Product Portfolio

Lithium Ion &NiMH
Battery Packs for Camcorders
Most digital camcorder users know by experience that a couple of spare battery packs would be most helpful for extended videomaking amusement. GP Batteries, one of the world's top battery manufacturers, has a comprehensive range of Lithium Ion and NiMH rechargeable battery packs to suit different models of the leading brands on the market. They have higher capacity and energy density to provide longer running hours. They are lighter in weight, higher in performance and work perfectly well as the original packs do. Long-lasting GP rechargeable battery packs - the smart tip to secure a longer-lasting customer relationship.

Lithium Ion Series

Sony VSL001

Capacity: 1800 mAh Fits:

CCD-SC5/E CCD-TR1/E CCD-TR56/E CCD-TR94/E CCD-TR350/E CCD-TR427 CCD-TR516/E CCD-TR717 CCD-TR840/E CCD-TR940 CCD-TR3100/E CCD-TRV3/E CCD-TRV15/E CCD-TRV35 CCD-TRV43 CCD-TRV51 CCD-TRV64/E CCD-TRV68 CCD-TRV85 CCD-TRV91/E CCD-TRV95/E CCD-TRV110/E CCD-TRV311/E CCD-TRV715 DCR-TRV4/E DCR-TRV51/E DCR-TRV82/E DCR-TRV310 DCR-TRV890/E DCR-TR7000 DCR-VX2000

Voltage: 7.4 V

CCD-SC7/E CCD-TR8/E CCD-TR64/E CCD-TR200 CCD-TR416 CCD-TR511/E CCD-TR555 CCD-TR810/E CCD-TR913/E CCD-TR2200/E CCD-TR3300/E CCD-TRV9 CCD-TRV23/E CCD-TRV36 CCD-TRV45K CCD-TRV58 CCD-TRV65/E CCD-TRV75 CCD-TRV89/E CCD-TRV93/E CCD-TRV99 CCD-TRV215 CCD-TRV510/E CCD-TRV3300 DCR-TRV7/E DCR-TRV62/E DCR-TRV110 DCR-TRV410/E DCR-TRV900/E DCR-TV900/E DCR-VX9000/E DCR-TRV9 DCR-TRV72/E DCR-TRV110/E DCR-TRV510 DCR-TRV7000/E DCR-VX700 GV-A500 DCR-VX1000 TR3300 DCR-TRV9/E DCR-TRV81/E DCR-TRV210/E DCR-TRV510/E CCD-TRV14 CCD-TRV25 CCD-TRV36/E CCD-TRV46 CCD-TRV61/E CCD-TRV66 CCD-TRV81 CCD-TRV90 CCD-TRV94/E CCD-TRV99/E CCD-TRV280PK CCD-TRV511/E CCD-TRV15 CCD-TRV27/E CCD-TRV41 CCD-TRV46/E CCD-TRV62 CCD-TRV66/E CCD-TRV82 CCD-TRV91 CCD-TRV95 CCD-TRV101/E CCD-TRV310/E CCD-TRV615 CCD-SC8/E CCD-TR18/E CCD-TR67 CCD-TR205 CCD-TR417 CCD-TR515/E CCD-TR713/E CCD-TR815 CCD-TR917 CCD-TR3000 CCD-SC55 CCD-TR54/E CCD-TR87 CCD-TR300 CCD-TR425/E CCD-TR516 CCD-TR716 CCD-TR820/E CCD-TR930 CCD-TR3000/E
CCD-SC6 CCD-TR3 CCD-TR57 CCD-TR97 CCD-TR411/E CCD-TR500 CCD-TR517 CCD-TR760/E CCD-TR910 CCD-TR1100/E CCD-TR3200/E CCD-TRV7 CCD-TRV16 CCD-TRV35/E CCD-TRV45/E CCD-TRV55/E CCD-TRV65 CCD-TRV72 CCD-TRV85K CCD-TRV93 CCD-TRV95K CCD-TRV120 CCD-TRV315 CCD-TRV815 DCR-TRV7 DCR-TRV61/E DCR-TRV103 DCR-TRV310/E DCR-TRV900 DCR-TR7100 DCR-VX9000
Replaces Batteries : Sony NP500, NP520, NP530, NP-F330, NP-F530, NP-F550 L series; Duracell DR5

VSL002

Capacity: 1300 mAh Fits: Voltage: 7.4 V
DCR-TRV6/E DCR-TRV20/E DCR-TRV140/E DCR-TRV940/E DCR-PC100/E DCR-PC101/E DCR-PC110/E DCR-TRV8/E DCR-TRV10/E DCR-TRV11/E DCR-TRV25/E DCR-TRV27/E DCR-TRV50/E DCR-TRV330 DCR-TRV340/E DCR-TRV740/E Replaces Batteries: Sony NP-FM50 M series; Duracell DR-SM50
VSL006 VSL007 VSL008 VSL009 VSL011

Capacity: 630 mAh Fits:

DCR-IP7E
DCR-IP5/E DCR-IP55E DCR-IP7BT Replaces Battery: Sony NP-FF50, NP-FF70 F series

Capacity: 3600 mAh Fits:

See VSL001 model list (for Sony) Replaces Batteries: Sony NP-F750 L series; Duracell DR-S750

Capacity: 2600 mAh Fits:

See VSL002 model list Replaces Batteries: Sony NP-FM70 M series; Duracell DR-SM70

Capacity: 3900 mAh Fits:

See VSL002 model list Replaces Batteries: Sony NP-FM90, NP-FM91 M series; Duracell DR-SM91

Capacity: 5400 mAh Fits:

See VSL001 model list (for Sony) Replaces Battery: Sony NP-F960 L series

VSL013 VSL014 VSL015

DCR-HC20, DCR-HC30, DCR-HC40, DCR-HC85 Replaces Battery: Sony NP-FP50P series

Capacity: 1260 mAh Fits:

DCR-HC30, DCR-HC40 Replaces Battery: Sony NP-FP70P series

Capacity: 1860 mAh Fits:

DCR-HC30, DCR-HC40 Replaces Battery: Sony NP-FP90P series
Capacity: VCL001 Fits: 1800 mAh
MV1 XM1 UC-V10Hi UC-V300 UC-X1Hi UC-X45Hi MV10 XM2 UC-V20Hi UC-X2Hi UC-X50Hi
MV100 XV1 UC-V30Hi UC-X20Hi UC-X55Hi MV200 UC-V100 UC-X30Hi Optura UC-V200 UC-X40Hi
Replaces Batteries: Canon BP-911, BP-914, BP-915, BP-924, BP-927; Duracell DR-C915 (DR2); Varta V274; Vivanco BL1367L, BP1367L, BP2767L
Capacity: VCL006 Fits: 1300 mAh

Elura 2 Elura 2MC

IXY DV2 MV3MC MV3iMC
Replaces Batteries: Canon BP-412; Duracell DR-C412
Capacity: VCL007 Fits: 2600 mAh

See VCL006 model list

Replaces Batteries: Canon BP-422; Duracell DR-C422
Capacity: VCL008 Fits: 2600 mAh
FV30 Optura Pi ZR-45 FV100 PV130 ZR-50
FV200 ZR-10 MVX1i ZR-20 Optura 100MC ZR-40

VL-NZ10E/U Replaces Batteries: Sharp BT-L225; Duracell DR-V225
VL-H860U VL-PD3U VL-ME10E/U VL-H870U
VL-H875U VL-H880U VL-H890U
VL-SD20U VL-WD250E/U VL-WD450E/U VL-WD650E/U VL-ME100E/U
Replaces Batteries: Sharp BT-L441; Duracell DR-V441

Hitachi

VSL001
VM-BPL13 VM-E330 VM-E555LA VM-H80E VM-H945 VM-BPL27 VM-E340 VM-E635 VM-H630A VM-H955LA VM-E530A VM-E645LE VM-H650A VM-E535 VM-E835 VM-H755 VM-E540

VM-H855LA

Replaces Batteries: Varta V270, V271 (for Hitachi Camcorder only)

NiMH Series

Sony / Panasonic
Capacity: 2100 mAh Fits Sony:

CCD-F550E

Voltage: 6.0 V

Fits Panasonic:

GR-AX500, GR-AX55U Replaces Batteries: Sony NP-68; Panasonic BP-15, BP-17
Capacity: 2200 mAh Fits Sony:
GR-AX500 / GR-AX55U Replaces Batteries: Sony NP-68; Panasonic BP-15, BP-17
Capacity: 4200 mAh Fits Sony:

VS151 VS152 VS153

Capacity: 2100 mAh Fits:
CCD-F550E Replaces Battery: Sony NP-68

Capacity: 2200 mAh Fits:

Replaces Battery: Sony NP-68

Capacity: 4200 mAh Fits:

VP151 VP152 VP153 VP154 VP155 VP157
GR-AX500, GR-AX55U Replaces Batteries: Panasonic BP-15, BP-17

GR-AX500, GR-AX55U

Replaces Batteries: Panasonic BP-15, BP-17

Voltage: 9.6 V

Replaces Battery: Panasonic VM-VBC2

NV-R65E, NV-S950PN

Voltage: 4.8 V
Replaces Batteries: Panasonic VM-VBS10, P-V211, HHR-V211
Replaces Batteries: Panasonic VW-VBS20E, P-V212, HHR-V212

VR151 VR152 VR153

Capacity: 3800 mAh Fits:
VL-E500U, VL-H420U Replaces Battery: Sharp BT-N1

Voltage: 3.6 V

Capacity: 2500 mAh Fits:

VL-E600U, VL-H800U

Replaces Batteries: Sharp BT-H21, BT-H22

Capacity: 5000 mAh Fits:

VL-A10M, VL-AH50E
Replaces Battery: Sharp BT-H32

VC151 VC152

ES-10V, ES-300, ES-750 Replaces Batteries: Canon BP-711, BP-714
Brand name of the camcorders and batteries is a registered trademark of the respective manufacturers.

ES-10V, ES-300, ES-750

Replaces Batteries: Canon BP-726, BP-729

VH151 VH152

VM-600M, VM-1280E Replaces Battery: Hitachi VM-BP22

VM-600M, VM-1280E

Replaces Battery: Hitachi VM-BP83

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GPPA4CC-A 10/04

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Math 229: Introduction to Analytic Number Theory How many points can a curve of genus g have over Fq ? Let k be a nite eld of q elements, and C/k a (smooth projective) curve of genus g = g(C). Let K = k(C) be its function eld. A prime (a.k.a. place, valuation) p of K is a Galois orbit of k-rational points of C. If that orbit has size d = dp (the degree of p) then we are dealing with dp conjugate points dened over the q dp -element eld (and no smaller eld intermediate between it and k), which is the residue eld of p. The zeta function K (s) = C (s) of this eld or curve may be dened as the Euler product C (s) =

1 = 1 (q dp )s

extending over all primes p, where Z = q s. Then

log C (s) =

dp = n n=1
But the inner sum is just the number Nn = Nn (C) of points of C rational over the eld of q n elements. Note1 that Nn C q n , whence the sum and thus the Euler product converge for |Z| < 1/q, i.e., for > 1. As in the number-eld case, C satises a functional equation relating its values at s and 1 s: C (1 s) = q (22g)( 2 s) C (s) = (qZ 2 )1g C (s); equivalently, C (s) := q (1g)( 2 s) C (s) is invariant under s 1 s. Moreover, C (s) is of the form C (s) = P (Z) (1 Z)(1 qZ)
for some polynomial P of degree 2g with P (0) = 1. It then follows from the functional equation that P (1/qZ) = P (Z)/(qZ 2 )g , which is to say that we can factor P (Z) as

P (Z) =

(1 j Z)(1 g+j Z)
for some complex numbers 1 ,. , 2g such that j g+j = q

instance, let f :

be any nonconstant function; then qn.
Nn (C) (deg f )Nn (P1 ) = (deg f )(q n + 1)
for j = 1,. , g. Comparing this with our formula for Nn we nd

Nn = q n + 1

(Fortunately this agrees with our known formula for Nn when g = 0.) The analogue of the Dirichlet class number formula is the fact that Jacobian JC (k) of C over k has size P (1) = (1 j ),
which is essentially the residue of C (s) at its pole s = 1. So far all this can be proved by more-or-less elementary means, and even extends to varieties over k of any dimension [Dwork 1960]. A much harder, but known, result is that the Riemann hypothesis holds: P (q s ) can vanish only for s such that = 1/2, i.e., |Z| = q 1/2 ; thus all the j have absolute value q 1/2 , and g+j = j. This theorem of Weil, and its generalization by Deligne to varieties of arbitrary dimension over nite elds, is at least to some tastes the strongest evidence so far for the truth of the original Riemann hypothesis and its various generalizations. The theorem |j |2 = q also has numerous applications. For instance, it follows immediately that the number N1 = N1 (C) of k-rational points on C is approximated by q + 1: (1) |N1 (q + 1)| 2g q. Equality can hold in this Weil bound at least for small g, though already for g = 1 there are surprises; for instance for q = 128 the bound (1) allows N1 to be as large as 151 and as small as 107, but in fact the maximum and minimum are 150 and 108. See [Serre 19824] for much more about this. We ask however what happens for xed q as g: How large can N1 (C) grow as a function of g? this is not only a compelling problem in its own right, but has applications to coding theory and similar combinatorial problems, see for instance [Goppa 1981, 1983; Tsfasman 1996; Elkies 2001]. We shall see that the bound N1 < 2g q + Oq (1) coming from (1) cannot be sharp, and obtain an improved bound, the DrinfeldVldut bound a N1 < ( q 1 + o(1)) g, (2) [Drinfeld-Vldut 1983], that turns out to be best possible for square q [Ihara a 1981, Tsfasman-Vldut-Zink 1982]. Moreover, we shall adapt Weyls equidistria bution argument to obtain the asymptotic distribution of the j on the circle ||2 = q for curves attaining that bound. The key idea is much the same as what we used to prove that (1 + it) = 0. To start with, note that if the Weil upper bound N1 q +1+2g q is attained then 2 each j = q. This can actually happen: for instance, let q = q0 and let C be q0 +1 q0 +1 the (q0 + 1)-st Fermat curve, the smooth plane curve x +y + z q0 +1 = 3 of degree q0 + 1 and hence of genus (q0 q0 )/2. Then C has q0 + 1 points over k,

the maximum allowed by (1) [check this! See Exercise 1]. But now consider this curve over the quadratic extension Fq2 of k: we have

N2 = q 2 + 1

2 = q 2 + 1 2gq = q 3/2 + 1 = N1 , j
which means that every point rational over Fq2 is already Fq -rational! [It is an amusing problem to verify this directly, without invoking the Riemann hypothesis for C.] It follows that if g were any larger than (q q0 )/2 and all the j were equal to q0 then N2 would actually be smaller than N1 , which is impossible. So, we have

0 N2 N1 = q 2 q +

(j 2 ), j
and likewise 0 Nn N1 = q n q +

(j n ) j

for each n = 2, 3, 4,. (We also have inequalities Ndn > Nn , but these do not help us asymptotically.) How to best combine them? For given q, g this is not an easy problem, but if we x q and only care about asymptotics as g then all we need do is use the inequality

(j / q)m

(M m)q m/2 (m + m ) j j+g
for each M. (This is the positivity of the Fejr kernel e Summing this inequality over j g we nd

for |z| = 1.)

0 Mg +

m=1 M 1

(M m)q m/2 (q m + 1 Nm ) (M m)q m/2 (q m + 1 N1 )

= M g + OM (1) N1

(M m)q m/2.

Thus N1 < For each

M 1 m=1 (1

m m/2 M )q

+ OM (1). of
> 0, the sum can be brought within

q m/2 = 1/( q 1)

by taking M large enough. We thus have for each > 0 N1 < ( q 1 + )g + O (1), from which (2) follows. What is required for asymptotic equality as C ranges over a sequence of curves with g? Let j = q 1/2 e(xj ) for xj R/Z with xj+g = xj. Then

Nn = q n/2

e(nxj ) + q n + 1.
Since Nn N1 is used for each n, we must have Nn = N1 + on (g), and thus

e(nxj ) = q

(1n)/2 j=1

e(xj ) + on (g).

Moreover
e(xj ) = (1 q 1/2 )g + o(g).
Adapting the Weyl equidistribution argument (see especially Exercise 2 of the Weyl handout), we conclude that the xj approach the distribution whose n-th Fourier moment (n = 0) is (1 q 1/2 )/2q (|n|1)/2 , that is, q (x) dx where the density q is e(nx) + e(nx) 1/2. q (1n)/(1 q ) 2 n=1 Since (1 q

q (1n)/2 = 1,

this density is nonnegative, so it can be attained and (2) is asymptotically the best inequality that can be obtained from Nn N1. In fact it is known [Ihara 1981, Tsfasman-Vldut-Zink 1982] that when q is a square2 there are curves a with arbitrarily large g for which N1 ( q 1) g; our proof of (2) gives the asymptotic distribution of j on the circle ||2 = q for any such sequence. It also 2g lets us compute the size #J = j=1 (1 j ) of the Jacobian in a logarithmic asymptotic sense:

g 1 log #J log q +

log |1 q 1/2 e(x)| q (x) dx.
The integral can be evaluated explicitly using the Taylor expansion of log(1 z) (see the Exercises). Such formulas are needed to determine the asymptotic
2 When q is not a square, lim sup g N1 (C)/g(C) is known to be positive (see for instance [Serre 19821984]), but its value is still a great mystery even for q = 2.
performance of families of codes or lattices constructed as in [Tsfasman 1996] from the curves of [Ihara 1981, Tsfasman-Vldut-Zink 1982]. a Remark The only families of curves known to attain the Drinfeld-Vldut bound consist a of modular curves of various kinds. Explicit formulas for some such families can be found in [Tsfasman-Vldut 1991, Garcia-Stichtenoth 1995] (Drinfeld moda ular curves), [Elkies 1998, 1998a] (elliptic and Shimura modular curves), and elsewhere. Exercises 1. Verify that if q0 is a prime power then the Fermat curve of degree q0 + 1 has q0 +1 rational points over the eld of q0 elements. It follows that each j = q0 , and thus that there are still q0 + 1 points rational over the eld of q0 elements. q0 +1 q0 +1 Can you prove directly that any nonzero solution of x +y + z q0 +1 = over the eld of q0 elements is proportional to one with all variables in the eld 2 of size q0 ? 2. What is the best upper bound that can be obtained on N1 using only the inequality N1 N2 ? Prove that the inequalities N1 Nn (n = 3, 4,.) further that improve this bound if and only if g > (q 2 q)/ 2q. [It is known if q = 22e+1 for some integer e 0 then there exists a curve of genus (q 2 q)/ 2q = 23e+1 2e with N1 = N2 = N3 = q 2 + 1. For instance, when e = 0 this is the elliptic curve with ane equation y 2 + y = x3 + x over the 2-element eld.] 3. Compute q (x) and the integral (3) in closed form. Generalize to obtain, for each s C of real part > 1/2, a closed form for limg g 1 log (1q 1s )C (s) as C ranges over a family of curves over Fq2 with N1 (C)/g(C) q 1. (For the answer and an application to error-correcting codes, see [Elkies 2001], already cited in the Exercises for Euler products.) References [Drinfeld-Vldut 1983] Drinfeld, V.G., Vldut, S.: Number of points of an algea a braic curve, Func. Anal. 17 (1983), 5354. [Dwork 1960] Dwork, B.M.: On the rationality of the zeta function of an algebraic variety. Amer. J. Math. 82 (1960), 631648. [Elkies 1998] Elkies, N.D.: Explicit modular towers. Pages 2332 in Proceedings of the Thirty-Fifth Annual Allerton Conference on Communication, Control and Computing, Univ. of Illinois at Urbana-Champaign, 1998. [Elkies 1998a] Elkies, N.D.: Shimura curve computations. Pages 147 in Proceedings of ANTS-3 (Lecture Notes in Computer Science 1423), Berlin: Springer, 1998. math.NT/0005160 at arXiv.org. [Elkies 2001] Elkies, N.D.: Excellent nonlinear codes from modular curves, pages 200208 in STOC01: Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, Hersonissos, Crete, Greece. Isomorphic with math.NT/0104115 at arXiv.org.

[Garcia-Stichtenoth 1995] Garcia, A., Stichtenoth, H.: A tower of Artin-Schreier extensions of function elds attaining the Drinfeld-Vladut bound. Invent. Math. 121 (1995), 211233. [Goppa 1981, 1983] Goppa, V.D.: Codes on algebraic curves, Soviet Math. Dokl. 24 (1981), 170172; Algebraico-geometric codes, Math. USSR Izvestiya 24 (1983), 7591. [Ihara 1981] Ihara, Y.: Some remarks on the number of rational points of algebraic curves over nite elds. J. Fac. Sci. Tokyo 28 (1981), 721724. [Serre 19824] Serre, J.-P.: Sur le nombre des points rationnels dune courbe algbrique sur un corps ni; Nombres de points des courbes algbriques sur Fq ; e e Rsum des cours de 19831984: reprinted as ##128,129,132 in his Collected e e Works III [O 9.86.1 (III) / QA3.S47] [Tsfasman 1996] Tsfasman, M.A.: Algebraic Geometry Lattices and Codes, pages 385390 in the proceedings of ANTS-II (second Algorithmic Number Theory Symposium), ed. H. Cohen, Lecture Notes in Computer Science 1122 [QA75.L4 #1122 in the McKay Applied Science Library]. [Tsfasman-Vldut 1991] Tsfasman, M.A., Vldut, S.G.: Algebraic-Geometric a a Codes. Dordrecht: Kluwer, 1991. [Tsfasman-Vldut-Zink 1982] Tsfasman, M.A., Vldut, S.G., Zink, T.: Modular a a curves, Shimura curves and Goppa codes better than the Varshamov-Gilbert bound. Math. Nachr. 109 (1982), 2128.