text: (cid:18) (cid:2) 7 dΩ 15π Ee sin z (cid:3) − − − = α(cid:15)4∆ − − (4 + 3e2) (11 10 cos z 45 cos2 z) cos(ω ω ) dτ 256 (1 e2)1/2(1 E2)5/2 sin ι (cid:19)3 − − (11 + 10 cos z 45 cos2 z) cos(ω + ω ) 3 − −(cid:18) − − − 35e2 [(1 + cos z)(1 3 cos z) cos((cid:2)3ω ω ) (1 cos z)(1 + 3 cos z) cos(3ω + ω )] , 3 3 − d(cid:36) −15π E(1 e2)1/2 (cid:3) − − = α(cid:15)4∆ − (4 + 9e2) (1 + cos z)(1 + 10 cos z 15 cos2 z) cos(ω ω ) dτ 256 e(1 E2)5/2 (cid:19) 3 − − − +(1 cos z)(1 10 cos z 15 cos2 z) cos(ω + ω ) 3 − (cid:18) − − 105e2 sin2 z [(1 + cos z) cos(3ω ω ) +(cid:2)(1 cos z) cos(3ω + ω )] , 3 3 dE 15π e (cid:3) − − = η(1 + α)1/2(cid:15)9/2∆ − (4 + 3e2) (1 + cos z)(1 + 10 cos z 15 cos2 z) sin(ω ω ) dτ 256 (1 E2)2 (cid:19) 3 − − − − (1 cos z)(1 10 cos z 15 cos2 z) sin(ω + ω ) 3 − − (cid:18) − − 35e2 sin2 z [(1 + cos z) sin(3ω ω ) (1 c(cid:2)os z) sin(3ω + ω )] , 3 3 dι 15π Ee (cid:3) − − 3 = η(1 + α)1/2(cid:15)9/2∆ − sin z (4 + 3e2) (1 + 10 cos z 15 cos2 z) sin(ω ω ) dτ 256 (1 E2)3 (cid:19)3 − − +(1 10 cos z 15 cos2 z) sin(ω + ω ) 3 − − (cid:18) − − 35e2 [(1 + cos z)(3 cos z) sin(3ω ω ) + (cid:2)(1 cos z)(3 + cos z) sin(3ω + ω )] , 3 3 d(cid:36) −15π e(1 + 4E2) (cid:3) − − 3 = η(1 + α)1/2(cid:15)9/2∆ − (4 + 3e2) (1 + cos z)(1 + 10 cos z 15 cos2 z) cos(ω ω ) dτ 256 E(1 E2)3 (cid:19) 3 − − − +(1 cos z)(1 10 cos z 15 cos2 z) cos(ω + ω ) 3 − − − 35e2 sin2 z [(1 + cos z) cos(3ω ω ) + (1 cos z) cos(3ω + ω )] . (2.27) 3 3 It is straightforward to verify that these results are completely equivalent to those of Ford et al. [5] and NFLRT [7]. In Appendix we provide a dictionary that translates between our osculating orbits language and the Delaunay variables language used in [5, 7]. Hexadecapole order − de −(cid:18)315π − e(1 e2)1/2 = α(cid:15)5(1 3η) (cid:2) − (cid:3) dτ 1024 (1 E2)7/2 × (cid:26) − − (2 + 3E2) sin2 z(cid:2) (4 + 2e2)(1 7 cos2 z) sin 2ω 21e2 sin2 z sin 4ω − − (cid:3) − E2 (4 + 2e2) (1 + cos z)2(1 7 cos z + 7 cos2 z) sin(2ω 2ω ) 3 (cid:27)(cid:19) − (cid:2) (cid:3) +(1 cos z)2(1 + 7 cos z + 7 cos2 z) sin(2ω + 2ω ) 3 − − + 21e2 sin2 z (1 + cos z)2 sin(4ω 2ω ) + (1 cos z)2 sin(4ω + 2ω ) , 3 3 dι 45(cid:18)π − sin z = α(cid:15)5(1 3η) − (cid:2) − (cid:3) dτ 2048 (1 e2)1/2(1 E2)7/2 × − − 14e2(2 + 3E2) cos z (4 + 2e2)(1 7 cos2 z) sin 2ω 21e2 sin2 z sin 4ω (cid:26) − (cid:2) + 2E2(1 7 cos2 z)(8 + 40e2 + 15e4) sin 2ω 3 − − + 7E2e2 4(2 + e2) (1 + cos z)(1 7 cos z + 7 cos2 z) sin(2ω 2ω ) 3 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: SPRINGER FIBERS FOR THE MINIMAL AND THE MINIMAL SPECIAL NILPOTENT ORBITS 18 ∩ B B dim(X ) = dim = 13 ∈ w N N w W red. exp. A w (0, 1, 0, 0), (0, 1, 2, 0), (0, 1, 2, 1), (1, 0, 0, 0), (1, 1, 0, 0), y s(3231234323123121) (1, 1, 2, 0), (1, 1, 2, 1), (1, 2, 2, 0), (1, 2, 2, 1), (1, 2, 3, 1), 1 (1, 2, 4, 2), (1, 3, 4, 2), (2, 3, 4, 2) (0, 0, 1, 0), (0, 1, 0, 0), (0, 1, 1, 0), (0, 1, 2, 0), (1, 0, 0, 0), y s(12342312343232) (1, 1, 0, 0), (1, 1, 1, 1), (1, 1, 2, 0), (1, 1, 2, 1), (1, 2, 2, 0), 2 (1, 2, 2, 1), (1, 2, 3, 1), (2, 3, 4, 2) (0, 1, 0, 0), (0, 1, 2, 0), (1, 0, 0, 0), (1, 1, 0, 0), (1, 1, 1, 1), y s(1234231234323121) (1, 1, 2, 0), (1, 1, 2, 1), (1, 2, 2, 0), (1, 2, 2, 1), (1, 2, 3, 1), 3 (1, 2, 4, 2), (1, 3, 4, 2), (2, 3, 4, 2) (0, 0, 1, 0), (0, 1, 0, 0), (0, 1, 2, 0), (1, 0, 0, 0), (1, 1, 0, 0), y s(123423123432321) (1, 1, 1, 1), (1, 1, 2, 0), (1, 1, 2, 1), (1, 2, 2, 0), (1, 2, 2, 1), 4 (1, 2, 3, 1), (1, 3, 4, 2), (2, 3, 4, 2) (0, 1, 0, 0), (0, 1, 2, 0), (1, 0, 0, 0), (1, 1, 0, 0), (1, 1, 1, 1), y s(123423123423121) (1, 1, 2, 0), (1, 1, 2, 1), (1, 2, 2, 0), (1, 2, 2, 1), (1, 2, 2, 2), 5 (1, 2, 3, 1), (1, 3, 4, 2), (2, 3, 4, 2) (0, 1, 0, 0), (0, 1, 1, 1), (0, 1, 2, 0), (1, 0, 0, 0), (1, 1, 0, 0), y s(231234323123121) (1, 1, 1, 1), (1, 1, 2, 0), (1, 2, 2, 0), (1, 2, 2, 1), (1, 2, 2, 2), 6 (1, 2, 3, 1), (1, 3, 4, 2), (2, 3, 4, 2) ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: SPRINGER FIBERS FOR THE MINIMAL AND THE MINIMAL SPECIAL NILPOTENT ORBITS 19 ∩ B B − dim(X ) = dim 1 = 12 ∈ w N N w W red. exp. A w (0, 1, 0, 0), (0, 1, 1, 0), (0, 1, 2, 0), (1, 0, 0, 0), (1, 1, 0, 0), (1, 1, 1, 1), z s(1234231234232) 1 (1, 1, 2, 0), (1, 1, 2, 1), (1, 2, 2, 0), (1, 2, 2, 1), (1, 2, 3, 1), (2, 3, 4, 2) (0, 1, 0, 0), (0, 1, 2, 0), (1, 0, 0, 0), (1, 1, 0, 0), (1, 1, 1, 1), (1, 1, 2, 0), z s(12342312342321) 2 (1, 1, 2, 1), (1, 2, 2, 0), (1, 2, 2, 1), (1, 2, 3, 1), (1, 3, 4, 2), (2, 3, 4, 2) (0, 1, 0, 0), (0, 1, 1, 1), (1, 0, 0, 0), (1, 1, 0, 0), (1, 1, 1, 1), (1, 1, 2, 0), z s(23123432312312) 3 (1, 2, 2, 0), (1, 2, 2, 1), (1, 2, 2, 2), (1, 2, 3, 1), (1, 3, 4, 2), (2, 3, 4, 2) (0, 1, 0, 0), (1, 0, 0, 0), (1, 1, 0, 0), (1, 1, 1, 1), (1, 1, 2, 0), (1, 1, 2, 1), z s(12342312342312) 4 (1, 2, 2, 0), (1, 2, 2, 1), (1, 2, 2, 2), (1, 2, 3, 1), (1, 3, 4, 2), (2, 3, 4, 2) (0, 1, 0, 0), (0, 1, 2, 0), (1, 0, 0, 0), (1, 1, 0, 0), (1, 1, 1, 1), (1, 1, 2, 0), z s(23123423123121) 5 (1, 2, 2, 0), (1, 2, 2, 1), (1, 2, 2, 2), (1, 2, 3, 1), (1, 3, 4, 2), (2, 3, 4, 2) (0, 1, 0, 0), (0, 1, 2, 0), (1, 0, 0, 0), (1, 1, 0, 0), (1, 1, 2, 0), (1, 1, 2, 1), z s(323123423123121) 6 (1, 2, 2, 0), (1, 2, 2, 1), (1, 2, 3, 1), (1, 2, 4, 2), (1, 3, 4, 2), (2, 3, 4, 2) (0, 0, 0, 1), (0, 1, 0, 0), (1, 0, 0, 0), (1, 1, 0, 0), (1, 1, 1, 1), (1, 1, 2, 0), z s(1234323123121) 7 (1, 1, 2, 1), (1, 1, 2, 2), (1, 2, 2, 0), (1, 2, 2, 1), (1, 2, 2, 2), (2, 3, 4, 2) (0, 0, 1, 0), (0, 1, 0, 0), (0, 1, 1, 1), (0, 1, 2, 0), (0, 1, 2, 1), (1, 1, 0, 0), z s(23423123432321) 8 (1, 1, 2, 0), (1, 2, 2, 0), (1, 2, 2, 1), (1, 2, 3, 1), (1, 3, 4, 2), (2, 3, 4, 2) (0, 1, 0, 0), (0, 1, 2, 0), (0, 1, 2, 1), (1, 1, 0, 0), (1, 1, 2, 0), (1, 1, 2, 1), z s(323123432312321) 9 (1, 2, 2, 0), (1, 2, 2, 1), (1, 2, 3, 1), (1, 2, 4, 2), (1, 3, 4, 2), (2, 3, 4, 2) (0, 0, 1, 0), (0, 1, 1, 0), (0, 1, 2, 0), (1, 0, 0, 0), (1, 1, 0, 0), (1, 1, 1, 1), z s(1234231234323) 10 (1, 1, 2, 0), (1, 1, 2, 1), (1, 2, 2, 0), (1, 2, 2, 1), (1, 2, 3, 1), (2, 3, 4, 2) (0, 1, 0, 0), (0, 1, 1, 0), (1, 0, 0, 0), (1, 1, 0, 0), (1, 1, 1, 1), (1, 1, 2, 0), z s(1234231234231) 11 (1, 1, 2, 1), (1, 2, 2, 0), (1, 2, 2, 1), (1, 2, 2, 2), (1, 2, 3, 1), (2, 3, 4, 2) (0, 0, 1, 0), (0, 1, 0, 0), (0, 1, 1, 1), (0, 1, 2, 0), (0, 1, 2, 1), (1, 1, 0, 0), z s(2342312343232) 12 (1, 1, 1, 0), (1, 1, 2, 0), (1, 2, 2, 0), (1, 2, 2, 1), (1, 2, 3, 1), (1, 3, 4, 2) (0, 1, 0, 0), (0, 1, 1, 1), (0, 1, 2, 0), (1, 1, 0, 0), (1, 1, 1, 1), (1, 1, 2, 0), z s(23123432312321) 13 (1, 2, 2, 0), (1, 2, 2, 1), (1, 2, 2, 2), (1, 2, 3, 1), (1, 3, 4, 2), (2, 3, 4, 2) (0, 1, 0, 0), (0, 1, 1, 1), (0, 1, 2, 0), (0, 1, 2, 1), (1, 1, 0, 0), (1, 1, 2, 0), z s(23423123423121) 14 (1, 2, 2, 0), (1, 2, 2, 1), (1, 2, 2, 2), (1, 2, 3, 1), (1, 3, 4, 2), (2, 3, 4, 2) (0, 1, 0, 0), (0, 1, 1, 1), (0, 1, 2, 0), (0, 1, 2, 1), (1, 1, 0, 0), (1, 1, 2, 0), z s(234231234323121) 15 (1, 2, 2, 0), (1, 2, 2, 1), (1, 2, 3, 1), (1, 2, 4, 2), (1, 3, 4, 2), (2, 3, 4, 2) (0, 0, 1, 1), (0, 1, 2, 0), (1, 0, 0, 0), (1, 1, 0, 0), (1, 1, 1, 1), (1, 1, 2, 0), z s(31234323123121) 16 (1, 1, 2, 1), (1, 1, 2, 2), (1, 2, 2, 0), (1, 2, 3, 1), (1, 2, 4, 2), (2, 3, 4, 2) (0, 1, 2, 0), (0, 1, 2, 1), (1, 0, 0, 0), (1, 1, 0, 0), (1, 1, 2, 0), (1, 1, 2, 1), z s(323123432312312) 17 (1, 2, 2, 0), (1, 2, 2, 1), (1, 2, 3, 1), (1, 2, 4, 2), (1, 3, 4, 2), (2, 3, 4, 2) (0, 0, 1, 0), (0, 1, 0, 0), (0, 1, 2, 0), (1, 0, 0, 0), (1, 1, 0, 0), (1, 1, 2, 0), z s(32312342312321) 18 (1, 1, 2, 1), (1, 2, 2, 0), (1, 2, 2, 1), (1, 2, 3, 1), (1, 3, 4, 2), (2, 3, 4, 2) (0, 1, 2, 0), (1, 0, 0, 0), (1, 1, 0, 0), (1, 1, 1, 1), (1, 1, 2, 0), (1, 1, 2, 1), z s(12342312343121) 19 (1, 1, 2, 2), (1, 2, 2, 0), (1, 2, 2, 1), (1, 2, 3, 1), (1, 2, 4, 2), (2, 3, 4, 2) (0, 1, 1, 0), (0, 1, 2, 0), (1, 0, 0, 0), (1, 1, 0, 0), (1, 1, 1, 1), (1, 1, 2, 0), z s(12342312343231) 20 (1, 1, 2, 1), (1, 2, 2, 0), (1, 2, 2, 1), (1, 2, 3, 1), (1, 2, 4, 2), (2, 3, 4, 2) (0, 1, 2, 0), (1, 0, 0, 0), (1, 1, 0, 0), (1, 1, 1, 1), (1, 1, 2, 0), (1, 1, 2, 1), z s(123423123432312) 21 (1, 2, 2, 0), (1, 2, 2, 1), (1, 2, 3, 1), (1, 2, 4, 2), (1, 3, 4, 2), (2, 3, 4, 2) (0, 0, 1, 0), (0, 1, 0, 0), (0, 1, 1, 0), (0, 1, 2, 0), (1, 0, 0, 0), (1, 1, 0, 0), z s(3231234231232) 22 (1, 1, 2, 0), (1, 1, 2, 1), (1, 2, 2, 0), (1, 2, 2, 1), (1, 2, 3, 1), (2, 3, 4, 2) (0, 1, 0, 0), (1, 0, 0, 0), (1, 1, 0, 0), (1, 1, 1, 1), (1, 1, 2, 0), (1, 1, 2, 1), z s(1234231234121) 23 (1, 1, 2, 2), (1, 2, 2, 0), (1, 2, 2, 1), (1, 2, 2, 2), (1, 2, 3, 1), (2, 3, 4, 2) ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## 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text: SPRINGER FIBERS FOR THE MINIMAL AND THE MINIMAL SPECIAL NILPOTENT ORBITS 5 1 3 4 5 6 Type E : 6 2 1 3 4 5 6 7 Type E : 7 2 1 3 4 5 6 7 8 Type E : 8 2 Type F : 1 2 3 4 4 Type G : 1 2 2 We deﬁne the minimal nilpotent orbit of g to be the nilpotent orbit of g which is minimal among O ≤ O(cid:48) ⇔ O ⊂ O(cid:48) non-trivial nilpotent orbits where the order is given by . Also we deﬁne the minimal special nilpotent orbit of g to be the nilpotent orbit which is minimal among non-trivial special nilpotent orbits with the same order. (Here the term “special” is in the sense of Lusztig.) These orbits are always well-deﬁned and coincide if and only if G is of simply-laced type, i.e. type A, D, E. Here we summarize some properties of these orbits and corresponding Springer ﬁbers. These can be read from subsequent sections of this paper, [DG84], [Car93], [CM93], etc. ∈ B | B | Type N g M/MS dim Irr C(N) − −N N A (1n 121) M=MS 1(n2 n) n 1 n − 2 − − (12n 322) M n2 2n + 2 n 1 1 B − − Z n (12n 231) MS n2 2n + 1 n + 1 /2 − − (12n 221) M n2 n 1 1 C − − − Z n (12n 422) MS n2 2n + 1 2n 1 /2 − − D (12n 422) M=MS n2 3n + 3 n 1 n E A M=MS 25 6 1 6 1 E A M=MS 46 7 1 7 1 E A(cid:102) M=MS 91 8 1 8 1 A M 16 2 1 F4 1 Z A MS 13 6 /2 1 A M 3 1 1 G 1 2 G (a ) MS 1 4 S 2 1 3 Here each column represents: • Type: the type of G ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: 16 BLOW-UP OF SOLUTIONS OF NLS ∈ R momentum, the decay assumption on u should not be too strong so that xu / L2( ) is satisﬁed. 0 0 1 This motivates us to consider a function u which involves one or more terms similar to . 0 (x2 + 1)12 We ﬁrst construct a concrete example of a compactly supported weight function ϕ. Note that the ≈ | | function ϕ given in (1.11) is assumed to stay smooth for 1.57735 1 + √1 < x < 2 but an explicit 3 deﬁnition on that portion was not given. Since, our aim here is to give a numerical example, we ﬁrst deﬁne this missing smooth portion of the function ϕ. In order to do this, we use two molliﬁers l and r, where l refers to the molliﬁer that we will apply to the the left hand side of the graph of ϕ whereas r refers to the molliﬁer that we will apply to the right hand side of the graph of ϕ. To (cid:40) (cid:16) (cid:17) this end, we deﬁne − | | Exp(1)Exp − 1 if x + 1.6 < .4, l(x) = (1 (0.4(x+1.6))4) (4.1) 0 otherwise (cid:40) (cid:16) (cid:17) and − | − | Exp(1)Exp − 1− if x 1.6 < .4, r(x) = (1 (0.4(x 1.6))4) (4.2) 0 otherwise ∓ The reason that we are using the fourth power in the term (x 1.6)4 is because when we apply these molliﬁers to smoothly join two pieces of ϕ, we want ϕ to have continuous derivatives at least up to order three. The graph of these molliﬁers are given in Figure 2. 1.0 0.8 ( ) 0.6 r x 0.4 ( ) 0.2 l x - - - 0.0 3 2 1 0 1 2 3 Figure 2. Molliﬁers l and r ≈ Now, we deﬁne (noticing that 1 + √1 1.57735 < 1.6):  3     | | ≤  x if x 1, √   − − ≤  x√ (x 1)3 √ if 1 < x √ 1 + 1/ 3, − − − ≤ x + 2(1 + 1/ 3) ( x + 1 + 2/ 3)3 if 1 + 1/ 3 < x 1.6  − −  (x (x 1)3)r(x) if 1.6 < x <√2, ϕ(x) =  − − ≤ − (4.3)  x√ (x + 1)3 √ if (1 + 1/ 3) x < √1,  − − − − − − − ≤ −  x 2(1 + 1/ 3) ( x 1 2/ 3)3 if 1.6 x < (1 + 1/ 3), − − − (x (x + 1)3)l(x) if 2 < x < 1.6, ≤ | | 0 if 2 x , This function is C3 and satisﬁes the desired co√ndition ϕ(cid:48)(x) ≤ 0 for |x| ≥ 1 + √1 . Note that we ∓ 3 have used vertical reﬂections at x = (1 + 1/ 3) to enforce the latter property. The graph of ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: BLOW-UP

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text: SOLUTIONS

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text: the

text: decay

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text: so

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text: This

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text: a

text: function

text: u

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text: We

text: ﬁrst

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text: example

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text: a

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text: supported

text: weight

text: function

text: ϕ.

text: Note

text: that

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text: function

text: ϕ

text: given

text: in

text: is

text: assumed

text: to

text: stay

text: smooth

text: for

text: √1

text: x

text: but

text: an

text: explicit

text: deﬁnition

text: on

text: that

text: portion

text: was

text: not

text: given.

text: Since,

text: our

text: aim

text: here

text: is

text: to

text: give

text: a

text: numerical

text: example,

text: we

text: ﬁrst

text: deﬁne

text: this

text: missing

text: smooth

text: portion

text: of

text: the

text: function

text: ϕ.

text: In

text: order

text: to

text: do

text: this,

text: we

text: use

text: two

text: molliﬁers

text: l

text: and

text: r,

text: where

text: l

text: refers

text: to

text: the

text: molliﬁer

text: that

text: we

text: will

text: apply

text: to

text: the

text: the

text: left

text: hand

text: side

text: of

text: the

text: graph

text: of

text: ϕ

text: whereas

text: r

text: refers

text: to

text: the

text: molliﬁer

text: that

text: we

text: will

text: apply

text: to

text: the

text: right

text: hand

text: side

text: of

text: the

text: graph

text: of

text: ϕ.

text: To

text: this

text: end,

text: we

text: deﬁne

text: if

text: x

text: l(x)

text: otherwise

text: and

text: 1−

text: if

text: x

text: r(x)

text: otherwise

text: The

text: reason

text: that

text: we

text: are

text: using

text: the

text: fourth

text: power

text: in

text: the

text: term

text: (x

text: is

text: because

text: when

text: we

text: apply

text: these

text: molliﬁers

text: to

text: smoothly

text: join

text: two

text: pieces

text: of

text: ϕ,

text: we

text: want

text: ϕ

text: to

text: have

text: continuous

text: derivatives

text: at

text: least

text: up

text: to

text: order

text: three.

text: The

text: graph

text: of

text: these

text: molliﬁers

text: are

text: given

text: in

text: Figure

text: r

text: x

text: l

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text: l

text: and

text: r

text: Now,

text: we

text: deﬁne

text: (noticing

text: that

text: √1

text: x

text: if

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text: (x

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text: if

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text: satisﬁes

text: the

text: desired

text: co√ndition

text: for

text: |x|

text: √1

text: Note

text: that

text: we

text: have

text: used

text: vertical

text: reﬂections

text: at

text: x

text: to

text: enforce

text: the

text: latter

text: property.

text: The

text: graph

text: of

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text: 10 BLOW-UP OF SOLUTIONS OF NLS Com(cid:90)bining (2.9)-(2.(cid:90)11), using (2.12), and integrating over the time interval (t , t), we get the 0 following identity: − (cid:61) (cid:90)(cid:61) (cid:20) (cid:90) (cid:48) (cid:90) (cid:90) (cid:21) ϕuu¯ dx + ϕu u¯ dx x 0 0 (cid:90) (cid:34) t (cid:48)(cid:90)| | − 2a(2Ωs) (cid:48)| | −(cid:90)1 (cid:48)(cid:48)(cid:48)| | (cid:35) = 2 ϕ u 2dx ϕ u 6dx ϕ u 2dx ds x 3 2 t0 t − (cid:90) −(cid:20) (cid:48) | (cid:90)| 2a(2Ωs) − (cid:90)(cid:48) | | (cid:21) = 2E(u ) 2 (1 ϕ ) u 2dx + (1 ϕ ) u 6dx ds 0 | |≥ x 3 | |≥ t0 x 1 x 1 t −2Ω s (cid:48) (cid:107) (cid:107) − 1 (cid:48)(cid:48)(cid:48)| | + a (2Ωτ) u(τ) 6 dτ ϕ u 2dx ds. (2.13) 3 L6 2 t0 t0 Multiplying the complex conjugate of (1.1) by Ψu, taking the imaginary parts, integrating over R × (cid:90) (cid:90) (cid:90) (cid:90) (0, t), we obtain | | | | − t (cid:61) Ψ u 2dx = Ψ u 2dx 2 ϕuu¯ dxds. 0 x t0 (cid:3) Now we will prove the estimate given by the lemma below. ∈ (cid:115) (cid:32) (cid:33) (cid:48) Lemma 2.14. Let u C(t , t + T ; H1) be a solution of(cid:26) (1.1)-(1.2) a(cid:27)nd a > 0 in [t , t + T ). 0 0 0 0 Then, for √ 1 3 (cid:107) (cid:48)(cid:48)(cid:48)(cid:107) 1(cid:107) (cid:48)(cid:48)(cid:107) 2 δ = ϕ ∞ + max 3, ϕ ∞ , 2 8A (t + T ) 2 Ω 0 (cid:90) (cid:90) one has − (cid:61) ≤ −(cid:61) (cid:48) − ϕuu¯ dx ϕu u¯ dx + (2E(u ) + δ)(t t ) (2.15) x 0 0 0 0 (cid:107) (cid:107) 3 on [t , t + T ) provided u(t) 4 | |≥ < . 0 0 L2( x 1) 8A (t + T ) Ω 0 ≥ Proof. For x 1 we have (cid:90) (cid:104) (cid:105) ∞ (cid:90) (cid:90) − (cid:48) | | − − (cid:48) | | (1 ϕ(x) )12 u(x) 2 = (1 ϕ(y) )12 u(y) 2 dy x y ≤ 1 − (cid:48) − | (cid:48)(cid:48)|| | − (cid:48) | || | 2 ≥ (1 ϕ ) 12 ϕ u 2dy + 2 ≥ (1 ϕ )12 u ux dy y 1 y 1 ≤ (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) − (cid:48) (cid:107) C0 u 2L2(|x|≥1) + 2 u L2(|x|≥1) (1 ϕ )12 ux L2(|x|≥1) (2.16) √ { 1(cid:107) (cid:48)(cid:48)(cid:107) } ≤ − where C = max 3, ϕ ∞ . (A similar estimate also holds for x 1; the details are omitted.) 0 2 Hence, one has (cid:107) − (cid:48) | | (cid:107) ≤ (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) − (cid:48) (cid:107) (1 ϕ ) u 4 L∞(|x|≥1) 2C02 u 4L2(|x|≥1) + 8 u 2L2(|x|≥1) (1 ϕ )12 ux 2L2(x≥1). (2.17) ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: BLOW-UP

text: OF

text: SOLUTIONS

text: OF

text: NLS

text: using

text: and

text: integrating

text: over

text: the

text: time

text: interval

text: (t

text: t),

text: we

text: get

text: the

text: following

text: identity:

text: ϕuu¯

text: dx

text: ϕu

text: u¯

text: dx

text: x

text: t

text: ϕ

text: u

text: ϕ

text: u

text: ϕ

text: u

text: ds

text: x

text: t

text: ϕ

text: u

text: ϕ

text: u

text: ds

text: x

text: x

text: x

text: t

text: −2Ω

text: s

text: a

text: u(τ)

text: dτ

text: ϕ

text: u

text: ds.

text: Multiplying

text: the

text: complex

text: conjugate

text: of

text: by

text: Ψu,

text: taking

text: the

text: imaginary

text: parts,

text: integrating

text: over

text: R

text: t),

text: we

text: obtain

text: t

text: Ψ

text: u

text: Ψ

text: u

text: ϕuu¯

text: dxds.

text: x

text: Now

text: we

text: will

text: prove

text: the

text: estimate

text: given

text: by

text: the

text: lemma

text: below.

text: Lemma

text: Let

text: u

text: C(t

text: t

text: T

text: be

text: a

text: solution

text: a

text: in

text: [t

text: t

text: T

text: Then,

text: for

text: δ

text: ϕ

text: max

text: ϕ

text: (t

text: T

text: Ω

text: one

text: has

text: ϕuu¯

text: dx

text: ϕu

text: u¯

text: dx

text: δ)(t

text: t

text: x

text: on

text: [t

text: t

text: T

text: provided

text: u(t)

text: x

text: (t

text: T

text: Ω

text: Proof.

text: For

text: x

text: we

text: have

text: ϕ(x)

text: u(x)

text: ϕ(y)

text: u(y)

text: dy

text: x

text: y

text: ϕ

text: ϕ

text: u

text: ϕ

text: u

text: ux

text: dy

text: y

text: y

text: u

text: u

text: ϕ

text: ux

text: where

text: C

text: max

text: ϕ

text: (A

text: similar

text: estimate

text: also

text: holds

text: for

text: x

text: the

text: details

text: are

text: omitted.)

text: Hence,

text: one

text: has

text: ϕ

text: u

text: u

text: u

text: ϕ

text: ux

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text:        N1 + 12 0 0 0  0 + 1 0 0 ∞ = N2 2 (46) σ 0 0 + 1 0 . J=0 N1 2 0 0 0 + 1 N2 2 For a non-zero coupling, the CM is not diagonal anymore, as the coupling establishes correlations amongst ? the two harmonic oscillators.The corresponding CM is given by the following matrix [, ]]Campbell2016,    D +−N1 + 12 +−δC+ 1 −0 γ0C  σ∞ = ζ 0δC D −N2 2 + γC+ 1 − , (47) γ0C D −N1 2 + δC+ 1 γC δC D N2 2 with ζ = 4J2γ+2+γ2δ+2 δ2 , D = 2J2(Nγ21++δN22+1), and C = J(γN21+−δN22). Next, we need to identify (i) the SLD associated to J, and (ii) the time evolution of the desired observable ( ) (47) B t , under the unperturbed map. One can ﬁnd (i) with the help of Eq. . We know that for such a Λ Gaussian state, 0 can be expressed as a linear combination of all the second order moments of the quadratures ? [ , ]]monras2013phase,PhysRevA.89.032128, namely (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) Λ = 2 − ( ∞) + 2 − ( ∞) + [ + ] 0 d1 x1 σ11 J=0 d2 p1 σ33 J=0 d3 x1p1 p1x1 + 2 − ( ∞) + 2 − ( ∞) + [ + ] d4 x2 σ22 J=0 d5 p2 σ44 J=0 d6 x2p2 p2x2 + + + + (48) c1 x1x2 c2 x1p2 c3 p1x2 c4 p1p2, (4) with d s and c s being coeﬃcients which are to be determined. To this end, we make beneﬁt of Eq. of j j the main text, which states that for any observable B we have: 1 | h i h i = Λ + Λ (49) ∂J J=0 B J 2 0B B 0 0 . ∈ (48) Next, we imply this relation to all of the quadratic observables appearing in , i.e., we choose B { } 2 2 + (49) x1, x2, x1p1 p1x1, . . . . With such choices of B, the left hand side of Eq. can be evaluated by taking ∞ derivative from the covariance matrix σ . Moreover, by using the Wick’s theorem for Gaussian distributions ? [ , ]]Louisell:103059, one can easily simplify the right hand side and write it down in terms of the elements of ∞ (cid:10) (cid:11) = 2 σ as well. We shall start by focusing on the local terms. As an example, for B x1 we have: 1 Λ 2 + 2Λ = ( ∞) 2 0x1 x1 0 0 ∂Jσ11 J=0 , ⇒ 2 ( ∞)2 − d2 = 0 (50) d1 σ11 J=0 2 , ∞ (46) = 2 where we used the fact that σJ=0, has no oﬀ-diago↔nal terms, as stated by Eq.= =. F0or B p1 one ﬁnds a similar equation, with the change of coeﬃcients d1 d2. This implies that, d1 d2 . In the same manner, = = 0 ∈ {1( + ) 1( + )} (49) one can ﬁnd that d4 d5 . For the othe(cid:20)r two local terms i.e., B(cid:21) 2 x1p1 p1x1 , 2 x2p2 p2x2 , Eq. leads to: (cid:20)2( ∞) ( ∞) + 1(cid:21) = 0 d3 σ11 J=0 σ33 J=0 2 , 1 2( ∞) ( ∞) + = 0 (51) d6 σ22 J=0 σ44 J=0 2 , = = 0 whence, d3 d6 as well. This conﬁrms that the coeﬃcients associated to local observables are zero, i.e., ∀ ∈ { } = 0 (49) di i. However, the non-local coeﬃcients are non-zero. For B x1x2, x1p2, p1x2, p1p2 , using Eq. yields: Accepted in Quantum 2018-04-23, click title to verify 9 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: equation

text: σ

text: For

text: a

text: non-zero

text: coupling,

text: the

text: CM

text: is

text: not

text: diagonal

text: anymore,

text: as

text: the

text: coupling

text: establishes

text: correlations

text: amongst

text: the

text: two

text: harmonic

text: oscillators.The

text: corresponding

text: CM

text: is

text: given

text: by

text: the

text: following

text: matrix

text: D

text: +−δC+

text: −0

text: σ∞

text: ζ

text: D

text: γC+

text: D

text: δC+

text: γC

text: δC

text: D

text: with

text: ζ

text: δ2

text: D

text: and

text: C

text: Next,

text: we

text: need

text: to

text: identify

text: (i)

text: the

text: SLD

text: associated

text: to

text: J,

text: and

text: (ii)

text: the

text: time

text: evolution

text: of

text: the

text: desired

text: observable

text: B

text: t

text: under

text: the

text: unperturbed

text: map.

text: One

text: can

text: ﬁnd

text: (i)

text: with

text: the

text: help

text: of

text: Eq.

text: We

text: know

text: that

text: for

text: such

text: a

text: Λ

text: Gaussian

text: state,

text: can

text: be

text: expressed

text: as

text: a

text: linear

text: combination

text: of

text: all

text: the

text: second

text: order

text: moments

text: of

text: the

text: quadratures

text: namely

text: Λ

text: with

text: d

text: s

text: and

text: c

text: s

text: being

text: coeﬃcients

text: which

text: are

text: to

text: be

text: determined.

text: To

text: this

text: end,

text: we

text: make

text: beneﬁt

text: of

text: Eq.

text: of

text: j

text: j

text: the

text: main

text: text,

text: which

text: states

text: that

text: for

text: any

text: observable

text: B

text: we

text: have:

text: h

text: i

text: h

text: i

text: Λ

text: Λ

text: ∂J

text: B

text: J

text: B

text: Next,

text: we

text: imply

text: this

text: relation

text: to

text: all

text: of

text: the

text: quadratic

text: observables

text: appearing

text: in

text: i.e.,

text: we

text: choose

text: B

text: With

text: such

text: choices

text: of

text: B,

text: the

text: left

text: hand

text: side

text: of

text: Eq.

text: can

text: be

text: evaluated

text: by

text: taking

text: derivative

text: from

text: the

text: covariance

text: matrix

text: σ

text: Moreover,

text: by

text: using

text: the

text: Wick’s

text: theorem

text: for

text: Gaussian

text: distributions

text: one

text: can

text: easily

text: simplify

text: the

text: right

text: hand

text: side

text: and

text: write

text: it

text: down

text: in

text: terms

text: of

text: the

text: elements

text: of

text: σ

text: as

text: well.

text: We

text: shall

text: start

text: by

text: focusing

text: on

text: the

text: local

text: terms.

text: As

text: an

text: example,

text: for

text: B

text: we

text: have:

text: Λ

text: 2Λ

text: where

text: we

text: used

text: the

text: fact

text: that

text: has

text: no

text: oﬀ-diago↔nal

text: terms,

text: as

text: stated

text: by

text: Eq.=

text: B

text: one

text: ﬁnds

text: a

text: similar

text: equation,

text: with

text: the

text: change

text: of

text: coeﬃcients

text: This

text: implies

text: that,

text: In

text: the

text: same

text: manner,

text: one

text: can

text: ﬁnd

text: that

text: For

text: the

text: two

text: local

text: terms

text: i.e.,

text: Eq.

text: leads

text: to:

text: whence,

text: as

text: well.

text: This

text: conﬁrms

text: that

text: the

text: coeﬃcients

text: associated

text: to

text: local

text: observables

text: are

text: zero,

text: i.e.,

text: di

text: i.

text: However,

text: the

text: non-local

text: coeﬃcients

text: are

text: non-zero.

text: For

text: B

text: using

text: Eq.

text: yields:

text: Accepted

text: in

text: Quantum

text: click

text: title

text: to

text: verify

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text: ( ∞) ( ∞) = ( ∞) ( ∞) ( ∞) = ( ∞) c1(σ1∞1)J=0(σ2∞2)J=0 = (∂Jσ1∞2)J=0, c2(σ1∞1)J=0(σ4∞4)J=0 = (∂Jσ1∞4)J=0, (52) c3 σ33 J=0 σ22 J=0 ∂Jσ32 J=0, c4 σ33 J=0 σ44 J=0 ∂Jσ34 J=0. By using the symmetry in the cova(cid:18)riance ma(cid:19)trix one can simplify t(cid:18)o arrive (cid:19)at: ∞ ∞ c1 = c4 = ∂∞Jσ1∞2 , c2 = −c3 = ∂∞Jσ1∞4 . (53) σ11σ22 σ11σ44 J=0 J=0 (48) Replacing in Eq. for the SLD yields: − Λ = ( + ) + ( ) 0 c1 x1x2 p1p2 c2 x1p2 p1x2 † = ( + ) + (54) c1 ic2 a1a2 h.c. ∈ { } 2 2 + The other key element (ii) is also easy to evaluate for any choice of B x1, x2, x1p1 p1x1, . . . , because the time evolution shall be evaluated under the unperturbed map. To begin with, we remind that the time ? evolution of the ladder operators are easy to ﬁnd (see [ , ]]breuer2002theory, for instance), and read as follow: a (t) = e(−iωj−γ20 )ta , a†(t) = e(iωj−γ20 )ta†, j j j j a†a (t) = e−γ0ta†a + N(1 − e−γ0t), a a†(t) = e−γ0ta a† + (N + 1)(1 − e−γ0t). (55) j j j j j j j j Therefore, by using the deﬁnition of x and p quadratures, we ﬁnd the ﬁrst moments to evolve as: j j 1 x (t) = √ (a†(t) + a (t)) = e−γ/2t (cos(ω t)x + sin(ω t)p ) , j 2 j j j j j j p (t) = √i (a†(t) − a (t)) = e−γ/2t (− sin(ω t)x + cos(ω t)p ) . (56) j 2 j j j j j j In addition, the local second(cid:18)moments are: (cid:19) x2j(t) = e−γt (cid:18)cos2(ωjt)x2j + sin2(ωjt)p2j + sin(22ωjt)(xjpj + pjxj)(cid:19) + (Nj + 12)(1 − e−γt), p2j(t) = e−γt (cid:0) sin2(ωjt)x2j + cos2(ωjt)p2j − sin(22ω(cid:0)jt)(xjpj(cid:1)+(cid:1) pjxj) + (Nj + 12)(1 − e−γt), (p x (t) + x p (t)) = e−γt cos(2ω t) (x p + p x ) − sin(2ω t) x2 − p2 . (57) j j j j j j j j j j j j Note that the evolution keeps locality of the quadratures. On this account, their correlations with the non- ∈ { } Λ 2 2 + ( ) = 0 local 0 vanishes at any time. In other words for B xj, pj, xjpj pjxj the response function φB t , hence they are blind to the perturbation. On the contrar(cid:16)y, for the non-local second moments we have: (cid:17) (cid:16) (cid:17) x1x2(t) = e−γt cos(ω1t) cos(ω2t)x1x2 + cos(ω1t) sin(ω2t)x1p2 + sin(ω1t) cos(ω2t)p1x2 + sin(ω1t) sin(ω2t)p1p2 , (cid:16) (cid:17) x1p2(t) = e−γt − cos(ω1t) sin(ω2t)x1x2 + cos(ω1t) cos(ω2t)x1p2 − sin(ω1t) sin(ω2t)p1x2 + sin(ω1t) cos(ω2t)p1p2 , (cid:16) (cid:17) p1x2(t) = e−γt − sin(ω1t) cos(ω2t)x1x2 − sin(ω1t) sin(ω2t)x1p2 + cos(ω1t) cos(ω2t)p1x2 + cos(ω1t) sin(ω2t)p1p2 , p1p2(t) = e−γt sin(ω1t) sin(ω2t)x1x2 − sin(ω1t) cos(ω2t)x1p2 − cos(ω1t) sin(ω2t)p1x2 + cos(ω1t) cos(ω2t)p1p2 , (58) Accepted in Quantum 2018-04-23, click title to verify 10 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: By

text: using

text: the

text: symmetry

text: in

text: the

text: one

text: can

text: simplify

text: arrive

text: ∂∞Jσ1∞2

text: ∂∞Jσ1∞4

text: Replacing

text: in

text: Eq.

text: for

text: the

text: SLD

text: yields:

text: Λ

text: h.c.

text: The

text: other

text: key

text: element

text: (ii)

text: is

text: also

text: easy

text: to

text: evaluate

text: for

text: any

text: choice

text: of

text: B

text: because

text: the

text: time

text: evolution

text: shall

text: be

text: evaluated

text: under

text: the

text: unperturbed

text: map.

text: To

text: begin

text: with,

text: we

text: remind

text: that

text: the

text: time

text: evolution

text: of

text: the

text: ladder

text: operators

text: are

text: easy

text: to

text: ﬁnd

text: (see

text: for

text: instance),

text: and

text: read

text: as

text: follow:

text: a

text: (t)

text: )ta

text: a†(t)

text: )ta†,

text: j

text: j

text: j

text: j

text: a†a

text: (t)

text: a

text: a†(t)

text: a†

text: (N

text: j

text: j

text: j

text: j

text: j

text: j

text: j

text: j

text: Therefore,

text: by

text: using

text: the

text: deﬁnition

text: of

text: x

text: and

text: p

text: quadratures,

text: we

text: ﬁnd

text: the

text: ﬁrst

text: moments

text: to

text: evolve

text: as:

text: j

text: j

text: x

text: (t)

text: (a†(t)

text: a

text: (t))

text: (cos(ω

text: t)x

text: sin(ω

text: t)p

text: j

text: j

text: j

text: j

text: j

text: j

text: j

text: p

text: (t)

text: √i

text: (a†(t)

text: a

text: (t))

text: sin(ω

text: t)x

text: cos(ω

text: t)p

text: j

text: j

text: j

text: j

text: j

text: j

text: j

text: In

text: addition,

text: the

text: local

text: are:

text: e−γt

text: (Nj

text: e−γt),

text: e−γt

text: pjxj)

text: (Nj

text: e−γt),

text: (p

text: x

text: (t)

text: x

text: p

text: (t))

text: e−γt

text: t)

text: (x

text: p

text: p

text: x

text: t)

text: j

text: j

text: j

text: j

text: j

text: j

text: j

text: j

text: j

text: j

text: j

text: j

text: Note

text: that

text: the

text: evolution

text: keeps

text: locality

text: of

text: the

text: quadratures.

text: On

text: this

text: account,

text: their

text: correlations

text: with

text: the

text: non-

text: Λ

text: local

text: vanishes

text: at

text: any

text: time.

text: In

text: other

text: words

text: for

text: B

text: xj,

text: pj,

text: xjpj

text: pjxj

text: the

text: response

text: function

text: φB

text: t

text: hence

text: they

text: are

text: blind

text: to

text: the

text: perturbation.

text: On

text: the

text: for

text: the

text: non-local

text: second

text: moments

text: we

text: have:

text: e−γt

text: e−γt

text: e−γt

text: e−γt

text: Accepted

text: in

text: Quantum

text: click

text: title

text: to

text: verify

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: WELL-POSEDNESS AND NUMERICAL APPROXIMATION OF TEMPERED FRACTIONAL TERMINAL VALUE PROBLEM1S1 the same solution trajectory. In order to obtain an approximation of y(0) we propose a shooting algorithm based on the bisection method. Let y and y be the solutions 1 2− of (36) with initial values y and y , such that y (a) < exp( λa)y < y (a), 01 02 1 a 2 the bisection method provide successive approximations for y(0) until the distance between the two last approximations does not exceed a given tolerance (cid:15). To evaluate the value of y(a) we need a numerical method to solve the initial value problems D ∈ (37) α,λ (y(t)) = f(t, y(t)), t (0, a], (38) y(0) = y . 0 This will be straightforward if we take relationship (3) into account. In fact, deﬁning the functions u and g as in (35), we can use any available solver for (non-tempered) Caputo-type initial value problems to determine the solution u of D ∈ (39) C,α (u(t)) = g(t, u(t)), t (0, a], 0 t (40) y(0) = y , 0 − and then the solution of (37)-(38) will be given by y(t) = e λtu(t). 5. Numerical results In this section we present some numerical examples to illustrate the eﬃciency of the numerical algorithm. In this subsection we present 3 examples and the method that we apply for each one, depends on the nature of the diﬀerential equation and regularity of the solution. The ﬁrst example is a linear fractional diﬀerential equation with a smooth solu- tion. (cid:18) (cid:18) (cid:19)(cid:19) Example 1. − − D − 3Γ(3)t2 α Γ(5)t4 α 3t2 − α,λ (y(t)) = e λt − + − + c t4 + c y(t), t > 0, 4Γ(3 α) Γ(5 α) α,λ 4 α,λ − e λ/2 y(0.5) = , 4 (cid:18) (cid:19) Γ(α + 1) where cα,λ = 21−αeλ/2 and whose analytical solution is given by 3t2 − y(t) = t4 + e λt . 4 The second example is a nonlinear fractional diﬀerential equation with a smooth solution deﬁned by: (cid:18) (cid:19) Example 2. − D − Γ(3)t2 α − − − α,λ (y(t)) = e λt − t2 α 3t4 exp( λt) + 3y2, t > 0, Γ(3 α) − e λ/2 y(0.5) = , 4 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: WELL-POSEDNESS

text: AND

text: NUMERICAL

text: APPROXIMATION

text: OF

text: TEMPERED

text: FRACTIONAL

text: TERMINAL

text: VALUE

text: the

text: same

text: solution

text: trajectory.

text: In

text: order

text: to

text: obtain

text: an

text: approximation

text: of

text: we

text: propose

text: a

text: shooting

text: algorithm

text: based

text: on

text: the

text: bisection

text: method.

text: Let

text: y

text: and

text: y

text: be

text: the

text: solutions

text: 2−

text: of

text: with

text: initial

text: values

text: y

text: and

text: y

text: such

text: that

text: y

text: (a)

text: exp(

text: λa)y

text: y

text: (a),

text: a

text: the

text: bisection

text: method

text: provide

text: successive

text: approximations

text: for

text: until

text: the

text: distance

text: between

text: the

text: two

text: last

text: approximations

text: does

text: not

text: exceed

text: a

text: given

text: tolerance

text: To

text: evaluate

text: the

text: value

text: of

text: y(a)

text: we

text: need

text: a

text: numerical

text: method

text: to

text: solve

text: the

text: initial

text: value

text: problems

text: D

text: α,λ

text: (y(t))

text: f(t,

text: y(t)),

text: t

text: a],

text: y

text: This

text: will

text: be

text: straightforward

text: if

text: we

text: take

text: relationship

text: into

text: account.

text: In

text: fact,

text: deﬁning

text: the

text: functions

text: u

text: and

text: g

text: as

text: in

text: we

text: can

text: use

text: any

text: available

text: solver

text: for

text: (non-tempered)

text: Caputo-type

text: initial

text: value

text: problems

text: to

text: determine

text: the

text: solution

text: u

text: of

text: D

text: C,α

text: (u(t))

text: g(t,

text: u(t)),

text: t

text: a],

text: t

text: y

text: and

text: then

text: the

text: solution

text: of

text: will

text: be

text: given

text: by

text: y(t)

text: e

text: λtu(t).

text: Numerical

text: results

text: In

text: this

text: section

text: we

text: present

text: some

text: numerical

text: examples

text: to

text: illustrate

text: the

text: eﬃciency

text: of

text: the

text: numerical

text: algorithm.

text: In

text: this

text: subsection

text: we

text: present

text: examples

text: and

text: the

text: method

text: that

text: we

text: apply

text: for

text: each

text: one,

text: depends

text: on

text: the

text: nature

text: of

text: the

text: diﬀerential

text: equation

text: and

text: regularity

text: of

text: the

text: solution.

text: The

text: ﬁrst

text: example

text: is

text: a

text: linear

text: fractional

text: diﬀerential

text: equation

text: with

text: a

text: smooth

text: solu-

text: tion.

text: Example

text: D

text: α

text: α

text: α,λ

text: (y(t))

text: e

text: λt

text: c

text: c

text: y(t),

text: t

text: α)

text: α)

text: α,λ

text: α,λ

text: e

text: Γ(α

text: where

text: cα,λ

text: and

text: whose

text: analytical

text: solution

text: is

text: given

text: by

text: y(t)

text: e

text: λt

text: The

text: second

text: example

text: is

text: a

text: nonlinear

text: fractional

text: diﬀerential

text: equation

text: with

text: a

text: smooth

text: solution

text: deﬁned

text: by:

text: Example

text: D

text: α

text: α,λ

text: (y(t))

text: e

text: λt

text: α

text: exp(

text: λt)

text: t

text: α)

text: e

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text: ##LTLine##

text: ##LTLine##

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text: WELL-POSEDNESS AND NUMERICAL APPROXIMATION OF TEMPERED FRACTIONAL TERMINAL VALUE PROBLEMS7 (cid:18)(cid:90) (cid:90) (cid:19) (cid:107) − (cid:107) ≤ | − | Leλa (cid:107) − (cid:107) (cid:18) a − (cid:19) − t − − y z y z + y z (a s)α 1ds + (t s)α 1ds a a Γ(α) 0 0 | − | Leλa (cid:107) − (cid:107) aα tα = y z + y z + a a Γ(α) α α ≤ | − | 2Leλaaα (cid:107) − (cid:107) y z + y z . a a Γ(α + 1) According to the upper bound on the Lipschitz condition L, established in The- orem 3, we have − 2Leλaaα (29) β = 1 > 0, Γ(α + 1) and therefore, we conclude that (cid:107) − (cid:107) ≤ 1 | − | y z y z , β a a and the Theorem is proved. Theorem 5. Let y and z be the unique solutions of problems (1)-(2) and (19)-(20), respectively, where in the later we assume that δ is such that − 0 < α δ < 1. Then (cid:107) − (cid:107) O y z = (δ) . ∈ Proof. Taking (16) and ((cid:12)26) into(cid:90)account, for any t [0, a], we have (cid:12) (cid:12) (cid:12) − | − | − e λt (cid:90)a − − y(t) z(t) = eλs(a s)α 1f(s, y(s))ds+ Γ(α) 0 − e λt (cid:90) a − − − + − eλs(a s)α δ 1f(s, z(s))ds Γ(α δ) 0 − e λt (cid:90)t − − (cid:12) + eλs(t s)α 1f(s, y(s))ds (cid:12) (cid:12) Γ(α) 0 (cid:12) − − e λt t − − − − eλs(t s)α δ 1f(s, z(s))ds Γ(α δ) 0 − ≤ ≥ ≤ ≤ Since e λt 1, for a(cid:18)ll(cid:90)t (cid:12)0 and λ > 0 and eλs eλa, for all 0 s, then(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) − − − − − | − | ≤ (cid:90) (cid:12) a (a s)α 1 − (a s)α δ 1 (cid:12) (cid:19) y(t) z(t) eλa (cid:12) f(s, y(s)) − f(s(cid:12), z(s)) ds+ (cid:12) (cid:12) (cid:12) 0 Γ(α) Γ(α δ) (cid:12) − − − − − t (t s)α 1 − (t s)α δ 1 (30) + f(s, y(s)) − f(s, z(s)) ds . Γ(α) Γ(α δ) 0 Let us look at the ﬁrst integral on the right-hand side of (30) ﬁrst: ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: WELL-POSEDNESS

text: AND

text: NUMERICAL

text: APPROXIMATION

text: OF

text: TEMPERED

text: FRACTIONAL

text: TERMINAL

text: VALUE

text: Leλa

text: a

text: t

text: y

text: z

text: y

text: z

text: y

text: z

text: (a

text: s)α

text: (t

text: s)α

text: a

text: a

text: Γ(α)

text: Leλa

text: aα

text: tα

text: y

text: z

text: y

text: z

text: a

text: a

text: Γ(α)

text: α

text: α

text: y

text: z

text: y

text: z

text: a

text: a

text: Γ(α

text: According

text: to

text: the

text: upper

text: bound

text: on

text: the

text: Lipschitz

text: condition

text: L,

text: established

text: in

text: The-

text: orem

text: we

text: have

text: β

text: Γ(α

text: and

text: therefore,

text: we

text: conclude

text: that

text: y

text: z

text: y

text: z

text: β

text: a

text: a

text: and

text: the

text: Theorem

text: is

text: proved.

text: Theorem

text: Let

text: y

text: and

text: z

text: be

text: the

text: unique

text: solutions

text: of

text: problems

text: and

text: respectively,

text: where

text: in

text: the

text: later

text: we

text: assume

text: that

text: δ

text: is

text: such

text: that

text: α

text: δ

text: Then

text: O

text: y

text: z

text: (δ)

text: Proof.

text: Taking

text: and

text: for

text: any

text: t

text: a],

text: we

text: have

text: e

text: λt

text: y(t)

text: z(t)

text: eλs(a

text: s)α

text: y(s))ds+

text: Γ(α)

text: e

text: λt

text: a

text: eλs(a

text: s)α

text: δ

text: z(s))ds

text: Γ(α

text: δ)

text: e

text: λt

text: eλs(t

text: s)α

text: y(s))ds

text: Γ(α)

text: e

text: λt

text: t

text: eλs(t

text: s)α

text: δ

text: z(s))ds

text: Γ(α

text: δ)

text: Since

text: e

text: λt

text: for

text: and

text: λ

text: and

text: eλs

text: eλa,

text: for

text: all

text: s,

text: a

text: (a

text: s)α

text: (a

text: s)α

text: δ

text: y(t)

text: z(t)

text: eλa

text: f(s,

text: y(s))

text: z(s))

text: ds+

text: Γ(α)

text: Γ(α

text: δ)

text: t

text: (t

text: s)α

text: (t

text: s)α

text: δ

text: f(s,

text: y(s))

text: f(s,

text: z(s))

text: ds

text: Γ(α)

text: Γ(α

text: δ)

text: Let

text: us

text: look

text: at

text: the

text: ﬁrst

text: integral

text: on

text: the

text: right-hand

text: side

text: of

text: ﬁrst:

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

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text: 7 Police ebanking.police 103.11.142.65 (A) This server’s certiﬁcate will be Bank Ltd bank.com.au distrusted by Google and Mozilla from (Australia) September 2018 8 Bank of ibanking.bankof 203.23.44.204 (A) This server does not support Melbourne melbourne.com.au Authenticated encryption (AEAD) (Australia) cipher suites. Grade will be capped to B from March 2018. 9 Bank www2.irakyat 1.9.61.247 (B) This server is vulnerable to the Return Kerjasama .com.my of Bleichenbacher’s Oracle Threat Rakyat (ROBOT) vulnerability. Grade will be Malaysia set to F from March 2018. This server Berhad supports weak Diﬃe-Hellman (DH) (Malaysia) key exchange parameters. Grade capped to B. This server uses SSL 3, which is obsolete and insecure. Grade capped to B. This server accepts RC4 cipher, but only with older protocols. Grade capped to B. This server does not support Forward Secrecy with the reference browsers. Grade will be capped to B from March 2018 10 National https://nbp.com. 167.114.191.212 (A+) This server’s certiﬁcate will be Bank of pk/login/ distrusted by Google and Mozilla from Pakistan September 2018. HTTP Strict (Pakistan) Transport Security (HSTS) with long duration deployed on this server. 11 Agrani www.agranibank.org 180.92.224.101 (B) This server supports weak Bank Diﬃe-Hellman (DH) key exchange Limited parameters. Grade capped to B. This (Bangladesh) server does not support Forward Secrecy with the reference browsers. Grade will be capped to B from March 2018. This server’s certiﬁcate chain is incomplete. Grade capped to B. This server’s certiﬁcate will be distrusted by Google and Mozilla from September 2018. Table 5: Security analysis of some non-Indian banks. This study is unique, as it’s the ﬁrst one of its kind in the context of Indian banks. However, the analysis performed is not deep enough. Consequently, it opens up the possibility of a deeper analysis in future (which will be reported elsewhere). As the adopted method is valid in general, it’s possible to use this simple approach to analyze the security of banks and other e-commerce sites of other countries, too. Finally, we conclude the article with an optimistic view and a hope that this article will be able to draw the attention of the concerned authorities, and necessary steps will be taken by them to perform regular checking of vulnerability of the encryption techniques adopted by then (as recommended by RBI) and to take corrective measures wherever required. This would help banks to provide higher security and thus to improve the conﬁdence level of the end-users (acceptability of the end-users), which the banks desire to improve. Further, we hope that the suggestions made from the perspectives of a cryptographer will be used by the banks in near future and that would help them to circumvent many attacks in the future. Acknowledgment: AP thanks Defense Research and Development Organization (DRDO), India for the support provided through the Project Number: ERIP/ER/1403163/M/01/1603. Authors also thank Abhishek Parakh and Kishore Thapliyal for their interest in this work and some useful technical comments. References 11 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Police

text: ebanking.police

text: (A)

text: This

text: server’s

text: certiﬁcate

text: will

text: be

text: Bank

text: Ltd

text: bank.com.au

text: distrusted

text: by

text: Google

text: and

text: Mozilla

text: from

text: (Australia)

text: September

text: Bank

text: of

text: ibanking.bankof

text: (A)

text: This

text: server

text: does

text: not

text: support

text: Melbourne

text: melbourne.com.au

text: Authenticated

text: encryption

text: (AEAD)

text: (Australia)

text: cipher

text: suites.

text: Grade

text: will

text: be

text: capped

text: to

text: B

text: from

text: March

text: Bank

text: (B)

text: This

text: server

text: is

text: vulnerable

text: to

text: the

text: Return

text: Kerjasama

text: .com.my

text: of

text: Bleichenbacher’s

text: Oracle

text: Threat

text: Rakyat

text: (ROBOT)

text: vulnerability.

text: Grade

text: will

text: be

text: Malaysia

text: set

text: to

text: F

text: from

text: March

text: This

text: server

text: Berhad

text: supports

text: weak

text: Diﬃe-Hellman

text: (DH)

text: (Malaysia)

text: key

text: exchange

text: parameters.

text: Grade

text: capped

text: to

text: B.

text: This

text: server

text: uses

text: SSL

text: which

text: is

text: obsolete

text: and

text: insecure.

text: Grade

text: capped

text: to

text: B.

text: This

text: server

text: accepts

text: cipher,

text: but

text: only

text: with

text: older

text: protocols.

text: Grade

text: capped

text: to

text: B.

text: This

text: server

text: does

text: not

text: support

text: Forward

text: Secrecy

text: with

text: the

text: reference

text: browsers.

text: Grade

text: will

text: be

text: capped

text: to

text: B

text: from

text: March

text: National

text: (A+)

text: This

text: server’s

text: certiﬁcate

text: will

text: be

text: Bank

text: of

text: pk/login/

text: distrusted

text: by

text: Google

text: and

text: Mozilla

text: from

text: Pakistan

text: September

text: HTTP

text: Strict

text: (Pakistan)

text: Transport

text: Security

text: (HSTS)

text: with

text: long

text: duration

text: deployed

text: on

text: this

text: server.

text: Agrani

text: (B)

text: This

text: server

text: supports

text: weak

text: Bank

text: Diﬃe-Hellman

text: (DH)

text: key

text: exchange

text: Limited

text: parameters.

text: Grade

text: capped

text: to

text: B.

text: This

text: (Bangladesh)

text: server

text: does

text: not

text: support

text: Forward

text: Secrecy

text: with

text: the

text: reference

text: browsers.

text: Grade

text: will

text: be

text: capped

text: to

text: B

text: from

text: March

text: This

text: server’s

text: certiﬁcate

text: chain

text: is

text: incomplete.

text: Grade

text: capped

text: to

text: B.

text: This

text: server’s

text: certiﬁcate

text: will

text: be

text: distrusted

text: by

text: Google

text: and

text: Mozilla

text: from

text: September

text: Table

text: Security

text: analysis

text: of

text: some

text: non-Indian

text: banks.

text: This

text: study

text: is

text: unique,

text: as

text: it’s

text: the

text: ﬁrst

text: one

text: of

text: its

text: kind

text: in

text: the

text: context

text: of

text: Indian

text: banks.

text: However,

text: the

text: analysis

text: performed

text: is

text: not

text: deep

text: enough.

text: Consequently,

text: it

text: opens

text: up

text: the

text: possibility

text: of

text: a

text: deeper

text: analysis

text: in

text: future

text: (which

text: will

text: be

text: reported

text: elsewhere).

text: As

text: the

text: adopted

text: method

text: is

text: valid

text: in

text: general,

text: it’s

text: possible

text: to

text: use

text: this

text: simple

text: approach

text: to

text: analyze

text: the

text: security

text: of

text: banks

text: and

text: other

text: e-commerce

text: sites

text: of

text: other

text: countries,

text: too.

text: Finally,

text: we

text: conclude

text: the

text: article

text: with

text: an

text: optimistic

text: view

text: and

text: a

text: hope

text: that

text: this

text: article

text: will

text: be

text: able

text: to

text: draw

text: the

text: attention

text: of

text: the

text: concerned

text: authorities,

text: and

text: necessary

text: steps

text: will

text: be

text: taken

text: by

text: them

text: to

text: perform

text: regular

text: checking

text: of

text: vulnerability

text: of

text: the

text: encryption

text: techniques

text: adopted

text: by

text: then

text: (as

text: recommended

text: by

text: RBI)

text: and

text: to

text: take

text: corrective

text: measures

text: wherever

text: required.

text: This

text: would

text: help

text: banks

text: to

text: provide

text: higher

text: security

text: and

text: thus

text: to

text: improve

text: the

text: conﬁdence

text: level

text: of

text: the

text: end-users

text: (acceptability

text: of

text: the

text: end-users),

text: which

text: the

text: banks

text: desire

text: to

text: improve.

text: Further,

text: we

text: hope

text: that

text: the

text: suggestions

text: made

text: from

text: the

text: perspectives

text: of

text: a

text: cryptographer

text: will

text: be

text: used

text: by

text: the

text: banks

text: in

text: near

text: future

text: and

text: that

text: would

text: help

text: them

text: to

text: circumvent

text: many

text: attacks

text: in

text: the

text: future.

text: Acknowledgment:

text: AP

text: thanks

text: Defense

text: Research

text: and

text: Development

text: Organization

text: (DRDO),

text: India

text: for

text: the

text: support

text: provided

text: through

text: the

text: Project

text: Number:

text: Authors

text: also

text: thank

text: Abhishek

text: Parakh

text: and

text: Kishore

text: Thapliyal

text: for

text: their

text: interest

text: in

text: this

text: work

text: and

text: some

text: useful

text: technical

text: comments.

text: References

text: ##LTLine##

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text: some of them are found to use keys of inadequate length, support RC4 cipher with older protocols, support TLS 1.0, TLS 1.1, SSL 3.0, etc., and to be vulnerable to Padding Oracle On Downgraded Legacy Encryption (POODLE) attack (Allahabad Bank, Canara Bank, Indusind Bank and Punjab National Bank) and to man-in-the-middle attack (MITM) (Bank of Baroda, Syndicate Bank, Indusind Bank). Here, it may be apt to note that POODLE is a particular type of MITM attack that exploits internet and security software clients’ fallback to SSL 3.0. These observations are illustrated through Fig. 1 (c), and summarized in Table A.II of Appendix A. S.No Name of the Website Server (Grade) Remarks bank (Pub/Pvt) 1 State Bank retail.onlinesbi.com 223.31.160.67 (A+) (i) HTTP Strict Transport of India 2405:a700:14:12c:0:0:0:148 Security (HSTS) with long (Pub) (A+) duration deployed on this server. 2 Axis Bank retail.axisbank.co.in 121.243.50.21 (A) (Pvt) 2403:0:500:11:0:0:0:160 (A) 3 Andhra www.online 103.196.117.1 (A) Bank (Pub) andhrabank.net.in 4 Catholic www.csbnet.co.in 115.112.233.198 (A) Syrian Bank (Pvt) 5 HDFC netbanking.hdfcbank.com 175.100.160.21 (A) (i) Intermediate certiﬁcate has Bank an insecure signature. (Pvt) (ii) HTTP Strict Transport Security (HSTS) with long duration deployed on this server. 6 Indian www.indianbank.net.in 115.110.173.99 (A-) (i) Intermediate certiﬁcate has Bank (Pub) an insecure signature. 7 Oriental www.obconline.co.in 220.226.206.37 (A-) (i) Forward Secrecy with the Bank of reference browsers is not Commerce supported. (Pub) (ii) HTTP Strict Transport Security (HSTS) with long duration deployed on this server. 8 UCO Bank www.ucoebanking.com 203.200.206.227 (A-) (i) Secure renegotiation is not (Pub) 2405:a080:0:0:0:0:0:30 supported. (A-) (ii) Forward Secrecy with the reference browsers is not supported. 9 United ebank.unitedbank 14.142.120.11 (A-) Bank of oﬁndia.com India (Pub) 10 ICICI Bank inﬁnity.icicibank.com 203.189.92.162 (A-) (i) Secure renegotiation is not (Pvt) 2001:df0:2fd:1:0:0:0:2 supported. (A-) (ii) Forward Secrecy with the reference browsers is not supported. 11 Karnataka moneyclick.karnataka 210.212.203.235 (A-) (i) Forward Secrecy with the Bank (Pvt) bank.co.in 125.16.142.105 (A-) reference browsers is not 2404:a800:3001:b:0:0:0:5 supported. (A-) 12 RBL Bank online.rblbank.com 180.179.110.171 (A-) (i) Secure renegotiation is not (Pvt) supported. (ii) Forward Secrecy with the reference browsers is not supported. 3 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: some

text: of

text: them

text: are

text: found

text: to

text: use

text: keys

text: of

text: inadequate

text: length,

text: support

text: cipher

text: with

text: older

text: protocols,

text: support

text: TLS

text: TLS

text: SSL

text: etc.,

text: and

text: to

text: be

text: vulnerable

text: to

text: Padding

text: Oracle

text: On

text: Downgraded

text: Legacy

text: Encryption

text: (POODLE)

text: attack

text: (Allahabad

text: Bank,

text: Canara

text: Bank,

text: Indusind

text: Bank

text: and

text: Punjab

text: National

text: Bank)

text: and

text: to

text: man-in-the-middle

text: attack

text: (MITM)

text: (Bank

text: of

text: Baroda,

text: Syndicate

text: Bank,

text: Indusind

text: Bank).

text: Here,

text: it

text: may

text: be

text: apt

text: to

text: note

text: that

text: POODLE

text: is

text: a

text: particular

text: type

text: of

text: MITM

text: attack

text: that

text: exploits

text: internet

text: and

text: security

text: software

text: clients’

text: fallback

text: to

text: SSL

text: These

text: observations

text: are

text: illustrated

text: through

text: Fig.

text: (c),

text: and

text: summarized

text: in

text: Table

text: A.II

text: of

text: Appendix

text: A.

text: S.No

text: Name

text: of

text: the

text: Website

text: Server

text: (Grade)

text: Remarks

text: bank

text: (Pub/Pvt)

text: State

text: Bank

text: retail.onlinesbi.com

text: (A+)

text: (i)

text: HTTP

text: Strict

text: Transport

text: of

text: India

text: Security

text: (HSTS)

text: with

text: long

text: (Pub)

text: (A+)

text: duration

text: deployed

text: on

text: this

text: server.

text: Axis

text: Bank

text: retail.axisbank.co.in

text: (A)

text: (Pvt)

text: (A)

text: Andhra

text: (A)

text: Bank

text: (Pub)

text: andhrabank.net.in

text: Catholic

text: (A)

text: Syrian

text: Bank

text: (Pvt)

text: HDFC

text: netbanking.hdfcbank.com

text: (A)

text: (i)

text: Intermediate

text: certiﬁcate

text: has

text: Bank

text: an

text: insecure

text: signature.

text: (Pvt)

text: (ii)

text: HTTP

text: Strict

text: Transport

text: Security

text: (HSTS)

text: with

text: long

text: duration

text: deployed

text: on

text: this

text: server.

text: Indian

text: (A-)

text: (i)

text: Intermediate

text: certiﬁcate

text: has

text: Bank

text: (Pub)

text: an

text: insecure

text: signature.

text: Oriental

text: (A-)

text: (i)

text: Forward

text: Secrecy

text: with

text: the

text: Bank

text: of

text: reference

text: browsers

text: is

text: not

text: Commerce

text: supported.

text: (Pub)

text: (ii)

text: HTTP

text: Strict

text: Transport

text: Security

text: (HSTS)

text: with

text: long

text: duration

text: deployed

text: on

text: this

text: server.

text: UCO

text: Bank

text: (A-)

text: (i)

text: Secure

text: renegotiation

text: is

text: not

text: (Pub)

text: supported.

text: (A-)

text: (ii)

text: Forward

text: Secrecy

text: with

text: the

text: reference

text: browsers

text: is

text: not

text: supported.

text: United

text: ebank.unitedbank

text: (A-)

text: Bank

text: of

text: oﬁndia.com

text: India

text: (Pub)

text: ICICI

text: Bank

text: inﬁnity.icicibank.com

text: (A-)

text: (i)

text: Secure

text: renegotiation

text: is

text: not

text: (Pvt)

text: supported.

text: (A-)

text: (ii)

text: Forward

text: Secrecy

text: with

text: the

text: reference

text: browsers

text: is

text: not

text: supported.

text: Karnataka

text: moneyclick.karnataka

text: (A-)

text: (i)

text: Forward

text: Secrecy

text: with

text: the

text: Bank

text: (Pvt)

text: bank.co.in

text: (A-)

text: reference

text: browsers

text: is

text: not

text: supported.

text: (A-)

text: RBL

text: Bank

text: online.rblbank.com

text: (A-)

text: (i)

text: Secure

text: renegotiation

text: is

text: not

text: (Pvt)

text: supported.

text: (ii)

text: Forward

text: Secrecy

text: with

text: the

text: reference

text: browsers

text: is

text: not

text: supported.

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text: 13 Yes Bank netbanking.yesbank.co.in 123.136.19.72 (A-) (i) Secure renegotiation is not (Pvt) supported. (ii) Forward Secrecy with the reference browsers is not supported. Table 1: List of banks that provide reasonably good security in the ∈ { −} sense that they obtain grades A+, A, A S. Name of the Website Server (Grade) Issues that reduced the No bank grade (Pub/Pvt) 1 Vijaya Bank www.vijayabankonline.in 210.212.204.30 (B) (i) Accepts RC4 cipher, but (Pub) 219.65.65.166 (B) only with older protocols. 2403:0:500:31:0:0:0:12 (ii) Forward Secrecy with the (B) reference browsers is not supported. 2 Bank of starconnectcbs.bank 61.246.202.9 (B) (i) Supports weak India (Pub) oﬁndia.com Diﬃe-Hellman (DH) key exchange parameters. (ii) Forward Secrecy with the reference browsers is not supported. 3 Indian www.iobnet.co.in 121.242.125.68 (B) (i) Uses SSL 3, which is obsolete Overseas and insecure. Bank (Pub) (ii) Accepts RC4 cipher, but only with older protocols. (iii) Forward Secrecy with the reference browsers is not supported. 4 IDBI Bank inet.idbibank.co.in 103.93.45.42 (B) (i) Accepts RC4 cipher, but Ltd (Pub) 2001:df1:3700:1:0:0:0:7 only with older protocols. (B) (ii) Secure renegotiation is not supported. (iii) Forward Secrecy with the reference browsers is not supported. 5 DCB Bank pib.dcbbank.com 202.56.244.126 (B) (i) Accepts RC4 cipher, but (Pvt) 203.196.200.194 (B) only with older protocols. (ii) Forward Secrecy with the reference browsers is not supported. 6 Fednet www.fednetbank.com 121.243.127.68 (B) (i) Accepts RC4 cipher, but Internet only with older protocols. Banking (ii) Forward Secrecy with the (Pvt) reference browsers is not supported. 7 IDFC Bank my.idfcbank.com 103.233.77.139 (B) (i) Supports weak (Pvt) Diﬃe-Hellman (DH) key exchange parameters. (ii) Forward Secrecy with the reference browsers is not supported. 4 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Yes

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text: the

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text: browsers

text: is

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label: caption

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text: (Pub)

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text: is

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text: (B)

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text: which

text: is

text: obsolete

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text: and

text: insecure.

text: Bank

text: (Pub)

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text: Accepts

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text: older

text: protocols.

text: (iii)

text: Forward

text: Secrecy

text: with

text: the

text: reference

text: browsers

text: is

text: not

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text: IDBI

text: Bank

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text: (B)

text: (i)

text: Accepts

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text: Ltd

text: (Pub)

text: only

text: with

text: older

text: protocols.

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text: (ii)

text: Secure

text: renegotiation

text: is

text: not

text: supported.

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text: Forward

text: Secrecy

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text: the

text: reference

text: browsers

text: is

text: not

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text: DCB

text: Bank

text: pib.dcbbank.com

text: (B)

text: (i)

text: Accepts

text: cipher,

text: but

text: (Pvt)

text: (B)

text: only

text: with

text: older

text: protocols.

text: (ii)

text: Forward

text: Secrecy

text: with

text: the

text: reference

text: browsers

text: is

text: not

text: supported.

text: Fednet

text: (B)

text: (i)

text: Accepts

text: cipher,

text: but

text: Internet

text: only

text: with

text: older

text: protocols.

text: Banking

text: (ii)

text: Forward

text: Secrecy

text: with

text: the

text: (Pvt)

text: reference

text: browsers

text: is

text: not

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text: IDFC

text: Bank

text: my.idfcbank.com

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text: (i)

text: Supports

text: weak

text: (Pvt)

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text: Forward

text: Secrecy

text: with

text: the

text: reference

text: browsers

text: is

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text: 15 Kotak www.kotak.com 203.196.200.28 (C) (i) Uses RC4 with modern Mahindra 2403:0:100:51:0:0:0:51 protocols. Bank (Pvt) (C) (ii) Uses 64-bit block cipher (3DES / DES / RC2 / IDEA) (iii) Forward Secrecy with the reference browsers is not supported. 16 South Indian sibernet.southindian 103.212.29.50 (C) (i) Uses 64-bit block cipher Bank (Pvt) bank.com (3DES / DES / RC2 / IDEA) with modern protocols. (ii) Forward Secrecy with the reference browsers is not supported. 17 Tamilnad www.tmbnet.in 124.124.113.139 (A) Mercantile 14.141.76.115 (A) Bank (Pvt) 115.254.82.230 (C) 14.142.204.70 (C) Table 2: List of banks that provide moderate security in the sense ∈ { that they obtain grades B, C}. S. Name of the Website Server(s) (Grade) Issues that reduced the No bank grade (Pub/Pvt) 1 Canara netbanking.canarabank.in 180.92.164.8 (F) (i) Vulnerable to MITM attacks Bank (Pub) 2404:a500:0:1000:0:0:60:e because it supports insecure (F) renegotiation. (ii) Forward Secrecy with the reference browsers is not supported. 2 Corporation www.corpretail.com 202.62.247.23 (F) (i) Supports anonymous Bank (Pub) (insecure) suites and other insecure cipher suites (ii) Supports weak DH key exchange parameters. 3 Dena Bank www.denaiconnect.co.in 103.224.110.30 (F) (i) Supports insecure cipher (Pub) suites (see below for details). (ii) Supports 512-bit export suites and might be vulnerable to the FREAK attack. (iii) Intermediate certiﬁcate contains an insecure signature. (iv) Supports older protocols, but does not support the current best TLS 1.2. (v) Forward Secrecy with the reference browsers is not supported. (vi) Accepts RC4 cipher, but only with older protocols. 6 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: Secrecy

text: with

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text: reference

text: browsers

text: is

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text: Secrecy

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text: browsers

text: is

text: not

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text: (F)

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text: FREAK

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text: insecure

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text: current

text: best

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text: (v)

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text: with

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text: reference

text: browsers

text: is

text: not

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text: (vi)

text: Accepts

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