Skip to main content

Full text of "Hilbert Flow Spaces with Operators over Topological Graphs"

See other formats


International J.Math. Combin. Vol.4(2017), 19-45 


Hilbert Flow Spaces with Operators over Topological Graphs 


Linfan MAO 


1. Chinese Academy of Mathematics and System Science, Beijing 100190, P.R.China 
2. Academy of Mathematical Combinatorics & Applications (AMCA), Colorado, USA 


E-mail: maolinfan@163.com 


Abstract: A complex system .Y consists m components, maybe inconsistence with m > 2, 
such as those of biological systems or generally, interaction systems and usually, a system 
with contradictions, which implies that there are no a mathematical subfield applicable. 
Then, how can we hold on its global and local behaviors or reality? All of us know that there 
always exists universal connections between things in the world, i.e., a topological graph G 
underlying components in .. We can thereby establish mathematics over graphs di, ©з, ee 
by viewing labeling graphs Gia Gh. +++ to be globally mathematical elements, not only 
game objects or combinatorial structures, which can be applied to characterize dynamic 
behaviors of the system .7 on time t. Formally, a continuity flow Grisa topological graph G 
associated with a mapping L : (v,u) — L(v, и), 2 end-operators Aj, : L(v,u) > L^9. (v, и) 
and Ay, : L(u,v) > ГАЧ (и, v) оп a Banach space Z over а field F with Г(о, и) = —L(u, v) 
and Aj;,(—L(v,u)) = ГА (о, и) for Ү(о, u) € E (c) holding with continuity equations 





У) 14% (vu) = 0), ^ WweV (c) 
u€Nq(v) 
The main purpose of this paper is to extend Banach or Hilbert spaces to Banach or Hilbert 
continuity flow spaces over topological graphs [6s Gs ees ] and establish differentials on 
continuity flows for characterizing their globally change rate. A few well-known results such 
as those of Taylor formula, L'Hospital's rule on limitation are generalized to continuity flows, 
and algebraic or differential flow equations are discussed in this paper. АП of these results 
form the elementary differential theory on continuity flows, which contributes mathematical 
combinatorics and can be used to characterizing the behavior of complex systems, particu- 


larly, the synchronization. 


Key Words: Complex system, Smarandache multispace, continuity flow, Banach space, 


Hilbert space, differential, Taylor formula, L'Hospital's rule, mathematical combinatorics. 


AMS(2010): 34A26, 35A08, 46B25, 92B05, 05C10, 05C21, 34D43, 51D20. 


§1. Introduction 


A Banach or Hilbert space is respectively a linear space & over a field R or C equipped with a 
complete norm ||- || or inner product ( - , - ), i.e., for every Cauchy sequence {£n} in æ, there 


1Received May 5, 2017, Accepted November 6, 2017. 


20 Linfan MAO 


exists an element x in & such that 


Jim ||z, — alle = 0 or Jim (tn — T, En 20) og = 0 
and a topological graph (С) is an embedding of a graph С with vertex set V(G), edge set 
E(G) in a space Z, i.e., there is a 1 — 1 continuous mapping р: С > (С) C Z with 
(р) 5 plq) if p Z q for Vp, q € G, i.e., edges of G only intersect at vertices in .7, an embedding 
of a topological space to another space. A well-known result on embedding of graphs without 
loops and multiple edges in IR" concluded that there always exists am embedding of G that all 
edges are straight segments in IR" for n > З (1221) such as those shown in Fig.1. 





Fig.1 


As we known, the purpose of science is hold on the reality of things in the world. However, 
the reality of a thing 27 is complex and there are no a mathematical subfield applicable unless 
a system maybe with contradictions in general. Is such a contradictory system meaningless 
to human beings? Certain not because all of these contradictions are the result of human 
beings, not the nature of things themselves, particularly on those of contradictory systems in 
mathematics. Thus, holding on the reality of things motivates one to turn contradictory systems 
to compatible one by a combinatorial notion and establish an envelope theory on mathematics, 
i.e., mathematical combinatorics ([9]-[13]). Then, Can we globally characterize the behavior of a 
system or a population with elements> 2, which maybe contradictory or compatible? 'The answer 
is certainly YES by continuity flows, which needs one to establish an envelope mathematical 
theory over topological graphs, i.e., views labeling graphs G^ to be mathematical elements 
([19]), not only a game object or a combinatorial structure with labels in the following sense. 


Definition 1.1 A continuity flow (с: L, A) is an oriented embedded graph © in а topological 
space Z associated with a mapping L : v > L(v), (v,u) > L(v,u), 2 end-operators Aj, : 
L(v,u) > ГА“ (v,u) and Aj, : L(u,v) > LAw (u,v) on a Banach space Z over a field F 


Go) А L(v, u) А}, T 


U u 
Fig.2 


Hilbert Flow Spaces with Differentials over Graphs 21 


+ => 
with L(v,u) = —L(u,v) and At,(—L(v,u)) = —L4e«(v,u) for V(v,u) € E (С) holding with 
continuity equation 

5 Ач (v,u) =L(v) for vv € V (С) 


иЄ Na(v) 


such as those shown for vertex v in Fig.3 following 


L(v, ил) fin) 





ид 
U5 
из Fig.3 ив 
with a continuity equation 
L^ (v, чл) + L^? (v, ua) + L4 (v, из) — L^*(v,u4) — L^*(v, us) — L^: (v, ug) = L(v), 


where L(v) is the surplus flow on vertex v. 
Particularly, if L(v) = t» or constants уо € V (С), the continuity flow (Gin. A) 


is respectively said to be a complex flow or an action A flow, and G-flow if A = ly, where 


Ly = dt, /dt, x, is a variable on vertex v and v is an element in Z for Vu € E (с f 


Clearly, an action flow is an equilibrium state of a continuity flow (G: L, A). We have 
shown that Banach or Hilbert space can be extended over topological graphs ([14],[17]), which 
can be applied to understanding the reality of things in [15]-[16], and we also shown that 
complex flows can be applied to hold on the global stability of biological n-system with n > 3 
in [19]. For further discussing continuity flows, we need conceptions following. 


Definition 1.2 Let 01,25 be Banach spaces over a field F with norms || ||у and ||- |l2, 
respectively. An operator T : 21 — Bo is linear if 


T (Avi + uv2) = AT (vi) + uT (уз) 


for А, п € F, and T is said to be continuous at a vector vo if there always exist such a number 


22 Linfan MAO 


б(=) for Ve > 0 that 
IT (v) - T (vo)lla < e 


if ||v — voll, < б(є) for Vv, vo, v1, V2 € By. 


Definition 1.3 Let 244,445; be Banach spaces over a field F with norms ||- |) and ||- |l2, 
respectively. An operator T : у — Bz is bounded if there is a constant M > 0 such that 


(У) 


ЕМ е. т 


<M 
for Vv € B and furthermore, T is said to be a contractor if 

IT (vi) — T (v3)Il € ellvi — va) 
for Yvı, v2 € Z with c € [0, 1). 


We only discuss the case that all end-operators Аў, А+, are both linear and continuous. 


vu? Uv 


In this case, the result following on linear operators of Banach space is useful. 


Theorem 1.4 Let 21, B2 be Banach spaces over a field F with norms ||| and ||-||2, respectively. 
Then, a linear operator T : у — Be is continuous if and only if it is bounded, or equivalently, 
|Tv) Ile 


IT] := sup ———= < +оо 
ozvez, l|vlh 


Let {Gi, Gs e ) be a graph family. The main purpose of this paper is to extend Ba- 
nach or Hilbert spaces to Banach or Hilbert continuity flow spaces over topological graphs 
Gi, бз, ee ) and establish differentials on continuity flows, which enables one to characterize 
their globally change rate constraint on the combinatorial structure. A few well-known results 
such as those of Taylor formula, L'Hospital's rule on limitation are generalized to continuity 
flows, and algebraic or differential flow equations are discussed in this paper. All of these 
results form the elementary differential theory on continuity flows, which contributes math- 
ematical combinatorics and can be used to characterizing the behavior of complex systems, 
particularly, the synchronization. 

For terminologies and notations not defined in this paper, we follow references [1] for 
mechanics, [4] for functionals and linear operators, [22] for topology, [8] combinatorial geometry, 
[6]-[7],[25] for Smarandache systems, Smarandache geometries and Smaarandache multispaces 
and [2], [20] for biological mathematics. 


82. Banach and Hilbert Flow Spaces 


2.1 Linear Spaces over Graphs 


TL 


> — => => => 
Let Gi, G2,::- , Gn be oriented graphs embedded in topological space Z with Я = |) Gi, 
1 


Hilbert Flow Spaces with Differentials over Graphs 23 


» G: is a subgraph of g for integers 1 < i € n. In this case, these is naturally an embedding 
— 
С. 


25 
t: Gir G. 


ie 


Let 7 be a linear space over a field F. A vector labeling L : G — ¥ is a mapping with 
=> => 


L(v), L(e) € У for vv € V(G),e € E(G). Define 


GH + Gua = (c; N da) U (CNC) 


LictLa 





=> —XL2 
U (G2\ G) (2.1) 
and 
=> => 
GY = Gre (2.2) 
F RE Ali RL d cu И " 
for VA € F. Clearly, if , and G^, Су, G5? are continuity flows with linear end-operators 
+ + R Tila R Lə AL S 
Aj, and Aj, for V(v,u) € E (С), Gi? + Gy? and à- С are continuity flows also. If we 


> >> a => = 
consider each continuity flow G7 a continuity subflow of 4L, where L : С; = L(G;) but 
a ә ә — 

L: € \ Gi — 0 for integers 1 € i € n, and define О: Я — 0, then all continuity flows, 


particularly, all complex flows, or all action flows on oriented graphs Gi, б», ees ‘Ga naturally 
ГА 

form а linear space, denoted by (s. 1<i< n) b, ) over a field F under operations (2.1) 

and (2.2) because it holds with: 


(1) A field .Z of scalars; 
WV 
(2) A set (Gi, 1<{< n) of objects, called continuity flows; 


(3) An operation “+”, called continuity flow addition, which associates with each pair of 


DEC ALY Tib —-— А А С Li E35 А v 
continuity lows С, Gz? ір ( С;,1<71 < п) acontinuity lows Су +G’ in( Gi,l<i<n 





ZL А»; 
called the sum of Су! and Gy’, in such a way that 





(a) Addition is commutative, Тї + G5? = GI? + G^ because of 





epee? S (G6) Wie ues 
ad eiu aea о 

= Gi Gh, 

2 1» 





let 


1,(а), itze Gi \ (GU Ga) 
(к), if хє Gj\(GiU Gx) 
Ly (x), if € G&N (GiU G;) 
Lj.(r)- 4 (2), жє GiG;) \ Ge (2.3) 








24 Linfan MAO 


and 
> > 
Li(a), if xe Gi\ Gj 
+ 2 => > 
(0) = є L;(2), ifrc Су \ С; (2.4) 
=> => 
L,(«) + L; (2), ifrc Gi[1G;j 


for integers 1 € i, j, k € n, then 








+ 
Li 


(6: UJ 63) ^ «Gf =(GUGU z^ 


Gp (GU dy = Gh. (Gf + GS) 


| 


(Gh + Gf) + Gi 








II 


(c) There is a unique continuity flow O on G hold with O(v,u) = О for Y(v, u) € E (4 and 





V (2) in (Gil <i< "у called zero such that GL 40 = GL for Gre (Gil <i< "у; 


=> => y 
(d) For each continuity flow G^ € (G 1<1< п) there is а unique continuity flow 
G- such that GL + б = O; 


“. called scalar multiplication, which associates with each scalar k in F 


(4) An operation 
. . AL . = . У . . AL . 
and a continuity flow G~“ in (Gi, 1<1< п) a continuity flow k- G^ in Y, called the product 


of k with GL, in such a way that 
7 
(а) 1: GL = GL for every GL in (Gil <4< п) З 
(b) (kiko): G^ = (в. G”); 
(c) k- (GIA + 12) =k. Gore. Gh, 
(d) (ky tho) GE = ky GE +k- GP. 


— Ж — Vv 

Usually, we abbreviate (s. 1<i< n) ; +, ) to (Gi, 1<i< n) if these operations 

+ and - are clear in the context. 
Д >L —L AL. e => А => > 

By operation (1.1), С + G3? # Ст if and only if С, A G2 with Lı : G1 \ G3 7 0 and 
AD, OT. Ls. NE => ў > > S ; 
Gi G4 G3? if and only if С» A Gy with Lz: G2\ Gi Æ 0, which allows us to introduce 

=> => => 

the conception of linear irreducible. Generally, a continuity flow family {G 41, G5?,--- , GL} 


is linear irreducible if for any integer i, 
G: РА U [in with Li : G, \ U rae y^ 0, (2.5) 
lxi 154 


where 1 <i < n. We know the following result on linear generated sets. 


. 2L AL AL 
Theorem 2.1 Let V be a linear space over a field F and let Mcr G3. Get be an 


EN 
linear irreducible family, Li : Gi — У for integers 1 € i < n with linear operators Aj, 


А+, for V(v,u) € E (С). Then, се ee I is an independent generated set of 


Hilbert Flow Spaces with Differentials over Graphs 25 


=> У 
(б, 1<i< n) , called basis, i.e., 
y 


ER 
dim (Gi, <i<n) =n. 


T р Й Я К = x А 
Proof Ву definition, G;*,1 < i < m is a linear generated of (Gia <i< n) with 
Li : G, = Y, i.e., 
— Ж 
dim (G;,1 ie n) Zur 


C 
We only need to show that GS 1 < 1 < nis linear independent, i.e., 
— y 
dim (G;,1 <1< n) 2n, 
which implies that if there are n scalars сі, со, · ·· , c, holding with 


=> = => 
е GP фоб +... pe Сі" = О, 








then a ғ C2 ‚+ = Cn = 0. Notice that {G 1, rm А G,)i is linear irreducible. We are easily 
know Gi \ Ш ©, hz 0 and find an element x € HE U Gi) such that c;L;(x) = О for integer 
1524 124 





1,1< 4 < п. However, L;(x) 5 0 by (1.5). We get that с; = 0 for integers 1 < i < n. 











=> У 
А subspace of (Gi, 1l<i< n) is called an Ao-flow space if its elements are all continuity 


flows GL with L(v,v € V (c) are constant v. The result following is an immediately 
conclusion of Theorem 2.1. 





Theorem 2.2 Let С, Сі, Сә, , Gn be oriented graphs embedded in a space S and Ў 
> > => => 
a linear space over a field F. If GY, GY, Gy^,--- , GY” are continuity flows with v(v) = 
=> 
v,vi(v) =v; € Y for Wu € V (С), 1<i<n, then 


(1) (Gv) is an Ag-flow space; 


=> 


у ze | - КЕЕ — 
(2) (Gy (G3 ТО is an Ag-flow space if and only if Gi = Gg = = Gn or 


Vi Уә P Vn 








=> => => => => 
Proof By definition, GY + G3? and АСУ are Ao-flows if and only if Gi = Сі or 
vı = V3 = 0 by definition. We therefore know this result. 














2.2 Commutative Rings over Graphs 


Furthermore, if Y is a commutative ring (2; --,-), we can extend it over oriented graph family 





=> > > 
{Gi, Go,--- , Gn} by introducing operation + with (2.1) and operation - following: 


Gh GP - (848) U (ENE) UAT) ew 





26 Linfan MAO 


where Гу. Lo : x — Li(x): Lo(x), and particularly, the scalar product for R”,n > 2 for 
=> => 
1Є С N Go. 


ЕА 
As we shown in Subsection 2.1, ((&. 1<1< п) +) is an Abelian group. We show 


= 2 
(Gi, 1<1< п) +, : | is а commutative semigroup also. 


In fact, define 


Li(x), ifrc Gi\ С; 
Lj,(z) = 4 Lj(x), ifr c С; \ ©, 





LX LS 
Then, we are easily known that GG = (GU G2) ш (6.063) = Gd 


Ali VAL i 7 A T : ЛИЕВ : 
for ҮСТ", Gy? € (Gi, l<i< n) ;: | by definition (2.6), i.e., it is commutative. 





Let 
Li(z), if zc G; \ (©, Обь) 
Lj(x), ifrc ©, \(Ф®ши дь) 

=> => 

L(a), fre d uU G;) 
Liz (2), if x€ (С, N Gr) wer 
L5x(2), if xe (С, N б) \ ©; 
Liz): L;(z)- Lel) if £e GNG NC 

Then, 

(GP dr) бн = (GUG) ”- c7 =(GUGUGs) ^ 
= Gu. (dg) ^ - db . (GP. Gb) 
Thus, 





(GP. GP).Gh ло о 
31 З AL => р 2 > Р s spo : 
for VG", Gy", Ga? є (Gi, 1<1< п) i], which implies that it is a semigroup. 


We are also need to verify the distributive laws, i.e., 








GS. (GP +67) =Gh Gh сигар (2.7) 


and 





(Gh +). Gr =GP- 0p +07. GP (2.8) 


Hilbert Flow Spaces with Differentials over Graphs 27 





R 
for VG3, G1, G2 € ((G..1<i<n) +). Clearly, 





| 


dy. (GUT) ^ = (d (dL) 0:)) ^ 


(UT) *U(GsU Gs) = Gh. GP «dr d, 


Gh. (db + Gh) 





II 


which is the (2.7). The proof for (2.8) is similar. Thus, we get the following result. 


: : Ali 112 AL г 
Theorem 2.3 Let (2; +, :) be a commutative ring and let {Gi „С5?2,:--, Gi be a linear 
irreducible family, Li : G: — & for integers 1 < i < n with linear operators АЎ,, АЎ, for 


=> => 24 
V(v,u)e E (С). Then, (s. 1<i< n) s+, ) is a commutative ring. 
2.3 Banach or Hilbert Flow Spaces 


HV B AL . => . ФА : . 
Let { Су, G5?,---, Gin} bea basis of (G;,1<i<n) , where Y is a Banach space with a 


=> => У 
norm || - |. For VG" є (д1 <i< n) , define 


[|= Уу; 120. (2.9) 


ec E(G) 





2 
'Then, for VG, ЄТ, Ge € (Gil <1< п) we are easily know that 
=> => => 
(1) [5] > 0 and [5] = 0 if and only if G^ = O; 


(2) 4 =f 15] for any scalar £; 


























(3) en + Ge < er + [бг because of 
[er +6) = X nol 
ec E( Gi\G2) 
+ P, [++ P; bol 
ecE( Gif G2) e€E(G2\G1) 
< У; uno УУ Wal 
ecE( GiNG3) ecE( Gif G2) 








Jer 








+| X kolt YS е | = |68 


= 


e€E(G2\G1) (С.П G2) 


for ||L1(e) + La(e)|| € ||Z£1(e)|| + ||D2(e)|| in Banach space 7. Therefore, || - || is also a norm 


28 Linfan MAO 
=> У 
on (G,1Xi&n) . 
А М х : А г. AL — . V 
Furthermore, if Ў is a Hilbert space with an inner product (-,-), іо УС, G3? € (Gi, 1<1< п) Р 
define 


(db, dP) = 5 (Lı (е), Li(e)) 


e€E(G1\G2) 


+ 5 (Тл (е), Lo(e)) + (La(e), Lo(e)). (2.10) 


ecE(GiN G2) e€B(G2\G1) 


Then we are easily know also that 


>, > 


and a Gr) — 0 if and only if GE = О. 


ГА 
(2) For VG41, б € (Gil <i< n) 


because of 
(dise = (a0), Li (6) + Ua (e), L2(e)) 

e€B(Gi\G2) ec (Gif) G2) 

+ У) (ale), Le) 

e€E(G2\G1) 

- E GoOno+ E BOLO 
e€B(Gi\G2) ec E( Gif) G2) 

+ Уу; 0520150) = (02,01) 
e€E(G2\G1) 


for (Гл (е), Го(е)) = (Го(е), Li(e)) in Hilbert space У. 





y 
(3) For Gr GM GT € (Giisisn) and A, u € F, there is 


(AGP e uGP,G") =a (GP, GU eu (GP. GU 


Hilbert Flow Spaces with Differentials over Graphs 29 


because of 





(GP + GP, GU = (Gi + di^ dr) 
(21) (апа) U (Ge e)" e). 
Define L1,5, : "za UG: — V by 


>.> 
AL, (2), ifízc G4 N Сә 
1лә„(®) = 4 uLa(a), тє Go \ Gi 
ALi (х) + uLa(x), if xe Ca N e 


'Then, we know that 


(лб rudpP,GU = (La, (е), Газ, (0)) 





and 


Notice that 


cc E( (6; U G2) VG) 
= (ALa (е), Ma (е)) + (uL2(e), шо(е)) 
ecz(G,\2) eez( dad) 
+ (Lina, (e), L(e)) 


30 Linfan MAO 


eeE( ci G) ecE( GaN G) 
2: (L(e), L(e)) 
ec E( GG) 
= (L(e), L(e)) + (L(e), L(e)) 
ecE(GNG) ecE(GNG3) 


We therefore know that 
(Adr Tui, Gt) = А (Gb, G^) + и (Gb, Gt) | 
=. 
Thus, G” is an inner space 


2L AL LG y => : У 1 
If (G5, G5?,--- , Су" } is a basis of space б„1<4<п) , we are easily find the exact 


formula on L by Ly.L2,--- , Ln. In fact, let 
GU EQ +MP i, die, 
where (41, 42,::: , An) Æ (0,0,---,0), and let 
^ : — => : 
L: (À Gn) \ U Са | > X An Lr 
1=1 ГИ 1=1 


for integers 1 < i € n. Then, we are easily knowing that L is nothing else but the labeling L 
=> 
on G by operation (2.1), a generation of (2.3) and (2.4) with 


х 


1 ec E(G.) 


25 У; Ne Li оу). (2.12) 
= ec E(G.) s=1 


УА, Lr (е), (2.11) 


1=1 


"| 


| 


[А 














P S 
Ql 
qu 
e 

2 

М7 
i 

[7]: 


1 1=1 


where СЇ” = X Gb 4G +..-+ GE” and G; = (^ 2.) \ U [гй 
l=1 Szk „К 


We therefore extend the Banach or Hilbert space Y over graphs Gi Ga eg Ga following. 


Theorem 2.4 Let G, Go, ee ‘Ga be oriented graphs embedded in a space F and Ў a Banach 
V 
space over a field F. Then (Gi, 1<1< п) with linear operators Аў, Ai, for V(v,u) € 


vu? 


y 
E (с) is a Banach space, and furthermore, if V is a Hilbert space, (Gi, 1<i< n) iS a 
Hilbert space too. 


Hilbert Flow Spaces with Differentials over Graphs 31 


y 

5 

Proof We have shown, (Gi, 1<i< n) is a linear normed space or inner space if V is a 
linear normed space or inner space, and for the later, let 


е e 


=> — 4 У m j y * a4. 8 
for Gh E(G,1<i< n) . Then (Gy 1<1< п) is а normed space and furthermore, it is 
a Hilbert space if it is complete. Thus, we are only need to show that any Cauchy sequence is 

=> 
converges in (Gi, 1<1< п) 
FL . A . á . 
In fact, let pss be а Cauchy sequence in (Gi, 1<1< п) ‚ i.e., for any number e > 0, 
there always exists an integer №(=) such that 


=> => 
| 2 = Н: 2g 


3 


= => => 
if n,m > N(e). Let €" be the continuity flow space on 4 = |) G;. We embed each HTI^ to 
i=1 


а di € d by letting 


Then 


Gin Gem] = Уу Wo — Male) - Em (Ol 


e€E(Gn\Gm) ec (Gs Gm) 
+ MX 100 = || BE - Er <e 


еєв(@„\@„) 
= = 
Thus, {9 ie is a Cauchy sequence also in 4 ". By definition, 
[o-n e [Po] 


i.e., {Ln(e)} is a Cauchy sequence for Ve € E (4). which is converges on in 7 by definition. 


Let 
L(e) = lim L,(e) 


=> =F = = 
for Ve € E (v). Then it is clear that lim 4L" = 42, which implies that (4 7"), i.e., 


& K S 
EP is converges to dic d. an element in (Gi, 1<i< n) because of L(e) € Y for 

















vee Е(@) ma = 0 G 
i=1 


i— 


32 Linfan MAO 


83. Differential on Continuity Flows 


3.1 Continuity Flow Expansion 


'Theorem 2.4 enables one to establish differentials and generalizes results in classical calculus in 
2 

врасе (Gi, 1<1<Я n) . Let L be kth differentiable to t on a domain 2 C R, where k > 1. 

Define 





'Then, we are easily to generalize Taylor formula in (Gs 1<i< n) following. 


КЕ EAS RxR” 
Theorem 3.1(Taylor) Let G^ € (Gi, 1<i< [A and there exist kth order derivative of 


L to t on a domain 9 C R, where k > 1. If AL, At, are linear for V(v,u) € E (С) ‚ then 
t-t ‚ —t 
Оте 4g biles ro(ü-8)* 8), €D 


for Vto € 2, where o (« — ict G) denotes such am infinitesimal term L of L that 


lim к Ш 


о р) — ж; (©). 


Particularly, if L(v,u) = f(t)cuu, where cy, is a constant, denoted by f(t)GLc with Lo : 
(v, ш) > cw for (о, и) € E (С) and 


Ft) = f) + E19) p (to) + 0 pag) usa CL 5 (06) do (a Li), 


then 


Proof Notice that L(v,u) has kth order derivative to t on 9 for V(v,u) € E (С). Ву 
applying Taylor formula on to, we know that 
L'(v, u)(to) Lu) (to) 


T @ — to) +: -+ =—— + o ((t — to)") 


L(v,u) = L(v,u)(to) + k! 


if t — to, where o ((t — to)*) is an infinitesimal term L(v,u) of L(v,u) hold with 


un ce 
t—to (t ан to)’ 


=0 


Hilbert Flow Spaces with Differentials over Graphs 33 


for Vv,u) € E (c). By operations (2.1) and (2.2), 





E 


Gh diac da and 1G" Ф? 


because AF 


vu? 


At. are linear for V(v,u) € E G). We therefore get 
uv g 


GL = бию) ц (t — to) GL (to) (t — to)” GLO (to) kG 
= p жыр e +o ((t— to) 


for to € 2, where o (« — to)” G) is an infinitesimal term L of L, i.e., 


L(v,u) 
im 7 
t—to (t = to) 





for Vv,u) e E (С). Calculation also shows that 


GI WLe wu) a(t Sie Fo) EP F (to) +0((t—t0)*) ) ev 


= Jie + P780, еу 


f (to) (t — to)" 
PSI 


фе 


Je cou G +0 ((t — to)") G 


(t= to) pi y. E509 gto у a.d 
f' (to) + + SP f (to) +o ((— t0)*) | eG 











This completes the proof. 





ER 
Taylor expansion formula for continuity flow G enables one to find interesting results on 

zy 

GŁ such as those of the following. 


Theorem 3.2 Let f(t) be ak differentiable function to t on a domain 9 C R with 0 € 2 and 
> => => 
f(0G) = f(0)G. If At, At, are linear for V(v,u) € E (6). then 


vau? 


f(0G-f (С) | (3.2) 


Proof Let tọ = 0 in the Taylor formula. We know that 


FO, FO) 5 £™ (0) 
f(t) = pL ERI РЕ e 


34 Linfan MAO 


Notice that 








Kod = (ло) sp FAO IL EDS S iue PO +o e) © 








1! 2! 
_ drop Pepe LO о) 
'(0)t (k) (O)e* 
= коб + ££ duo EOG owe 


and by definition, 

















£(8) = 1(06) +4? 02) +5? iy 
++ GR) +0((@)') 
f (ос) + LONG + ro Pd. PO wg - o (t^) d 





because of Gy = Gt = t! G for any integer 1 < i < k. Notice that f(0G) = f(0)G. We 
therefore get that 











096 =f (С) ; 





Theorem 3.2 enables one easily getting Taylor expansion formulas by f (rc) . For example, 
let f(t) = e*. Then 
— im 
et G = ete (3.3) 


by Theorem 3.5. Notice that (e*)' = et and е? = 1. We know that 











а t p 13 
tG _ ota = po 277 L ПЕТ аы Era 
€ =eG=G 1 1! G T 21 G T kl G 1 (3.4) 
and 
ete . ese 25 Ge | Ge =: Gee NS Get 22, pure (3.5) 


where ¢ and s are variables, and similarly, for a real number a if |t| < 1, 





a роби 1)#" 
(d id) 0.910 +... Mod Orns DP gu (3.6) 


n! 


3.2 Limitation 


=; SL => , ЖЕ. . 
Definition 3.3 Let GF, Gy є (Gil <i< n) with L, Lı dependent om a variable t € 
[a,b] C (~œ, +оо) and linear continuous end-operators АЎ, for V(v,u) € E G). For to € 
[a,b] and any number € > 0, if there is always a number (=) such that if |t — to| < d(e) 
p AL Aly . d AL 
then |Су — G | < =, then, Сү? is said to be converged to G^ as t — to, denoted by 








2 AL А GRL. m . — 
lim Сү = С”. Particularly, if G^ is a continuity flow with a constant L(v) for Vu € V (c) 


t—to "P | = | 
and tg = +оо, Gy is said to be G -synchronized. 


Hilbert Flow Spaces with Differentials over Graphs 35 


Applying Theorem 1.4, we know that there are positive constants cy, € R such that 
=> 
ПАХ, < ef, for Y(v, u) € E (С). 


By definition, it is clear that 
үке 


-|(e ^ 








eno" 








(61) 



































1+ n + + 
= 5 Le vu (v, u)|| + 5 (2 vu — i (v,u)! + 5 |-24% (о, а) 
чЄМ№су\с (о) чЄ№суп Gav) ueNa@\a,(v) 
< 5 с 112 (®, u)]| HS 5 ciu ll (Lı = L) (v,u)|| + »9 Cull = L(v, u)||. 
uENG yaQ) ue NG, nel) u€ Naya, (v) 


and |L(v,u)|| > 0 for (v, u) € E (С) апа Е (1). Let 


max + a 
ces = max с, max Cp. 
GiG Lamm (v,u)e E(G1) } 
ALi AL : max R R 
If |6: -G | < e, we easily get that ||Li(v,u)| < c@®Ge for (v,u) € E (GN а), 


lla — 2) (о, ш) < се for (о, ш) € B(GiNG) and || — L(v,u)| < се for (v,u) € 
> — 
B(G\ Gi). 


Conversely, if ||L1(v,u)|| < € for (v,u) € E (G1 \ ©), lai — L)(v, ч) < € for (v,u) € 
E (c; NG) and || — L(v,u)| < = for (v,u) € E (с \ б), we easily know that 
„+ 1+ 
[еге = X meje у |) oa 
u€ Naiva(v) u€Na,na(v) 
+ 5 |-2^- (а) 
u€ Nava, (v) 
< chew У 10-2) (0) 
иЄЇЇсү\с(%®) u€ Nai nael) 
E 5 Chull m L(v,u)| 
u€ Nava, (v) 
=> => => => >. > => => 
« [dix С | ense + an Gi стах + IG \ Gi стах е = aU G| cese, 








: Cs so umb cs FAL : 
'Thus, we get an equivalent condition for Jim Ст = G~“ following. 
— to 


. ^L AL. . 7 
Theorem 3.4 Шш Ср = G~ if and only if for any number = > 0 there is always a number д (=) 
—t0 
such that if |t — tol < б(є) then ||Li(v,u)|| < є for (v,u) € E (c; \ G), Ia- L)(v, ч) < e 
for (vu) € E (inc) and || — L(v,u)|| < е for (v,u) € E (б\д), Gh — GF is an 
infinitesimal or lim (Gb — Gt) = О. 
i—1to 


36 Linfan MAO 


— — 
If lim GL ; im G," and m G2"? exist, the formulas following are clearly true by defi- 
tto — to 











t—to 
nition: 
= = 
lim (c; x Gy!) = = lim Gib + lim Go", 
—>to 0 
= = 
lim (c; Gs) = = lim Gi. lim Go", 
1—10 t—to 1-10 
=> = => = 
lim ( L. (aie + бз!) = lim G+ lim G1% + lim С^. lim С»*?, 
1—10 1—10 t—to t—to —to 
— => = => 
lim (Gs + бә?) . G") -im-G . lim Co + lim G3”? . lim G} 
1—10 1—10 1-10 1-10 1—10 


. . FALE: 
and furthermore, if Jim G2”? Æ O, then 
—t0 





Theorem 3.5(L'Hospital's rule) If lim Gi =O, jim Gal = O and Lı, Lo are differentiable 
ЖА. | — to 
respect to t with Jim Li(v,u) = 0 for (v,u) € E (Gi G2), Jim L5(v,u) # 0 for (v,u) € 
— — E => > M5 
E (Gin Са) and Jim L5(v,u) = 0 for (v,u) € E (G2\ Gi), then, 
—to 


=? lim G1: 
im 1 
j Gh 2202210 1 
lim | = mdi 
кою V Gola lim Got 2 


1—10 





Proof By definition, we know that 








ma (ZE) - m (em) 
- iy (8^ (608) (ла) 
- fy GG)" = m (Gies) 


lim, L'i 


saN = (à, па)" 


РЕТ РА lim ZL’; «lim L';! уу | —, X lim L5 
= (G:\ G2)" (С. NG- y to toto (G2\ Gyr" 


li gn 

А 1 ; /—1 101 

= iim, L = jim, Е 2 —to 1 

= di 20 = 2a 
lim Gi 


t—to 











This completes the proof. 





Hilbert Flow Spaces with Differentials over Graphs 37 


Corollary 3.6 If Jm Gl — O, lim GL = O and Li, L2 are differentiable respect to t with 
—t0 


? >to 


Е, 
Jim L5(v,u) £0 for (vu) € E (С), then 
—t0 





= lim G4 
im 1 
li G Lı 1—10 
im 
t—to С 


^ lim GU 
Generally, by Taylor formula 


ito 


n asper Bec (to) (t — to)" SLO (to) -k R 
С== С + G + 2—6 +o((t—to) С}, 


1! k! 

if _ Tl ar ee (k—1) e d zc ENS (k—1) = b 
H Li(to) = Гл (to) = = Li (to) = 0 an La(to) = Li (to) = = Ls (to) = 0 but 
L$” (to) # 0, then 

ZL (t— fo)" 19 (to) -k R 

1 = —H C1 +о(@-%) Gi). 
k 
t-t () E 
бе = ЧЮ) 0) Ge 09) 40 ((t= to) * Ga). 


We are easily know the following result. 








Theorem 3.7 If lim Gi =O, lim G2 = O and Dili) xL (to) = oe LE? (to) = 0 
—to —to 
and La(to) = Lh(to) =--- = LET” (ty) = 0 but LY (to) #0, then 
. RLP (to) 
lim eh = e NN a 
ito GL uw die 


t—to 


=> => => 
Example 3.8 Let Gi = С = Cn, Аў, = 1, Аў, = 2 and 
h- fi + (2-1 — 1) F(z) " n! 
m 2i-1 (2n + 1)et 
for integers 1 <i € n in Fig.4. 
U1 fi U2 
fr fa 
fi fs 
Un Vi41 Ui U3 


Fig.4 


38 Linfan MAO 


Calculation shows That 


ht (E = NRE) А+ (21-1) Fa) 


L(vi) = 2fii-fi-2x 9i 9i-1 


n! 


S AEE (2n + let 


Calculation shows that jim L(v;) = F(T), i.e., jim CL = CL. where, L(v;) = F(T) for 
integers 1 € i € n, i.e., CL is G-synchronized. 
§4. Continuity Flow Equations 


A continuity flow GL is in fact an operator L : G — 4 determined by L(v,u) € Z for 
=> 
V(u,u) є E (с). Generally, let 


Li Li = Lin 

Ш _ | La La +e Lan 
mxn — 

Limi Lm2 ыс Lynn 


with Lij : G — Ф іо1<1< т,1 < j< п, called operator matrix. Particularly, if for integers 
1xizm,lzjzEn,Li;: G — К, we can also determine its rank as the usual, labeled the 
edge (v, и) by Rank|L]mxn for V(v,u) € E (c) and get a labeled graph Сак). Then we 
get a result following. 








Theorem 4.1 A linear continuity flow equations 








zı Giu 1. a3 G Di2 esc tm GL = Gl 
ту Сї AR a3 G L22 ork т G 2 = Giz 
(4.1) 
туб! + a Gin фф £n С" = Gin 
is solvable if and only if 
GRank(L] _ GRank(Z] (4.2) 
where 
Ly, La ++: Lin Lair Гә c: Lin da 
y= Гәр Lo +++ Lo "m Z] = Loi Log +++ Lan Le 


Lui Lng Es Linn Lui Lng "PT Linn Ln 


Hilbert Flow Spaces with Differentials over Graphs 39 


Proof Clearly, if (4.1) is solvable, then for V(v,u) € E (С), the linear equations 





z1Lii(v, и) + zaLis(v, u) +--+ 2, Го (0,0 = Li (v, и) 














ay Loi(v, u) + za Las(v, и) +--+ 2. Loi1(v, u0 = La(v, и) 


d1Lgai(v, и) + xaLas(v, u) -- + 2. Lss(v,u0 = Г.(о, u) 


is solvable. By linear algebra, there must be 


Li(v,u) Lye(v,u) +++ Lin(v,u) 
Rank Loi(v,u) Гәә(о, и) +++ Lon(v,u) _ 
Lai(v,u) Lne(v,u) +++ Г.(о, и) 
Lii(v,u) Lye(v,u) +++ Li&(v,u) Li(v,u) 
Loi(v,u) Loo(v,u) +++ Lon(v,u) Le(v,u) 
Rank | 
Lai(v,u) Lne(v,u) +++ Lnnv, u) Г.(о, и) 


which implies that 
Q Rank[L] = GG Rank[Z]. 


Conversely, if the (4.2) is hold, then for V(v,u) € E (С), the linear equations 


z1Lii(v, и) + zaLis(v, и) +++ 2, Lai(v,u0 = Li (v, и) 








zı Loi (v, u) + хә 1,әо(®, и) +. + Zn Loi(v, u0 = La(v, u) 


d1Lgai(v, и) + zxaLas(v, u) +--+ 2, Las(v, u0 = Г.(о, u) 











is solvable, i.e., the equations (4.1) is solvable. 





'Theorem 4.2 A continuity flow equation 


Gh ане Ен а. ОТВ О (4.3) 
— — 
always has solutions Сх with Ly : (v,u) € E (c) — (AT, А5", ARMS, where M", 1 <1< s 
are roots of the equation 
o2  A* + ө?“ M71 +... aD" =0 (4.4) 


with Li : (v,u) — o^, a?” £0 a constant for (v,u) € E (c) and lxi s. 


For (v,u) € E (С), if п“ is the maximum number i with L;(v,u) Æ 0, then there are 


40 Linfan MAO 


II т“ solutions Gh, and particularly, if Г.(о, и) 2 0 for V(v,u) € E (С), there are 
(v,u yeg(G G) 
sl®()| 


solutions GE of equation (4.3). 


Proof By the fundamental theorem of algebra, we know there are s roots AY“, A9", ... , A?" 
=> 
for the equation (4.3). Whence, Là С is a solution of equation (4.2) because of 


—\ sS NÉ —\s-1 AT —\ 0 AL 
(3) Ge « (ad) d 4-4 Gd) d» 
= х1. GNU La p... GOP Lo — GN Lat T Locas Lo 
and 
ARD КЭА he А Ез ДЕД; (v,u) — aA? + a A97 ү T = 


for Vv,u) € E (С), i.e., 





(xa) JB (x3) de uuu (х9). 99-06 =O. 


Count the number of different Ly for (о, и) € E G). It is nothing else but just n”” 
Therefore, the number of solutions of equation (4.3) is II nu. 
(v,u)eE(G) 














Theorem 4.3 A continuity flow equation 





dG 
Fila , QL 
= С. 4.5 
2 G- G (4.5) 
=> => 
with initial values С^ = Св always has a solution 
t=0 
e 


di c dro. (0) 
where La : (v, и) > avu, Lg : (v, u) > Bou are constants for V(v,u) € E (С). 
Proof А calculation shows that 


(GL 
_ dG Glo. GL = Qet 


d 


S 
G 





II 


which implies that 
— = а, (4.6) 


for Vv,u) € E (С). 


Solving equation (4.6) enables one knowing that L(v,u) = @„е'®”“ for V(v,u) € E (С). 


Hilbert Flow Spaces with Differentials over Graphs 41 


Whence, the solution of (4.5) is 


=> 


GE - duet _ Сів. (d) 


and conversely, by Theorem 3.2, 





R гера etLa 
dG “se = rius ЖОГ 
dt 
= Gla. Glser 
i.e., 
ZL 
dG = Gila аг 
dt 


=> 


if G4 = Сів. (6-0). This completes the proof. 














Theorem 4.3 can be generalized to the case of L : (v,u) > К", п> 2 for Vv,u) € E (©). 


Theorem 4.4 A complex flow equation 





dG 
=> => 

—— = Gre. G} 4.7 
di (4.7) 

Я . Der m» = . 
with initial values С^ = С 8 always has a solution 

t=0 
Gt = Glo. (ed), 

where La : (v,u) > (alu, o2,,--- u) La : wu) > (Blua Bus Bu) with constants 


; ; => 
oi, Bu 1<i<n for V(v,u) € E (С). 


Theorem 4.5 A complex flow equation 


nA L 
Filan. d"G AL 


асі 
an—1. 


AL AL 
" ao. Е 4. 
"т + dni +G G O (4.8) 





+ 


with La, : (v,u) — a” constants for V(v,u) € E (С) and integers 0 € i < n always has a 


general solution G^ with 


8 
Ly:(v,u) > | 0, S nte | 
i=1 
for(v,u)€ E (с) , where hm, (t) is a polynomial of degree< m;—1 on t, mi-- ma: -+M =n 
and Aq", А", +++ , A" are the distinct roots of characteristic equation 


o A^ + ad ҮА" E... Боа“ = 0 


42 Linfan MAO 


with о?“ £0 for (v,u) € E (С). 


Proof Clearly, the equation (4.8) on an edge (v,u) € E (c) is 


vu d" L(v, u) 
ý dt” 


ш. PAL (yu) 
Ont apd o Fo 00 =O. (4.9) 


As usual, assuming the solution of (4.6) has the form GL = eG. Calculation shows that 





XC = ЛС = AG, 
E. 
2с E eG Е XG, 
=> 
а" L 
< = ем = АС 


Substituting these calculation results into (4.8), we get that 
(v dre pAn IG ena p. Geo) G^ =0, 


i.e., 
(уп т 1 
GO -Lan tr -La 143+): О 


, 


= 
which implies that for V(v,u) € E (С), 


А? ой p АТ“ 4... tay — 0 (4.10) 
Or 
L(v,u) — 0. 
Let АТ“, A2", --- , A?" be the distinct roots with respective multiplicities mi", m5", +--+ , m$?" 


of equation (4.8). We know the general solution of (4.9) is 
L(v,u) — 5 hi(t)e™"t 
i=1 


with hm; (t) a polynomial of degree< m; — 1 on t by the theory of ordinary differential equations. 
Therefore, the general solution of (4.8) is GL» with 


Ly: (v, u) ^ | 0, Уһ (ем"* | 














for (v,u) € E (С). 


Hilbert Flow Spaces with Differentials over Graphs 43 


85. Complex Flow with Continuity Flows 


'The difference of a complex flow GL with that of a continuity flow GL is the labeling L on a 


vertex is L(v) = ty or Ly. Notice that 


d + d + 
тт 5 L^» (v, u) | = 5 m 


ue Na(v) u€Nao(v) 
=> => => 
for w € V (c) . There must be relations between complex flows G^ and continuity flows С2. 
We get a general result following. 


t t 
Theorem 5.1 If end-operators Аў, Aj, are linear with / As = / As — 0 and 
0 0 


dt vu 
me RxR” Ay 
(Gi, 1<i< n) is a continuity flow with a constant L(v) for Vu € V (С) if апа only if 


E 4 = Е л = 0 for V(v,u) € E (c). and particularly, АЎ, = ly, then GL. c 


t 
J Саі is such a continuity flow with a constant one each verter v, v € V (С). 
0 


t 
Proof Notice that if АЎ, = 1y, there always is Jj As — 0 and E Ai. — 0, and by 
0 


t t t 

A =0 = y oA, = Áu o f , 
0 0 0 
d 


Has Т Д1 | 


FEES =0 < dt vu 9 dt 


definition, we know that 








=> => 
If GF is a continuity flow with a constant L(v) for Wv € V (С), i.e., 
5 ГА“ (v,u) =v for Vv cV (c) , 
u€Nao(v) 
we are easily know that 


|: | L^ 2 dt = 5 (f At.) L(v,u)dt = 


u€Nqa(v) 


= 5 at ( [ tooa) = f var 


uENg(v) 


(4%. о [) L(v, u)dt 


u€Nqa(v) u€Nq(v) 


t t 
for Ww € V (С) with a constant vector / vdt, i.e., J Са is a continuity flow with a 
0 0 


x 
constant flow on each vertex v, v € V (С). 


t 
NS 
Conversely, if / Саі is a continuity flow with a constant flow on each vertex v, v € 
0 


44 Linfan MAO 


5 
V(G), 1.е., 
©) | | 
У) Ano f L(v,u)dt=v for меу (c), 
u€Nao(v) 0 
then j 
_ «(f ба) 
Т 0 
SUE dt 


"ES 
is such a continuity flow with a constant flow on vertices in G because of 


( x Zo) 


u€Nao(v) ЕТ а 3 d 
x^ = X foshe "i foad 
u€Nao(v) 
d 5 dv 
ET E = Aja = 
= 5 Alu о Zef L(v, u)dt = 5 L(v,u)^v» = = 
u€Nga(v) ue Na(v) 














dv 
with a constant flow ph on vertex v, v c V (С). 'This completes the proof. 


t 
If all end-operators А, and АЎ, are constant for V(v, u) € E (С) ‚ the conditions / ; at 
0 


t 
d 
/ As — 0 and E Ai. = E A = 0 are clearly true. We immediately get а conclu- 
0 
sion by Theorem 5.1 following. 


Corollary 5.2 For V(v,u) € E (С), if end-operators АЎ, and АЎ, are constant c, Cuv for 
=> => => RxR” 
V(v,u)e E (С), then G^ € (Gil <4< п) is а continuity flow with a constant L(v) 
t 
for vv € V (С) if апа only it | Саі is such a continuity flow with a constant flow on each 
0 


za 
vertex v, v € V (с). 


References 


1] R.Abraham and J.E.Marsden, Foundation of Mechanics (2nd edition), Addison-Wesley, 
Reading, Mass, 1978. 

2| Fred Brauer and Carlos Castillo-Chaver, Mathematical Models in Population Biology and 
Epidemiology(2nd Edition), Springer, 2012. 

3| G.R.Chen, X.F.Wang and X.Li, Introduction to Complex Networks — Models, Structures 
and Dynamics (2 Edition), Higher Education Press, Beijing, 2015. 

4] John B.Conway, A Course in Functional Analysis, Springer-Verlag New York,Inc., 1990. 
5] Y.Lou, Some reaction diffusion models in spatial ecology (in Chinese), Sci.Sin. Math., 
Vol.45(2015), 1619-1634. 


6] Linfan Mao, Automorphism Groups of Maps, Surfaces and Smarandache Geometries, The 








11 


12 


13 


14 


15 


16 


17 


18 


19 


20 


21 





[22 








Hilbert Flow Spaces with Differentials over Graphs 45 


Education Publisher Inc., USA, 2011. 

Linfan Mao, Smarandache Multi-Space Theory, The Education Publisher Inc., USA, 2011. 
Linfan Mao, Combinatorial Geometry with Applications to Field Theory, The Education 
Publisher Inc., USA, 2011. 

Linfan Mao, Global stability of non-solvable ordinary differential equations with applica- 
tions, International J.Math. Combin., Vol.1 (2013), 1-37. 

Linfan Mao, Non-solvable equation systems with graphs embedded in R”, Proceedings of 





the First International Conference on Smarandache Multispace and Multistructure, The 
Education Publisher Inc. July, 2013. 

Linfan Mao, Geometry on G4-systems of homogenous polynomials, International J. Contemp. 
Math. Sciences, Vol.9 (2014), No.6, 287-308. 

Linfan Mao, Mathematics on non-mathematics - A combinatorial contribution, Interna- 
tional J.Math. Combin., Vol.3(2014), 1-34. 

Linfan Mao, Cauchy problem on non-solvable system of first order partial differential equa- 
tions with applications, Methods and Applications of Analysis, Vol.22, 2(2015), 171-200. 
Linfan Mao, Extended Banach G-flow spaces on differential equations with applications, 
Electronic J.Mathematical Analysis and Applications, Vol.3, No.2 (2015), 59-91. 

Linfan Mao, A new understanding of particles by G-flow interpretation of differential 
equation, Progress in Physics, Vol.11(2015), 193-201. 

Linfan Mao, А review on natural reality with physical equation, Progress in Physics, 
Vol.11(2015), 276-282. 

Linfan Mao, Mathematics with natural reality-action flows, Bull. Cal. Math.Soc., Vol.107, 
6(2015), 443-474. 

Linfan Mao, Labeled graph — A mathematical element, International J.Math. Combin., 
Vol.3(2016), 27-56. 

Linfan Mao, Biological n-system with global stability, Bull. Cal. Math. Soc., Vol.108, 6(2016), 
403-430. 

J.D.Murray, Mathematical Biology I: An Introduction (3rd Edition), Springer-Verlag Berlin 
Heidelberg, 2002. 

F.Smarandache, Paradoxist Geometry, State Archives from Valcea, Rm. Valcea, Romania, 
1969, and in Paradozist Mathematics, Collected Papers (Vol. II), Kishinev University 
Press, Kishinev, 5-28, 1997. 

J.Stillwell, Classical Topology and Combinatorial Group Theory, Springer-Verlag, New 
York, 1980.