02777pt${V}$}}\kern
19.98901pt{\hbox{\kern-3.95901pt${U}$}}\@arraycr\end{array}\ \ =\ \
\begin{array}[c]{l}\par\par\kern 9.9945pt{\hbox{\kern-3.95901pt${U}$}}\kern
19.98901pt{\hbox{\kern-4.02777pt${V}$}}\kern
19.98901pt{\hbox{\kern-5.41667pt${W}$}}\@arraycr\kern
9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\par\par\par\hbox{\put(1500.0,1000.0){\oval(3000.0,2000.0)}\put(0.0,0.0){\makebox(3000.0,2000.0)[cc]{\scriptsize$C_{V,W}$}}}\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\par\kern
9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\par\kern
9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\hbox{\put(1500.0,1000.0){\oval(3000.0,2000.0)}\put(0.0,0.0){\makebox(3000.0,2000.0)[cc]{\scriptsize$C_{U,W}$}}}\@arraycr\par\par\kern
9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\par\hbox{\put(1500.0,1000.0){\oval(3000.0,2000.0)}\put(0.0,0.0){\makebox(3000.0,2000.0)[cc]{\scriptsize$C_{V,W}$}}}\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\par\par\kern
9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\kern
9.9945pt{\hbox{\kern-5.41667pt${W}$}}\kern
19.98901pt{\hbox{\kern-4.02777pt${V}$}}\kern
19.98901pt{\hbox{\kern-3.95901pt${U}$}}\@arraycr\end{array}\ \
\stackrel{{\scriptstyle\mbox{by Hexagon Axiom
}}}{{=}}\begin{array}[c]{l}\par\par\kern
9.9945pt{\hbox{\kern-14.09785pt${U\otimes V}$}}\kern
39.97803pt{\hbox{\kern-5.41667pt${W}$}}\@arraycr\kern
9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
39.97803pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\par\par\par\hbox{\put(1500.0,1000.0){\oval(3000.0,2000.0)}\put(0.0,0.0){\makebox(3000.0,2000.0)[cc]{\scriptsize$C_{V,W}$}}}\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\par\kern
9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 29.98352pt\kern
19.98901pt\@arraycr\par\hbox{\put(2000.0,1000.0){\oval(4000.0,2000.0)}\put(0.0,0.0){\makebox(4000.0,2000.0)[cc]{\scriptsize$C_{V\otimes
U,W}$}}}\@arraycr\par\par\par\par\par\kern
9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\@arraycr\kern 9.9945pt{\hbox{\kern-5.41667pt${W}$}}\kern
29.98352pt{\hbox{\kern-14.09785pt${V\otimes U}$}}\kern
19.98901pt\@arraycr\end{array}\ \ $

$\ \ \stackrel{{\scriptstyle\mbox{by naturality
}}}{{=}}\begin{array}[c]{l}\par\par\kern
9.9945pt{\hbox{\kern-14.09785pt${U\otimes V}$}}\kern
29.98352pt{\hbox{\kern-5.41667pt${W}$}}\@arraycr\kern
9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\hbox{}\@arraycr\par\par\hbox{\put(2000.0,1000.0){\oval(4000.0,2000.0)}\put(0.0,0.0){\makebox(4000.0,2000.0)[cc]{\scriptsize$C_{U\otimes
V,W}$}}}\kern 19.98901pt\@arraycr\par\kern
9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\@arraycr\par\kern
9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
9.9945pt\hbox{\put(1500.0,1000.0){\oval(3000.0,2000.0)}\put(0.0,0.0){\makebox(3000.0,2000.0)[cc]{\scriptsize$C_{U,V}$}}}\@arraycr\par\par\par\par\par\kern
9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\@arraycr\kern 9.9945pt{\hbox{\kern-5.41667pt${W}$}}\kern
29.98352pt{\hbox{\kern-14.09785pt${V\otimes U}$}}\kern
19.98901pt\@arraycr\end{array}\ \ \stackrel{{\scriptstyle\mbox{by Hexagon
Axiom }}}{{=}}\begin{array}[c]{l}{\hbox{\kern-3.95901pt${U}$}}\kern
19.98901pt{\hbox{\kern-4.02777pt${V}$}}\kern
19.98901pt{\hbox{\kern-5.41667pt${W}$}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\hbox{}\@arraycr\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
<EMAIL_ADDRESS>19.98901pt{\hbox{\kern-4.02777pt${V}$}}\kern
19.98901pt{\hbox{\kern-3.95901pt${U}$}}\@arraycr\end{array}\ \ \ .$

(ii) We have

$\begin{array}[c]{l}{\hbox{\kern-3.95901pt${U}$}}\kern
19.98901pt{\hbox{\kern-4.02777pt${V}$}}\kern
9.9945pt{\hbox{\kern-5.41667pt${W}$}}\@arraycr\hbox{\put(400.0,1500.0){\oval(800.0,800.0)[lb]}\put(1600.0,1500.0){\oval(800.0,800.0)[rb]}\put(0.0,1500.0){\line(0,1){500.0}}\put(2000.0,1500.0){\line(0,1){500.0}}\put(1000.0,0.0){\line(0,1){500.0}}\put(1000.0,1100.0){\oval(1200.0,1200.0)}\put(1000.0,1100.0){\makebox(0.0,0.0)[cc]{\scriptsize$f$}}}\kern
9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\kern
9.9945pt\hbox{}\@arraycr\kern 9.9945pt{\hbox{\kern-5.41667pt${W}$}}\kern
19.98901pt{\hbox{\kern-3.90451pt${P}$}}\@arraycr\end{array}\kern
9.9945pt=\kern 9.9945pt\begin{array}[c]{l}{\hbox{\kern-14.09785pt${U\otimes
V}$}}\kern
19.98901pt{\hbox{\kern-5.41667pt${W}$}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){400.0}}\put(0.0,1600.0){\line(0,1){400.0}}\put(0.0,1000.0){\oval(1200.0,1200.0)}\put(0.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$f$}}}\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\<EMAIL_ADDRESS>19.98901pt{\hbox{\kern-3.90451pt${P}$}}\@arraycr\end{array}\ \
\stackrel{{\scriptstyle\mbox{by naturality }}}{{=}}\kern
9.9945pt\begin{array}[c]{l}{\hbox{\kern-14.09785pt${U\otimes V}$}}\kern
19.98901pt{\hbox{\kern-5.41667pt${W}$}}\@arraycr\hbox{}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
<EMAIL_ADDRESS>19.98901pt{\hbox{\kern-3.90451pt${P}$}}\@arraycr\end{array}$
$\stackrel{{\scriptstyle\mbox{by Hexagon Axiom
}}}{{=}}\begin{array}[c]{l}{\hbox{\kern-3.95901pt${U}$}}\kern
19.98901pt{\hbox{\kern-4.02777pt${V}$}}\kern
19.98901pt{\hbox{\kern-5.41667pt${W}$}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\hbox{}\@arraycr\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
<EMAIL_ADDRESS>29.98352pt{\hbox{\kern-3.90451pt${P}$}}\@arraycr\end{array}\ \ \ .$

Similarly, we can show the second equation. $\Box$

Now we give some concepts as follows: Assume that $H,A\in ob\ {\cal C},$ and

$\displaystyle\alpha:H\otimes A\rightarrow$ $\displaystyle A$
$\displaystyle,\hbox{ \ \ \ \ }\beta:H\otimes A\rightarrow H,$
$\displaystyle\phi:A\rightarrow H\otimes$ $\displaystyle A$
$\displaystyle,\hbox{ \ \ \ \ }\psi:H\rightarrow H\otimes A,$ $\displaystyle
m_{H}:H\otimes H\rightarrow$ $\displaystyle H$ $\displaystyle,\hbox{ \ \ \ \
}m_{A}:A\otimes A\rightarrow A,$ $\displaystyle\Delta_{H}:H\rightarrow
H\otimes$ $\displaystyle H$ $\displaystyle,\hbox{ \ \ \ \
}\Delta_{A}:A\rightarrow A\otimes A,$ $\displaystyle\eta_{H}:I\rightarrow$
$\displaystyle H$ $\displaystyle,\hbox{ \ \ \ \ }\eta_{A}:I\rightarrow A,$
$\displaystyle\epsilon_{H}:H\rightarrow$ $\displaystyle I$
$\displaystyle,\hbox{ \ \ \ \ }\epsilon_{A}:A\rightarrow I.$

are morphisms in ${\cal C}$.

$(A,m_{A},\eta_{A})$ is called an algebra living in ${\cal C}$, if the
following conditions are satisfied:

$\begin{array}[c]{l}{\hbox{\kern-3.75pt${A}$}}\kern
9.9945pt{\hbox{\kern-3.75pt${A}$}}\kern
19.98901pt{\hbox{\kern-3.75pt${A}$}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
9.9945pt\hbox{\put(400.0,1500.0){\oval(800.0,800.0)[lb]}\put(1600.0,1500.0){\oval(800.0,800.0)[rb]}\put(0.0,1500.0){\line(0,1){500.0}}\put(2000.0,1500.0){\line(0,1){500.0}}\put(1000.0,0.0){\line(0,1){500.0}}\put(1000.0,1100.0){\oval(1200.0,1200.0)}\put(1000.0,1100.0){\makebox(0.0,0.0)[cc]{\scriptsize$m$}}}\@arraycr\hbox{\put(400.0,1500.0){\oval(800.0,800.0)[lb]}\put(1600.0,1500.0){\oval(800.0,800.0)[rb]}\put(0.0,1500.0){\line(0,1){500.0}}\put(2000.0,1500.0){\line(0,1){500.0}}\put(1000.0,0.0){\line(0,1){500.0}}\put(1000.0,1100.0){\oval(1200.0,1200.0)}\put(1000.0,1100.0){\makebox(0.0,0.0)[cc]{\scriptsize$m$}}}\@arraycr\kern
9.9945pt{\hbox{\kern-3.75pt${A}$}}\@arraycr\end{array}\kern 9.9945pt=\kern
9.9945pt\begin{array}[c]{l}{\hbox{\kern-3.75pt${A}$}}\kern
19.98901pt{\hbox{\kern-3.75pt${A}$}}\kern
9.9945pt{\hbox{\kern-3.75pt${A}$}}\@arraycr\hbox{\put(400.0,1500.0){\oval(800.0,800.0)[lb]}\put(1600.0,1500.0){\oval(800.0,800.0)[rb]}\put(0.0,1500.0){\line(0,1){500.0}}\put(2000.0,1500.0){\line(0,1){500.0}}\put(1000.0,0.0){\line(0,1){500.0}}\put(1000.0,1100.0){\oval(1200.0,1200.0)}\put(1000.0,1100.0){\makebox(0.0,0.0)[cc]{\scriptsize$m$}}}\kern
9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\kern
9.9945pt\hbox{\put(400.0,1500.0){\oval(800.0,800.0)[lb]}\put(1600.0,1500.0){\oval(800.0,800.0)[rb]}\put(0.0,1500.0){\line(0,1){500.0}}\put(2000.0,1500.0){\line(0,1){500.0}}\put(1000.0,0.0){\line(0,1){500.0}}\put(1000.0,1100.0){\oval(1200.0,1200.0)}\put(1000.0,1100.0){\makebox(0.0,0.0)[cc]{\scriptsize$m$}}}\@arraycr\kern
19.98901pt{\hbox{\kern-3.75pt${A}$}}\@arraycr\end{array}\kern 9.9945pt\kern
9.9945pt,\kern 9.9945pt\kern 9.9945pt\begin{array}[c]{l}\kern
19.98901pt{\hbox{\kern-3.75pt${A}$}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){400.0}}\put(0.0,1000.0){\oval(1200.0,1200.0)}\put(0.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$\eta_{A}$}}}\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\hbox{\put(400.0,1500.0){\oval(800.0,800.0)[lb]}\put(1600.0,1500.0){\oval(800.0,800.0)[rb]}\put(0.0,1500.0){\line(0,1){500.0}}\put(2000.0,1500.0){\line(0,1){500.0}}\put(1000.0,0.0){\line(0,1){500.0}}\put(1000.0,1100.0){\oval(1200.0,1200.0)}\put(1000.0,1100.0){\makebox(0.0,0.0)[cc]{\scriptsize$m$}}}\@arraycr\kern
9.9945pt{\hbox{\kern-3.75pt${A}$}}\@arraycr\end{array}\kern 9.9945pt=\kern
9.9945pt\begin{array}[c]{l}{\hbox{\kern-3.75pt${A}$}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){400.0}}\put(0.0,1000.0){\oval(1200.0,1200.0)}\put(0.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$\eta_{A}$}}}\@arraycr\hbox{\put(400.0,1500.0){\oval(800.0,800.0)[lb]}\put(1600.0,1500.0){\oval(800.0,800.0)[rb]}\put(0.0,1500.0){\line(0,1){500.0}}\put(2000.0,1500.0){\line(0,1){500.0}}\put(1000.0,0.0){\line(0,1){500.0}}\put(1000.0,1100.0){\oval(1200.0,1200.0)}\put(1000.0,1100.0){\makebox(0.0,0.0)[cc]{\scriptsize$m$}}}\@arraycr\kern
9.9945pt{\hbox{\kern-3.75pt${A}$}}\@arraycr\end{array}\kern 9.9945pt=\kern
9.9945pt\begin{array}[c]{l}{\hbox{\kern-3.75pt${A}$}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\<EMAIL_ADDRESS>
In this case, $\eta_{A}$ and $m_{A}$ are called unit and multiplication of $A$
respectively.

$(H,\Delta_{H},\epsilon_{H})$ is called a coalgebra living in ${\cal C},$ if
the following conditions are satisfied:

$\begin{array}[c]{l}\kern
9.9945pt{\hbox{\kern-4.56248pt${H}$}}\@arraycr\hbox{\put(400.0,500.0){\oval(800.0,800.0)[lt]}\put(1600.0,500.0){\oval(800.0,800.0)[rt]}\put(0.0,500.0){\line(0,-1){500.0}}\put(2000.0,500.0){\line(0,-1){500.0}}\put(1000.0,2000.0){\line(0,-1){500.0}}\put(1000.0,900.0){\oval(1200.0,1200.0)}\put(1000.0,900.0){\makebox(0.0,0.0)[cc]{\scriptsize$\Delta$}}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
<EMAIL_ADDRESS>9.9945pt{\hbox{\kern-4.56248pt${H}$}}\kern
19.98901pt{\hbox{\kern-4.56248pt${H}$}}\@arraycr\end{array}\kern
9.9945pt=\kern 9.9945pt\begin{array}[c]{l}\kern
19.98901pt{\hbox{\kern-4.56248pt${H}$}}\@arraycr\kern
9.9945pt\hbox{\put(400.0,500.0){\oval(800.0,800.0)[lt]}\put(1600.0,500.0){\oval(800.0,800.0)[rt]}\put(0.0,500.0){\line(0,-1){500.0}}\put(2000.0,500.0){\line(0,-1){500.0}}\put(1000.0,2000.0){\line(0,-1){500.0}}\put(1000.0,900.0){\oval(1200.0,1200.0)}\put(1000.0,900.0){\makebox(0.0,0.0)[cc]{\scriptsize$\Delta$}}}\@arraycr\hbox{\put(400.0,500.0){\oval(800.0,800.0)[lt]}\put(1600.0,500.0){\oval(800.0,800.0)[rt]}\put(0.0,500.0){\line(0,-1){500.0}}\put(2000.0,500.0){\line(0,-1){500.0}}\put(1000.0,2000.0){\line(0,-1){500.0}}\put(1000.0,900.0){\oval(1200.0,1200.0)}\put(1000.0,900.0){\makebox(0.0,0.0)[cc]{\scriptsize$\Delta$}}}\kern
<EMAIL_ADDRESS>19.98901pt{\hbox{\kern-4.56248pt${H}$}}\kern
9.9945pt{\hbox{\kern-4.56248pt${H}$}}\@arraycr\end{array}\kern 9.9945pt\kern
9.9945pt,\kern 9.9945pt\kern 9.9945pt\begin{array}[c]{l}\kern
9.9945pt{\hbox{\kern-4.56248pt${H}$}}\@arraycr\hbox{\put(400.0,500.0){\oval(800.0,800.0)[lt]}\put(1600.0,500.0){\oval(800.0,800.0)[rt]}\put(0.0,500.0){\line(0,-1){500.0}}\put(2000.0,500.0){\line(0,-1){500.0}}\put(1000.0,2000.0){\line(0,-1){500.0}}\put(1000.0,900.0){\oval(1200.0,1200.0)}\put(1000.0,900.0){\makebox(0.0,0.0)[cc]{\scriptsize$\Delta$}}}\@arraycr\hbox{\put(0.0,1600.0){\line(0,1){400.0}}\put(0.0,1000.0){\oval(1200.0,1200.0)}\put(0.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$\varepsilon_{H}$}}}\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\kern
19.98901pt{\hbox{\kern-4.56248pt${H}$}}\@arraycr\end{array}\kern
9.9945pt=\kern 9.9945pt\begin{array}[c]{l}\kern
9.9945pt{\hbox{\kern-4.56248pt${H}$}}\@arraycr\hbox{\put(400.0,500.0){\oval(800.0,800.0)[lt]}\put(1600.0,500.0){\oval(800.0,800.0)[rt]}\put(0.0,500.0){\line(0,-1){500.0}}\put(2000.0,500.0){\line(0,-1){500.0}}\put(1000.0,2000.0){\line(0,-1){500.0}}\put(1000.0,900.0){\oval(1200.0,1200.0)}\put(1000.0,900.0){\makebox(0.0,0.0)[cc]{\scriptsize$\Delta$}}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\hbox{\put(0.0,1600.0){\line(0,1){400.0}}\put(0.0,1000.0){\oval(1200.0,1200.0)}\put(0.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$\varepsilon_{H}$}}}\@arraycr{\hbox{\kern-4.56248pt${H}$}}\@arraycr\end{array}\kern
9.9945pt=\kern
9.9945pt\begin{array}[c]{l}{\hbox{\kern-4.56248pt${H}$}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\<EMAIL_ADDRESS>
In this case, $\epsilon_{H}$ and $\Delta_{H}$ are called counit and
comultiplication of $H$ respectively.

If $A$ is an algebra and $H$ is a coalgebra, then $Hom_{\cal C}(H,A)$ becomes
an algebra under the convolution product

$f*g=\quad\begin{array}[c]{l}\kern
9.9945pt{\hbox{\kern-4.56248pt${H}$}}\@arraycr\hbox{\put(400.0,500.0){\oval(800.0,800.0)[lt]}\put(1600.0,500.0){\oval(800.0,800.0)[rt]}\put(0.0,500.0){\line(0,-1){500.0}}\put(2000.0,500.0){\line(0,-1){500.0}}\put(1000.0,2000.0){\line(0,-1){500.0}}\put(1000.0,900.0){\oval(1200.0,1200.0)}\put(1000.0,900.0){\makebox(0.0,0.0)[cc]{\scriptsize$\Delta$}}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){400.0}}\put(0.0,1600.0){\line(0,1){400.0}}\put(0.0,1000.0){\oval(1200.0,1200.0)}\put(0.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$f$}}}\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){400.0}}\put(0.0,1600.0){\line(0,1){400.0}}\put(0.0,1000.0){\oval(1200.0,1200.0)}\put(0.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$g$}}}\@arraycr\hbox{\put(400.0,1500.0){\oval(800.0,800.0)[lb]}\put(1600.0,1500.0){\oval(800.0,800.0)[rb]}\put(0.0,1500.0){\line(0,1){500.0}}\put(2000.0,1500.0){\line(0,1){500.0}}\put(1000.0,0.0){\line(0,1){500.0}}\put(1000.0,1100.0){\oval(1200.0,1200.0)}\put(1000.0,1100.0){\makebox(0.0,0.0)[cc]{\scriptsize$m$}}}\@arraycr\kern
9.9945pt{\hbox{\kern-3.75pt${A}$}}\@arraycr\end{array}\ \ \ .$

and its unit element $\eta=\eta_{A}\epsilon_{H}.$ If $S$ is the inverse of
$id_{H}$ in $Hom_{\cal C}(H,H)$, then $S$ is called antipode of $H$.

If $(H,m_{H},\eta_{H})$ is an algebra, and $(H,\Delta_{H},\epsilon_{H})$ is a
coalgebra living in ${\cal C}$, and the following condition is satisfied:

$\begin{array}[c]{l}{\hbox{\kern-4.56248pt${H}$}}\kern
19.98901pt{\hbox{\kern-4.56248pt${H}$}}\@arraycr\hbox{\put(400.0,1500.0){\oval(800.0,800.0)[lb]}\put(1600.0,1500.0){\oval(800.0,800.0)[rb]}\put(0.0,1500.0){\line(0,1){500.0}}\put(2000.0,1500.0){\line(0,1){500.0}}\put(1000.0,0.0){\line(0,1){500.0}}\put(1000.0,1100.0){\oval(1200.0,1200.0)}\put(1000.0,1100.0){\makebox(0.0,0.0)[cc]{\scriptsize$m$}}}\<EMAIL_ADDRESS>19.98901pt{\hbox{\kern-4.56248pt${H}$}}\@arraycr\end{array}\kern
9.9945pt=\kern 9.9945pt\begin{array}[c]{l}\kern
9.9945pt{\hbox{\kern-4.56248pt${H}$}}\kern
29.98352pt{\hbox{\kern-4.56248pt${H}$}}\@arraycr\hbox{\put(400.0,500.0){\oval(800.0,800.0)[lt]}\put(1600.0,500.0){\oval(800.0,800.0)[rt]}\put(0.0,500.0){\line(0,-1){500.0}}\put(2000.0,500.0){\line(0,-1){500.0}}\put(1000.0,2000.0){\line(0,-1){500.0}}\put(1000.0,900.0){\oval(1200.0,1200.0)}\put(1000.0,900.0){\makebox(0.0,0.0)[cc]{\scriptsize$\Delta$}}}\kern
9.9945pt\hbox{\put(400.0,500.0){\oval(800.0,800.0)[lt]}\put(1600.0,500.0){\oval(800.0,800.0)[rt]}\put(0.0,500.0){\line(0,-1){500.0}}\put(2000.0,500.0){\line(0,-1){500.0}}\put(1000.0,2000.0){\line(0,-1){500.0}}\put(1000.0,900.0){\oval(1200.0,1200.0)}\put(1000.0,900.0){\makebox(0.0,0.0)[cc]{\scriptsize$\Delta$}}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\hbox{\put(400.0,1500.0){\oval(800.0,800.0)[lb]}\put(1600.0,1500.0){\oval(800.0,800.0)[rb]}\put(0.0,1500.0){\line(0,1){500.0}}\put(2000.0,1500.0){\line(0,1){500.0}}\put(1000.0,0.0){\line(0,1){500.0}}\put(1000.0,1100.0){\oval(1200.0,1200.0)}\put(1000.0,1100.0){\makebox(0.0,0.0)[cc]{\scriptsize$m$}}}\kern
9.9945pt\hbox{\put(400.0,1500.0){\oval(800.0,800.0)[lb]}\put(1600.0,1500.0){\oval(800.0,800.0)[rb]}\put(0.0,1500.0){\line(0,1){500.0}}\put(2000.0,1500.0){\line(0,1){500.0}}\put(1000.0,0.0){\line(0,1){500.0}}\put(1000.0,1100.0){\oval(1200.0,1200.0)}\put(1000.0,1100.0){\makebox(0.0,0.0)[cc]{\scriptsize$m$}}}\@arraycr\kern
9.9945pt{\hbox{\kern-4.56248pt${H}$}}\kern
29.98352pt{\hbox{\kern-4.56248pt${H}$}}\@arraycr\end{array}\kern
9.9945pt,\kern 9.9945pt\begin{array}[c]{l}{\hbox{\kern-4.56248pt${H}$}}\kern
19.98901pt{\hbox{\kern-4.56248pt${H}$}}\@arraycr\hbox{\put(400.0,1500.0){\oval(800.0,800.0)[lb]}\put(1600.0,1500.0){\oval(800.0,800.0)[rb]}\put(0.0,1500.0){\line(0,1){500.0}}\put(2000.0,1500.0){\line(0,1){500.0}}\put(1000.0,0.0){\line(0,1){500.0}}\put(1000.0,1100.0){\oval(1200.0,1200.0)}\put(1000.0,1100.0){\makebox(0.0,0.0)[cc]{\scriptsize$m$}}}\@arraycr\kern
9.9945pt\hbox{\put(0.0,1600.0){\line(0,1){400.0}}\put(0.0,1000.0){\oval(1200.0,1200.0)}\put(0.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$\varepsilon$}}}\@arraycr\end{array}\kern
9.9945pt=\kern 9.9945pt\begin{array}[c]{l}{\hbox{\kern-4.56248pt${H}$}}\kern
19.98901pt{\hbox{\kern-4.56248pt${H}$}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\hbox{\put(0.0,1600.0){\line(0,1){400.0}}\put(0.0,1000.0){\oval(1200.0,1200.0)}\put(0.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$\varepsilon$}}}\kern
19.98901pt\hbox{\put(0.0,1600.0){\line(0,1){400.0}}\put(0.0,1000.0){\oval(1200.0,1200.0)}\put(0.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$\varepsilon$}}}\@arraycr\end{array}\kern
9.9945pt,\kern 9.9945pt\begin{array}[c]{l}\kern
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9.9945pt=\kern
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9.9945pt,\kern
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9.9945pt=\kern
9.9945pt\begin{array}[c]{l}{\hbox{\kern-2.59027pt${I}$}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\<EMAIL_ADDRESS>9.9945pt.\kern 9.9945pt$

then $H$ is called a bialgebra living in ${\cal C}$. If $H$ is a bialgebra and
there is an inverse $S$ of $id_{H}$ under convolution product in $Hom_{\cal
C}(H,H)$, then $H$ is called a Hopf algebra living in ${\cal C},$ or a braided
Hopf algebra.

When $H$ is a Hopf algebra algebra in braided tensor category
$({\mathcal{C}},C)$, then the condition above is equivalent to

(YD):

$\phi\alpha=\begin{array}[c]{l}\kern
29.98352pt{\hbox{\kern-4.56248pt${H}$}}\kern
69.96155pt{\hbox{\kern-5.39583pt${M}$}}\@arraycr\par\kern
9.9945pt\hbox{\put(2000.0,0.0){\oval(4000.0,2000.0)[ct]}\put(2000.0,1000.0){\line(0,1){1000.0}}}\kern
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29.98352pt\hbox{\put(0.0,0.0){\line(0,1){400.0}}\put(0.0,1600.0){\line(0,1){400.0}}\put(0.0,1000.0){\oval(1200.0,1200.0)}\put(0.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$S$}}}\kern
29.98352pt\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\par\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\kern 19.98901pt\kern
39.97803pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\par\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\kern 9.9945pt\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
39.97803pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\par\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\kern 19.98901pt\kern
29.98352pt\hbox{}\@arraycr\par\hbox{\put(1000.0,2000.0){\oval(2000.0,2000.0)[cb]}\put(1000.0,0.0){\line(0,1){1000.0}}}\kern
19.98901pt\kern
19.98901pt\hbox{\put(400.0,1500.0){\oval(800.0,800.0)[lb]}\put(1600.0,1500.0){\oval(800.0,800.0)[rb]}\put(0.0,1500.0){\line(0,1){500.0}}\put(2000.0,1500.0){\line(0,1){500.0}}\put(1000.0,0.0){\line(0,1){500.0}}\put(1000.0,1100.0){\oval(1200.0,1200.0)}\put(1000.0,1100.0){\makebox(0.0,0.0)[cc]{\scriptsize$\alpha$}}}\@arraycr\par\kern
9.9945pt\hbox{\put(1000.0,2000.0){\oval(2000.0,2000.0)[cb]}\put(1000.0,0.0){\line(0,1){1000.0}}}\kern
39.97803pt\kern 9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\kern
19.98901pt{\hbox{\kern-4.56248pt${H}$}}\kern
59.96704pt{\hbox{\kern-5.39583pt${M}$}}\end{array}\ \ \ \ .$

Let ${}^{H}_{H}{\mathcal{Y}D(C)}$ denote the category of all Yetter-Drinfeld
$H$-modules in ${\mathcal{C}}$. If $({\mathcal{C}},C)={\mathcal{V}}ect(k)$, we
write ${}^{H}_{H}{\mathcal{Y}D(C)}=$ ${}^{H}_{H}{\mathcal{Y}D}$, called
Yetter-Drinfeld category. It follows from [RT93] and [BD98, Theorem 4.1.1]
that ${}^{H}_{H}{\mathcal{Y}D}({\mathcal{C}})$ is a braided tensor category
with ${}^{YD}C_{U,V}=(\alpha_{V}\otimes id_{U})(id_{H}\otimes
C_{U,V})(\phi_{U}\otimes id_{V})$ for any two Yetter-Drinfeld modules
$(U,\phi_{U},\alpha_{U})$ and $(V,\phi_{V},\alpha_{V})$ when $H$ has an
invertible antipode. In this case,
${}^{YD}C_{U,V}^{-1}=(id_{V}\otimes\alpha_{U})(C_{H,V}^{-1}\otimes
id_{U})(S^{-1}\otimes C_{U,V}^{-1})(C_{U,H}^{-1}\otimes
id_{V})(id_{U}\otimes\phi_{V}).$ Algebras, coalgebras and Hopf algebras and so
on in ${}^{H}_{H}{\mathcal{Y}D}$ are called Yetter-Drinfeld ones or YD ones in
short.

Let $\mathbb{Z},$ $\mathbb{N}$ and $\mathbb{C}$ denote integer set, natural
number set and complex field, respectively. Throughout basic field is a
complex field $\mathbb{C}$, which is denoted by $k$ sometimes. If $V$ is a
vector space with a basis $v_{1},v_{2},\cdots,v_{n}$ and
$q_{ij}\in\mathbb{C}^{*}$ for $1\leq i,j\leq n$ such that map
$C:\left\\{\begin{array}[]{lll}V\otimes V&\rightarrow&V\otimes V\\\
v_{i}\otimes v_{j}&\mapsto&q_{ij}v_{j}\otimes v_{i}\end{array}\right.,$ then
$(V,C)$ is called a braided vector space of diagonal type. Denote by
$(q_{ij})_{n\times n}$ the braiding matrix of $(V,C)$ under the basis
$v_{1},v_{2},\cdots,v_{n}$. Then $(V,C)$ is also written as
$(V,(q_{ij})_{n\times n}).$ We can get Nichols algebra $\mathfrak{B}(V)$ (see
[He06b, He06a] and [ZZC04]). Let $\mathcal{B}$ denote the set of all
generators of restricted PBW basis and $\mathcal{P}$ denote restricted PBW
basis of $\mathfrak{B}(V)$.

Let $v_{1},v_{2},\cdots,v_{n}$ be a basis of $V\in^{kG}_{kG}{\mathcal{Y}D}$
with comodule operation and module operation $\delta(v_{i})=g_{i}\otimes
v_{i},$ $h\cdot v_{i}=\chi_{i}(h)v_{i}$, where $\chi_{i}$ is a multiplication
character of $G$, i.e. a homomorphism from $G$ to $\mathbb{C}^{*}$, for $1\leq
i\leq n$. It is clear that $(V,C)$ is a braided vector space of diagonal type.
Otherwise, for any $q_{ij}\in\mathbb{C}$, $1\leq i,j\leq n$ and a basis
$v_{1},v_{2},\cdots,v_{n}$ of $V$, $V$ can become a Yetter-Drinfeld module
over group $\mathbb{Z}^{n}$ by defining $\delta(v_{i})=e_{i}\otimes v_{i}$ and
$e_{j}\cdot v_{i}=q_{ji}v_{i}$, $1\leq i,j\leq n$, where
$e_{1}=(1,0,\cdots,0),\cdots,e_{n}=(0,0,\cdots,1)\in\mathbb{Z}^{n}.$

##  2 Quantum Design of Industry

In this section we give some examples to design by means of braiding. That is,
we design by means of quantum theory.

######  Example 2.1.

(Category of quantum design grammars) Let $\\{D_{i}\mid 1\leq i\leq m\\}$ be
the set of all shapes in some design $D$. Let $v_{1},v_{2},\cdots,v_{n}$ be a
basis of $V\in^{kG}_{kG}{\mathcal{Y}D}$ with $\delta(v_{i})=g_{i}\otimes
v_{i},$ $h\cdot v_{i}=\chi_{i}(h)v_{i}$. Assume that $g_{i}=g_{j}$ and
$\chi_{i}=\chi_{j}$ when there exists rule $R_{ij}:D_{i}\rightarrow D_{j}$.
Define a morphism $f_{ij}$ from $kv_{i}$ to $kv_{j}$ in
${}^{kG}_{kG}{\mathcal{Y}D}$ such that $f_{ij}(v_{i})=R_{ij}(D_{i})$. It is
clear that $f_{ij}$ is a morphism in ${}^{kG}_{kG}{\mathcal{Y}D}$ and $kv_{i}$
and $V$, are in ${}^{kG}_{kG}{\mathcal{Y}D}$. Define a map
$\psi:\mathcal{P}\rightarrow\\{D_{i}\mid 1\leq i\leq m\\}$ such that
$\psi(v_{i})=D_{i}$. $\psi(u)$ denote the combination of shapes of
$D_{i_{1}}$, $D_{i_{1}}$, $\cdots,$ $D_{i_{r}}$ when
$u=v_{i_{1}}v_{i_{2}}\cdots v_{i_{r}}\in\mathcal{P}.$

######  Example 2.2.

( Quantum design grammars of Coca-Cola) Let $D_{1},D_{2}$, $D_{3}$ and $D_{4}$
be initial shapes; $D_{5},D_{6}$, $D_{7}$ and $D_{8}$ be cap shape, above
shape, middle shape and below shape of battle of Coca-Cola; Rule
$R_{15}:D_{1}\rightarrow D_{5}$, Rule $R_{26}:D_{2}\rightarrow D_{6}$, Rule
$R_{37}:D_{3}\rightarrow D_{7}$ and $R_{48}:D_{4}\rightarrow D_{8}$.

(i) Let $G=\mathbb{Z}_{7}$ be cycle group and $C(v_{s}\otimes
v_{t})=e^{\frac{2\pi ig_{s}g_{t}}{7}}=\omega^{g_{s}g_{t}}$, where
$\omega:=e^{\frac{2\pi i}{7}}$ and $i:=\sqrt{-1}$. Assume that $g_{s}=s$,
$\chi_{s}(g_{t})=\omega^{st}$ when $1\leq s,t\leq 4;$ $g_{4+p}=g_{p}$,
$\chi_{4+p}=\chi_{p}$ for $1\leq p\leq 4.$

$\begin{array}[c]{clr}\kern 9.9945pt{\hbox{\kern-4.003pt${v_{1}}$}}\kern
19.98901pt{\hbox{\kern-4.003pt${v_{2}}$}}\kern
19.98901pt{\hbox{\kern-4.003pt${v_{3}}$}}\kern
19.98901pt{\hbox{\kern-4.003pt${v_{4}}$}}\@arraycr\kern
9.9945pt\hbox{\put(0.0,0.0){\line(0,1){400.0}}\put(0.0,1600.0){\line(0,1){400.0}}\put(0.0,1000.0){\oval(1200.0,1200.0)}\put(0.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$f_{15}$}}}\kern
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9.9945pt{\hbox{\kern-16.5417pt${\hbox{figure }1}$}}\@arraycr\end{array};\ \ \
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9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\kern
9.9945pt\hbox{\put(1000.0,2000.0){\oval(2000.0,2000.0)[cb]}\put(1000.0,0.0){\line(0,1){1000.0}}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
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9.9945pt\hbox{}\@arraycr\kern 9.9945pt\hbox{}\@arraycr\kern
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9.9945pt{\hbox{\kern-9.54456pt${\omega^{18}u^{\prime}}$}}\@arraycr\kern
9.9945pt{\hbox{\kern-16.5417pt${\hbox{figure }2}$}}\@arraycr\end{array}\ \ \ \
\ .$

Here $u=v_{6}v_{7}v_{8}$ and $\psi(u)$ is the battle of Coca-Cola without cap,
$u^{\prime}=v_{5}v_{6}v_{7}v_{8}$ and $\psi(u^{\prime})$ is the battle of
Coca-Cola with cap. Figure 1 is an ordinary design and Figure 2 is a quantum
design. Define that
$\omega^{1}u^{\prime},\omega^{2}u^{\prime},\cdots,\omega^{7}u^{\prime}$ means
that the color of the battle of Coca-Cola is Red, orange, yellow, green, cyan,
blue, purple, respectively. Therefore,
$\omega^{18}u^{\prime}=\omega^{4}u^{\prime}$ and the color of the battle of
Coca-Cola is green.

(ii) Let $G=\mathbb{Z}$ and $\chi_{1}(g_{1})=q$; $1\not=q$ be a positive real
number; $g_{i}=g_{1}$ and $\chi_{i}=\chi_{1}$ for $1\leq i\leq 8$. Then
$C(v_{s}\otimes v_{t})=q(v_{t}\otimes v_{s})$. Similarly, we can get figure 3
by replacing $\omega^{18}$ by $q^{6}$ in figure 2. Define that
$q^{6}u^{\prime}$ means that the size of the battle of Coca-Cola is $q^{6}$
times of original battle $\psi(u^{\prime})$ of Coca-Cola.

######  Example 2.3.

Let $D_{0},D_{1},D_{3}$, $D_{5}$, $D_{7}$ and $D_{9}$ be frame, left steering
wheel, left front gate, left back gate, left front lamp and left back lamp of
car. We shall obtain a design of car by means of these left components of car
as follows. Let $G=\mathbb{Z}^{10}$ and $C(v_{0}\otimes v_{t})=qv_{t}\otimes
v_{0}$, $C(v_{s}\otimes v_{t})=v_{t}\otimes v_{s}$, where $q\not=1$ is
positive real number for $s\not=0$, $s,t=1,3,5,7,9.$

$\begin{array}[c]{lll}\kern 9.9945pt{\hbox{\kern-4.003pt${v_{9}}$}}\kern
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19.98901pt{\hbox{\kern-4.003pt${v_{5}}$}}\kern
19.98901pt{\hbox{\kern-4.003pt${v_{3}}$}}\kern
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19.98901pt{\hbox{\kern-4.003pt${v_{1}}$}}\kern
19.98901pt{\hbox{\kern-4.003pt${v_{3}}$}}\kern
19.98901pt{\hbox{\kern-4.003pt${v_{5}}$}}\kern
19.98901pt{\hbox{\kern-4.003pt${v_{7}}$}}\kern
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99.94507pt{\hbox{\kern-6.67374pt${q^{4}u}$}}\@arraycr\par\kern
99.94507pt{\hbox{\kern-16.5417pt${\hbox{figure
}4}$}}\@arraycr\par\end{array},$

where $u=:v_{0}v_{1}v_{3}^{2}v_{5}^{2}v_{7}^{2}v_{9}^{2}$ and
$q^{4}u=v_{0}v_{1}(qv_{3})v_{3}(qv_{5})v_{5}(qv_{7})v_{7}(qv_{9})v_{9}$.
Define that $(qv_{1}),$ $(qv_{3}),$ $(qv_{5}),$ $(qv_{7}),$ $(qv_{9})$ denote
right steering wheel, right front gate, right back gate, right front lamp and
right back lamp of car, respectively. Figure 4 is a braiding diagram which
represents a combination with left components of car and whole car consists of
them. Notice that the steering wheel is in left side. If we require that the
steering wheel is in right side of car, we must place the $v_{1}$ in the left
hand side of $v_{0}$ in braiding diagram and have braiding $C(v_{1}\otimes
v_{0}).$

######  Example 2.4.

Let $D_{4},D_{1}$, $D_{2}$ and $D_{3}$ be the first floor, the second floor
and the third floor of ship; $D_{5}$ the brightness of ship. We shall obtain a
design about the brightness of ship by means of braiding diagram as follows.
Let $G=\mathbb{Z}^{5}$ and $C(v_{s}\otimes v_{t})=q_{st}(v_{t}\otimes v_{s})$,
where $q_{st}$ is positive real number for $1\leq s,t\leq 5.$

$\begin{array}[c]{lll}\kern 9.9945pt{\hbox{\kern-4.003pt${v_{3}}$}}\kern
19.98901pt{\hbox{\kern-4.003pt${v_{2}}$}}\kern
19.98901pt{\hbox{\kern-4.003pt${v_{1}}$}}\kern
19.98901pt{\hbox{\kern-4.003pt${v_{4}}$}}\kern
19.98901pt{\hbox{\kern-4.003pt${v_{5}}$}}\@arraycr\par\kern
9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\hbox{}\@arraycr\par\kern
9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\par\kern
9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\par\kern
9.9945pt\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\par\kern
9.9945pt{\hbox{\kern-4.003pt${v_{5}}$}}\kern
19.98901pt{\hbox{\kern-4.26228pt${u_{3}}$}}\kern
19.98901pt{\hbox{\kern-4.26228pt${u_{2}}$}}\kern
19.98901pt{\hbox{\kern-4.26228pt${u_{1}}$}}\kern
19.98901pt{\hbox{\kern-4.26228pt${u_{4}}$}}\@arraycr\par\kern
49.97253pt{\hbox{\kern-16.5417pt${\hbox{figure 5}}$}}\@arraycr\par\end{array}\
\ \ \ ,$

where $u_{3}:=q_{35}v_{3}$, $u_{2}:=q_{25}v_{2}$, $u_{1}:=q_{15}v_{1}$,
$u_{4}:=q_{45}v_{4}.$ Define that the brightness in the first floor, the
second floor, the third floor and negative first floor are $q_{15}$ unit,
$q_{25}$ unit, $q_{35}$ unit and $q_{45}$ unit according to $q_{35}v_{1}$,
$q_{25}v_{2}$, $q_{35}v_{2}$ and $q_{45}v_{4}$ in braiding diagram. We can
choose the value of $q_{st}$ according to the physical truth and requirement
of customer.

In fact, we also provide the brightness of every components of car be means of
braiding diagram.

##  3 Quantum Design of Codes

In this section we obtain a code by means of braiding.

######  Example 3.1.

We need send a word $w$ from $X$ to $Y$ by internet and require to keep
secret, where $X$ and $Y$ are two persons or two companies. Let $\\{D_{i}\mid
1\leq i\leq n\\}$ be a vocabulary and $Q=(q_{ij})_{n\times n}$ its quantum
matrix.

$\begin{array}[c]{lll}\kern 39.97803pt{\hbox{\kern-4.003pt${v_{1}}$}}\kern
19.98901pt{\hbox{\kern-4.003pt${v_{2}}$}}\kern
19.98901pt{\hbox{\kern-4.003pt${v_{3}}$}}\kern
19.98901pt{\hbox{\kern-4.003pt${v_{4}}$}}\kern
19.98901pt{\hbox{\kern-4.003pt${v_{5}}$}}\kern
19.98901pt{\hbox{\kern-4.003pt${v_{6}}$}}\kern
19.98901pt{\hbox{\kern-4.003pt${v_{7}}$}}\kern
19.98901pt{\hbox{\kern-4.003pt${v_{8}}$}}\kern
19.98901pt{\hbox{\kern-4.003pt${v_{9}}$}}\kern
<EMAIL_ADDRESS>39.97803pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
<EMAIL_ADDRESS>39.97803pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\kern
19.98901pt\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
<EMAIL_ADDRESS>39.97803pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\kern
19.98901pt\kern
19.98901pt\hbox{\put(1000.0,2000.0){\oval(2000.0,2000.0)[cb]}\put(1000.0,0.0){\line(0,1){1000.0}}}\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
<EMAIL_ADDRESS>39.97803pt\kern 19.98901pt\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\kern
19.98901pt\kern 9.9945pt\kern
<EMAIL_ADDRESS>39.97803pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\kern
19.98901pt\kern 9.9945pt\kern
9.9945pt\hbox{}\<EMAIL_ADDRESS>39.97803pt\hbox{\put(1000.0,2000.0){\oval(2000.0,2000.0)[cb]}\put(1000.0,0.0){\line(0,1){1000.0}}}\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\kern
19.98901pt\hbox{\put(1000.0,2000.0){\oval(2000.0,2000.0)[cb]}\put(1000.0,0.0){\line(0,1){1000.0}}}\kern
9.9945pt\kern<EMAIL_ADDRESS>49.97253pt\kern 19.98901pt\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\kern
9.9945pt\kern<EMAIL_ADDRESS>59.96704pt\hbox{\put(1000.0,2000.0){\oval(2000.0,2000.0)[cb]}\put(1000.0,0.0){\line(0,1){1000.0}}}\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern
<EMAIL_ADDRESS>69.96155pt\kern 19.98901pt\kern
19.98901pt\hbox{\put(1000.0,2000.0){\oval(2000.0,2000.0)[cb]}\put(1000.0,0.0){\line(0,1){1000.0}}}\kern
<EMAIL_ADDRESS>79.95605pt\hbox{\put(1000.0,2000.0){\oval(2000.0,2000.0)[cb]}\put(1000.0,0.0){\line(0,1){1000.0}}}\kern
19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\kern
<EMAIL_ADDRESS>89.95056pt\kern 19.98901pt\kern
<EMAIL_ADDRESS>99.94507pt\hbox{\put(1000.0,2000.0){\oval(2000.0,2000.0)[cb]}\put(1000.0,0.0){\line(0,1){1000.0}}}\kern
9.9945pt\kern<EMAIL_ADDRESS>109.93958pt\hbox{\put(1000.0,2000.0){\oval(2000.0,2000.0)[cb]}\put(1000.0,0.0){\line(0,1){1000.0}}}\kern
<EMAIL_ADDRESS>119.93408pt\hbox{\put(1000.0,2000.0){\oval(2000.0,2000.0)[cb]}\put(1000.0,0.0){\line(0,1){1000.0}}}\@arraycr\par\kern
129.92859pt{\hbox{\kern-5.27374pt${qu}$}}\@arraycr\par\kern
119.93408pt{\hbox{\kern-16.5417pt${\hbox{figure
6}}$}}\@arraycr\par\end{array}\ \ \ \ ,$

where $u=v_{5}v_{6}v_{4}v_{7}v_{3}v_{8}v_{2}v_{9}v_{1}v_{10}$.

Let $Q$ and figure 6 are the private key cryptography. i.e. $X$ and $Y$ have
$Q$ and figure 6 without using internet. Assume $w=D_{s}$. We choose $p$ such
that $s\in\\{p,p+1,\cdots,p+9\\}$ and positive real numbers
$a_{1},\cdots,a_{10}$ randomly with $a_{1}\not=1,\cdots,a_{10}\not=1$. Let
$b_{i}=a_{i}$ when $i\not=s$ and $b_{s}=1.$ Let $v_{i}:=D_{p+i-1}$ and
$\widetilde{q_{ij}}:=q_{p+i-1,p+j-1}$ for $1\leq i,j\leq 10.$ It is clear
$C^{-1}(v_{i}\otimes v_{j})=\widetilde{q_{ji}}^{-1}(v_{j}\otimes v_{i})$ and
$C^{-1}(v_{i}\otimes u)=\prod_{j=1}^{10}\widetilde{q_{ji}}^{-1}(u\otimes
v_{i})$. Compute the value $c_{i}$ of
$b_{i}\prod_{j=1}^{10}\widetilde{q_{ji}}^{-1}$ and send $c_{i}$ to $Y$ for
$i=1,2,\cdots,10$ and send $p$ to $Y$. In $Y$ compute the value $d_{i}$ of
$c_{i}\prod_{j=1}^{10}\widetilde{q_{ij}}$ and $d_{i}=1$ if and only if
$s=i+p-1$ since coefficient of $C^{-1}(v_{i}\otimes u)$ is
$\prod_{j=1}^{10}\widetilde{q_{ji}}^{-1}$ and coefficient of $c(u\otimes
v_{i})$ is $\prod_{j=1}^{10}\widetilde{q_{ji}}$. Consequently, we can get the
word $w$ in $Y.$ Furthermore, we can send a article from $X$ to $Y$ by means
of internet and keeping secret because they consist of some single words. We
also can send file from $X$ to $Y$ by means of internet and keeping secret
because the password consist of some single words.

Otherwise, we can modify the method above as follows.

Remark. (i) In case above, we choose the last layer 14-th layer and left line
$u$. In fact, we also can choose $t$-th layer with $1\leq t\leq 14$ and left
line $v$ which need be sent to $Y$. That is, private key cryptography contains
$Q$, figure 6, $p$, $t$-th layer and left line $v$. For example, set $t=7$,
i.e. $v=v_{5}v_{6}.$ It is clear $C^{-1}(v_{i}\otimes
v)=\prod_{j=5}^{6}\widetilde{q_{ji}}^{-1}(u\otimes v_{i})$ and $c(v\otimes
v_{i})=\prod_{j=5}^{6}\widetilde{q_{ji}}(v_{i}\otimes v)$. Compute the value
$c_{i}$ of $b_{i}\prod_{j=5}^{6}\widetilde{q_{ji}}^{-1}$ and send $c_{i}$ to
$Y$ for $i=1,2,\cdots,10$. In $Y$ compute the value $d_{i}$ of
$c_{i}\prod_{j=5}^{6}\widetilde{q_{ij}}$ and $d_{i}=1$ if and only if $s=i$.

(ii) figure 6 and vocabulary $\\{D_{i}\mid 1\leq i\leq n\\}$ can become public
key cryptography. The order of vocabulary $\\{D_{i}\mid 1\leq i\leq n\\}$ can
be the same as Xinhua dictionary.

(iii) Vocabulary $\\{D_{r+i}\mid 1\leq i\leq n\\}$ can become public key
cryptography and $r$ can become the private key cryptography.

## References

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  * [OCP08] S. Orsborn, J. Cagan and P. Boatwright, Automating the Creation of Shape Grammar Rules, Design Computing and Cognition DCC 08. J.S. Gero and A. Goel (eds), Press Springer 2008, pp. 3-22.
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# Multiscale modelling of animal movement
with persistent dynamics

Théo Michelot111Email<EMAIL_ADDRESS>Dalhousie University

###### Abstract

Wild animals are commonly fitted with trackers that record their position
through time, to learn about their behaviour. Broadly, statistical models for
tracking data often fall into two categories: local models focus on describing
small-scale movement decisions, and global models capture large-scale spatial
distributions. Due to this dichotomy, it is challenging to describe
mathematically how animals’ distributions arise from their short-term movement
patterns, and to combine data sets collected at different scales. We propose a
multiscale model of animal movement and space use based on the underdamped
Langevin process, widely used in statistical physics. The model is convenient
to describe animal movement for three reasons: it is specified in continuous
time (such that its parameters are not dependent on an arbitrary time scale),
its speed and direction are autocorrelated (similarly to real animal
trajectories), and it has a closed form stationary distribution that we can
view as a model of long-term space use. We use the common form of a resource
selection function for the stationary distribution, to model the environmental
drivers behind the animal’s movement decisions. We further increase
flexibility by allowing movement parameters to be time-varying, e.g., to
account for daily cycles in an animal’s activity. We formulate the model as a
state-space model and present a method of inference based on the Kalman
filter. The approach requires discretising the continuous-time process, and we
use simulations to investigate performance for various time resolutions of
observation. The approach works well at fine resolutions, though the estimated
stationary distribution tends to be too flat when time intervals between
observations are very long.

##  1 Introduction

Wild animals are commonly tracked using satellite tags, typically resulting in
two-dimensional locations collected at discrete time intervals. Various
statistical methods have been developed to analyse such data and answer
increasingly complex ecological questions (Hooten et al.,, 2017). Those
methods can broadly be divided into two classes, depending on the spatio-
temporal scale of focus. Global models describe large-scale patterns of space
use by animals, and local models focus on small-scale movement patterns. These
roughly correspond to the distinction in mathematical biology between Eulerian
approaches, which describe the temporal evolution of the spatial distribution
of animals, and Lagrangian approaches, which describe the movement dynamics of
a single animal (Turchin,, 1998). Although the large-scale patterns emerge
from small-scale decisions, most existing models ignore this mechanism. The
main approach to bridge this gap has been to develop analytical and numerical
methods to simulate the large-scale distribution that emerges from a fitted
animal movement model (Barnett and Moorcroft,, 2008; Whitehead and Jonsen,,
2013; Potts and Börger,, 2023; Signer et al.,, 2024). However, those methods
do not provide an explicit model for the large-scale distribution. Recently,
it has been proposed that stationary processes could be used to define
movement models that scale up to a known long-term distribution (Michelot et
al., 2019a, ; Michelot et al., 2019b, ; Michelot et al.,, 2020). Here, we
present the underdamped Langevin diffusion process, a model from statistical
physics, as a multiscale model of animal space use.

##  2 Model overview

###  2.1 The underdamped Langevin process

Let $(\bm{Z}_{t})$ denote the continuous-time process for the location $X_{t}$
and velocity $V_{t}$ of an animal at time $t\geq 0$, i.e.,
$\bm{Z}_{t}=(X_{t},V_{t})^{\intercal}$. The underdamped Langevin process is
described by the system of SDEs,

$\begin{cases}dX_{t}=V_{t}\ dt\\\ dV_{t}=-\gamma V_{t}\
dt+\sigma^{2}\nabla\log[\pi(X_{t})]\ dt+\sqrt{2\gamma}\sigma\
dW_{t}\end{cases},$ (1)

with initial condition $Z_{0}=(x_{0},v_{0})^{\intercal}$, where $\pi$ is a
differentiable function from $\mathbb{R}$ to $\mathbb{R}^{+}$, $\nabla$ is the
gradient with respect to location, $(W_{t})$ is a standard Wiener process, and
$\gamma>0$ and $\sigma>0$ are parameters (Cheng et al.,, 2018). In Equation 1,
the location is defined as the time integral of the velocity, and the velocity
is a mean-reverting process pulled towards higher values of $\pi$ (with some
stochasticity). The location process $(X_{t})$ therefore combines directional
persistence, because velocity is autocorrelated, and attraction to high values
of $\pi$. We focus the mathematical exposition on the one-dimensional case
because Equation 1 can be used directly to model each dimension of a multi-
dimensional _isotropic_ process (e.g., two or three dimensions for animal
movement). Under mild regularity conditions, the marginal stationary
distribution of the process $(X_{t})$ defined in Equation 1 is proportional to
$\pi$ (Cheng et al.,, 2018; Eberle et al.,, 2019). In later sections, we will
use this property to define a multiscale model where an animal’s movement
decisions, described by the SDE, are directly linked to the emergent
distribution.

In physics, Equation 1 is used to describe the motion of a particle under an
external force measured by $F=\nabla\log\pi$ (Risken,, 1996, Chapter 10). In
that context, the parameter $\gamma$ is often called the friction coefficient,
as larger values imply stronger reversion of the velocity to zero (Erdmann et
al.,, 2000). Using the Fokker-Planck approach, the underdamped Langevin
process has also been formulated through a partial differential equation for
evolution of the probability density function of $(\bm{Z}_{t})$, known as the
Kramers equation (Kramers,, 1940; Erdmann et al.,, 2000; Hadeler et al.,,
2004; Gardiner,, 2004).

Because $(X_{t})$ has $\pi$ as its stationary distribution, this process has
also been used to build Markov Chain Monte Carlo (MCMC) algorithms to sample
from $\pi$ (Chen et al.,, 2014). It is a special case of the general method of
Ma et al., (2015, 2019) for specifying stochastic differential equations with
a given stationary distribution, which they use to define a general family of
samplers. In particular, Equation 1 defines an irreversible stochastic
process, in contrast with more common reversible processes such as the
_overdamped_ Langevin diffusion (Roberts and Tweedie,, 1996).

###  2.2 A multiscale animal movement model

If the process $(X_{t})$ defined in Equation 1 describes the position of an
animal, the function $\pi$ represents its long-term distribution over space,
analogous to species distribution models widely used in ecology (Aarts et
al.,, 2012). Then, $\pi$ may be determined by habitat suitability and other
factors that affect the animal’s space use (e.g., corridors it needs to travel
through). We propose using the underdamped Langevin process as a multi-scale
model of animal movement, which links the small-scale movement dynamics
described by the stochastic differential equations (Equation 1) to the large-
scale spatial distribution $\pi$. Michelot et al., 2019b  proposed a similar
model based on the _overdamped_ Langevin diffusion, but the underdamped
process presented herein has the advantage of describing realistic small-scale
movement dynamics, including a finite speed and directional persistence.

####  2.2.1 Modelling the utilisation distribution

To ensure that it is positive, a natural choice for the utilisation
distribution $\pi$ is of the form

$\pi(x)\propto\exp\left(\sum_{k=1}^{K}\beta_{k}\psi_{k}(x)\right),$ (2)

where the $\psi_{k}$ are known differentiable functions over space. Due to the
linearity of the gradient operator, the gradient term in the SDE for the
velocity process can then be written as a linear combination of the gradients
of the functions $\psi_{k}$, i.e.,
$\nabla\log\pi(X_{t})=\sum_{k=1}^{K}\beta_{k}\nabla\psi_{k}(X_{t})$. We
suggest two possible choices for the functions $\psi_{k}$, either as spatial
covariates that can be used to predict space use, or as basis functions in a
non-parametric approach.

The log-linear form for the utilisation distribution $\pi$ (Equation 2) is
often called a resource selection function when the functions $\psi_{k}$ are
spatial covariates of interest (Johnson et al.,, 2006; Aarts et al.,, 2012).
The $\beta_{k}$ coefficients then measure the association between an animal’s
space use and the measured environmental features, and often interpreted in
terms of strength of “selection” or “avoidance” (Fieberg et al.,, 2021). This
approach can directly be integrated into the underdamped Langevin process: the
velocity $V_{t}$ tends to be “pulled” towards higher values (if $\beta_{k}>0$)
or lower values (if $\beta_{k}<0$) of the covariate $\psi_{k}$. This provides
an intuitive description of an animal’s movement decisions, based on the local
environmental features captured by the gradient.

Alternatively, if spatial covariates are not available, or if they do not
fully capture patterns of space use, we can use a non-parametric model of
$\pi$ where the $\psi_{k}$ are basis functions. In this context, the weights
$\beta_{k}$ cannot be interpreted separately, and only the resulting function
$\pi$ is of interest. This is for example similar to the approaches of
Preisler et al., (2004), Russell et al., (2018) and Brückner et al., (2020).
In the non-parametric approach, we assume that the vector of basis
coefficients follows $\bm{\beta}\sim N(\bm{0},\bm{\Sigma})$, which can be
viewed as a prior distribution or as a smoothness penalty to constrain the
shape of the function $\pi$ (Wood,, 2017). In the two-dimensional case of
animal movement, one could for example use thin plate regression splines, for
which both the basis functions $\psi_{k}$ and the smoothness penalty matrix
$\bm{\Sigma}$ can be obtained using the R package mgcv (Wood,, 2003, 2017). A
model could combine parametric and non-parametric terms, as is common in
spatial modelling where penalised splines are used to capture patterns that
cannot be explained by available covariates.

Note that, in practice, environmental covariates are almost always measured
and stored over a discrete spatial grid (a “raster”). The corresponding
functions $\psi_{k}$ are then piecewise constant, and their gradients cannot
be used directly in the model (because the slope is zero within each grid
cell, and undefined at the boundaries). In this case, we instead consider the
bilinear interpolation of the raster, for which the gradient can easily be
computed analytically (see e.g., Michelot et al., 2019b, ). Other smoothing or
interpolation methods may be considered.

####  2.2.2 Parameter interpretation

The parameter $\gamma$ is called the friction coefficient in physics, and
larger values lead to less smooth-looking trajectories. Its inverse
$1/\gamma$, sometimes called the relaxation time of the process (Risken,,
1996), is the time interval it takes for the autocorrelation of the velocity
to decrease by a factor $e$, such that $3/\gamma$ can be interpreted as the
time scale over which the autocorrelation of the velocity decreases by
approximately 95% (Michelot and Blackwell,, 2019). The parameter $\sigma^{2}$
is most directly interpreted as the variance of the limiting distribution of
the velocity process. As a result, $\sqrt{\pi/2}\times\sigma$ is the mean
speed of movement of the animal (Michelot and Blackwell,, 2019). Figure 1
shows simulations from the underdamped Langevin process for different
combinations of values for $\gamma$ and $\sigma$, all with the same underlying
stationary distribution $\pi$. It illustrates the flexibility of the model to
capture a wide range of movement dynamics, all leading to the same large-scale
emergent patterns. In Section 2.3, we extend this further by allowing $\gamma$
and $\sigma$ to depend on time-varying covariates, e.g., to account for
cyclical patterns in an animal’s activity.

Figure 1: Simulations from the underdamped Langevin process for different
values of the movement parameters $\gamma$ (columns) and $\sigma$ (rows). The
background colour shows the stationary distribution, which was identical for
all simulations.

####  2.2.3 Links to other movement models

The underdamped Langevin process includes many widely-used continuous-time
movement models as special or limiting cases. The continuous-time correlated
random walk (or integrated Ornstein-Uhlenbeck process) of Johnson et al.,
(2008) is obtained when $\nabla\log\pi=0$, i.e., when the stationary
distribution $\pi$ is flat. The model presented here can therefore be viewed
as an extension of that model to the case where movement is driven by habitat,
similar to the approach of Russell et al., (2018). When both $\gamma\to\infty$
and $\sigma\to\infty$ such that $\sigma^{2}/\gamma$ remains constant, the
underdamped Langevin process reduces to the overdamped (first-order) Langevin
process (proof in Appendix A). Michelot et al., 2019b  used the overdamped
process for animal movement, but it has no directional persistence and
infinite speed, making it a less realistic model for the fine-scale movement
dynamics. Michelot et al., (2020) described a reversible stochastic process
inspired by Markov chain Monte Carlo samplers as the basis for a multiscale
movement model. Similarly to the overdamped Langevin process, that model did
not include directional persistence, leading to random walk-like movement
patterns.

###  2.3 Time-varying dynamics

The dynamics of an animal’s movement often change through time, as a result of
changes in its behavioural state (e.g., resting, foraging, exploring; Morales
et al.,, 2004). To account for this, two common approaches are to allow for
regime-switching through a latent continuous-time Markov chain (Blackwell,,
1997; Michelot and Blackwell,, 2019), or to specify the movement parameters as
functions of time-varying covariates (Michelot et al.,, 2021). Here, we adopt
the latter approach, which is relatively easy to implement, and can relate
movement patterns to temporal variables. Following the varying-coefficient
stochastic differential equation approach of Michelot et al., (2021), the
movement parameters can be specified using a combination of linear, non-
linear, and random effects of covariates. A common example would be to model
the speed or directionality of movement with a cyclical effect of time of day.

As long as the movement parameters do not depend on space (e.g., do not depend
on spatial covariates), the resulting process $(X_{t})$ still has the
stationary distribution $\pi$. Indeed, it can be viewed as an infinite mixture
of processes that all share the same stationary distribution. This provides a
very flexible framework to relate an animal’s time-varying movement dynamics
to a explicit long-term emerging distribution.

It is in principle straightforward to apply a similar method to write the
parameters of $\pi$ as functions of time-varying covariates. Although this
could be used to capture behaviour-dependent habitat preferences (similar to
Nicosia et al.,, 2017), the time-varying function $\pi$ would not be a
stationary distribution for $(X_{t})$ any more, and the large-scale
distribution of space use may not be available in closed form.

##  3 Statistical inference and implementation

Given observations from the location process $(X_{t})$, we wish to estimate
the dynamics of the velocity process, i.e., the movement parameters $\gamma$
and $\sigma$, and the parameters of the stationary distribution $\pi$. Russell
et al., (2018) proposed a similar model for the movements of ants, and
implemented a Bayesian approach based on the Euler-Maruyama discretisation of
the position and velocity processes. Brückner et al., (2020) used an
alternative method to derive estimators of the force field and noise in
underdamped systems, while correcting for discretisation and observation
errors.

In this section, we propose another framework of inference based on the Kalman
filter. The key challenge is that the velocity process is not observed
directly, so we treat it as a latent variable in a state-space model
formulation, similar to the approach of Johnson et al., (2008) for the
integrated Ornstein-Uhlenbeck process.

###  3.1 Transition density of discretised process

We first consider a realisation from the joint process $(\bm{Z}_{t})$
(location and velocity), with values
$(\bm{z}_{1},\bm{z}_{2},\dots,\bm{z}_{n})$ at observation times
$t_{1}<t_{2}<\dots<t_{n}$, and we define $\Delta_{i}=t_{i+1}-t_{i}$ as the
$i$-th time interval. There is no general solution to the system of SDEs in
Equation 1 because the dynamics of the process depend on the possibly complex
function $\pi$, but an approximate solution can be found based on the
assumption that the force term $\nabla\log\pi(X_{t})$ is constant over each
time interval $[t_{i},t_{i+1})$. Cheng et al., (2018) derived the transition
density of a special case of this discretised underdamped Langevin process
(for a particular value of the parameters $\gamma$ and $\sigma$), and we use
similar calculations to find the following general formulas:

$\bm{Z}_{t_{i+1}}\mid\\{\bm{Z}_{t_{i}}=\bm{z}_{i}\\}\sim N(\bm{\mu}_{i},\
\bm{Q}_{i}),$

where

$\bm{\mu}_{i}=\begin{pmatrix}x_{i}+\frac{v_{i}(1-e^{-\gamma\Delta_{i}})}{\gamma}+\frac{\sigma^{2}\nabla\log\pi(x_{i})}{\gamma}\left(\Delta_{i}-\frac{1-e^{-\gamma\Delta_{i}}}{\gamma}\right)\\\
v_{i}e^{-\gamma\Delta_{i}}-\frac{\sigma^{2}\nabla\log\pi(x_{i})}{\gamma}(1-e^{-\gamma\Delta_{i}})\end{pmatrix}$

and

$\bm{Q}_{i}=\begin{pmatrix}\sigma^{2}\left(\frac{2\Delta_{i}}{\gamma}-\frac{e^{-2\gamma\Delta_{i}}}{\gamma^{2}}-\frac{3}{\gamma^{2}}+\frac{4e^{-\gamma\Delta_{i}}}{\gamma^{2}}\right)&\frac{\sigma^{2}}{\gamma}\left(1-2e^{-\gamma\Delta_{i}}+e^{-2\gamma\Delta_{i}}\right)\\\
\frac{\sigma^{2}}{\gamma}\left(1-2e^{-\gamma\Delta_{i}}+e^{-2\gamma\Delta_{i}}\right)&\sigma^{2}(1-e^{-2\gamma\Delta_{i}})\end{pmatrix}$

Over very short time intervals, the slope of $\log\pi$ will typically be
almost constant, leading to a good approximation, but the error will be
greater at coarser time resolutions. We illustrate this in simulations in
Section 4.

###  3.2 Inference using state-space model formulation

In practice, the velocity process is not observed, and we may only observe the
location at discrete times, so we take a state-space modelling approach. A
state-space model jointly describes the dynamics of an unobserved state
process, and the mechanism that connects the state to the observed data
(Durbin and Koopman,, 2012). In this context, the underlying state is the
joint process for the location and velocity of the animal, and the observation
are just the locations. The model can be rewritten in the form

Observation Equation: |  | $X_{t_{i}}=\bm{A}\bm{Z}_{t_{i}}$
---|---|---
State Equation: |  | $\bm{Z}_{t_{i+1}}=\bm{T}_{i}\bm{Z}_{t_{i}}+\bm{B}_{i}\bm{h}_{i}+\bm{\eta}_{i}$

where $\bm{\eta}_{i}\sim N(\bm{0},\bm{Q}_{i})$ is the process error (with
$\bm{Q}_{i}$ as defined in Section 3.1),
$\bm{h}_{i}=\nabla\log\pi(X_{t_{i}})$, and

$\displaystyle\bm{A}=\begin{pmatrix}1&0\end{pmatrix},\quad\bm{T}_{i}=\begin{pmatrix}1&\frac{1-e^{-\gamma\Delta_{i}}}{\gamma}\\\
0&e^{-\gamma\Delta_{i}}\end{pmatrix},\quad\text{and
}\bm{B}_{i}=\begin{pmatrix}\frac{\sigma^{2}}{\gamma}\left(\Delta_{i}-\frac{1}{\gamma}(1-e^{-\gamma\Delta_{i}})\right)\\\
\frac{\sigma^{2}}{\gamma}(1-e^{-\gamma\Delta_{i}})\end{pmatrix}.$

The observation equation simply represents the fact that velocity is not
observed. Note that, although the state-space formulation could in principle
be used to account for measurement error with an additional term in the
observation equation, this would not be sufficient to correct for the
corresponding errors in $\nabla\log\pi(X_{t})$. Therefore, we assume that
locations are observed with negligible error, and leave the problem of noisy
locations open. The state equation is directly derived from the transition
density of the discretised underdamped Langevin process given in Section 3.1.
For two-dimensional data such as animal movement tracks, the model takes a
similar formulation, but where the matrices given above are combined into
block matrices.

For a state-space model in this form, the likelihood can be evaluated using
the Kalman filter (Durbin and Koopman,, 2012, Section 7.2). We use maximum
likelihood estimation, but other likelihood-based inference methods could also
be implemented directly using this approach. Using large-sample
approximations, we derive uncertainty estimates based on the inverse of the
Hessian matrix of all model parameters. Model parameters can be estimated from
multiple independent time series, e.g., movement tracks collected on different
animals, by multiplying their likelihoods. If random effect and penalised
terms are included in the model for the utilisation distribution $\pi$ (e.g.,
spatial random field) or in the movement parameters (e.g., cyclical effect of
time of day), the model can be fitted using the marginal likelihood approach
described by Michelot et al., (2021).

##  4 Simulations

We assessed the performance of the method of inference presented in Section 3
using simulations. We defined the true stationary distribution as
$\pi(x)=\exp(\beta_{1}\psi_{1}(x)+\beta_{2}\psi_{2}(x)+\beta_{3}\psi_{3}(x))$
where $\psi_{1}$ and $\psi_{2}$ were generated at random on a raster grid, to
resemble spatial environmental covariates, and where $\psi_{3}$ was the
squared distance to the centre of the study region. We used the coefficients
$(\beta_{1},\beta_{2},\beta_{3})=(2,5,-10)$, representing positive “selection”
for the two environmental variables, and a tendency to remain close to the
centre (similar to home range behaviour of animals). For the simulation, the
movement parameters were set to $\gamma=1$ and $\sigma=1$. We simulated 100
trajectories from the discretised underdamped Langevin process on this map,
using a very fine time resolution of $\Delta=0.01$ to avoid any discretisation
error in the simulation. Then, we downsampled the data at random to (average)
time intervals of $\Delta\in\\{0.02,0.05,0.1,0.2,0.5,1,2\\}$, and fitted the
model on the thinned data.

The results are shown in Figure 2. All model parameters were recovered well at
short time resolutions. The friction parameter $\gamma$ displayed some
moderate positive bias at coarse time resolutions, corresponding to a slight
underestimation of the time scale of autocorrelation of the velocity process.
The speed parameter $\sigma$ was estimated well even for the longer time
intervals. In the model for the stationary distribution $\pi$, the estimates
for the coefficients $\beta_{1}$ and $\beta_{2}$ (measuring the association
between the distribution and two artificial environmental variables) decreased
as the time interval of observation increased; for very long intervals, they
were estimated close to zero. This is because those parameters are estimated
based on the association between the animal’s movement and the gradient of the
covariates at the start point of each observed step. As the time intervals get
longer, the association becomes weaker, i.e., the coefficients are
underestimated. Interestingly, the coefficient for the squared distance
$\psi_{3}$ did not seem to be biased even at coarse resolutions, although the
estimation variance increased somewhat. This is probably because $\psi_{3}$
does not change quickly over space, i.e., the assumption that its gradient
does not change between two observed locations is approximately correct even
for coarser data. As a result, even though $\beta_{1}$ and $\beta_{2}$ could
not be recovered at long intervals, the overall shape of the utilisation
distribution $\pi$ was estimated quite well; the correlation between the true
and estimated $\pi$ (measured on a raster grid) was around 0.85 on average
even at the coarsest resolution.

Figure 2: Results of simulation study. Parameter estimates from 100
simulations, for data thinned to different time resolutions $\Delta$. The
bottom-left panel shows the correlation between the estimated and true
utilisation distributions, measured on a raster grid. The other five panels
show the five parameters of the model, with the true values shown as
horizontal red lines.

##  5 Discussion

It is a common challenge to carry out inference for stochastic differential
equations from discrete data, and most approaches have to use approximations
such as the Euler-Maruyama discretisation (Iacus,, 2008). Similarly, the
approach we presented requires the assumption that the gradient of the
stationary distribution $\nabla\log\pi(X_{t_{i}})$ is approximately constant
between two observation times $t_{i}$ and $t_{i+1}$, and simulations revealed
that this leads to bias for coarse temporal resolutions. It may be possible to
use higher-order approximations to improve inference. We tried replacing
$\nabla\log\pi(X_{t_{i}})$ by
$(\nabla\log\pi(X_{t_{i}})+\nabla\log\pi(X_{t_{i+1}}))/2$, or by
$\nabla\log\pi((X_{t_{i}}+X_{t_{i}})/2)$, to account for dependence on the
shape of $\pi$ along the whole step (rather than only at the start point), but
this did not yield noticeably better results. It would be interesting to
explore other methods of inference for second-order stochastic differential
equations, such as that described by Pilipovic et al., (2024).

The Kramers equation is a partial differential equation describing the
evolution of the joint probability density of the position and velocity
through time, for the underdamped Langevin process (Hadeler et al.,, 2004). It
could be used to simulate the spatial distribution of an animal over
intermediate time steps, to learn about its transient movement dynamics (Potts
and Painter,, 2024). This could for example provide insights into the time
scale over which animals explore their environment, and the effects of habitat
fragmentation on their ability to efficiently travel across their range.

The underdamped Langevin diffusion process is a promising approach to analyse
animal tracking data, with great flexibility to capture small-scale movement
patterns, and with a direct link to the emerging utilisation distribution.
Other stationary processes within the general class described by Ma et al.,
(2015, 2019) might provide even better descriptions of animal movement, and we
hope that this paper will stimulate more work in this area.

### Acknowledgements

I want to thank Paul Blackwell for discussions on this topic, and for bringing
Ma et al., (2019) to my attention.

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Appendices

##  Appendix A Overdamped Langevin process as a limiting case

We first define $\tilde{\sigma}=\sigma\sqrt{\varepsilon}$ and
$\tilde{\lambda}=\lambda\varepsilon$ for some $\epsilon>0$, and we consider
the situation where $\varepsilon\rightarrow 0$ (i.e.,
$\sigma\rightarrow\infty$, and $\sigma^{2}/\lambda$ is fixed). The SDE for
$V_{t}$ (Equation 1) can be rewritten

$dV_{t}=-\frac{\tilde{\lambda}}{\varepsilon}V_{t}\
dt+\frac{\tilde{\sigma}^{2}}{\varepsilon}\nabla\log[\pi(X_{t})]\
dt+\frac{\sqrt{2\tilde{\lambda}}\tilde{\sigma}}{\varepsilon}\ dW_{t}$

and, if we multiply both sides by $\varepsilon$,

$\varepsilon dV_{t}=-\tilde{\lambda}V_{t}\
dt+\tilde{\sigma}^{2}\nabla\log[\pi(X_{t})]\
dt+\sqrt{2\tilde{\lambda}}\tilde{\sigma}\ dW_{t}$

In the limit $\varepsilon\rightarrow 0$, the left-hand side tends to zero, and
this equation becomes

$\tilde{\lambda}V_{t}\ dt=\tilde{\sigma}^{2}\nabla\log[\pi(X_{t})]\
dt+\sqrt{2\tilde{\lambda}}\tilde{\sigma}\ dW_{t}$

We plug this into the equation $dX_{t}=V_{t}dt$ (Equation 1), and we get

$dX_{t}=\frac{\tilde{\sigma}^{2}}{\tilde{\lambda}}\nabla\log[\pi(X_{t})]\
dt+\frac{\sqrt{2}\tilde{\sigma}}{\sqrt{\tilde{\lambda}}}\ dW_{t}.$

This equation defines an overdamped (first-order) Langevin diffusion process,
where the parameter $\tilde{\sigma}/\sqrt{\tilde{\lambda}}$ measures the
diffusion speed (Michelot et al., 2019b).