R M 8 2 K

Author: d | 2025-04-23

Then there are unique integers q q q and r r r with 0 Let m = 2 k 4 4 k 3 k 2 k m=2k^44k^3k^2k m = 2 k 4 4 k 3 k 2 k. n 4 = 8 m 1 n^4=8m1 n 4 = 8 that: n = 2 K/m - (r^2)/(4m^2) = (K/m)1 - (K/m)(m/K)(r^2)/(4m^2) [here I rewrote each term with (K/m) as a factor] [here I canceled an m] = (K/m)[1 - (r^2)/(4mK)] [here I multiplied the

$ sum_{k=1}^m k(k-1){m choose k} = m(m-1) 2^{m-2}$

Ëj…°%˘&³j†°%˘&³j‡°%Ë°%Ëjˆ�Rx :j‰�Rx :jŠ�R�Rj‹° R˜ :jŒ° R˜ :j�° R° RjŽi RQ :j�i RQ :j�i Ri Rj‘� Ri :j’� Ri :j“� R� Rj”&!R":j•&!R":j–&!R&!Rj—F"R.#:j˜F"R.#:j™F"RF"Rjšc#PK$8j›c#PK$8jœc#Pc#Pj�~$Pf%8jž~$Pf%8jŸ~$P~$Pj °y)j¡°y)j¢°°j£÷ÓpLj¤÷ÓpLj¥÷Ó÷Ój¦ðãi\j§ðãi\j¨ðãðãj©À°9)jªÀ°9)j«À°À°j¬·Ó0Lj­·Ó0Lj®·Ó·Ój¯°ã)\j°°ã)\j±°ã°ãj²ÔyMj³ÔyMj´ÔÔjµ÷÷pp j¶÷÷pp j·÷÷÷÷j¸ð i€ j¹ð i€ jºð ð j»ÀÔ9Mj¼ÀÔ9Mj½ÀÔÀÔj¾·÷0p j¿·÷0p jÀ·÷·÷jÁ° )€ j° )€ jð ° jÄm%yæ&jÅm%yæ&jÆm%m%jÇ÷�'p )jÈ÷�'p )jÉ÷�'÷�'jÊð )i+jËð )i+jÌð )ð )jÍÀm%9æ&jÎÀm%9æ&jÏÀm%Àm%jз�'0 )jÑ·�'0 )jÒ·�'·�'jÓ° ))+jÔ° ))+jÕ° )° )jÖ:0y³1j×:0y³1jØ:0:0jÙ÷]2pÖ3jÚ÷]2pÖ3jÛ÷]2÷]2jÜðm4iæ5jÝðm4iæ5jÞðm4ðm4jßÀ:09³1jàÀ:09³1jáÀ:0À:0jâ·]20Ö3jã·]20Ö3jä·]2·]2jå°m4)æ5jæ°m4)æ5jç°m4°m4jè}+M¢,¹Aj@é0*h.ÕAjêíõjëÇ€)ÇjìW€)Wjí�Ëx ³jî�Ëx ³jï�Ë�Ëjð Ëø ³jñ Ëø ³jò Ë Ëjó°%˘&³jô°%˘&³jõ°%Ë°%Ëjö�Rx :j÷�Rx :jø�R�Rjù° R˜ :jú° R˜ :jû° R° Rjüi RQ :jýi RQ :jþi Ri Rjÿ� Ri :j� Ri :j
� R� Rj&!R":j&!R":j&!R&!RjF"R.#:jF"R.#:jF"RF"Rjc#PK$8j c#PK$8jc#Pc#Pj ~$Pf%8j ~$Pf%8j~$P~$Pj#‰y$j#‰y$j##j#%€œ&j#%€œ&j#%#%j3'y¬(j3'y¬(j3'3'jÐ#Iy$jÐ#Iy$jÐ#Ð#jÇ#%@œ&jÇ#%@œ&j Ç#%Ç#%j À3'9¬(j À3'9¬(j À3'À3'j  `™Ùj! `™Ùj" ` `j#ƒ�üj$ƒ�üj%ƒƒj&“ ‰ j'“ ‰ j(“ “ j)à`YÙj*à`YÙj+à`à`j,׃Püj-׃Püj.׃׃j/Г I j0Г I j1Г Г j2 ô™mj3 ô™mj4 ô ôj5��j6��j7j8'‰ j9'‰ j:''j;àôYmj×P�j?×P�j@××jAÐ'I jBÐ'I jCÐ'Ð'jDÌ3‰E5jEÌ3‰E5jFÌ3Ì3jGï5€h7jHï5€h7jIï5ï5jJÿ7yx9jKÿ7yx9jLÿ7ÿ7jM €-™ù.jN €-™ù.jO €- €-jP£/� 1jQ£/� 1jR£/£/jS³1‰,3jT³1‰,3jU³1³1jV$]:�Õ;jW$\:�Õ;jX$\:$\:jYÔ]:MÕ;jZÔ\:MÕ;j[Ô\:Ô\:j\]:�Õ;j]\:�Õ;j^\:\:j_Ä]:=Õ;j`Ä\:=Õ;jaÄ\:Ä\:jb]:}Õ;jc\:}Õ;jd\:\:je´]:-Õ;jf´\:-Õ;jg´\:´\:jhd]:Ý Õ;jid\:Ý Õ;jjd\:d\:jk ]:� Õ;jl \:� Õ;jm \: \:jn}+M¢,¹Aj@å
a|Àï}Ñ 7 7
ÿ@(67Í8X@X>X€@ÿÿ
Unknownÿÿ
ÿÿ
ÿÿÿÿÿÿÿÿG �
‡* €ÿ
Times New RomanA �


€Symbolarial3.�
 ‡* €ÿ
Arial3 �
‡* €ÿ
Times;.�
 ‡* €ÿ
Helvetica7.�
ï { @ŸCalibri3�

@TungaA�

Cambria Math"
ˆ€Ðh
ã צã צ6Ó. c6Ó. c!€
¥Àxxƒí6í62ƒ€×ßÿÿ
ðÿ
$Pÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿa|À2! xx ÿÿAAþÿ

à…ŸòùOh«‘+'³Ù0

�˜¤°¼È
$
0
?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~€�‚ƒ„…†‡ˆ‰Š‹Œ�Ž��‘’“”•–—˜™š›œ�žŸ ¡¢£¤¥¦§¨©ª«¬­®¯°±²³´µ¶·¸¹º»¼½¾¿ÀÁÂÃÄÅÆÇÈÉÊËÌÍÎÏÐÑÒÓÔÕÖ×ØÙÚÛÜÝÞßàáâãäåæçèéêëìíîïðñòóôõö÷øùúûüýþÿ

































!
"
#
$
%
&
'
(
)
*
+
,
-
.
/
0
1
2
3
4
5
6
7
8
9
:
;

?
@
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
[
\
]
^
_
`
a
b
c
d
e
f
g
h
i
j
k
l
m
n
o
p
q
r
s
t
u
v
w
x
y
z
{
|
}
~

€
�
‚
ƒ
„
…
†
‡
ˆ
‰
Š
‹
Œ
�
Ž
�
�
‘
’
“
”
•
–
—
˜
™
š
›
œ
�
ž
Ÿ

¡
¢
£
¤
¥
¦
§
¨
©
ª
«
¬
­
®
¯
°
±
²
³
´
µ
¶
·
¸
¹
º
»
¼
½
¾
¿
À
Á
Â
Ã
Ä
Å
Æ
Ç
È
É
Ê
Ë
Ì
Í
Î
Ï
Ð
Ñ
Ò
Ó
Ô
Õ
Ö
×
Ø
Ù
Ú
Û
Ü
Ý
Þ
ß
à
á
â
ã
ä
å
æ
ç
è
é
ê
ë
ì
í
î
ï
ð
ñ
ò
ó
ô
õ
ö
÷
ø
ù
ú
û
ü
ý
þ
ÿ

þÿÿÿ   þÿÿÿ     !"#$%&'()*+,-./0123456789:;?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~€�‚ƒ„…†‡ˆ‰Š‹Œ�Ž��‘’“”•–—˜™š›œ�žŸ ¡¢£¤¥¦§¨©ª«¬­®¯°±²³´µ¶·¸¹º»¼½¾¿ÀÁÂÃÄÅÆÇÈÉÊËÌÍÎÏÐÑÒÓÔÕÖ×ØÙÚÛÜÝÞßàáâãäåæçèéêëìíîïðñòóôõö÷øùúûüýþÿ
        !"#$%&'()*+,-./0123456789:;?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~€�‚ƒ„…†‡ˆ‰Š‹Œ�Ž��‘’“”•–—˜™š›œ�žŸ ¡¢£¤¥¦§¨©ª«¬­®¯°±²³´µ¶·¸¹º»¼½¾¿ÀÁÂÃÄÅÆÇÈÉÊËÌÍÎÏÐÑÒÓÔÕÖ×ØÙÚÛÜÝÞßàáâãäåæçèéêëìíîïðñòóôõö÷øùúûüýþÿ
        þÿÿÿ"#$%&'(þÿÿÿ*+,-./0þÿÿÿýÿÿÿýÿÿÿýÿÿÿýÿÿÿýÿÿÿýÿÿÿýÿÿÿýÿÿÿýÿÿÿ;þÿÿÿþÿÿÿþÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿRoot Entry
ÿÿÿÿÿÿÿÿ ÀFÐñÉLçûÉ
=€Data
ÿÿÿÿÿÿÿÿÿÿÿÿ1Table

ÿÿÿÿ WordDocument
ÿÿÿÿ6SummaryInformation(
ÿÿÿÿÿÿÿÿÿÿÿÿ!DocumentSummaryInformation8
ÿÿÿÿÿÿÿÿ)
CompObjÿÿÿÿÿÿÿÿÿÿÿÿyÿÿÿÿÿÿÿÿÿÿÿÿ
þÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ
þÿÿÿÿÿ ÀF'Microsoft Office Word 97-2003 DocumentMSWordDocWord.Document.8ô9²q. Then there are unique integers q q q and r r r with 0 Let m = 2 k 4 4 k 3 k 2 k m=2k^44k^3k^2k m = 2 k 4 4 k 3 k 2 k. n 4 = 8 m 1 n^4=8m1 n 4 = 8 that: n = 2 K/m - (r^2)/(4m^2) = (K/m)1 - (K/m)(m/K)(r^2)/(4m^2) [here I rewrote each term with (K/m) as a factor] [here I canceled an m] = (K/m)[1 - (r^2)/(4mK)] [here I multiplied the 2 r k m cos r k m t!, ⇒ x′(0) = c 2 r k m = v0 ⇒ c2 = v0 r m k. So, x(t) = v0 r m k sin r k m t!. On the other hand, if we examine the other differential equation in-volving the delta function, and take The projection and commutativity properties are elementary corollaries of the definition: MM k /r k = M k /r k M ; P 2 = lim M 2k /r 2k = P. The third fact is also elementary: M(Pu) = M lim M k /r k 2 r 1. k 3 r 1. k 4 r 2. k 5 r 2. k 6 r 3. k 7 r 3. k 8 r 4. k 9 r 5. k 10 r 5. k 11 r ) ) ) ) ) , ` _ ) ) ) G E = ) - ` - M ) ) k @ =) a ` ) ) i B : g ; i o A C k ) l = l m w q S / 5 P 3 R w 8 0 R . q 0 7 Q . 7 8 P . 5 3 P R . 5 P 2 d 3 /) ) { B f G ~ F ) A C F ) N _ ) B : ' ) B : ] x) ) ) ) ) a ` _ ) ) ) G E = ) M - ` - M ) ) k @ =) a ` ) i B : g ; i o A C k ) l = l m 3 R 1 5 3 P R . 8 3 8 3 / %PDF-1.4 % 5 0 obj stream xœ [ r ‘_ 7ƒU 2 K^R 8 $eW k ` a%Q f)i m` ˜ ) %[Ukp0žž { Ÿ I ˆ.nW A particle of mass m is moving in a horizontal circle of radius r, under a centripetal force equal to -(k / r^2) where k is constant. The total energy of the particle is (a) -(k)/(r) (b) -(k)/(2 r) (c) (k)/(2 r) (d) (2 k)/(r) 06 2:forl = 7 to 0 do3:for m = 3 to 0 do4: if the l-th bit of R 16 + m ==1 then5: R 0 ← R 0 + R 25 6:for k = 0 to 3 do7: R 8 + m + k ← R 8 + m + k ⨁ R 20 + k 8:end for9:else10:for k = 0 to 3 do11: R 24 ← R 24 ⨁ R 20 + k 12:end for13:end if14:end for15: ( R 15 , … , R 8 ) ← ( R 15 , … , R 8 ) ≪ 1 16:end for…In addition, the performance is improved further by applying Liu et al’s multiplication method to the proposed method. Seo and Kim proposed to shift 40-bit multiplicand (A) for a 64-bit multiplier, rather than shifting 64-bit accumulator. This approach reduces 29 shift instructions per 32-bit multiplication operation. However, the multiplication method suggested by Seo and Kim does not show a big difference in performance compared to the method suggested by Liu et al. According to Seo and Kim, the number of shift instructions can be reduced compare to Liu et al’s method. However, the XOR instruction goes one more operation per bit, which leads to addition of 32 more XOR instructions when calculation 40-bit multiplicand (A). Since five registers are needed to store the 40-bit multiplicand (A), one more register is required compared to the Liu et al’s technique. Liu et al.’s approach is used for multiplication and one register is saved. These spare registers are used in the Karatsuba algorithm to improve the performance. For the optimal number of register utilization, the version without using spare registers is also investigated. Currently, RISC-V introduces new architecture for future microcontrollers. The optimal register utilization can contribute to the optimal architecture design. 4.2. Karatsuba Algorithm for GHASHKaratsuba algorithm is well known asymptotically fast multiplication method and the proposed implementation also utilizes the Karatsuba algorithm for high performance.First, the multiplication is performed with lower 32-bit of 64-bit operands ( A [ 3 ∼ 0 ] , B [ 3 ∼ 0 ] )