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Comments to date: 7. Page 1 of 1. Average Rating:
keesje76 8:39am on Friday, August 20th, 2010 
Still abit laggy due to WM, but it is the best of all. Hope the market place will have more app choices. Huge screen for web browsing, wifi router. This is the most power packed smartphone you can find currently. Big screen, good multitouch, extremely responsive Poor battery life, no more wm 7
ahpin 3:44am on Friday, July 9th, 2010 
There is no practical way to change the language. Also advertisement did not say phone OS was German. Attractive Design Difficult Navigation Probably a great phone, but w/o 3G signal in U.S. phone can not show off all its stuff. Sense UI is best I have seen, WinMo 6.5 is OK.
razoruser 5:19am on Friday, June 25th, 2010 
So far, T-mobile has replaced this phone 3 times under warranty. Its defects are so numerous as to defy listing in under 5000 characters. This. Overall rating: 4 out of 5 Screen, processor, apps, camera Interface, limited apps, battery life
ctopher916 9:53am on Tuesday, June 8th, 2010 
The HTC HD2 is a tough phone to review. On the one hand, the hardware is top of the line. Reviewing smart phones can be very tricky these days since there are typically two or three main aspects of the product that can make or break the dev...
The_Game 12:45pm on Friday, April 9th, 2010 
THIS PHONE IS GREAT WHEN TRAVELING... Attractive Design","Brilliant Display","Good Interface w/email","QWERTY Keypad Crashes or Freezes".
arnechaa 11:29am on Wednesday, March 31st, 2010 
Good for a Windows phone but... The HTC Touch HD2 is a great phone the problem is with Windows mobile. HTC HD2 The phone is everything I expected, excellant display and good operating software. No regrets buying this inovative phone.
chdoula 8:13pm on Thursday, March 11th, 2010 
This charger was as described on the Overstock site and was received promptly. If you lookin for cheap caese then this is a good purchase if it under 10 dollar, itz better cases out there. itz slippery, hard and fragile.

Comments posted on www.ps2netdrivers.net are solely the views and opinions of the people posting them and do not necessarily reflect the views or opinions of us.

 

Documents

doc0

tuncUons

corresponding 10

circuli elements.

Our aim Is 10 support. struclured hierarchical design
method by allowing the.resIgner 10 determine (by reasonlngl whether. gIven comblnellon 01 precisely specified componenls has lhe required behaviour.
includes. IlFP mllnual. II also Includes 8 Siudy 01 the application
of IlFP and a discussion 01 our requirements for a VLSI design language. Ch<1pler 2 Is an In1roducllon 10 11'18 basic lorm 01 IJ,FP. The illS! pari 01
the chapler gives. relalively Inlorma' description of the

language.

second part gl. es Ihe lormlll semenllcs. We show how algebraic Idenlilies In ItFP ere pro.ed and we demonslrete some trensformellon!
prognms. In chepler 3. we g,.. some erampte! of the use 01 IIFP. We mOtlvale lhe addition 01 some new combIning forms Which are apprczr1lte 10 SySlOliC arrays. Chapler. Conlalns further e.amples 01 the use al lhe language. one 0' "hlch Is SAOcell. Ihe ba!lc cell In a chip WhIch we ha.e designed. In chapler S. we present aur main e.ample. We gl.e a slep by step derl.atlon of a systolIc correlator cirCUit. We Introduce 1I0me lechnlques for the analysis of clrculls In "hleh SO. af the processOrs are acllve al
any time. Chaplet e describes hO" IlFP may be ",un" 10 gl.e a slmuta\or. We also d~crlbe a program "hlch prOduces Ihe 1I00r-pfan 01 a gIven ~FP presslon. The nrSl pari 01 chapler 7 Is review 01 design lool~. ranging '(am "automated graph paper' systems 10 Silicon compilers. In the secDnd part of thl chapler. "e lurn our IUenllon to design language! the Important properties 01 a high ltNel VlSI We can~il!er and we

design language

comparl our apprOach 10 thai of olhers "ho have applied lormal tectmlques 10 the problems 01 VLSI design. In chapter. "e presenl our cancluslons and aur plans lor future work.

Chapter 2: Simple IIFP

Intfoducllon
.FP Is 8 ",arlant 01 FP. lhe Functional Progremmlng language In IBackus 16]. We..111 nol Introduced
give. dOlalled description 01 FP Qui will Include
only SUCh dotalls 8!io lire nocesury 10 'he undorSlandlng of Jl,rp. In the IIrsl
perl 01 this chapter.8 will glvo a roUItI'f'Oly Informar descrlplion 01
piecing the empha,,!o on the geomclr1c Interprotatlon
01 ,he chapler. e..111 give the formal

the ser.ond

We will

provt!

semanTIcs of J,lFP.
thai some FP aJlloms hold In pFP end..e will demonstrate

transformation,

of IlFP programs
A brief Inlroducllon 10 FP " program In r-p I, ,Imply an elfprtlss10n representing 8 luncUon thai maps

ob/eCI, In'o ObJeCI' IBackus 78: program rapresenllng I

Williams

61l. For 8ltamp1c. I' en
luncllon..hlch maps a pair of numbers onto their undellned Ul. We
sum. The otljects on.. hich our programs operate can tie
Iltoma or aoquonces of oDJects. Note Ihal Ihls Is e recursive defInition
shan lalte lhe &el of aloms 10 tie Ihe Inlegers. Ith e "don'l cere" valuo. '7' Some po,,!Dle obJecls ere.1.42
<) Uha empty sequence)

q.o(l.t1.1~)).

denolos a sequence. We will represenl 'don't caro although we ShOuld. strictly.

sequences by lor every

'7' also.

heve a dllJerenl symbol

po"IDle shaDA
Nc.-!..e need a SCI 01 primitIve functions 10 operate on our ObjeclS divide Into lhree main calegoues {' denOles function aDplication)
Functions lor manipulallng sequences ego selecior funclions 1.2., Ippend 10 lhe lell. apndl append to tho right e g. 1:<x1.11'2.xn) eg.

= 11'1 II n;'1.

other..Jse.

epndl:<3.(4.5.6n

= (3.4.5.6).
apndr: t tl.2).3.: lip = la1. a2.

<1.2.3~.

matrix ,ransposillon. rip length of oach of Ihe

an) where n Is Ihe

sUbs~uencos
01 Ihe matrix 10 be Irensposed
ArllnmBllc funcllons e.g. -t. ":<1.2). 3 -t:<"'2>

"':<3.4.5'

3) Predicollies
(we denole true by 1. 'oIIlse by 0)

e.g. greeler than

(. 1>

1 (lrue)

Cts map functions Inlo
Flne"y. we need e sel 01 combIning lorms CCFs>
'uncllons and ao allow ua 10 build up Ihe lunCIJons (or progr.ms) Iholll we reqUire. Wo use Ihe lollowlng CFs Composllion ConSlfuCllon Apply Condlllonel Const.nl Insert lell Insert rlghl

tlf:x

"

'nh. tf I :_. 12:x.

cf:Jll.l:x2. I:.n> If.--=<.1.x2.n>. 1 otherwise
1:11 If p: l. g:x II P;XII:O .1 otherwise

(p_f;g):11

., : y C/l.n: t xl) (jAO:<.h

r If y xl

1.1 olherwlse

(/,-D:<xl,. xn)

I: ((/I 0: <x1.xn-1).xnJ
I: <.l.lfllO: <x2.lIn))

.1. (fllO:<x1.n.

use lhe

CFs and

prlmillye lunctlons to write

now lunctlons or

programs. To edd
1 to each 01 the elemenls of a lequence. we lIll'lIply
wrne--cra T. ----wnere----.r Is lfiT- aadOOno runclfon~ oIIcra-a~1 t~e'emeiliS of 011 sequence. wo wrlle /1,. Cor /I"'. A typical FP program I' thai com pules Ihe lenglh 01 a sequence lenglh

ot numbers and

Before considering tho olher CFs. lei us describe a half-adder. ha. whIch (repealedly) lakes two btls and produces a sum bit and a carry bll. The sum bll Is Iho exclusIve or bor) of Ihe two Input b/ls. while Ihe carry bl! Is the and of Ihe two blls. Thus. our halt-adder Is simply ha =- bcor. andl whore
lfor '"' end lor. not. and!. This dellnlllon uses live 'oalcs' lind can be reprosenled as lollows

l~1tRY

FlO 2.2
A half-adder wUh S oates"
By applying Ihe aloebralc laws of In Ihlll chapler),
In a vory simple way (as shown Iller
we can Iransform our dellnlllon Inlo 21. 21. lor, andJ.
ha. land 11. nol which has only lour "gales',

S'"

'-"

CJ\klr

'<:Nor

011 I(

FlO 2.3
A halHllddcr with 4 oates,
NO.le Ihal 'I)Qlh definitions o' Ihe half-adder have the sarno semanlles, but dillarent layouts,
We now Introduce lour more CFt., ,It.nd ,.,. can be used 10 'spre.d InpUIS.Iong
or Idenllc.1 cells. 8S shown In FlO 2.

FIG 2.-4

1"":
(11:1. 11:2_ x3. 1(-4. 11:5)

CII:l.

.2. 11:3. 11:4. 11:5>
This form Is commonly used In clrcullS Combined wUh
concept at slale are allen used
11 llI'tll.1I0w us to descrIbe the tlnear syslollc IIrrays which

',n- a1on.l-processlng

Th. Inlerprot.,lon of Ihe condition.' CF Is shown In nG

FIG 2,5

Conditional
The swilCh chooses between Inpuls a 1Jnd c,.ccordlng 10 lhe. lue 01 b. So. lunCllon whIch gt.1:2. (repealedly) selecls. Ihe I.rget 01 IwO nurnbers Is
fln.lly. the const.nt luncllon. f. Is Jusl
source 01 rs. representod by

+---ill.

In ttl. 101l0winO secllon, we Inlrg~uce ffew CF. -. which mllows UI 10 deal _I.ttt Ihe concepl or s'.le.
oeallng wtlh 5"1.The basic 'unctions en., combining lorm, Introchlee" In the previOus sectlCln can onty be use" 10 repre,ent "comb'nelorlaI
Juncllon,. There I, no way
Of represenllng the concepl 01 slate..hlch I. central to.,lgUal.,eslgn. Since mOst "'glla' clrcu"s. 'rom shill reglslers to microprocessors, have some
"memory,.0 aro 'arced to add another combining form. 1/. I/. lakcs a r 'unCllon and pro"uccs a lunctlon-.hlch has Inl43.na, 5tale. FIG 26 sho., the geometric tnierprelllllion of 1/.

FIG 2.6

----or theru,'-CiIon
We use a I.lch" 10 hoi., the slale. Thus, the current state Is supplied to the lunctlon as II' sec on., Inpul and Ihe second outpui
relre,hes the "ale. The Initial 'Ielue In the lelch Is e!Sumod 10 be '1', Ihe "don't care" SIale, So. If

A, parallel Inl serial OUI shill reg Isler. with serial dala
The Input I~ In Ihe torm (shilllload. se,lal elala. parallel inpulS>. "

sh"'ng

Is enabled. the circuit beheves IIkp. e 5150 51'11" register. O1he'wlse. the perallel In pUiS overw,lle Ihe sla'8. So. the ~FP elescrlplion Is PISO? =

- 111-2. 2-11: 3-11

CLEA,A. we gel

~1l-2 -1

CPI$01
epnd, - 111-2. 2-11: 3-1]
At In e.en more abslract level. lei us consider very simple

mUIU~)ICl(er

dGmulllpleJlet clrculls. A MUX has as In puiS a swltch" value and a pair 01
dal' In puis ~s'*ltch.(dl.d2. " lhe ,,,,,Uch Is high. the MUlf
dall Inpul through. Otherwise. II passes Iho second daIS MUX

passes the lirs.

Input. Thus.

1-2: 2-2.

dala outputs

Ihe lirSI

A demulilplOllOt has a '.. iICh. "ngle da'" Input and two

sW'lch

III mpUI

through

OUIPUI

'cMnncl" and SCIS

olher OUlpUI to zero

II the

s.. itch

Is lOW.

passes the InpUI Ihrouph on lhe second output channel and sets lhe first

10 zero

_ _ _ _ m

9EMUll'

We..oul(J

12. II: lU. 21.

In FIG is the idenlily
like 10 check thai lhe circuit shown

on the dill Input 2.

I DE~

FlO 3.2

MUX (I. DEMUXI

totux

11. DEMUXI 12; 22) [1 11. 12.0); n.21l
A2,A6.A7 AUl Slalelcss).A6.A;

111 -+ 12.01

ro.21J: 2(1 -+ 12.al: lo.211

-+ 2. 0).

This means Ihal our MUX and DEMUX elreulls "maleh"

Olhcr correelly.

AnOlher familiar DlCemple 's ltle buHdlng Dl a lull-adder IrDm hall-adders.

~~hQ_c;"

FIG 3.3
A full-adder made IrDm 2 hall-adders
A half-adder which produces a <carry-bll. sum-bit. pair ha
land. xorl In FlO 3.3 is II. 1-21. 2-21 -

g over separ8tely 8nd compoSlno them properly. Anolher law about I_g.!Il1ows us 10 mo.e "detachable' CirculI seomenlS Irom Ihe I part 10 the g parI. or vIce versa.

1------.1 V

FlO 3.:24
f[l.y2.31"n-<lu1.2.3JO
WIl.u2.3U.-(g(l,2.y31l 88
transformation may allow us to simplify the clrculls on either sIde
01 the -. Our rlnal CF. \\. 'orms I grid 0' cel/s. as shown In FIG 3.25.

I.I: ~

lor Inputs of the form
8.b.c).<d.e>.<g.hn,
outputs are 01 the form UI.j>.<k.l>.<m.n.o

FlO 3.25

As with \. we.111 add \\ IIrst to FP. and then to IlFP, We ha.-e already

denned 1)

In FP. and
Is defined In FP exaclly a, II Is In J,LFP. Then.

\\I;lP.Q'H

alJdl:Ux.y.z.
\\1: texl,.xn. ~ l.ym>. <I: 1. ,zmH
Iconca'-1.concal-2.31 (\\1 t \\f)-ll.ll' J.t11-2.lIlJ.II]- 3J:
xl. ,In>. <yl.ym>. (ll,.zm

If m )

(1.2.concal-3J (\\f-\\O -(Imsl.[Iastll-l.2.3):Hxl,.xn> (y 1.ym). <zl. zm)~
The second definition use!> a recursive "divide and conquer technique. as lIIuSlrated In FlO

<vi>

)"

'"~

~YM !.

\\1 In terms 01 t

FIG 3.26

\\1 In terms 01.

The lemantle equallon defining \\ In p.FP Is

HInt). zIp

(nip - \\(zIP Hil)

azlp lip

<IV
This has eX8ctly Ihe 58me 'orm

the equation lor."

As usual. we use
zip 10 ensure thai each subprogram gets Input 01 the correcl type. lhe algetualc law B7. Gelrev Our final 'or \\. Is analagous to 84 lor

aorev releles

= \\(orev

\ and \\.

aigebrlllic

Illustrated

----MHtt-:-rtw~'fl
-1.211) -11.121.13 11-\\ 1 -
'~.g1-ll.12.3111_-lL[2J.[3UBB

FlO 3.27

By ellmlnlliing Ihe IwO reverses. we have Inlroduced bldlreclional dala HOw.
il Is II question 0' whelher we dlvlde- the cirCuit up horllontally or.erllcally.
In Ihe systolic correlator cirCuli 01 chapler 5. we lind that consider 1he IWO horizontal slices Of 1he us 10 In'roduce I'llie dimensional dala

clrcul

110'111

88slest 10

separately at first This allows In II re'81lvely p81n1ess way. the
Howevpr. In Ihe second stage 01 1he design. II Is userul

10 consider

CirCUli as a row 01 ldenlrcal cells. 1hls freedom 10 decompose the

clrcUIl

numDer 01 d.Uerenl

useful lealure 01 the

no. combining
forms. In Ihe IOllowing \'1110 chap,ers.
we give some ellamples of Ihe
01 simple jJ.FP lind 01 U'le new com Dining torms.
Ch.pter 4: C.se Studies: Some Smell EK8mpres The Telty Circuli The laity funclion from 0)

All or these cirCuits tl8". Itlfo-bit 11 ate. one all lOr eacll tnveTf1tr-;- AM--of 1M outpulS.r. Of the form C1 21. ttlal Is. Ihe Inye,.e 01 Ihe lelthand ala Ie
bit rwe tlor. In'orm.llon on lhe Input oat8 capacltence 01 Iny.rlers,) The
four clrculls diller, however. In their

tie.' sf.'.

functions

When RF Is high.

"'. can see Ihat the

1'1."- sti'r.

Iun-clldn Is--t-l~or--l2t ,Inee lhe tlQflltllrn:t
Inye"e, I, refreshed by ttle lefthand one. When RA Is high, the

n."

"'ar.

function I' 11-2.

OTln) ,ince
dala bit ), ,.ed" Inlo Ihe Input ot lhe
tlghlhand In-rerter, Simlla"y. when AA Is

II." lira'. I,

(1-2. OAln}.
FlnaUy. when _2 Is high. Ihe n6rl st" . Ino'-2-2. 2-2]. FOIlO"lng Ihl:::
analysl,. wrl'. SADcell as

pllnol-1-2. nol-l-2I

j1f2-lnot-2-2.2-21;AF -11 -2.nol-1-21:AA-11-2.DTlnJ:(1 -2.DAlnJl
Oltltously. we wOuld like to check thai 0"'
two phase clOCk version Is an
acceptable Imp1emenl8110n of the original SADcst!. One clock cycle consists 01
JIl' going high lwllh "2 10wL lolloweel
by "2 going high (wilh "1

Since we know thel ,I'

and "2 always havl!! this pallern.
our squauO" lor SAQce" aCcordingly. We make
we essume Ihat Bach tim"

"2

we can mOdify
our 'master" clock and

goes high Immediately

allerwards. Thus. W8 make "2 "'nYlslble- and since we had the condllional

1"~-lnOI22.22n.

we compose Inol 2. 2) with

of the remaining

Ihue pOSSibilities. (The 2.hlCh has been dropped lust slands lor the siale J

SADcell the"

becom8~

:dlnol-1-2. nol-I-2).

nol-EN-lnOI- 2.2) (1 -2.nOI-1 -2]:lInd -{EN.SAI-Ina! 2.21 {l -2.0ftn}; r'lot-SA-Inot- 2,21- [l - 2.DAlnJ.2J
pllr'lot-1-2. nOI-1-21. not-EN-O-2.nol-1-2LSA-tnOI-OTln.OTln): nol-SA-(nOI- ORln.DRlnJ:2) which reduces to p1l2. 21. EN _ eSA _ OTln. ORin): 21 Implemenlalion 01 'he pass
This 15 our original SADcell and we have shown lhal our using a Iwo phase clock 15 a sallslaclory one. The
'rar'lsisior as a sImple ,wllch 15. of course. lar Irom Ihe HOJrlllwer. 'he mell'1od whiCh we hawe lusl used could be circuit!: WhiCh are SImple combinallons 01 Inwerters and

because

,,,.e shape of the slate of a Il cannOI always be deCluCe<1

from the conlelll.

The abstract synlall Iree constructed In thiS.ay Is then translormed. usmg
Ihl'! inlrHmallon c::oniarneCl In the "'sl Input 10 the

program. to eliminate

all us and
program containing as or Is cannot be

soIICll1len

Into liS linal lorm unlll we kno. the shape of liS inputs 11-1. ,- 2. '31 lor

3-el~menl

Thus. al represents Inpuls which are The

Inputs.

and II 1. f 21 lor
pairs. ILl represents '"'''"'11.21.3J rOr 3-elemenl InpulS. and f lor palrs
funclion Irans lake$ a Itee and an Inpul and replaCe! all as ano Is by thft appropriate some examples combinaiions 01 conslruclion and composition to shOw how Irans works. T represents lrans Let us take

TMNSfol:tMED

ton!:ltntd

Cllpha

- - - - - - - - 7 ' - t<il) tl, - _':n>

Tlf,i,) TII,it).

-=---=

Til, to)

tOl"lposi.

c.omposlt

T (f, / appl~. (TI~,,),,))

Tlj, ,)

T(g, i)

c.onsfrlK!

lon)f.

""t

1 If. i)

ellecl of

/"-.

FIG 6.::!

.!!.!!!.!
on abslract SyRia. tree! lor G. lind 11

',anslorm.Uon

relleets

0:1=11-1.1-2,.f-nl

n-8I8m'nI Inpuis. We must call Irans recursively on each

Ihe elemer,iS

01 till, new construcllon as I might. lise If. conleln as and Is. Tran510rmlng

co,.,.,ouU.g) ",lIh

Input I I,. slightly

more complleeled.

The resu/l Is a
composlllon whon right branch Is tren, Ig.l>. The left branch must be Irans of I wllh some input. To calevl.1e Ihal Inpul. we compule the QulPUI which liars.
Ita.!) Dives lor input I. __u.s.ing IheJuncUon

earlier work.

SliCk diagrams. which were IIrSI proposed In {Williams 771 and which
described In (Mead. Conway e01. can be used to express lhe topOlogy or
connecllvi1y 01 a clreui. STICKS (Williams 78) Is a graphical compiler for
high level VlSI deSign. REST IMo!:.relier ellis the protOTypical leal cell design system. using Ihe connecllvHy approach. The Input 10 REST Is Just a slmp'e colour sketCh or rough shck diagram 01 the leal cell. Tha sketChing Is done on a graphics 'ermln,L The REST prOCess then produces a compacted slicks representallon. which has a unique physical representation The leat cells
Ihus designed are then composed lexlually. USing the SPAM language (Segal
90). The compOsition cells whiCh are used to combine olher cells conlaln
behavioural as _ell as structural dala and a mUlll-valued junction a' simuialor

-------.s-~- clieelt

IAa 'yoc"?"?1 correctness of the Chip.
The Daedalus/OPt design envlronmenl ISh robe 831 combines bOlh text and graphics. The OPt layout language IBalatl. Mayfe. Shrobe. Sussman. Weise
pSt.melrlc _ ceiL de1inillQ.!ls
_~nd _$ym!l0lic~ de~Ctip1iOns
manipulales a hierarchical Object-oriented dlla base
The whole syslem is Daedalus
embedded in LISP. allowing the user to define his own lunClions
1$ a graphiCS editor whIch allows one to edil graphiCS objecl,. but which OUlputs DPl code. The whole system was used In the desIgn 01 the Scheme Chip al M.I.T. (Steele. Sussman 901.
SliCkS & Stones (Cardeili. PIOlkin 6ll 15 language which Is designed

ell~ress

Ihe hierarchical ,'rUCture and lopologlcal properties 01 VlSI circuits.

slick diagrams In

Thl language can be used to specify and communicate
lerual torm. A more concrele form, wl'1ICh Includes the necessary geome1r1c
de,115. has been Imptemenled [Cardelll 811. The

a~5lracl

picture"
anc (he algebraic operalors. on It 1'18\18 been embedded In a general purpose
_p,llC.,.ve programming langu~~n.g " powerlul Chlp assembly 1001. (We
wUI rfllum to the work of Cardelli In the nelll seclion l
POJh {Whllney. Mead 831 Is a symbolic 'slich-lIke" represenlallon lor circuit
fe.et designs and a sel of algorithms which operale Tht Pooh syslem mainlains conneclivity. circuli
this representall0n. and port

scheme lie

placement Inlormalion. II defines
an automalically Ctlecked

represenlalion

lor cirCUli level designs and II allows mask geomelry 10

be automatically

ge,eraled. The user Inlerlace may be either graphlcel or IImguage based.

In~lvldual

slruclures.re "forced" to Obey Ihe layoul rules

by construction.

Sone analysis Is needed 10 ensure Ihe correct spacing and angles belween

.d~cenl

structures. The Pooh languege system has been

embedded In the

programming language Mainsail <TMl. The syslem has been used as Ihe ba,e-Ievel representallon S111er. Clow. Whllney 821. In lhe SiC lOps silicon Compiler proJecl IHedges.
The ASTRA CAD syslem [Reven. Ivey B3J being developed by Brl1lsh elms to support Ihe design 01 hfgh-comple_ity Integrated

~elecom

circuitS.

It eim

encourages and supports a structured
approaCh to design. wilh the
of managing In'erconneclion elltC1cnlly from the eerhesl stages 01 the deSIgn The system uses floor-plans 10 dehne Ihe topOlogy 01 Ihe layout and Ihe hierarchical slruCture 01 Ihe design. A lorm of symboliC layoul Is used 10
design low level cells and geomelrlc layoul Is aUlometically generaled. The floor-plans have Ihe rote 01 composl1ion-cells when Ihe leal-cells are finally pllced. When the Chip Is beIng assembled. leal-cells are The eutomallcally e.perlmenlsl
SlrelChed 10 ensure Ihal pilCh

matChing Is malnlalned.

syslem Is currenlly being epplied 10 a realistic design example.

Berkeley

Building-Block

syslem

(Chen.
hierarchical aulQmated layout system for Ie design.hlch Is designed to Inlerface to other design aids through a general purpose dala ba$e. The
SIUcon Assembler lUBAICK lSchoelikopl 83J has been de.eloped 81 Grenoble
as part of a silicon complier project. II allows the hierarchical design of lunctlonal celts using basic Interconneetlon slruClures. The

CHIPMASON

automatic layout II troe-like
program IWu. Parker. Conner 831 opera1es recursively on whIch represents Ihe deSign floor-plan inCluding

slruclure

lmplemenlallon al1ernallves. The OUlput is mask dale.
Two systems which combine te.lual and graphical represeniaHons wil" Ihe use of both connecllvily and geomelry ,re Eleclrlc (Aubin 831 and MULGA (Wesle. Ackland e 1; ACkland. Weste 83]. MULGA Is 8 symbolic deSign 5yslem which uses Ihe notion of. vlnual or 'opologlcal grid on which symbOlic
"cirCuit Component, are praced. The grid dehne, a rei' live layOut tOPOlogy wilhout spectlying the aClual dlslances belween components Aller 50me
cheCking lor circuli Inconslslencles. the cell 15 compacled by "moving" Ihe virtual grid tlne5. A detailed symbolic 1I00r plan I" used to give a StruclUral deflnltlon of thB ChIp. This floor-plan. togelher wllh a number 01 struclural

combining 101m. we can place lhose Signals

Ja~~_h

'0 remember in a

re:quired

size. The

signals

are Ihen available to the function on the nelll clock cyCle way ot intrOdUCing We

~l(lte

This Is a restricled only lor one
since the signalS are remembered

Clock Cycle

Me nOl prOVIded With varJables Inlo whiCh we can place
value5 lor sale-keeping. until lhey 3re needed laler In ttle compulallon. SUCh a language mIght be able 10 ellpless some compulaUons more nSlurally and conCIsely. bur the Increa.o;e In expressIve power would be far oulwelghed
by the Increase in complekily 01 the formal semanlics. wl1h lis consequent decrease In our abltlly 10 reason about Ihe language. In #lFP. a circuli wllh stale Is expres$ed liS a linlle Slale machIne. with n." outplJ' and ned sfate tunc.liom: Thu$_ we c.an express any lunClion which Is suitable lor
implemenlalion 01 the orioinal
on SilIcon. While retainIng most 01 Ihe algebraic prOperties apphcallve I.]nouaoe. FP
Olter workers have (lor similar reasons) placed restrictions on theIr nOllons of Siale. IBablker, Fleming, Milne 831. In t"elr Il'nguage lTS, ('layout and Timing lor Siructures'>' use I' non-procedurl" style of descrlpllon. with
backwara-Iooklng synchronous Ireatmenl 01 time The behaviOural de scrip lion of circuit Is given In a lunctlonal language. wl1" a signal being viewed as a malhemallcal lunCllon taking Umes to values. They Introduce state by u.irIg lac.t a

lu-nc~-Om ~gnall.

10 slgnala. LUllxl la- -. signal which, al
Jln, Im. "nl, has Ihe value Ihal 11 had al Ihe previous Inslant The lormal senanlics 01 L TS has nOI yet been published but we expect thai {as in IJ.FPl the combinaUon 01 an appUcative language and a restricted notion 01 slale

wUJ---g.we -a._ simple

p1egilol seman1jc-'

GOlaon IOoraon

similarly

restricled

nOli on
chooses 10 work wilh sequential behaviours, ralher Ihan wllh machines. The dOlllalna ComlX; '1'] end SeQIX: '\'1 reresenl comblnalorial and seQvenlial
behaviour. reapeC!lvely. The comblnatorlal behaviour 01 a device. whose sel of Inpul line, Is X and who,e sel 01 oUlput lines Is Y, Is lunctlon Irom 81\j(Xl 10 6IgIY). Membera 01 SlglXI Cand SlgIY) are called signalS.

Call/X; '1']

= SlglX) -
SlgIY) Is Ihe domain 01 combinatorial behaviours Irom
X n Y. Seqlx; '1'1. tho domain 01 sequential behavlouriS hom X tOY, Is Ihe

solvllon

the domain equallon

realised by a large number or

proce~ses.

be The

operaUng

Challenge 01 VLSI Is 10 lind ways 01 harnessing this C0'lurrency. wl1hout being overwhelmed by il. Merely 10 Implement bigger lag leT- sequential machines is 10 lail 10 take lull advanlage 01 Ihe pOlenllal. New arctlilectures must be Ol5covlned, opllmal

~y Re~earch

concurrer111y.
InlO systoliC archlleclures. which are designed to make and parallelism 10 give high performance.

ptpohnlng

,advanced Work
Ihls Area IKllog 79' BrAn(

Kung 82;.

FosiOr. Kung 79: Lelserson 81; Kr8mer, van Leeuwen 83:

Evans. McWhIrter,

W()od. McCanny. McCabe 83: McCabe. McCabe. Arambepola. Robinson. Corry
M()tlvaleo by the Imporlance of concurrency.
mosl researchers choose to
charaClerlze the circulls lhemselves as collecrlons 01 sequential processes.-hlCh communlcale with Ihrough named POtls. network Of

processe~ Thl~

named lines

channels

Is an unconSlrained lorm 01 communication. ,he lorm an arbitrarily complex graph. with
sJlaghetlHlkO communicaiion lines. This causes prOblems. First. the layoul, in two dimensions. Of an arbiuary graph Is a hopelessly difficult task. Second, Ihls approaCh 'onores an elltremely Imporlanl properly of VLSt. the lac! lhat.-hlle local communicaiion is Cheap. glObal communicallon Is e. pen~lve. 10 A

~ys'em

enormou~ly
In whiCh communicallon Is through and local communlcallons.

POrl~ ~

(l"tingtllSh

global

Third,

unconstrained when

C(lmmumcalion oreally complicales Ihe prools whiCh must be done reil50njng aboul Circuits which are composed 01 subclrcuil$

rea~onlng

aboul circuit behaviours. the most common form 01 Identity
whiCh we need 10 prove Is that "the composillon of A and B has the same behaviour as C'.
In a design language. the way In WhiCh !he composliion defined (which determines the allowed forms oj

subbloCks

communication)
a great bearing on Ihe ease with which proots aboul
the language can be performed. One 01 our require men IS lor IlFP wes !hat II shoulCl allow the designer 10 reason about hiS circuits. using simple
algebraic laws layout. ddhcuH
Another requiremenl was thai II communicaUon resirici makes

~hould

algorithm~

Invesligale
non-systolIC jrilplementa1l0ns 01 algorithms InlO systolic ones.

References

IAbrial 82J J.-R. Abrlal: "SpecIfication and
Construction 01 Machines 1982.

note~.

semanlics workshop. WoUson College. OxloreJ, Sept.
lAckland. Wesle B3J B. Ackland, N Virtual Grid Symbolic Norlh-Hotland. 1983 Layout". In
Weste: "An Aulomatlc Assembly Tool lor "VLSI 83". F. Anceau, E J. A8s (eds).

{Ayre~

SpeclfiC<lllon 19 79

L<lngu<lge".

Design
Automation Conlerence (IE EE).

IEiabiker.

Flemmg.

Babiker.

"A

'Tulorlal lor lTS

St<lndard Telecommunlcallon laboratOries UmlleeJ. Inlernfll 224833 Commercial In Confidence. 1983

Technical Memorandum Nn

IBackus

Backu!.

"C<ln

Programming

liberated
Neumann Slyle?", Communications or the A.C M. VOl 21. No 8. pp 613-641. Aug. 1978.
IB<lckuS 811 J. Backus. "'The Algebra 01 Funcllonal Programs: Function Leve-I Reasoning. linear

Equallon~.

f_tended
Delinitlons". f>roc

Symposium June 1981

on FunctIOnal Languages and Computer Archl'ecture. GOlhenberg

IB,lrbaccl

fl.aro,lLrl

SymbOliC

ManipulatIon

Computer

D(!~.Cflpllon:-;
An IntrocJUClIon 10 ISPS-. Technical Report. Dept. ot Computer Aug 1978.
SCience. Carnegie-Mellon Unl. er~,1ty
ISat. 11. Mayle, Shrobe. Sussm<ln. Weise 811 J. Balall. N. Mayle. H D. Weise: "The
Shrobe. C; 50%"'4'1 Gray
DPUDaeeJatus Design Emironmenl". In VLSI 81-. J. 1981

led>. Academic P'ess.

(Baudel 82) G.t.4. BaucJel. "Design aneJ Comple_lty 01 VLSI ALgorilhmS" "Foundalions of Leeuwen (edsl, Compuler Science IV: Pan 1", J,W. de 1983 Bakker,

J, yan

Malhemahsch

Cenlrum, AmSlerdam.

In "VlSI 81".

J. Gray (edL Academic Press. 1981.
IRwn 821 M. Rem: "Partially Ordered CompulaliOM. llh Apphcallons 10 VlSI DMll1n". Technical Report MR 8213. Dept.
Malhemallcs and Com pUling
Scence. Eindhoven Uruversily 01 Technology. 1982.
(Rim. van de SnepsCheul. Udding 831 M. Rem. J. van de SnepsCheul. J T. UcjcUng: "Trace Theory and Ihe Oellnltton of Hterarchlcal Componenls'
Technical Report. Depl. ot Malhemallcs and Com puling Science. Eindhoven Unverslly of Technology. 1983
IAIovel1. Ivey 831 M.C. Revetl. PA Ivey: "ASTRA - A CAD Syslem 10 Suppori Structured Approach to IC Design". In "VlSI 83". F. Anceau. E. J.

(e~sJ.

North-Holland.
81) J.P. Rolh: "Automalic Synthesis. Verllication and 1981.

Tesllng"

In "VLSI
81". J. Oray (edJ. Academic Press.
(Rawson 80l J A. Rowson. "Understanding Hierarchical Design". Ph. D. Thesl!:. Cllilornla Instilule 01 Technology. 1980.
lA.rbm 831 S M In "VLSI 83". F
Rubm "An Inlegraled Aid lor Top-Down Anceau.
Eleclrlcal Design" 1983
E. J. Aas ledsl. Norlh-Holland.
fF\.lpp 811 C.R. Rupp "Components of a Silicon Complier 81". J. Gray ledJ. Academic Press. 1981.
System". In "VLSI

fSchoeltkopl 83) J -P

SChoellkopl
"lUBRICK: A SWcon Assembler and
ApplicalJOn 10 Dala-Path DesIgn lor FISC". In 'VlSI 83". F Aas (e"sl. NOrlh-Hollanct. 1983

Anceau. E. J.

(Segal

Segar:

"SPAM.

CalUornla

Instltule

Technology

SIlicon

Struclures Projecl.

SSP "EMQ 4029.
[Sheeran layoul". Oxlord.

811 Sc

Sheeran Dissenahon.

Geomelry

lnlegraled

Circuit

Research Group.

Universlly 01

(Shrobe 83J HE
Shrobe "AI Meels CAD'

In "VlSI 83'. F

Anceau. E
Aas (eCls). North-Holl"nd. 1983
(Shute 83} M J. Shute: "The Role 01 Simulalion In lhe Siudy 01 Conlrol Flow. and DIUI Flow the Universlly 01

Sy~tems".

Mul1lproces~or.
Ph. D. Thesis. WesHield COllege 01

london. 1983

ISlsklnd.
SOUlh.r". CrOUCh 82al J.M. Siskind. J.R. Soulhard.

doc1

1 ln(1 ) ln(2 ) ln(k )

where the (nite) sum is over all integer multiplicative compositions n = k and each j 2, then 1 lim N N

Fk (n) 1+ k! n=2 k=1

= exp( 0 ln(ln(2)) = 1.2429194164.
where 0 = 0.4281657248. is the analog of Eulers constant when 1/x is replaced by 1/(x ln(x)) (see Table 1.1). See [76] for a dierent generalization of. 1.6. Aprys Constant. The famous alternating central binomial series for (3) e dates back at least as far as 1890, appearing as a special case of a formula due to Markov [77, 78, 79]:
1 X (1)n (n!)6 2(x 1)2 + 6(n + 1)(x 1) + 5(n + 1)=. n=0 (2n + 1)! [x(x + 1) (x + n)]4 n=0 (x + n) X
Ramanujan [80, 81] discovered the series for (3) attributed to Grosswald. Ploue [82] uncovered remarkable formulas for 2k+1 and (2k + 1), including = 72

1 n(en 1 1) 1) 1)

1 n(e2n 1)
1 n(e4n 1) 1 n3 (e4n 1 n5 (e4n

= 720

n3 (en 1 n5 (en

1 n3 (e2n 1) 1 n5 (e2n

5 = 7056

(3) = 28 (5) = 24 (7) =

1 n3 (en 1 1)
X 103 X 19 X +. 7 (en 1) 7 (e2n 1) 7 (e4n 1) 13 n=1 n 4 n=1 n 52 n=1 n

n5 (en 1)

259 X 1 X , 10 n=1 n5 (e2n 1) 10 n=1 n5 (e4n 1)

1 n3 (e2n 1)

n3 (e4n
1.7. Catalans Constant. Rivoal & Zudilin [83] proved that there exist innitely many integers k for which (2k) is irrational, and that at least one of the numbers (2), (4), (6), (8), (10), (12), (14) is irrational. More double integrals (see section 1.5 earlier) include [84, 85, 86, 87]
1 Z Z ln(xy) dx dy (3) = , xy dx dy 1Z Z q G=. 8 (1 xy) x(1 y) 0 0
Zudilin [86] also found the continued fraction expansion 1040| 42322176| 15215850000| 13 =7+ + + +. 2G |10699 |434871 |4090123 where the partial numerators and partial denominators are generated according to the polynomials (2n 1)4 (2n)4 (20n2 48n + 29)(20n2 + 32n + 13) and 3520n6 + 5632n5 + 2064n4 384n3 156n2 + 16n + 7. 1.8. Khintchine-Lvy Constants. Let m(n, x) denote the number of partial e denominators of x correctly predicted by the rst n decimal digits of x. Lochs result is usually stated as [88] lim 6 ln(2) ln(10) m(n, x) = = 0.9702701143. n 2 = (1.0306408341.)1 = [(2)(0.5153204170.)]1
for almost all x. In words, an extra 3% in decimal digits delivers the required partial denominators. The constant 0.51532. appears in [89] and our entry [2.17]. A corresponding Central Limit Theorem is stated in [90, 91]. If x is a quadratic irrational, then its continued fraction expansion is periodic; hence limn M(n, x) is easily found and is algebraic. For example, limn M(n, ) = 1, where is the Golden mean. We study the set of values limn ln(Qn )/n taken over all quadratic irrationals x in [92]. Additional references include [93, 94, 95].
1.9. Feigenbaum-Coullet-Tresser Constants. Consider the unique solution of (x) = T2 [](x) as pictured in Figure 1.6: (x) = 1 (1.5276329970.)x2 + (0.1048151947.)x4 +(0.0267056705.)x6 (0.0035274096.)x8 + The Hausdor dimension D of the Cantor set {xk } [1, 1], dened by x1 = 1 k=1 and xk+1 = (xk ), is known to satisfy 0.53763 < D < 0.53854. This set may be regarded as the simplest of all strange attractors [96, 97, 98]. In two dimensions, Kuznetsov & Sataev [99] computed parameters = 2.502907875., = 1.505318159., = 4.669201609. for the map

d(k2 ) =

0 ln(n)2 + 4 (2) ln(n) + c n + O n1/2+
as n , where the expression for c is complicated. It is easily shown that d(n2 ) is the number of ordered pairs of positive integers (i, j) satisfying lcm(i, j) = n. The best known result for r(n) is currently [192]
r(k) = n + O n 416 ln(n) 8320.
Dene R(n) to be the number of representations of n as a sum of three squares, counting order and sign. Then

R(k) =

4 3/2 n + O n3/4+ 3

for all > 0 and [193]

R(k)2 =

2 n + O n14/9. 21(3)

The former is the same as the number of integer ordered triples falling within the ball of radius n centered at the origin; an extension of the latter to sums of m squares, when m > 3, is also known [193]. Let (n) denote the number of square divisors of n, that is, all positive integers d for which d2 |n. It is known that [194]

(n) (2)x + (1/2)x1/2

as x . Analogous to various error-term formulas in [195], we have
(m) (2)y (1/2)y 1/2 dy C x4/3
21/3 X X d . C = n=1 d2 |n n5/6
This supports a conjecture that the error in approximating nx (n) is O(x1/6+ ). 2.11. Abundant Numbers Density Constant. The denition of A(x) should be replaced by |{k n : (k) x k}| A(x) = n lim. n If K(x) is the number of all positive integers m that satisfy (m) x, then [196] lim x
Y K(x) + 1 = 1 + x p p j=1 i=1 !
2.12. Linniks Constant. In the denition of L, lim should be replaced by limsup. Clearly L exists; the fact that L < was Linniks important contribution. 2.13. Mills Constant. Caldwell & Cheng [197] computed C to high precision. Let q1 < q2 <. < qk denote the consecutive prime factors of an integer n > 1. Dene ! k1 k1 X X qj qj F (n) = 1 = (n) 1 qj+1 j=1 j=1 qj+1

X . + (p 1) j+p p j=1

1 pi
if k > 1 and F (n) = 0 if k = 1. Erds & Nicolas [198] demonstrated that there exists o q 0 a constant C = 1.70654185. such that, as n , F (n) ln(n) C 0 + o(1), with equality holding for innitely many n. Further, C 0 = C 00 + ln(2) + 1/2, where [198, 199] C =

00 X i=1

pi+1 ln pi

pi 1 pi+1

= 0.51339467.,

3n + 1 n/2

if n is odd. if n is even
Let f k denote the kth iterate of f. The 3x + 1 conjecture asserts that, given any positive integer n, there exists k such that f k (n) = 1. Let (n) be the rst k such that f k (n) < n, called the stopping time of n. If we could demonstrate that every positive integer n has a nite stopping time, then the 3x + 1 conjecture would be proved. Heuristic reasoning [289, 290, 291] provides that the average stopping time over all odd integers 1 n N is asymptotically

lim Eodd ((n)) =

ln(3) ln(2)
j cj 2b ln(2) j c = 9.4779555565.
where cj is the number of admissible sequences of order j. Such a sequence {ak }m k=1 Q satises ak = 3/2 exactly j times, ak = 1/2 exactly m j times, m ak < 1 but k=1 Ql k=1 ak > 1 for all 1 l < m [292]. In contrast, the total stopping time (n) of n, the rst k such that f k (n) = 1, appears to obey (n) lim E N ln(n)
= 6.9521189935. = (8.0039227796.). 2 ln(2) ln(3) ln(10)
2.31. Freimans Constant. New proofs of the Markov unicity conjecture for prime powers w appear in [293, 294, 295, 296]. See [297] for asymptotics for the number of admissible triples of Diophantine equations such as u2 + v 2 + 2w2 = 4uvw, u2 + 2v2 + 3w2 = 6uvw, u2 + v 2 + 5w2 = 5uvw and [298] for mention of the constant 3.29304. 2.32. De Bruijn-Newman Constant. Further work regarding Lis criterion, which is equivalent to Riemanns hypothesis and which involves the Stieltjes constants, appears in [255, 256]. A dierent criterion is due to Matiyasevich [257, 258]; the constant ln(4) + + 2 = 0.0461914179. = 2(0.0230957089.) comes out as a special case. See also [259, 260]. As another aside, we mention the unboundedness of (1/2 + i t) for t (0, ), but that a precise order of growth remains open [299, 300, 301, 302]. In contrast, there is a conjecture that [303, 304, 305]

t[T,2T ]

max |(1 + i t)| = e (ln(ln(T )) + ln(ln(ln(T ))) + C + o(1)) ,
1 6e = 2 (ln(ln(T )) + ln(ln(ln(T ))) + C + o(1)) t[T,2T ] |(1 + i t)| as T , where max C = 1 ln(2) +

Z ln(I0 (t)) ln(I0 (t)) t dt + dt = 0.0893. 2 t t2
and I0 (t) is the zeroth modied Bessel function. These formulas have implications for |(i t)| and 1/|(i t)| as well by the analytic continuation formula. Looking at the sign of Re((1 + i t)) for 0 t 105 might lead one to conjecture that this quantity is always positive. In fact, t 682112.92 corresponds to a negative value (the rst?) The problem can be generalized to Re((s + i t)) for arbitrary xed s 1. Van de Lune [306] computed that = sup {s 1 : Re((s + i t)) < 0 for some t 0} = 1.1923473371. is the unique solution of the equation

arcsin (p ) = /2,

where the summation is over all prime numbers p. 2.33. Hall-Montgomery Constant. Let be the unique solution on (0, ) of the equation sin() cos() = /2 and dene K = cos() = 0.3286741629. Consider any real multiplicative function f whose values are constrained to [1, 1]. Hall & Tenenbaum [307] proved that, for some constant C > 0,
and that, moreover, the constant K is sharp. (The latter summation is over all prime numbers p.) This interesting result is a lemma used in [308]. A table of values of sharp constants K is also given in [307] for the generalized scenario where f is complex, |f | 1 and, for all primes p, f (p) is constrained to certain elliptical regions in C. 3.3. Landau-Kolmogorov Constants. For L2 (0, ), Bradley & Everitt [309] were the rst to determine that C(4, 2) = 2.9796339059. = 8.8782182137.; see also [310, 311, 312]. Ditzian [313] proved that the constants for L1 (, ) are the same as those for L (, ). Phng [311] obtained the following best possible o inequality in L2 (0, 1):

f (n) CN exp K

1 f (p) p

for suciently large N,

|f 0 (x)| dx
1 Z Z(6.4595240299.) |f (x)|2 dx + |f 00 (x)| dx
where the constant is given by sec(2)/2 and is the unique zero satisfying 0 < < /4 of sin()4 e2 sin() 1
cos(2)4 [1 + e4 sin() 2e2 sin() cos(2 cos())][1 + e4 sin() 2e2 sin() cos(2 cos())] 2 cos()2 sin()2 [cos(2 cos())](1 e2 sin() ) e2 sin() 1. We wonder about other such additive analogs of Landau-Kolmogorov inequalities. 3.4. Hilberts Constants. Borwein [314] mentioned the case p = q = 4/3 and = 1/2, which evidently remains open. Peachey & Enticott [315] performed relevant numerical experiments. 3.5. Copson-de Bruijn Constant. Ackermans [316] studied the recurrence {un } in greater detail. We hope to discuss such results later. Let be a domain in R n and let p > 1. A multidimensional version of Hardys inequality is [317]

(j + k)

in particular, e1 = 1.7052111401. (Nivens constant) for the cyclic case and = r er = 1 + lim

(k)1 = 2.118456563.

in general. It is remarkable that this limit is nite! Let also =
then for the multiplicative group Z of integers relatively prime to n, n 1 and 2 < n l mod 8} = +1
(k)1 = 1.742652311., if if if if l = 1, 3, 5 or 7, l = 2 or 6, l = 4, l = 0.

sup {e(G) : G = Z n

We emphasize that l, not n, is xed in the supremum (as according to the right-hand side). The constant A1 = 0.4357570767. makes a small appearence (as a certain 1 best probability corresponding to nite nilpotent groups). See [370, 371] for more on nonabelian group enumeration. 5.3. Rnyis Parking Constant. Expressions similar to those for M(x), m and e v appear in the analysis of a certain stochastic fragmentation process [372]. 5.4. Golomb-Dickman Constant. Let P + (n) denote the largest prime factor of n and P (n) denote the smallest prime factor of n. We mentioned that
ln(P + (n)) N ln(N) (1 )N,
ln(P (n)) e N ln(ln(N)) + cN
as N , but did not give an expression for the constant c. Tenenbaum [373] found that c = e (1 + ) +
where the sum over p and product over q are restricted to primes. A numerical evaluation is still open. Another integral [374]
X (t) e 1 ln(p) Y 1 dt + e ln 1 + 1 , t p p 1 qp q p

(x) dx = (1.916045.)1 x

deserves closer attention (when the denominator is replaced by x2 , 1 emerges). The longest tail L(), given a random mapping : {1, 2,. , n} {1, 2,. , n}, is called the height of in [375, 376, 377] and satises
X k2 x2 L() lim P x = (1)k exp n n 2 k=
for xed x > 0. For example,

L() lim Var n n

ln(2)2. 3
The longest rho-path R() is called the diameter of in [378] and has moments !j Z R() j lim E = j/2 xj1 (1 eEi(x)I(x) ) dx n n 2 ((j + 1)/2)
for xed j > 0. Complicated formulas for the distribution of the largest tree P () also exist [376, 377, 379]. A permutation p Sn is an involution if p2 = 1 in Sn. Equivalently, p does not contain any cycles of length > 2: it consists entirely of xed points and transpositions. Let tn denote the number of involutions on Sn. Then tn = tn1 + (n 1)tn2 and [380, 381] n/2 n 1 tn 1/2 1/4 e n 2 e e as n . The equation pd = 1 for d 3 has also been studied [382]. A permutation p Sn is a square if p = q2 for some q Sn ; it is a cube if p = r3 for some r Sn. For convenience, let = (1 + i 3)/2 and 1 exp(x) + 2 exp(x/2) cos( 3x/2). (x) = 3 The probability that a random n-permutation is a square is [383, 384, 385, 386, 387] 21/(1/2) n1/2 =

of two elements chosen uniformly at random, with replacement, from {1, 2,. , n}. Dene d(t ) to be the distance from I at time t, that is, the minimum number of transpositions required to return to I. For any xed c > 0, [612] 1 kk2 d(c n/2 ) 1 (c ec )k n c k! k=1 in probability as n . The coecient simplies to c/2 for c < 1 but is < c/2 otherwise. It is similar to the expansion
X 1 k k1 + W (c ec ) = 1 (c ec )k , c c k! k=1
diering only in the numerator exponent. Consider the spread of a rumor though a population of n individuals. Assume that the number of ignorants is initially n and that the number of spreaders is (1 )n, where 0 < < 1. A spreader-ignorant interaction converts the ignorant into a spreader. When two spreaders interact, they stop spreading the rumor and become stiers. A spreader-stier interaction results in the spreader becoming a stier. All other types of interactions lead to no change. Let denote the expected proportion of initial ignorants who never hear the rumor, then as decreases, increases (which is perhaps surprising!) and [613, 614, 615, 616, 617, 618, 619] 0.2031878699. = (1 ) < () < (0+ ) = 1/e = 0.3678794411. as n . The inmum of is the unique solution of the equation ln()+2(1) = 0 satisfying 0 < < 1, that is, = W (2e2 )/2. As with the divergent alternating factorial series on p. 425, we can assign meaning to [620]
(1)n nn Z n x Z ex n n dx = 0.7041699604. (1) n = x e dx = n! 1 + W (x) n=0 n=0 X X 0 0
which also appears on p. 263. A variation is [621]

(1)n+1 (2n)2n1

(1)n+1 (2n)2n1 Z 2n x = x e dx (2n)! n=1
ex dx = 0.3233674316. ln q W (i x)W (i x)
which evidently is the same as [622]
W (x) cos(x) dx = 0.3233674316. x(1 + W (x))
although a rigorous proof is not yet known. The only two real solutions of the equation xx1 = x + 1 are 0.4758608123. and 2.3983843827., which appear in [623]. Another example of striking coincidences between integrals and sums is [624, 625]
Z Z X sin(n) X sin(n)2 sin(x) sin(x)2 dx = =. dx = = x x2 n n2 n= n=

The integral [626] lim

2N Z 1
eix x1/x dx = 0.0707760393. (0.6840003894.)i
2N +1 Z 2 eix x1/x dx = i + lim N 1
is analogous to the alternating series on p. 450 (since (1)x = eix ). 6.12. Conways Constant. A biochemistry based on Conways chemistry appears in [627]. 7.1. Bloch-Landau Constants. In the denitions of the sets F and G, the functions f need only be analytic on the open unit disk D (in addition to satisfying f (0) = 0, f 0 (0) = 1). On the one hand, the weakened hypothesis doesnt aect the values of B, L or A; on the other hand, the weakening is essential for the existence of f G such that m(f ) = M. The bounds 0.62 < A < 0.7728 were improved by several authors, although they studied the quantity A = A instead (the omitted area constant). Barnard & Lewis [628] demonstrated that A 0.31. Barnard & Pearce [629] established that A 0.240005, but Banjai & Trefethen [630] subsequently computed that A = (0.2385813248.). It is believed that the earlier estimate was slightly in error. See [631, 632, 633] for related problems. The spherical analog of Blochs constant B, corresponding to meromorphic functions f mapping D to the Riemann sphere, was recently determined by Bonk & Eremenko [634]. This constant turns out to be arccos(1/3) = 1.2309594173. A proof as such gives us hope that someday the planar Bloch-Landau constants will also be exactly known.

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