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An isomorphism theorem between the extended generalized balanced ternary numbers and the P-adic integers

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An isomorphism theorem between the extended generalized balanced ternary numbers and the P-adic integers
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Kitto, Wei Z., 1955-
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English
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vi, 78 leaves : ill. ; 28 cm.

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Algebra ( jstor )
Cell aggregates ( jstor )
Cells ( jstor )
Conceptual lattices ( jstor )
Hexagons ( jstor )
Image processing ( jstor )
Integers ( jstor )
Isomorphism ( jstor )
Mathematical theorems ( jstor )
Polynomials ( jstor )
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bibliography ( marcgt )
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Thesis:
Thesis (Ph. D.)--University of Florida, 1991.
Bibliography:
Includes bibliographical references (leaves 75-77).
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Wei Z. Kitto.

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AN ISOMORPHISM THEOREM BETWEEN
THE EXTENDED GENERALIZED BALANCED TERNARY NUMBERS
AND THE P-ADIC INTEGERS



By


WEI Z. KITTO


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


1991
















ACKNOWLEDGEMENTS


I would like to thank my advisors, Dr. David C. Wilson and Dr. Gerhard X.

Ritter, for introducing me to the exciting research area of image processing and for

their insights into the process of researching. I would like to especially thank Dr.

Wilson for his constant patience and encouragement.Without him this dissertation

would not have been possible. I would like to thank the other members on my

graduate committee, especially Dr. Andrew Vince, for all their help.

Most of all, I thank my parents and my husband for their support during the

years of my graduate study.


- ii -
















TABLE OF CONTENTS

ACKNOWLEDGEMENTS ...................... ii

ABSTRACT . v

CHAPTERS

1. INTRODUCTION ...... ....... .. ... 1

2. THE 7 adic INTEGERS AND THE RING EGBT2 ...... 8

2.1. Introduction .. .. .. .. 8
2.2. Inverse Limits and p-adic Integers .... 11
2.3. The 2-Dimensional Extended Generalized Balanced Ternary Numbers 15
2.4. The 2-dimensional Generalized Balanced Ternary Numbers 23
2.5. 15-adic Integers and the Ring EGBT3 .. 24
3. THE p adic INTEGERS AND THE RING EGBTn 28

3.1. Introduction . 28
3.2. The Carry Tables of EGBTn ............... 28
3.3. The ring EGBTn and q-adic integers ... 35
3.4. Examples .. ... .. 39

4. ANOTHER APPROACH TO EGBTn AND THE q adic INTEGERS 41

4.1. Introduction . .. .. 41
4.2. The Structure of Ra .. . 44

5. THE MATRIX Aa ...................... ... 449

5.1. The Algebraic Properties of the Matrix A. 50

6. IMAGE ALGEBRA IN HEXAGONAL LATTICE ... 57










6.1. A Brief Review of the Image Algebra .... .57
6.2. Hexagonal Images and Polynomial Rings .. 62
6.3. GBT2 Circulant Templates .. 64
6.4. Another Representation of GBT2 Circulant Templates 68

7. FINAL REMARKS ....... ................ 74

REFERENCES . 75

BIOGRAPHICAL SKETCH ......... .. .......... 78
















Abstract of Dissertation Presented to
the Graduate School of the University of Florida
in Partial Fulfillment of the Requirements for the Degree
of Doctor of Philosophy


AN ISOMORPHISM THEOREM BETWEEN
THE EXTENDED GENERALIZED BALANCED TERNARY NUMBERS
AND THE p-ADIC INTEGERS

By

Wei Zhang Kitto

December 1991


Chairman: Dr. David C. Wilson
Cochairman: Dr. Gerhard X. Ritter
Major Department: Mathematics


The Generalized Balanced Ternary Numbers (GBT) were developed by L. Gib-

son and D. Lucas (1982), who describe them as a hierarchical addressing system

for Euclidean space that has a useful algebraic structure derived from a hierarchy

of cells. At each level the cells are constructed of cells from the previous level ac-

cording to a rule of aggregation. For each dimension the most basic cell is different.

In dimension two it is a hexagon and in dimension three a truncated octahedron.

The basic cell in dimension n is an n + 1-permutohedron. A finite sequence of the

GBT digits 0, 1,..., 2n+ 2 can be used to identify any cell in n-dimensional Eu-

clidean space. Under the addition and multiplication which are defined in a manner

analogous to decimal arithmetic, the set of GBT addresses forms a commutative

ring GBTn. The Extended Generalized Balanced Ternary ring EGBTn consists

V -










of all such infinite sequences. The primary goal of this research is to prove that

if 2n+1 1 and n + 1 are relatively prime, then EGBTn is isomorphic as a ring

to the (2n+1 1)- adic integers. Extensions of this result are also given. The sec-

ondary goal of this research is to discuss template decomposition and inversion over

hexagonally sampled images.


- vi -















CHAPTER 1


INTRODUCTION


The 2-dimensional Generalized Balanced Ternary Numbers (GBT2) were de-
veloped by L. Gibson and D. Lucas as a method to address a hexagonal tiling of

Euclidean 2-space [6,7,8,17,18]. They describe the hexagonal tiling as a hierarchy of

cells, where at each level in the hierarchy new cells are constructed according to a

rule of aggregation. The most basic cell in this tiling is a hexagon. A hexagon and

its six neighbors form a cell called a first level aggregate. (The first level aggregates

obviously also tile 2-space and also have the uniform adjacency property that the

hexagonal tiling possesses.) A first level aggregate and its six neighbors form a sec-

ond level aggregate. The hierarchy continues in the obvious way. Figures 1, 2, and

3 illustrate a first, second and third level aggregate, respectively.

The GBT2 addressing method of the tiling above is based on the following
scheme. A first level aggregate L1 is chosen and labeled with the integers 0 through

6 as shown in Figure 4. The six first level aggregates neighboring L1 are labeled

with two digits as shown in Figure 5 and form a second level aggregate L2; each digit

is some integer from 0 to 6. Reading from right to left, the first digit corresponds

to where the labeled hexagon is in its first level aggregate L1 and the second digit

corresponds to where L1 is in the second level aggregate L2. Figure 6 shows the

labeling of the third level aggregate centered at L1. Continuing in this manner,

every hexagon in the tiling corresponds to a unique finite sequence, an address,

with entries integers from 0 to 6.








-2-

Most gridded representations used in image processing use rectangular pixels

corresponding to tiling the plane with squares. Reasons for interest in a hexagonal

grid in image processing are that natural scenes in low resolution images look more

"natural" when presented in a hexagonal rather than square grid, hexagons can be

grouped into aggregates and each pixel in a hexagonal grid has six equal neighbors,

thus avoiding the 4-neighbor/8-neighbor problem.

B. H. McCormick [21] in 1963 proposed the hexagonal array as a possible

gridded representation for planar images. M. J. E. Golay [9] in 1969 applied the

hexagonal array in a parallel computer and developed the hexagonal pattern trans-

formation. K. Preston [22] in 1971 developed "a special purpose computer system

which uses hexagonal pattern transformations to perform picture processing at high

speed." D. Lucas and L. Gibson [6,7,8,17,18] in 1982 exploited the geometric ad-

vantages of the hexagonal representation in their applications to automatic target

recognition. N. Ahuja [1] in 1983 investigated polygonal decomposition for such hier-

archical image representations as triangular, square, and hexagonal. D. K. Scholten

and S. G. Wilson [24] in 1983 showed that the hexagonal lattice outperforms the

usual square lattice as a basis for performing the chain code quantization of line

drawings. J. Serra [25] in 1988 discussed the properties of the hexagonal grid.

The ring GBT2 can be thought of as the set of all finite sequences with entries

from the set {0, 1,2, 3,4,5, 6}. One uses EGBT2, the 2-dimensional Extended Gen-

eralized Balanced Ternary, to denote the set of all infinite sequences with entries

from the set {0, 1,2,3,4,5, 6}. Motivation for this dissertation originated in a ques-

tion posed by D. Lucas to my cochairman G. X. Ritter. Lucas wondered if EGBT2

is isomorphic as a ring to the 7-adic integers. (It will be shown in Chapter 2 that

EGBT2 can be made into a ring.) The answer is that EGBT2 is isomorphic as a

ring to the 7-adic integers and this is shown in Chapter 2.








-3-
Lucas and Gibson [6] have defined an n-dimensional Generalized Balanced

Ternary (GBTn) ring for each n > 1 as the set of all finite sequences with en-

tries from the set {0,1,..., q 1}, where q = 2n+l 1. As an addressing method,

the GBTn addresses an (n + 1)-permutohedron tiling or packing of n-space. (As

mentioned by Lucas [6], the permutohedrons in 2-space are hexagons; in 3-space

they are truncated octahedrons.) The n-dimensional Extended Generalized Bal-

anced Ternary Numbers EGBTn are the set of all infinite sequences with entries

from the set {0,1,...,q 1}, where q = 2n+l 1. It is natural to ask for what

values of n, if any, other than 2 is the EGBTn isomorphic as a ring to the q-adic

integers. In fact Lucas wrote to the author posing this question [16]. In Chapter 3

it is shown that for certain values of n EGBTn is isomorphic as a ring to the q-adic

integers.

In Chapter 4 another proof is given of the main result in Chapter 3. The proof
is due to A. Vince and is algebraic in nature.

In Chapter 5, the algebraic properties of a special linear transformation which

takes the hexagonal lattice associated with GBT2 into itself are investigated.

In Chapter 6, the inversion and decomposition of the templates over the hexag-

onally sampled images are discussed.








-4-


Figure 1 : The First Level Aggregate.


Figure 2: The Second Level Aggregate.









-5-


Figure 3 : The Third Level Aggregate.








-6-


Figure 4: The GBT Address of First Level Aggregate.


Figure 5 : The GBT Address of Second Level Aggregate.









-7-


Figure 6 : The GBT product of 255 and 25 over the Third Level Aggregate.















CHAPTER 2


THE 7 adic INTEGERS AND THE RING EGBT2


In this Chapter we will prove that the 7-adic and 15-adic integers are isomorphic

to EGBT2 and EGBT3, respectively. These particular case studies will lead to a

more general result in Chapter 3.


2.1. Introduction

As stated in Chapter 1, one of the goals of this chapter is to prove the existence

of a ring isomorphism from the 7-adic integers onto the EGBT2, a ring defined by

an unusual remainders and carries tables.

Recall that the p-adic integers, Zp, can be thought of as the set of all series al

+ a2p + a3p2 ... + ... + + ..., 0 < ak < p where addition and multiplication are
performed with the usual "carries rules" for arithmetic modulo p. Since the integer

p is primarily a place holder, the elements of Zp can be thought of as the set of

all infinite sequences a = (ak) = (al,a2,..., ak,...), where 0 < ak < p. In this

setting, if a = (ak) and b = (bk) are elements of Zp, then the sum a + b = (sk),

where al + bl = c2p + sl and ak + bk + ck = ck+1p + sk. Similarly, the product

ab is defined by ab = (tk), where alb1 = d2p+t1 and albk + a2bk-1 + ... + akbl +

dk = dk+lp + tk. Here the variables ck and dk are the carries, and the variables sk
and tk are the remainders. (These familiar carry and remainder rules are presented

in Table 1.) Note that the carries are uniquely determined by the fact that each

sk and tk is between 0 and p. Recall that the basic rules of arithmetic imply that
-8-








-9-
these addition and multiplication operations satisfy the associative, commutative,

and distributive laws. Moreover, the sequence (0,0,...) is the additive zero element

and the sequence (1,0,0,...) is the multiplicative identity. If p is a prime, then an

element has a multiplicative inverse if and only if the first coordinate is nonzero.

We use Z7 to denote this description of the p-adic integers for the case when p = 7.

(The above discussion of the p-adic integers is essentially the same as the one given

on page 43 of Kaplansky [11].)

Let H denote the addressed hexagonal tiling of the plane described in Chapter

1. Guided by algebraic rules associated with H, L. Gibson and D. Lucas [6,7]

defined new rules for addition and multiplication of the elements in EGBT2, the

set of all infinite sequences a = (ak) = (al, a2, a3, ...) with integer entries 0 < ak < 6,

that makes EGBT2 into a ring. If a and b are elements in EGBT2, then define

a +b = (sk) by ak +bk +ck = ck+17+ sk, where the rules for determining the carries

ck are given in Table 2, and; define ab = (tk) by albk + ... + akbl +dk = dk+l17 + tk,
where the carries dk are given in Table 2. The rules for the remainders remain the

same as in Z7 for both addition and multiplication. It is straightforward to show
that under these operations EGBT2 is made into a ring. It is worth noting here

that an element (ak) has a multiplicative inverse if and only if a1 # 0. Thus, it will

be shown that EGBT2 with its very odd carry rules has the same ring structure as

Z7 with its very familiar carries rules.

It is well known that Z7 is isomorphic to the inverse limit of the system

{Z/(7k); k = 1,2,...}. The fundamental idea of the proof that EGBT2 is iso-
morphic to Z7 is to show that if Ik is the ideal in EGBT2 defined by Ik =

{(0,...) k+1,Xk+2,...): xi E Z/(7)}, then EGBT2 is isomorphic to the inverse
k
limit of the system {EGBT2/Ik; k = 1,2,... }. This is done in Section 2.3. The








10-
ring GBT2 is the subring of EGBT2 consisting of all the finite sequences (ak) of

EGBT2; that is, (ak) is an element of GBT2 if and only if ak 5 0 for a finite

number of integers k. If I' is the ideal of GBT2 consisting of all sequences whose

first k entries are zero, then, as is shown in Section 2.4, the inverse limit of the

system {GBT2/I4; k = 1,2,... } is also isomorphic to Z7.

Remember that GBT2 is an addressing method. Gibson and Lucas had good
geometric reasons to define their carry table for addition in GBT2 as they did.

Consider Figure 5 and recall the standard rules for the addition of two planar vectors.

If the hexagon with address 0 is centered at the origin of the plane, then 1 + 2 = 3,

3 + 6 = 2 and 5 + 6 = 4 are all "vectors" inside this first level aggregate. But 3 + 2,

for example, should equal 25 because of the way vectors add. As can be quickly

checked, Table 2 shows that 3 + 2 has a remainder of 5 and a carry of 2, as desired.

Consider Figure 6, a third level aggregate, and add 416 to 346. One adds 6 to 6 to

get 65, then carries the 6 to the next column and adds to get 1 + 4 + 6 = 4, then

carries the 0 to the next column to get 3 + 4 + 0 = 0. Thus, 416 + 346 = 45, which

is "vectorially" correct.

The rules of multiplication in GBT2 are best explained with an example. Mul-
tiply 255 by 25.

255 x 25:

255
x 25
344 (= 5x255)
433 (= 2 x 255)
604 (= the GBT sum)

Note that there is no carrying during the two modulo 7 multiplications; the

only carrying is done in the addition. Figure 6 illustrates this product. Consider

Figure 4, and let the origin of the complex plane C be at the center of the hexagon








11 -
with address 0. Let the positive real axis pass through the center of the hexagon

with address 1, and let the positive imaginary axis pass over the boundary common

to the hexagons with addresses 4 and 5. Let C be coordinatized so that the centers
of hexagons with addresses 1,5,4,6,2,3 are at the 6th roots of unity 1,e2Ri/6, e4ri/6,
e67ri/6, e87i/6, e10i/6, respectively. Then the remainder table for multiplication

given in Table 2 is just complex multiplication of these roots of unity. There are

no carries since a product of two complex numbers each of modulus one is itself a

complex number of modulus one.



2.2. Inverse Limits and p-adic Integers


In this section, a number of definitions and propositions stating the elementary
facts concerning inverse limits are presented. These facts will be needed in our proof
that Z7 is isomorphic to EGBT2.

DEFINITION 2.2.1. If Ai(i E I) is a system of groups, indexed by a directed set I,
and for each pair i,j E I with i < j there is given a homomorphism 4 : Aj -

Ai(i < j) such that

(1) 0- is the identity map of Ai, for each i E I, and

(2) for all i j < k, we have 0 o = ,
then the system A = {Ai(i E I); ;} is called an inverse system. The inverse
limit of this system, A* = lim(Ai; O), is defined to consist of all vectors a =
(..., a, ...) in the direct product A = IEIJ Ai for which 'aj = ai(i < j) holds.

It is a routine exercise to show that A* is a subgroup of A. A similar definition
can be given for the inverse limit of rings, where it will be the case that A* is a
subring of A.








12-
If Z* is defined to be the inverse limit of the inverse system {Z/(pk)(k E I); p)},
where 1kal = ak is defined as ak = al mod(pk),at E Z/(pt),ak E Z/(pk), then
the ring Z* is isomorphic to Zp. In particular, Z* is isomorphic to Z7. The next
Proposition makes this last statement more precise.

PROPOSITION 2.2.2. If Z = limZ/(pk), then for each a = (al,a2,...,an,...) E Z
we can associate a with a uniquely determined p-adic integer Sl +s2p+...+snpn- +
..., where for all positive integers n, 0 < Sn < p and a1 = sl,a2 = s8 + s2p,...,
an = s1 + s2p + ... + SnP-l,..

PROOF: Let (pk) be the principal ideal of multiples of pk for k = 1,2... and any
p in Z. Consider the sequence of the rings Z/(p), Z/(p2),..., Z/(pn),.... Define the
map 1: Z/(p1) -+ Z/(pk) by the congruence relation: lka1 = ak means ak = al
mod pk. We have #i is the identity map and bJt = O if i < j 5 k. Therefore,
we have the inverse system {Z/(pk)(k E I); 01} and we call its inverse limit the
ring of p-adic integers. An element of Zp is a sequence of residue classes (or costs)
(al + (p), a2 + (p2), a3 + (p3),...) where the ai's are integers and for I > k, ak = at
mod pk. We can represent this element by the sequence of integers (al, a2,...),
where ak = at mod pk for k < I. Two such sequences (al,a2,...) and (bl, b2,...)
represent the same element if and only if ak = bk mod pk, k = 1, 2,.... Addition
and multiplication of such sequences is component-wise. If a E Z, we can write
a = s1 + s2p + ... + Snpn-1 where 0 < si < p. We can replace the representative
(al1,a2,...) in which ak = al mod pk if k < 1 by a representative of the form
(Si, 81 +2p, sl +s2P+s3p2, ...) where 0 < si < p. In this way we can associate with
any element of Zp a uniquely determined p-adic number sl + s2p + s3p2 +..., where
0 < si < p. Addition and multiplication of these series corresponding to these








-13-
compositions in Zp are obtained by applying these compositions on the si and "car-
rying". [10] E

PROPOSITION 2.2.3. If Cn =< Cn > is the cyclic group of order pn generated by
Cn, and n+1 : Cn+l -=- Cn acts as n+n+ = Cn, then {Cn(n = 1,2,...); nm} is
an inverse system such that C* = lim Cn is isomorphic to Zp as an Abelian group.

PROOF: If On denotes the canonical map C* -+ Cn, and if we define on : Zp F-+ Cn
by on(l) = Cn (1 E Zp), then there is a unique a : Zp F- C* such that Ono = an.
Since no none zero element of Zp belongs to every Keran, Kerr = 0. If c =
(c,...,I n,...) E C* with cn = kncn (kn E Z), then by the choice of n+1 we have

kn+i = kn mod pn, and there is a p-adic integer r such that r = kn mod pn for
every n. Thus a(r) = cn, and a must be epic [4]. O

COROLLARY 2.2.4. If Cn is a ring with multiplicative identity In, which has the
property that Cn is generated(under addition) by In and has order pn (n=,2,...),
and 0n+1 : Cn+l --+ Cn is defined by Ofn+1n+1 = In (i.e. generator goes to
generator), then On+1 is a ring homomorphism and {Cn(n = 1,2,...); nm} is an
inverse system of rings such that C* = limCn is isomorphic to Zp.

The Corollary 2.2.4 follows immediately from Proposition 2.2.3 since the mul-
tiplicative structure is essentially additive.
We can also think of Zp as the completion of Z with respect to the absolute
value JI p [2,13,26]. Here,

InIp = p-ordp(n)

where ordp(n) is the highest exponent to which p divides n. The idea is that two
integers are close if their difference is 0 modulo a high power of p. The completion
contains the subring of Q known as the p-integral numbers (rational numbers whose
denominators are not divisible by p).








14-
If m is a positive integer, m has a finite base p expansion


m = mi + m2p + m3p2 + m4p3 + ... + mrpr-1


where the mix's are integers between 0 and p-1. This expansion (the p-adic expansion

for m) will be denoted by its digits, and we will write


m = mlm2m3...mr.


It's easy to see that ordp(n) is the smallest integer k such that mk > 0. It follows

that two integers are close if their p-adic expansions agree for many places. In

particular, the sequence

1,11,111,1111,11111,...

is Cauchy, and its limit in Zp can be calculated from the usual formula for the limit

of a convergent geometric series


1111111 ... = 1 + p2 +p3 +p4 + 1
p-l

(Notice that the common ratio in this series is p, and Iplp = 1.) Since every p-

adic integer is the limit of some Cauchy sequence of integers, and since the p-adic

expansions for these integers agree for arbitrarily long initial strings, we can think

of a p-adic integer as an infinite p-adic expansion, denoted by an infinite string of

digits

8182s3s845 *. 81 + 82p + 83p2 + s4p3 + p4 + ....

where each si is an integer between 0 and p 1. As with all completions, I [p

extends to a valuation on Zp, and its value can be calculated by the same formula








15 -
that defines it on Z (ordp(a) is the smallest integer k such that ak > 0). So, just

as in Z, two p-adic integers are close if their representations agree for many digits

(that is, if their difference is 0 modulo a high power of p). An element of Zp is a

non-negative integer if and only if its digits are eventually 0; an element of Zp is in

Q precisely when its digits eventually repeat.

Given a p-adic integer a, a fundamental system of neighborhoods for a is the

sequence

{a +pnZp: n = 0,1,2,3,...}.

Indeed, given an integer n, Zp splits up into p" disjoint disks of diameter ,p namely

the costs in Zp/pnZp. Two p-adic integers x and y are within r of each other if

and only if they belong to the same disk.

The metric d defined by I| p satisfies a stronger condition than the triangle

inequality; if a, b and c are in Zp, then


d(a, b) < max{d(a, c), d(b, c)}


(equality holds if d(a, c) and d(b, c) are unequal). This non-archimedean property

of d implies that every triangle is isosceles and that every point interior to a circle

is its center.


2.3. The 2-Dimensional Extended Generalized Balanced Ternary Numbers


PROPOSITION 2.3.1. The subring Ik = {(0,...,0, xk+1,k+2,...) : i E Z/(7) for
k
all i > k +1} is an ideal in EGBT2 for all k = 1,2,....

PROOF: If y = (Y1,y2,...,yn,...) E EGBT2 and x = (0,...0,k+l, xk+2,...) E Ik,
k
then by the rules of multiplication and the carries rules given in Table 2 xy =








-16-

(0 ,.,Oxk+lYl, Xk+2Y1 + k+lY2 +ck+2, ...). Therefore, xy E Ik, yx E Ik and Ik is
k
an ideal in EGBT2. M

PROPOSITION 2.3.2. For each positive integer k the cardinality of EGBT2/Ik is

7k.

PROOF: The set EGBT2/Ik = {(al,a2,..., ak, ,...) + Ik : ai E Z/(7) for all i =

1,2,...}. Since there are 7 choices for each ai in each of the first k components, there
are 7k choices for (al, a2,..., ak, 0,...) + Ik. Therefore, the cardinality of EGBT2/Ik

is 7k. D

Note: In the following lemmas and propositions an arbitrary value will be

denoted by the symbol *. It may be the case that will represent one value on one

side of an equation or expression and another on the other side.

LEMMA 2.3.3. In the ring EGBT2, the following relations hold.


1. (1,*) + ... + (1,) =
1



2. (2,*) + ... + (2,*) =
l


3. (3,*) +. + (3,,) =
4. (4, (



4. (4, *) + ... + (4, *) =
1 l


(0, 5, ),

(X1,*),



(0, 3,),

(X1,*),



(0,1,*),

(x1, *),



(0, 6, *),

(Xl,*),


ifl=7

if I < 7, where xi is a nonzero element in Z/(7)



if =7

if I < 7, where x1 is a nonzero element in Z/(7)



if I =7

if I < 7, where xl is a nonzero element in Z/(7)



if I =7

if I < 7, where xl is a nonzero element in Z/(7)








- 17-


5. ,)+ ...+(5,= (0,4,), if I =7

S(xl,*), if I < 7, where xi is a nonzero element in Z/(7)




f (0,2,*), ifl=7
6. (6,) +... + (6,) =
S(X1,*), if I <7, where x1 is a nonzero element in Z/(7)


PROOF: It is a routine (but lengthy) computation to verify these identities. We

found a simple Fortran program to be a convenient tool to check that these calcu-

lations are correct. O

LEMMA 2.3.4. In the ring EGBT2, if I = 7n, then


S(0, 0, 5,*),
n
(0, ., 0,4, *),
n
(00, 6, ),
n
(0,..., 0, 2,*),
n
(0, 3, ),

(01, 1, *),


if n = 1 mod(6)

if n = 2 mod(6)

if n = 3 mod(6)

if n = 4 mod(6)

if n = 5 mod(6)

if n =0 mod(6)


PROOF:

Case 1. If n = 1 and I = 7, then by Lemma 2.3.3.1



(1, + ... + (1,) = (0, 5, )
7


(1,*)+...+(1,*)=
l








- 18-


Case 2. If n = 2 and 1 = 72, then by Case 1 and Lemma 2.3.3.5




(1, ) + ... + (1, ) = (1, ) + ... + (1, ) + (1, ) + ... + (1, ), +... + (1, ) + ... +(1, )
72 7 7 7


= (0, 5,*) + (0, 5, *) + ... + (0, 5,*) = (0, 0, 4, )
7
Case 3. If n = 3 and I = 73, then by Case 2 and Lemma 2.3.3.4




(1, *) + ... + (1, ) = (1 )+...(1, *) + (1, *) + ... + (1, *)+... + (1, *) + ... + (1, )
73 72 72 72


= (0, 0,4, *)+ (0,0,4,*)+... + (0, 0,4,) = (0, 0, 0,6,*)
7
Case 4. If n = 4 and I = 74, then by Case 3 and Lemma 2.3.3.6




(1, *) + + (1,*) = (1,*)+...(1, *)+(1,*) + ... + (1, +*) +... +(1, *) + ... + (1, *)
74 73 73 73

= (0, 0, 0, 6, ) + (0,0, 0, 6, *)+ ... + (0,0, 0, 6,) = (0,0,0,0,2,)
7








- 19 -


Case 5. If n = 5 and I = 75, then by Case 4 and Lemma 2.3.3.2




(1,*) +...+ (1,*) = (1,*) + ...(1,) + (1,) +...+ (1,+...+(1, *)+.. .+(1,*)
75 74 74 74

= (0,0,0,0,2, ) + (0,0,0,0,2, *) + ... + (0, 0,0, 0, 2,) = (0, 0,0, 0, 0, 3,*)
7
Case 6. If n = 6 and I = 76, then by Case 5 and Lemma 2.3.3.3




(1, ) + ... + (1, ) = (1,*) + ...(1,) (1,) + ...(1,) ...+(1, ) + ... +(1,)
76 (1)75 (2)75 (7)75

= (0,0, 0, 0, 0, 3,) +(0, 0, 0, 0, 0, 3, *)+... +(0, 0, 0,0, 0, 3,)
(1) (2) (7)
= (0,0,0,0,0,0, 1,*)

By inductively repeating the six steps in the process indicated above, we have

the conclusion of the Lemma. O

COROLLARY 2.3.5. If = 7n for some integer n, then (1, *) + ... + (1,*) = (0,.0,n+l, *),
I n
where Xn+1 is some nonzero element in Z/(7).

PROOF: From the six patterns in Lemma 2.3.4, we know that if I = 7n, then

(1, *) + ... + (1,*) = (0, ...,0,n+1, *), where Xn+1 is different from zero. 0
7" n
LEMMA 2.3.6. In the ring EGBT2, if I = 7", then (1,*) +... + (1, *) = ( Xn+1,*)
I n
where x,+1 is different from zero, and if l < 7n, then (1, *) + ... + (1, *) = (x1i,.., Xn, *),

where xi is different from zero for some integer i between 1 and n.


PROOF: The proof will be by induction on the integer n.








20 -

If n = 1, then by Lemma 2.3.3.1 (1,*) + ... + (1, *) (0,5, *) for I = 7 and
1
(1, ) + ... + (1, *) = (xl,*) for I < 7, where xl is different from zero. Therefore,
1
the inductive step is true for n = 1.

Assume that the inductive step is true for all integers less than or equal n (i.e.

I < 7n).

If I = 7n+1, then by Corollary 2.3.5



(1,*) + ... (1, *) = (0,1 .0n+2, *), where Xn+2 is different from zero.
I n+1


If I < 7n+1 (i.e. l = h7n + 1' where h < 7 and 1' < 7n), then by Corollary 2.3.5

we have



(1, ) + ...(1, ) = (1,*) + ... + (1,) + (1,*) + ... + (1, *
I h7" I'

= (1, *)+... + (1, *)+... + (1, *)+ ... (1, *)+(1, )+ ... (1, )
7" 7" 1'

= (0,...,(hn+l) mod 7,*) + (l,*)+... + (1,*).
n I,

By the induction assumption we have (1, *) + ... + (1, *) = (x1, x2,...n, *), where
I'
some xi is different from zero if and only if 1' is different from zero.

Thus, if 1' 5 0, then by induction xi 0 0 for some i < n. If 1' = 0, then by

induction Xn+1 7 0. Since h # 0, (hxn+l) mod 7 5 0. Therefore, the induction is

true for the integer n+1. 0

PROPOSITION 2.3.7. The ring EGBT2/Ik is generated (under addition) by the

element ik = (1,0,0,...)+ Ik E EGBT2/Ik.







- 21 -


PROOF: By Lemma 2.3.6, we have:

k + + 1k = (1, 0, 0, ...) + I + ... + (1, 0, 0,...)+ Ik


= (1, 0, 0,...)+... + (1, 0, 0, ...) +Ik

f (0,.,1rk+,*)4 Ik, if = 7k
k
S(Xl,...,k,*)+ Ik, if I < 7k, where xi # 0 for some i E {1,...k}
Thus, 1k +... + 1k = k if and only if I = 7k. Therefore, the order of Ik is 7k
1
By Proposition 2.3.2, 1k is a generator of the ring EGBT2/Ik under addition. O

PROPOSITION 2.3.8. If k1 : EGBT2/1 -*- EGBT2/Ik is defined by 1 =
ik, then {EGBT2/Ik(k = 1,2,...); q} is an inverse system. If EGBT2* =
lim(EGBT2/Ik; 1), then EGBT2* is isomorphic to Z7 and EGBT2* = { ( (x1,0,...)+

I1, (x1,2, 0,...) + 2, ...); zk E Z/(7) for k =1,2,... }.
PROOF: By Proposition 2.3.2 we know that EGBT2/Ik has order 7k for all positive
integers k. By Proposition 2.3.7 we know that EGBT2/Ik is generated (under ad-
dition) by 1k for all positive integers k. Thus, by Corollary 2.2.4, {EGBT2/Ik(k =
1, 2,...); 1 } is an inverse system, and EGBT2* = lim(EGBT2/Ik; /I) is isomor-
phic to Z7.
Let S denote the set { ( (x1,0,...)+ I, (x1,x2,0,...) +12, ... ); k E Z/(7)
for k = 1,2,... }. It is easy to see that S C EGBT2*. Let t = (G1,~ 2..., k, ...) E
EGBT2*, where 'k = (xai, 2, ...k, 0,...) + Ik E Z/(Ik) (k=1,2,...). By Proposition
3.7, 5k = mik where m E Z/(7k). Therefore, kzI = mik = m((1,0,...)+ Ik) =
(x1, 2,...,xk,*) + Ik = (l,x2,..., xk, 0,...) + Ik. This last equation shows that
Xk = (x1, 2,...,k, O,...)+ Ik, and t = ( (xi,0,...) + II, (xi,x2,0,...) + 12, .. ,
(X1, X2, ..., k, 0, ...) + Ik, ... ). Therefore, EGBT2* C S and EGBT2* = S. O








22 -
PROPOSITION 2.3.9. The ring EGBT2 is isomorphic to EGBT2*.

PROOF:
Define the function r7: EGBT2 -+ EGBT2* by l(xl, x2,...) = ((xi,0,...) +

II, (xl, X2, 0, ...) 2, ...) for all (l,x2,...) E EGBT2.
We will now show that 7 is an isomorphism.
(1) It is straightforward to show that 7 is well defined and surjective.
(2) The function 7r is injective.
Let x = (xl, X2, ...) and y = (Yi Y2, ...) be elements in EGBT2. If rq(x) = rl(y),
then ( (xl,0,...) + II, (xl, X2, 0, ...)+12, ... ) = ( (yl, O, ...)+Ii, (Yi, Y2, 0, ...)+12, ...).
This equation implies that (xl,0,...) + II = (i, 0,...) + II, (xl,2,0,...) + 12 =

(Y, Y2, ..) + 2, ... (xl, x2,...xn,0,...) + In = (Yl, Y2,...Yn,0,...) + In, ..., which
implies that (xi y1,0,...) E II, (xl yl,x2 2,0, ...) E 12, (x., ( Yl yl,2 -

Y2, ..., n yn, 0, ...) E In, ... for all integers n. Thus, we have zx = y1, x2 = Y2, **.,
xn = yn, ... for all integers n. Therefore, x = y and I7 is injective.
(3) The function r7 is a ring homomorphism.
If x = (xl, 2,...) and y = (Yl, Y2,...) are elements in EGBT2, then x + y =

(s, s2, ..-, sn, ...), where xl + yl = c27+si, ..., Xn + Yn + n = Cn+17 + n, ... for all
integers n, and xy = (tl,t2, ...tn, ...), where xlyl = c27 + tl, ..., Xlyn + X2Yn-l +
... + nyil + cn = cn+17 + tn for all integers n. Thus, 7(x + y) = ( (s1,0,...) + I, ...,

(sl, s2, ..., Sn, ...) + In, ... ), and 77(xy) = ( (tl, O,...) + I, ..., (tl, t2, ..., tn, O,...) + In,
... ). Since iq(x) + Y(y) = ( (xi,0,...) + II, ..., (xl,X2,...xn,0,...) + In, ... ) +

( (yi,0,...) + II, ..., (yi,y2, ...n, O,...) + In, ... ) = ( (xli,0,... ) + (yl,0,...) +
II, ..., (x1,X2,...,axn,0,...)+(yl, 2, ... n, 0,...) + In, ... ) = ( (a1,0,...) + II, ...
(sl,s2,...sn,.....) + In, ... ), we have y(x) + q(y) = 7r(x + y). Since q(x)7(y) =
( (x,0, ..) + I, .., (x, .. n, 0,...) + In, ... )( (yl, 0,...) + II, (1, ( --., Yn, 0,...) +
In, ... ) = ( (x l, 0, ..)(y1, 0,...)II, ..., ( l, ...,X n, 0,...) (yI, ..., Yn, 0,...) + In, ... ) =








23 -

( (t,, ...) + Ii, ..., (t, ..., n, 0,...) + In, ... ), we have r (x))(y) = rq(xy). Therefore,
7 is a ring homomorphism.
Thus by (1),(2) and (3), the map 17 is a ring isomorphism from EGBT2 to
EGBT2*. Ol

THEOREM 2.3.10. EGBT2 is isomorphic to Z7.

PROOF: By Proposition 2.3.9, EGBT2 EGBT2*. By Proposition 2.3.8, EGBT2* -

Z7. Therefore, EGBT2 Z7. O


2.4. The 2-dimensional Generalized Balanced Ternary Numbers

The 2-dimensional Generalized Balanced Ternary Numbers, denoted by GBT2,
are the subring of EGBT2 consisting of all the finite sequences of EGBT2. For
convenience of notion we will use G to denote GBT2.
Lucas proved that G/I' is isomorphic to Z/(7k), where I' is the ideal of G
consisting of all those sequences whose first k digits are zero. We will now show
that the inverse limit of G/I' is isomorphic to EGBT2.

PROPOSITION 2.4.1. For each positive integer k the cardinality of G/II is 7k.

PROOF: The proof is the same as the proof of Proposition 2.3.2. [

PROPOSITION 2.4.2. The ring G/II is generated (under addition) by the element

ik = (1,0,0,...) + IEk G/I'.

PROOF: Since the results of Lemmas 2.3.3, 2.3.4, 2.3.5 and 2.3.6 are true in the
subring G, we can follow the same proof given for Proposition 2.3.7. O

THEOREM 2.4.3. If 01: G/I -- G/II is defined by 0II = 1k, then {G/I'(k =
1,2,...); 1} is an inverse system. If G* = lim(G/I,; k1), then G* is isomorphic to
both Z7 and EGBT2.








24 -
PROOF: By Proposition 2.4.1, G/Ik has order 7k for any positive integer k. By
Proposition 2.4.2, G/Ik is generated by 1k for any positive integer k. Thus, by Corol-
lary 2.2.4, {G/I,(k = 1,2,...); k} is an inverse system, and G* = lim(G/I, 4) is
isomorphic to Z7. By Theorem 2.3.10, Z7 is isomorphic to EGBT2. Therefore, G*
is isomorphic to EGBT2. O



2.5. 15-adic Integers and the Ring EGBT3


Guided by algebraic rules motivated by a truncated octahedral tiling in 3-
dimensional space, L. Gibson and D. Lucas [6] defined rules for addition and mul-
tiplication for the 3-dimensional Generalized Balanced Ternary Numbers. (For a
diagram of the truncated octahedron see Figure (4,6,6) in Toth [27].) The carry
rules have been modified to those presented in Table 3 while the remainder rules
remain the same as the rules known for the ring of 15-adic integers. The extended
3-dimensional Generalized Balanced Ternary Numbers are the set of all sequences

{(al, a2, a3,...) : 0 < ak < 15} with the same addition and multiplication as defined
for the 3-dimensional Generalized Balanced Ternary Numbers. It can be shown that
the resulting structure forms a commutative ring with unity. We use EGBT3 to
denote this ring. The following lemmas and proposition can be proved in a way sim-
ilar to the proofs in section 2.3. Therefore, we conclude that EGBT3 is isomorphic
to the ring of 15-adic integers.

LEMMA 2.5.1. In the ring EGBT3, the following relations hold.



f (0,2,*), if 1=15
1. ) + < ... + (1, *) elementt in
S(xl,*), if I < 15, where xi is a nonzero element in Z/(15)








- 25 -


(0, 4)
2. (2,) + ... + (2,) (0 ),
I t (xl,*),



i (Xl,*),
3. (4, *)+...+ (4,*)= (0,,*),
-(a1,*),


if I = 15

if I < 15, where xl is a nonzero element in Z/(15)



if = 15

if I < 15, where x1 is a nonzero element in Z/(15)


{ (0,1,*), ifl=15
4. (8, *) + ...+ (8,)= f 15
I (xl,*), if I < 15, where xz is a nonzero element in Z/(15)

LEMMA 2.5.2. In the ring EGBT3, if I = 15n, then

(01 0,,2,*), ifn=l mod(4)
n
(0,... ,4,), ifn=2 mod (4)
(1,*) + ...+ (1,*)= n
S(0,...,0,8,*), ifn=3 mod (4)
n
(0 ,,,1,*), ifn=0 mod (4)
COROLLARY 2.5.3. If = 15n for some integer n, then (1, *) + ...+ (1, *) = (0, 1+, *),
1 n
where Xn+1 is some nonzero element in Z/(15).

LEMMA 2.5.4. In the ring EGBT3, if = 15n, then (1, *)+ ... + (1,*) = (0,-0n+l,*),
1 n
where Xn+1 is different from zero, and if I < 15n, then (1, *) + ... + (1, *) = (xl, ...,n, *),
I
where xi is different from zero for some integer i between 1 and n.

PROPOSITION 2.5.5. The ring EGBT3/Ik is generated (under addition) by the
element 1k = (1,0,0,...) + Ik E EGBT3/Ik.








-26-


Table 1. Digitwise Operations on Z7


Remainder
+ 1 2 3 4 5 6
1 2 3 4 5 6 0
2 3 4 5 6 0 1
3 4 5 6 0 1 2
4 5 6 0 1 2 3
5 6 0 1 2 3 4
6 0 1 2 3 4 5


Remainder
x 1 2 3 4 5 6
1- 1 2 3 4 5 6
2 2 4 6 1 3 5
3 3 6 2 5 1 4
4 4 1 5 2 6 3
5 5 3 1 6 4 2
6 6 5 4 3 2 1


Carry
+ 1 2 3 4 5 6
1 0 0 0 0 0 1
2 0 0 0 0 1 1
3 0 0 0 1 1 1
4 0 0 1 1 1 1
5 0 1 1 1 1 1
6 1 1 1 1 1 1


Carry
x 1 2 3 4 5 6
1 0 0 0 0 0 0
2 0 0 0 1 1 1
3 0 0 1 1 2 2
4 0 0 1 2 2 3
5 0 1 2 2 3 4
6 0 1 2 3 4 5


Table 2. Digitwise Operations on EGBT2

Remainder Carry


+ 1 2 3 4 5 6
T 2 3 4 5 6 0
2 3 4 5 6 0 1
3 4 5 6 0 1 2
4 5 6 0 1 2 3
5 6 0 1 2 3 4
6 0 1 2 3 4 5


Remainder
x 1 2 3 4 5 6
T 1 2 3 4 5 6
2 2 4 6 1 3 5
3 3 6 2 5 1 4
4 4 1 5 2 6 3
5 5 3 1 6 4 2
6 6 5 4 3 2 1


+ 1 2 3 4 5 6
1 1 0 3 0 1 0
2 0 2 2 0 0 6
3 3 2 3 0 0 0
4 0 0 0 4 5 4
5 1 0 0 5 5 0
6 0 6 0 4 0 6


Carry
x 1 2 3 4 5 6
i 0 0 0 0 0 0
2 0 0 0 0 0 0
3 0 0 0 0 0 0
4 0 0 0 0 0 0
5 0 0 0 0 0 0
6 0 0 0 0 0 0








-27-


Table 3. The carry tables for the 3-dimensional GBT.
Carry
+ 1 2 3 4 5 6 7 8 9 A B C D E
1 1 0 3 0 1 0 7 0 1 0 3 0 1 0
2 0 2 2 0 0 6 6 0 0 2 2 0 0 E
3 3 2 3 0 7 7 6 0 3 2 3 0 0 0
4 0 0 0 4 4 4 4 0 0 0 0 C D C
5 1 0 7 4 5 4 7 0 1 0 0 D 0
6 0 6 6 4 4 6 6 0 0 E 0 C 0 E
7 7 6 7 4 7 6 7 0 0 0 0 0 0 0
8 0 0 0 0 0 0 0 8 9 8 B 8 9 8
9 1 0 3 0 1 0 0 9 9 B B 9 9 0
A 0 2 2 0 0 E 0 8 B A B 8 0 E
B 3 2 3 0 0 0 0 B B B B 0 0 0
C 0 0 0 C D C 0 8 9 8 0 C D C
D 1 0 0 D D 0 0 9 9 0 0 D D 0
E 0 E 0 C 0 E 0 8 0 E 0 C 0 E
Carry
x 1 2 3 4 5 6 7 8 9 A B C D E
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
--T- o o0-oDo-o-o -0o-o-o-o--o-o-o0o
2 0 0 0 0 0 0 0 0 0 0 0 0 0 0
3 0 0 6 0 0 C 0 0 3 0 0 9 0 0
4 0 0 0 0 0 0 0 0 0 0 0 0 0 0
5 0 0 0 0 5 0 0 0 0 A 0 0 0 0
6 0 0 C 0 0 9 0 0 6 0 0 3 0 0
7 0 0 0 0 0 0 0 0 0 0 0 0 0 0
8 0 0 0 0 0 0 0 0 0 0 0 0 0 0
9 0 0 3 0 0 6 0 0 9 0 0 C 0 0
A 0 0 0 0 A 0 0 0 0 5 0 0 0 0
B 0 0 0 0 0 0 0 0 0 0 0 0 0 0
C 0 0 9 0 0 3 0 0 C 0 0 6 0 0
D 0 0 0 0 0 0 0 0 0 0 0 0 0 0
E 0 0 0 0 0 0 0 0 0 0 0 0 0 0















CHAPTER 3


THE p adic INTEGERS AND THE RING EGBTn



3.1. Introduction


Recall that for integer n > 2 the GBTn is the set of all finite sequences

(al, a2,..., ak), k = 1,2,..., with entries from the set of integers {0, 1,... ,2n+1 -
2}. The EGBTn is the set of all infinite sequences (al, a2,...) with entries from

{0, 1,...,2n+1 2}. Since the integer 2n+l 1 may not necessarily be a prime

number, from now on we will denote 2n+1 1 by q and will refer to the q-adic

integers.

Lucas has defined an addition and multiplication upon GBTn that makes it

into a commutative ring with unity [15]. These definitions will be presented in

Section 3.2 once enough notation has been developed to express these definitions in

a simple manner. Extending these operations in a natural way makes EGBTn also

into a commutative ring with unity [15].

As previously stated, the main result of this chapter is that EGBTn is isomor-

phic as a ring to the q-adic integers for certain values of n.


3.2. The Carry Tables of EGBTn


DEFINITION 3.2.1. Let Sn denote the set of all sequences of the form Sn...s such

that si equals 0 or 1 for all i = 0, 1,...n. The sequence of all ones is identified with

28 -








29 -
the sequence of all zeros. Let Bq be the function from the set Z/(q) to the set Sn
defined by the rule that if x = sn2n + ... + s12 + so, then Bq(x) = Sn...So.

It is clear that Bq is a bijective map and that the inverse of Bq, denoted by
By1, is the map from Sn to Z/(q) defined by By1(sn...so)=sn2n+...+s12+so.

DEFINITION 3.2.2. Let T be the function from Sn to Sn defined by T(sn...so) =

Sn-1_...-- n, where Sn...so is any element in Sn.

The composition of T with itself i times is denoted by T'. For any Sn...so E Sn,
Ti(sn...so) = si...SOsn...si+1. The inverse of T is denoted by T-1, and is defined by

T-1(sn...so) = SOSn...s1. The function T is a twist(or shift) to the left of the binary
sequences and the function T-1 is a twist to the right.

DEFINITION 3.2.3. Let E be the function from Sn xSn to Sn defined by E(rn...ro, Sn...so) =

tn...to, where ti = (ri + si) mod 2 for i = 0, 1, ..., n.

Note that E is the well known exclusive or function.

Since the associative law holds for the binary operation E in Sn, it is understood
that E(rn...ro, n..so, tn...to) = E(E(rn...ro, sn...so),tn...to).

In the n-dimensional algebraic structure GBT, the addition of any two digits
x and y E Z/(q) yields a remainder r defined as the residue of x + y modulo q. Note
that this is analogous to the usual rules for base 10 arithmetic. The carry C(x, y)

defined by D. Lucas [6,7] is the following.

DEFINITION 3.2.4. Let C denote the function from Z/(q) x Z/(q) to the set Z/(q)
defined by C(x,y) = Bl1(T-1(E(Bq(x),Bq(y),Bq(r)))), where r is the remainder
of x + y mod q.

Heuristically, the carry C(x, y) is defined by first converting x, y and r to
binary sequences, second, adding using the exclusive or, third twisting the resulting








30 -
sequence one unit to the right, and, finally, converting the resulting binary sequence
back to an element in Z/(q).
It is easy to see that the associative law holds for the carry function C. There-
fore, it is understood that C(x, y, z) = C(C(x, y), z).
One is now in the position to define the operations of addition and multiplica-
tion that make GBTn into a ring.
Let a = (al,..., ak) and b = (bl,..., bl) be elements of GBTn where, without
loss of generality, k < I. Define the "carries" cj in the following recursive manner:
cl = 0 and cj = C(aj_1,bj_1) + C((aj1 + bj-l) mod q, cj-_), for j = 2,...,1.
The sum a + b is now defined to be the finite sequence r = (rl,...,rm), where
ri = (al + b) mod q and rj = (aj + bj + cj) mod q for j > 2. (Note that the
author is assuming that aj = 0 for j = k + 1,..., 1.)
To define the multiplication requires a little more preparation. Let y and z be
elements from the set {0,1,..., q 1} with Bq(z) = Zn ... zo. Then,

Bq(y)Bq(z) = 2nBq(y)zn + + 2Bq(y)zl + Bq(y)zo =

= zn(2nBq(y)) + + zl(2Bq(y)) + zo(Bq(y)) =

= znTn(Bq(y)) + .. + zlTl(Bq(y)) + zoBq(y).

Set w0 = B-l(Bq(y)) = y and wi = B-1(Ti(Bq(y))) for i = 1,...,n. With this
notation in place, the following definition is made.

DEFINITION 3.2.5. Let D denote the function from Z/(q) x Z/(q) into Z/(q) defined

by

D(y, z) = C(znWn, Zn-lWn-) + C((znwn + zn-lWn-1) mod q, zn-2Wn-2)+
+ C((znwn + + zlWl) mod q, zowo).

As above, let a = (al,...,ak) and b = (bl,...,bl),k < 1, be from GBTn.
Define the "carries" dj in the following recursive manner: dl = 0 and dj =








-31 -
D(ai,bj_-) + D(a2,bj-2)+ ** + D(aj-l,bl) + C(albj-l,a2bj-2)+C((albj-l +

a2bj-2) mod q, a3bj-3)+ +C((albj-l+. a+aj-2b2) mod q, aj-lb)+ C((albj-l+
S.* +aj-lbl) mod q, dj-1), for j = 2,..., m. The product ab is now defined as the fi-

nite sequence s = (sl,..., sm), where sl = (albl) mod q and sj = (albj +a2bj-1
+ ajba +dj) mod q for j > 2. Here one assumes aj = 0 for j = k + 1,..., m,

and, bj = 0 for j = I +1,..., m. See Section 3.4 for worked examples of an addition
and a multiplication.
With this addition and multiplication, the GBTn is made into a commutative
ring with unity [15]. (The multiplicative identity is the sequence a = (1).) These
operations can be extended in the most natural way to make the EGBTn into a
ring. That is, if a = (al, a2,...) and b = (bl, b2,...) are from EGBTn, then the
entries for the sum a + b = (rl, r2,...) and the product ab = (1s, s2,...) are given
by the same rules as in the GBTn. The EGBTn is also a commutative ring with
unity a = (1, 0,0,...) [15].
Let x and y be be any two elements in Z/(q) and denote Bq(x) by xn...x0
and Bq(y) by Yn...yo. Let C(x,y) be the carry of x + y and denote Bq(C(x,y)) by

Cn(x, y)...Co(x, y).

LEMMA 3.2.6. If there is a carry of 1 from the ith position of the binary sum of

Xn...xo and Yn...YO, then Ci(x,y) = 1; if there is a carry of 0, then Ci(x,y) = 0.

PROOF: If we add xn...x0 and yn...yo, then for (i + l)th position we have ri+ =

(xi+l + Yi+1 + Ci) mod 2, where ci is the carry from the ith position.(Notice that
since we use modulo p = 2n+l 1, the carry from the nth position will go to the 0th
position.) Let rn...ro denote Bq(r) and let zn...zo denote E(Bq(x),Bq(y), Bq(r)). If
ci = 0, we have the following four cases:








32 -
case 1 case 2 case 3 case 4

i+1 = 0 0 1 1

Yi+l 0 1 0 1
ri+ = 0 1 1 0

i+1 = 0 0 0 0.
If ci = 1, we have the following four cases:

case 1 case 2 case 3 case 4

xi+1 = 0 0 1 1
Yi+ = 0 1 0 1
ri+1 = 1 0 0 1
zi+ = 1 1 1 1.
From all the possible cases we conclude that the exclusive or zn...zi+l...zo
has a 1 in the (i + 1)th position. (i.e. zi+l = 1, if and only if ci = 1.) Since

Cn(x,y)...Co(x,y) = T-1(zn...zo), we have Cn(x,y) = zo, Cn-l(x,y) = n, ...,

Ci(x,y) = zi+1, ..., Co(x,y) = zl. Therefore, Ci(x,y) = 1 if and only if ci = 1.1

LEiMh. 3.2.7. Let x and y be any two elements in Z/(q). Let Un...uo denote

Ti(Bq(x)) and vn...vo denote Ti(Bq(y)). Let u denote B1l(un...uo) and v denote
Bq1(vn...vo). If Cn(x,y)...Co(x,y) denotes Bq(C(x,y)) and Cn(u,v)...Co(u,v) de-
notes Bq(C(u,v)), then Ci(x,y) = Cn(u,v).

PROOF: Since the carry Cn(x, y)...Co(x,y) of Bq(x) + Bq(y) is circular, the carry

Cn(u, v)...Co(u, v) of Bq(u)+ Bq(v) is T'(Cn(x, y)...Co(x, y)). Therefore, Cn(u, v) =

Ci(x, y).D

LEMMA 3.2.8. If x is a fixed element in Z/(q) and Bq(C(x,y)) is denoted by

Cn(x, y)...Co(x,y) for each element y in Z/(q), then there are x- 1 digits y e Z/(q)
such that Cn(x,y) = 1.








33 -
PROOF: Let x be a fixed element in Z/(q). If y E Z/(q) and y > q x, then
x + y > q. Thus, Bq(x) + Bq(y) has a carry of 1 from the nth position. By Lemma
3.2.6, Cn(x,y) = 1. Since y is less than or equal q 1, there are x 1 choices for y
such that Cn(x, y) = 1.0

LEMMA 3.2.9. If x is an element in Z/(q) and Bq(C(x, y)) is denoted by Cn(x, y)...Co(x, y)
for each y in Z/(q), then there are u 1 digits y E Z/(q) such that Ci(x,y) = 1,
where u = B-1(Ti(Bq())).

PROOF: Let un...uO equal Ti(Bq(x)) and vn-...v equal Ti(Bq(y)). Let u denote
B-l(un...ug) and v denote Bl1 (vn...vo). Let Cn(x,y)...Co(x,y) denote Bq(C(x,y))
and Cn(u,v)...Co(u,v) denote Bq(C(u,v)). By Lemma 3.2.7, Ci(x,y) = Cn(u,v).
By Lemma 3.2.8, there are u- 1 digits v E Z/(q) such that Cn(u, v) = 1. Therefore,
there are u -1 digits y E Z/(q) such that Ci(x,y) = 1.l

PROPOSITION 3.2.10. If x E Z/(q), then C(x,O) + C(x, 1) + ... + C(x,q 1) =
[(n + 1)x2n] mod q.

PROOF: Let tX = x, x1 = Bql(T(Bq(x))), x2 = Bq-(T2(Bq())), ..., tn =
Bql(Tn(Bq(x))). Notice that t1 = 2x mod q, E2 = 22x mod q, ..., Xn = 2nx
mod q. Consider the binary sequences Bq(C(x, 0)), Bq(C(x, 1)),..., Bq(C(x, q 1)).
By Lemma 3.2.9, there are to 1 digits j E Z/(q) such that Cn(x,j) = 1. There
are a1 1 digits j E Z/(q) such that Cn-l(x,j) = 1, ..., and there are Xn 1 digits
j E Z/(q) such that Co(x,j) = 1, where j = 0, ...,q 1. Therefore, the sum of the
carries C(x, 0), C(x, 1), ..., C(x,q 1) denoted by [~Y= C(x, y)] mod q can be
calculated by the following sequence of qualities.








- 34 -


q-1
[ C(x,)] modq [( 1)2 mod=1)2] modq +... + [(n -1)] mod q} mod q
y=l


= [(xo 1)2n + (li 1)2n-1 + ... + (Xn-1 1)2 + (xn 1)] mod q

= [(x 1)2n + (2x 1)2n-1 + ... + (2n-lx 1)2 + (2n 1)] mod q

= [(x2 +... + x2n) (2n + 2n-1 +... + 2 + 1)] mod q
n+l
= [(n + l)x2" (2n+l 1)] mod q

= [(n + 1)x2n] mod q.0

COROLLARY 3.2.11. If x = 1, then [E-i1 C(x,y)] mod q = [(n + 1)2n] mod q.

PROPOSITION 3.2.12. If x is an element in Z/(q) relatively prime to q, let x,..., )
q
denote the carry of the result of adding the element x to itself p times, then the carry
...( ) equals [x(n + 1)2n] mod q.
q
PROOF: Since F(( )=C(x, x) + C(2x mod q, x) + ... + C((q 1)x mod q, z),
q
and (mx) mod q 5 (lx) mod q if m 5 1, where 0 < 1,m < q, .(, ...,x) =
q
[C(x, 1)+C(x,2)+...+C(x,q-1)] mod q = [--i C(x, y)] mod q. By Proposition
3.2.10, (, = [x(n + 1)2"] mod q.0
q
COROLLARY 3.2.13. If we add the digit one p times, then the carry F(Q., 1 equals

[(n )2] mod q.
[(n + 1)2n] mod q.








35 -
3.3. The ring EGBTn and q-adic integers

The ring EGBTn is the set {(al, a2, a3, ...) : 0 < ak < q} with addition and
multiplication as defined in Section 3.2 [15]. It can be shown that the subset Ik of
EGBTn defined by Ik={(0,..., ,k+l, Xk+2,...) : xi EZ/(q)} is an ideal of EGBTn.
k
PROPOSITION 3.3.1. For each positive integer k the cardinality of EGBTn/Ik is

k
q.

PROOF: Let q = 2n+1 1. The set EGBTn = {(al,a2,...,ak,0...) +Ik : ai E

Z/(q) for all i = 1,2,...}. Since there are q choices for each ai in each of the first
k components, there are qk choices for (al, a2, ..., k, 0, ...) + Ik. Therefore, the
cardinality of EGBTn/Ik is qk.[

Note: In the following lemmas and propositions an arbitrary value will be
denoted by the symbol *. It may be the case that will represent one value on one
side of an equation or expression and another on the other side.

LEMMA 3.3.2. Let x be an element in Z/(q) which is relatively prime to q. If n+1
and q are relatively prime, then the following relation holds in the ring EGBTn.

(0, x2, *), if I = q, where x2 = [x(n + 1)2n] mod q

(x, )+ ...+ (x,*) = and gcd(x2, q) = 1;

(x1, *), if I < p, where x1 is a nonzero element in Z/(q).


In particular,


S(0, x2, *), if I = q,where X2 = [(n + 1)2n] mod q
(1, 1 (1, ), if I
PROOF: If I = q, then (x, *)+...+(x, *) = ((qx) mod q, .F( ,)) = (0, (x, ..., x), *).
q q
By Lemma 3.2.12, F(x, = [x(n+1)2n] mod q. Since x and n+1 are both rela-
q








36 -
tively prime to q, x(n + 1)2n is relatively prime to q. Therefore, (x, *) + ... + (x, *) =
q
(0, x2, *), where x2 = [x(n + 1)2n] mod q and x2 is relatively prime to q. If I < q,
then (x, *) + ... + (x, *) = ((lx) mod q, *). Since x and q are relative prime and
1
1 < q, (lx) mod q 5 0. If xl denotes (lx) mod q, then we have (x, *) + ... + (x, *) =

(xl, *), where x1 $ 0.0I

LEMMA 3.3.3. If n + 1 and q are relatively prime, and if I = qk for some positive

integer k, then (1,*) + ... + (1, *) = (0,..., 0, Xk+1, *), where Xk+1 is relatively prime
I k
to q and xk+1 = [(n + 1)2n]k mod q.

PROOF: The proof will be by induction on the integer k.

If k = 1, (i.e. I = q), then by Lemma 3.3.2, (1, *) + ... + (1, *)=(0, x2, *), where
1=q
X2 = [(n + 1)2n] mod q. By assumption, X2 is relatively prime to q. Therefore, the
inductive step is true for k = 1.

Assume that the inductive step is true for all integers less than or equal k.

By the inductive assumption, we have (1, ) +... + (1, *)=(0 x0,k+i,*),
qk k
where xk+1 is relatively prime to q and xk+1 = [(n + 1)2n]k mod q.

If = qk+l, then by Lemma 3.3.2



(1, *)+... + (1,*)=(1,)+...+(1,)+... +(1,)+... +(1,)
qk+1 qk qk








- 37 -


S(0 0,Xk+i, *)+ ... + (0,...,0,k+1, *)
k k
q
= (0, k+2,*),
k+1
where sk+2=[zk+l(n + 1)2n] mod q.

Since the integers xk+1 and (n + 1)2n are both relatively prime to q, xk+2 is

relatively prime to q, and xk+2 = {[(n + 1)2n]k(n + 1)2n} mod q = [(n + 1)2n]k+1

mod q. Therefore, the induction is true for the integer k + 1.0

LEMMA 3.3.4. If n + 1 and q are relatively prime, and if I = qk, then

(1, *) + ... + (1, *) = (0,...,0,k+, *), where xk+1 is relative prime to q. If I < qk,
I k
then (1, *) + ... + (1,) = (l, ..., xk, *), where xi is a nonzero element in Z/(q) for
N-
I
some integer i between 1 and k.

PROOF: The proof will be by induction on integer k.

If k = 1, then by Lemma 3.3.2 (1, *) + ... + (1, *) =(0, 2,*) for 1 = q, where
I
x2 is relatively prime to q. Also, if I < q, then (1, *) + ... + (1, *)=(xl, *), where xl
1
is a nonzero element in Z/(q).

Assume that the inductive step is true for all integers less than or equal k (i.e.

S< qk).

If I = qk+, then by Lemma 3.3.3



(1, ) + ... + (1, *) = (0, .., k+2, *), where xk+2 is relatively prime to q.
I k+1








38 -

If I < qk+l (i.e. 1 = hqk + I' where h < q and I' < qk), then by Lemma 3.3.2
we have


(1, + .. + (1, ) = (1,)... + ... + (1,)
I hqk I'

= (1, ) + ... + (1, *) +... +,(1, ) + ... + (1, )(1,*) + ... + (1,)


h
= ( ...0,+*)+ ... + (0, ...,0, k+,*) + (1, +)+... + (1,*)
k k I'
h
= (0 ,(hzk+i) modq,) + (1,*)+...+(1,*).
k
By the induction assumption we have (1, *) + ...+ (1, ) = (xl, 2,... k, *),
I'
where some xi is a nonzero element in Z/(q) if and only if 1' is different from zero.
Thus, if 1' 5 0, then, by induction, there is an integer i < k such that xi is a
nonzero element in Z/(q). If I' = 0, then by induction, xk+1 is relatively prime to
q. Since h < q, (hxk+l) mod q 5 0. Therefore, the induction is true for the integer
k + 1.

PROPOSITION 3.3.5. If n +1 and q are relatively prime, then the ring EGBTn/Ik
is generated (under addition) by the element 1k = (1,0,0,...) +Ik E EGBTn/Ik.

PROOF: By Lemma 3.3.4, we have:

1k ...= (1, 0, 0, ...)+ I + + (1,0, 0,...) + Ik
1 1



S(0...0, k+1,*) Ik, ifl=qk
S
I (Xi,...,xk7,*) + Ik, if l < qk, where xi E Z/(q),xia; 0 for someiE{1,..k}.








39 -
Thus, 1k +... + ik=Ik if and only if I = pk. Therefore, the order of Tk is qk.

By Proposition 3.3.1, 1k is the generator of EGBTn/Ik under addition.J

THEOREM 3.3.6. If n + 1 and q are relatively prime, where q = 2n+l 1, then the
ring EGBTn is isomorphic to the ring of q-adic integers.

PROOF: By Proposition 3.3.1 we know that EGBTn/Ik has order qk for all pos-
itive integers k. By Proposition 3.3.5, if n + 1 and q are relatively prime, then
EGBTn/Ik is generated (under addition) by Tk for all positive integers k. Thus, by
Corollary 2.2.4 {EGBTn/Ik(k = 1,2,...); 01k} is an inverse system, and EGBT* =
lim(EGBTn/Ik; 1) is isomorphic to Zq.It is easy to show that EGBTn = {((x, 0,...)+

II, (1, x2,0,...) + 12, ...); k E Z/(q) for k = 1,2,...}.
Now if we define the function 7 : EGBTn t- EGBT* by 7(1, x2, ...) =

((i, 0,...) +I1,(, (x,2,0,...) +12,...) for all (x1, x2,...) E EGBTn. It can be proved
in a way similar to the proof in Proposition 2.3.9 that 97 is an isomorphism from
EGBTn to EGBT*. Since EGBT* is isomorphic to Zq, we have that EGBTn is
isomorphic to Zq.0

COROLLARY 3.3.7. If n + 1 is prime, then the ring EGBTn is isomorphic to the
ring of q-adic integers, where q = 2n+l 1.

For the case when n + 1 and q = 2n+l 1 are not relative prime, this proof
can not be used to establish the existence of the isomorphism.



3.4. Examples

In this section some examples are given which illustrates that the addition and
multiplication operations between the addresses can be carried out using simple
operations on bit strings.








- 40 -


EXAMPLE 3.4.1.

In 3-dimensional space, q = 15, the sum of the addresses a = (6) = (0110)
and b = (12) = (110) has remainder r. Here, r1 = (6 + 12) mod 15 = 3 = 0011.

The carry c2 = C(6,12) = B -(T-1(E(0110, 1100,0011))) = B 1(T-1(1001)) =
B (1100) = 12. Thus, r2 = 0 + 0 + 12 = 12. Therefore, the sum of the addresses
a and b is r = (r, r2) = (3,12).

EXAMPLE 3.4.2.

The product of the addresses a = (6) = (0110) and b = (3) = (0011) in

3-dimensional space has remainder s. Here, sl = (6 x 3) mod 15 = 3 = 0011.

Since 3 = 2 + 1, D(6,3) = C((6 x 2) mod 15,6) = C(12,6). From Example 3.4.1,
C(12,6) = 12. The carry d2 = D(6,3) = C(12,6) = 12 = 1100. Thus, s2 =
6 x 0+0 x 3+12 = 12. Therefore, the product of the addresses a and b is s = (3,12).















CHAPTER 4
ANOTHER APPROACH TO EGBTn AND THE q adic INTEGERS


4.1. Introduction

The material presented in this section is a modification of the material described
by A. Vince [12, Section 3].
Let a be an arbitrary element of a ring R and consider the inverse system

fi f 2 A-f
(1) R/aR R/ac2R -- ... A- R/akR ...,

where the ring homomorphisms fk are defined so that fk(f3) is equal to the equiv-
alence class of p (mod ak). The inverse limit Ra of this system consists of all
sequences {0/3 1,... } such that fk(0k) = ~k-1. The definition and notation is
analogous to that of the p-adic integers Zp. Addition and multiplication in Ra are
defined in the usual manner for inverse systems. If {o0/1,*... } is an element in
Ra and S is a set of coset representatives for R/aR, then it follows from the defi-

nition of the homomorphisms fk that there exists a unique sequence (so, si,2 s...)

of elements of S such that


/P0 so (mod a)

31 so + s$ a (mod a2)



fk sO + S10 + + + skak (mod ak+1)


- 41








42-
The element 80 + sla + s2a2 +... in Ra will be abbreviated (ssls2 ...), where a
is understood.

Let n be a positive integer and consider the special case where R is the quotient
ring



R = Z[x]/(f),

with f(x) = xn + an-.n-1 +. + ali + ao. Let w = x + (f), the coset containing
x. Note that f(w) = 0. As a free abelian group, R has basis {1,w,... ,wn-1}. Vince
[28] discussed the ring structure of R and indicated that R can be realized as a
lattice in Rn by embedding the n basis elements as n linearly independent vectors
in R". In Section 4.2, we will discuss the algebraic structure of Ra, where a = 7- w
and I is a non-zero integer.

When one chooses f(x) = xn+ x"n-1 + + x + 1 and a = 2 w, R is
isomorphic to EGBTn as defined in Chapter 3. The isomorphism 0 is defined as
follows. Vince has shown that the set S = {o + eW +- + enwn : Ei {0, 1}, not all
ei = 1} is a set of coset representatives for R/aR [12]. Thus, any element s E Ra
can be expressed as s = so + sla + s2a2 + ..., where si = l o wi E S, i =
0,1,... and e E {0, 1}. Recall from Chapter 3 that the elements a E EGBTn are
infinite sequences (aO, al, a2,...), where ai is an integer from the set {0,..., q 1},
q = 2n+1 1. Define a map 0 from Ra to EGBTn by O(s) = (aO, al, a2,...), where
ai = E =o e=2'. Clearly 0 is a bijection from Ra onto EGBTn.

That ((s + t) = O(s) + 0(t), where s,t E Ra, can be seen as follows. Let
si = 6j=o w and ti = j=0 r w be two elements from S. Use the fact that
2 = w + a and wn+l = 1 mod (wn + n-1 + + w + 1) to express si + ti as
vi + wia. This is done formally as follows:








- 43 -


Define u = -n+l 6~wj by the following rules


60 = (e + 77I) mod 2,

6b = (,e + r+ + 7j) mod2,j=1,...,n + 1,

where o = 0, j. = 1 if and only of e -_ + '-1 j-1 2 and 'n+1 = 0 = +
For example, si = 1 + w added to ti = w + w2 yields ui = 1 + w3. Now define

n
Vi= Epwi,
j=0
ui = (7_ + 7^_ + 1iw2 + iwn-1 +- 7~wn)

by the following rules.


3 = ( + n + 7r ) mod 2,

S= (E, + j + +7) mod 2,j=,...,n,

where -y = n+1 and 7j = 1 if and only if cj_- + j-1_ + 7+-1 > 2. If s =
so + slc + + sia' +... and t = to + tla + -. + tioi + ... are elements of
Ra, then ai = o =0 ~2 and bi = j=0 r4j2i are the entries in the sequences O(s)
and 0(t). It can be checked that the definition in Chapter 3 used to define the sum
a, + bi corresponds exactly to the definition for si + ti = vi + wia. In other words,
the remainder and carry rules are preserved under the map 0. This implies that
O(s + t) = O(s) + 0(t). To convince oneself that q(st) = ((s)9(t) takes a little more
work and is left to the reader.








44 -
4.2. The Structure of Ra


The material presented in this section is a generalization of the results presented
by Vince [12,Section 4].

The q-adic integers Zq are defined as the inverse limit of the inverse system


qZ91 Z/q2Z 92 gk-1 k
Z/qZ .- Z/qZ -... +----- Z/qkZ -- ...,


where the homomorphisms gk take an integer j (mod qk+l) to the integer j (mod
qk). If f(x) is any monic polynomial, we have the following lemmas which lead us
to an isomorphism between Ra and Zq.

LEMMA 4.2.1. If R is a ring, a E R and S C R is a set of coset representatives of
R/aR, then

SeaS -... ak-lS R

is a set of coset representatives of R/akR.

PROOF: Since S is a set of coset representatives of R/aR


R = S aR = S a(S aR) = S aS EC a2R



= S e aS E a2S .. ED a k-1S $ R.



LEMMA 4.2.2. If m is any integer, then m is divisible by a in R = Z[x]/f(x) if
and only if m is divisible by q = f(1) in Z, where I E Z, a = w and f(x) =
xn + an-1xn-1 + + alx + ao.








45 -
PROOF: Let q = f(l) = In + an_-1n-1 + .. + all + ao and suppose m is divisible
by q. Let g(x) = f(l x). Since g(a) = g(l w) = f(w) = 0,

n
0 = g(a) = ai(l- a)i = q ah(a),
i=0

where an = 1 and h(x) is some polynomial in Z[x]. The above equation implies that
q is divisible by a. Therefore, since m is divisible by q, m is divisible by a.
Conversely, suppose m is divisible by a. Note that for any ~ E R, i = g(w), for
some g(x) E Z[z]. Since a = w, if we let hi(x) = g(l x), hi(x) E Z[x], then we
have hl(a) = g(l a) = g(w) = a. Since a divides m, m = aa, for some E R. But
a = hl(a), for some hi(x) E Z[z]. Therefore, mi = ahl(a), for some hl(x) E Z[x].

Let ki(x) = xhl(x) m. Let d be the greatest common divisor of the coefficients of

kl(x) and let k(x) = kil(x). Since k(a) = 0 and g(x) is the polynomial of minimum
degree in the ring R = Z[x]/(f) satisfied by a (since f(x) is the polynomial of
minimum degree in R such that f(w) = 0 and g(x) = f(l x)), it must be the case
that k(x) = g(x)q(x), where q(x) E Q[z]. Since the greatest common divisor of the
coefficients of k(x) (and also of g(x)) is 1, it follows that q(x) E Z[x]. The constant
terms in k(x) and g(x) are a and q, respectively. Therefore, since q divides q, q
divides m. O

Remark. The polynomial f(x) may not be minimal over Z[x]. For example, if

f(x) = x3 + 2 + + + 1 = (x2 + 1)(x + 1), then f(x) is not the polynomial with
minimum degree such that f(w) = f(v'TI) = 0.

LEMMA 4.2.3. If q = f(l) = In + anlln-1 + + all + ao and a = 7 w, then

IR/a RI = qk.

PROOF: Lemma 4.2.3 follows from Lemma 4.2.1 once it is shown that IR/aRI = q.
Since every element of R can be represented by a polynomial in w with coefficients








46 -
in Z, every element of R can be written as a polynomial in a with coefficients in Z.
This last fact implies that every element of R/aR can be represented as ~fi for some
integer m. Now IR/aRI = q follows from Lemma 4.2.2. El

THEOREM 4.2.4. Let f(x) = xn + an-lxn-1 + + aix + ao and a = 7 w,
where 1 E Z. Let q = f(l). If f(l) and f'(l) are relative prime, then there is a ring
isomorphism Ra = limR/akR Zq.

PROOF: An isomorphism will be constructed by finding vertical isomorphisms that
make the following diagram commute.

R/aR R/a2 ... R/aR ...

I 1 I I
Z/qZ Z/q2Z ... --- ZqkZ -- ...

Since each vertical map is to be a ring isomorphism, each of these vertical maps
must take the multiplicative identity 1 in R/akR to the 1 in Z/qkZ. By Lemma
4.2.3 the order of the additive group R/akR is qk. Therefore, these isomorphisms
exist if and only if the additive order of the element 1 in R/akR is qk. This fact
will be proved by induction on k. The case k = 1 is exactly Lemma 4.2.2. By way
of induction assume that the order of 1 in R/ak-1R is qk1. With the polynomial

g(x) defined exactly as it was in the proof of Lemma 4.2.2

n
0 = g(a) = ai(l = q aa + a2h(a),
i=0

where an= 1, h(x) E Z[x], q = In+ an-ln-1 +. *+all+ao and a= nln-+ (n-
l)an-1in-2 + + 2a21 + al. The above equation implies that


qk-1 = k- k-1 +hl(a)
q +ahica








47 -

where hi(x) E Z[z]. The order of 1 in R/akR must be a multiple cqk-1 of the order
of 1 in R/ak-lR. It now suffices to show that q is the least positive integer c such
that cqk-1 is divisible by ak. From (4)


cqk-1 = cak-lk-1 + ckhl(a).


This equation implies that cqk-1 is divisible by ak if and only if cak-1 is divisible
by a. By Lemma 4.2.2 this equivalence is the case if and only if cak-1 is divisible
by q. Since gcd(a, q) = 1 (i.e. f(l) and f'(1) are relative prime), q divides c. Since c
is the least such integer, c = q. Therefore, the induction is true for the integer k. O

One particular case of Theorem 4.2.4 is when f(x) = xn + xn-1 + -- + + 1.
In this case, one needs the following lemmas to prove Corollary 4.2.7.

LEMMA 4.2.5. Let q = n + ln-1 + ... + 1+ 1 and let a = nln-1 + (n 1)n-2 +
+ 21++ 1. If gcd(n+ 1,q)= 1, then gcd(n + 1,a) = 1.

PROOF: Since


q = In + In-1 + ...+ I +1 = (n + 1)n (nIn-1 +...- + 21 + 1)(1- 1),


we have q = (n + 1)ln a(l 1). Therefore, if hla and hl(n + 1) for some h E Z,
then hlq, i.e., if gcd(n + 1, a) # 1, then gcd(n + 1,q) 5 1. Thus, if gcd(n + 1,q) = 1,
then gcd(n + 1, a) = 1. O

LEMMA 4.2.6. If gcd(n + 1,q) = 1, then gcd(q,a) = 1.

PROOF: Suppose h is prime and h divides gcd(q, a). Since q = (n+l)ln-a(l-1), we
have h (n+1)ln. If hll, then hlq- n n-1 I = 1, i.e., h = 1. This contradicts
the fact that h is prime and thus greater than one. Therefore, h \ 1. Since h is prime,








48 -
h i In. Write n + 1 = q. qk, where qi is a prime for i = 1,..., k. Thus, we have

h = qi for some i E {1,...,k}. By Lemma 4.2.5, we have gcd(n + 1,a) > qi and

gcd(n + 1, q) > 1 for some i E {1,...k}. This fact contradicts the assumption that

gcd(n + 1,q) = 1. E

COROLLARY 4.2.7. Let a = w, where I E Z and q = In + In-1 +... + 1. If q

and n + 1 are relatively prime, then there is a ring isomorphism from Ra into Zq.

PROOF: By Lemma 4.2.6 we know that q and a = nln-1 + (n 1)n-2 + ... + 21+ 1

are relatively prime. Since q = f(1) and a = f'(1), the result follows from Theorem

4.2.4. O















CHAPTER 5

THE MATRIX Aa


Define the vector pi from Rn by


0o xl xi-1
,Li = (- 1 ..., t 0i, O, 0),
n n-_ 1 n-i+1


where xi = ( n-i+l ) for i = 0,...,n. The set {I0,1,.. -,n-1} is a basis

of Rn [15]. Denote by An the set of all integer linear combinations of the pi's.

An is an n-dimensional lattice of Rn and the elements of An are the centers of

the (n + 1)-permutohedron packing of Rn mentioned in Chapter 1. The particular

vector a = 2/po pl from An is at the center of a first level aggregate of the second

level aggregate centered at the origin. The vector a defines a linear transformation

Aa from Rn into Rn given by Aa(x) = xa. The linear transformation Aa maps

the centers of the kth level aggregates onto the centers of (k + 1)th level aggregates,

where k = 0,1,.... Relative to the ordered basis {O, l,... ,[n-l}, the linear

transformation Aa is represented by the n x n matrix

2 0 0 ... 0 1

-1 2 0 ... 0 1

0 -1 2 ... 0 1


0 0 0 ... 2 1

0 0 0 ... -1 3/x

49 -








50 -

By abuse of notation, this matrix will also be called Aa.

D. Wilson suggested that the author investigate the matrix A0 more closely.

In this chapter some of the properties of Aa are presented.




5.1. The Algebraic Properties of the Matrix An


Since there are 2n+1 -1 cells in the first level aggregate and (2n+1 -1)(2+1 -1)

cells in the second level aggregate, the volume "stretching factor" is 2n+1 1. This

fact is expressed algebraically in the next proposition.


PROPOSITION 5.2.1. det(Aa) = 2n+1 1.


PROOF: The proof will be by induction on the size n of the matrix A.

If n = 2,
(2 1\
Aa = = 7 = 23 1
-1 3

Therefore, the inductive step is true for n = 2.

Assume that the inductive step is true for all integers less than or equal k.

If n = k + 1, then


2 0 0 ... 0 1

-1 2 0 ... 0 1

0 -1 2 ... 0 1
Aa= .


0 0 0 ... 2 1

0 0 0 ... -1 3/ (k+l)x(k+l)









- 51 -


=2x


2

-1

0



0

0


+ (-1)1+k+ -1


-1 kxk


kxk


Thus,

det(Aa) = 2 x (2k+l 1) + (-1)k+2 (_-)k = 2k+2 -1

Therefore, the induction is true for the integer k + 1. O

PROPOSITION 5.2.2. Let Pn(A) be the characteristic polynomial of the n x n matrix

Aa, then Pn(A) = (2 A)Pn-(A) + 1 = (2 A)n+(2 A)n-l+... + (2 A) + 1.

PROOF:


/2-A

-1

0
Pn(A) = det

0

0

/2-A

-1

0
= (2 A)det


0

0


0 0

2-A 0

-1 2-A



0 0

0 0

0 0

2-A 0

-1 2-A .



0 0

0 0


S 0 1

0 1

0 1



S2-A 1

-1 3- A nn

0 1

0 1

0 1



2-A 1

-1 3-A n-lxn-1








52 -
-1
+(-l)n+l(1)det
-1
=(2- A)Pn-1(A) + (-1)n+1(-l)n-1

=(2- A)Pn-(A) +1

=(2 A)Pn-2(A) + (-1)n-1+1(-)n-2) + 1

=((2 A)Pn-2 + (2 A) + 1


=(2 A)" + ... + (2 A) + 1.0
COROLLARY 5.2.3. If n is odd and n > 3, then 3 A is a factor of Pn-2(A).

PROOF: By Proposition 5.2.2,


Pn(A) = (2 A)n + + (2 A) + 1

=(2 A)n-1[(2 A) + 1] + (2 A)n-3[(2 A) + 1] + ...

+ (2 A)2[(2 A) + 1] + [(2 A) + 1]

=(2 A)n-1(3 A) + (2 A)n-3(3 A) + ... + (2 A)(3 A) + (3 A)

=(3 A)[(2 A)n-1 + (2 A)n-3 + ... + (2 A) + 1].
Therefore, (3 A) I Pn(A).0

PROPOSITION 5.2.4. Let wl,w2, ... ,w be (n + 1)th roots of unity with wi 5 1 for
i = 1,...,n. The eigenvalues of Aa are 2 w,2-w 2,...,2-wn.

PROOF: By Proposition 5.2.2, Pn(A) = 0 implies that (2 A)" + (2 A)n-1 +... +
(2 A) + 1 = 0. Since if (2 w) 1 0, (2 A)n + (2 A)n-1 +... + (2 A) + 1 =
(2 A)+1 1 = 0. Therefore, if A $ 1, then (2 A)n+l = 1. Let W1,...,wn
denote the (n + 1)th roots of unity with wi # 1, for all i = 1,...,n we have
Al = 2 -w,...,An = 2 -n. D









53 -

PROPOSITION 5.2.5. The eigenvectors V1, V2,...,Vn of Aa are the following:

U1

u2
vi = ,


\un/

where un could be chosen as 1, and un- = 1+(2-A), un-2 = 1+(2-A)+(2-A)...,

ul = 1 + (2 A) + ... + (2 A)"-1, for i = 1,..., n.

PROOF: For each eigenvalue Ai, we assume that the eigenvector

1ul

Vi =

Un


and we have


2 0 0 ... 0

-1 2 0 ... 0

0 -1 2 ... 0



0 0 0 ... 2

0 0 0 ... -1

which produces the following system

2ul + Un

-ul + 2u2

-u2 + 2u3


1\ / ul

1 U2
1 u2

1 u3



1 un-1

3/nxn \ Un

of equations,

=A

+ Un = A

4- u. = A


-un-2 + 2n-1 + n =

-un-1 + 3un =


/ 1

u2

U3



un-1

\ Un /


ijl

.iu2

jiu3



iun-1

iun


A

A








- 54 -


Therefore,


(2 A)u1 + Un

-ul + (2 A)u2 + un,

-u2 + (2 A)u3 + un


-Un-2 + (2 A)un-1 +

I -Un-1 + (3 A)un
The solutions of this system of equations are


=0

=0

=0



=0

=0


Thus, if we choose


ul = [

u2 = [



Un-2 = [1

SUn-1 = [1

Un = 1, then


(2-

(2-


S+ (2 A) + (2 A)2]un

+ (2 A)]un.


u = 1 + (2 A)+...+(2 A)-1

u2 = 1 + (2 A) + ... + (2 A)-2



Un-2 = 1 + (2 A) + (2 A)2

Un-1 = 1 + (2 A).0


PROPOSITION 5.2.6. The inverse of Aa is Aa-1 = 1(aij), where aij = 2n-i+j -

2J-1 ifi > j, aij = 2-1 + 2j-i-1 if i < j and q = det(Aa).


A)n-1]un

A)"-2]un


1 + (2 A) +... +

L + (2 A) + ... +









- 55 -


PROOF: Let

/2 0 0 ... 0 1

-1 2 0 ... 0 1

0 -1 2 ... 0 1
Aa= .


0 0 0 ... 2 1

0 0 0 ... -1 3
and let

2n 1 -2 + 1 -22 +2 ... -2-2 + 2n-3 -2n-1 + 2n-2

2n-1 1 2n 2 -22 + 1 ... -2n-2 + 2n-4 -2n-1 + 2n-3

1 2n-2- 1 2n-1 2 2n 22 ... -2n-2 + 2n-5 -2n-1 + 2-4
B=-


22 1 23-2 24 22 ... 2n 2n-2 -2n-1 +1

21 1 22-2 23-22 ... 2n-1 2n-2 2n -2n-1

Let C = AaB, cij = n=1 aikbkj- If i > j, then

ci = aiilbilj + aiibij + ainbnj

= 1[(-1)(2n-(i-1)+j 2j-1) + (2)(2n-i+j 2j-1) + (1)(2n-n+j 2J-1)]
q

S(-2n-i+l+j + 2j-1 + 2n-i+j+l 2 2 + 2j-1)
q
=0.

If i < j, then


cij = [(-1)(-2j-1 + 2j-(i-1)+1) + (2)(-2j-1 + 2j-i-1) + (1)(2n-n+j 2J-1)]


q
S(21 2i 2 + 2-i + 2 2-1

=0.









- 56 -


If i = j, then


ii= aii-lbi-li + aiibii + ainbni

= [(-1)(-2i-1 + 2i-(i-1)-1) + (2)(2n 2i-1) + (1)(2n-n+i 2i-1)]
q
= 1(2i- 1 + 2n+1 2i + 2i 2i-1)

2n+1 1
--1.
q


Therefore, C = I, which is the identity matrix. If we let D = BAa, by the

similar calculation, we get D = I. Thus, we have AaB = BAa = I, i.e., B = Aa-1

D


PROPOSITION 5.2.7. Aa = LU, where L =


2

0

U=



0
PROOF:


1

1+


S1
1

0



0

0


and


0 ... 0

2 ... 0



0 ... 2

0 ... 0
Trivial. M


1 + +...+(1)n-1

3+ +...+( + + )n
















CHAPTER 6


IMAGE ALGEBRA IN HEXAGONAL LATTICE


In this chapter, we present some results concerning the decomposition and

invertibility of circulant templates over the hexagonal sampled images under the

generalized convolution operation (E) of image algebra [23]. Many of the results

are due to D. Lucas and L. Gibson [19]. Two types of polynomial representation for

hexagonal images are also discussed.



6.1. A Brief Review of the Image Algebra


In this section, a brief review of the fundamental concepts and notation of the

image algebra will be given.

The image algebra is an heterogeneous algebra structure specially designed

for image processing [23]. It has been demonstrated that many commonly used

image processing transformations, such as generalized convolutions, Discrete Fourier

Transform, edge detectors, and morphological operations, can be easily expressed

in terms of the image algebra.

An image algebra is an algebra whose operands are images and subimages (or
neighborhoods). It deals with six basic type of operands, namely, value sets, point

sets, the elements of the value sets and point sets, images, and templates.

A value set can be any semi-group. The most commonly used value sets in

image processing are the set of positive integers, integers, rational numbers, real

57 -








58 -

numbers, positive real numbers, or complex numbers. These sets will be denoted by

Z+, Z, Q,R, R+, C, respectively. The value set will be denoted by F.

A a point set is a topological space, in particular a subset of an n-dimensional

Euclidean space, Rn, for some n. Point sets are commonly denoted by the symbols

X and Y. The elements of such sets are denoted by lower case letters. Familiar

point sets include the rectangular and hexagonal arrays.

DEFINITION 6.1.1. Let X, and F be a point set and a value set, respectively. An

F valued image a on X is a function a: X -+ F.

Thus, the graph of an F valued image a on X is of the form


a = {(x,a(x)) : a(x) E F, for all x E X}.


The set X is called the set of image coordinates of a, and the range of the

function a is called the set of image values of a. The pair (x, a(x)) is called a

picture element or a pixel, x the pixel location, and a(x) the pixel (or gray) value.

We will denote the set of all F valued images on X by FX. We make no distinction

between an image and its graph.

DEFINITION 6.1.2. An image a: X -+ F has finite support on X ifa(x) 5 0 for

only a finite number of elements x E X.

Another basic, but very powerful tool of the image algebra, is the generalized

template.

DEFINITION 6.1.3. Let X and Y be two coordinate sets, and let F be a value set.

A generalized F valued template t from Y to X is a function t: Y -* FX.








59 -

Thus, for each y E Y, t(y) E FX, or equivalently, t(y) is an F-valued image

on X. For notational convenience, we define ty = t(y). Thus,


ty = {(x,ty(x)) :x E X}.


The sets Y and X are called the domain and range space of t, respectively. The
point y is called the domain point of the template t, and the values ty(x) are called
the weights of the template t at y. Note that the set of all F-valued templates from
Y to X can be denoted by (FX)Y.
If t is a template from Y to X, then the set


S(ty) = {x E X: ty(x) # 0}


is called the support of ty.
If t is an F valued template from X to X, and X is a subset of R", then
t is called translation invariant (or shift-invariant ) if and only if for each triple

x, y, z E Rn, with x + z and y + z E X, we have that


ty(x) = ty+z(x + z).


Note that a translation invariant template must be an element of (FX)X. Invariant
operators on Z x Z are commonly expressed in terms of polynomials of two variables.
A template which is not necessarily translation invariant is called translation variant
or, simply, a variant template. Translation invariant templates occur naturally in
digital image processing.
The basic operations on and between F valued images are naturally derived
from the algebraic structure of the value set F.








60 -

Let X be a subset of Rn. Suppose a E RX and t E (RX)X.

Addition on images is defined as follows: If a, b E FX, then


a + b {(x,c(x)): c(x) = a(x) + b(x),x E X}.


Higher level operations are the ones that involve operations between templates

and images, and between templates only.

The addition of two templates is defined pointwise. If s and t E (RX)X, then
we have

(s + t)y(x) = sy(x) + ty(x).

DEFINITION 6.1.4. The generalized convolution of an image a together with a tem-
plate t is defined by


aEt = {(y,b(y)): b(y) = a(x)ty(x),y E X}.
xEX

Linear convolution plays a fundamental role in image processing. It is involved
in such important examples as the Discrete Fourier Transform, the Laplacian, the
mean or average filter and the Gaussian mean filter.

DEFINITION 6.1.5. Ifs and t are templates on X, then we define the generalized
convolution of the two templates as the template r = s D t by defining each image
function ry by the rule


ry = {(z,ry(z)) : ry(z) = ty(x)sx(z), where z E X}.
xeX

Note that r can be viewed as a generalization of the usual notion of the composi-
tion of two convolution operators. If the templates s and t are translation invariant,








61 -
then, except for values near the boundary, the previous definition agrees with the
usual definition of polynomial product.
Note also that if s and t are two invariant templates, then r would be an
invariant template too. Computing r at just any one y E Y is sufficient to define
the template everywhere.
Many other image operations are described in detail in Ritter et. al. [23]. A
precise investigation of the linear convolution can also be found in Gader [5], and an
extensive study of other non-linear template operations can be found in Davidson

[3], Li [14] and Manseur [20].
If X is a finite rectangular subset of the plane with m rows and n columns,
then it can be linearly ordered left to right and row by row. Thus, we can write

X= {X1, X2, ...,Xmn}.
Let (Mmn, +,*) denote the ring of mn x mn matrices with entries from F
under matrix addition and multiplication. For any template t, we define a matrix

Mt = (mij) where mij = tx((xi). For the sake of notational convenience we will
write tij for tx (xi).
Define the mapping 0: (FX)X --+ Mmn by 0(t) = Mt.
The next Theorem was proved by Ritter and Gader [5]. It shows that there is
an embedding of the linear algebra in the image algebra.

THEOREM 6.1.6. The mapping q is an isomorphism from the ring ((FX)X, +, E)
onto the ring (Mmn,+,*). That is, if s,t E (FX)X, then

(1) q(s + t) = O(s) + 0(t) or Ms+t = Ms + Mt
(2) O(s t) = O(s)O(t) or Mset = MsMt

(3) is one-to-one and onto.








62 -

This theorem clearly states that template inversion or deconvolution is equiv-

alent to matrix inversion. Actually, a more powerful implication of this theorem is

that any tool available in linear algebra is directly applicable to any problem in the
image algebra.

DEFINITION 6.1.7. Let X be an m x n rectangular point set. We say that the

mapping 0 : X -- X is a circulant translation if and only if 01 is of the form

O(x) = (x + y) mod (m, n), for some y E X.

DEFINITION 6.1.8. We say that t E (FX)X is circulant if and only if for every

circulant translation i, the equation tx(y) = t (x)(0(y)) holds.

Remark: This last definition shows that a circulant template is completely deter-
mined if it is defined at only one point.


6.2. Hexagonal Images and Polynomial Rings

The point set X for hexagonal arrays based on the level k GBT2 address is the

quotient ring GBT2/Ik as indicated in Section 2.3. We use Ak to denote the ring

GBT2/I1 in this Section. A function a from Ak to the real numbers R is an image

as defined in Section 6.1, Since we have the one-to-one correspondence between Ak

and the hexagons in the level k aggregate, for each GBT2 address v, a(v) is the

pixel value of the hexagon grid [19].

The set FAk of images on a level k aggregate is itself a ring in two distinct
ways. The first is pointwise. Given two images a and b, one can define


(a + b)(v) = a(v) + b(v)


and


(a b)(v) = a(v) x b(v)








63 -

for all v in Ak. These operations result in a commutative ring structure on FAk. The

other ring structure on FAk is a convolution ring. Addition in this ring is pointwise

as above. Multiplication is convolution, defined by


(a b)(v) = conv(a, b)(v) = E a(w)b(v w)
WEAk

where v w is an operation in Ak. This means that as pixels from the image move

across the boundary of the image a they wrap around and reenter a from the other

side.

Let 77(1) = 1 in Ak. Define the function 7 : Z/(7k) Ak by 77(i) = 7r(1) + .. + q(1),
i
the sum of i 1's in Ak. Thus, in A2, 77(0) = 0, 7(1) = 1, 77(2) = 12, 77(3) = 13,

77(4) = 44, etc.
With each image a in Ak, one can associate a polynomial fa(x) defined by


fa(x) = a(v)zF(')
vEAk

where 7 = r-1 is the isomorphism from Ak to Z/(7k). For two images a and b we
have

fa(x) fb(x) = c(v)x(V), where
vEAk
c(v) = a(w)b(v w), the convolution of a and b.
wEAk
This last equality results from the fact that


a(w)x(w) b(v w)x(v-") = a(w)b(v w)xa() for all w E Ak.


Addition of polynomials likewise corresponds to image addition. Therefore, the

set of images on Ak with the convolution ring structure is isomorphic to the quotient

ring of polynomials with exponents taken modulo 7k








64-

6.3. GBT9 Circulant Templates


In general, an F valued template t on a level k GBT2 aggregate Ak is a

mapping from an index set Y to the set of functions from Ak to F. The generalized

convolution between a level k GBT2 image and a template t can be considered as

the image convolution. Especially, the template t is a circulant template since the

ring structure on Ak causes cells which cross the aggregate boundary to reenter the

aggregate at another location [19].

In Section 6.2, we saw that image convolution is equivalent to polynomial mul-

tiplication in the quotient ring of polynomials whose exponents are taken modulo

7k. In this ring

X7k 0 = 1


and each ring element is uniquely represented by a polynomial of degree less than

7k. Therefore questions concerning the decomposition or invertibility of a circu-

lant template under the circle plus operation can be posed as questions about the

corresponding polynomials.

The next Theorem was proved by D. Lucas and L. Gibson [19].

THEOREM 6.3.1. A circulant template is invertible if and only if each of its linear

factors is invertible.

PROOF: The Fundamental Theorem of Algebra states that any polynomial with

real or complex coefficients can be factored into linear factors in the field of complex

numbers. The template corresponding to a linear factor (x r) has a one at GBT2

address 1 and -r at GBT2 address 0. It follows that any circulant template on Ak

can be written as a E of these simple templates. Therefore, a circulant template is

invertible if and only if all of these simple templates are invertible. l









65 -

COROLLARY 6.3.2. Any template in first level aggregate can be decomposed into the

D of the templates with the shape





I ag I


and with the shape


where ai E R.


Any template in second or higher level aggregates can be decomposed into the

( of the templates with the shape





Salg
I a0 I


and the templates with the shape









- 66


a2

al

a0 I


The next theorem was also proved by D. Lucas and L. Gibson.

THEOREM 6.3.3. Any circulant template in a GBT2 level k aggregate is invertible

if and only if none of the roots of its corresponding polynomial are (7k)th roots of

one.

PROOF: We can derive this characterization of invertibility for circulant templates

by defining
7k
tr(x) = rn-lx7Kn
n=1
By multiplying tr(x) by (x r) one sees that



(x r)*tr(x)= x7k r7k = r7k


Therefore, if r7k is not equal to 1, then the polynomial



tr(x)/(1- r7k)


is the inverse of (x r). Otherwise (x r) is a zero divisor and not invertible. D








67 -
Using level one aggregate as an example, the template


has for its corresponding polynomial



-x5 _- 4 + X3 2 = _-2(x 1)(x + 1)2



one of whose roots is 1. Thus, it is not invertible. But the template


has polynomial



x6 + 5 + 4 + 3 + x2 + x = x(x + 1)(x2 +x+ 1)(x2- x + 1)



and is invertible since the roots of the last two quadratics are the cube and sixth

roots of 1 respectively. In fact, its inverse template is one sixth of the integer

template








- 68 -


6.4. Another Representation of GBT2 Circulant Templates

The family of real valued circulant templates on n x m rectangular images is
isomorphic to a quotient ring of the ring of real polynomials in two variables.
We can define a polynomial for hexagonal sampled images in a similar way.
As shown in Figure 8, if r is the dimension of a side of one of the six equilateral
triangles that make up a basic hexagon ( the hexagon radius) and r is chosen as 2,
then we can construct the polynomial


p(x, y) = a00(x3)0(yJN)0 + a02(x3)0(yv)2 + ao-2(x3)0(v)-2 + a11(x3)(h)1+


al-l(x3)1(y,)-1 + a-ll(x3)-1(yv')1 + a-1-1(z3)-(y)-1,

according to the center of the hexagons.
If we let u = 3 and v = yV then


p(x, y) = q(u, v) = aoo+a02v 2+ao-2-2+alluv+al-luv-+a-llu-lv+a--u-l1.


Notice that for each term umvn of the polynomial q(u, v), we have m and n are
either both odd or both even.









- 69 -


Figure 8: First level aggregate over the Cartesian Plane.

For the polynomial q(u, v), we can set up a 5 x 5 matrix as follow:


0

a-1-1

0

a-11

0


a0-2

0

a00

02

a02


0

al-1

0

all

0


Manseur [201 discussed the decomposition of polynomial with 2 variables. For

a size 5 x 5 template t, she proved that t = tlt2 + t3t4 + t5, where tl,t2,t3,t4

and t5 are size 3 x 3 templates. This decomposition method does not work well for

hexagonally sampled templates since after the decomposition, a size 3 x 3 template

corresponds the template with the shape









- 70 -


which does not provide the smaller template as we wanted.



PROPOSITION 6.4.1. If the template


where ai E R, for i = 0, 1,..., 18, and a8 = a7 + a9, a10 = a9 + all, a12 = all + a13,

a14 = a13 + a15, a16 = a15 + a17, a18 = a17 + a7, then, t = s1 E s2 + r, where


s1 =









- 71 -


s2=


1



and



rl
r6 r2
r r 0

r4



where r = -a7 ag all a13 a15 a17, rl = a ao a7 ag a17,

r2 = a2-a0-a7-ag-all, r3 = a3-ao-a9-all-a13, r4 = a4-a-all-a13-a15,

r5 = a5 ao a13 a15 a17, r6 = a6 a a15 a17 a7.

PROOF: The polynomials correspond to the templates sl and s2 are



sl(u, v) = ao + a7v2 + aguv + alluv-1 + a13v-2 + al5u-1v-1 + a174?lv,








- 72 -


and

s2(u,v) = 1 + v2 + u + v + -1 -2 + u--1 + u-1v,

respectively. Therefore,

sl(u, v)s2(u, v) = (aO + a7 + ag + all + a13 + a15 + a17)+

+ (ao + a7 + 9a + al7)v2 + (aO + a7 + ag + all)uv+

+ (ao + 9a + all a)+ + (ao + all + a13 + al5)v-2+

+ (ao + a13 + 1a5 + a17)u-v1-1 + (aO + al5 + a17 + a7)u-1v

+ a7v4 + (a7 + ag)uv3 + agu2v2 +

+ (a9 + all)u2 + allu2-2 + (all + al3)uv-3+

+ al3v-4 + (a13 + al5)u-l -3 + al5u-2V-2+

+ (al5 + a17)U-2 + al7u-2V2 + (a17 + a7)u-v3.

The polynomial corresponds to the template r is

r(u,v) = (-a7 ag all a13 a15)+

+ (al a0 a7 a9 al7)v2 + (a2 ao a7 a9 all)uv+

+ (a3 ao ag all al3)uv-1 + (a4 a0 all a13 al5)V-2+

+ (a5 a0 a13 al5 al7)u-lv-1 + (a6 ao -a5 a17 a7)u-1l.

Therefore, sl(u, v)s2(u, v) + r(u, v) = t(u, v), where


t(u, v) = a0 + alv2 + a2uv + a3uv-1 + a4v-2 + a5u--1 + a6u-l1+

+ a7v4 + aguv3 + agu2v2 + al0u2 + allu2 -2 + a2uv-3 +

+ a13v-4 + al4u-1 -3 + a15U-2V-2 + a16u-2 + al7u-2 2 + al8u-1v3

and t(u, v) is the polynomial of the template t. E








73 -

The author is still working on the decomposition of a template with the shape

















and arbitrary gray values into sl E S2 + 83, where si's are templates with a first level

aggregate shape.

The question raised here is that whether we can decompose a template t with

a second level aggregate shape (See Figure 2) into the form sl ( s2 E S3 s s4 + 85,

where si's are the templates with a first level aggregate shape. Obviously, if we

convolute a template with a first level shape 4 times, we can not get the template

with a second level aggregate shape. The author is presently working to determine

the possible decompositions of t.















CHAPTER 7


FINAL REMARKS

This dissertation is not of the sort where some conclusion can be drawn. In

short summary, the focal result of the dissertation was the proof that EGBTn is

isomorphic as a ring to the (2n+1 1)-adic integers if n+1 and 2n+ 1 are relatively

prime. This naturally leads to the question of whether EGBTn and the (2n+1 -1)-

adic integers are isomorphic if n + 1 and 2n+1 1 are not relatively prime. The

author hopes to investigate this question at a later date.


- 74 -
















REFERENCES


1. N. Ahuja, On approaches to polygonal decomposition for hierarchical image rep-
resentation, Computer Vision, Graphics Image Processing 24 (November, 1983),
200-214.

2. Z. I. Borevich and I. R. Shafarevich, "Number Theory," Academic Press, New
York, 1966.

3. J. L. Davidson, Lattice Structure in the Image Algebra and their Applications,
Ph.D. Dissertation, University of Florida, Gainesville, FL (1989).

4. L. Fuchs, "Infinite Abelian Groups I," Academic Press, New York, 1970, p. 62.

5. P. D. Gader, Image Algebra Techniques for Parallel Computation of Discrete
Fourier Transforms and General Linear Transforms, Ph.D. Dissertation, Uni-
versity of Florida, Gainesville, Fl (1986).

6. L. Gibson and D. Lucas, Spatial data processing using Generalized Balanced
Ternary, Proceedings of the IEEE Conference on Computer Vision and Pattern
Recognition (June, 1982), 566-571.

7. L. Gibson and D. Lucas, Vectorization of raster images using hierarchical meth-
ods, Computer Graphics and Image Processing 20 (1982), 82-89.

8. L. Gibson and D. Lucas, Pyramid algorithms for automated target recognition,
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CON, Dayton, Ohio, (1986), 215-219.

9. M. J. E. Golay, Hexagonal Parallel Pattern Transformations, IEEE Trans. Com-
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10. N. Jacobson, "Basic Algebra II," W. H. Freeman and Company, San Francisco,
1980, p. 74.

11. I. Kaplansky, "Infinite Abelian Groups," University of Michigan Press, Ann
Arbor, 1956, p. 43.


- 75 -








- 76 -


12. W. Z. Kitto, A. Vince and D. C. Wilson, An Isomorphism Between the p-adic
Integers and a Ring Associated with a Tiling of N-space by Permutohedra ,
submitted.

13. Neal Koblitz, "p-adic Numbers, p-adic Analysis, and Zeta-Functions," Springer
Verlag, New York, 1977.

14. D. Li, Recursive Operations in Image Algebra and their Applications to Image
Processing, Ph.D. Dissertation, University of Florida, Gainesville, Fl (1990).

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16. D. Lucas, Personal Communication.

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Conference on Pattern Recognition, Montreal, Canada, 1 (1984), 230-233.

18. D. Lucas and L. Gibson, Techniques to exploit the relation between polynomial
representations and moments of pictures, Proceedings of the IEEE Conference on
Computer Vision and Pattern Recognition, San Francisco, CA (1985), 138-143.

19. D. Lucas and Laurie Gibson, Template decomposition and inversion over hexag-
onally sampled images, Proceedings of the 1991 SPIE Image Algebra and Mor-
phological Image Processing II, San Diego, CA (1991), 157-163.

20. Z. Z. Manseur, Decomposition and Inversion of Convolution Operators, Ph.D.
Dissertation, University of Florida, Gainesville, FL (1990).

21. B. H. McCormick, The Illinois pattern recognition computer ILLIAC III, IEEE
Trans. Electron. Comput. vol. EC-12 (Dec. 1963), 791-813.

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77 -

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28. A. Vince, Aggregate Tessellations, submitted.
















BIOGRAPHICAL SKETCH


Wei Zhang Kitto was born on July 8, 1955, in Beijing, China. She received a

bachelor's degree in mathematics from East China Institute of Textile Science and

Technology in 1982, and a master's degree in mathematics from University of Florida

in 1986. Her research interests include applied mathematics, image processing, and

computer vision.


- 78 -










I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a thesis for the degree of Doctor of Philosophy.


David C. Wilson, Chairman
Professor of Mathematics

I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a thesis for the degree of Doctor of Philosophy.


ard X. Ritter, Cochairman
ofessor of Computer and
Information Sciences

I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a thesis for the degree of Doctor of losophy


drew Vince
Associate Professor of Mathematics

I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a thesis for the degree of Doctor of Philosophy.



Li-Chien Shen
Associate Professor of Mathematics










I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a thesis for the degree of Doctor of Philosophy.



Arun K. Varma
Professor of Mathematics

I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a thesis for the degree of Doctor of Philosophy.



ose N. Wilson
(Assistant Professor of Computer and
Information Sciences

This dissertation is submitted to the Graduate Faculty of the Department of
Mathematics in the College of Liberal Arts and Sciences and to the Graduate School
and was accepted as partial fulfillment of the requirements for the degree of Doctor
of Philosophy.

December, 1991


Dean, Graduate School























































UNIVERSITY OF FLORIDA


3 1262 08285 438 0




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