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).
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