数理解析研究所講究録
1169 巻 2000 年 1-14

1

Why such a meeting ?
Atsushi Yoshikawa
Graduate School of Mathematics
Kyushu University

1

Introduction

Since some time I have been interested in such topics as how to engineer
genuineon the computer. However, even a proper
formulation of the question itself is not yet clear, for there must be a kind of
spirals of what we should do, what we have done and what we can do.
$\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}’ \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{p}\mathrm{t}\mathrm{s}$The idea comes from very impressive recent developments of mathematical softwares. I do not deny its naiveness. We have numerical computations,symbolic comuptations, and lots of things are now carried out on computer.
In fact, we know most modern sophisticated activities are de facto carried out
computationally, based upon mathematical or pseudo-mathematical principles. Nevertheless, you must have observed plenty of discrepancies in themathematics-computer relations. We thus have to build a synthesized worldof mathematical reasoning and computations, hoping that its impact does
not remain within a mathematico-technical sphere.

As the first step, I have been trying to show relevance of developing acomputer version of classical mathematical analysis, in which crucial roles
are played by various concepts of infinities and topologies, crystalized in theform of inequalities (See, e.g., [10, 11]). However, such an endeavor can only
become possible with cooperations of brave people, amateurs of adventures.
Please pay your particular attention to the contributions of these courageous
mathematicians in these Proceedings.
Finally, to add some academic flavor to my notes, I show a further exampleof transferrability of good mathematical reasoning into a computer-related
world (For another example, see $[1l]$ with respect to Calder\’on’s work [3]).
In the classical measure theory, a regularity property of integration isexpressed in terms of the so called differentiation property or density theorem.
technique employed therein is a class of covering theorems (SeeThe

standar.d

2. Covering theorems of Vitali [8] and Besicovitch $[1, 2]$ are typicalinstances. While Vitali’s covering theorem is proven by a transfinite means
such as Hausdorff’s maximum principle, Besicovitch’s theorem admits a veryalgorithmic demonstration. This fact is, in the context of philosophy, say,very interesting, for Vitali’s theorem is valid for the Lebesgue measure andBescovitch’s theorem for the Radon measure.
Here we show in fact the customary proof of Bescovitch’s theorem admits
an algorithmic interpretation. We in fact implement such algorithms into
Maple V procedures, and are ready for experimenting in the case of finite
coverings, though possibly and hopefully with high enough cardinalities. In\S 5, we only compute a family of 1000 disks. The efficiency also depends on
the preassigned ranges of values of radii and how they are scattered.
$\mathrm{G}\mathrm{u}\mathrm{z}\mathrm{m}\acute{\mathrm{a}}\mathrm{n}[4])$Note that the idea of covering theorems contains a reduction to a simpler
structure from a very complicated one 1, of any cardinality. A most general
discussion involved therein requires the Axiom of Choice. It should thus
be worth emphasizing that actual computer experiments can be carried out
retaining essential algorithmic parts of the proof.

2

The Besicovitch Covering theorem

Let $A$ be a family of closed disks in the Euclid plane . For a disk $D\in A$ wedenote by center $(D)$ and radius $(D)$ respectively its center and radius. Thus,
,
and
when
$\mathbb{R}^{2}$$c=\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}(D)$$r=\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{i}\mathrm{u}\mathrm{s}(D)$$D=D_{r}(c)=\{x\in \mathbb{R}^{2} ; |x-c|\leq r\}$.

stands for the Euclid distance. When $A$ is a countable family, theHeredisks therein can be numbered:$||$$A=\{D_{n} ; n=.1,2, \cdots\}$ ,whence, for each $n=1,2,$$\mu_{n}(A)=\mathrm{t}\mathrm{h}\mathrm{e}$Thus,$\mu_{n}(A)’ \mathrm{s}$$\cdots$, we may speak ofcardinality of $\{D_{j} ; D_{j}\cap D_{n}\neq\emptyset, j<n\}$ .

(1)depend on the numbering of $A$ . However,$(\leq\infty)$$\gamma(A)=\sup_{n\geq 1}\mu_{n}(A)$is whyI am tempted to cite Watts’$\overline{\mathrm{l}\mathrm{T}\mathrm{h}\mathrm{i}\mathrm{s}}$$\mathrm{b}\mathrm{o}\mathrm{o}\mathrm{k}[9]$

.

(2)

3

is independent of the numbering of $A$ .
is no more valid. HowIf $A$ is not countable, then this definiton of
ever, the Besicovitch covering theorem ensures a certain uniform boundsuch that any family of closed disks has a countable subfamily $D$ with
. More precisely, we have the following$\gamma(A)$$\beta$$B$$\gamma(D)\leq\beta$

Theorem.
SupposeLet$B$

be a family of closed disks (of any cardinality $>0$ ).

$R= \sup\{\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{i}\mathrm{u}\mathrm{s}(B) ; B\in B\}<+\infty$

.

(3)

Let

(4)

$C=\{\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}(B) ; B\in B\}$be the set of the centers of the disks in B. There exists a countableandsubfamilyof such that
$D$

$B$

$\gamma(D)\leq\beta$$C \subset\bigcup_{B\in D}$

(5)

B.

and is independent of the choice of
Here$n=1,$
there are at most $\beta+1$ subfamilies$\beta\in \mathbb{N}$$D_{n},$$D= \bigcup_{n=1}^{\beta+1}D_{n}$Moreover, each

$D_{n}$$B$

and of R. Furthermore,$\beta+1$ , such that$\cdots,$.

(6)

consists of disjoint closed disks.

For a proof, see, e.g., Besicovitch $[1, 2]$ ,.
also$\mathrm{Z}\mathrm{i}\mathrm{e}\mathrm{m}\mathrm{e}\mathrm{r}[12]$or Evans et$\mathrm{a}1.[5]$. See$\mathrm{G}\mathrm{u}\mathrm{z}\mathrm{m}\acute{\mathrm{a}}\mathrm{n}[4]$In the present note, we are mainly interested in an algorithm for extractfrom $D$ . Actually,ing a subfamily from and another for selecting
the proofs cited above are basically carried out along similar algorithms. In
implementing such algorithms, we assume is finite, though its cardinality
can be sufficiently high compared to the size of .
$D$

$B$

$D_{n}’ \mathrm{s}$$B$$\neq B$$\beta$is known. A more detailed discussion
Remark. A rough estimate
on its estimate will be given elsewhere. We also note that analogous resultsfor other regular geometricto Theorem hold in the case ofobjects than closed disks.
$\beta\leq 289$$\mathbb{R}^{n}\mathrm{a}\mathrm{n}\mathrm{d}/\mathrm{o}\mathrm{r}$4

3

Ideas of algorithms

Suppose we are given a finite family$\mathcal{E}$of closed disks :$\mathcal{E}=\{E_{n} ; E_{n}=D_{\rho_{n}}((\xi_{n}, \eta_{n})), n=1, \cdots, N\}$

Here, the cardinality of is thus
$F$ from
and then assign a subfamily.

(7). We have to choose a closed diskof . Thus,$N=\neq \mathcal{E}$$\mathcal{E}$$\mathcal{G}$$\mathcal{E}$::$\mathrm{P}_{0}$$\mathrm{P}_{1}$$\mathcal{E}$

(8)
(9)

$\mathcal{E}arrow F$$\mathcal{E}arrow \mathcal{G}$and their recipes will be specified below.
Here are some basic ingredients. Let be a positive integer$D=D_{r}((u, v))$ a closed disk. We will require the set$n$$\leq\neq \mathcal{E}$and$S^{*}(D, n, \mathcal{E})=\{1\leq j<n ; (\xi_{j}-u)^{2}+(\eta_{j}-v)^{2}>(\rho_{j}+r)^{2}\}$of the indices $j(<n)$ of disks in$\mathcal{E}$

not intersecting with $D$ and the set. $C^{*}(D, \mathcal{E})=\{j ; (\xi_{j}-u)^{2}+(\eta_{j}-v)^{2}>r^{2}\}$of the indices of centers of disks ina certain complementary set of$j$$\mathcal{E}$outside the disk $D$ . We will also need
, i.e.,$S^{*}(D, n, \mathcal{E})$$S_{*}(D, n, \mathcal{E})=\{1\leq j<n ; (\xi_{j}-u)^{2}+(\eta_{j}-v)^{2}\leq(\rho_{j}+r)^{2}\}$,the set of the indices $<n$ of disks in intersecting with $D$ . For $n=1$ , bothare empty. Since verifications of equalities are
andfrom the view pointis less convenient thaninvolved,).
of computer analysis (See Pour-El et$\mathcal{E}$$S_{*}(D, n, \mathcal{E})$$S^{*}(D, n, \mathcal{E})$$S^{*}(D, n, \mathcal{E})$$S_{*}(D, n, \mathcal{E})$$\mathrm{a}1.[7]$The procedure

PO1$\mathrm{P}_{0}$goes as follows:

Find$R= \max\{\rho_{i}=\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{i}\mathrm{u}\mathrm{s}(E_{i}) ; E_{i}=D_{\rho_{i}}((\xi_{i}, \eta_{i}))\in \mathcal{E}\}$

P02 Let $m$ be the smallest index such that
P03 LetRemark.

$F=D_{\beta m}((\xi_{m}, \eta_{m}))$$\rho_{m}>\frac{3}{4}R$

.

.

.

Note that we may adopt another selection principle in Step P02.

The procedure$\mathrm{P}_{1}$then goes as follows:

5

Pll

Let

P12

Let

$F=\mathrm{P}_{0}(\mathcal{E})$$\mathcal{E}’$$C^{*}(F, \mathcal{E})$.

be the set of the disks in.

P13 Renumber
family be .

$\mathcal{E}’$$\mathcal{E}$with indices belonging to the set

for the sake of normalizing purpose, and let the resulting$\mathcal{G}$. If the set
Note
is empty in Step P12, thenThus, the procedures
cannot be repeated indefinitely.
$C^{*}(F, \mathcal{E})$$\neq \mathcal{G}<\neq \mathcal{E}$$\mathcal{G}$is empty.

$\mathrm{P}_{1}$Therefore, combining the procedures, we can extract a seandquence of closed disks from a given family of closed disks. Namely, wehave$\mathrm{P}_{1}$$\mathrm{P}_{0}$$\mathcal{E}$$F_{m}=\mathrm{P}_{0}(\mathrm{P}_{1})^{m-1}(\mathcal{E})$

,

(10)

$m=1,$ $\cdots,$ $M$ ,. This means that we have a procedure which produces
wherefrom a given family of closed disks a subfamily $F$ of closed disks
asgiven by (10):$(\mathrm{p}_{1})^{M}(\mathcal{E})=\emptyset$$\mathrm{B}$$F_{m}$$\mathcal{E}$:$\mathrm{B}$$\mathcal{E}arrow$

F.

(11)

a Besicovitch procedure, for $F$ is an extracted subfamily of
We callsatisfying the requirement of the Besicovitch Covering Theorem. In fact,
and their repeated applications are key steps in thethe proceduresclassical proof of this theorem. Observe thatmakes sense for of any
cardinality by the Axiom of Choice, that the set
is defined conceptually, and that, allowing $M=\infty,$
can be defined for any family ofclosed disks (with uniformly bounded radii).
$\mathrm{B}$$\mathcal{E}$$\mathrm{P}_{0},$$\mathrm{P}_{1}$$\mathcal{E}$$\mathrm{P}_{0}$$C^{*}(F, \mathcal{E})$$\mathrm{B}$$\mathcal{E}$producing a nonnegative integer
Here is another procedure:
given family of closed disks and a positive integer$\mathrm{Q}_{0}$$\mathrm{Q}_{0}$$\mathrm{Q}_{0}$$q$from a$n<\#\mathcal{E}$$\mathcal{E}$:$(\mathcal{E}, n)arrow q$

.

(12)

works in the following way.

QO1

Let

$D=E_{n}\in \mathcal{E}$

Q02

Let

$S_{*}=S_{*}(D, n, \mathcal{E})$.

Remark. If $n=1$ , thenrenders $q=0$ . Noteand

$S_{*}$$q$

be its cardinality: $q=\neq S_{*}$ .

in Step Q02 is empty, and(See (1)).

$\mathrm{Q}_{0}(\mathcal{E}, n)=\mu_{n}(\mathcal{E})$$\mathrm{Q}_{0}$automatically6$\mathrm{Q}_{0}$naturally induces the procedure$\mathrm{Q}$as follows :$\mathrm{Q}(\mathcal{E})=\max\{\mathrm{Q}_{0}(\mathcal{E}, n), n=1, \cdots, \neq \mathcal{E}\}$

Note is defined for any finite family of closed disks and, it is known(2) . However, for a Besicovitch subfamily$F$$\beta\leq289$.
cardinalityofregardless of thefor some$\mathrm{Q}(\mathcal{E})=\gamma(\mathcal{E})$$\mathcal{E}$$\mathrm{Q}$$F=\mathrm{B}(\mathcal{E})$

$)$

(13)

.

(See

$\mathrm{Q}(\mathcal{F})\leq\beta$

Now we define a procedure , which returns a positive integer $m$ for a:family of closed disks and a positive integer$\mathrm{R}$$n\leq\#\mathcal{E}$$\mathcal{E}$$\mathrm{R}$$\mathrm{R}$:$(\mathcal{E}, n)arrow m$

(14)

.

works in the following way:

Rl

Let

$\ell=\mathrm{Q}(\mathcal{E})+1$.

R2 Let $m=n$ for $n=1,$
R3Let$\ell+1\leq n\leq\neq \mathcal{E}$$\ell$$\cdots,$.

.

be defined for $k<n$ .

R31 Let$\mathrm{R}(\mathcal{E}, k)$

R32 Let$D=E_{n}\in \mathcal{E}$R33 Let$T_{*}=\{1, \cdots, \ell\}\backslash \{\mathrm{R}(\mathcal{E}, k) ; k\in S_{*}\}$

and$S_{*}=S_{*}(D, n, \mathcal{E})$.
.

R34 Let $m= \min T_{*}$ .
Remark. Note Step R3 is actually an iteration procedure. Note also thatthe choice of in Step Rl is to ensure the iteration procedure in Step R3, andin this respect we can replace the definition of by any integer larger than
, we can modify. Thus, for the Besicovitch subfamily
by choosing $\ell=\beta+1$ or, e.g., $\ell=290$ , in Step Rl to get
the procedure, thus bypassing entirely the procedure Q.
$\ell$$\ell$$F=\mathrm{B}(\mathcal{E})$$\mathrm{Q}(\mathcal{E})+1$$\mathrm{R}$$\mathrm{R}(F, n)$

The procedure$\delta$$\delta$defines a mapping$\mathrm{R}$:$\mathcal{E}\ni E_{n}\vdasharrow \mathrm{R}(\mathcal{E}, n)\in\{1, \cdots, \ell\}$

gives a classification of$\mathcal{E}$

.

(15)

by its inverse images$\delta^{-1}(n)=\mathcal{E}_{n}$,$n=1,2,$$\ell$$\cdots,$.

Moreover, the construction, particularly Step R33, ensures that$E’\cap E’’=\emptyset$

(16)

7

for distinct elements $E’,$:certain procedure$E”\in \mathcal{E}_{n},$$n=1,$ $\cdots,$$\ell$. We have thus arrived at a$\mathrm{B}\mathrm{D}$BD :$(\mathcal{E}, n)arrow \mathcal{E}_{n},$$n=1$ ,$\cdot$.. ,$\ell$.

(17)

is given by (16).
Now the procedures and BD realize a Besicovitch covering for a given
finite family of closed disks. In fact,
where each$\mathcal{E}_{n}$$\mathrm{B}$$B$$D=\mathrm{B}(B)$$\{D_{1}$,$\cdot$.. ,and the set of centers$D_{\ell}\}=\{\mathrm{B}\mathrm{D}(D, n), n=1, \cdots, \ell\}$$C=\{\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}(B) ; B\in B\}$$C \subset\bigcup_{D\in D}D$

4

,

,

is covered by disks in$D= \bigcup_{n=1}^{l}D_{n}$$D_{n}$:.

Proofs of algorithms and related discussions

That the validity of the procedure is clear. For a family of closed disks, an
require operations of order. The resultingor
execution of
subfamily
certainly contains the set of the centers of the disks in, which is crucial in designing B.
because of Step P12 in the procedure
See [12] or [5]. When the cardinality of is finite, the combined procedurefinishes in finite steps. However, operations required thento get.
are of order$\mathrm{B}$$\mathcal{E}$$(\neq \mathcal{E})^{2}$$\mathrm{P}_{1}(\mathcal{E})$$\mathrm{P}_{0}(\mathcal{E})$$\mathcal{F}=\mathrm{B}(\mathcal{E})$$\mathcal{E}$$\mathrm{P}_{1}$$\mathrm{B}$$\mathcal{E}$$\mathcal{F}=\mathrm{B}(\mathcal{E})$$(\neq \mathcal{E})^{3}$

andconcern with the numbers of the closed disks
The procedures
which intersect with the given disk. The procedures rely on an enumeration
is simple, but its execution requiresof elements of . The procedure
operations of order.
is quite complicated. The choice of 7 is crucial for asThe procedure, 7}
belongs to {1,suring the feasibility of Step R3. In fact, eachdoes not exceed $\ell-1$ . That is, Steps R33
and the cardinality ofand R34 are executable.
The Besicovitch covering theorem is valid for any family of closed disks(satisfying (3)). If the set of the centers (4) is bounded, it is known thatin a Besicovitch subfamily of . This meansforthat we will not detect disks with radii below the limit of the resolution.
Thus, it is suggested that a certain finiteness is in fact natural. The present
trial will then make quite a sense provided those undetected disks can beevaluated in advance by any mathematical means.
$\mathrm{Q}_{0}$$\mathrm{Q}$$\mathcal{E}$$\mathrm{Q}$$(\#\mathcal{E})^{2}$$\mathrm{R}$$\cdots$$\mathrm{R}(\mathcal{E}, n)$$S_{*}(E_{n}, n, \mathcal{E})$$B$$\lim_{narrow\infty}\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{i}\mathrm{u}\mathrm{s}(D_{n})=0$$C$$D_{n}$$B$

8

Implementations of Besicovitch proceduresin Maple V

5

We represent the family$\mathcal{E}$of closed disks given by (7) as a list $E$ of lists $E[i]$ :$E=[E[1], E[2],$$\cdots,$$E[N]],$$N=\neq \mathcal{E}$,where:$E[i]=[[x_{i}, y_{i}],$ $r_{i}],$$r_{i}=E[i][2]$$[x_{i}, y_{i}]=E[i][1],$.

and the radius .
thus stands for a closed disk with the center
disks,
we appeal to random
In order to generate examples of families of closed
numbers generated by the following procedure (see Managan et al. [6] \S 2.2):$(x_{i}, y_{i})$$E[i]$$>.$ .
$>$

$>$

$>$

$r_{i}$( : : constant. . constant)uniform:
local intrange, :$\mathrm{r}$$=\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{c}$$\mathrm{f}$intrange:$=\mathrm{m}\mathrm{a}\mathrm{p}$($\mathrm{x}->\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}(\mathrm{x}*10^{arrow}\mathrm{D}.\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{t}\mathrm{s})$$\mathrm{f}:=\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}(\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e})$$>$(evalf@eva1

$>$

end:

$(\mathrm{f})$), evalf$(\mathrm{r})$)::$/10^{-}\mathrm{D}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{t}\mathrm{s}$

:

Let
$>$

$>$

$\mathrm{x}:=\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}(-10.0.

$\mathrm{r}:=\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}(0.1.

.2)$.10.0):\mathrm{y}:=\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}(-10.0.

.10.0)$::

Then centers $(x, y)$ and radii chosen randomly in $-10\leq x,$ $y\leq 10$ and$0.1\leq r\leq 2$ . For instance, a family of 1000 closed disks is now generated in
the following way:
$r$

$>$

$\mathrm{E}:=[\mathrm{s}\mathrm{e}\mathrm{q}^{(}[[\mathrm{x} () , \mathrm{y} ()] , \mathrm{r} ()] , \mathrm{i}=1.

.

1000)]$:

Now the following is a Maple V implementation of the procedure$>$$\mathrm{P}\mathrm{O}:=\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{c}$

$>$

local

$>$

$\mathrm{R}:=\max$$>$$\mathrm{C}^{\cdot}1:=\{\}$

$>$

$>$

$>$

$>$

$>$

for
if
$\mathrm{o}\mathrm{d}$$\mathrm{i}$($\mathrm{E}$: : list),CI,C2:(seq$\mathrm{i},\mathrm{R}$$(\mathrm{E}[\mathrm{i}][2]$,$\mathrm{i}=1$$(\mathrm{E})$))::from 1 to nops(E) dothen Cl:$\mathrm{E}[\mathrm{i}][2]>3/4*\mathrm{R}$:C2:
C2 [1] :. . nops$=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{t}$(Cl, list):$=\mathrm{C}1$

union$\{\mathrm{E}[\mathrm{i}]\}\mathrm{f}\mathrm{i}$:$\mathrm{P}_{0}$

:

9

$>$

end:

Here is ajuxtaposed plot of the family
as a bold circle.
and the result of$\mathcal{E}$of closed disks given in the above$\mathrm{P}0(E)$Here is a Maple V implementation of the procedure
$>$

$>$

Pl:
local

$=\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{c}$$\mathrm{i}$($\mathrm{E}$, C,F,r:$>$$>$$\mathrm{r}:=\mathrm{P}0(\mathrm{E})[2]$$>$$\mathrm{c}:=\{\}$$\succ$

$>$

$>$

$>$

$>$

$>$

$>$

for
do
if
then

$\mathrm{i}$.

: : list)$\mathrm{F}:=\mathrm{P}0(\mathrm{E})[1]$$>$$\mathrm{P}_{1}$:::from 1 to nops(E)$(\mathrm{E}[\mathrm{i}][1][1]-\mathrm{F}[1])^{arrow}2+(\mathrm{E}[\mathrm{i}][1][2]-\mathrm{F}[2])^{\wedge}2>\mathrm{r}^{\wedge}2$$\mathrm{f}\mathrm{i}$$\mathrm{o}\mathrm{d}$$\mathrm{C}:=\mathrm{C}$union$\{\mathrm{E}[\mathrm{i}]\}$::

convert ( , list) :end:$\mathrm{C}$The following is the procedure$>$$\mathrm{B}:=\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{c}$

$>$

local

(

$\mathrm{E}$$\mathrm{G},\mathrm{L}$: : list):$\mathrm{B}$in the Maple V way:

10

$>$

$\mathrm{G}:=\{\}$$>$$\mathrm{L}:=\mathrm{E}$

$>$

$>$

:

:

while nops(L) $>0$ do
union {PO}:$(\mathrm{L})$$\mathrm{G}:=\mathrm{G}$$>$$\mathrm{L}:=\mathrm{P}1(\mathrm{L})$$>$$\mathrm{o}\mathrm{d}$

$>$

$>$

:

:

convert ( , list):end:$\mathrm{G}$Here is a Maple V procedure corresponding to a construction of the set.

$S_{*}(D, n, \mathcal{E})$$>$$\mathrm{S}:=\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{c}$

$>$

local

$>$

$\mathrm{s}:=\{\}$

$>$

$>$

for
if

$\mathrm{j}$($\mathrm{D}$$\mathrm{j},$: : list,:: : posint, : : list)$\mathrm{E}$$\mathrm{n}$$\mathrm{s}$:from 1 to n-l do$(\mathrm{E}[\mathrm{j}][1][1]-\mathrm{D}[1][1])^{\wedge}2$$>$$+(\mathrm{E}[\mathrm{j}][1][2]-\mathrm{D}[1][2])^{-}2$$>$$<=(\mathrm{E}[\mathrm{j}][2]+\mathrm{D}[2])^{arrow}2$

$>$

then

$>$

$\mathrm{f}\mathrm{i}$$>$$\mathrm{o}\mathrm{d}$$>$$\mathrm{s}$:::$\mathrm{s}:=\mathrm{s}$union$\{\mathrm{j}\}$

11

$>$

end:

Now the following is a Maple V version of the procedure$>$($\mathrm{Q}\mathrm{O}:=\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{c}$$\mathrm{E}$: : list,$>$

local D, s:$>$$\mathrm{D}:=\mathrm{E}[\mathrm{n}]$$>$$\mathrm{s}:=\mathrm{S}(\mathrm{D},\mathrm{n},\mathrm{E})$nops (s):: : posint)

$>$

end:

:

Thus, the procedure$\mathrm{R}$(see (12)).

:

$>$

(

$\mathrm{Q}:=\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{c}$

$>$

local

$>$

$\mathrm{N}:=\mathrm{n}\mathrm{o}\mathrm{p}\mathrm{s}(\mathrm{E})$$>$$\max$$\mathrm{E}$(see (13)) is implemented as follows:$\mathrm{Q}$: : list)
:
:

$>$

$>$

$\mathrm{n}$$\mathrm{Q}_{0}$$\mathrm{N}$$\mathrm{i},$(seq (QO
end:$(\mathrm{E},$$\mathrm{i})$,$\mathrm{i}=1$.

$.\mathrm{N}$)):Now we are ready for giveing a Maple V implementation of the procedure(see (14)).
$>$$\mathrm{R}:=\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{c}$($\mathrm{E}$: : list, : : posint)$\mathrm{n}$

$>$

local

$>$

$\mathrm{K}:=\mathrm{n}\mathrm{o}\mathrm{p}\mathrm{s}(\mathrm{E})$$>$$\mathrm{L}:=\mathrm{Q}(\mathrm{E})+1$$>$$\mathrm{N}:=\{\mathrm{s}\mathrm{e}\mathrm{q}^{(}\mathrm{i}, \mathrm{i}=1.

$>$

$>$

for
for

$\mathrm{i}$$\mathrm{k}$$\mathrm{i},$$\mathrm{j}$,$\mathrm{k},\mathrm{M}$

, MM,

:

MM:

$>$

$\mathrm{T}:=\mathrm{N}$$>$$\mathrm{m}[\mathrm{k}]:=\min$$\mathrm{o}\mathrm{d}$$\mathrm{m}[\mathrm{i}]$$\mathrm{L}+1$

$>$

$>$

:

from 1 to
do
from
to
do

$\mathrm{M}:=\mathrm{S}(\mathrm{E}[\mathrm{k}],\mathrm{k},\mathrm{E})${.\mathrm{L})\}$$\mathrm{L}$$\mathrm{s}\mathrm{e}\mathrm{q}$

(

:

:

$>$

$=$

$\mathrm{N},\mathrm{T},\mathrm{K},\mathrm{L},\mathrm{m}$:$\mathrm{K}$:$\mathrm{m}[\mathrm{M}[\mathrm{i}]]$minus MM:(seq ($:=\mathrm{i}\mathrm{o}\mathrm{d}$,$\mathrm{i}=1$$\mathrm{T}[\mathrm{j}]$,. . nops$\mathrm{j}=1$$(\mathrm{M})$. . nops)}:$(\mathrm{T})$

)):

:

$>$

$\mathrm{m}[\mathrm{n}]$

$>$

end:

:

Finally, we have the following Maple V version of the procedures BD $(E, n)$(See (17)).
$>$

$>$

$>$

$>$

$\mathrm{B}\mathrm{D}:=\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{c}$

local
if

$\mathrm{N},$($\mathrm{E}$$\mathrm{i},$: : list, : : posint)
:
then
fi :
:
$\mathrm{n}$$\mathrm{j},$$\mathrm{n}>\mathrm{Q}(\mathrm{E})+1$$\mathrm{N}:=\mathrm{n}\mathrm{o}\mathrm{p}\mathrm{s}(\mathrm{E})$$\mathrm{s}$

$[]$

12

$>$

$>$

$>$

$>$

$>$

$>$

$>$

$\mathrm{s}:=\{\}$:

from 1 toforthen
if R(E, i)
union$\mathrm{i}$$\mathrm{N}$do$=\mathrm{n}$$\{\mathrm{E}[\mathrm{i}]\}\mathrm{f}\mathrm{i}$$\mathrm{s}:=\mathrm{s}$$\mathrm{o}\mathrm{d}$::convert ( , list) :end:$\mathrm{s}$Here are plots of$\mathrm{B}\mathrm{D}(\mathrm{B}(E),n)$for $n=1,2$ .

13Finally, we juxtapose these two outputs:

14

References[1] A. S. Besicovitch. A general form of the covering principle and relative
differentiation of additive functions. Proc. Cambridge Phil. Soc., Vol. 41,
pp. 103-110, 1945.
[2] A. S. Besicovitch. A general form of the covering principle and relative
differentiation of additive functions. II. Proc. Cambridge Phil. Soc.,
Vol. 42, pp. 1-10, 1946.

[3] A. P. Calder\’on. Intermediate spaces and interpolation, the complex
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[4] Miguel de Guzm\’an. Differentiation of Integrals in
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