Hilbert Flow Spaces with Operators over Topological Graphs

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 S 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 S . We can thereby establish mathematics over graphs G 1 , G 2 , · · · → − 1 → − L2 by viewing labeling graphs G L 1 , G 2 , · · · to be globally mathematical elements, not only game objects or combinatorial structures, which can be applied to characterize dynamic → − → − behaviors of the system S on time t. Formally, a continuity flow G L is a topological graph G + Avu associated with a mapping L : (v, u) → L(v, u), 2 end-operators A+ (v, u) vu : L(v, u) → L + A+ and Auv : L(u, v) → L uv (u, v) on a Banach space B over a field F with L(v, u) = −L(u, v) → − A+ vu (v, u) for ∀(v, u) ∈ E (−L(v, u)) = −L G holding with continuity equations and A+ vu X

+

LAvu (v, u) = L(v),

u∈NG (v)

∀v ∈ V

→ − G .

The main purpose of this paper is to extend Banach or Hilbert n→ o spaces to Banach or Hilbert − → − continuity flow spaces over topological graphs G 1 , G 2 , · · · 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. All 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, particularly, 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 A over a field R or C equipped with a complete norm k · k or inner product h · , · i, i.e., for every Cauchy sequence {xn } in A , there 1 Received

May 5, 2017, Accepted November 6, 2017.

20

Linfan MAO

exists an element x in A such that lim kxn − xkA = 0

n→∞

or

lim hxn − x, xn − xiA = 0

n→∞

and a topological graph ϕ(G) is an embedding of a graph G with vertex set V (G), edge set E(G) in a space S , i.e., there is a 1 − 1 continuous mapping ϕ : G → ϕ(G) ⊂ S with ϕ(p) 6= ϕ(q) if p 6= q for ∀p, q ∈ G, i.e., edges of G only intersect at vertices in S , an embedding of a topological space to another space. A well-known result on embedding of graphs without loops and multiple edges in Rn concluded that there always exists an embedding of G that all edges are straight segments in Rn for n ≥ 3 ([22]) 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 T 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 GL 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

−

→ − → G ; L, A is an oriented embedded graph G in a topological

space S associated with a mapping L : v → L(v), (v, u) → L(v, u), 2 end-operators A+ vu : + A+ uv (u, v) on a Banach space B over a field F L(v, u) → LAvu (v, u) and A+ : L(u, v) → L uv A+ vu L(v)

v

-

L(v, u)

Fig.2

A+ uv

L(u)

u

21

Hilbert Flow Spaces with Differentials over Graphs

− → A+ vu (v, u) for ∀(v, u) ∈ E with L(v, u) = −L(u, v) and A+ (−L(v, u)) = −L G holding with vu continuity equation − X + → LAvu (v, u) = L(v) f or ∀v ∈ V G u∈NG (v)

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

L(u1 )

-

-

L(v, u4 )

L(u1 , v)

u1

u4 A1

L(u2 )

-

L(u2 , v)

u2

A4

A2

A5

L(v)

A3

v

- L(v, u ) 5

A6

L(u5 )

u5

-

-

L(v, u6 )

L(u3 , v) L(u3 )

L(u4 )

u3

Fig.3

L(u6 )

u6

with a continuity equation LA1 (v, u1 ) + LA2 (v, u2 ) + LA3 (v, u3 ) − LA4 (v, u4 ) − LA5 (v, u5 ) − LA6 (v, u6 ) = L(v), where L(v) is the surplus flow on vertex v.

− −

→ → G , the continuity flow G ; L, A − → is respectively said to be a complex flow or an action A flow, and G -flow if A = 1V , where − → ẋv = dxv /dt, xv is a variable on vertex v and v is an element in B for ∀v ∈ E G .

Particularly, if L(v) = ẋv or constants vv , v ∈ V

−

→ 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 B1 , B2 be Banach spaces over a field F with norms k · k1 and k · k2 , respectively. An operator T : B1 → B2 is linear if T (λv1 + µv2 ) = λT (v1 ) + µT (v2 )

for λ, µ ∈ F, and T is said to be continuous at a vector v0 if there always exist such a number

22

Linfan MAO

δ(ε) for ∀ǫ > 0 that

kT (v) − T (v0 )k2 < ε

if kv − v0 k1 < δ(ε) for ∀v, v0 , v1 , v2 ∈ B1 .

Definition 1.3 Let B1 , B2 be Banach spaces over a field F with norms k · k1 and k · k2 , respectively. An operator T : B1 → B2 is bounded if there is a constant M > 0 such that kT (v)k2 ≤ M kvk1 ,

i.e.,

kT(v)k2 ≤M kvk1

for ∀v ∈ B and furthermore, T is said to be a contractor if kT (v1 ) − T (v2 )k ≤ c kv1 − v2 )k for ∀v1 , v2 ∈ B with c ∈ [0, 1). + We only discuss the case that all end-operators A+ vu , Auv are both linear and continuous.

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

Theorem 1.4 Let B1 , B2 be Banach spaces over a field F with norms k·k1 and k·k2 , respectively.

Then, a linear operator T : B1 → B2 is continuous if and only if it is bounded, or equivalently, kTk :=

sup 06=v∈B1

kT(v)k2 < +∞. kvk1

n− o → → − Let G 1 , G 2 , · · · be a graph family. The main purpose of this paper is to extend Banach or Hilbert n o spaces to Banach or Hilbert continuity flow spaces over topological graphs − → − → G 1 , G 2 , · · · 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 mathematical 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.

§2. Banach and Hilbert Flow Spaces 2.1 Linear Spaces over Graphs n − S − → − → − → → − → Let G 1 , G 2 , · · · , G n be oriented graphs embedded in topological space S with G = G i, i=1

Hilbert Flow Spaces with Differentials over Graphs

23

− → − → i.e., G i is a subgraph of G for integers 1 ≤ i ≤ n. In this case, these is naturally an embedding − → − → ι : Gi → G . − → Let V be a linear space over a field F . A vector labeling L : G → V is a mapping with − → − → L(v), L(e) ∈ V for ∀v ∈ V ( G ), e ∈ E( G ). Define

and

− →L1 − → 2 − → − → L1 [ − → \− → L1 +L2 [ − → − → L2 G1 + GL G1 G2 G2 \ G1 2 = G1 \ G2

(2.1)

− → − → λ · G L = G λ·L (2.2) − → − → 1 − →L2 for ∀λ ∈ F . Clearly, if , and G L , G L 1 , G 2 are continuity flows with linear end-operators − → − → − →L2 − →L L1 + A+ are continuity flows also. If we vu and Auv for ∀(v, u) ∈ E G , G 1 + G 2 and λ · G − →Lb − →L → − → b : − consider each continuity flow G i a continuity subflow of G , where L G i = L( G i ) but − → − → − → b : G \ G i → 0 for integers 1 ≤ i ≤ n, and define O : G → 0, then all continuity flows, L − → − → → − particularly, all complex flows, or Dall action flowsEon oriented graphs G 1 , G 2 , · · · , G n naturally V − → form a linear space, denoted by G i, 1 ≤ i ≤ n ; +, · over a field F under operations (2.1) and (2.2) because it holds with:

(1) A field F of scalars; D− EV → (2) A set G i , 1 ≤ i ≤ n of objects, called continuity flows;

(3) An operation “+”, called continuity flow addition, which associates with each pair of EV EV − → 1 − →L2 D− → − → 1 − →L2 D− → continuity flows G L a continuity flows G L , 1 , G 2 in G i , 1 ≤ i ≤ n 1 + G 2 in G i , 1 ≤ i ≤ n − → L1 − →L2 called the sum of G 1 and G 2 , in such a way that − → 1 − →L2 − →L2 − →L1 (a) Addition is commutative, G L 1 + G 2 = G 2 + G 1 because of − →L1 − → 2 G1 + GL 2

− → − → L2 → L1 +L2 [ − → \− → − → L1 [ − G2 − G1 G2 G1 G1 − G2 − → − → L2 [ − → \− → L2 +L1 [ − → − → L1 = G2 − G1 G1 G2 G1 − G2 − → 2 − →L1 = GL 2 + G1 ;

=

(b) Addition is associative, let

− →L1 − → 2 − →L3 − →L1 − →L2 − →L3 G1 + GL + G = G + G + G3 because if we 2 2 3 1

  Li (x),       Lj (x),       L (x),    k + Lijk (x) = L+ ij (x),      L+  ik (x),      L+  jk (x),     Li (x) + Lj (x) + Lk (x)

− → − → S− → if x ∈ G i \ G j G k − → − → S− → if x ∈ G j \ G i G k − → − → S− → if x ∈ G k \ G i G j − → T− → − → if x ∈ G i G j \ G k − → T− → − → if x ∈ G i G k \ G j −

→ T− → − → if x ∈ G j G k \ G i − → T− → T− → if x ∈ G i G j G k

(2.3)

24

and

Linfan MAO

    Li (x), + Lij (x) = Lj (x),    Li (x) + Lj (x),

for integers 1 ≤ i, j, k ≤ n, then − →L1 − → 2 − → 3 G1 + GL + GL 2 3

=

− → − → if x ∈ G i \ G j − → − → if x ∈ G j \ G i − → T− → if x ∈ G i G j

(2.4)

+ + − → [− → L12 − → 3 − → [− → [− → L123 G1 G2 + GL = G G G 1 2 3 3

− →L1 − → [− → L23 − → 1 − →L2 − →L3 G1 + G2 G3 = GL ; 1 + G2 + G3 +

=

− − → → (c) There is a unique continuity flow O on G hold with O(v, u) = 0 for ∀(v, u) ∈ E G and − EV D− EV → D− → − → − → − → → V G in G i , 1 ≤ i ≤ n , called zero such that G L + O = G L for G L ∈ G i , 1 ≤ i ≤ n ;

D− EV − → → (d) For each continuity flow G L ∈ G i , 1 ≤ i ≤ n there is a unique continuity flow − →−L − →L − →−L G such that G + G = O; (4) An operation “·, called scalar multiplication, which associates with each scalar k in F EV D− → − → − → a continuity flow k · G L in V , called the product and a continuity flow G L in G i , 1 ≤ i ≤ n − → of k with G L , in such a way that D− EV − → − → − → → (a) 1 · G L = G L for every G L in G i , 1 ≤ i ≤ n ;

− → − → (b) (k1 k2 ) · G L = k1 (k2 · G L ); − → 1 − →L2 − →L1 − →L2 (c) k · ( G L 1 + G2 ) = k · G1 + k · G2 ; − → − → − → (d) (k1 + k2 ) · G L = k1 · G L + k2 · G L .

D EV EV D− → − → if these operations G i, 1 ≤ i ≤ n ; +, · to G i , 1 ≤ i ≤ n Usually, we abbreviate

+ and · are clear in the context. − → 1 − →L2 − →L1 − → − → − → − → By operation (1.1), G L 1 + G 2 6= G 1 if and only if G 1 6 G 2 with L1 : G 1 \ G 2 6→ 0 and − →L1 − → 2 − →L2 − → − → − → − → G1 + GL 2 6= G 2 if and only if G 2 6 G 1 with L2 : G 2 \ G 1 6→ 0, which allows us to introduce − → 1 − →L2 − →Ln the conception of linear irreducible. Generally, a continuity flow family { G L 1 , G2 , · · · , Gn } is linear irreducible if for any integer i, [− − → → G i 6 Gl l6=i

with

− → [− → Li : G i \ G l 6→ 0,

(2.5)

l6=i

where 1 ≤ i ≤ n. We know the following result on linear generated sets. n− → 1 − →L2 − →Ln o Theorem 2.1 Let V be a linear space over a field F and let G L , G , · · · , Gn be an 1 2 − → linear irreducible family, Li : G i n→ V for integers 1 o ≤ i ≤ n with linear operators A+ vu , − → − →L1 − →L2 − →Ln + Auv for ∀(v, u) ∈ E G . Then, G 1 , G 2 , · · · , G n is an independent generated set of

Hilbert Flow Spaces with Differentials over Graphs

25

D− EV → G i , 1 ≤ i ≤ n , called basis, i.e.,

D− EV → dim G i , 1 ≤ i ≤ n = n.

D− EV − → i → , 1 ≤ i ≤ n is a linear generated of G , 1 ≤ i ≤ n Proof By definition, G L with i i − → Li : G i → V , i.e., D− EV → dim G i , 1 ≤ i ≤ n ≤ n. − → i We only need to show that G L i , 1 ≤ i ≤ n is linear independent, i.e., D− EV → dim G i , 1 ≤ i ≤ n ≥ n, which implies that if there are n scalars c1 , c2 , · · · , cn holding with − → 1 − →L2 − →Ln c1 G L 1 + c2 G 2 + · · · + cn G n = O, − → − → − → then c1 = c2 = · · · = cn = 0. Notice that { G 1 , G 2 , · · · , G n } is linear irreducible. We are easily − → S− → − → S− → know Gi \ G l 6= ∅ and find an element x ∈ E(Gi \ G l ) such that ci Li (x) = 0 for integer l6=i

l6=i

i, 1 ≤ i ≤ n. However, Li (x) 6= 0 by (1.5). We get that ci = 0 for integers 1 ≤ i ≤ n.

2

D− EV → A subspace of G i , 1 ≤ i ≤ n is called an A0 -flow space if its elements are all continuity − − →L → flows G with L(v), v ∈ V G are constant v. The result following is an immediately conclusion of Theorem 2.1.

− → − → − → − → Theorem 2.2 Let G , G 1 , G 2 , · · · , G n be oriented graphs embedded in a space S and V − → − → − → − → a linear space over a field F . If G v , G v1 1 , G v2 2 , · · · , G vnn are continuity flows with v(v) = − → v, vi (v) = vi ∈ V for ∀v ∈ V G , 1 ≤ i ≤ n, then

D− →v E G is an A0 -flow space; D− → → − − → E − → − → − → (2) G v1 1 , G v2 2 , · · · , G vnn is an A0 -flow space if and only if G 1 = G 2 = · · · = G n or v1 = v2 = · · · = vn = 0.

(1)

− → − → − → − → − → Proof By definition, G v1 1 + G v2 2 and λ G v are A0 -flows if and only if G 1 = G 1 or 2 v1 = v2 = 0 by definition. We therefore know this result.

2.2 Commutative Rings over Graphs Furthermore, if V is a commutative ring (R; +, ·), we can extend it over oriented graph family − → − → − → { G 1 , G 2 , · · · , G n } by introducing operation + with (2.1) and operation · following: − →L1 − → 2 − → − → L1 [ − → \− → L1 ·L2 [ − → − → L2 G1 · GL G1 G2 G2 \ G1 , 2 = G1 \ G2

(2.6)

26

Linfan MAO

where L1 · L2 : x → L1 (x) · L2 (x), and particularly, the scalar product for Rn , n ≥ 2 for − → T− → x ∈ G 1 G 2.

D ER − → As we shown in Subsection 2.1, G i, 1 ≤ i ≤ n ; + is an Abelian group. We show D

ER − → G i, 1 ≤ i ≤ n ; +, · is a commutative semigroup also.

In fact, define L× ij (x) =

    Li (x),   

Lj (x), Li (x) · Lj (x),

− → − → if x ∈ G i \ G j − → − → if x ∈ G j \ G i − → T− → if x ∈ G i G j

× × − → 1− → L2 − → S− → L12 − → S− → L21 − → 2− →L1 Then, we are easily known that G L · G = G G = G G = GL 1 2 1 2 1 2 2 ·G1 D

ER − → 1 − →L2 − → for ∀ G L , G ∈ G , 1 ≤ i ≤ n ; · by definition (2.6), i.e., it is commutative. i 1 2

Let

L× ijk (x) =

Then,

  Li (x),       Lj (x),       L (x),    k

Lij (x),      Lik (x),       Ljk (x),      Li (x) · Lj (x) · Lk (x)

− →L1 − → 2 − → 3 G1 · GL · GL 2 3

Thus,

− → − → 1 − →L2 for ∀ G L , G L 1 , G2 ∈

→ S− → − → − if x ∈ G i \ G j G k − → − → S− → if x ∈ G j \ G i G k → S− → − → − if x ∈ G k \ G i G j − → T− → − → if x ∈ G i G j \ G k

− → T− → − → if x ∈ G i G k \ G j − → T− → − → if x ∈ G j G k \ G i − → T− → T− → if x ∈ G i G j G k

× × − → L123 → [− → [− → 3 − → L12 − → [− G G G · GL = G2 G1 3 2 1 3 ×

[ L − → 23 − → 1 − →L2 − →L3 → − → 1 − G3 = GL . = GL 1 · G2 · G3 1 · G2

=

− → 2 − → 3 − →L1 − →L2 − →L3 →L1 − · GL G1 · GL 2 3 = G1 · G2 · G3

D ER − → G i, 1 ≤ i ≤ n ; · , which implies that it is a semigroup.

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

and

− →L3 − → 1 − →L2 − → 3 − →L1 − →L3 − →L2 G3 · GL + G2 = GL 1 3 · G1 + G3 · G2

(2.7)

− →L1 − → 2 − → 3 − →L1 − →L3 − →L2 − →L3 G1 + GL · GL 2 3 = G1 · G3 + G2 · G3

(2.8)

27

Hilbert Flow Spaces with Differentials over Graphs

− → − → − → for ∀ G 3 , G 1 , G 2 ∈

D

ER − → G i, 1 ≤ i ≤ n ; +, · . Clearly,

− →L3 − → 1 − →L2 G3 · GL 1 + G2

= =

+ × − →L3 − → [− → L12 − → − → [− → L3(21) G3 · G1 G2 = G3 G1 G2

× × − → [− → L31 [ − → [− → L32 − → 3 − →L1 − →L3 − →L2 G3 G1 G3 G2 = GL 3 · G1 + G3 · G2 ,

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

n− → 1 − →L2 − →Ln o Theorem 2.3 Let (R; +, ·) be a commutative ring and let G L be a linear 1 , G2 , · · · , Gn − → + + irreducible family, Li : G 1 ≤ iD → R for integers i ≤ n with linear operators Avu , Auv for −

E R → − → ∀(v, u) ∈ E G . Then, G i, 1 ≤ i ≤ n ; +, · is a commutative ring.

2.3 Banach or Hilbert Flow Spaces D− EV − → 1 − →L2 − →Ln → Let { G L , where V is a Banach space with a 1 , G 2 , · · · , G n } be a basis of G i , 1 ≤ i ≤ n EV → − →L D− norm k · k. For ∀ G ∈ G i , 1 ≤ i ≤ n , define −

→L G =

X

→ − e∈E G

kL(e)k .

(2.9)

EV → − → − → 1 − →L2 D− we are easily know that Then, for ∀ G , G L 1 , G 2 ∈ G i, 1 ≤ i ≤ n

− − − → → → (1) G L ≥ 0 and G L = 0 if and only if G L = O; −

−

→ → (2) G ξL = ξ G L for any scalar ξ;

→ − − →L2 →L2 − → 1 − L1 (3) G L 1 + G 2 ≤ G 1 + G 2 because of

− → 2 →L1 −

G1 + GL 2 =

≤

X

→ − → − e∈E G 1 \ G 2

X

+

  

kL1 (e)k

→ − T→ − e∈E G 1 G2

X

→ − → − e∈E G 1 \ G 2



 +

X

kL1 (e) + L2 (e)k +

kL1 (e)k +

→ − → − e∈E G 2 \ G 1

→ − → − e∈E G 2 \ G 1

X

→ − T→ − e∈E G 1 G2

kL2 (e)k +

X

X

kL2 (e)k



 kL1 (e)k

→ − T→ − e∈E G 1 G2



→ −

 − →L2 1 kL2 (e)k = G L 1 + G2 .

for kL1 (e) + L2 (e)k ≤ kL1 (e)k + kL2 (e)k in Banach space V . Therefore, k · k is also a norm

28

on

Linfan MAO

D− EV → G i, 1 ≤ i ≤ n .

EV − → 1 − →L2 D− → Furthermore, if V is a Hilbert space with an inner product h·, ·i, for ∀ G L , 1 , G 2 ∈ G i, 1 ≤ i ≤ n define D− →L1 − → 2E G1 , GL 2

X

=

→ − → − e∈E G 1 \ G 2

hL1 (e), L1 (e)i

X

+

→ − T→ − e∈E G 1 G2

hL1 (e), L2 (e)i +

X

→ − → − e∈E G 2 \ G 1

hL2 (e), L2 (e)i .

Then we are easily know also that D− EV − → → (1) For ∀ G L ∈ G i , 1 ≤ i ≤ n ,

D− →L − → E G , GL =

and

X

→ − e∈E G

hL(e), L(e)i ≥ 0

D− →L − → E − → G , G L = 0 if and only if G L = O.

EV D− → → − → − (2) For ∀ G L1 , G L2 ∈ G i , 1 ≤ i ≤ n ,

D− →L1 − → 2 E D− → 2 − →L1 E G1 , GL = GL 2 2 , G1

because of D− →L1 − → 2E G1 , GL 2

X

=

→ − → − e∈E G 1 \ G 2

+

hL1 (e), L1 (e)i +

X

→ − → − e∈E G 2 \ G 1

X

=

→ − → − e∈E G 1 \ G 2

+

X

hL2 (e), L2 (e)i

hL1 (e), L1 (e)i +

→ − → − e∈E G 2 \ G 1

X

hL1 (e), L2 (e)i

X

hL2 (e), L1 (e)i

→ − T→ − e∈E G 1 G2

→ − T→ − e∈E G 1 G2

D− → 2 − →L1 E hL2 (e), L2 (e)i = G L , G1 2

for hL1 (e), L2 (e)i = hL2 (e), L1 (e)i in Hilbert space V .

EV − → → − 1 → − L 2 D− → (3) For G L , G L and λ, µ ∈ F , there is 1 , G 2 ∈ G i, 1 ≤ i ≤ n

D − D− D− → 1 − →L2 → − LE → 1 − →L E → 2 − →L E λGL = λ GL + µ GL 1 + µG2 , G 1 ,G 2 ,G

(2.10)

29

Hilbert Flow Spaces with Differentials over Graphs

because of D − → 1 − →L2 − →L E D− → 1 − → 2 − → E λGL = G λL + G µL , GL 1 + µG2 , G 1 2

− → − → λL1 [ − → \− → λL1 +µL2 [ − → − → µL2 − →L = G1 \ G2 G1 G2 G2 \ G1 ,G .

− → S− → Define L1λ 2µ : G 1 G 2 → V by

    λL1 (x), L1λ 2µ (x) = µL2 (x),    λL1 (x) + µL2 (x),

Then, we know that

D − → 1 − →L2 − →L E λGL 1 + µG2 , G

X

= e∈E

+

→ − S→ − → − G1 G2 \G

e∈E

+

− → − → if x ∈ G 1 \ G 2 − → − → if x ∈ G 2 \ G 1 − → T− → if x ∈ G 2 G 1

X

L1λ 2µ (e), L1λ 2µ (e)

L1λ 2µ (e), L(e)

→ − S→ − T→ − G1 G2 G

X

→ − → − S→ − e∈E G \ G 1 G2

hL(e), L(e)i .

and D− D− → 1 − →L E → 2 − →L E λ GL + µ GL 1 ,G 2 ,G X = hλL1 (e), λL1 (e)i + → − → − e∈E G 1 \ G

X

+

+

→ − → − e∈E G \ G 1

X

hL(e), L(e)i +

→ − T→ − e∈E G 2 G

X

→ − T→ − e∈E G 1 G

X

→ − → − e∈E G 2 \ G

hµL2 (e), L(e)i +

hλL1 (e), L(e)i

hµL2 (e), µL2 (e)i

X

→ − → − e∈E G \ G 2

hL(e), L(e)i .

Notice that

e∈E

=

X

→ − S→ − → − G1 G2 \G

X

→ − → − e∈E G 1 \ G

+

e∈E

L1λ 2µ (e), L1λ 2µ (e)

hλL1 (e), λL1 (e)i +

X

→ − S→ − T→ − G1 G2 G

X

→ − → − e∈E G 2 \ G

L1λ 2µ (e), L(e)

hµL2 (e), µL2 (e)i

30

Linfan MAO

X

=

→ − T→ − e∈E G 1 G

X

+

=

→ − → − e∈E G \ G 2

X

→ − → − e∈E G \ G 1

hλL1 (e), L(e)i + hL(e), L(e)i

hL(e), L(e)i +

X

→ − T→ − e∈E G 2 G

X

→ − → − e∈E G \ G 2

hµL2 (e), L(e)i

hL(e), L(e)i .

We therefore know that D − D− D− → 1 − →L2 − →L E →L1 − →L E →L2 − →L E λGL + µ G , G = λ G , G + µ G , G .

1 2 1 2

− → Thus, G V is an inner space

D− EV − → 1 → − L2 − →Ln → If { G L , G , · · · , G } is a basis of space G , 1 ≤ i ≤ n , we are easily find the exact i n 1 2 formula on L by L1 .L2 , · · · , Ln . In fact, let − →L − → 1 − →L2 − →Ln G = λ1 G L 1 + λ2 G 2 + · · · + λn G n , where (λ1 , λ2 , · · · , λn ) 6= (0, 0, · · · , 0), and let b: L

i \ − → G kl

l=1

!  \

[

s6=kl ,··· ,ki

 i X − →  Gs → λkl Lkl l=1

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

X →L G =

D− → ′E →L − G , G′L

X

− i=1 e∈E → Gi

=

n X

X

i X

λkl Lkl (e) ,

− i=1 e∈E → Gi

(2.11)

l=1

*

i X

λkl L1kl (e),

l=1

− → ′ − → 1 →L2 →Ln − → ′− ′ − where G′L = λ′1 G L 1 + λ2 G 2 + · · · + λn G n and G i =

i X

λ′ks L2ks

s=1

i \ − → G kl

l=1

+

!  \

,

(2.12)

[

s6=kl ,··· ,ki



− →  Gs .

− → − → − → We therefore extend the Banach or Hilbert space V over graphs G 1 , G 2 , · · · , G n following. − → − → − → Theorem 2.4 Let G 1 , G 2 , · · · , G n be oriented graphs embedded in a space S and V a Banach D− EV → + space over a field F . Then G i , 1 ≤ i ≤ n with linear operators A+ vu , Auv for ∀(v, u) ∈ −

D EV → − → E G is a Banach space, and furthermore, if V is a Hilbert space, G i , 1 ≤ i ≤ n is a Hilbert space too.

31

Hilbert Flow Spaces with Differentials over Graphs

D− EV → Proof We have shown, G i , 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 − rD− →L − → E →L G , GL G =

D− EV D− EV − → → → for G L ∈ G i , 1 ≤ i ≤ n . Then G i , 1 ≤ i ≤ n is a 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 D− EV → converges in G i , 1 ≤ i ≤ n .

n− D− EV → no → In fact, let H L be a Cauchy sequence in G , 1 ≤ i ≤ n , i.e., for any number ε > 0, i n there always exists an integer N (ε) such that − → m →Ln −

H n − H L m < ε n − S − → → − → n if n, m ≥ N (ε). Let G V be the continuity flow space on G = G i . We embed each H L n to i=1 − →b − → a G L ∈ G V by letting   Ln (e), if e ∈ E (Hn ) b n (e) = − L → − →  0, if e ∈ E G \ H n .

Then

− →b →Lb n −

− G Lm = G

X

→ − → − e∈E G n \ G m

+

X

kLn (e)k +

→ − → − e∈E G m \ G n

Thus,

X

→ − T→ − e∈E G n Gm

kLn (e) − Lm (e)k

− →Lm

→ n − k−Lm (e)k = H L − H m ≤ ε. n

n− →Lb n o − → G is a Cauchy sequence also in G V . By definition,

− − → b →

b m (e) Ln (e) − L ≤ G Ln − G Lm < ε,

− → i.e., {Ln (e)} is a Cauchy sequence for ∀e ∈ E G , which is converges on in V by definition.

Let b b n (e) L(e) = lim L n→∞

− → − →b − →b − →b for ∀e ∈ E G . Then it is clear that lim G Ln = G L , which implies that { G Ln }, i.e., n→∞ n− D− EV − →b − → →Ln o → b is converges to G L ∈ G V , an element in G i , 1 ≤ i ≤ n Hn because of L(e) ∈ V for −

n S − → − → → ∀e ∈ E G and G = G i.

2 i=1

32

Linfan MAO

§3. Differential on Continuity Flows

3.1 Continuity Flow Expansion Theorem 2.4 enables one to establish differentials and generalizes results in classical calculus in D− EV → space G i , 1 ≤ i ≤ n . Let L be kth differentiable to t on a domain D ⊂ R, where k ≥ 1.

Define Rt Zt − → dGL − → dL − →L − →0 Ldt = G dt and G dt = G . dt 0

Then, we are easily to generalize Taylor formula in

D− EV → G i, 1 ≤ i ≤ n following.

D− ER×R − → → Theorem 3.1(Taylor) Let G L ∈ G i , 1 ≤ i ≤ n and there exist kth order derivative of − → + L to t on a domain D ⊂ R, where k ≥ 1. If A+ vu , Auv are linear for ∀(v, u) ∈ E G , then n

k

− →L − → →L′ (t0 ) →L(k) (t0 ) → t − t0 − (t − t0 ) − −k − G = G L(t0 ) + G + ···+ G + o (t − t0 ) G , 1! k!

− → b of L that for ∀t0 ∈ D, where o (t − t0 )−k G denotes such an infinitesimal term L

lim

t→t0

b u) L(v,

(t − t0 )

k

=0

(3.1)

− → ∀(v, u) ∈ E G .

f or

− →LC Particularly, if L(v, u) = f (t)c with LC : vu , where cvu is a constant, denoted by f (t) G −

→ (v, u) → cvu for ∀(v, u) ∈ E G and f (t) = f (t0 ) +

(t − t0 ) ′ (t − t0 )2 ′′ (t − t0 )k (k) f (t0 ) + f (t0 ) + · · · + f (t0 ) + o (t − t0 )k , 1!

2! k!

then

− → − → f (t) G LC = f (t) · G LC .

− → Proof Notice that L(v, u) has kth order derivative to t on D for ∀(v, u) ∈ E G . By applying Taylor formula on t0 , we know that L(v, u) = L(v, u)(t0 ) +

L′ (v, u)(t0 ) L(k)(v,u)(t0 ) (t − t0 ) + · · · + + o (t − t0 )k 1! k!

b u) of L(v, u) hold with if t → t0 , where o (t − t0 )k is an infinitesimal term L(v, lim

t→t0

b u) L(v,

t

(t − t0 )

=0

33

Hilbert Flow Spaces with Differentials over Graphs

− → for ∀(v, u) ∈ E G . By operations (2.1) and (2.2), − →L1 − → − → G + G L2 = G L1 +L2

− − → − → → and λ G L = G λ L

− → + because A+ vu , Auv are linear for ∀(v, u) ∈ E G . We therefore get

k

(t − t0 ) − − → − →L − → (t − t0 ) − →L′ (t0 ) →L(k) (t0 ) G + ···+ G + o (t − t0 )−k G G = G L(t0 ) + 1! k!

→ −k − b of L, i.e., for t0 ∈ D, where o (t − t0 ) G is an infinitesimal term L

lim

t→t0

b u) L(v, t = 0 (t − t0 )

− → for ∀(v, u) ∈ E G . Calculation also shows that − → f (t) G LC (v,u)

k

0 ) f ′ (t )···+ (t−t0 ) f (k) (t )+o (t−t )k ( ) cvu − → − → f (t0 )+ (t−t 0 0 0 1! k! = G f (t)LC (v,u) = G − → f ′ (t0 )(t − t0 ) − → f ”(t0 )(t − t0 )2 − → = f (t0 )cvu G + cvu G + cvu G 1!

2!

→ − → f (k) (t0 ) (t − t0 )k k − +···+ cvu G + o (t − t0 ) G k! ! k

− → (t − t0 ) ′ (t − t0 ) (k) k = f (t0 ) + f (t0 ) · · · + f (t0 ) + o (t − t0 ) cvu G 1! k! − → − → = f (t)cvu G = f (t) · G LC (v,u) ,

i.e.,

− → − → f (t) G LC = f (t) · G LC .

This completes the proof.

2

− → Taylor expansion formula for continuity flow G L enables one to find interesting results on − →L G such as those of the following.

Theorem 3.2 Let f (t) be a k differentiable function to t on a domain D ⊂ R with 0 ∈ D and − → − → − → + f (0 G) = f (0) G . If A+ vu , Auv are linear for ∀(v, u) ∈ E G , then − − → → f (t) G = f t G .

Proof Let t0 = 0 in the Taylor formula. We know that f (t) = f (0) +

f ′ (0) f ′′ (0) 2 f (k) (0) k t+ t + ···+ t + o tk .

1!

2! k!

(3.2)

34

Linfan MAO

Notice that − → f (t) G = = = and by definition, − → f tG =

=

− → f ′ (0) f ′′ (0) 2 f (k) (0) k k f (0) + t+ t + ···+ t +o t G 1!

2! k! − →f (0)+ f ′1!(0) t+ f ′′2!(0) t2 +···+ f (k)k!(0) tk +o(tk ) G − − → f ′ (0)t − → f (k) (0)tk − → → f (0) G + G + ··· + G + o tk G 1! k!

− → f ′ (0) − → f ′′ (0) − → 2 tG + tG f 0G + 1!

2!

→ k − → k f (k) (0) − +···+ tG + o tG k! −

− → → f ′′ (0) 2 − → → → f ′ (0) − f (k) (0) k − f 0G + tG + t G + ···+ t G + o tk G 1!

2! k!

− → i − →i − → − → − → because of t G = G t = ti G for any integer 1 ≤ i ≤ k. Notice that f (0 G) = f (0) G . We therefore get that − − → → f (t) G = f t G .

2

− → Theorem 3.2 enables one easily getting Taylor expansion formulas by f t G . For example,

let f (t) = et . Then

→ − − → et G = et G

(3.3)

′

by Theorem 3.5. Notice that (et ) = et and e0 = 1. We know that

and

→ − − → − → → t2 − → → t− tk − et G = et G = G + G + G + · · · + G + · · · 1!

2! k!

(3.4)

→ − → − → − − →t − →s − →t s − → t+s et G · es G = G e · G e = G e ·e = G e = e(t+s) G ,

(3.5)

where t and s are variables, and similarly, for a real number α if |t| < 1, − → − → α − → αt − → α(α − 1) · · · (α − n + 1)tn − → G + tG =G+ G + ···+ G + ··· 1! n!

(3.6)

3.2 Limitation D− EV − → − → 1 → Definition 3.3 Let G L , G L ∈ G , 1 ≤ i ≤ n with L, L1 dependent on a variable t ∈ i 1 − → [a, b] ⊂ (−∞, +∞) and linear continuous end-operators A+ for ∀(v, u) ∈ E G . For t0 ∈ vu [a, b] and any number ε > 0, if there is always a number δ(ε) such that if |t − t0 | ≤ δ(ε) →L1 − →L − →L1 − →L −

then G 1 − G < ε, then, G 1 is said to be converged to G as t → t0 , denoted by − − → 1 − → − → → lim G L = G L . Particularly, if G L is a continuity flow with a constant L(v) for ∀v ∈ V G 1 t→t0 − → 1 − → and t0 = +∞, G L 1 is said to be G -synchronized.

35

Hilbert Flow Spaces with Differentials over Graphs

Applying Theorem 1.4, we know that there are positive constants cvu â&#x2C6;&#x2C6; R such that â&#x2C6;&#x2019; â&#x2020;&#x2019; + kA+ vu k â&#x2030;¤ cvu for â&#x2C6;&#x20AC;(v, u) â&#x2C6;&#x2C6; E G .

By definition, it is clear that

â&#x2020;&#x2019; â&#x2C6;&#x2019; L â&#x2C6;&#x2019; L1 â&#x2020;&#x2019;

G1 â&#x2C6;&#x2019; G

â&#x2020;&#x2019; â&#x2020;&#x2019; â&#x2020;&#x2019; â&#x2C6;&#x2019; â&#x2020;&#x2019; â&#x2C6;&#x2019; L1 â&#x2C6;&#x2019; \â&#x2020;&#x2019; â&#x2C6;&#x2019; L1 â&#x2C6;&#x2019;L â&#x2C6;&#x2019; â&#x2020;&#x2019; â&#x2C6;&#x2019; â&#x2C6;&#x2019;L

= G \ G + G G + G \ G 1 1 1

X X

Aâ&#x20AC;² + +

Aâ&#x20AC;² + vu L1 vu (v, u) + L1 vu â&#x2C6;&#x2019; LA + = (v, u) vu

uâ&#x2C6;&#x2C6;NG \G (v) 1

â&#x2030;¤

X

c+ vu kL1 (v, u)k

+

uâ&#x2C6;&#x2C6;NG T G (v) 1

X

c+ vu k (L1

=

max

(v,u)â&#x2C6;&#x2C6;E(G1 )

1

c+ vu k

A+ â&#x2C6;&#x2019;L vu (v, u)

â&#x2C6;&#x2019; L(v, u)k.

uâ&#x2C6;&#x2C6;NG\G (v)

â&#x2C6;&#x2019; â&#x2C6;&#x2019; â&#x2020;&#x2019; â&#x2020;&#x2019; and kL(v, u)k â&#x2030;¥ 0 for (v, u) â&#x2C6;&#x2C6; E G and E G 1 . Let cmax G1 G

uâ&#x2C6;&#x2C6;NG\G (v)

X

â&#x2C6;&#x2019; L) (v, u)k +

uâ&#x2C6;&#x2C6;NG T G (v) 1

uâ&#x2C6;&#x2C6;NG \G (v) 1

X

c+ vu ,

max

(v,u)â&#x2C6;&#x2C6;E(G1 )

c+ vu

1

.

â&#x2C6;&#x2019; â&#x2C6;&#x2019; â&#x2020;&#x2019; â&#x2C6;&#x2019; â&#x2020;&#x2019; â&#x2020;&#x2019;L

â&#x2020;&#x2019; 1 â&#x2C6;&#x2019; If G L â&#x2C6;&#x2019; G < ε, we easily get that kL1 (v, u)k < cmax G1 G ε for (v, u) â&#x2C6;&#x2C6; E G 1 \ G , 1 â&#x2C6;&#x2019; â&#x2020;&#x2019; Tâ&#x2C6;&#x2019; â&#x2020;&#x2019; k(L1 â&#x2C6;&#x2019; L)(v, u)k < cmax ε for (v, u) â&#x2C6;&#x2C6; E G G and k â&#x2C6;&#x2019; L(v, u)k < cmax 1 G1 G G1 G ε for (v, u) â&#x2C6;&#x2C6;

â&#x2C6;&#x2019; â&#x2020;&#x2019; â&#x2C6;&#x2019; â&#x2020;&#x2019; E G \ G1 . â&#x2C6;&#x2019; â&#x2020;&#x2019; â&#x2C6;&#x2019; â&#x2020;&#x2019; Conversely, if kL1 (v, u)k < ε for (v, u) â&#x2C6;&#x2C6; E G 1 \ G , k(L1 â&#x2C6;&#x2019; L)(v, u)k < ε for (v, u) â&#x2C6;&#x2C6; â&#x2C6;&#x2019; â&#x2C6;&#x2019; â&#x2020;&#x2019; â&#x2C6;&#x2019; â&#x2020;&#x2019; â&#x2020;&#x2019; Tâ&#x2C6;&#x2019; â&#x2020;&#x2019; E G 1 G and k â&#x2C6;&#x2019; L(v, u)k < ε for (v, u) â&#x2C6;&#x2C6; E G \ G 1 , we easily know that â&#x2C6;&#x2019; â&#x2020;&#x2019; â&#x2020;&#x2019;L1 â&#x2C6;&#x2019;

G 1 â&#x2C6;&#x2019; G L

=

X

uâ&#x2C6;&#x2C6;NG1 \G (v)

+

X

uâ&#x2C6;&#x2C6;NG\G1 (v)

â&#x2030;¤ +

X

uâ&#x2C6;&#x2C6;NG1 \G (v)

X

uâ&#x2C6;&#x2C6;NG\G1 (v)

â&#x20AC;²+

A vu

L1 (v, u) +

A+

â&#x2C6;&#x2019;L vu (v, u)

c+ vu kL1 (v, u)k +

c+ vu k

â&#x2C6;&#x2019; L(v, u)k

X

â&#x20AC;²+

A vu A+

â&#x2C6;&#x2019; Lvuvu (v, u) L1

X

c+ vu k (L1 â&#x2C6;&#x2019; L) (v, u)k

uâ&#x2C6;&#x2C6;NG1 T G (v)

uâ&#x2C6;&#x2C6;NG1

T

G (v)

â&#x2C6;&#x2019;

â&#x2C6;&#x2019;

â&#x2C6;&#x2019;

â&#x2C6;&#x2019; â&#x2020;&#x2019;

â&#x2020;&#x2019;

max â&#x2020;&#x2019;

max â&#x2020;&#x2019;

max

â&#x2020;&#x2019; â&#x2C6;&#x2019;

â&#x2020;&#x2019; \ â&#x2C6;&#x2019;

â&#x2020;&#x2019; â&#x2C6;&#x2019;

â&#x2020;&#x2019; [ â&#x2C6;&#x2019; < G 1 \ G cmax ε + G G c ε + G \ G c ε = G G cG1 G ε.

1 1 G1 G 1 G1 G G1 G

â&#x2C6;&#x2019; â&#x2020;&#x2019; 1 â&#x2C6;&#x2019; â&#x2020;&#x2019;L Thus, we get an equivalent condition for lim G L 1 = G following. tâ&#x2020;&#x2019;t0

â&#x2C6;&#x2019; â&#x2020;&#x2019; 1 â&#x2C6;&#x2019; â&#x2020;&#x2019;L Theorem 3.4 lim G L 1 = G if and only if for any number ε > 0 there is always a number δ(ε) tâ&#x2020;&#x2019;t0 â&#x2C6;&#x2019; â&#x2020;&#x2019; â&#x2C6;&#x2019; â&#x2020;&#x2019; such that if |t â&#x2C6;&#x2019; t0 | â&#x2030;¤ δ(ε) then kL1 (v, u)k < ε for (v, u) â&#x2C6;&#x2C6; E G 1 \ G , k(L1 â&#x2C6;&#x2019; L)(v, u)k < ε â&#x2C6;&#x2019; â&#x2C6;&#x2019; â&#x2020;&#x2019; Tâ&#x2C6;&#x2019; â&#x2020;&#x2019; â&#x2020;&#x2019; â&#x2C6;&#x2019; â&#x2020;&#x2019; â&#x2C6;&#x2019; â&#x2020;&#x2019; 1 â&#x2C6;&#x2019; â&#x2020;&#x2019;L for (v, u) â&#x2C6;&#x2C6; E G 1 G and k â&#x2C6;&#x2019; L(v, u)k < ε for (v, u) â&#x2C6;&#x2C6; E G \ G 1 ,i.e., G L is an 1 â&#x2C6;&#x2019; G â&#x2C6;&#x2019; â&#x2020;&#x2019;L1 â&#x2C6;&#x2019; â&#x2020;&#x2019;L infinitesimal or lim G 1 â&#x2C6;&#x2019; G = O. tâ&#x2020;&#x2019;t0

36

Linfan MAO

− → − → − → If lim G L , lim G1 L1 and lim G2 L2 exist, the formulas following are clearly true by defit→t0

t→t0

t→t0

nition: − → − → − → − → lim G1 L1 + G2 L2 = lim G1 L1 + lim G2 L2 , t→t0 t→t0 t→t0 − → − → − → − → lim G1 L1 · G2 L2 = lim G1 L1 · lim G2 L2 , t→t0 t→t0 t→t0 − → − → − → − → − → − → − → lim G L · G1 L1 + G2 L2 = lim G L · lim G1 L1 + lim G L · lim G2 L2 , t→t0 t→t0 t→t0 t→t0 t→t0 − →L1 − →L2 − →L − →L1 − →L − → L2 − → lim G1 + G2 ·G = lim G1 · lim G + lim G2 · lim G L

t→t0

t→t0

t→t0

t→t0

t→t0

− → and furthermore, if lim G2 L2 6= O, then t→t0

lim

t→t0

− → − → ! lim G1 L1 −

→ L1 − →L−1 G1 L1 t→t0 = lim G1 · G2 2 = − → − → . t→t0 G2 L2 lim G2 L2 t→t0

− → − → Theorem 3.5(L’Hospital’s rule) If lim G1 L1 = O, lim G2 L2 = O and L1 , L2 are differentiable t→t0 t→t 0 − → − → respect to t with lim L′1 (v, u) = 0 for (v, u) ∈ E G 1 \ G 2 , lim L′2 (v, u) 6= 0 for (v, u) ∈ t→t0 t→t0 − − → T− → → − → ′ E G 1 G 2 and lim L2 (v, u) = 0 for (v, u) ∈ E G 2 \ G 1 , then, t→t0

lim

t→t0

− → ′ − → ! lim G1 L1 G1 L1 t→t0 = − → ′. − → lim G2 L2 G2 L2 t→t0

Proof By definition, we know that lim

t→t0

− → !

G1 L1 − → G2 L2

= = =

= = =

− → − → −1 lim G1 L1 · G2 L2

t→t0

lim

t→t0

lim

t→t0

−1 − → − → L1 − → \− → L1 ·L2 − → − → L2 G1 \ G2 G1 G2 G2 \ G1 L1 −1 − − → \− → L1 ·L2 → \− → L−1 G1 G2 = lim G 1 G2 2

− lim → \− → t→t 0 G1 G2

t→t0

L1 −1 L 2

lim L′ 1 t→t

0 − −1 lim L′ → \− → t→t 2 0 = G1 G2

′ − lim L′ 1 → − → lim L1 − → \− → t→t 0 G 1 \ G 2 t→t0 G1 G2 − →L′ ′ ′−1 lim G 1 1 lim L lim L 1 2 − →t→t0 − →t→t t→t0 G1 · G2 0 = − →L′ . lim G 2 2

· lim L′−1 2 t→t0

′ − → − → lim L2 G 2 \ G 1 t→t0

t→t0

This completes the proof.

2

37

Hilbert Flow Spaces with Differentials over Graphs

â&#x2C6;&#x2019; â&#x2020;&#x2019; â&#x2C6;&#x2019; â&#x2020;&#x2019; Corollary 3.6 If lim G L1 = O, lim G L2 = O and L1 , L2 are differentiable respect to t with tâ&#x2020;&#x2019;t0 tâ&#x2020;&#x2019;t 0 â&#x2C6;&#x2019; â&#x2020;&#x2019; lim Lâ&#x20AC;²2 (v, u) 6= 0 for (v, u) â&#x2C6;&#x2C6; E G , then

tâ&#x2020;&#x2019;t0

â&#x2C6;&#x2019; â&#x2020;&#x2019; â&#x20AC;² â&#x2C6;&#x2019; â&#x2020;&#x2019;L1 ! lim G L1 G tâ&#x2020;&#x2019;t0 â&#x2C6;&#x2019; â&#x2020;&#x2019;L2 = â&#x2C6;&#x2019; â&#x2020;&#x2019; â&#x20AC;².

G lim G L2

lim

tâ&#x2020;&#x2019;t0

tâ&#x2020;&#x2019;t0

Generally, by Taylor formula

â&#x2C6;&#x2019; â&#x2020;&#x2019; â&#x2C6;&#x2019; â&#x2020;&#x2019;L â&#x2C6;&#x2019; â&#x2020;&#x2019; t â&#x2C6;&#x2019; t0 â&#x2C6;&#x2019; â&#x2020;&#x2019;Lâ&#x20AC;² (t0 ) (t â&#x2C6;&#x2019; t0 )k â&#x2C6;&#x2019; â&#x2020;&#x2019;L(k) (t0 ) G + ··· + G + o (t â&#x2C6;&#x2019; t0 )â&#x2C6;&#x2019;k G , G = G L(t0 ) + 1! k! (kâ&#x2C6;&#x2019;1)

if L1 (t0 ) = Lâ&#x20AC;²1 (t0 ) = · · · = L1 (k) L2 (t0 ) 6= 0, then â&#x2C6;&#x2019; â&#x2020;&#x2019;L1 G1

=

â&#x2C6;&#x2019; â&#x2020;&#x2019;L2 G2

=

(kâ&#x2C6;&#x2019;1)

(t0 ) = 0 and L2 (t0 ) = Lâ&#x20AC;²2 (t0 ) = · · · = L2

(t0 ) = 0 but

k

â&#x2020;&#x2019;L(k) â&#x2020;&#x2019; (t â&#x2C6;&#x2019; t0 ) â&#x2C6;&#x2019; (t ) â&#x2C6;&#x2019;k â&#x2C6;&#x2019; G 1 1 0 + o (t â&#x2C6;&#x2019; t0 ) G 1 , k! k

(t â&#x2C6;&#x2019; t0 ) â&#x2C6;&#x2019; â&#x2020;&#x2019;L(k) â&#x2C6;&#x2019; â&#x2020;&#x2019; (t ) G 2 2 0 + o (t â&#x2C6;&#x2019; t0 )â&#x2C6;&#x2019;k G 2 . k!

We are easily know the following result. â&#x2C6;&#x2019; â&#x2020;&#x2019; â&#x2C6;&#x2019; â&#x2020;&#x2019; (kâ&#x2C6;&#x2019;1) Theorem 3.7 If lim G1 L1 = O, lim G2 L2 = O and L1 (t0 ) = Lâ&#x20AC;²1 (t0 ) = · · · = L1 (t0 ) = 0 tâ&#x2020;&#x2019;t0

tâ&#x2020;&#x2019;t0 (kâ&#x2C6;&#x2019;1) (t0 )

and L2 (t0 ) = Lâ&#x20AC;²2 (t0 ) = · · · = L2

(k)

= 0 but L2 (t0 ) 6= 0, then

â&#x2C6;&#x2019; â&#x2020;&#x2019;L(k) (t ) â&#x2C6;&#x2019; â&#x2020;&#x2019;L1 lim G 1 1 0 G1 tâ&#x2020;&#x2019;t0 lim â&#x2020;&#x2019;L = . â&#x2C6;&#x2019; â&#x2020;&#x2019;L(k) (t ) 2 tâ&#x2020;&#x2019;t0 â&#x2C6;&#x2019; G2 lim G 2 2 0 tâ&#x2020;&#x2019;t0

â&#x2C6;&#x2019; â&#x2020;&#x2019; â&#x2C6;&#x2019; â&#x2020;&#x2019; â&#x2C6;&#x2019; â&#x2020;&#x2019; + Example 3.8 Let G 1 = G 2 = C n , A+ vi vi+1 = 1, Avi viâ&#x2C6;&#x2019;1 = 2 and

f1 + 2iâ&#x2C6;&#x2019;1 â&#x2C6;&#x2019; 1 F (x) n! + fi = 2iâ&#x2C6;&#x2019;1 (2n + 1)et for integers 1 â&#x2030;¤ i â&#x2030;¤ n in Fig.4.

fn

-f

v1

v2

1

?f

6 vn

2

vi+1

fi

Fig.4

f3 vi

v3

38

Linfan MAO

Calculation shows That L(vi )

f1 + 2i − 1 F (x) f1 + 2i−1 − 1 F (x) − = 2fi+1 − fi = 2 × 2i 2i−1 n! = F (x) + .

(2n + 1)et

− → − →Lb b Calculation shows that lim L(vi ) = F (x), i.e., lim C L n = C n , where, L(vi ) = F (x) for t→∞ t→∞ − →L − → integers 1 ≤ i ≤ n, i.e., C n is G -synchronized. §4. Continuity Flow Equations − → − → A continuity flow G L is in fact an operator L : G → B determined by L(v, u) ∈ B for − → ∀(v, u) ∈ E G . Generally, let

[L]m×n



L11

  L  21 =  ···  Lm1

L12

···

L22

···

··· Lm2

L1n



 L2n    ··· ···   · · · Lmn

− → with Lij : G → B for 1 ≤ i ≤ m, 1 ≤ j ≤ n, called operator matrix. Particularly, if for integers − → 1 ≤ i ≤ m, 1 ≤ j ≤ n, Lij : G → R, we can also determine its rank as the usual, labeled the − → − → edge (v, u) by Rank[L]m×n for ∀(v, u) ∈ E G and get a labeled graph G Rank[L] . Then we get a result following.

Theorem 4.1 A linear continuity flow equations  − → − → − → − →  x1 G L11 + x2 G L12 + · · · + xn G Ln1 = G L1    − → − → − → →L21   x − + x2 G L22 + · · · + xn G L2n = G L2 1G  ························     − → − → − → − →  x1 G Ln1 + x2 G Ln2 + · · · + xn G Lnn = G Ln

is solvable if and only if

(4.1)

− →Rank[L] − → G = G Rank[L] ,

(4.2)

where 

L11

  L  21 [L] =   ···  Ln1

L12 L22 ··· Ln2

···

L1n



 L2n    ··· ···   · · · Lnn ···

and



L11

   L21 L =  ···  Ln1

L12

···

L1n

L22

···

L2n

···

···

···

Ln2

· · · Lnn

L1



 L2   .   Ln

39

Hilbert Flow Spaces with Differentials over Graphs

− → Proof Clearly, if (4.1) is solvable, then for ∀(v, u) ∈ E G , the linear equations   x1 L11 (v, u) + x2 L12 (v, u) + · · · + xn Ln1 (v, u0 = L1 (v, u)      x L (v, u) + x L (v, u) + · · · + x L (v, u0 = L (v, u) 1 21 2 22 n 21 2   ························     x1 Ln1 (v, u) + x2 Ln2 (v, u) + · · · + xn Lnn (v, u0 = Ln (v, u)

is solvable. By linear algebra, there must be 

L11 (v, u)

L12 (v, u)

  L (v, u) L (v, u) 22  21 Rank   ··· ···  Ln1 (v, u) Ln2 (v, u)  L11 (v, u) L12 (v, u)   L (v, u) L (v, u) 22  21 Rank   ··· ···  Ln1 (v, u) Ln2 (v, u) which implies that

···

L1n (v, u)

···

L1n (v, u)



 L2n (v, u)   =  ··· ···  · · · Lnn (v, u)

···

L1 (v, u)



 L2 (v, u)   ,  ··· ···  · · · Lnn (v, u) Ln (v, u)

···

L2n (v, u)

− →Rank[L] − → G = G Rank[L] .

− → Conversely, if the (4.2) is hold, then for ∀(v, u) ∈ E G , the linear equations   x1 L11 (v, u) + x2 L12 (v, u) + · · · + xn Ln1 (v, u0 = L1 (v, u)      x L (v, u) + x L (v, u) + · · · + x L (v, u0 = L (v, u) 1 21 2 22 n 21 2  ························      x1 Ln1 (v, u) + x2 Ln2 (v, u) + · · · + xn Lnn (v, u0 = Ln (v, u)

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

2

Theorem 4.2 A continuity flow equation − → − → − → λs G Ls + λs−1 G Ls−1 + · · · + G L0 = O

− → always has solutions G Lλ are roots of the equation

(4.3)

− → vu vu vu with Lλ : (v, u) ∈ E G → {λvu 1 , λ2 , · · · , λs }, where λi , 1 ≤ i ≤ s s vu s−1 αvu + · · · + αvu s λ + αs−1 λ 0 =0 − → 6= 0 a constant for (v, u) ∈ E G and 1 ≤ i ≤ s.

(4.4)

vu with Li : (v, u) → αv,u i , αs − → For (v, u) ∈ E G , if nvu is the maximum number i with Li (v, u) 6= 0, then there are

40

Linfan MAO

Q

− − →Lλ → vu n solutions G , and particularly, if L (v, u) = 6 0 for ∀(v, u) ∈ E G , there are s

→ − (v,u)∈E G

→

−

E G

s

− → solutions G Lλ of equation (4.3).

vu vu Proof By the fundamental theorem of algebra, we know there are s roots λvu 1 , λ2 , · · · , λs − → for the equation (4.3). Whence, Lλ G is a solution of equation (4.2) because of

− − − → s − → → s−1 − → → 0 − → λ G · G Ls + λ G · G Ls−1 + · · · + λ G · G L0 s−1 − → s − → s−1 − → 0 − → s = G λ Ls + G λ Ls−1 + · · · + G λ L0 = G λ Ls +λ Ls−1 +···+L0 and s vu s−1 λs Ls + λs−1 Ls−1 + · · · + L0 : (v, u) → αvu + · · · + αvu s λ + αs−1 λ 0 = 0, −

→ for ∀(v, u) ∈ E G , i.e.,

− − − → s − → → s−1 − → → 0 − → − → λ G · G Ls + λ G · G Ls−1 + · · · + λ G · G L0 = 0 G = O.

− → Count the number of different Lλ for (v, u) ∈ E G . It is nothing else but just nvu .

Q Therefore, the number of solutions of equation (4.3) is nvu .

2

→ − (v,u)∈E G

Theorem 4.3 A continuity flow equation − → → − → dGL − = G Lα · G L dt − →

with initial values G L

t=0

(4.5)

− → = G Lβ always has a solution

− →L − → − → G = G Lβ · etLα G ,

− → where Lα : (v, u) → αvu , Lβ : (v, u) → βvu are constants for ∀(v, u) ∈ E G .

Proof A calculation shows that − → − → dL → − → − → dGL − G dt = = G Lα · G L = G Lα ·L , dt which implies that − → for ∀(v, u) ∈ E G .

dL = αvu L dt

(4.6)

− → Solving equation (4.6) enables one knowing that L(v, u) = βvu etαvu for ∀(v, u) ∈ E G .

Hilbert Flow Spaces with Differentials over Graphs

41

Whence, the solution of (4.5) is

− →L − → tLα − → − → G = G Lβ e = G Lβ · etLα G and conversely, by Theorem 3.2, − → tLα d G Lβ e dt

= =

i.e.,

tL tLα − → − → d(Lβ e α ) dt = G Lα Lβ e G − →Lα − → tLα G · G Lβ e ,

− → → − → dGL − = G Lα · G L dt

− → − → − → if G L = G Lβ · etLα G . This completes the proof.

2

− → Theorem 4.3 can be generalized to the case of L : (v, u) → Rn , n ≥ 2 for ∀(v, u) ∈ E G .

Theorem 4.4 A complex flow equation − → → − → dGL − = G Lα · G L dt − →

with initial values G L

t=0

(4.7)

− → = G Lβ always has a solution

− →L − → − → G = G Lβ · etLα G ,

1 2 n where Lα : (v, u) → α1vu , α2vu , · · · , αnvu , Lβ : (v, u) → βvu , βvu , · · · , βvu with constants

− → i i αvu , βvu , 1 ≤ i ≤ n for ∀(v, u) ∈ E G .

Theorem 4.5 A complex flow equation − → − → →Lαn−1 dn−1 G L − → − → − →Lαn dn G L − +G · + · · · + G L α0 · G L = O (4.8) G · n n−1 dt dt − → with Lαi : (v, u) → αvu constants for ∀(v, u) ∈ E G and integers 0 ≤ i ≤ n always has a i − → general solution G Lλ with Lλ : (v, u) →

(

0,

s X i=1

hi (t)e

λvu i t

)

− → for (v, u) ∈ E G , where hmi (t) is a polynomial of degree≤ mi −1 on t, m1 +m2 +· · ·+ms = n vu vu and λvu 1 , λ2 , · · · , λs are the distinct roots of characteristic equation n vu n−1 αvu + · · · + αvu n λ + αn−1 λ 0 = 0

42

Linfan MAO

− → with αvu = 6 0 for (v, u) ∈ E G . n

− → Proof Clearly, the equation (4.8) on an edge (v, u) ∈ E G is αvu n

dn L(v, u) dn−1 L(v, u) + αvu + · · · + α0 = 0. n−1 n dt dtn−1

(4.9)

− → − → As usual, assuming the solution of (4.6) has the form G L = eλt G . Calculation shows that − → dGL = dt − → d2 G L = dt2 ······ ··· − → dn G L = dtn

− → − → λeλt G = λ G , − → − → λ2 eλt G = λ2 G , ··············· ,

− → − → λn eλt G = λn G .

Substituting these calculation results into (4.8), we get that

− → − → − → − → λn G Lαn + λn−1 G Lαn−1 + · · · + G Lα0 · G L = O,

i.e.,

− →(λn ·Lαn +λn−1 ·Lαn−1 +···+Lα0 )·L G = O, −

→ which implies that for ∀(v, u) ∈ E G , n−1 vu λn αvu αn−1 + · · · + α0 = 0 n +λ

(4.10)

or L(v, u) = 0. vu vu vu vu vu Let λvu 1 , λ2 , · · · , λs be the distinct roots with respective multiplicities m1 , m2 , · · · , ms of equation (4.8). We know the general solution of (4.9) is

L(v, u) =

s X

vu

hi (t)eλi

t

i=1

with hmi (t) a polynomial of degree≤ mi −1 on t by the theory of ordinary differential equations. − → Therefore, the general solution of (4.8) is G Lλ with Lλ : (v, u) → − → for (v, u) ∈ E G .

(

0,

s X i=1

hi (t)e

λvu i t

)

2

43

Hilbert Flow Spaces with Differentials over Graphs

§5. Complex Flow with Continuity Flows − → − → The difference of a complex flow G L with that of a continuity flow G L is the labeling L on a vertex is L(v) = ẋv or xv . Notice that 

d  dt

X



+

u∈NG (v)

X

LAvu (v, u) =

u∈NG (v)

d A+ L vu (v, u) dt

− → − → − → for ∀v ∈ V G . There must be relations between complex flows G L and continuity flows G L .

We get a general result following.

Z t

Z t

+ + + + Theorem 5.1 If end-operators Avu , Auv are linear with , Avu = , Auv = 0 and 0 0

d + d + − → − →L ,A = ,A = 0 for ∀(v, u) ∈ E G , and particularly, A+ ∈ vu = 1V , then G dt vu dt uv n D− E

R×R → − → G i, 1 ≤ i ≤ n is a continuity flow with a constant L(v) for ∀v ∈ V G if and only if Z t − − →L → G dt is such a continuity flow with a constant one each vertex v, v ∈ V G .

0

Proof Notice that if A+ vu = 1V , there always is definition, we know that Z

t

0

, A+ vu

=0

d + ,A =0 dt vu

Z

⇔

Z

0

t

0

t

d + , A+ = 0 and , A = 0, and by vu dt vu

+ ◦A+ vu = Avu ◦

Z

t

,

0

d d + ◦ A+ . vu = Avu ◦ dt dt

⇔

− − → → If G L is a continuity flow with a constant L(v) for ∀v ∈ V G , i.e., X

u∈NG (v)

+

LAvu (v, u) = v for ∀v ∈ V

− → G ,

we are easily know that Z

t 0

 

X

u∈NG (v)

A+ vu

L



(v, u) dt

=

X

u∈NG (v)

=

X

u∈NG (v)

Z

t 0

A+ vu

◦A+ vu L(v, u)dt =

Z

t

L(v, u)dt 0

=

X

u∈NG (v)

Z

t

A+ vu ◦

Z t

L(v, u)dt

0

vdt 0

Z t Z t − → − →L for ∀v ∈ V G with a constant vector vdt, i.e., G dt is a continuity flow with a 0 0 − → constant flow on each vertex v, v ∈ V G .

Z t − →L Conversely, if G dt is a continuity flow with a constant flow on each vertex v, v ∈ 0

44

V

Linfan MAO

− → G , i.e.,

X

A+ vu

u∈NG (v)

◦

Z

t

0

L(v, u)dt = v for ∀v ∈ V

then − →L G =

d

Z

t

− →L G dt

0

dt

− → G ,

− → is such a continuity flow with a constant flow on vertices in G because of d

P

L

A+ vu

!

(v, u)

u∈NG (v)

dt

=

X

u∈NG (v)

=

X

u∈NG (v)

d ◦ A+ vu ◦ dt A+ vu ◦

d ◦ dt

Z Z

t

L(v, u)dt 0 t

L(v, u)dt = 0

X

u∈NG (v)

+

L(v, u)Avu =

dv dt

− dv → 2 on vertex v, v ∈ V G . This completes the proof. dt Z t

− → + + + If all end-operators Avu and Auv are constant for ∀(v, u) ∈ E G , the conditions , Avu = 0 Z t

d d + , A+ , A+ , Auv = 0 are clearly true. We immediately get a concluuv = 0 and vu = dt dt 0 sion by Theorem 5.1 following. with a constant flow

− → + Corollary 5.2 For ∀(v, u) ∈ E G , if end-operators A+ vu and Auv are constant cvu , cuv for n −

D E R×R → − → − → ∀(v, u) ∈ E G , then G L ∈ G i , 1 ≤ i ≤ n is a continuity flow with a constant L(v) Z t −

→ − →L for ∀v ∈ V G if and only if G dt is such a continuity flow with a constant flow on each 0 − → vertex v, v ∈ V G .

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

Hilbert Flow Spaces with Differentials over Graphs

45

Education Publisher Inc., USA, 2011.

[7] Linfan Mao, Smarandache Multi-Space Theory, The Education Publisher Inc., USA, 2011.

[8] Linfan Mao, Combinatorial Geometry with Applications to Field Theory, The Education Publisher Inc., USA, 2011.

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[10] Linfan Mao, Non-solvable equation systems with graphs embedded in Rn , Proceedings of the First International Conference on Smarandache Multispace and Multistructure, The Education Publisher Inc. July, 2013.

[11] Linfan Mao, Geometry on GL -systems of homogenous polynomials, International J.Contemp.

Math. Sciences, Vol.9 (2014), No.6, 287-308.

[12] Linfan Mao, Mathematics on non-mathematics - A combinatorial contribution, International J.Math. Combin., Vol.3(2014), 1-34.

[13] Linfan Mao, Cauchy problem on non-solvable system of first order partial differential equations with applications, Methods and Applications of Analysis, Vol.22, 2(2015), 171-200. − → [14] 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. − → [15] Linfan Mao, A new understanding of particles by G -flow interpretation of differential equation, Progress in Physics, Vol.11(2015), 193-201.

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