Yu-Hao Liang Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan
|
|
- Adelia Willis
- 5 years ago
- Views:
Transcription
1 Detail proofs of some results in the article: Classification and evolution of bifurcation curves for the one-dimensional perturbed Gelfand equation with mixed boundary conditions Yu-Hao Liang Department of Applied Mathematics, National Chiao Tung University, Hsinchu 3, Taiwan Shin-Hwa Wang Department of Mathematics, National Tsing Hua University, Hsinchu 3, Taiwan In this article, we give detail proofs of Lemma 3.1(I)(ii) (I)(iv), (I)(vi) (I)(viii), (I)(x) (I)(xv) and (III)(v) (III)(ix) in the article Classification and evolution of bifurcation curves for the one-dimensional perturbed Gelfand equation with mixed boundary conditions 1], where these proofs are not given. To start with, we recall some definitions of functions and Lemma 3.1 therein. Remark that following labels (3.5) and (3.7) (3.19) are consistent with those in 1]. Define f(s) = exp ( ) as s a+s, F(s) = f(t)dt, θ(s) = 2F(s) sf(s) and P 1 (ρ,s) = ρf(ρ) sf(s) F(ρ) F(s), P 2(ρ,s) = ρ2 f (ρ) s 2 f (s) ρf(ρ) sf(s). (3.5) Lemma 3.1. Define M 1 (ρ) = ρf(ρ) F(ρ) for ρ >, (3.7) M 2 (ρ) = ρf (ρ) f(ρ) = a2 ρ for ρ, (3.8) (a+ρ) 2 Q(ρ,s) = θ(ρ) θ(s)+m 1 (ρ)f(ρ) F(s)] for s ρ, (3.9) This work is partially supported by the Ministry of Science and Technology of the Republic of China. Corresponding author. addresses: moonsea.am96g@g2.nctu.edu.tw (Y.-H. Liang), shwang@math.nthu.edu.tw (S.-H. Wang). 1
2 November 4, 215 R(ρ,s) = 2θ(ρ) θ(s)] f(ρ) F(ρ) ρ F(ρ) F(s)] 3/2 ρ s for s < ρ, (3.1) 8 3 K 1 (ρ) = ρ2 + 2ρ ρ (16ρ 3) for ρ, (3.11) ρ ( 8 K 2 (ρ) = 3 132a+165 ) ρ ρ+1 for ρ, (3.12) 5a+25 3 ] K 3 (ρ) = M 1 (ρ)+2ρ f(ρ) f (ρ) for ρ >, (3.13) f(ρ) L 1 (ρ,s) = 8 25 s M 2(s) 8 F(ρ) F(s) +M 1 (ρ) 1 for s ρ, (3.14) 25 f(s) L 2 (ρ,s) = P 1 (ρ,s) 8 25 s M 1(ρ) for s < ρ, (3.15) L 3 (ρ,s) = P 2 (ρ,s) 3 2 P 1(ρ,s) Then the following assertions (I) (III) hold: (I) For a >, the following assertions (i) (xv) hold: for s < ρ. (3.16) (i) f(ρ,a)(= f(ρ)) is a strictly increasing function of ρ on, ) and of a on (, ) when the other variable is fixed. Moreover, 1 < f(ρ) < min{e ρ,e a for ρ,a >. (ii) M 2 (ρ) < M 1 (ρ) for ρ >. (iii) 1 < 1 2 M 2(ρ)+2] < M 1 (ρ) for ρ >. (iv) M 1 (ρ) < M 2 (ρ)+1 for < ρ a. (v) M 1 (ρ) is strictly increasing on (,a]. (vi) M 2 (ρ) satisfies M 2(ρ) = a2 (a ρ) (a+ρ) 3 and M 2 (ρ) M 2 (a)(= a/4) for ρ >. > when ρ (,a), = when ρ = a, < when ρ (a, ), (3.17) (vii) P 1 (ρ,) = M 1 (ρ), lim s ρ P 1 (ρ,s) = M 2 (ρ)+1 for ρ > and lim ρ s +P 1 (ρ,s) = M 2 (s)+1 for s. (viii) P 1 (ρ,s) > 1 for s < ρ. (ix) For < ρ a, P 1 (ρ,s) is a strictly increasing function of s on,ρ). Moreover, M 1 (ρ) P 1 (ρ,s) < M 2 (ρ)+1 for s < ρ a, where the equality holds if and only if s =. (x) For s < a, P 1 (ρ,s) is a strictly increasing function of ρ on (s,a]. Moreover, P 1 (ρ,s) > M 2 (s)+1 for s < ρ a. (xi) P 2 (ρ,) = M 2 (ρ) and lim s ρ P 2 (ρ,s) = 2ρf (ρ)+ρ 2 f (ρ) f(ρ)+ρf (ρ) for ρ >. 2
3 November 4, 215 (xii) M 2 (ρ) < P 2 (ρ,s) for < s < ρ a. (xiii) Q(ρ,s) > for s < ρ. ] f(ρ) (xiv) + M 2(ρ)+1 f(ρ) > for < ρ a. F(ρ) F(ρ) 2ρ (xv) For ρ >, M 1 (ρ,a) and M 2 (ρ,a) are both strictly increasing functions of a on (, ). (II) For < a a (.51), R(ρ,s) > for s < ρ. Here a is the unique positive zero of a/2 e a a/ (3.18) (III) For a 4, the following assertions (i) (ix) hold: (i) R(ρ,s) < for ρ a and s < ρ. (ii) For < ρ < a, R(ρ,s) > for < s < ρ if ρ satisfies 21 M 2 (ρ)] f(ρ) M 1 (ρ) > ; while R(ρ,s) < for < s < ρ if ρ satisfies 22 M 1 (ρ)] f(ρ) <. (iii) R(ρ,s) > for s < ρ ρ 1. Here ρ 1 (.286) > 7/25 =.28 is the unique positive zero of 2(1 ρ) ρe 3ρ /(e ρ 1). (iv) R(1/2,s) < for s < 1/2. (v) K 1 (ρ) < M 1 (ρ) < K 2 (ρ) for 1/4 ρ 1/2. (vi) K 3 (ρ) < M 1 (ρ)+ 132a+165 5a+25 ρ2 for 1/4 ρ 1/2. (vii) For any ρ 1/4,1/2], there exists a unique positive s (= s (ρ,a)) on (,ρ) such that > when s (,s ), L 1 (ρ,s) = when s = s, < when s (s,ρ]. (viii) L 2 (ρ,s) for 1/4 ρ 1/2 and s < ρ. (ix) L 3 (ρ,s) for 7/25 ρ 1/2 and s < ρ. (3.19) Proof of Lemma 3.1. In the following, we would give detail proofs of Lemma 3.1 for parts (I)(ii) (I)(iv), (I)(vi) (I)(viii), (I)(x) (I)(xv) and (III)(v) (III)(ix). Proof of Lemma 3.1(I)(ii). Let n 1 (ρ) (a+ρ) 2F(ρ) ρ = (a+ρ) 2F(ρ) ρ = (a+ρ) 2 f(ρ) a 2 F(ρ) M 1 (ρ) M 2 (ρ)] ] ρf(ρ) F(ρ) a2 ρ (a+ρ) 2 (by (3.7) and (3.8)) 3
4 November 4, 215 for ρ >. Then we compute that lim ρ + n 1 (ρ) = a 2 > and n 1(ρ) = 2(a+ρ)f(ρ) > for ρ >. It implies that, for ρ >, n 1 (ρ) > and hence M 1 (ρ) > M 2 (ρ). Hence the proof of Lemma 3.1(I)(ii) is complete. Proof of Lemma 3.1(I)(iii). First, for ρ >, 1 2 M 2(ρ)+2] > 1 since M 2 (ρ) > by (3.8). Next, we show that 1 2 M 2(ρ)+2] < M 1 (ρ) for ρ >. Let n 2 (ρ) for ρ >. Then we compute that n 2 () = and ρf(ρ) 1 M 2 2(ρ)+2] F(ρ) = ρf(ρ) 1 2 ρf (ρ)/f(ρ)+2] F(ρ) n 2 (ρ) = a 2 ρ 2 (4ρ+a 2 +4a) (2ρ 2 +a 2 ρ+4aρ+2a 2 ) 2f(ρ) > forρ >. Itimpliesthat,forρ >,n 2 (ρ) > andhence 1 2 M 2(ρ)+2] < M 1 (ρ) (= ρf(ρ)/f(ρ)) by (3.7). Hence the proof of Lemma 3.1(I)(iii) is complete. Proof of Lemma 3.1(I)(iv). Let n 3 (ρ) ρf(ρ) M 2 (ρ)+1 F(ρ) = for ρ >. Then we compute that n 3 () = and ρf(ρ) ρf (ρ)/f(ρ)+1 F(ρ) n 3 (ρ) = a 2 ρ(a 2 ρ 2 ) (ρ 2 +a 2 ρ+2aρ+a 2 ) 2f(ρ) < for < ρ < a. It implies that, for < ρ a, n 3 (ρ) < and hence M 1 (ρ) (= ρf(ρ)/f(ρ)) < M 2 (ρ)+1 by (3.7). Hence the proof of Lemma 3.1(I)(iv) is complete. Proof of Lemma 3.1(I)(vi). The proof is trivial and hence it is omitted. Proof of Lemma 3.1(I)(vii). The first equality in the statement of Lemma 3.1(I)(vii) is clear while the second and the third ones follow from the L Hôspital s rule directly. Proof of Lemma 3.1(I)(viii). By integration by parts, we have that, for s < ρ, F(ρ) F(s) = ρ f(t)dt = ρf(ρ) sf(s) ρ s s tf (t)dt < ρf(ρ) sf(s)sincef (t) = a2 f(t) > (a+t) 2 for t >. Hence P 1 (ρ,s) > 1 by (3.5), which completes the proof of Lemma 3.1(I)(viii). Proof of Lemma 3.1(I)(x). Fix s on,a). Then we compute that ρ P 1(ρ,s) = f(ρ){m 2(ρ)+1] P 1 (ρ,s) > F(ρ) F(s) for s < ρ a by Lemma 3.1(I)(ix). Hence, for s < a, P 1 (ρ,s) is a strictly increasing function of ρ on (s,a]. It implies that P 1 (ρ,s) > lim ρ s +P 1 (ρ,s) = M 2 (s) + 1 for s < ρ a by Lemma 3.1(I)(vii). Hence the proof of Lemma 3.1(I)(x) is complete. Proof of Lemma 3.1(I)(xi). The first equality in the statement of Lemma 3.1(I)(xi) is clear while the second one follows from the L Hôspital s rule directly. Proof of Lemma 3.1(I)(xii). By (3.5) and (3.8), for < s < ρ a, we compute that P 2 (ρ,s) M 2 (ρ) = ρ2 f (ρ) s 2 f (s) ρf(ρ) sf(s) ρf (ρ) f(ρ) = sf(s)m 2(ρ) M 2 (s)] >, ρf(ρ) sf(s)] 4
5 November 4, 215 since M 2 (ρ) is strictly increasing on (,a] by (3.17). Hence the proof of Lemma 3.1(I)(xii) is complete. Proof of Lemma 3.1(I)(xiii). Weprovethat Q(ρ,s) = θ(ρ) θ(s)+m 1 (ρ)f(ρ) F(s)] > for s < ρ in the following three cases. Case 1: s < ρ a. We compute that Q(ρ,s) = F(ρ) F(s)]M 1 (ρ)+2 P 1 (ρ,s)] F(ρ) F(s)]{M 1 (ρ)+2 M 1 (ρ)+1] (by Lemma 3.1(I)(ii) and (I)(ix)) = F(ρ) F(s) >. (3.2) Case 2: s a < ρ. Note first that, since Q(a,s) > for s < a by (3.2) and Q(a,a) = by (3.9), we have that Q(a,s) for s a. Moreover, we compute that ρ F(ρ)Q(ρ,s)] = f(ρ)4f(ρ) 3F(s)+sf(s) M 2(ρ)F(s)] f(ρ)4f(a) 3F(s)+sf(s) M 2 (a)f(s)] (by (3.17)) f(ρ)n 4 (s), wheren 4 (s) = 4F(a) 3F(s)+sf(s) M 2 (a)f(s). Sincen 4 (a) = F(a)1+M 1 (a) M 2 (a)] > by (3.7) and Lemma 3.1(I)(ii) and since n 4 (s) = f(s) 2+M 2(s) M 2 (a)] 2f(s) < for < s < a by (3.17), we have that n 4 (s) > for s a. It follows that F(ρ)Q(ρ,s)] > for s a < ρ. Combining the fact with F(a)Q(a,s) for ρ s a as claimed above, we have that Q(ρ,s) > for s a < ρ. Case 3: a < s < ρ. Fix ρ > a. Suppose it is not true that Q(ρ,s) >, then there exists some s on (a,ρ) such that Q(ρ,s ). Note first that, by the result from Case 2 with s = a, we have that Q(ρ,a) >. On the other hand, since Q(ρ,ρ) = by (3.9) and lim s ρ s Q(ρ,s) = f(ρ)1+m 1(ρ) M 2 (ρ)] < by Lemma 3.1(I)(ii), there exists some s 1 on (s,ρ) such that Q(ρ,s 1 ) >. Hence Q(ρ,s ) < min{q(ρ,a), Q(ρ,s 1 ).It implies that Q(ρ,s) on a,s 1 ] has its global minimum at some point s 2 on (a,s 1 ). However, we compute that, for a < s < ρ, s Q(ρ,s) = f(s)1+m 1(ρ) M 2 (s)], 2 s 2Q(ρ,s) = f (s)1+m 1 (ρ) M 2 (s)]+f(s)m 2(s) = a2 (s+a) 2f(s)1+M 1(ρ) M 2 (s)] f(s) a2 (s a) (by (3.17)) (a+s) 3 a 2 (s a) = (s+a) 2 s Q(ρ,s) f(s)a2 (a+s), 3 which implies that 2 s 2 Q(ρ,s 2 ) = f(s 2 ) a2 (s 2 a) (s 2 +a) 3 < since s Q(ρ,s 2) =. Consequently, s 2 can not be a local minimum point, and we get a contradiction. Hence Q(ρ,s) > for a < s < ρ. 5
6 November 4, 215 By Cases 1 3, we complete the proof of Lemma 3.1(I)(xiii). Proof of Lemma 3.1(I)(xiv). For < ρ a, we compute that, by (3.7) and (3.8), ] f(ρ) + M 2(ρ)+1 F(ρ) 2ρ Hence the proof of Lemma 3.1(I)(xiv) is complete. f(ρ) = f(ρ) F(ρ) 2ρ F(ρ) 3M 2(ρ)+1 M 1 (ρ)] > f(ρ) 2ρ F(ρ) {3M 2(ρ)+1 M 2 (ρ)+1] (by Lemma 3.1(I)(iv)) = f(ρ) ρ F(ρ) M 2(ρ) >. Proof of Lemma 3.1(I)(xv). For ρ >, we compute that ρ2 f(ρ,a) = f(ρ,a) and a (ρ+a) 2 a M 1(ρ,a) = ρf(ρ,a) a ρ 2 f(ρ,a) (ρ+a) = ρ 2 F(ρ,a) = ρ ] ρ = ρf(ρ,a) F(ρ,a)] 2 = aρf(ρ,a) F(ρ,a)] 2 ρ ρ a M 2(ρ,a) = 2aρ2 (ρ+a) 3 >. f(ρ,a)] F(ρ,a) f(ρ,a) F(ρ,a)] a a F(ρ,a)] 2 f(s,a)ds f(ρ) ρ F(ρ,a)] 2 ] ρ 2 (ρ+a) 2 s 2 (s+a) 2 f(s,a)ds s 2 (s+a) 2 f(s,a)ds (ρ s)(aρ+2sρ+as) (ρ+a) 2 (s+a) 2 f(s,a)ds >, Hence, for ρ >, M 1 (ρ,a) and M 2 (ρ,a) are both strictly increasing functions of a on (, ), which completes the proof of Lemma 3.1(I)(xv). Proof of Lemma 3.1(III)(v). (I). We prove that K 1 (ρ) < M 1 (ρ) for 1/4 ρ 1/2 and a 4. Note first that, for 1/4 ρ 1/2, ] K 1 (ρ) = 32ρ3 +198ρ ρ (2ρ 2 +5ρ+153) > ρ2 ( 32/2+198) 5(2ρ 2 +5ρ+153) >. Hence, for 1/4 ρ 1/2 and a 4, to prove that K 1 (ρ) < M 1 (ρ,a)(= ρf(ρ,a)/f(ρ,a)) is equivalent to prove that Indeed, we compute that, ρ n 5(ρ,a) = n 5 (ρ,a) ρf(ρ,a) K 1 (ρ) F(ρ,a) >. (3.21) n 6 (ρ,a) (ρ+a) 2 (32ρ 3 198ρ 2 272ρ 7959) 2f(ρ,a), (3.22) where n 6 (ρ,a) n 6,2 (ρ)a 2 +n 6,1 (ρ)a+n 6, (ρ) 6
7 November 4, 215 and n 6,2 (ρ) = 4224ρ ρ ρ ρ ρ ρ , n 6,1 (ρ) = 248ρ ρ ρ ρ ρ ρ ρ, n 6, (ρ) = 124ρ ρ ρ ρ ρ ρ ρ 2. Then it can be verified easily (but tediously and hence omitted) that, for 1/4 ρ 1/2 and a 4, n 6 (ρ,4) = 124ρ ρ ρ ρ ρ ρ ρ ρ n 7 (ρ) >, (See Fig. 1(a).) (3.23) a n 6(ρ,4) = 248ρ ρ ρ ρ ρ ρ ρ n 8 (ρ) >, (See Fig. 1(b).) (3.24) 2 a 2n 6(ρ,a) = 8448ρ ρ ρ ρ ρ ρ n 9 (ρ) >. (See Fig. 1(c).) (3.25) It implies that, for 1/4 ρ 1/2 and a 4, n 6 (ρ,a) > and hence, by (3.22), ρ n 5(ρ,a) >. (3.26) n 7 (ρ) n 8 (ρ) n 9 (ρ) (a) ρ (b) ρ.25.5 (c) ρ Fig. 1. Function n 7 (ρ) (resp., n 8 (ρ) and n 9 (ρ)) defined in (3.23) (resp., (3.24) and (3.25)) is positive on the interval 1/4, 1/2]. 7
8 November 4, 215 Next, we show that n 5 (1/4,a) > for a 4. It is easy to see that 1/4 (6b1 48b +192)(s 1/4) 2 +b 1 (s 1/4)+b ] ds = 1 for any b,b 1 >. Thus, if we let b 9822 and b s f(1/4,a), then 2225 f(1/4,a) n 5 (1/4,a) = 1/ f(1/4,a) F(ρ,a) = K 1 (1/4) 9822 f(1/4,a) F(1/4,a) 1/4 { 2225 = 9822 f(1/4,a) (6b 1 48b +192)(s 1/4) 2 ] +b 1 (s 1/4)+b f(s,a) ds 1/4 n 1 (s,a)ds. (3.27) Note that n 1 (1/4,a) =, s n 1(1/4,a) = and we compute that, for < s < 1/4 and a 4, 2 s 2n 1(s,a) = (6b 1 48b +192)f(1/4,a) 2 s 2f(s,a) = a a 461 f(1/4,a) (4a+1) 2 s 2f(s,a) > 32 ] 2446a a 461 a 2 (a 2 2a 2s) 1637 (4a+1) 2 (a+s) 4 e 1/4 s= (since 2 s 2f(s,a) = a2 (a 2 2a 2s) f(s,a) < a2 (a 2 2a 2s) e 1/4 by Lemma 3.1(I)(i) (a+s) 4 (a+s) 4 and s a 2 (a 2 2a 2s) (a+s) 4 = 2a2 (2a 2 3a 3s) (a+s) 5 2a2 (2a 2 3a 3/4) (a+s) 5 < ) = ( e1/4 )a 3 ( e 1/4 )a 2 +(24555e 1/ )a+3274e 1/4 1637a(4a+1) 2 > a2 ( e 1/4 )a ( e 1/4 ) ] +3274e 1/4 1637a(4a+1) 2 (since 24555e 1/ ( 16777) > ) a2 4( e 1/4 ) ( e 1/4 ) ] +3274e 1/4 1637a(4a+1) 2 > >. (since e 1/4 ( 4464) > ) 3274e 1/4 1637a(4a+1) 2 (since 4( e 1/4 ) ( e 1/4 ) = e 1/4 ( 11994) > ) So n 1 (s,a) > for < s < 1/4 and a 4 and hence n 5 (1/4,a) > for a 4 by (3.27). 8
9 November 4, 215 Combining this fact with (3.26), we conclude that, for 1/4 ρ 1/2 and a 4, n 5 (ρ,a) > and hence K 1 (ρ) < M 1 (ρ,a) by (3.21). (II). We prove that M 1 (ρ) < K 2 (ρ) for < ρ 1/2 and a 4. Note first that, for ρ 1/2, K 2 (ρ,a) a+165 ] ρ ρ+1 = 31 5a+25 a= ρ2 + 2 ρ+1 >, 3 since ρ2 + 2 ρ+1] = 123/1 >. Hence, for < ρ 1/2 and a 4, to prove that 3 ρ=1/2 M 1 (ρ,a) (= ρf(ρ,a)/f(ρ,a)) < K 2 (ρ,a) is equivalent to prove that n 11 (ρ,a) F(ρ,a) ρf(ρ,a) >. (3.28) K 2 (ρ,a) By direct computations, we have that n 11 (,a) =, n ρ 11(,a) = and 2 n ρ 2 11 (,a) = 1/3 >. Hence, if we can prove that 3 n ρ 3 11 (ρ,a) < for < ρ < 1/2 and a 4 and that n 11 (1/2,a) > for a 4, then n 11 (ρ,a), a 4, is either strictly increasing or first strictly increasing and then strictly decreasing on,1/2]. Consequently, n 11 (ρ,a) > min{n 11 (,a),n 11 (1/2,a) = for < ρ < 1/2 and a 4, which completes the proof. Indeed, we compute that, for < ρ < 1/2 and a 4, where and 3 n 12 (ρ,a) ρ 3n 11(ρ,a) = (a+ρ) 6 (4ρ 2 +1ρ+15)a (295ρ 2 5ρ 75)] 4f(ρ,a), (3.29) n 12 (ρ,a) n 12,1 (ρ)a 1 +n 12,9 (ρ)a 9 +n 12,8 (ρ)a 8 +n 12,7 (ρ)a 7 +n 12,6 (ρ)a 6 +n 12,5 (ρ)a 5 +n 12,4 (ρ)a 4 +n 12,3 (ρ)a 3 +n 12,2 (ρ)a 2 +n 12,1 (ρ)a+n 12, (ρ) n 12,1 (ρ) = 256ρ 8 16ρ 7 372ρ ρ ρ ρ ρ ρ+2565 ρ 4( 256ρ 4 16ρ 3 372ρ ρ ) +( 19575ρ ) ρ 4 256ρ 4 16ρ 3 372ρ ρ ] ρ=1/2 +( 19575/ ) = ρ , n 12,9 (ρ) = 512ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ 8 ( 512ρ+24832)+ρ 4 ( ρ ) +( ρ ) ρ 8 ( 512/ )+ρ 4 ( / ) +( / ) = 24576ρ ρ , 9
10 November 4, 215 n 12,8 (ρ) = 256ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ 9 ( 256ρ ) ρ 9 ( 256/ ) = ρ 9, n 12,7 (ρ) = 7756ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ] ρ=1/2 = ρ 9 /4, n 12,6 (ρ) = 512ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ 9 ( ρ )+ρ 5 ( ρ ) ρ 9 ( / )+ρ 5 ( / ) = ρ ρ 5 / , n 12,5 (ρ) = 1514ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ 1 ( 1514ρ )+ρ 8 ( ρ ) + ( ρ ) ρ 1 ( 1514/ )+ρ 8 ( / ) +ρ 6 ( / ) = ρ ρ ρ , n 12,4 (ρ) = 16788ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ] ρ=1/2 = /124 > 12511, 1
11 November 4, 215 n 12,3 (ρ) = ρ ρ ρ ρ ρ ρ ρ ρ ρ 3 ρ 1 ( ρ )+ρ 8 ( ρ ) ρ 1 ( / )+ρ 8 ( / ) ρ ρ 8, n 12,2 (ρ) = ρ ρ ρ ρ ρ ρ ρ ρ ρ 3 ρ 9 ( ρ ) ρ 9 ( / ) = ρ 9, n 12,1 (ρ) = ρ ρ ρ ρ ρ ρ 5 ρ 5 ( ρ ) ρ 5 ( / ) = ρ 6, n 12, (ρ) = ρ ρ ρ ρ 6 ρ 6( ρ ρ ) ρ ρ ρ ] ρ=199125/( ) = ρ 6. Above inequalities imply that, for a 4 and ρ 1/2, n 12 (ρ,a) > ( a a a a 4) = a ( a 9 a 6) ( a 9 a 5) (a 9 a 4 ) > and hence, by (3.29), 3 ρ 3n 11(ρ,a) <. (3.3) Next, we show that n 11 (1/2,a) > for a 4. It is easy to see that 1/2 (3b1 12b +24)(s 1/2) 2 +b 1 (s 1/2)+b ] ds = 1 11
12 November 4, 215 for any b,b 1 >. If we let b 268a+35 and b 1a a+35 s f(1/2), then 1a+5 f(1/2) n 11 (1/2,a) 1 1a+5 = F(1/2,a) f(1/2,a) = F(1/2,a) 2K 2 (1/2,a) 268a+35 f(1/2,a) 1/2 { = f(s,a) 1a+5 268a+35 f(1/2,a) (3b 1 12b +24)(s 1/2) 2 ] +b 1 (s 1/2)+b 1/2 n 13 (s,a)ds. ds (3.31) Note that n 13 (1/2,a) =, s n 13(1/2,a) = and we compute that, for < s < 1/2 and a 4, 2 s 2n 13(s,a) = 2 2a+1 s2f(s,a) 268a+35 (3b 1 12b +24)f(1/2,a) = 2 24(4a3 23a 2 192a 65) s 2f(s,a) f(1/2,a) (268a+35)(2a+1) ] 2 a 2 (a 2 2a 2s) 24(4a 3 23a 2 192a 65) > f(s,a)] (a+s) 4 a=4 + a=4 (268a+35)(2a+1) 2 (by Lemma 3.1(I)(i) and since 2 a2 (a 2 2a 2s) s 2f(s,a)= (a+s) 4 a 2 (a 2 2a 2s) = 2a(1+2s)a2 sa 2s 2 ] a (a+s) 4 (a+s) ] 5 d 24(4a 3 23a 2 192a 65) da (268a+35)(2a+1) 2 = 16( 2s+8) e 4s 28 4+s (s+4) 4 ] 16( 2s+8) e 4s (s+4) 4 4+s 28 s=1/2 117 e1/2 (since d ] 16( 2s+8) e 4s ds (s+4) 4 4+s = e4/ e1/2 (.9) >. 117 e1/2 ] e 1/2 a=4 f(s,a), 2a ( a 2 1 a 1 2 2) >, (a+s) 5 = a3 +14a a+99 (268a+35) 2 (2a+1) 3 > ) = 32(3s2 24s 16) e 4s (s+4) 6 32(3/4 16) 4+s < (s+4) 6 e 4s 4+s < ) So n 13 (s,a) > for < s < 1/2 and a 4 and hence n 13 (1/2,a) > by (3.31). Combining this fact with (3.3), we conclude that, for 1/4 ρ 1/2 and a 4, n 11 (ρ,a) > and hence M 1 (ρ,a) < K 2 (ρ,a) by (3.28). Hence the proof of Lemma 3.1(III)(v) is complete. Proof of Lemma 3.1(III)(vi). Let n 14 (ρ,a) 1 { M 1 (ρ,a)+ 132a+165 ] ρ 5a+25 ρ2 K 3 (ρ,a) = 132a+165 5a+25 ρ+2 a 2 aρ 2ea+ρ (a+ρ) 2 12
13 November 4, 215 for < ρ 1/2 and a 4. Then lim ρ +n 14(ρ,a) =, (3.32) and n 14 (1/2,a) = (664a2 +462a+165) 1(4a 2 +4a+1)e a 2a+1 n 15 (α), = 8α (2a+1) 2 25 α eα whereα a 2a+1 satisfies4/9 α 1/2fora 4. Wehavethatn 15(α) > for4/9 α 1/2 since n 15 (4/9) = 12637/45 2e 4/9 (.1) >, n 15 (4/9) = 79/225 2e4/9 (.32) > and n 15(α) = 16 2e α > 16 2e 1/2 ( 12.73) > for 4/9 α 1/2. Hence n 14 (1/2,a) > (3.33) for a 4. Moreover, we compute that, for < ρ 1/2 and a 4, 2 ρ 2n 14(ρ,a) = 2a2 (a+ρ) 4 (a 2 2a 2ρ)e aρ a+ρ 6 ] 2a2 (a+ρ) 4 { a 2 2a 2/2 ] a=4 6 = 2a2 (a+ρ) 4 <. Combing this fact with (3.32) and (3.33), we conclude that, for < ρ 1/2 and a 4, n 14 (ρ,a) > and hence K 3 (ρ) < M 1 (ρ)+ 132a+165 5a+25 ρ2. Hence the proof of Lemma 3.1(III)(vi) is complete. Proof of Lemma 3.1(III)(vii). We first show that (3.19) holds for function L 1 (ρ,s) at the two endpoints s = and ρ when ρ is fixed on 1/4,1/2] and a 4. That is, L 1 (ρ,) > and L 1 (ρ,ρ) < for 1/4 ρ 1/2 and a 4. Indeed, we compute that, for 1/4 ρ 1/2 and a 4, L 1 (ρ,) = 8 25 F(ρ,a)+M 1(ρ,a) 1 > 8 ρ e s ds+k 1 (ρ) 1 (by Lemma 3.1(I)(i) and (III)(v)) 25 = (eρ 3 1)+ ρ2 + 2ρ (16ρ 3) 1 (by (3.11)) ρ ρ n 16 (ρ) >. (see Fig. 2.) (3.34) Remark that the proof of (3.34) is easy but tedious and hence we omit it here. On the other hand, for 1/4 ρ 1/2 and a 4, since L 1 (ρ,ρ) = 8 25 ρ M 2(ρ,a) + M 1 (ρ,a) 1 = ρf(ρ,a) F(ρ,a) M 2 (ρ,a) ρ] by (3.7) and M 2 (ρ,a) ρ M 2(ρ,a) = M 2(ρ,a) >, we have that L 1(ρ,ρ) < if and only if F(ρ,a) ρf(ρ,a) M 2 (ρ,a)+1 8 >. (3.35) ρ 25 13
14 November 4, n 16 (ρ) Fig. 2. Function n 16 (ρ) defined in (3.34) is positive on the interval 1/4,1/2]. ρ By direct computation, we have that F(ρ,a) a = ρ ρf(ρ,a) M 2 (ρ,a) ρ ] ( 17ρ+25)a 2 +16ρ 2 a+(8ρ 3 25ρ 2 ) a f(ρ,a)ds+25ρ3 f(ρ,a) (17ρ+25)a 2 (16ρ 5)ρa (8ρ 25)ρ 2 ] >, 2 since a f(ρ,a) by Lemma 3.1(I)(i) and since ( 17ρ+25)a2 + 16ρ 2 a + (8ρ 3 25ρ 2 ) 33 2 a2 + a 21 4 > for 1/4 ρ 1/2 and a 4. Hence L 1(ρ,ρ) < for 1/4 ρ 1/2 and a 4 if (3.35) holds for 1/4 ρ 1/2 and a = 4 or equivalently, L 1 (ρ,ρ)] a=4 < for 1/4 ρ 1/2. Indeed, by Lemma 3.1(III)(v), we have that L 1 (ρ,ρ)] a=4 = 8 25 ρ M 2(ρ)+M 1 (ρ) ρ M 2(ρ)+K 2 (ρ) 1 ρ ( = 31ρ ρ 2 96ρ+16 ) ρ ( < 174ρ 2 96ρ+16 ) 75(4+ρ) 2 75(4+ρ) 2 ρ 174ρ 2 96ρ+16 ] 75(4+ρ) = 8ρ 2 ρ=96/(2 174) 2175(4+ρ) < 2 for 1/4 ρ 1/2. Hence we have that for 1/4 ρ 1/2 and a 4. L 1 (ρ,ρ) < (3.36) Now we show that (3.19) holds in the following two cases, which completes the proof. 14
15 November 4, 215 Case 1: 4 a 5. Fix 1/4 ρ 1/2. Then we compute that, for < s < ρ, 2 s 2L 1(ρ,s) = 8 { a 2 (a 2 +2a+2s) 4a 2 +( 8s+5)a (4s 2 +25s) 25 (a+s) 4 f(s,a) 4(a 2 +2a+2s) 8 { ] a 2 (a 2 +2a+2s) 4a 2 +( 8s+5)a (4s 2 +25s) 25 (a+s) 4 f(s,a) 4(a 2 +2a+2s) (since ] 4a 2 +( 8s+5)a (4s 2 +25s) a 4(a 2 +2a+2s) f(s,a) F(ρ,a) F(s,a)] f(s,a) f(s,a)(ρ s) a=5 = (29 4s)a2 (4s 2 +17s)a+(4s 2 75s) 27a2 19a (a 2 +2a+2s) 2 2(a 2 +2a+2s) < and 2 by the Mean Value Theorem for some s s,ρ],1/2]) 8 ] a 2 (a 2 +2a+2s) 4s 2 65s+15 e 1/2 (1/2 s) 25 (a+s) 4 f(s) 8s+14 (since 4s 2 65s+15 4s 2 65s+15 ] = 233/2 > and s=1/2 1 f(s,a) < e 1/2 by Lemma 3.1(I)(i)) 2a 2 (a 2 +2a+2s) = (8e 1/2 4)s 2 +(136e 1/2 65)s+(15 7e 1/2 ) ] >, 25(2s+35)(a+s) 4 f(s) since 8e 1/2 4( 9) >, 136e 1/2 65( 159) > and 15 7e 1/2 ( 35) >. Combining this fact with (3.34) and (3.36), we conclude that (3.19) holds for 4 a 5. Case 2: a > 5. Fix 1/4 ρ 1/2. Then we compute that, for < s < ρ, s L 1(ρ,s) { = 8 25 a2 (a s) (a+s) 8 1 f(s,a) s 3 25 f(s,a)] 2 F(ρ,a) F(s,a)] = 8 a { 2 9a3 73sa 2 48s 2 a 16s 3 f(s,a)+f(ρ,a) F(s,a)] 25(a+s) 2 f(s,a) 8a 2 (a+s) 8 ] a { 2 9a3 73sa 2 48s 2 a 16s 3 f(s,a)+f(s 25(a+s) 2 f(s,a) 8a 2,a)(ρ s) (a+s) a=5 (since ] 9a3 73sa 2 48s 2 a 16s 3 a 8a 2 (a+s) = s(41a3 +48sa 2 +48s 2 a+16s 3 ) 4a 3 (a+s) 2 < and by the Mean Value Theorem for some s s,ρ],1/2]) 8 ] a 2 16s3 24s s e 1/2 (1/2 s) 25(a+s) 2 f(s,a) 2(5+s) (since 16s 3 24s s s 3 24s s+1125 ] s=1/2 = 31/2 > and 1 f(s,a) < e 1/2 by Lemma 3.1(I)(i)) = a2 16s 3 (2e 1/2 24)s 2 +(1825 9e 1/2 )s (1125 5e 1/2 ) ] 625(5+s)(a+s) 2 f(s,a) 15
16 a2 (2e 1/2 24)s 2 +(1825 9e 1/2 )s ( 16/ e 1/2 ) ] 625(5+s)(a+s) 2 f(s,a) a2 (1825 9e 1/2 )/2 (1123 5e 1/2 ) ] 625(5+s)(a+s) 2 f(s,a) (since 2e 1/2 24 ( 9) > and (1825 9e 1/2 ) ( 341) > ) = a2 421/2 5e 1/2] 625(5+s)(a+s) 2 f(s,a) <, November 4, 215 since 421/2 5e 1/2 ( 128) >. Combining this fact with (3.34) and (3.36), we conclude that (3.19) holds for a > 5. The proof of Lemma 3.1(III)(vii) is now complete. Proof of Lemma 3.1(III)(viii). Fix a 4 and 1/4 ρ 1/2. Let s on (,ρ) be defined in (3.19). We claim first that L 2 (ρ,s) > for < s < s. Suppose it is not true. Then there exists some s on (,s ) such that L 2 (ρ,s ) =. Consequently, since L 2 (ρ,) = M 1 (ρ) M 1 (ρ) = and lim s + s L 2(ρ,s) = M 2 (ρ) ρ M 1(ρ) = L 1 (ρ,ρ) > by Lemma 3.1(I)(vii), we have that there exists some t on (,s ] such that L 2 (ρ,t ) = and L s 2(ρ,t ). Note that, by the definition of L 2 (ρ,s) in (3.15), we have that P 1 (ρ,t ) = 8 t 25 +M 1 (ρ). Moreover, we compute that s L 2(ρ,t ) = f(t ) P 1 (ρ,t ) 1 M 2 (t ) 8 ] F(ρ) F(t ) F(ρ) F(t ) 25 f(t ) f(t ) 8 = F(ρ) F(t ) 25 t +M 1 (ρ) 1 M 2 (t ) 8 25 f(t ) = F(ρ) F(t ) L 1(ρ,t ) >, F(ρ) F(t ) f(t ) by (3.14) and (3.19), which makes a contradiction with s L 2(ρ,t ). Hence L 2 (ρ,s) > for < s < s. By similar arguments, we can prove that L 2 (ρ,s) > for s < s < ρ. Hence L 2 (ρ,s) for s < ρ, which completes the proof of Lemma 3.1(III)(viii). Proof of Lemma 3.1(III)(ix). We would show that L 3 (ρ,s) > for 7/25 ρ 1/2, < s < ρ and a 4 by the following strategy. For fixed ρ on (7/25,1/2) and a 4, suppose that there exists some s on (,ρ) such that L 3 (ρ,s ) =. Then we claim that it must hold that s L 3(ρ,s ) <. However, it makes a contradiction with lim s ρ L 3 (ρ,s) > as claimed below. Claim 1: lim s ρ L 3 (ρ,s) > for 7/25 ρ 1/2 and a 4. Proof of Claim 1. By Lemma 3.1(I)(vii) and (I)(xi), for 7/25 ρ 1/2 and a 4, we ] 16
17 November 4, 215 have that lim L 3(ρ,s) = ( 25ρ2 +24ρ 1)a 4 (52ρ 2 +4ρ)a 3 (76ρ 3 +6ρ 2 )a 2 (4ρ 3 )a ρ 4 s ρ 5(a+ρ) 2 (ρ+1)a 2 +2ρa+ρ 2 ] > ( 25ρ2 +24ρ 1)a 4 (52ρ 2 +4ρ)a 3 11a 2 a 1 5(a+ρ) 2 (ρ+1)a 2 +2ρa+ρ 2 ] n 17 (ρ,a) 5(a+ρ) 2 (ρ+1)a 2 +2ρa+ρ 2 ] >, since n 17 (ρ,a) > which follows from the facts that, for 7/25 ρ 1/2, n 17 (ρ,4) = 9728ρ ρ 449 a n 17(ρ,4) = 8896ρ ρ a 2n 17(ρ,4) = 648ρ ρ a 3n 17(ρ,4) = 2712ρ ρ 96 4 a 4n 17(ρ,a) = 6ρ ρ 24 So Claim 1 holds. ( min 9728ρ ρ 449 ) = 63 >, ρ {7/25,1/2 ( 8896ρ ρ 348 ) = 44 >, min ρ {7/25,1/2 min ρ {7/25,1/2 min ρ {7/25,1/2 min ρ {7/25,1/2 ( 648ρ ρ 214 ) = 53 >, ( 2712ρ ρ 96 ) = 26112/625 >, ( 6ρ ρ 24 ) = 2256/25 >. Claim 2: For 7/25 ρ 1/2 and a 4, if there exists some s on (,ρ) such that L 3 (ρ,s ) =, then s L 3(ρ,s ) >. Proof of Claim 2. Fix ρ on (7/25,1/2) and a 4. Suppose s be an arbitrary point on (,ρ) such that L 3 (ρ,s ) =. Then, by (3.5) and (3.16), we have that ρ 2 f (ρ) s 2 f (s ) = 3 ρf(ρ) s f(s )] F(ρ) F(s ) 25 ρf(ρ) s f(s )]. 17
18 November 4, 215 Moreover, where s L 3(ρ,s ) = 3 f(s )ρf(ρ) s f(s )] 2 F(ρ) F(s )] f(s )+s f (s ) 2s f (s )+s 2 f (s ) 2 F(ρ) F(s ) ρf(ρ) s f(s ) + f(s )+s f (s )]ρ 2 f (ρ) s 2 f (s )] ρf(ρ) s f(s )] 2 = 3 f(s )ρf(ρ) s f(s )] 2 F(ρ) F(s )] { + f(s )+s f (s ) ρf(ρ) s f(s )] f(s )+s f (s ) F(ρ) F(s ) ρf(ρ) s f(s )] 2 F(ρ) F(s ) = 3 f(s )ρf(ρ) s f(s )] 2 F(ρ) F(s )] 2 +3 f(s )+s f (s ) F(ρ) F(s ) = 2s f (s )+s 2 f (s ) ρf(ρ) s f(s ) ρf(ρ) s f(s )] 2s f (s )+s 2 f (s )+ 37 f(s 25 )+s f (s )] ρf(ρ) s f(s ) { f(s ) 3 ρf(ρ) s f(s ) 2 P 1(ρ,s )] 2 3M 2 (s )+1]P 1 (ρ,s )+ s 2 f (s )+ 87s 25 f (s )+ 37f(s 25 ) (by (3.5) and (3.8)) f(s ) f(s ) ρf(ρ) s f(s ) n 18(ρ,s ), (3.37) n 18 (ρ,s) 3 2 P 1(ρ,s)] 2 3M 2 (s)+1]p 1 (ρ,s)+ s2 f (s) sf (s)+ 37 f(s) 25. (3.38) f(s) Hence to prove that s L 3(ρ,s ) <, it suffices to prove n 18 (ρ,s) > for < s < ρ, 7/25 ρ 1/2, and a 4. We show it by the following Claims 2(a) (c), which completes the proof of Claim 2. Claim 2(a): For < s < 1/2 and a 4, n 18 (ρ,s) is a strictly increasing function of ρ on (s,1/2). Proof of Claim 2(a). Fix s on (,1/2) and a 4. Then we compute that, for s < ρ < 1/2(< a), ρ n 18(ρ,s) = 3{P 1 (ρ,s)] M 2 (s)+1] ρ P 1(ρ,s) = 3{P 1 (ρ,s)] M 2 (s)+1] f(ρ){m 2(ρ)+1] P 1 (ρ,s) F(ρ) F(s) >, by Lemma 3.1(I)(ix) and (I)(x). So Claim 2(a) holds. By Claim 2(a), for a 4, to prove n 18 (ρ,s) > for < s < ρ and 7/25 ρ 1/2, it suffices to prove that lim ρ s +n 18 (ρ,s) > for 7/25 s < 1/2, and n 18 (7/25,s) > for < s < 7/25. We show them by the following Claims 2(b) and 2(c), respectively. Claim 2(b): lim ρ s +n 18 (ρ,s) > for 7/25 s < 1/2 and a 4. 18
19 November 4, 215 Proof of Claim 2(b). By Lemma 3.1(I)(vii), we compute that lim ρ s +n 18(ρ,s) = 3 2 M 2(s)+1] 2 3M 2 (s)+1] 2 + s2 f (s) sf (s)+ 37f(s) 25 f(s) = ( 25s2 +24s 1)a 4 (52s 2 +4s)a 3 (76s 3 +6s 2 )a 2 4s 3 a s 4 5(a+s) 4 > for a 4. Above inequality holds by the exactly same arguments given in Claim 1. So Claim 2(b) holds. Claim 2(c): n 18 (7/25,s) > for < s < 7/25 and a 4. Proof of Claim 2(c). Let k = 3/25. Then, by (3.38), we have that { 3 n 18 (7/25,s) = 2 P 1(7/25,s)] 2 3(k +1)P 1 (7/25,s) + {3k M 2 (s)]p 1 (7/25,s)+ s2 f (s) sf (s)+ 37f(s) 25 f(s) 3 2 (1+k)2 + {3k M 2 (s)]p 1 (7/25,s)+ s2 f (s) sf (s)+ 37f(s) 25 f(s) (since 3 ] 3 2 x2 3(k +1)x 2 x2 3(k +1)x x=k (1+k)2 for any k,x R) { 3 = 3k M 2 (s)]p 1 (7/25,s) 2 (1+k)2 s2 f (s) sf (s)+ 37 f(s) 25 f(s) n 19 (s)p 1 (7/25,s) n 2 (s) n 21 (s). (3.39) Note first that n 19 () = 3k = 9/25 >, n 19 (7/25) = 3 25a2 15a 147 < for a 4, and 25(25a+7) 2 n 19 (s) is strictly decreasing on (,7/25) by Lemma 3.1(I)(vi). Thus there exists some s 1 on (, 7/25) such that > when s (,s 1 ), n 19 (s) = when s = s 1, < when s (s 1,7/25]. Moreover, since n 19 (s 1 ) = 3k M 2 (s 1 )] =, i.e., M 2 (s 1 ) = k, we have that, for a 4, n 2 (s 1 ) = s2 1f (s 1 ) f(s 1 ) = f (s 1 )f(s 1 ) f (s 1 )] 2 87 s 1 f (s 1 ) f(s 1 ) (1+k)2 s1 f (s 1 ) f(s 1 ) ] 2 87 s 1 f (s 1 ) f(s 1 ) (1+k)2 = a2 2a 2s 1 M a 2 2 (s 1 )] M 2(s 1 ) (1+k)2 (by (3.8)) ] a 2 2a 2s k 2 87 a 2 25 k (1+k)2 (since < s 1 < 7/25) s=7/25 = 475a2 45a a 2 475a2 45a 126] a= a 2 = a 2 <. 19 (3.4)
20 November 4, 215 It follows, by (3.39), that n 18 (7/25,s 1 ) = n 2 (s 1 ) >. Next, we show that n 18 (7/25,s) > for s (,s 1 ) (s 1,7/25) and a 4. By (3.39), it suffices to show that n 21 (s) = n 19 (s) P 1 (7/25,s) n ] 2(s) > (3.41) n 19 (s) for s (,s 1 ) (s 1,7/25) and a 4. We first compute that ] d n2 (s) 5a 2 n 22 (s,a) = ds n 19 (s) 3(a+s) 3 (25s 3)a 2 6sa 3s 2 ] 2, where n 22 (s,a) = ( 125s 2 +3s+2)a 5 +(125s 3 +28s 2 54s)a 4 +(22s 3 146s 2 )a 3 (3s 4 +94s 3 )a 2 +24s 4 a+28s 5 a 2 ( 125s 2 +3s+2)a 3 +(28s 2 54s)a 2 +( 146s 2 )a /78125 ] (since 28s 2 54s ] s=1/2 = /78125) a 2 n 23 (s,a) >, since, for < s < 7/25 and a 4, n 23 (s,4) = 414s s / s s /78125 ] = /78125 >, s=7/25 a n 23(s,4) = 396s 2 +18s s 2 +18s+96 ] = 456/625 >, s=7/25 2 a 2n 23(s,4) = 244s s s s+48 ] = 358/125 >, s=7/25 3 a 3n 23(s,a) = 75s 2 +18s+12 75s 2 +18s+12 ] = 18/5 >. s=7/25 ] Hence d n2 (s) ds n 19 < for s (,s (s) 1 ) (s 1,7/25) and a 4. Moreover, since P 1 (7/25,s) is strictly increasing on,7/25] by Lemma 3.1(I)(x), P 1 (7/25,s) n 2(s) n 19 is strictly increasing (s) on (,s 1 ) and (s 1,7/25). Hence by (3.4) and (3.39), n 18 (7/25,s) > for s (,s 1 ) if n 21 () >. Similarly, n 18 (7/25,s) > for s (s 1,7/25) if lim s (7/25) n 21 (s) >. Indeed, we 2
21 November 4, 215 compute that n 21 () = 3kM 1 (7/25) ] 2 (1+k)2 > 3kK 1 (7/25) ] 2 (1+k)2 (by Lemma 3.1(III)(v)) = 3k = >, and, by Lemma 3.1(I)(vii), (1+k)2 = ( 1+ 7 ) 2 25 lim s (7/25) n 21(s) = a4 175a a a (25a+7) 4 So Claim 2(c) holds. > a4 ( )a 3 625(25a+7) 4 ( a )a3 625(25a+7) 4 >. By Claims 1 and 2, we have that L 3 (ρ,s) > for < s < ρ, 7/25 ρ 1/2 and a 4, which completes the proof of Lemma 3.1(III)(ix). References 1] Y.-H. Liang, S.-H. Wang, Classification and evolution of bifurcation curves for the onedimensional perturbed Gelfand equation with mixed boundary conditions, submitted. 21
Exact multiplicity results for a p-laplacian problem with concave convex concave nonlinearities
Nonlinear Analysis 53 (23) 111 137 www.elsevier.com/locate/na Exact multiplicity results for a p-laplacian problem with concave convex concave nonlinearities Idris Addou a, Shin-Hwa Wang b; ;1 a Ecole
More informationAP Exercise 1. This material is created by and is for your personal and non-commercial use only.
1 AP Exercise 1 Question 1 In which of the following situations, does the list of numbers involved make an arithmetic progression, and why? (i) The taxi fare after each km when the fare is Rs 15 for the
More informationAIMS OES - Author - Decision
AIMS OES - Author - Decision http://www.aimsciences.org/oes/ar/arppmsgpop.jsp?mode=16&jid=4&pid=945 頁 1 / 1 212/9/1 Close -----------------------------------------------------------------------------------
More informationExact multiplicity of boundary blow-up solutions for a bistable problem
Computers and Mathematics with Applications 54 (2007) 1285 1292 www.elsevier.com/locate/camwa Exact multiplicity of boundary blow-up solutions for a bistable problem Junping Shi a,b,, Shin-Hwa Wang c a
More informationStochastic integral. Introduction. Ito integral. References. Appendices Stochastic Calculus I. Geneviève Gauthier.
Ito 8-646-8 Calculus I Geneviève Gauthier HEC Montréal Riemann Ito The Ito The theories of stochastic and stochastic di erential equations have initially been developed by Kiyosi Ito around 194 (one of
More informationTurán s problem and Ramsey numbers for trees. Zhi-Hong Sun 1, Lin-Lin Wang 2 and Yi-Li Wu 3
Colloquium Mathematicum 139(015, no, 73-98 Turán s problem and Ramsey numbers for trees Zhi-Hong Sun 1, Lin-Lin Wang and Yi-Li Wu 3 1 School of Mathematical Sciences, Huaiyin Normal University, Huaian,
More informationOptimal Control. Macroeconomics II SMU. Ömer Özak (SMU) Economic Growth Macroeconomics II 1 / 112
Optimal Control Ömer Özak SMU Macroeconomics II Ömer Özak (SMU) Economic Growth Macroeconomics II 1 / 112 Review of the Theory of Optimal Control Section 1 Review of the Theory of Optimal Control Ömer
More informationarxiv: v1 [math.ap] 29 May 2018
Non-uniqueness of admissible weak solution to the Riemann problem for the full Euler system in D arxiv:805.354v [math.ap] 9 May 08 Hind Al Baba Christian Klingenberg Ondřej Kreml Václav Mácha Simon Markfelder
More informationON THE MULTIPLICITY OF EIGENVALUES OF A VECTORIAL STURM-LIOUVILLE DIFFERENTIAL EQUATION AND SOME RELATED SPECTRAL PROBLEMS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 127, Number 1, Pages 2943 2952 S 2-9939(99)531-5 Article electronically published on April 28, 1999 ON THE MULTIPLICITY OF EIGENVALUES OF A VECTORIAL
More informationDiscrete Mathematics, Spring 2004 Homework 9 Sample Solutions
Discrete Mathematics, Spring 00 Homework 9 Sample Solutions. #3. A vertex v in a tree T is a center for T if the eccentricity of v is minimal; that is, if the maximum length of a simple path starting from
More informationPLUMBING NOTES: ALL NATURAL GAS APPURTENANCES. COMPONENT IMPORTANCE FACTOR (Ip) NOTES FLOOR MOUNTED RESTRAIN ALL RESTRAIN ALL - 1, 2
II I QUI UI Y I-00 / -0. QUI, I (IU U & I) X I I I I U I I I II UII. II I I QUI, I II U U. II II I I UU.... I II II II Y. QUI U II I II. U I I II I QUI I. QUII II I, I Y I I I I U I. UI U IU I I UI. II
More informationINTRODUCTION TO ALGEBRAIC TOPOLOGY. (1) Let k < j 1 and 0 j n, where 1 n. We want to show that e j n e k n 1 = e k n e j 1
INTRODUCTION TO ALGEBRAIC TOPOLOGY Exercise 7, solutions 1) Let k < j 1 0 j n, where 1 n. We want to show that e j n e k n 1 = e k n e j 1 n 1. Recall that the map e j n : n 1 n is defined by e j nt 0,...,
More informationTHE EXISTENCE OF GLOBAL ATTRACTOR FOR A SIXTH-ORDER PHASE-FIELD EQUATION IN H k SPACE
U.P.B. Sci. Bull., Series A, Vol. 8, Iss. 2, 218 ISSN 1223-727 THE EXISTENCE OF GLOBAL ATTRACTOR FOR A SIXTH-ORDER PHASE-FIELD EQUATION IN H k SPACE Xi Bao 1, Ning Duan 2, Xiaopeng Zhao 3 In this paper,
More informationName (please print) Mathematics Final Examination December 14, 2005 I. (4)
Mathematics 513-00 Final Examination December 14, 005 I Use a direct argument to prove the following implication: The product of two odd integers is odd Let m and n be two odd integers Since they are odd,
More informationBetter bounds for k-partitions of graphs
Better bounds for -partitions of graphs Baogang Xu School of Mathematics, Nanjing Normal University 1 Wenyuan Road, Yadong New District, Nanjing, 1006, China Email: baogxu@njnu.edu.cn Xingxing Yu School
More informationBrownian Motion and Conditional Probability
Math 561: Theory of Probability (Spring 2018) Week 10 Brownian Motion and Conditional Probability 10.1 Standard Brownian Motion (SBM) Brownian motion is a stochastic process with both practical and theoretical
More informationTheory of Computation
Thomas Zeugmann Hokkaido University Laboratory for Algorithmics http://www-alg.ist.hokudai.ac.jp/ thomas/toc/ Lecture 13: Algorithmic Unsolvability The Halting Problem I In the last lecture we have shown
More informationTopological properties
CHAPTER 4 Topological properties 1. Connectedness Definitions and examples Basic properties Connected components Connected versus path connected, again 2. Compactness Definition and first examples Topological
More informationON HOW TO CONSTRUCT LEFT SEMIMODULES FROM THE RIGHT ONES
italian journal of pure and applied mathematics n. 32 2014 (561 578) 561 ON HOW TO CONSTRUCT LEFT SEMIMODULES FROM THE RIGHT ONES Barbora Batíková Department of Mathematics CULS Kamýcká 129, 165 21 Praha
More information201B, Winter 11, Professor John Hunter Homework 9 Solutions
2B, Winter, Professor John Hunter Homework 9 Solutions. If A : H H is a bounded, self-adjoint linear operator, show that A n = A n for every n N. (You can use the results proved in class.) Proof. Let A
More informationSequences and Series
CHAPTER Sequences and Series.. Convergence of Sequences.. Sequences Definition. Suppose that fa n g n= is a sequence. We say that lim a n = L; if for every ">0 there is an N>0 so that whenever n>n;ja n
More informationAn idea how to solve some of the problems. diverges the same must hold for the original series. T 1 p T 1 p + 1 p 1 = 1. dt = lim
An idea how to solve some of the problems 5.2-2. (a) Does not converge: By multiplying across we get Hence 2k 2k 2 /2 k 2k2 k 2 /2 k 2 /2 2k 2k 2 /2 k. As the series diverges the same must hold for the
More informationEstimation of parametric functions in Downton s bivariate exponential distribution
Estimation of parametric functions in Downton s bivariate exponential distribution George Iliopoulos Department of Mathematics University of the Aegean 83200 Karlovasi, Samos, Greece e-mail: geh@aegean.gr
More informationNOTES ON VECTOR-VALUED INTEGRATION MATH 581, SPRING 2017
NOTES ON VECTOR-VALUED INTEGRATION MATH 58, SPRING 207 Throughout, X will denote a Banach space. Definition 0.. Let ϕ(s) : X be a continuous function from a compact Jordan region R n to a Banach space
More informationLimiting behaviour of moving average processes under ρ-mixing assumption
Note di Matematica ISSN 1123-2536, e-issn 1590-0932 Note Mat. 30 (2010) no. 1, 17 23. doi:10.1285/i15900932v30n1p17 Limiting behaviour of moving average processes under ρ-mixing assumption Pingyan Chen
More informationSolutions Final Exam May. 14, 2014
Solutions Final Exam May. 14, 2014 1. Determine whether the following statements are true or false. Justify your answer (i.e., prove the claim, derive a contradiction or give a counter-example). (a) (10
More informationLemma 15.1 (Sign preservation Lemma). Suppose that f : E R is continuous at some a R.
15. Intermediate Value Theorem and Classification of discontinuities 15.1. Intermediate Value Theorem. Let us begin by recalling the definition of a function continuous at a point of its domain. Definition.
More informationA NOTE ON A DISTRIBUTION OF WEIGHTED SUMS OF I.I.D. RAYLEIGH RANDOM VARIABLES
Sankhyā : The Indian Journal of Statistics 1998, Volume 6, Series A, Pt. 2, pp. 171-175 A NOTE ON A DISTRIBUTION OF WEIGHTED SUMS OF I.I.D. RAYLEIGH RANDOM VARIABLES By P. HITCZENKO North Carolina State
More informationASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN MODULAR FUNCTION SPACES ABSTRACT
ASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN MODULAR FUNCTION SPACES T. DOMINGUEZ-BENAVIDES, M.A. KHAMSI AND S. SAMADI ABSTRACT In this paper, we prove that if ρ is a convex, σ-finite modular function satisfying
More informationPartial cubes: structures, characterizations, and constructions
Partial cubes: structures, characterizations, and constructions Sergei Ovchinnikov San Francisco State University, Mathematics Department, 1600 Holloway Ave., San Francisco, CA 94132 Abstract Partial cubes
More informationNONTRIVIAL SOLUTIONS TO INTEGRAL AND DIFFERENTIAL EQUATIONS
Fixed Point Theory, Volume 9, No. 1, 28, 3-16 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.html NONTRIVIAL SOLUTIONS TO INTEGRAL AND DIFFERENTIAL EQUATIONS GIOVANNI ANELLO Department of Mathematics University
More informationTechnical Appendix to "Sequential Exporting"
Not for publication Technical ppendix to "Sequential Exporting" acundo lbornoz University of irmingham Héctor. Calvo Pardo University of Southampton Gregory Corcos NHH Emanuel Ornelas London School of
More information2019 Spring MATH2060A Mathematical Analysis II 1
2019 Spring MATH2060A Mathematical Analysis II 1 Notes 1. CONVEX FUNCTIONS First we define what a convex function is. Let f be a function on an interval I. For x < y in I, the straight line connecting
More informationarxiv: v1 [math.pr] 6 Jan 2014
On Cox-Kemperman moment inequalities for independent centered random variables P.S.Ruzankin 1 arxiv:141.17v1 [math.pr] 6 Jan 214 Abstract In 1983 Cox and Kemperman proved that Ef()+Ef() Ef(+) for all functions
More informationGeorgian Mathematical Journal 1(94), No. 4, ON THE INITIAL VALUE PROBLEM FOR FUNCTIONAL DIFFERENTIAL SYSTEMS
Georgian Mathematical Journal 1(94), No. 4, 419-427 ON THE INITIAL VALUE PROBLEM FOR FUNCTIONAL DIFFERENTIAL SYSTEMS V. ŠEDA AND J. ELIAŠ Astract. For a system of functional differential equations of an
More informationBOUNDS OF MODULUS OF EIGENVALUES BASED ON STEIN EQUATION
K Y BERNETIKA VOLUM E 46 ( 2010), NUMBER 4, P AGES 655 664 BOUNDS OF MODULUS OF EIGENVALUES BASED ON STEIN EQUATION Guang-Da Hu and Qiao Zhu This paper is concerned with bounds of eigenvalues of a complex
More informationDecomposing oriented graphs into transitive tournaments
Decomposing oriented graphs into transitive tournaments Raphael Yuster Department of Mathematics University of Haifa Haifa 39105, Israel Abstract For an oriented graph G with n vertices, let f(g) denote
More informationREAL AND COMPLEX ANALYSIS
REAL AND COMPLE ANALYSIS Third Edition Walter Rudin Professor of Mathematics University of Wisconsin, Madison Version 1.1 No rights reserved. Any part of this work can be reproduced or transmitted in any
More informationMulti-Term Linear Fractional Nabla Difference Equations with Constant Coefficients
International Journal of Difference Equations ISSN 0973-6069, Volume 0, Number, pp. 9 06 205 http://campus.mst.edu/ijde Multi-Term Linear Fractional Nabla Difference Equations with Constant Coefficients
More informationExistence of homoclinic solutions for Duffing type differential equation with deviating argument
2014 9 «28 «3 Sept. 2014 Communication on Applied Mathematics and Computation Vol.28 No.3 DOI 10.3969/j.issn.1006-6330.2014.03.007 Existence of homoclinic solutions for Duffing type differential equation
More informationa i,1 a i,j a i,m be vectors with positive entries. The linear programming problem associated to A, b and c is to find all vectors sa b
LINEAR PROGRAMMING PROBLEMS MATH 210 NOTES 1. Statement of linear programming problems Suppose n, m 1 are integers and that A = a 1,1... a 1,m a i,1 a i,j a i,m a n,1... a n,m is an n m matrix of positive
More informationTABLE -I RAINFALL RECORDED AT PORT BLAIR (MM) FROM 1949 TO 2009
A. RAINFALL TABLE -I RAINFALL RECORDED AT PORT BLAIR (MM) FROM 1949 TO 2009 MONTH/YEAR 1949 1950 1951 1952 1953 1954 1955 1956 1957 1958 JANUARY 0.0 0.8 82.5 0.0 26.9 37.3 71.4 46.2 10.2 28.7 FEBRUARY
More informationGlobal Solutions for a Nonlinear Wave Equation with the p-laplacian Operator
Global Solutions for a Nonlinear Wave Equation with the p-laplacian Operator Hongjun Gao Institute of Applied Physics and Computational Mathematics 188 Beijing, China To Fu Ma Departamento de Matemática
More informationSolutions for Homework Assignment 2
Solutions for Homework Assignment 2 Problem 1. If a,b R, then a+b a + b. This fact is called the Triangle Inequality. By using the Triangle Inequality, prove that a b a b for all a,b R. Solution. To prove
More informationNon equilibrium thermodynamic transformations. Giovanni Jona-Lasinio
Non equilibrium thermodynamic transformations Giovanni Jona-Lasinio Kyoto, July 29, 2013 1. PRELIMINARIES 2. RARE FLUCTUATIONS 3. THERMODYNAMIC TRANSFORMATIONS 1. PRELIMINARIES Over the last ten years,
More informationPRODUCT OF OPERATORS AND NUMERICAL RANGE
PRODUCT OF OPERATORS AND NUMERICAL RANGE MAO-TING CHIEN 1, HWA-LONG GAU 2, CHI-KWONG LI 3, MING-CHENG TSAI 4, KUO-ZHONG WANG 5 Abstract. We show that a bounded linear operator A B(H) is a multiple of a
More informationFuzzy and Non-deterministic Automata Ji Mo ko January 29, 1998 Abstract An existence of an isomorphism between a category of fuzzy automata and a cate
University of Ostrava Institute for Research and Applications of Fuzzy Modeling Fuzzy and Non-deterministic Automata Ji Mo ko Research report No. 8 November 6, 1997 Submitted/to appear: { Supported by:
More informationAbstract. In the paper, a particular class of semimodules typical for additively idempotent
italian journal of pure and applied mathematics n. 37 2017 (361 376) 361 CHARACTERISTIC SEMIMODULES Barbora Batíková Department of Mathematics CULS Kamýcká 129, 165 21 Praha 6 Suchdol Czech Republic e-mail:
More information!"#$%&'(&)*$%&+",#$$-$%&+./#-+ (&)*$%&+%"-$+0!#1%&
!"#$%&'(&)*$%&",#$$-$%&./#- (&)*$%&%"-$0!#1%&23 44444444444444444444444444444444444444444444444444444444444444444444 &53.67689:5;978?58"@A9;8=B!=89C7DE,6=8FG=CD=CF(76F9C7D!)#!/($"%*$H!I"%"&1/%/.!"JK$&3
More informationOnline Appendix for Multiproduct-Firm Oligopoly: An Aggregative Games Approach
Online Appendix for Multiproduct-Firm Oligopoly: An Aggregative Games Approach Volker Nocke Nicolas Schutz December 20 207 Contents I The Demand System 4 I. Discrete/Continuous Choice..........................
More informationJUNXIA MENG. 2. Preliminaries. 1/k. x = max x(t), t [0,T ] x (t), x k = x(t) dt) k
Electronic Journal of Differential Equations, Vol. 29(29), No. 39, pp. 1 7. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu POSIIVE PERIODIC SOLUIONS
More informationarxiv: v3 [math.dg] 19 Jun 2017
THE GEOMETRY OF THE WIGNER CAUSTIC AND AFFINE EQUIDISTANTS OF PLANAR CURVES arxiv:160505361v3 [mathdg] 19 Jun 017 WOJCIECH DOMITRZ, MICHA L ZWIERZYŃSKI Abstract In this paper we study global properties
More informationGiaccardi s Inequality for Convex-Concave Antisymmetric Functions and Applications
Southeast Asian Bulletin of Mathematics 01) 36: 863 874 Southeast Asian Bulletin of Mathematics c SEAMS 01 Giaccardi s Inequality for Convex-Concave Antisymmetric Functions and Applications J Pečarić Abdus
More informationFactorization of weighted EP elements in C -algebras
Factorization of weighted EP elements in C -algebras Dijana Mosić, Dragan S. Djordjević Abstract We present characterizations of weighted EP elements in C -algebras using different kinds of factorizations.
More informationCOMPLETION OF A METRIC SPACE
COMPLETION OF A METRIC SPACE HOW ANY INCOMPLETE METRIC SPACE CAN BE COMPLETED REBECCA AND TRACE Given any incomplete metric space (X,d), ( X, d X) a completion, with (X,d) ( X, d X) where X complete, and
More information(2.1) m p (L- s ) sin 0 ds = rn + dm, (2.2) m -EI ds"
SIAM J. MATH. ANAL. Vol. 19, No. 4, July 1988 1988 Society for Industrial and Applied Mathematics 007 ANALYSIS OF LARGE DEFORMATION OF A HEAVY CANTILEVER* SZE-BI HSUt AND SHIN-FENG HWANGt Abstract. In
More informationIDEAL AMENABILITY OF MODULE EXTENSIONS OF BANACH ALGEBRAS. M. Eshaghi Gordji, F. Habibian, and B. Hayati
ARCHIVUM MATHEMATICUM BRNO Tomus 43 2007, 177 184 IDEAL AMENABILITY OF MODULE EXTENSIONS OF BANACH ALGEBRAS M. Eshaghi Gordji, F. Habibian, B. Hayati Abstract. Let A be a Banach algebra. A is called ideally
More informationNETWORK THEORY (BEES2211)
LECTURE NOTES On NETWORK THEORY (BEES2211) 3 rd Semester ETC Engineering Prepared by, Manjushree Jena Jemimah Digal Monalisha Nayak INDIRA GANDHI INSTITUTE OF TECHNOLOGY, SARANG NETWORK THEORY Ms. Manjushree
More informationc i r i i=1 r 1 = [1, 2] r 2 = [0, 1] r 3 = [3, 4].
Lecture Notes: Rank of a Matrix Yufei Tao Department of Computer Science and Engineering Chinese University of Hong Kong taoyf@cse.cuhk.edu.hk 1 Linear Independence Definition 1. Let r 1, r 2,..., r m
More informationExistence Theory for the Isentropic Euler Equations
Arch. Rational Mech. Anal. 166 23 81 98 Digital Object Identifier DOI 1.17/s25-2-229-2 Existence Theory for the Isentropic Euler Equations Gui-Qiang Chen & Philippe G. LeFloch Communicated by C. M. Dafermos
More informationNatural Boundary Integral Method and Its Applications
Natural Boundary Integral Method and Its Applications By De-hao Yu State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing
More informationGlobal Asymptotic Stability of a Nonlinear Recursive Sequence
International Mathematical Forum, 5, 200, no. 22, 083-089 Global Asymptotic Stability of a Nonlinear Recursive Sequence Mustafa Bayram Department of Mathematics, Faculty of Arts and Sciences Fatih University,
More informationResearch Article Frequent Oscillatory Behavior of Delay Partial Difference Equations with Positive and Negative Coefficients
Hindawi Publishing Corporation Advances in Difference Equations Volume 2010, Article ID 606149, 15 pages doi:10.1155/2010/606149 Research Article Frequent Oscillatory Behavior of Delay Partial Difference
More informationMoore-Penrose-invertible normal and Hermitian elements in rings
Moore-Penrose-invertible normal and Hermitian elements in rings Dijana Mosić and Dragan S. Djordjević Abstract In this paper we present several new characterizations of normal and Hermitian elements in
More informationarxiv: v2 [math.ap] 14 May 2016
THE MINKOWSKI DIMENSION OF INTERIOR SINGULAR POINTS IN THE INCOMPRESSIBLE NAVIER STOKES EQUATIONS YOUNGWOO KOH & MINSUK YANG arxiv:16.17v2 [math.ap] 14 May 216 ABSTRACT. We study the possible interior
More informationMath 421, Homework #7 Solutions. We can then us the triangle inequality to find for k N that (x k + y k ) (L + M) = (x k L) + (y k M) x k L + y k M
Math 421, Homework #7 Solutions (1) Let {x k } and {y k } be convergent sequences in R n, and assume that lim k x k = L and that lim k y k = M. Prove directly from definition 9.1 (i.e. don t use Theorem
More informationSeries of Error Terms for Rational Approximations of Irrational Numbers
2 3 47 6 23 Journal of Integer Sequences, Vol. 4 20, Article..4 Series of Error Terms for Rational Approximations of Irrational Numbers Carsten Elsner Fachhochschule für die Wirtschaft Hannover Freundallee
More informationAll Ramsey numbers for brooms in graphs
All Ramsey numbers for brooms in graphs Pei Yu Department of Mathematics Tongji University Shanghai, China yupeizjy@16.com Yusheng Li Department of Mathematics Tongji University Shanghai, China li yusheng@tongji.edu.cn
More informationFormal Groups. Niki Myrto Mavraki
Formal Groups Niki Myrto Mavraki Contents 1. Introduction 1 2. Some preliminaries 2 3. Formal Groups (1 dimensional) 2 4. Groups associated to formal groups 9 5. The Invariant Differential 11 6. The Formal
More informationAN APPROXIMATION ALGORITHM FOR COLORING CIRCULAR-ARC GRAPHS
AN APPROXIMATION ALGORITHM FOR COLORING CIRCULAR-ARC GRAPHS Wei-Kuan Shih 1 and Wen-Lian Hsu 2 Key Words: graph, clique, coloring, matching, algorithm ABSTRACT Consider families of arcs on a circle. The
More informationMA651 Topology. Lecture 10. Metric Spaces.
MA65 Topology. Lecture 0. Metric Spaces. This text is based on the following books: Topology by James Dugundgji Fundamental concepts of topology by Peter O Neil Linear Algebra and Analysis by Marc Zamansky
More informationAnalytical formulas for calculating the extremal ranks and inertias of A + BXB when X is a fixed-rank Hermitian matrix
Analytical formulas for calculating the extremal ranks and inertias of A + BXB when X is a fixed-rank Hermitian matrix Yongge Tian CEMA, Central University of Finance and Economics, Beijing 100081, China
More informationGluing linear differential equations
Gluing linear differential equations Alex Eremenko March 6, 2016 In this talk I describe the recent work joint with Walter Bergweiler where we introduce a method of construction of linear differential
More informationarxiv: v1 [math.ca] 12 Feb 2010
YOUNG S INTEGRAL INEQUALITY WITH UPPER AND LOWER BOUNDS DOUGLAS R. ANDERSON, STEVEN NOREN, AND BRENT PERREAULT arxiv:12.2463v1 [math.ca] 12 Feb 21 Abstract. Young s integral inequality is reformulated
More informationOn weak solution approach to problems in fluid dynamics
On weak solution approach to problems in fluid dynamics Eduard Feireisl based on joint work with J.Březina (Tokio), C.Klingenberg, and S.Markfelder (Wuerzburg), O.Kreml (Praha), M. Lukáčová (Mainz), H.Mizerová
More informationA diamagnetic inequality for semigroup differences
A diamagnetic inequality for semigroup differences (Irvine, November 10 11, 2001) Barry Simon and 100DM 1 The integrated density of states (IDS) Schrödinger operator: H := H(V ) := 1 2 + V ω =: H(0, V
More informationPositive solutions for a class of fractional boundary value problems
Nonlinear Analysis: Modelling and Control, Vol. 21, No. 1, 1 17 ISSN 1392-5113 http://dx.doi.org/1.15388/na.216.1.1 Positive solutions for a class of fractional boundary value problems Jiafa Xu a, Zhongli
More informationConsequences of Continuity and Differentiability
Consequences of Continuity and Differentiability We have seen how continuity of functions is an important condition for evaluating limits. It is also an important conceptual tool for guaranteeing the existence
More informationLaplace Transforms Chapter 3
Laplace Transforms Important analytical method for solving linear ordinary differential equations. - Application to nonlinear ODEs? Must linearize first. Laplace transforms play a key role in important
More informationand monotonicity property of these means is proved. The related mean value theorems of Cauchy type are also given.
Rad HAZU Volume 515 013, 1-10 ON LOGARITHMIC CONVEXITY FOR GIACCARDI S DIFFERENCE J PEČARIĆ AND ATIQ UR REHMAN Abstract In this paper, the Giaccardi s difference is considered in some special cases By
More informationMEAN VALUE THEOREMS FOR SOME LINEAR INTEGRAL OPERATORS
Electronic Journal of Differential Equations, Vol. 2929, No. 117, pp. 1 15. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu MEAN VALUE THEOREMS FOR
More informationFUNDAMENTALS OF REAL ANALYSIS by. IV.1. Differentiation of Monotonic Functions
FUNDAMNTALS OF RAL ANALYSIS by Doğan Çömez IV. DIFFRNTIATION AND SIGND MASURS IV.1. Differentiation of Monotonic Functions Question: Can we calculate f easily? More eplicitly, can we hope for a result
More informationHyperbolic Problems: Theory, Numerics, Applications
AIMS on Applied Mathematics Vol.8 Hyperbolic Problems: Theory, Numerics, Applications Fabio Ancona Alberto Bressan Pierangelo Marcati Andrea Marson Editors American Institute of Mathematical Sciences AIMS
More informationONLINE APPENDIX TO: NONPARAMETRIC IDENTIFICATION OF THE MIXED HAZARD MODEL USING MARTINGALE-BASED MOMENTS
ONLINE APPENDIX TO: NONPARAMETRIC IDENTIFICATION OF THE MIXED HAZARD MODEL USING MARTINGALE-BASED MOMENTS JOHANNES RUF AND JAMES LEWIS WOLTER Appendix B. The Proofs of Theorem. and Proposition.3 The proof
More information(i) The optimisation problem solved is 1 min
STATISTICAL LEARNING IN PRACTICE Part III / Lent 208 Example Sheet 3 (of 4) By Dr T. Wang You have the option to submit your answers to Questions and 4 to be marked. If you want your answers to be marked,
More informationMATH 5640: Fourier Series
MATH 564: Fourier Series Hung Phan, UMass Lowell September, 8 Power Series A power series in the variable x is a series of the form a + a x + a x + = where the coefficients a, a,... are real or complex
More informationu( x) = g( y) ds y ( 1 ) U solves u = 0 in U; u = 0 on U. ( 3)
M ath 5 2 7 Fall 2 0 0 9 L ecture 4 ( S ep. 6, 2 0 0 9 ) Properties and Estimates of Laplace s and Poisson s Equations In our last lecture we derived the formulas for the solutions of Poisson s equation
More informationThis is a submission to one of journals of TMRG: BJMA/AFA EXTENSION OF THE REFINED JENSEN S OPERATOR INEQUALITY WITH CONDITION ON SPECTRA
This is a submission to one of journals of TMRG: BJMA/AFA EXTENSION OF THE REFINED JENSEN S OPERATOR INEQUALITY WITH CONDITION ON SPECTRA JADRANKA MIĆIĆ, JOSIP PEČARIĆ AND JURICA PERIĆ3 Abstract. We give
More informationNON-MONOTONICITY HEIGHT OF PM FUNCTIONS ON INTERVAL. 1. Introduction
Acta Math. Univ. Comenianae Vol. LXXXVI, 2 (2017), pp. 287 297 287 NON-MONOTONICITY HEIGHT OF PM FUNCTIONS ON INTERVAL PINGPING ZHANG Abstract. Using the piecewise monotone property, we give a full description
More informationMATH 51H Section 4. October 16, Recall what it means for a function between metric spaces to be continuous:
MATH 51H Section 4 October 16, 2015 1 Continuity Recall what it means for a function between metric spaces to be continuous: Definition. Let (X, d X ), (Y, d Y ) be metric spaces. A function f : X Y is
More informationMATH 131A: REAL ANALYSIS (BIG IDEAS)
MATH 131A: REAL ANALYSIS (BIG IDEAS) Theorem 1 (The Triangle Inequality). For all x, y R we have x + y x + y. Proposition 2 (The Archimedean property). For each x R there exists an n N such that n > x.
More information2.4 Algebra of polynomials
2.4 Algebra of polynomials ([1], p.136-142) In this section we will give a brief introduction to the algebraic properties of the polynomial algebra C[t]. In particular, we will see that C[t] admits many
More informationON THE OSCILLATION OF THE SOLUTIONS TO LINEAR DIFFERENCE EQUATIONS WITH VARIABLE DELAY
Electronic Journal of Differential Equations, Vol. 008(008, No. 50, pp. 1 15. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp ON THE
More informationThe WhatPower Function à An Introduction to Logarithms
Classwork Work with your partner or group to solve each of the following equations for x. a. 2 # = 2 % b. 2 # = 2 c. 2 # = 6 d. 2 # 64 = 0 e. 2 # = 0 f. 2 %# = 64 Exploring the WhatPower Function with
More informationPM functions, their characteristic intervals and iterative roots
ANNALES POLONICI MATHEMATICI LXV.2(1997) PM functions, their characteristic intervals and iterative roots by Weinian Zhang (Chengdu) Abstract. The concept of characteristic interval for piecewise monotone
More informationOn complete convergence and complete moment convergence for weighted sums of ρ -mixing random variables
Chen and Sung Journal of Inequalities and Applications 2018) 2018:121 https://doi.org/10.1186/s13660-018-1710-2 RESEARCH Open Access On complete convergence and complete moment convergence for weighted
More informationStrongly Consistent Multivariate Conditional Risk Measures
Strongly Consistent Multivariate Conditional Risk Measures annes offmann Thilo Meyer-Brandis regor Svindland January 11, 2016 Abstract We consider families of strongly consistent multivariate conditional
More informationAnalysis Comprehensive Exam Questions Fall F(x) = 1 x. f(t)dt. t 1 2. tf 2 (t)dt. and g(t, x) = 2 t. 2 t
Analysis Comprehensive Exam Questions Fall 2. Let f L 2 (, ) be given. (a) Prove that ( x 2 f(t) dt) 2 x x t f(t) 2 dt. (b) Given part (a), prove that F L 2 (, ) 2 f L 2 (, ), where F(x) = x (a) Using
More informationOn Strongly Prime Semiring
BULLETIN of the Malaysian Mathematical Sciences Society http://math.usm.my/bulletin Bull. Malays. Math. Sci. Soc. (2) 30(2) (2007), 135 141 On Strongly Prime Semiring T.K. Dutta and M.L. Das Department
More informationAppendix A. Math Reviews 03Jan2007. A.1 From Simple to Complex. Objectives. 1. Review tools that are needed for studying models for CLDVs.
Appendix A Math Reviews 03Jan007 Objectives. Review tools that are needed for studying models for CLDVs.. Get you used to the notation that will be used. Readings. Read this appendix before class.. Pay
More information