Papers
International Journal of Bifurcation and Chaos, Vol. 11, No. 8 (2001) 2085–2095c
World Scientific Publishing Company
SPATIAL DISORDER OF CNN
WITH ASYMMETRIC
OUTPUT FUNCTION
JUNG-CHAO BAN, KAI-PING CHIEN and SONG-SUN LIN∗ Department of Applied Mathematics,
National Chiao Tung University, Hsinchu, Taiwan CHENG-HSIUNG HSU†
Department of Mathematics,
National Central University, Chungli, Taiwan Received June 29, 2000; Revised November 15, 2000
This investigation will describe the spatial disorder of one-dimensional Cellular Neural Net-works (CNN). The steady state solutions of the one-dimensional CNN can be replaced as an iteration map which is one dimensional under certain parameters. Then, the maps are chaotic and the spatial entropy of the steady state solutions is a three-dimensional devil-staircase like function.
1. Introduction
Following their introduction by Chua and Yang [1988a, 1988b], Cellular Neural Networks have been extensively studied and applied mainly in image processing and pattern recognition [Thiran et al., 1995; Chua & Roska, 1993]. An important class of solutions of one-dimensional CNN
dxi
dt =−xi+ z + αf (xi−1) + af (xi) + βf (xi+1) , (1) is the steady state solutions, thus necessitating the study of the complexity of steady state solu-tions of (1). Juang and Lin [1998, 2000] and Hsu and Lin [1999a, 1999b, 2000] recently considered some mathematical results about the complexity of steady state solutions and multiplicity of traveling wave solutions. Hsu and Lin [1999a] considered the
output function of (1) with
f (x) = rx + 1− r if x ≥ 1 , x if|x| ≤ 1 , rx + r− 1 if x ≤ −1 . (2)
They described the spatial entropy of steady state solutions as a devil-staircase like function.
The investigation elucidates the complexity of a set of bounded steady state solutions of (1). Herein f (x) is a piecewise-linear output function defined by f (x) = rx + m− r if x ≥ 1 , mx if|x| ≤ 1 , lx + l− m if x≤ −1 , (3)
where r, m, l ∈ R+\ {0} are constants and the quantity z is called threshold, which is related to independent voltage sources in electric circuits. The ∗Work partially supported by NSC under Grant No. 89-2115-M-009-003, The Lee and MTI Center for Networking Research
and National Center for Theoretical Sciences Mathematics Division, R.O.C.
†Work partially supported by NSC under Grant No. 89-2115-M-008-012 and National Center for Theoretical Sciences Mathematics Division, R.O.C.
2085
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coefficients of f (x) are real constants and called the space invariant A-template denoted by
A≡ [α, a, β] . (4)
For simplicity, let m = 1 in (3). That is,
f (x) = rx + 1− r if x ≥ 1 , x if|x| ≤ 1 , lx + l− 1 if x≤ −1 . (5)
Let output v = f (x) be taken as the unknown variable, i.e.
vi= f (xi) , (6)
and let F be the inverse function of f . When α = 0, β6= 0, the steady state solutions of (1) can be writ-ten as a one-dimensional iteration map:
T (v) = 1
β(F (v)− z − av) . (7) For this map, each bounded trajectory corresponds to the outputs of bounded steady state solutions. If the maps are chaotic, then the steady state solu-tions of (1) are of spatial disorder. However, only steady state solutions of (1) should be considered. Therefore, in addition to considering the set of all stable bounded orbits of T , the entropy h of T on the set must be computed as well. If the entropy is positive, then the steady state solutions of (1) are of spatial disorder. In fact, we have the following main theorem:
Main Theorem. Assume that α = 0, β > 0,
z = 0, a > β + 1 and h(r, l) is the entropy function
of T with F = f−1, r, l > 0. Denote r∞= l∞= a− β − 1
a(a− 1) + β(a − 2), (8) then there exists strictly decreasing sequences {rp},
{lq}, p, q = 2, 3, . . . , with lim p→∞rp = r∞, (9) lim q→∞lq= l∞, (10) such that
(i) If 0 < r ≤ r∞ and 0 < l ≤ l∞, then h(r, l) = ln 2.
(ii) If rp ≤ r < rp−1 and lq ≤ l < lq−1, for
p, q = 3, 4, 5, . . . , then h(r, l) = ln λ(p,q), where
λ(p,q) is the maximum root of
xp+q−2− pX−2 i=0 xi q−2 X j=0 xj = 0 . (11)
(iii) If r2 ≤ r < (1)/(a + β) and l2 ≤ l <
(1)/(a + β), then h(r, l) = 0.
A table can be constructed based on the re-sults of the above theorem to contrast the entropy between different r, l as in Fig. 1. Moreover, a three-dimensional graph can be designed as shown in Fig. 2.
The paper is organized as follows. In Sec. 2, we will consider the basic propositions of T and study the steady state solutions of (1) for some range of
, (13,13) (13,12) (13,11) (13,10) (13,9) (13,8) (13,7) (13,6) (13,5) (13,4) (13,3) (13,2) (12,12) (12,11) (12,10) (12,9) (12,8) (12,7) (12,6) (12,5) (12,4) (12,3) (12,2) (11,11) (11,10) (11,9) (11,8) (11,7) (11,6) (11,5) (11,4) (11,3) (11,2) (10,10) (10,9) (10,8) (10,7) (10,6) (10,5) (10,4) (10,3) (10,2) (9,9) (9,8) (9,7) (9,6) (9,5) (9,4) (9,3) (9,2) (8,8) (8,7) (8,6) (8,5) (8,4) (8,3) (8,2) (7,7) (7,6) (7,5) (7,4) (7,3) (7,2) (6,6) (6,5) (6,4) (6,3) (6,2) (5,5) (5,4) (5,3) (5,2) (4,4) (4,3) (3,3) (3,2) (2,2) Entropy is larger (4.2) Entropy is larger Fig. 1.
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) r,l ( h r r4 r3 r ln2 l r2 l2 l3 l4 l r1 l1 Fig. 2. r1= l1= (1)/(α + β).
parameters. Section 3 will prove the main theorem and construct the table in Fig. 1.
2. Iteration Map
This section considers the one-dimensional map (7). If α = 0, a > 1, β > 0, z = 0 and m = 1, then according to (5), the inverse function F of f is
F (v) = 1 rv− 1 r + 1 if v≥ 1 , v if|v| ≤ 1 , 1 lv− 1 + 1 l if v≤ −1 , (12)
and, according to (7), the map T is
T (v) = 1 β 1 rv− 1 r + 1− av if v≥ 1 , 1 β(1− a)v if|v| ≤ 1 , 1 β 1 lv− 1 + 1 l − av if v≤ −1 . (13)
By elementary computation, the fixed points of T are A = (A1, A2) = 1− r 1− r(a + β), 1− r 1− r(a + β) , O = (O1, O2) = (0, 0) , D = (D1, D2) = l− 1 1− l(a + β), l− 1 1− l(a + β) . (14) Let B and C be the points (1, T (1)), (−1, T (−1)), i.e. B = (B1, B2) = 1, 1− a β , (15) C = (C1, C2) = −1, a− 1 β . (16)
Therefore, some graphs of T are shown in the following figures.
Now, we define the interval L by L≡ v∈ R| l− 1 1− l(a + β) ≤ v ≤ 1− r 1− r(a + β) . (17)
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(I) 1 < a < β + 1 v T(v) v=T(v) O A B C D 1 Fig. 3. Graph of T . (II) a = β + 1 v T(v) v=T(v) A B C D O 1 Fig. 4. Graph of T .
It is easy to see the stability of the fixed points of T as follows.
Proposition 2.1
(1) If a ≥ 1 with (1 − r)/(1 − r(a + β)) ≥ (a− 1)/β and (l− 1)/(1 − l(a + β)) ≤ (1− a)/β then Ω ≡ {(s, t) ∈ R2|s ∈ L and t∈ L} is an invariant region of T .
(2) If 1 < a < β + 1, then both A and D are unsta-ble, but O is stable.
(3) If a = β + 1, then every point in L is eventually periodic with period 2 except for the fixed points. (4) If a > β + 1, then O, A, D are all unstable.
By applying Proposition 2.1, the chaotic behavior of trajectories of T only occurs when a > β + 1. Next, the stability results of steady state solutions of (1) are studied. (III) a > β + 1 v=T(v) T(v) v A D C B O 1 Fig. 5. r > r∞ and l > l∞. T(v) v=T(v) v A B C D O 1 Fig. 6. r > r∞and 0 < l < l∞. v=T(v) T(v) v A B O 1 C D
Fig. 7. Graph of T with 0 < r < r∞and 0 < l < l∞.
Definition 2.2. By letting v = {vi}i=i=∞−∞ be the
steady state solutions of (1), the linearized opera-tor at v is defined by
(L(v)ζ)i=−ζi+ af
0 (vi)ζi
+ βf0(vi+1)ζi+1 for ζ ∈ `2. (18)
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v is called stable if all real parts of eigenvalues of L are negative with eigenvectors in `2 and unstable
otherwise.
Since the function f is not differentiable at |vi| = 1, (18) may not be well defined.
There-fore, only|vi| 6= 1 is considered herein, subsequently
leading to the following stability results.
Proposition 2.3. Let v ={vi}i=i=∞−∞ be the steady
state solutions of (1). Assuming that a > 1, α = 0 and β > 0 leads to
(i) If |vi| < 1 for some i ∈ Z, then v is unstable.
(ii) If r(a + β) < 1, l(a + β) < 1 and |vi| > 1 for
all i∈ Z, then v is stable.
Proof. The assertion holds by Definition 2.2
di-rectly. For details, see [Juang & Lin, 2000; Hsu & Lin, 1999a].
3. Proof of Main Theorem
According to Propositions 2.1 and 2.3, we only have to consider{Ti(v)}i=∞
i=−∞for some v ∈ L that satisfy
Ti(v)∈ L and |Ti(v)| > 1, for all i ∈ Z. (19) The entropy function h can be computed to express whether the map has chaotic behavior. In particu-lar, if the entropy is positive, then the map is called chaotic. Therefore, in this section, we attempt to compute the entropy of T at the set of all bounded stable orbits and see how the entropy h of T varies as r, l change.
We recall some definitions and some results of entropy for a dynamical system.
Definition 3.1. [Robinson, 1995]
(i) Let H : X → X be a continuous map on the space X with metric d. A set S ⊂ X is called (n, ε)-separated for H for a positive integer n and ε > 0 provided for every pair of distinct points x, y ∈ S, there is at least one k with 0≤ k < n such that d(Hk(x), Hk(y)) > ε. (ii) The number of different orbits of length n (as
measured by ε) is defined by
γ(n, ε, H) = max{](S)|S ⊂ X is (n, ε) — separated set for H} (20) where ](S) is the number of elements in S.
(iii) The topological entropy of H is defined as
(H) = lim
ε→0,ε>0lim supn→∞
ln γ(n, ε, H)
n . (21)
(iv) An interval J1H-covers an interval J2 provided
that H(J1)⊃ J2. We write J1 → J2.
Proposition 3.2. [Robinson, 1995]. Let A be a
transition matrix on N symbols. Let H : X → X be a continuous map on the space X with metric d and σA: ΣA→ ΣA be a subshift of finite type. If H
is topologically conjugate to σA, then the entropy of
H is equal to
h(H) = ln λ1 (22)
where λ1 is the real eigenvalue of A such that λ1 ≥
|λj| for all other eigenvalues λj of A.
Proposition 3.2 indicates that a subshift of fi-nite type can be found such that T is topologically conjugate to the subshift. The subshift can be con-structed by finding some subintervals of L\(−1, 1) with covering relation as in the proof of the main theorem later.
Proof of main theorem. Assume that β = 1 and
the general cases can be similarly discussed. If 0 < r≤ r∞and 0 < l≤ l∞, then C and B are not in L. Under these circumstances, the behavior of map T resembles that of the logistic map. Therefore, there exists an invariant Cantor set in L such that T is topologically conjugate to a one-side Bernoulli shift of two symbols. The entropy of the one-side Bernoulli shift of two symbols is ln 2, according to why the entropy of the map T is ln 2. That is, if 0 < r≤ r∞ and 0 < l≤ l∞ then h(r, l) = ln 2.
To prove the case r > r∞ and l > l∞. Let R+(r) = (R+1(r), R2+(r)) and R−(r) = (R−1(r), R2−(r)) be the intersecting points of AB with T (v) = 1 and T (v) = −1, respectively. Let L+(l) = (L+1(l), L+2(l)) and L−(l) = (L−1(l), L−2(l)) be the intersecting points of CD with T (v) = 1 and T (v) = −1, respectively. By simple computation, we have R+1(r) = 1 1− ra, R − 1(r) = 1− 2r 1− ra, (23) L+1(l) = 2l− 1 1− la and L − 1(l) = −1 1− la. (24)
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Then, the continuity of T (v; r, l) with respect to r, l makes it easy to prove that for any positive inte-gers p, q with p≥ 2, q ≥ 2, there exists an unique rp > 0 and lq > 0 such that{Ti(a− 1; rp, lq)}i=−i=∞∞
is a p + q periodic orbit. Indeed, rp and lq satisfy
Tp−1(a− 1; rp, lq) = 1 , (25)
Tp+q−1(a− 1; rp, lq) =−1 . (26)
Restated, (v, T (v; rp, lq)) maps C to B after p
it-erations; (v, T (v; rp, lq)) maps B to C after q
iter-ations. When p =∞, (v, T (v; r∞, lq)) maps C to
A. When q = ∞, (v, T (v; rp, l∞)) maps B to D,
where r∞and l∞are given by (8), i.e. A2 = C2 and
B2= D2. Since {T (v) = C2 = a− 1} ∩ AB = (ra− 2r − 1 1− ra , a− 1) , and {T (v) = B2= 1− a} ∩ CD = 2l− la − 1 1− la , 1− a , denote Ωr,l by Ωr,l={(v, T (v))| 2l− la − 1 1− la ≤ v ≤ ra− 2r − 1 1− ra and |T (v)| ≤ a − 1} . (27) Obviously, Ωr,l⊂ Ω. By Proposition 2.1, every
tra-jectory of T on Ω\Ωr,l will tend to A or D
back-wards. Therefore, trajectories are all that need to be considered of T on Ωr,l. Figure 8 illustrates the
5-periodic orbit of T (v; r3, l2). The 2p-periodic
or-bit of T (v; rp, lp) with p≥ 2 is given in [Hsu & Lin,
1999a]. I5 I4 I3 I2 I1 B D -1 O v T(v) A v=T(v) C
Fig. 8. Graph of T (a− 1; r3, l2) and its stable subintervals.
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Since the characteristic polynomial of the tran-sition matrix A(p, q) with q < p and p < q are the same, assume that q < p in the following process.
We define the (p + q)-stable subintervals with p, q≥ 3, p > q and r = rp, l = lq by Iq = [−1, L+1], Iq−k1 = [T−k 1−1(L+ 2), T−k1(L−2)] for k1 = 1, 2, . . . , q− 1 , Iq+1 = [−1, R−1], Iq+k2 = [T−k 2+1(R+ 2), T−k2(R−2)] for k2 = 2, 3, . . . , p.
Obviously, the (p + q)-stable subintervals have the following covering relation:
I1 → I2 → I3· · · → Iq,
Iq → Iq+ˆk1 for ˆk1 = 1, . . . , p− 1 ,
Iq+1→ Iq−ˆk2 for kˆ2 = 0, 1, . . . , q− 2
Ip+q → Ip+q−1→ · · · → Ip→ Ip−1→ · · · → Iq+1.
Therefore, the transition matrix A(p, q) of the stable subintervals is given by
A(p, q = [aij(p, q)] = 0 1 0 0 · · · 0 0 0 0 0 0 1 0 · · · 0 0 0 0 .. . ... 0 0 0 0 · · · 0 1 0 0 · · · 0 0 0 0 0 0 0 0 · · · 0 0 1 1 · · · 1 1 1 0 0 1 1 1 · · · 1 1 0 0 . . . 0 0 0 0 0 0 0 0 · · · 0 0 1 0 . . . 0 0 0 0 .. . ... 0 0 0 0 · · · 0 1 0 0 0 0 0 0 · · · 0 0 1 0 (28) Lemma 3.3. If rp < r < rp−1 and lq < l < lq−1,
then the corresponding transition matrix is the same as (28).
Proof. Pulling back from B and C to find other
points C0 and B0, see Fig. 9. T will map CC0 and BB0 into {v| |Ti(v)| < 1 for some i ∈ Z}, so CC0 and BB0 are not considered. Then the subinter-vals have the same covering relations as rp and lq.
Therefore, the corresponding transition matrix is the same as (28). The proof is complete. This study defines spaces Σp+q and ΣA by
Σp+q ={1, 2, . . . , q, q + 1, . . . , p + q}N, (29) ΣA={s ∈ Σp+q|asksk+1 = 1 for k = 0, 1, 2, . . .} (30) with a metric Σp+q by d(s, t) = ∞ X k=0 δ(sk, tk) 3k (31) for s = (s0, s1, s2, . . .) and t = (t0, t1, t2, . . .), where δ(i, j) = ( 0 if i = j , 1 if i6= j . (32) Define a subshift map on ΣA by σA(s) = t, where
tk = sk+1, i.e. σA(s0, s1, . . .) = (s1, s2, . . .). Then,
by Proposition 3.2, we have
Lemma 3.4. If rp ≤ r < rp−1 and lq ≤ l < lq−1,
then there exists an invariant set Λp+q ⊆ Ωr,l such
that T is topologically conjugate to the subshift of p + q symbols with transition matrix as in (28). Restated, T is topologically conjugate to the space (ΣA, σA).
Lemma 3.5. The characteristic polynomial P (x;
p, q) of the transition matrix A(p, q) is
P (x; p, q) = xp+q−2− pX−2 i=0 xi q−2 X j=0 xj. (33)
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T(v) v=T(v) A C C' 1 O -1 B B' v D I I1 2 I3 I4 I5 I6 Fig. 9. r3< r < r2 and l3< l < l2.
Proof. Only the special case is computed when (p, q) = (6, 4). For other p, q, P (x; p, q) can be computed
by induction. det[A(6, 4)] = −x 1 0 0 0 0 0 0 0 0 0 −x 1 0 0 0 0 0 0 0 0 0 −x 1 0 0 0 0 0 0 0 0 0 −x 1 1 1 1 1 0 0 1 1 1 −x 0 0 0 0 0 0 0 0 0 1 −x 0 0 0 0 0 0 0 0 0 1 −x 0 0 0 0 0 0 0 0 0 1 −x 0 0 0 0 0 0 0 0 0 1 −x 0 0 0 0 0 0 0 0 0 1 −x (34) = −x 1 0 0 0 0 0 0 0 0 0 −x 1 0 0 0 0 0 0 0 0 0 −x 1 0 0 0 0 0 0 0 0 0 −x 1 1 1 1 0 x 0 1 1 1 −x 0 0 0 0 0 0 0 0 0 1 −x 0 0 0 0 0 0 0 0 0 1 −x 0 0 0 0 0 0 0 0 0 1 −x 0 0 0 0 0 0 0 0 0 1 −x 0 0 0 0 0 0 0 0 0 1 −x (35)
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=−xP (5, 4) + x −x 1 0 0 0 0 0 0 0 0 −x 1 0 0 0 0 0 0 0 0 −x 1 0 0 0 0 0 0 1 1 1 −x 0 0 0 0 0 0 0 0 1 −x 0 0 0 0 0 0 0 0 1 −x 0 0 0 0 0 0 0 0 1 −x 0 0 0 0 0 0 0 0 1 −x 0 0 0 0 0 0 0 0 1 (36) =−xP (5, 4) + x(−x) −x 1 0 0 −x 1 1 1 1 (37) Let M4 = −x 1 0 0 −x 1 1 1 1 = x 2+ x + 1 , then (36) is P (x; 6, 4) = xP (x; 5, 4) + x2|M4| . (38)
Repeat the same process from (34) to (37). It is easy to see that P (x; 5, 4) = det[A(5, 4)] = −xP (x; 4, 4) + x2|M 4|. Hence, P (x; 6, 4) = x2P (x; 4, 4)− (x3+ x2)|M4| . (39) Induction produces P (x; p, 4) = xp−4P (x; 4, 4)− p−3 X i=2 xi|M4| . (40) Again, by induction P (x; p, q) = xp−qP (x; q, q)− p−q+1X i=2 xi|Mq| , (41) where Mq= (−1)q Pq−2
j=0xj. By [Hsu & Lin, 1999a],
P (x; q, q) = x2q−2−(Pqi=0−2xi)2, then by elementary computation (33) is proven.
By Proposition 3.2, the entropy of T is h(rp, lq) = ln λ(p,q), where λ(p,q) is the maximum
root of P (x; p, q).
Remark 3.6. Adjusting r when rp ≤ r < rp−1 but
0 < l≤ l∞ such that T maps C to B0 after p itera-tion. Then, new subintervals with special covering
relations and transition matrix can be found. Sim-ilar to Lemmas 3.4 and 3.5, the entropy function h can be computed. It can be discussed similarly when 0 < r≤ r∞ but lq ≤ l < lq−1.
Corollary 3.7. Let p, q ≥ 2 and p1, q1 ≥ 2
then: (1) If p + q = p1 + q1 and p− q < p1 − q1 then h(rp, lq) > h(rp1, lq1). (2) If p− q = p1 − q1 and p + q > p1 + q1 then h(rp, lq) > h(rp1, lq1). (3) If q = q1 and p > p1 then h(rp, lq) > h(rp1, lq1). Proof
(1) We may assume (p− q) + 2 = p1 − q1, then
q− q1 = 1 and P (x; p, q) = xp−qP (x; q, q)− p−q+1X i=2 xi|M1| = xp1−q1−2P (x; q, q) − p1−qX1−1 i=2 xi|Mq| .
Similar to the process from (34) to (37) and by induction, we have
P (x; q, q) = x2P (x; q− 1, q − 1)
− (x2+ 2x3+· · · + 2xq) . (42)
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Then, by elementary computation P (x; p, q) = P (x; p1, q1)− n X i=q xi. Obviously, P (λ(p1,q1); p, q) < 0, so λ(p,q) > λ(p1,q1). Hence h(rp, lq) > h(rp1, lq1).
(2) By assuming that q1 = q−1 and p1−q1 = p−q,
then
P (x; p, q) = xp−qP (x; q, q)−
p−q+1X
i=2
xi|Mq| .
By (42) and elementary computation P (x; p, q) = xp−qP (x; p1, q1) − p X i=q xi− x2|Mq1|(1 + x) . Obviously, P (λ(p1,q1); p, q) < 0, so λ(p,q) > λ(p1,q1). Hence h(rp, lq) > h(rp1, lq1).
(3) By assuming that p1 = p− 1 and q = q1 leads
to p(x; p, q) = xp−qP (x; q, q)− p−q+1X i=2 xi|Mq| = xP (x; p1, q1)− q X i=2 xi. Obviously, P (λ(p1,q1); p, q) < 0, so λ(p,q) > λ(p1,q1). Hence h(rp, lq) > h(rp1, lq1).
According to Corollary 3.7, a table can be constructed as shown in Fig. 1 to contrast the en-tropy between different p, q. The proof of the main theorem is complete.
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Appendix
This study demonstrates that{rp} and {lq} are
de-creasing sequences in p and q, respectively.
Figure 10 reveals that d = a− 2. Since d = R1+− 1, then
r2
1− r2a
= a− 2, i.e. r2(a2− a) = a − 2 .
On the other hand, Fig. 11 reveals that d(1 + r3) =
a− 2, hence r3 1− r3a (1 + r3) = a− 2 , i.e. r23a + r3(a2− a) = a − 2 . By induction, we have a p−1 X i=2 rpi + rp(a2− a) = a − 2 ,
Int. J. Bifurcation Chaos 2001.11:2085-2095. Downloaded from www.worldscientific.com
d 2
T(v) v=
d
Fig. 10. T (v) with slope 1/r2− a.
T(v) v= 2 -a d 2 d d r3
Fig. 11. T (v) with slope 1/r3− a.
and a p X i=2 rip+1+ rp+1(a2− a) = a − 2 = a pX−1 i=2 rPi + rp(a2− a) . Therefore, rpp+1a + [rp+1− rp]a· η(rp+1, rp) = 0 , where η(rp+1, rp) = p−1X i=2 iX−1 j=0 rjp+1rpi−j−1 + a− 1 . Since rpp+1> 0 and η(rp+1, rp) > 0, so rp+1−rp< 0.
That is, rp+1< rp for all p≥ 2. Similarly, lq+1 < lq
for all q≥ 2. The proof is complete.
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