Hilbert Flow Spaces with Operators over Topological Graphs

Transcription

1 International J.Math. Combin. Vol.4217, Hilbert Flow Spaces with Operators over Topological raphs infan MAO 1. Chinese Academy of Mathematics and System Science, Beijing 119, P.R.China 2. Academy of Mathematical Combinatorics & Applications AMCA, Colorado, USA 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 underlying components in S. We can thereby establish mathematics over graphs 1, 2, by viewing labeling graphs 1, 2 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 is a topological graph associated with a mapping : v, u v, u, 2 end-operators A vu : v, u A vuv, u and A uv : u, v A uvu, v on a Banach space B over a field F with v, u u, v and A vu v, u A vuv, u for v, u E holding with continuity equations u N v A vu v, u v, v V. The main purpose of this paper is to extend Banach { or Hilbert spaces to Banach or Hilbert 1, continuity flow spaces over topological graphs } 2, and establish differentials on continuity flows for characterizing their globally change rate. A few well-known results such as those of Taylor formula, 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, Hospital s rule, mathematical combinatorics. AMS21: 34A26, 35A8, 46B25, 92B5, 5C1, 5C21, 34D43, 51D2. 1. Introduction A Banach or Hilbert space is respectively a linear space A over a field R or C equipped with a complete norm or inner product,, i.e., for every Cauchy sequence {x n } in A, there 1 Received May 5, 217, Accepted November 6, 217.

2 2 infan MAO exists an element x in A such that lim x n x A or lim x n x, x n x n n A and a topological graph ϕ is an embedding of a graph with vertex set V, edge set E in a space S, i.e., there is a 1 1 continuous mapping ϕ : ϕ S with ϕp ϕq if p q for p, q, i.e., edges of 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 R n concluded that there always exists an embedding of that all edges are straight segments in R n 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 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, A is an oriented embedded graph in a topological space S associated with a mapping : v v, v, u v, u, 2 end-operators A vu : v, u A vu v, u and A uv : u, v A uv u, v on a Banach space B over a field F v A vu v, u ¹ A uv u v Fig.2 u

3 Hilbert Flow Spaces with Differentials over raphs 21 with v, u u, v and A vu v, u A vu v, u for v, u E continuity equation A vu v, u v for v V u N v such as those shown for vertex v in Fig.3 following holding with u 1 ¹ u 1, v ¹ v, u 4 u 4 u 1 u 4 A 1 A 4 u 2 u 2, v ¹ A 2 v A 5 ¹ v, u 5 u 5 u 2 A 3 v A 6 u 5 u 3 u 3, v ¹ v, u 6 ¹ u 6 u 3 Fig.3 u 6 with a continuity equation A1 v, u 1 A2 v, u 2 A3 v, u 3 A4 v, u 4 A5 v, u 5 A6 v, u 6 v, where v is the surplus flow on vertex v. Particularly, if v ẋ v or constants v v, v V, the continuity flow ;, A is respectively said to be a complex flow or an action A flow, and -flow if A 1 V, where ẋ v dx v /dt, x v is a variable on vertex v and v is an element in B for v E. ; Clearly, an action flow is an equilibrium state of a continuity flow, 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 et B 1, B 2 be Banach spaces over a field F with norms 1 and 2, respectively. An operator T : B 1 B 2 is linear if Tλv 1 µv 2 λtv 1 µtv 2 for λ, µ F, and T is said to be continuous at a vector v if there always exist such a number

4 22 infan MAO δε for ǫ > that Tv Tv 2 < ε if v v 1 < δε for v,v,v 1,v 2 B 1. Definition 1.3 et B 1, B 2 be Banach spaces over a field F with norms 1 and 2, respectively. An operator T : B 1 B 2 is bounded if there is a constant M > such that Tv 2 M v 1, i.e., Tv 2 v 1 M for v B and furthermore, T is said to be a contractor if for v 1,v 2 B with c [, 1. Tv 1 Tv 2 c v 1 v 2 We only discuss the case that all end-operators A vu, A uv are both linear and continuous. In this case, the result following on linear operators of Banach space is useful. Theorem 1.4 et B 1, B 2 be Banach spaces over a field F with norms 1 and 2, respectively. Then, a linear operator T : B 1 B 2 is continuous if and only if it is bounded, or equivalently, T : Tv 2 sup v B 1 v 1 <. { 1 et, 2, } be a graph family. The main purpose of this paper is to extend Banach { or Hilbert spaces to Banach or Hilbert continuity flow spaces over topological graphs 1, 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, 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], [2] for biological mathematics. 2. Banach and Hilbert Flow Spaces 2.1 inear Spaces over raphs et 1, 2,, n be oriented graphs embedded in topological space S with n i, i1

5 Hilbert Flow Spaces with Differentials over raphs 23 i.e., i is a subgraph of for integers 1 i n. In this case, these is naturally an embedding ι : i. et V be a linear space over a field F. A vector labeling : V is a mapping with v, e V for v V, e E. Define \ \ and λ λ 2.2 for λ F. Clearly, if, and, 1 1, 2 2 are continuity flows with linear end-operators A vu and A uv for v, u E, and λ are continuity flows also. If we consider each continuity flow i a continuity subflow of, where : i i but : \ i for integers 1 i n, and define O :, then all continuity flows, particularly, all complex flows, or all action flows on oriented graphs 1, 2,, n naturally form a linear space, denoted by 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; i 2 A set, 1 i n of objects, called continuity flows; 3 An operation, called continuity flow addition, which associates with each pair of continuity flows 1 1, i 2 2 in, 1 i n a continuity flows 1 i V 2 2 in, 1 i n, called the sum of 1 1 and 2 2, in such a way that a Addition is commutative, because of let ; b Addition is associative, ijk x because if we i x, if x j i \ k j x, if x i j \ k k x, if x k \ i j ij x, if x i j \ k 2.3 ik x, if x i k \ j jk x, if x j k \ i i x j x k x if x i j k

6 24 infan MAO and i x, if x i \ j ij x j x, if x j \ i 2.4 i x j x, if x i j for integers 1 i, j, k n, then V c There is a unique continuity flow O on hold with Ov, u for v, u E and i V in, 1 i n, called zero such that O for i V, 1 i n ; ; d For each continuity flow such that O; i, 1 i n there is a unique continuity flow 4 An operation, called scalar multiplication, which associates with each scalar k in F and a continuity flow in i, 1 i n a continuity flow k in V, called the product of k with, in such a way that a 1 for every in b k 1 k 2 k 1 k 2 ; c k k 1 1 k 2 2 ; d k 1 k 2 k 1 k 2. i, 1 i n ; i i Usually, we abbreviate, 1 i n ;, to, 1 i n if these operations and are clear in the context. By operation 1.1, if and only if 1 2 with : 1 \ 2 and if and only if 2 1 with 2 : 2 \ 1, which allows us to introduce the conception of linear irreducible. enerally, a continuity flow family { 1 1, 2 2,, n n } is linear irreducible if for any integer i, i l i l with i : i \ l i l, 2.5 where 1 i n. We know the following result on linear generated sets. Theorem 2.1 et V be a linear space over a field F and let { 1, 2 2,, } n n be an linear irreducible family, i : i V for integers 1 i n with linear operators A vu, { A uv for v, u E. Then, 1 1, 2 2,, } n n is an independent generated set of

7 Hilbert Flow Spaces with Differentials over raphs 25 i, 1 i n, called basis, i.e., dim i, 1 i n n. Proof By definition, i i, 1 i n is a linear generated of i : i V, i.e., dim i, 1 i n n. We only need to show that i i, 1 i n is linear independent, i.e., i, 1 i n with dim i, 1 i n n, which implies that if there are n scalars c 1, c 2,, c n holding with c 1 1 c c n n n O, then c 1 c 2 c n. Notice that { 1, 2,, n } is linear irreducible. We are easily know i \ l and find an element x E i \ l such that c i i x for integer l i i, 1 i n. However, i x by 1.5. We get that c i for integers 1 i n. ¾ i V A subspace of, 1 i n is called an A -flow space if its elements are all continuity flows with v, v V are constant v. The result following is an immediately conclusion of Theorem 2.1. l i Theorem 2.2 et, 1, 2,, n be oriented graphs embedded in a space S and V a linear space over a field F. If v, v1 1, v2 2,, vn n are continuity flows with vv v,v i v v i V for v V, 1 i n, then 1 v is an A -flow space; v 2 1 1, v2 2,, vn n is an A -flow space if and only if 1 2 n or v 1 v 2 v n. Proof By definition, v1 1 v2 2 and λ v are A -flows if and only if 1 1 or v 1 v 2 by definition. We therefore know this result. 2.2 Commutative Rings over raphs Furthermore, if V is a commutative ring R;,, we can extend it over oriented graph family { 1, 2,, n } by introducing operation with 2.1 and operation following: ¾ \ \ 1 2, 2.6

8 26 infan MAO where 2 : x x 2 x, and particularly, the scalar product for R n, n 2 for x 1 2. i R As we shown in Subsection 2.1,, 1 i n ; is an Abelian group. We show i R, 1 i n ;, is a commutative semigroup also. In fact, define i x, ij x j x, i x j x, if x i \ j if x j \ i if x i j Then, we are easily known that for 1 1, i R 2 2, 1 i n ; by definition 2.6, i.e., it is commutative. et Then, ijk x i x, if x i \ j k j x, if x i j \ k k x, if x i k \ j ij x, if x i j \ k ik x, if x i k \ j jk x, if x j k \ i i x j x k x if x i j k Thus, for, 1 1, i R 2 2, 1 i n ;, which implies that it is a semigroup. We are also need to verify the distributive laws, i.e., and

9 Hilbert Flow Spaces with Differentials over raphs 27 for 3, 1, i R 2, 1 i n ;,. Clearly, , which is the 2.7. The proof for 2.8 is similar. Thus, we get the following result. Theorem 2.3 et R;, be a commutative ring and let { 1, 2 2,, } n n be a linear irreducible family, i : i R for integers 1 i n with linear operators A vu, A uv for i R v, u E. Then,, 1 i n ;, is a commutative ring. 2.3 Banach or Hilbert Flow Spaces et { 1 1, 2 2,, i n n } be a basis of, 1 i n, where V is a Banach space with a norm. For i, 1 i n, define e. 2.9 Then, for, 1 1, i V 2 2, 1 i n we are easily know that 1 and if and only if O; 2 ξ ξ for any scalar ξ; because of \ \ 2 e 2\ 1 e 2 e e 2 e \ 1 e 2 e 2 e 1 2. for e 2 e e 2 e in Banach space V. Therefore, is also a norm 1 2

10 28 infan MAO on i, 1 i n. Furthermore, if V is a Hilbert space with an inner product,, for 1 1, 2 2 i, 1 i n, define 1, 2 2 1\ Then we are easily know also that e, e e, 2 e 2\ 1 2 e, 2 e For i, 1 i n,, and, if and only if O. e, e 2 For 1, 2 i, 1 i n, 1, , 1 1 because of 1, 2 2 1\ 2 2\ 1 1\ 2 2\ 1 e, e 2 e, 2 e e, e e, 2 e 2 2, 1 1 e, 2 e 2 e, e for e, 2 e 2 e, e in Hilbert space V. 3 For, 1 1, 2 2 i, 1 i n and λ, µ F, there is λ 1 1 µ 2 2, λ 1 1, µ 2 2,

11 Hilbert Flow Spaces with Differentials over raphs 29 because of λ 1 1 µ 2 2, λ 1 1 µ2 2, 1 \ λ1 1 2 λ1µ \ µ2 1,. Define λ 2 µ : 1 2 V by λ x, if x 1 \ 2 λ 2 µ x µ 2 x, if x 2 \ 1 λ x µ 2 x, if x 2 1 Then, we know that λ 1 1 µ 2 2, 1 2 \ \ 2 1λ 2 µ e, λ 2 µ e 1λ 2 µ e, e e, e. and λ 1 1, µ 2 2, λ e, λ e 1\ \ 1 2 e, e µ 2 e, e 1 2\ \ 2 λ e, e µ 2 e, µ 2 e e, e. Notice that 1λ 2 µ e, λ 2 µ e 1 2 \ 1\ λ e, λ e 1 2 2\ 1λ 2 µ e, e µ 2 e, µ 2 e

12 3 infan MAO 1 We therefore know that Thus, V is an inner space \ 2 \ 1 λ e, e e, e e, e 2 \ 2 µ 2 e, e e, e. λ 1 1 µ 2 2, λ 1 1, µ 2 2,. If { 1 1, 2 2,, n n } is a basis of space i, 1 i n, we are easily find the exact formula on by. 2,, n. In fact, let where λ 1, λ 2,, λ n,,,, and let λ 1 1 λ λ n n n, i : kl \ l1 s k l,,k i s i λ kl kl for integers 1 i n. Then, we are easily knowing that is nothing else but the labeling on by operation 2.1, a generation of 2.3 and 2.4 with, n i1 i n i1 i l1 i λ kl kl e, 2.11 l1 i λ kl k l e, l1 where λ 1 1 λ λ n n n and i i λ k s 2 k s, 2.12 s1 i kl \ l1 s k l,,k i s We therefore extend the Banach or Hilbert space V over graphs 1, 2,, n following.. Theorem 2.4 et 1, 2,, n be oriented graphs embedded in a space S and V a Banach i space over a field F. Then, 1 i n with linear operators A vu, A uv for v, u E is a Banach space, and furthermore, if V is a Hilbert space, i, 1 i n is a Hilbert space too.

13 Hilbert Flow Spaces with Differentials over raphs 31 i Proof We have shown,, 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, for V i V i, 1 i n. Then, 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 converges in i, 1 i n. { H } i V In fact, let n n be a Cauchy sequence in, 1 i n, i.e., for any number ε >, there always exists an integer Nε such that H n n H m m < ε if n, m Nε. et V be the continuity flow space on n a V by letting i1 n e, if e E H n n e, if e E \ H n. i. We embed each H n n to Then n m n\ m m\ n n e m e n m H n n n e m e H m m ε. { } n Thus, is a Cauchy sequence also in V. By definition, n e m e n m < ε, i.e., { n e} is a Cauchy sequence for e E, which is converges on in V by definition. et e lim n e n for e E. Then it is clear that lim n, which implies that { n }, i.e., n { H } n n is converges to V i, an element in, 1 i n because of e V for e E and n i. i1 ¾

14 32 infan 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 i space, 1 i n. et be kth differentiable to t on a domain D R, where k 1. Define d dt d dt and Then, we are easily to generalize Taylor formula in t dt t dt. i, 1 i n following. Theorem 3.1Taylor et i R R n, 1 i n and there exist kth order derivative of to t on a domain D R, where k 1. If A vu, A uv are linear for v, u E, then t t t t t t k k t o t t k, 3.1 1! for t D, where o t t k denotes such an infinitesimal term of that v, u lim t t t t k for v, u E. Particularly, if v, u ftc vu, where c vu is a constant, denoted by ft C with C : v, u c vu for v, u E and then ft ft t t f t 1! t t2 f t 2! ft C ft C. t tk f k t o t t k, Proof Notice that v, u has kth order derivative to t on D for v, u E. By applying Taylor formula on t, we know that v, u v, ut v, ut t t kv,ut o t t k 1! if t t, where o t t k is an infinitesimal term v, u of v, u hold with lim t t v, u t t t

15 Hilbert Flow Spaces with Differentials over raphs 33 for v, u E. By operations 2.1 and 2.2, 2 12 and λ λ because A vu, A uv are linear for v, u E. We therefore get t t t t t t k k t o t t k 1! for t D, where o t t k is an infinitesimal term of, i.e., v, u lim t t t t t for v, u E. Calculation also shows that ft Cv,u ftcv,u ft t t 1! f t t t k f k t ot t k c vu f t t t f t t t 2 ft c vu c vu c vu 1! 2! fk t t t k c vu o t t k ft t t 1! f t t t k f k t o t t k c vu ftc vu ft Cv,u, i.e., This completes the proof. ft C ft C. ¾ Taylor expansion formula for continuity flow enables one to find interesting results on such as those of the following. Theorem 3.2 et ft be a k differentiable function to t on a domain D R with D and f f. If A vu, A uv are linear for v, u E, then ft f t. 3.2 Proof et t in the Taylor formula. We know that ft f f 1! t f 2! t 2 fk t k o t k.

16 34 infan MAO Notice that and by definition, f ft t because of therefore get that f f f f 1! t f 2! f f 1! t f 2! t 2 fk t k ot k f f t 1! f 1! fk f 1! t 2 fk t k o t k f k t k o t k t f 2! t k o t t f 2! t 2 k t 2 fk t k o t k t i t i t i for any integer 1 i k. Notice that f f. We ft f t. ¾ Theorem 3.2 enables one easily getting Taylor expansion formulas by f let ft e t. Then e t e t by Theorem 3.5. Notice that e t e t and e 1. We know that t. For example, 3.3 e t e t t t 2 t k 3.4 1! 2! and e t e s et es et e s ets e ts, 3.5 where t and s are variables, and similarly, for a real number α if t < 1, 3.2 imitation t α αt 1! αα 1 α n 1t n 3.6 n! Definition 3.3 et, 1 1 i, 1 i n with, 1 dependent on a variable t [a, b], and linear continuous end-operators A vu. for v, u E For t [a, b] and any number ε >, if there is always a number δε such that if t t δε then 1 1 < ε, then, 1 is said to be converged to as t t, denoted by lim t t 1. Particularly, if is a continuity flow with a constant v for v V and t, 1 1 is said to be -synchronized.

17 Hilbert Flow Spaces with Differentials over raphs 35 Applying Theorem 1.4, we know that there are positive constants c vu R such that A vu c vu for v, u E. By definition, it is clear that 1 1 \ \ 1 1 v, u A vu 1 A vu vu v, u vu A u N 1 \ v u N 1 \ v c vu v, u u N 1 v u N 1 v c vu v, u 1 and v, u for v, u E and E. et If { c max 1 max max v,u E c vu, 1 v,u E c vu 1 u N \1 v }. u N \1 v A vu v, u c vu v, u. 1 1 < ε, we easily get that 1 v, u < c max ε for v, u E 1 1 \, ε for v, u E 1 1 v, u < c max E \ 1. and v, u < c max 1ε for v, u 1 Conversely, if v, u < ε for v, u E \, v, u < ε for v, u 1 E and v, u < ε for v, u E \ 1, we easily know that 1 1 u N 1 \v u N \1 v u N 1 \v u N \1 v A vu 1 v, u A vu v, u c vu v, u c vu v, u u N 1 v u N 1 v A vu 1 A vu vu v, u c vu v, u < 1 \ c max ε 1 1 c max ε 1 \ 1 c max ε c max 1 1 Thus, we get an equivalent condition for lim t t 1 following. 1 ε. Theorem 3.4 lim t t 1 if and only if for any number ε > there is always a number δε 1 such that if t t δε then v, u < ε for v, u E \ 1 for v, u E and v, u < ε for v, u E infinitesimal or lim t t 1 O., v, u < ε \ 1,i.e., 1 1 is an

18 36 infan MAO If lim, lim 1 1 and lim 2 2 exist, the formulas following are clearly true by definition: t t t t t t 1 lim lim 1 1 lim 2 2, t t t t t t 1 1 lim lim 1 lim 2 2, t t t t t t lim t t lim t t lim t t 1 lim t t lim t t 2 2, 1 lim lim 1 lim 2 lim 2 lim t t t t t t t t t t and furthermore, if lim t t 2 2 O, then lim t t lim t t lim t t 1. lim 2 2 t t Theorem 3.5 Hospital s rule If lim 1 1 O, lim 2 2 O and, 2 are differentiable t t t t 1 \ 2, lim t t 2v, u for v, u respect to t with lim t t 1v, u for v, u E 1 E 2 and lim t t 2 v, u for v, u E 2 \ 1, then, lim t t Proof By definition, we know that lim t t lim t t 1 lim 1 1 t t. 2 2 lim t t lim \ \ 2 1 t t lim lim t t 1 2 lim t t t t lim t t 1 lim 1 t t 2 1 \ lim t t 1 2 lim 1 lim 1 t t 2 1 t t 2 2 \ lim t t 2 1 This completes the proof. lim t t 1 1 lim 1 t t lim t t 1. lim 2 t t 2 ¾

19 Hilbert Flow Spaces with Differentials over raphs 37 Corollary 3.6 If lim O, lim 2 O and, 2 are differentiable respect to t with t t t t lim t t 2v, u for v, u E, then enerally, by Taylor formula t t t 1! lim t t 2 t t t k lim 1 t t. lim 2 t t k t o t t k, if t 1 t k 1 1 t and 2 t 2 t k 1 2 t but k 2 t, then 1 t t k 2 2 t t k We are easily know the following result. k 1 t 1 o t t k 1, k 2 t 2 o t t k 2. Theorem 3.7 If lim 1 1 O, lim 2 2 O and t t t t t 1 t k 1 1 t and 2 t 2t k 1 2 t but k 2 t, then 1 lim t t 2 2 lim t t 1 lim t t 2 k 1 t k 2 t. Example 3.8 et 1 2 C n, A v iv i1 1, A v iv i 1 2 and f i f 1 2 i 1 1 Fx n! 2 i 1 2n 1e t for integers 1 i n in Fig.4. v 1 v 2 ¹ f 1 f n f 2 v n v i1 f i Fig.4 v i f 3 v 3

20 38 infan MAO Calculation shows That v i 2f i1 f i 2 f 1 2 i 1 Fx 2 i f 1 2 i 1 1 Fx 2 i 1 n! Fx 2n 1e t. Calculation shows that lim v i Fx, i.e., lim C t t n C n, where, v i Fx for integers 1 i n, i.e., C n is -synchronized. 4. Continuity Flow Equations A continuity flow is in fact an operator : B determined by v, u B for v, u E. enerally, let [] m n 1 2 n n m1 m2 mn with ij : B for 1 i m, 1 j n, called operator matrix. Particularly, if for integers 1 i m, 1 j n, ij : R, we can also determine its rank as the usual, labeled the edge v, u by Rank[] m n for v, u E and get a labeled graph Rank[]. Then we get a result following. Theorem 4.1 A linear continuity flow equations x 1 1 x 2 2 x n n1 1 x 1 21 x 2 22 x n 2n 2 x 1 n1 x 2 n2 x n nn n 4.1 is solvable if and only if Rank[] Rank[], 4.2 where [] 1 2 n n and [ ] 1 2 n n 2. n1 n2 nn n1 n2 nn n

21 Hilbert Flow Spaces with Differentials over raphs 39 Proof Clearly, if 4.1 is solvable, then for v, u E, the linear equations x 1 1 v, u x 2 2 v, u x n n1 v, u v, u x 1 21 v, u x 2 22 v, u x n 21 v, u 2 v, u x 1 n1 v, u x 2 n2 v, u x n nn v, u n v, u is solvable. By linear algebra, there must be 1 v, u 2 v, u n v, u 21 v, u 22 v, u 2n v, u Rank n1 v, u n2 v, u nn v, u 1 v, u 2 v, u n v, u v, u 21 v, u 22 v, u 2n v, u 2 v, u Rank n1 v, u n2 v, u nn v, u n v, u, which implies that Rank[] Rank[]. Conversely, if the 4.2 is hold, then for v, u E, the linear equations x 1 1 v, u x 2 2 v, u x n n1 v, u v, u x 1 21 v, u x 2 22 v, u x n 21 v, u 2 v, u x 1 n1 v, u x 2 n2 v, u x n nn v, u n v, u is solvable, i.e., the equations 4.1 is solvable. ¾ Theorem 4.2 A continuity flow equation λ s s λ s 1 s 1 O 4.3 always has solutions λ with λ : v, u E {λ vu 1, λvu are roots of the equation α vu s λ s α vu s 1λ s 1 α vu with i : v, u α v,u i For v, u E 2,, λvu s }, where λvu i, 1 i s 4.4, α vu s a constant for v, u E and 1 i s., if n vu is the maximum number i with i v, u, then there are

22 4 infan MAO nvu solutions λ, and particularly, if s v, u for v, u E, there are v,u E E s solutions λ of equation 4.3. Proof By the fundamental theorem of algebra, we know there are s roots λ vu 1, λvu 2,, λvu s for the equation 4.3. Whence, λ is a solution of equation 4.2 because of λ s s λ s 1 s 1 λ λs s λs 1 s 1 λ λs sλ s 1 s 1 and λ s s λ s 1 s 1 : v, u α vu s λs α vu s 1 λs 1 α vu, for v, u E, i.e., λ s s λ s 1 s 1 λ O. Count the number of different λ for v, u E. It is nothing else but just n vu. Therefore, the number of solutions of equation 4.3 is nvu. ¾ v,u E Theorem 4.3 A continuity flow equation d dt α 4.5 with initial values t β always has a solution β e tα, where α : v, u α vu, β : v, u β vu are constants for v, u E. Proof A calculation shows that d dt d dt α α, which implies that for v, u E. d dt α vu 4.6 Solving equation 4.6 enables one knowing that v, u β vu e tαvu for v, u E.

23 Hilbert Flow Spaces with Differentials over raphs 41 Whence, the solution of 4.5 is and conversely, by Theorem 3.2, βe tα β e tα d βe tα dt d β e tα dt α βe tα α βe tα, i.e., d α dt if β e tα. This completes the proof. ¾ Theorem 4.3 can be generalized to the case of : v, u R n, n 2 for v, u E. Theorem 4.4 A complex flow equation d dt α 4.7 with initial values t β always has a solution β e tα, where α : v, u α 1 vu, α2 vu,, αn vu, β : v, u β 1 vu, β2 vu,, βn vu with constants α i vu, βi vu, 1 i n for v, u E. Theorem 4.5 A complex flow equation αn d n dt n d n 1 α n 1 dt n 1 α O 4.8 with αi : v, u α vu i constants for v, u E and integers i n always has a general solution λ with λ : v, u {, s i1 h i te λvu i t for v, u E, where h mi t is a polynomial of degree m i 1 on t, m 1 m 2 m s n and λ vu 1, λ vu 2,, λ vu s are the distinct roots of characteristic equation } α vu n λn α vu n 1 λn 1 α vu

24 42 infan MAO with α vu n for v, u E. Proof Clearly, the equation 4.8 on an edge v, u E is i.e., α vu d n v, u n dt n α vu d n 1 v, u n 1 dt n 1 α. 4.9 As usual, assuming the solution of 4.6 has the form e λt. Calculation shows that d λe λt λ, dt d 2 dt 2 λ 2 e λt λ 2,, d n dt n λ n e λt λ n. Substituting these calculation results into 4.8, we get that which implies that for v, u E, λ n αn λ n 1 α n 1 α O, λ n αn λ n 1 αn 1 α O, λ n α vu n λ n 1 α vu n 1 α 4.1 or v, u. et λ vu 1, λ vu 2,, λ vu s be the distinct roots with respective multiplicities m vu 1, m vu 2,, m vu s of equation 4.8. We know the general solution of 4.9 is v, u s i1 h i te λvu i t with h mi t a polynomial of degree m i 1 on t by the theory of ordinary differential equations. Therefore, the general solution of 4.8 is λ with λ : v, u {, s i1 h i te λvu i t for v, u E. ¾ }

25 Hilbert Flow Spaces with Differentials over raphs Complex Flow with Continuity Flows The difference of a complex flow with that of a continuity flow is the labeling on a vertex is v ẋ v or x v. Notice that d A vu v, u dt u N v u N v d dt A vu v, u for v V. There must be relations between complex flows and continuity flows. We get a general result following. [ t ] [ t ] Theorem 5.1 If end-operators A vu, A uv are linear with, A vu, A uv and [ ] [ ] d d dt, A vu dt, A uv for v, u E, and particularly, A vu 1 V, then i R R n, 1 i n is a continuity flow with a constant v for v V if and only if t dt is such a continuity flow with a constant one each vertex v, v V. [ t ] Proof Notice that if A vu 1 V, there always is, A vu and definition, we know that [ t ], A vu [ ] d dt, A vu t t A vu A vu, d dt A vu A vu d dt. If is a continuity flow with a constant v for v V u N v A vu v, u v for v V,, i.e., [ ] d dt, A vu, and by we are easily know that t u N v A vu v, u dt u N v u N v t A vu A vu t v, udt v, udt u N v t vdt A vu t v, udt for v V with a constant vector constant flow on each vertex v, v V. t vdt, i.e., t dt is a continuity flow with a Conversely, if t dt is a continuity flow with a constant flow on each vertex v, v

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