Manufacturer ABIT

Motherboard IC7

Update (Date Available) Merits 17 (2003-09-03) Fixes Maxtor SATA / ICH5R RAID 0 boot issue, many others Fixes Maxtor SATA / ICH5R RAID 0 boot issue, many others Fixes Maxtor SATA / ICH5R RAID 0 boot issue, many others Fixes Maxtor SATA / ICH5R RAID 0 boot issue, many others Fixes Maxtor SATA / ICH5R RAID 0 boot issue, many others Fixes BIOS Setup / USB keyboard hang; other fixes Solves boot hang after CMOS clearing, other issues Adds CPU fan stop detection and shutoff; other items Speeds up BIOS EEPROM saving, etc. Fixes overclocking inability in linear mode Implements DRAM timing; other fixes Supports 32X multiplier, USB device bootup Supports 32X multiplier

IC7-G Max II Advance

17 (2003-09-03)

IC7-MAX3

12 (2003-08-21)

16 (2003-08-26)

11 (2003-09-04)

11 (2003-09-03)

12 (2003-09-03)

AOpen AOpen AOpen

AK79D Max AX4GE-N AX4SP-G / AX4SPE-G
R1.06 (2003/09/09) R1.03 (2003/09/06) R1.01 (2003/09/06)

MX46 U2

R1.15 (2003/08/30)

MX46-533GN

R1.09 (2003/08/30)

MX46-533V

R1.07 (2003/08/30)

AX4SP-N

R1.05 (2003/08/30)

AX4SPE-N

R1.05 (2003/08/23)
Fixes SVSB value error in Silent BIOS/HW Monitor, etc. Adds DRAM timing values, etc. Fixes boot from LAN issue Supports 32X multiplier, etc. Updates VGA BIOS; fixes WinMe BIOS R1.04 issue Fixes FSB 800MHz and Corsair DDR 400 CL2 stability issue; many others Replaces unstable Promise PDC20375 driver; others Fixes Corsair DDR400 256MB compatibility; etc. Fixes SATA boot issue, many others Speeds up BIOS EEPROM saving, etc. Supports PXE/RPL function Fixes SATA boot issue, many others Supports PXE/RPL function Patches some DDR instabilities; other fixes Patches some DDR instabilities; other fixes

MX4GVR

R1.14 (2003/08/23)

MX46U2-GN

R1.07 (2003/08/23)

MX36CE

AX4C Max II

R1.03 (2003/08/23)

AK79D-400 Max

R1.04 (2003/08/23)

AK77-600 Max

R1.01 (2003/08/23)

AX4SG Max

AK79G Max

R1.11 (2003/08/23)

MX4GVR-GN

AX4SG-N

R1.02 (2003/08/23)

AX4PER-GN
MK79G-1394 / MK79G- R1.05 (2003/08/19) N AK79D-400V R1.02 (2003/08/10)

P4C800 Deluxe

1011 (2003/09/04)
Fixes POST hang, etc. Read fine print

P4P800 Deluxe / P4P800

1010 (2003/08/28)

Fixes POST hang, etc.

P4C800-E Deluxe

P4T-EM

1005 (2002/08/15)
Supports P4 Northwood 28X multiplier, etc.

P4G800-V

1004 (2003/08/26)
Enhances memory compatibility; other fixes

P4C800

P4S8X-X

1003 (2003/08/19)

Fixes Win2000 boot / USB disabled issue, others Supports new CPUs

A7N266-VM

1007 (2003/08/20)

A7N8X-VM

1008 (2003/08/18)
Supports new CPUs, etc. Read fine print

A7N266-VM/AA

Supports new CPUs

A7N8X-X

1006 (2003/08/20)

1013 (2003/08/21)

Fixes audio/modem codec ID disappearance after resuming from S3 Supports new CPUs
A7N8X rev. 1.03, 1.04, and 1.06
A7N8X rev. 2.0 and later C1006 (2003/08/20)

A7V8X-MX

1005 (2003/08/19)
A7N8X Deluxe rev. 1.03, 1006 (2003/08/20) 1.04, and 1.06 A7N8X Deluxe rev. 2.0 and later U8668-D V7 C1006 (2003/08/20)

Biostar

u8668dr11.exe (2003 / 08 / 13) u8668dr10.exe (2003 / 08 / 13)
Improves memory compatibility Updates new BIOS code

U8668-D V6

viz0901bs.exe (2003 / 09 Supports new Duron with / 05) CPUID 680/681 1.1c (08/18/03) Improves stability of 1.5V CPU; fixes USB storage hang Fixes power issue Improves DDR400 memory stability Supports Duron 1400, 1600, 1800 Patch code for USB2.0

L4VXA2 PCB 1.X

ECS ECS ECS ECS
L7VTA PCB 1.0x 748-A K7S5A Pro PCB 5.0 K7SOM+ PCB 5.X
1.7d (08/25/03) 030820 (08/25/03) 030811 (08/25/03) 030807 (08/25/03)

K7SOM+ PCB 7.5C

030807 (08/22/03)

Patch code for USB2.0

ECS EPoX

P4VMM2 PCB 7.3 EP-4PCA3+

030703 (08/25/03) 08/19/2003
Supports P4M266a CE Solves power button failure after loading failsafe defaults; other fixes Solves power button failure after loading failsafe defaults; other fixes Solves power button failure after loading failsafe defaults; etc.

EP-4PCA3I

08/15/2003

EP-4PDA

08/14/2003

EP-4PDA2+

EP-4PDA2V

EP-4PDAEI

EP-4PDAI

EP-4PGAI

Solves power button failure after loading failsafe defaults; etc. Solves power button failure after loading failsafe defaults; etc. Solves power button failure after loading failsafe defaults; etc. Solves power button failure after loading failsafe defaults; etc. Solves power button failure after loading failsafe defaults; other fixes Shows AppleBred CPU information; other fixes Shows AppleBred CPU information; other fixes Shows AppleBred CPU information; other fixes Adds Auto Precharge Selectable option for DDR stability; other fixes Addresses several RAID issues

EP-8KRA2+

08/28/2003

EP-8KRA2I

EP-8KRAI

GIGABYTE

EP-8RDA / EP-8RDA+ / 08/27/2003 EP-8RDA3 / EP-8RDA3+ / EP-8RDA3G / EP8RDAE GA-8KNXP Ultra F6 (2003/8/18)

GA-8KNXP

F6 (2003/8/18)
Addresses several RAID issues

GA-8IK1100

F7 (2003/8/18)

GA-7N400-L1

F6 (2003/8/13)

Fixes NV-LAN issue

GA-7N400E

F3 (2003/8/26)

Addresses DDR stability in DIMM slots 3 & 4

GA-7N400E-L

GA-7N400V Pro2

GA-7N400V-L

GA-7N400V

GA-7NNXPV

GA-7N400V Pro

F7 (2003/8/13)

GA-7NNXP

F13 (2003/8/13)

GIGABYTE GIGABYTE

GA-7N400 Pro2 GA-7N400

F3 (2003/8/26) F3 (2003/8/26)
Addresses DDR stability in DIMM slots 3 & 4 Addresses DDR stability in DIMM slots 3 & 4 Addresses DDR stability in DIMM slots 3 & 4 Supports Duron w/ 64KB cache Supports Duron w/ 64KB cache Supports Duron w/ 64KB cache; other fixes Supports future processors; other fixes Fixes several hangs, etc. Fixes a hang and many other issues

GA-7N400-L

GA-7VA-C

F11 (2003/8/29)

GA-7VA

GA-7VT600 1394

F2 (2003/8/29)

Intel Intel Intel

D875PBZ D865PERL D865GLC
P12 (22 Aug 2003) P09 (05 Sep 2003) P09 (27 Aug 2003)

D865GBF

P09 (27 Aug 2003)
Fixes a hang and many other issues Updates Silicon Image ROM, but also drops UDMA-6 support Fixes ALC655 audio issue

D845PEBT2

P10 (21 Aug 2003)

P4SE / P4SE-GOLD

08.13.2003 (8/27/2003)

DPX2-S320

1.07a, 1.07b (8/22/2003) Supports 1MB L3 cache; read fine print 1.28 (8/22/2003) Supports 1MB L3 cache

DPL533

K7S3-N

08.18.2003 (8/20/2003)

Read fine print

648 Max-V (MS-6585v5) 5.1 (8/15/2003)

648F_Neo-L

7.3 (8/12/2003)

MSI MSI MSI

845PE_Max / 845GE_Max-L 848P Neo (MS-6788) 865PE_Neo2-S / 865G Neo2-S (MS-6728G) 875P Neo (MS-6758) K7N2 (MS-6570) / K7N2G (MS-6570) K7N2 Delta-ILSR(MS6570-030)
3.8 (9/3/2003) 1.1 (9/3/2003) 1.4 (8/26/2003)

MSI MSI

1.7 (8/29/2003) 3.7 (8/15/2003)
Fixes WinME/Win2000 LAN failure after CMOS clearing Fixes WinME/Win2000 LAN failure after CMOS clearing Fixes L1/L2 cache issue, others Fixes CMOS clearing issue Fixes Samsung 1GB DDR inability to install Windows XP, others Fixes MegaRAID BIOS 1.66 issue, others Fixes Nvidia FX5900 Ultra compatibility issue Fixes Nvidia FX5900 Ultra compatibility issue Fixes Nvidia FX5900 Ultra compatibility issue Fixes Nvidia FX5900 Ultra compatibility issue Fixes Nvidia FX5800 & FX5900 compatibility issue; other fixes Fixes AMI Diag 6.21 test / USB keyboard or mouse issue

7.4 (8/15/2003)

K7N2 Delta-L (MS-6570- 5.4 (8/15/2003) 020) K7N2G-ILSR 1.6 (8/15/2003)

K7N2G-L

3.6 (8/14/2003)

K7N2GM-L (MS-6777)

1.2 (9/2/2003)

1.9 (9/2/2003)

Supports boot from onboard LAN; other fixes Supports boot from onboard LAN; other fixes

Shuttle

SN41G2 (FN41)
fn41s021.bin (9/4/2003) N/A
fb54s00d.bin (8/25/2003) N/A

ak38s00h.exe (8/25/2003)

SB51G (FB51)
fb51s036.bin (8/25/2003) N/A

MV43V (V7.3)

mv43v006.zip (8/19/2003) av49vs04.exe (8/12/2003) F1.3L (09-08-2003)
AV49V / AV49VN (AV49V V1.0b) SL-B7A-F

Soltek

SL-NV400-L64

G1.2L (2003/08/29)

SL-NV400-64

G1.2 (2003/08/29)

SL-86SPE / SL-86SPE-L AM1.1L (2003/08/19)

SL-75FRN2

D1.5L (2003/08/19)

Soltek Soltek SOYO

SL-86MP / SL-86MP-L 86MP-L11.zip (08-19(PCB ver. T6) 2003) SL-KT400-C / SL-KT400- AW2.1 (2003/08/14) A4C / SL-KT400A-C SY-P4VGA 2AA2 (2003-08-29)
Fixes some USB2.0 HDD compatibility problems, etc. Fixes some USB2.0 HDD compatibility problems, etc. Fixes some USB2.0 HDD compatibility problems, etc. Supports hardware Turbo Mode (for PCB T6 version) Fixes some USB2.0 HDD compatibility problems, etc. N/A Fixes ATI 9200 3DMark2001 issue, etc. Supports MX BIOS chip
SY-K7V DRAGON Plus! 2BA9 (2003-09-08)

Supports Duron Applebred

SY-K7VEM Pro
K7VEM PRO_2DA2 (2003-08-15)

SY-K7VTA Pro

Supports Duron Applebred FSB 133MHz L2 64KB Duron 1400, 1600, 1800 K7VTA PRO V1.0_2BA2 Supports Duron (2003-09-03) Applebred K7VTA PRO_2BA3 (2003-09-03) Supports Duron Applebred Supports Duron Applebred Supports Duron Applebred Supports Duron Applebred Add Opteron Model 246 support; other fixes Adds new clock chip support, etc. Updates DDR400 compatibility, SATA Changes default value for power on after power failure, etc.

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77 R$

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X ~e11k0v5k~, H. N~me1jer/5y5tem5 C0n~01 L ~

27 (1996) 135-144

have recNved a 10t 0f attent10n. 0ne rea50n that the tr1an9u1ar ~ru~ure ha5 6een 1nve5f19med 15 the c105e c0nnect10n 6Nween ~f1an9u1af12a611~ and ~ed6ack 11neaf12at10n (5ee [8, 10]). An0ther m0t1vat10n f0rm5 the 5tudy 0f ~ed6ack ~ a 6 ~ a t n 0f 5y~em5 0f the f0rm (1), 5ee e.9. [17-19], where f0r 50me 5pec1f1c 7F-5y~em5 a ~a61112at10n the0rY ha5 6een dev~0pe~ 0r [6]. 1n 60th ca5e5 a pa~1cu1ar r01e 15 p1ayed 6y 7F~y~em5 1n wNch each 0f the c0mp0nent5 f1, 1 = 1. n - 1, 15 a 10cN 61ject10n 1n k5 1a~ ar9ument and 9,(0) 0 (5ee f0r a prec15e def1Nt10n 1~er 0n); 1n ~h15 ca5e we re~r t0 a 61ject~e ~ n 9 u 1 a r f0rm (87F). A 5uff1c~nt c0n~t10n f0r the a60ve 61ject1~ty pr0pe~y 15 thm 9,(x) 0, (~f~/~x1+1)(0) 0, V1 = 1,., n - 1; tN5 ca5e 15 r e ~ e d a5 the re9u~r ca5e. 7he pr061em 0fthe e4u1vNence t0 the 7F wa5 c0mp1em1y 501ved 1n the re9u1ar ca5e 6y Jaku6c2yk and Re5p0ndek 1n [8] where the 501ut10n Nve5 c0n~t10n5 f0r exact ~ed6ack 11neaf12at10n 0f (1). F0r a d ~ N ~ d 5urvey 0f the re9u1ar ca5e 5ee N50 the m0n09raph5 [10, 7]. N e v e ~ h e ~ , 0n1y ~ w re5uk5 are kn0wn f0r the Nn9u1ar ca5e (1.e. the ca5e that 15 n0t re9u1ar) and the5e re5u1~ u5uN1y Nve 5pec1N tr1an9u1ar f0rm~ 5ee e.9. [12] and fu~her ~ r e n c e 5 1n there. We Nve here nece55ary and 5uff1c~nt c0n~t10n5 f0r the e4u1vMence 0f (1) t0 the tr1an9u1ar f0rm and N50 nece55ary and 5uff1Nent c0n~t10n5 f0r th15 f0rm t0 6e 61ject1ve. 7he 6a51c re5uR 15 th~ the5e c0n~t10n5 N50 c0ver the 51n9u1ar ca5e and are Nven ~ a c r N n ~ e e f0rm. 0 u r ~udy 0f the pr061em 0f the e4u1v~ence t0 51n9u1ar tr1an9u1ar f0rm wa5 pa~1N~ m0t1v~ed 6y 1tc r ~ t 1 m e ve~10n 1nve5t19~ed 1n [14] 1n c0nnect10n w1th the n0n5m00th ~ed6ack 5m611~at10n and 11neaf12at10n 0f d15crN~f1me 5y~em5. H0weve~ that paper d0e5 n0t ~udy c00rd1n~e5-~ee c0n~t10n5 f0r the 7F t0 6e 87F. 7he paper 15 0r9aN2ed a5 f0H0w5. 5ect10n 2 1n~0duce5 n0tat10n and Nve5 nece55ary def1n1t10n5 t09~her w~h 50me pre11m1nary re5u1t5.7he mNn re5u16 are f0rmu1ated and ~1u5trated ~ 5ect10n 3 wN1e thek pr00~ are pr0~ded 1n 5 e ~ n 4. F1nN~, app1~at10n t0 the n0n5m00th ~ a N 1 ~ n pr061em 15 c0n51dered 1n 5ect10n 5. We c0nc1ude the paper 6y 91v1n9 50me remark5 and 0ut~0k5 1n 5 e ~ n 6.

1f 9n(0) 0 and there ex15t5 a p05~1ve re~ num6er e 5uch that V1 = 1.. n - 1 , V x k ~ ( - e , e L k = 1. ~ the f0110w1n9 c0nd1t10n h01d5:
~1(x~,x~. x~,.) : ( - ~ , e )
~ f1(x~,x~.. x1, ( - ~ , e)) 15 a 61ject10n. ~
2. Def1~f10n5 and pre11m1nary re5u1~
F~5t, we 1ntr0duce fu~her nece55ary n0tat10n and term1n0109~ We 5ay that the 7F-5y5mm (2) 15 10cM1y ar0und the 0r191n 1n 6~ect~e ~1an9u1arf0rm ( F )
N0t~e that 1f the F f0r the ~n91e-1nput aff1ne 5y5tem (1) ex1~5, 0ne ha5 af1er a 5u1ta6~ (10ca1) c00rd1nate chan9e 9n ~ 1. Rec~1 that 6y the a60ve ment10ned re9u1ar ca5e we under~and the ca5e when 1n (4) a11 61ject10n5 are 10c~ d1ffe0m0rph15m5, 1.e. V1 (~J1/~x1+~)(0) 0. 8 y the 51n9u1ar ca5e we under5tand the ca5e wh1ch 15 n0t re9u1a~ 1.e. (~J1/~x1+~)(0) = 0 f0r 50me 1. We u5e thr0u9h0ut the paper ~andard d1fferent1a19e0metr1c n0t10n5 and n0tat10n (5ee e.9. [10,7]). 51nce 0ur re5uR5 w111 6e 10c~, we re~f1ct 0ur5e1ve5 t0 5y5tem5 ev01v1n9 0n ~n. 8e51de5 the fre4uent1y u5ed re9u1ar d1~r16ut10n5 we c0n~der 51n9u1ar d15~16ut10n5: a d15tr16ut10n A(x) 15 f0r ~1 x E E" a 5u65pace 0f ~ E " w1th d1men510n p055161y varY1n9 w1th re5pect t0 x. A d15tr16ut10n 15 c ~ d 5m00th (an~yt1c) 1f~ 15 ~panned 6y a (p055161y 1nf1n1~) 5et 0f 5m00th (ana1yt1c) vect0r f1e1d5. We c~1 a 5et 0f 5m00th vect0r f1e1d5.~ t0 6e 10c~1y f1n1~1y 9enerated ( L F 6 ) 1~ 10ca11y ar0und eachx E ~n, there ex1~5 a f1n~e 5et 0fvect0r f1e1d5 ~5uch that any vect0r f1e1d fr0m ~ may 6e expre55ed a5 ~-~=~ ~ ( x ) ~ ( x L r~ ~ ,Y-, ~ ~ C ~ ( ~ ) , 5 < ~x~. A 5et 0f vect0r f1e1d5 ~- 15 c~1ed 1nv01ut1ve 1f 1t 15 c105ed w1th re5pect t0 tak1n9 L1e 6racket, 1.e. Vr~, ~2 ~ ~, 22] ~ ~Y-.N0t~e that a n0n1nv01ut1ve 5et 0f vect0r f1e1d5 may 5pan an 1nv01ut1ve 51n9u1ar d1~r16ut10n A, 51nce the 1a~ pr0pe~y mean5 that V2~,r2 E ,Y5uch th~ r~,~(x) ~ A(xL Vx, we have that ~ , 2~](x) ~ A(x), Vx. 7ake f0r examp1e #~ = {(0, 1)~,(x~,0)~} then [(0, 1)~,(x~,0) ~] = (2x2,0) ~ ~ , 1.e. ~ 15 n0t an 1nv01ut1ve 5et 0f vect0r f1e1d5. 0 n the 0ther hand, the d1~f16uf10n A(x) -- 5pan{(~ 1V,(x~,0)~} 15 an 1nv01ut1ve 51n9u1ar d1~r16ut10n 0n ~2 51nce f0r ~1 x ~ ~2{(0, 1y,(x~,0)~] = (2x~,0)~ A(x). F0r a 91ven 5~ 0fvect0r f1~d5 ~- we den0te 6y [~-] ~5 1nv01ut1ve ~05ur~ 1~. the 5 m ~ (w1th re5pect t0 the 1nc1u510n) 1nv01ut1ve 5et 0fvect0r f1~d5 c0nta1n1n9 1t. 06v10u~y, 3- 15 1nv01ut1ve 1f and 0n1y 1f [~-] = ~-. Rec~11hat 1n the a n ~ y t k ca5e a f1n1t~y 9ener~ed 5et 0f vect0r f1e1d5 ha5 ~way5 a f1n1te1y 9enerated

~~ ~ d~ ~ ~e ~m~10n a h0me0m0~5m. 0f 5m0~hn~5 ~ map~n9 15
~e11k0v5k~, H. N~me1jer15y5tem5 C0ntr01 L e t ~ 27 (1996) 135-144
~v0~f1ve c105ure; 1n the C ~ ca5e th15 may n0t 6e true (5ee [7] f0r fuaher d ~ ). 70 51mp11fy and 5h~aen ~ e m e n t 5 we 0Ren a110w a 5119ht a6u5e 0f n0tat10n5: where n0 c0nfu~0n ar15e5 x may ~and f0r 5ever~ ~ffe~nt c00r~n~e5 5y~em5. 7hr0u9h0ut the p~3er we w111 u5e the f0110w1n9 ne~ed 5e4uence 0f ~5tr16ut~n5 9ener~ed 6y the 5y~em (1),
A1( X ) = 5 p a n { 9 ( x ~ ad f 9( x ). a d ) , 9 ( x ~ ,
and the c0~e5p0n~n9 5u6ma~f01d5 ~ each p~nt ~ e M50 ne~ed. Fk~, we f0rmu1ate and pr0ve a c0nt~u0u5-t1me anM09ue 0f C0rMhry 4.6 0f [14]. 7 h e 0 ~ m 2. C0n51der ~ e 5m00~ 5y5tem (1). 7he f0110w~9 ~ a ~ m e n ~ a ~ e4u~a~n~ 1. 7 h ~ e ex~t ~ca1 c00rd1nate5 ar0und ~ e 0 r ~ m k ~ 9 5y5~m (1) ~ m ~ e 7 F (2)-(3). 2. 7here e x ~ a ne~h60urh00d ~V0 0f ~ e 0 r ~ and an 0pen 5u65et ~V" w~h ~ = ~V6 5uch ~at: (a) ~ e d15~16ut10n5 (5) are c0n5tant d1men510n~ and ~ v 0 ~ 0n dV w1~ ~ m A n - ~ ) = n Vx ~ ~V~, (6) 0n ~V0 ~ e e x ~ a ne5~d 5e4uence 0f ~ 9 u ~ r d15~16ut10n5 { ~ } ~ ~uch ~ a t ~)=A1~) VxE~V ~, 1 = 0 , 1. , n - 1. (9)

1=0. ,n-1,

and def1ne the m~r1ce 5

1 = 0. n - 1.

7he f0110w1n9 1emn~a 15 1mp0rtant 1n the paper (5ee [8, 11]).
~ m m a 1. ~ d ~ r a~ d ~ ~ ~ ~t~e c0n5t~t d1men510na1 d15tr~ut10n5
8 e ~ m pm~n9 Me a60ve Me~em, we ~ve an e4~v~em ~rmu1at10n 0f 2.(6) M~ ~ c a ~ h0w m c h ~ k th15 c0n~f10n ~ r a pa~1cuhr ~ e m. C0r0~ary 1. 5upp05e that 7he0rem 2.2.(a) ~ v a ~ 7hen 7he0rem 2.2.(6) # e4u~a~nt t0: 2(6*) F0r aH 1 = 0,1. n - 1 there ex~t an (n (1 1)) matr1x ~ - ( x ) andan ((1+ 1) (1+ 1)) ma~1x ~(x) 5uch that
vx ~ ~ 5 : ~ ( x ) = ~(x)~(x~ (10)

A0 C A, C. A~,

7hen 10ca11y there ex~t 5m00th c00rd1nate5 x 5uch that A1(x) = 5pan { ~x~1.. 0x~1,} , 11=d1mA1, 1 = 0 , 1.. ~ (8)
7he ~110w1n9 Me0::em 15 an adaptat10n 0f the wd1kn0wn re5u1t 0n e~mt ~ed6ack 11neaf12af10n (5ee [8, 10]) and c 0 m ~ d y 5dve5 the pr0~em 0f 5tate e4u1va1ence m the m9u1~ tr1an9u1~ ~rm. 7 h e 0 ~ m 1. ~ e ~ w ~ 9 ~ e m e n ~ a ~ e4u~a~t: 1. ~ e ~ ~ c ~ ar0und ~ e 0 r ~ ~ 0 r ~ a ~ mk9 ~ 5 ~ m (1) W ~ e ~ 9 u ~ r ~ u ~ r ~rm (2~4). 2. ~ r aH 1 = 0.. n - 1, ~ e ~ u ~ n A1 9~en ~ (5) ~ ~ v 0 ~ t ~ e and ~ 9 u ~ r ar0und ~ e 0 r ~ w1~ ~ m A1 = 1+ 1. 3. ~ e ~ ~n~r ~5~m ~ ~ e ~ k ~ ~ ~9 a ~00~ ~ a ~ ~ ~d a 5m00~ 5mt~ 5m~ f 1 ~ a c ~ Fu~herm~e, ~ an~09y wkh 7 h e ~ e m 2.1.7 0f [7], ~ r a ne~ed ~4uence 0f 5m0mh f 1 n 9 ~ ~ f m ~ n 5 (7) w1th e ~ h A1 6e~9 LF6, we have th~ each ~ r 1 6ut10n A1 ha5 the max1m~ 1nte9ra1 m a n 1 ~ pmperty

where ~ e matr~ ~ ) 9~en 6y (6) ha5 ~ rank 0n ~ , ~ e m a t r ~ ~ ) ha5 ~ H rank 0n ~ and ~ e 54uare m a t r ~ ~ ) ha5 ~ 1 rank 0n ~. 5 ~ ~ ~e 5~9u~ ~ n 6e ~ 0m ~ 0 m ~ e m ~ ~ ~ ~ 0 ~ ~ ~ (10) m ~ 6e 5e~c~d ~ 5uch a w ~ ~ a t a~ v1~ ) are ~wer ant# ~ n 9 u ~ r ~ and v1, ~ ) ~ a ~ 6 m a ~ ~ ) ~ r ~ 1~ < ~.
Pr00f 0f C0r011ary 1. 5upp05e 7he0rem 2.2.(6) and 2.2.(a) 15 vM1~ then 6y 7he0rem 2 there e~5t tr1an9 ~ c00rd1nate5 and ~ Me5e c00r~nm~ ~ m ~ > w~d c0mpm~10n5 9Ne ~//1 = [ 0 ( / + ~ ) ( , ~ 1 ~ ) ] ~ , wh~e ~ ) 15 an (1+ 1) x (1+ 1 ) 10w~ ant1-tr1an9u1~ matr1x w1M v1,~) 6e1n9 a 5u6m~f1x 0f ~ ) ~ r N1 1~ < ~. 7he N~0f12at10n (10) N Men 06~0u5, n a m d ~ take ,/ff1 = [ 0 ( 1 + 1 ) ~ - - 1 ) ] 1 ( / + 1 ) ~ + 1 ~. 7he 9e0m~f1c mea~n9 0f (10) 15 the f01~w1n9: the c0~mn5 0f 6~h ,A1 and ~//1 ~ e ve~0r f1e~5 w~1e demem5 0f v1 are C ~ ~ncf10n5 needed t0 expre55 any vect0r f1e~
2 7hat 1~ ~1 dement5 a60ve k5 a n t ~ a n ~ are e4u~ t0 2er0.
~e11k0v5k~,H. N~me~er/5y5tem5 C0n~01Let~r527 (1996) 13~144
[0{k+~)~ ~ ~) Dk(x)]~, then Rk+~ = ~ " - ~ - ~ + ~ R k where ~a+~ 15 5uch that ~-+~ ~ = [0.. 0, f1~+~d~+~ ], d~+~(0) 0 (rec~1 that ~/f1~+~ 15 5m00th and n0n2er0 at the 0r191n), and ~n-~-1 = (~1,j), ~J = 1,.,n, 15 the permutat10n matr1x w1th 2er0 entr1e5 except ~,~. ~ - ~ - ~ - ~ - - 2 = ~,.~-~ ~ = ~,~t.~u,,~. ~,~ = 1. M0re0ve~ the 0n1y matr1ce5 t0 6e 1nve~ed are e~her d1a90nM 0r repre5ent r0w 0perat10n5. 1n 0ther w0rd5, 0n1y tw0 ~mp1e t0015 are nece55ary ~ther t0 06tMn 10wer ant1Ar1an9u1ar fact0~ 12af10n5 (10) 0r t0 exc1ude the1r ex1~ence, name1y: f1nd1n9 a c0mm0n 5ca1ar fact0r 0f a c01umn vect0r and r0w 0perat10n5. Pr00f 0f 7he0rem 2. 1 ==~ 2. 5upp05e 5y~em ( 1 ) 15 1n 7F (2). 1ntr0du~n9 the f0110w1n9 n0tat10n~ ~ = ( 0. 0,1~,,0. 0) ~, d1(x) = 1 = 1. n,
cMumn 0f./h1 - ~ a th05e 0f ~/ff1. A5 an 1 m m e ~ e c0n5e4uence we have that 1f the 10wer ant1-tr1an9~ar fa~0r12at10n (10) (hav1n9 the 5u6mmr1x pr0peay 0f ~ 5 ) ex1~5 1n 50me paa1cu1ar c r ~ n ~ e ~ then 1t ex15t5 1n a11 c00r~n~e5, 1.e. C0rMhry 1.2.(6*) 15 vM1d. C0nve~e1y, 1f 7he0rem 2.2.(a) 15 va11d t09~her w1th C0rMhry 1.2.(6*) then the ~r16ut10n5 ~1 1n 7he0rem 2.2.(6) ~ 0 u 5 ~ e ~ , name1y, they are 5panned 6y the c01umn5 0f,/ff1. [] Remark 1 (1mp0rtant c0mputat10na1 065ervat10nL 7he ex1~ence 0f ~0r12at10n5 (10) w1th 10wer ant1tf1an9~ar mMr1ce5 ~ may 6e checked 1n the f01~w~9 c0n~rucf1ve way ( ~ c u ~ N e ~ w1th 1 = 0. n - 1): 1. F0r 1 = 0 we have t0 f1nd a 5m0~h 5cMar funct10n v0 5uch that 9/v5m0~h and n0n2er0 at the 0r19~. 1f 5uch a v0 d0e5 n0t ex15L we c0nc~de that the ~0r12at10n (10) d0e5 n0t e~5t a5 we11. 0 t h e v w15e, we have ~ e Mwer a n t 1 ~ r 1 a n 9 ~ ~0r12af10n (10) f0r 1 = 0, nam~y: ~(x) = (9(x~(x)~0(x). 2. R e c u ~ e ~ep. 5upp05e we have M~ady 06tMned 10wer ant1-tr1an9u1ar ~c~f12at10n5 f0r 1 = 0.. ~ 0 ~ k ~ n - 1. Fu11 rank 0f , ~ ( x ) 1m#~5 ~ R~(x)~/~(x) = [0~-+1)~-~-~)1D~(x~ where D~(x) 15 a ((k + 1) (k + 1)) n 0 n ~ n 9 ~ a r ~ a90nM matr1x and R~(x) 15 an (n n) n0n5~9u1ar m~r1x ~ p ~ 5 e m ~ 9 r0w 0perat10n5 needed t0 take ~/ff~(x) 1nt0 the f0rm [ 0 ~ + ~ ) ~. ~ ) 1 D ~ ( x ) ] ~. Let R ~ ( x ~ d ~ 9 = [7~(xL. ~,(x)] ~ and ~ = ~. , ~ k ~ ] , , ~2 = [~n~. ~,]~. We have R~(x)~9~+~(x) = [R~(x)0/h~(x)1R~(x)ad~ ~], 1.e. u5~9 (10) and the n0tat10n j u ~ 1ntr0duced:

9n(x), d1+1(x) = d 1 ( x ) ~ ( x ) ,

1 = 1. n - 1, x ~ ,fr0,

we have, a~er 5trM9htf0rward c0mputat10n5, f0r M1 x ~ ,U
A0(x) = 5pan{(0. ~ d 1 ( x ) f } , A ~ ( x ) = 5pan{A1(xLen~1~d1+2(x~,

1=0.. n-2, (12)

= [{0~+1),~) 1Dk(x)]~(x ) 1f there e ~ 5
a 5m0~h funct10n f1k+~(x) 5uch that ~(x~f1~+~(x) 15 5m00th and n0n2er0 at the 0r1~n, then (10) 15 va11d ~50 f0r 1 = k + 1 w1th
50 thm, 06~0u51y, A1(x) ha5 the pr0pe~y 2.(a) wh11e ~1(x) = 5pan{en~1. e,}, 1 = 0. n - 1, are the ~r16ut10n5 re4u1red 6y 2.(6). 2 ~ 1. App1y1n9 Lemma 1 t0 the ne~ed 5e4uence 0f re9u1ar 1nv01ut1ve ~r16uf10n5 ~1(x) hav1n9 pr0pe a y (9) we 06tMn the1r f1at c00r~n~e5 x f0r wh1ch ~1(x) = 5pan{#/#x,~1. ~/~x~} and theref0re due t0 2.(6) --(x)=0

~+~=R~

1~1/f1~+~ D~0] , 0

Vx~,U, V1,j~{1.. n},

vk+1 = [vk Dff1~2f1~+]
F1nM~, 1t may 6e 5h0wn that 1f n0 f1k+~ e~5t5 f0r w ~ c h ~(x~f1~+~(x) 15 5m00th and n0n2er0 m the 0ff~n, then the ~0r12at10n (10) d0e5 n0t ex15t a5 we11. 7he ~ c u ~ N e ~ep 15 c 0 m # ~ e. N~e that the m~r1ce5 R 0 ,. , R ~ may 6e a150 ea51~ c0mputed r e c u ~ N e ~ : ff R~(x)~ff~(x) =
5uch th~ j ~ 1 + 2. 7he v~1d1ty 0f the ~5t re1at10n5 extend5 6y a 11m1t ar9ument t0 ~1 x ~.A% =. ~ , 1.e. the 5y~em 15 1n the 7F (2). N0w, c0n~f10n 2.(a) 9uarantee5 th~ th15 7F ha5 pr0pe~y (3). [] 7he c0nd1t10n 2.(6) 0f the prev10u5 the0rem 15 actuM1y cruc1~ f0r the e4u1v~ence t0 the 7F 1n the ca5e
~e11k0v5k~, H, N~me1jer/5y5tem5 C0n~01Let~r(1996) 135-144
0f ~n9u1ar d1~f16ut10n5 (5). 0ne can eaN1y check R5 va11d1ty f0r a pa~1cu1ar 5e4uence 0f d1~f16uf10n5. R can 6e v101~ed ~t tw0 d1fferent way5. Perhap5, R 15 n0t 5urpf1f1n9 that f0r 5m00th n0nanNyt1c 5y5~m5 0ne may ea511y c0n5tru~ an examp1e v101at1n9 7he0rem 2.2.(6) (u51n9 the fam11hr funct10n 4(x) = exp(-1/x), x > 0, ~ ( x ) = 0 , x ~ 0 ). A m0re 1ntere5t~ 9 fact 15 that even an anNyt1c 5y~em hav1n9 pr0pe~y 2.2.(a) 0f 7he0rem 2 need nct c 0 m p ~ w1th 7he0rem 2.2.(6), ~f11u~rau~ 6y ~he f0110w1n9 examp~.
man1f01d5.7he 1a5t pr0pe~y 15 an 06v10u5 c0ntrad1ct10n, 1.e. 7he0rem 2.2.(6) 15 n0t va11d. 8 e ~ r e N~her ~ve~9at1n9 the e ~ e n c e 0f the F we Nve an e4~vNem ~rm~at10n 0f 7 h e ~ e m 2 ~ w1H 6e u5ed ~ ~ e ~ 4 u d. M ~ a R pm~d~ a r ~ h ~ c ~ ~ 0 m e ~ c ~f19N.
Exam~e 1. C0nf1d~" ~ e ~ m e ~ n ~

~p~ ~em

21 : X2 -~- X1X3,

51n91e-

~2 := 2X~ + 2X~X3,

23 : U.

7he0rem 2*. C0n51der ~ e 5m00~ 5y5~m (1). 7he f0110w~ m e n ~ a ~ e4u~a~n~ 1. 7here ex~t ~ca1 c00rd1nate5 ar0und ~ e 0 r ~ m k ~ 9 5y5~m (1) ~ ~e 7F(2)-(3~ 2. 0 n a ne~h60urh00d ~ff J ~ e 0 r ~ ~ e ex~t 5m00~ exact 0ne-f0rm5 ~ , 1 = 0, 1. n - 1, 5uch ~ a t f0r a H 1 = 0,1,. , n - 1 : (a) (~,ad~9) 0 f0r aH x ~ ~f~, ~3~ 0pen,
= ~V0, (6) ~ m ~ a n { 0 0. ~ } = 1 + 1, (~,ad~.9) = 0, f0r aH j < 1 and f0r aH x ~ ~/U0. Pr00~ 1 ~ 2.7h15 1mp11cat10n 15 06v10u5: take tr1an9u1ar c00rd1nate5 x f0r the 5y~em (1), then ~1 = dx~-1, 1 = 0.. n - 1, are exact f0rm5 5at15fy1n9 2. 2 =~ 1.0ne can eaf11y 5ee (uf1n9 a dun f0rm 0fthe Fr06en1u5 the0rem - 5ee e.9. [10]) that the d1~f16uf10n5 ~1,1 = 0. n - 1, 91ven 6y (5) are 1nv01ut1ve and c0naant d1menf10na1 0n ~V~. M0re0veL the d1~ tr16ut10n5 ~ : (5pan{~,~1. ~1+~})~, 1 = 0. n - 2, and A,1(x) = 7x~" 5at15fy 7he0rem 2.2.(6). App11cat10n 0f 7he0rem 2 c0mp1ete5 the pr00 [] ~

H ~ e 7 h e ~ e m 2.2.(a) 15 06~0u5~ va11d 5~ce A0(x) = 5pan{(0,0, 1)},
A1~ ) = 5pan{(0,0, 1~,(x1,2x2,0~}, A2~) = ~ a n { ( ~ 0 , 1 ~ , ~ 1 , 2 ~ , 0 ~ , ( - - ~ , 2 x 1 , 0 ~ } ,
6ut the~ d0e5 n0t e ~ a m9u1~ N~r16ut10n ~ ~ ) w1~ pr0peay (9). A~uN~, ~ e ~ d0e5 n ~ e ~ 5m0~h ~ncf10n f1~) 5uch that (x~,2x2,0)~/~(x) 15 n0n2er0 and 5m0~h m h e 0r1Nm 1.e. 6y Remark h e 0 ~ m 2.2.(6) 15 n0t v N ~ and the ~ve5t19~ed 5y~em 15 n0t e4u1va1e~ t0 7F. A1ternat1ve1~ the m ~ n N~w5 ~ e f01~w1n9 c1e~ 9e0metr1cN de~f1pt10n. ~ r v e ~ ~ e ~f16uf10n A1~) ha5 ~ e m a ~ m N 1nte9rN man1f01d pr0peay (u5~9 7he0rem 2.1.7 0f ~ ] ) and the c0~e5p0n~n9 m a N ~ 5 have a very pecMhr t0p01091c~ ~ru~ure ar0und h e 0f1Nn: r0u8h ~ e p01m ~ , x ~ , x ~ ~ e m a ~ m ~ 1nte9rN m a N ~ N 15 91ven a5

3. Ma1n re5u1t

7he mmn re5u1t5 0fth15 paper ~ h e f01~w~9 ~e0rem a60m 10cN ~ate e4~vNence m the F (4) pr0~ n 9 a natura1, c00rd1nate-free c0n5tmct1ve cf1~r10n ~ r 61ject1v1t~ R mm~ ~ m have 5uch a cf1mf10n we ha~e ~ ~ f 1 c t 0 u ~ e ~ m h e an~yt1c c~e. 7~5 mNn m5u1t 15 pa~cu1ar1y 6a5ed 0n the f01~w1n9 L1e N9e6rNc c0~tmcf10n (c N50 [12, 13]).
{x = (x,,x~,x3y ~ ~3 1x~ : x~ exp(r~L
X2 = X~ eXp(2~L )~ = 72,71,2 ~ ~}, 1.e. there 15 a 6und1e 0f para60f1c 5ur~c~ cr0551n9 e ~ h ~ h ~ ~ ~ e 1~e x~ = ~ = 0. E ~ h 5uch p ~ a 6 N ~ 5urface ~ c0mp0~d fr0m ~w0 t w ~ m e n ~ 0 n N m~x1mN 1 ~ e 9 N m a N ~ 5 and ff0m ~ e f1ne x~ = ~ = 0, R~1f 6Nn9 a 0n~NmenN0nM m a ~ m N 1me9m1 m a ~ 1~1d. 5~ce h e ~ 6 u t n ~ ) 5h0~d 6e m 9 ~ and 1nv0~f1ve w1~ pmpe~y (9), k~ e ~ N m a ~ ~1d5 5h0~d 6e everywhem tw0-d1men5~na1 and ~ e y 5h0~d 6e everywhem exceN the 1~e x1 = x~ = 0 cNn~de w1~ ~ e Nmv~d~cf16ed m a ~ m N 1nte9rN
A190~thm 1. 1. 5upp05e 9(0) 0. Put k1 : 1, h~ = 9 and 1etj~ 6e 5uch that 9j, (0) 0. 2. F0r 1 > 1 we def1ne ~ a5 the 5ma11e~ 1n1e9er 5uch that a d ~ , f ( 0 ) 15 f1neady 1ndependent 0f
h1. h1-1 and we put h1 = a d ~ , f. F1na11y, ~ 15 def1ned a5 the p0~f1ve 1nte9er 5uch that the r0w5 j1. ,~ 0f the matr1x [h1[. 1h1](0) are f1neady 1ndependent.

X ~0~k~,

~ N1jme~er/5y5tem5 C0n~01 L ~

27 (19~) ~ 1 ~

7he0rem 3. C0n51der the anMyt1c 5y5~m (1). 7he f0110w 9 5ta~men~ are e4u~a~nt: 1. L0ca11y ar0und the 0 r ~ the n0n11near 5y5~m (1) ~ e4u~a~nt t0 ~ e F (2 )-( 4). 2. A~0r1thm 1 ~ app11ca61e t0 5y5~m (1) 9 ~ 9 the ~ 9 e r 5 ~.. k,; j~.. j~ and vect0r f1e1d5 h~.. h~ 5uch that: (a) ~1 = 5pan{h~ h1+1}, 1 = 1.. n - 1, are re9u1ar ~ v 0 ~ v e d15~16ut10n5 w1th ~ m ~n-~ = n. M0re0ver, the5e d15~16ut10n5fu1f11 7he0rem 22.(6L (6) F0r a~1 = 0.. n - 1 ~ e (1+ 1)~ 0rder m~0r ~ ( x ) 0f the matr1x ,//91 (cf (6)) 9~en 6y the r0w5 j~.. ~+~ ~ ~ther n0np051t~e 0r n0nne9a~ve def1n~e 0n a ne~h60urh00d ~V0 0f the 0~9~. 3. C0nd1t~n 2 0f 7he0rem 2* ~ va11d and f0r aH 1 = 0.. n - 1 thef0110w~ c0nd1t~n5 h01d (recaH that A1~ are ~ e n 6y (5)): (a) Let ~1 6e an ar61trary d15~ut10n hav~9 the max1ma1 ~ 9 r a 1 man~01d pr0perty w~h

M ~ 60th the5e ~ e m e n t 5 are 1 ~ ~ , a5 f 1 ~ r ~ e d 6y the ~ c0unt~examp1e5. Fff~, ~ r ~e 87F 5y~em A~ = ~ ) , A~ = u, ~ ) = ~ 9 n ~ ) e x p ( - 1 / ~ [ ) 7he0~m 3.2 06~0u5~ d0e5 n0t h0~. 5 e c 0 n ~ c0n51der ~ e 5m0~h ~ncf10n ~ ) , ff(0) = 0, ~ ) ~ 0 5uch ~at
( 1 ~ ) ) ~ 1 ~ (1/k+ 1))) ~5ewh~e.k1x1= ~ 1,3,~5,.,~) ~
N0w, ~e 5y~em 2~ = ~ + 0 ~ ) , 2~ = u ~f115 7he0rem 3.3, 1t 15 ~ 7F ~ 3 ) , 6ut 1t 15 n0t 1n 87F

(2~4).

7he0~m 3.2, affh0u9h c 0 m ~ e ~ 91ve5 an ~90f1thm 0f h0w t0 check 6~ect1vRy 1n 9enera1 (p055~ Ny n0ntr1an9Mar) c00rNn~e5.7he0rem 3.3 then pr0v1de5 a ~ 0 m ~ , c ~ n a t e - ~ e verN0n 0f 7he0rem 3.2.7he0~m 3 ~ 6e pr0ved 1n ~ e ne~ ~cf10n. 70 111u5trate ~ we Nve 50me typ~N exam~e5 where e c0nNt10n5 2. ~ 6 ) are ~ 0 ~. Exam~e 2. 7he p1anar 7F 5y~em

~1 ~ X~, ~2 ~ R

A1(x) C ~1(x) C (5pan{6 a.. ~1+~}) Vx ff ~V~0, then ~1~) = ( ~ a n { 6 - ~.. ~+~ }) Vx ~U0. (6) 7he f u n ~ n (~, ad11,9) ~ e1ther n0np051t1ve 0r n0nne9at1ve def1n1te 0n ~ e ne~h60urh00d ~ff 0.
Remark 2. 7he c0nd1t10n 2.(a) 15 a 5tr~9htf0rward re1axat10n 0f a c0nd1t10n c0n~dered 1n [12] f0r the e4u1va1ence t0 tr1an9u1ar p01yn0m1a1 f0rm 0f a 5pec~1 type a5 we11 a5 5u99e~ed 6y [13]. A55um1n9 th~ k~.. kn are 0dd, var10u5 c0ntr0Ha6111ty re5u1~ were 06t~ned (c~ [2, 16, 12]). Let u5 n0te that th15 0ddne55 pr0peay 15 nece55ary f0r the v~1d1ty 0f 2.(6) 0f 7he0rem 3, th0u9h, k 15 n0t 5uff1c1ent. 7he rea50n 15 that A190r1thm 1 may pr0duce f0r d1fferent 5y~em5 the 5ame 1nte9e~ ~ and vect0r f1e1d5h1.0ur 0f191n~ c0ntr16ut10n here 15 the c0mputat10n 0f 1nte9er5 j~.. jn t0 u5e them 1n check1n9 2.(6). N0t1ce, ~50, that the c0mputat10n 0f 1nte9er5 j~.. j~ 15, 1n fact, nece55ary ~ready f0r check1n9 the ~near 1ndependence 0f h~,., hn. Remark 3. 7he0rem n0t v~1d 1n the 5m00th ca5e. 7hpred5e1y due t0 the fact that the 1mp11cat10n 3 ~ 2 d0e5 n0t h01d f0r the 9ener~ 5m00th ca5e. Namdy, a5 w1116e 5een dur1n9 the pr00f 0f 7he0rem 3, the f0110w1n9 ~atement5 are v~1d: 1. 5upp05e 7he0rem 3.va11d then the 5m00th 5y~em (1) 15 10ca11y 5tate e4u1va1ent t0 the 87F. 2. 5upp05e the 5m00th 5y~em (1) 15 10ca11y 5tate e4u1v~ent t0 the 87F then 7he0rem 3.va11d.

15 c1eady n0t 1n the 87F and 1t 15 n0t e4~v~ent t0 87F 51nce 7he0rem 3.2.(6) 15 n0t v~1d. Acm~1y, we have
and d~ ~/d~(x) = -2x~. N0t1ce that 7he0~m 3.2.(a) 15 v ~ h ~ 5~ce a d ~ y ( x ~ , 0 y = ( 2 , 0 L Exam~e 3. 7he p~nar 7F 5y~em

~1 ~ X~X2, ~2 ~ ~

15 n~ e4uNf1ent ~ 87F 51nce c0nd1t10n 2Xa) 0f7he0rem 3 15n ~ v ~. A~u~1~ we have ad(x~m~y(0, 1)t = f7he0rem 3.3: ~the (x~ 0 and a~ 151~a~ ~5(t~)t~10~ A01" 5 ~ 7 ~ a ~ ~ 1m~ 1 n t e ~ ma~f01d pr0pe~y; n0t1ce th~ the 0ne men5~n~ m a ~ x~ = ~var1ant w1th ~5pe~ t0 the 5y~em. 80th ~ e c0n~f10n5 2.(6) and 3.(6) 0f 7he0rem 3 are v~1d ~ r t~5 5y~em 5~ce
X ~e11k0v5k~, H. N~me1jer/5y5tem5 C0n~01 L ~ r (1996) 13~144
Remark An 1mp0~ant 06~rvat10n ~110w1n9 fr0m 7he0~m 3.that 1f a Nven anN~1c n 0 N ~ e ~ 5y5tem ~ N~ady ~ 7F and th15 ~ r m 15 n ~ 61ject1ve, then a11 7F 0f ~ 5y~em ~ e n0t 61ject1ve. 7N6 ~ r v ~ f10n ~em5 m 6e at f1~t ~ance 06v10u~ neveahde55, R 15 n0t at N1 d e a l h0w m pr0ve 1t ~re~1y, w1~0ut u51n9 7he0~m 3.3. 7he0rem 3.3 1n Nct c1a1m5 that e 6 ~ e ~ e n ~ ~ r ~ n a ~ ~var~nL Ac~M~, 6y Remark 3, tNvN1d N50 ~ the 5m0~h c~e. We c0nc1ude th15 5ect10n 6y ~v1n9 the f0110w1n9 n0ntf1v1~ (1.e. n0t 6e1n9 ~ready 1n the 7F) threed1men510n~ examp~ 0f a 5y~em e4u1v~ent t0 the 61ject1ve tr1an9u1ar f1~rm t0 111u~r~e that 7he0rem 3 re~1y may w0rk 1n 9ener~ c00r~n~e5. E x a m ~ e 4. C0nNder the n 0 N ~ e ~ 5y~em 2~ = x2 - 2 ~

~2 = ~ =U. --

0f the 0r1~11n,th15f0rm 1587F(2)-(4) 1f and 0n1y 1f

90(0) 0,

W = 1.. n - 1 : ~ef0, ~ ) ) 0 ) (13)

~)~0),

and there ex15t5 ~ > 05uch that f0r a111 = 1.. n - 1,(x,.. x1) ~ ( - 6 e ) x -. - ( - e , ~ ) :

--~,.. ~,~+,) 0

~r a~t

aH~+~ ~ ( - ~ ).

+ ~ + 4x2x3u,
Pr00f1 7he c0nd1t10n5 0f the 1emma mean that f0r a11 (x~.. x,) ~ E ~/~0 and 1 = 1.. n - 1 the funct10n f1(x1. x1,x1+~ ) 15 5tr1ct1y m0n0t0n0u5 w1th re5pect t0 x1+~ ~ (-~, ~) ~nce R ha5 a1m05t everywhere n0n2er0 n0np051t1ve (0r n0nne9at1ve) der1vat1ve. 7he 1a5t pr0perty 15 06v10uNy e4u1va1ent t0 (4). []

We have

Lemma 3. C0n51der the 5m00th n0n11near5y5tem ( 1) 1n 7 F ( 2 ) - ( 3 ). 7hen the c0nd1t10n (13) 0f Lemma 2 h01d5 1f and 0n1y 1f 7he0rem 3.3.(6) h01d5. Pr00f1 5tra19htf0rward. []
N0w, 1t 15 an ea5y exerc15e t0 check th~ the c0r~5p0n~n9 d19f16ut10n5 may 6e e~ended t0 d15tr16ut10n5 w1th c0n5tant ~men5~n, n a m ~ the ~n9uNr1W ~ may ~e Ncm~d 0 ~ ~ r A~. Here, ~ e cmc1a1Nct 15~ e Ne5ence 0fth15 5~9daf1W N50 ~ ~ e f1r5t c0mp0ne~ 0f adf9. E4~va1ent1~ u51n9 7he0rem 3.3:k1 = ~ = 1,~ = 3 ; j ~ = 1,~ = 2 , f 1 = 3 and Ne c0~e~0nNn9 m~0r5 are e4uM t~ and -9x~, 1.e. the 5y9em 15 ~ e e4u1va1ent t7 F (2)-(4). Let u5 undef1~e that we were a6~ t0 effect1ve~ pr0ve 61ject1vene~ de5p~e the Nct th~ tr1an9u1ar c r d ~ e 5 were uNm0wn. 4. Pr00f 0f 7he0rem 3 We f1r5t 5tate and pr0ve 5evera1 1emma5 nece55ary f0r the pr00f 0f 7he0rem 4. 7he f1r5t 0f them 91ve5 nece55ary and 5uff1c1ent c0nd1t10n5 f0r the 7F (2) t0 6e the F (4) 1n c00rd1nate-dependent f0rm. Lemma2. C0n51der 10ca11y ar0und the 0~9~ 5m00th n0n11near ~5tem (1) and 5upp05e 1t a# ready 1n 7F (2)-(3). 7hen, 1n a ne19h60urh00d.A~0

Lemma 4. C0n51derthe 5m00th n0n11near5y5tem ( 1) 1n 7F (2-3). 1f c0nd1t~n (14) 0f Lemma 2 h01d5, then c0nd1t10n 3.(a) 0f 7he0rem 3 h01d5. Pr00f1 5upp05e 7he0rem 3.3(a) d0e5 n0t h01d. 7h15 mean5 that there ex15t5 ~ E {0.. n - 2} 5uch th~ 7he0rem 3.3.(a) h01d5 f0r ~ 6ut d0e5 n0t h01d f0r ~ + 1 (reca11 th~ A~way5 r~9u1ar at the 0r1~n). Pa~u1af1~ the d1~f16ut10n A~+1 th~ ha5 the max1m~ 1nte9r~ man1f01d pr0pe~y, e4u~5 A~+~ 0n ~A~, A = ~ 0 , and ~mA~+~(0) = ~ + 1 (06~0u51~, 1f ~ m ~ + ~ ( 0 ) = ~ + 2 = d1mA~+~(0L then A~ w 0 d d 6e re9u1ar and 7he0rem 3.3.(a) w0u1d 6e v~1d ~50 f0r ~ + 1). N0w, 06v10u51y, ~+~ = 5pan{~,(~f~/~x~/~xn-~-1)} w1th ~ 6e1n9 re9u1ar 1n a ne19h60urh00d 0f the 0ffNn. 7heref0~ ~ + ~ ha5 0n a ne19h60urh00d 0f the 0r191n 1n N~ the max1mN 1nte9rN man1f01d pr0pe9y w1th d1m ~ + ~ ( 0 ) = 6 + 1 1fand 0n1y 1f

~6-~-, (~. ~ ) ~ 0

X ~e11k0v5k~, H. N1jme~er15y5tem5 C0ntr01Lette~ 27 (1996) 135-144
7 h e 0 r e m 4. C0n51der the 5m00th n0n1~ear c0ntr01 5y5~m (1) and At 7he0rem 3.2.(a) 6e va~d w1th aH k1.. k~ 6e~9 0dd. 7hen th~ 5y5tem ~ ~ca11y a5ympt0t1ca11y 5ta61112a6A u5~9 c0nt~u0u5 feed6ac~
0n a ne19h60urh00d 0fthe 0r191n 1n N. 7h15 c0n~ad~t5 (14) and theref0re 7he0rem 3.3.(a) h01d5 f0r a= 0. n - 1. [] 3. 7han 1mmeN~e c0n5e4uence 0f Lemma5 2-4. N0t1ce, thm th15 1mp11cat10n 15 vNN a150 f0r the 5m00th ca5e. 3 ~ 2. 7he0rem 3.3 9uarantee5, u51n9 7he0rem 2*, the ex1~ence 0ftr1an9u1ar c00rd1n~e5.5~N9htf0~ ward c0mputat10n5 5h0w that 1n the tr1an9u1ar c00rNnate5 7he0rem 3.3 1mphe5 7he0rem 3.2.70 c0nc1ude th15 pa~ 0fthe pr00f 0ne ha5 t0 5h0w thm the va11d1ty 0f 7he0rem 3 2. ( a ) - ( 6 ) 1n the tr1an9u1ar c00rd1n~e5 1mp11e5 R5 vN1Nty 1n N1 c00r~n~e5.7h15 fact 15 06v10u5 f0r 7he0rem 3.2.(a), 1.e. pa~1cu1ar1y we may a55ume th~ A190r1thm apphca61e. N0w, 1~ 7he0rem 3.2.(6) 6e vN1d 1n the tr1an9u1ar c00rd1nate5 x and ~t 2 = ~ ( x ) 6e ar61trary anNyt~ c00r~nme5. We have ~/~1(2) = ~x(C~3-1(2))~//~1(~-1(2)~ Where ~~1(2) = [9(2)1. 1ad~-~9(2~ (5ee a150 (6)). N0W, app1y 1n the new C00rd1n~eA190r1thm 1 t0 06tNn the 1nte9e~ j1. jn and Ve~0r f1dd5 h~. h~. 51nCej~. f1+~r0W5 0fthe matr1X [h~(0)1. 1h~+~(0)] are 11neady 1ndependent and the mmf1x ~1(x)1. 1h1~(x)] 15 10wer ant14r1an9u1ar we have th~ the m1n0r 0f ~x 91ven 6y R5 1a~ 1 + 1 c01umn5 and j~. ~+~-r0wn0n2er0 ~ a n~9h60urh00d 0f the 0r191n. A5 a c0n5e4uence, 519n-def1n1tene55 0f the m1n0r 0f ~/1 Nven 6y ~5 j~.. f1+~-r0wn0t chan9ed dur1n9 the c00rd1n~e chan9e and theref0re 7he0rem 3.2.(6) 15 vN1d N50 1n the new c00rd1nate5 2. 2 ~ 1. 1f 7he0rem ( a ) - ( 6 ) 15 vN1d 1n 50me c00rNn~e~ then 6y 7he0rem 2 there 06~0uNy ex1~ tr1an9u1ar c00rd1n~e5 and 1n the5e C00rd1n~e5 7he0rem 3.2.(a)-(6) 15 vN1d a5 we11 (5ee the pre~0u5 pa~ 0f the pr00~. M0re0ve~ 5trN9htf0rward c0mput~10n5 5h0w th~ f0r the 7F-5y~em 7he0rem 3.2 1mp1~5 (13)-(14) and theref0re th15 7F 15 the F (2)-(4). N0t1ce, th~ the ~5t pa~ 0f the pr00f d0e5 n0t re4u1re the anNyt~1ty a55umpt10n. ~

Pr00f 0f 7 h e 0 r e m 3. 1 ~
Pr00f1 Fk~, n0te that 7he0rem 3.2.(a) 9uarantee5 that the n0nf1near 5y~em 15 5tate e4u1vMent t0 the 7F and theref0re w1th0ut 1055 0f 9enera1Ry we may a55ume th~ 0ur 5y~em 15 Nready 1n 7F (2)-(3). N0w, c0mput1n9 appr0pr1ate L1e 6racket5 0ne can 5ee that the 1nte9er5 k2. k. 91ven 6y A190r1thm 1 are 5uch that f0r N= 2. n

~/;-1-~ (0) # 0

~kfn-1-1

(0)= 0, ~ < ~.

App1y1n9 [6] - C0r011ary 1, we 06t~n that 0ur 5y~em ~ 10c~1y a5ympt0t1ca11y ~a611~a6~ at the 0r191n u51n9 c0nt1nu0u5 feed6ack. ~ F~a11~ 1et u5 6r1ef1y ~ u ~ an apN~at10n 0f the F t0 n0n5m0~h ~a6ff12at10n. 1n t~5 ca5e, a c0n~rucf1ve pr0cedu~ t0 f1nd a ~a~1121n9 ~ed6ack wa5 p m ~ d e d and te5ted 1n [3,5]. 7 ~ 5 pr0cedure 1~ m ~cL 6a5ed 0n the ~110w1n9 p m p ~ 0 ~ Pr0p0~f10n 1. C0n51der ~ e 5m00~ n0n1~ear 5y5tem ( 1) that 5m~ e4~vaAnt ~ the F a ne~h60urh00d ~ff 0 0f ~ e 0 r ~. 7hen, ~ere ex~t 1. an 0pen 5et J~, ~A~ = ~U0, 2. an 0pen 5et 7 , ~ = 70, 70 6e~9 a ne~h60u~ h00d 0 f ~ e 0 r ~ ~ ~n, 3. 30, 5e0 - ne~h60urh00d5 0f ~ e 0 r ~ ~ ~, 4. ~ E C ~ ( 0 / V 0 , ) ~ D1FF(~V~,7), 3 ~ E
C~(~0~V0,~0hVx ~ JV,~(x,.) ~ D1FF(~0,,~0~
5uch ~ a t f0r any p~cew~e c0nt~u0u5 1nput u(t) ~ ~0 Vt ~ [~, t~] C ~ and the c0rre5p0nd1n9 ~ajec~ry 0f x(t) ~ ~ 0 Vt ~ [~,h] C ~ 1t h01d5

~1=Y2, ~2=Y3,

5. App11caf10n5 t0 the n0n5m00th ~a~f12af10n

~n~1=yn. ~ n = V ,

Where y = (y~.. yn)V ~ ~n, V ~ ~ and
Here we 5h0~1y ~ u p 0 ~ app11cat10n5 0fthe tr1an9~ar f0nn t0 the (n0n5m0mh) ~a61112at10n pr061em. A5 wa5 5tre55ed at the 6 e N n ~ n ~ var10u5 r e 5 ~ dea11n9 w1th the 5ta611~at10n 0f 5pec1a1 tr1an9Mar f0rm 5y~em5 are avNNNe. 7he f01~w1n9 ~ a 6 ~ a t n re5dt 15 6a5ed 0n ~]. y=~),

v=~(x,u),

x~V~0, u~0.
M0re0ver, At x 6e ~ e c0rre5p0nd1n9 ~ n 9 u ~ r c00rd1nate5, ~en ~e 5et ~ 0 ~V~ = U]21 ~ J and
3 7he 5~ 0f ~ ~0m0~h15m5 6~ween ~ and ~.
X ~e11k0v5k~, H. N1jme1jer/5y5tem5 C0n~01 L e t ~
the 5et5.A#j = {x ~ ~ff 01(~fj/~xj+~)(x) = 0}, j = 1,., n - 1, are n0t ~ v a r ~ n t w~h re5pect t0 the 0r19~a1 n0n11near 5y5~m.

E x a m # e 5. C 0 n ~ d ~ ~ e F ~ y ~ e m
~1 = X 1 + ~ , ~ =~, ~ = U.
Pr00L 5upp05e w1f1mm ~55 0f 9 e n ~ 1 ~ th~ 0ur ~y~em ~ 1n 87F. N ~ n ~ and 7 0ne may pmceed ~ a ~andard way:

Y~ = x1, Y2 = f1~1,X2),

H(~fj/~xj+~) + ~),

1 = 3. n,

and f1n~1y
~ 15 n0t 5 m 0 ~ y ~a6f112a61e (1t ha5 an un5ta6~ appr0~mme h n e a f ~ n ) 6m ff ~ c0nt1nu0u51y 5ta61112a6~ uf1nh e ~ e m 4. Neve~he~5~ ~15 ~e0rem d 0 ~ n ~ exp11cff1y pr0v1de a ~aNf1~n9 ~ed6acL U5~9 Pr0p051t10n 1 we have ~ r ya = x~, y2 = x~ + ~ , Y3 = x, + ~ + 3 x ~ , ~ : X 1 + x ~ + 3 x ~ + 6 x ~ + 3 x ~ u th~ ~1 = Y~, f1~ = Y3, ~3 = V. N0t1Ce that rdaf1cn5 6~ween x and y have a11 pr0pe91~ ~ 4 ~ d 6y Pr0p0~f10n 1, name~, ~ e y pr0~de a Nffe0m0~N5m 6~ween {x E ~31~ 0} and {y E ~ 3 [ y 1 # Y2}. An anN090U5 ~ e m e m 15 tme ~ r U, V. Pr0p05N0n n d ~ e 5 ~ r a 5y~em 1n 6~ecf1ve t r 1 a n 9 ~ ~ r m a 5trN9N~rw~d appr0ach t0 R5 ~a61112at10m Name~, 0ne may e ~ 1 ~ c0mp~e a ~a611121n9 ~ e d 6 ~ k th~ 15 n0t def1ned ~ 51n9daf1t1~ and 15 un60unded, then th15 ~ed6ack may 6e ~e9u1af12eC t0 06tNn a c0nt1nu0u5 ~a611121n9 ~ed6ac~ 7h15 appmach w ~ numer1ca11y 5ucc~5N1~ ~5ted ~ r t w ~ and ~ N m e n 5 ~ n N 5y~em5 ~ [3, 5].
H e ~ the ~ 5 ~and f0r 50me C ~ funct10n5. N0w, the c1a1m5 1-m m e ~ e ~ f01~w fr0m the fa~ th~ each 0fthe der1vat1ve5 ( ~ f ~ / ~ x j + ~ j = 1. n - 1, d0e5 n0t chan9e the ~9n 0n dV0 and 15 n0n2er0 0n ~ (u~n9 Lemma 2). F1na11~ 5upp05e ~j = ~. n - 1 5uch th~ D~ 15 mvaf1ant wRh re5pect t0 the 87F-5y~em (2)-(4) and 1~ ~ 6e the m a ~ m ~ 5uch j. C0n~der the f0110w1n9 au~f1ary 5y~em,
6. C0nc1u~0n5 and 0uf100k5
N e c e ~ a ~ and 5uff1dem 9e0m~f1c c0nd1t10n5 ~ r e ~ate e4u1va1ence ~ the ~ e c t 1 v e ) t r 1 a n 9 ~ ~ r m w~e p~de~ 5~9~rw~d p~cedure5 ~ r ~ d r chec~n9 ~ any f1xed c ~ e 5y~em were ~50 pr0v1ded. 0 ~ y ~ n 9 ~ n p m n0n11near 5y~em5 were c0n~dered. At p r ~ e m ~ 5eem5 t0 6e ~ff1cu1t t0 ~ v e e4~v~em5 ~ r ~ e ~ c0n~f10n5 ~ ~ e mu1t1-1np~ ca5e. P0~ app11cat10n5 ~ r ~ e n0n5m0~h ~a6~2at10n w ~ e ~ N c ~ e d. 7N5 N~ct10n ce~NNy d ~ e ~ e 5 N~her ~ve5t19at10n.

f ,(x,,x2L. ~

= f f(X1. Xf+1), ~]+1 = ~,
then th15 5y~em ha5 ~V~, a5 1~ ~var1ant 5et. 5~ce 0 ~ ~V~],we have th~9 (~/~xf+1)fj(0. 0,xj+~ ) ~ 0 1n a ne~h60urh00d 0f the 0f1Nn and theref0m (14) d0e5 n0t h01d. U5h~ Lemma 2 we have th~ the 5y~em 1n 4ue5t10n 15 nm ~ F - 1.e. 1he 06~0u5 c0ntra~ct10n. []

Remark 5. De5pke 1t5 ~n9thy f0rmu1at10n, the 1dea 0f the pr0p05k10n 15 very ~mp~: c0n51der the 0f191n~ n 0 ~ e a r 5y5tem 0n the m a ~ f 0 ~ ~/V (w1th ~17 = ~ 0 ) , then 1t 15 e 4 ~ v ~ e n t (u~n9 a 5m00th c00rd1nate chan9e ~ad a 5m00th ~ed6ack) t0 a f1near 5y~em def1ned 0n an 0pen 5~ ~ (w1th c ~ 5 u ~ 6e~ 9 a9~n a ne19h60arh00d 0f the 0f19~). M0re0ve~ 0 ~ y ~ e ~ 0 f 1 e 5 pr0duced 6y very 5pec1a1 1nput5 may 11e 1n51de the 51n9u1ar 5et ~V~0 ~A/~. 1t can 6e even 5h0wn that 1n the an~yt1c ca5e 0 ~ y the 2er0 1nput may pr0duce a ~ e ~ 0 r y 6 e ~ n ~ n 9 ~0 the 5~9u1ar 5et. We f11u~rme ~ 0 ~

Re~nc~

[1] D. Aeyd~ L0c~ and ~06~ c 0 m ~ 1 ~ f 1 ~ ~ r n0n11near ~ e m ~ ~ e m 5 ~ n ~ Let~ 5 (1984) 19-26, [2] D. Aeyd~ 5m6f1~m~n 0f a c1a55 0f n0n11near 5y~em5 6y a 5m0~h ~ e ~ k c 0 m r ~ ~ 5 ~ m 5 ~ n ~ L e ~ 5 0985) 2 ~. [3] 5. ~ e 1 ~ 0 v ~ 70p0109~a1 ~ 2 ~ 0f n 0 ~ e ~ 5y5~m5: ~ 1 c ~ t0 the n0n5m0~h ~ 2 ~ , Pr0~ 2nd ECC~3, 6 m ~ n 9 e ~ 7he N~hedand5 (1993) 41-44. ~ 5. ~ 0 v ~ 70pd09~ e~N~ ~d ~p~09~ 11neaf12af10n 0f c0mm1~d d y n a m ~ ~ e m ~ ~ ~ 31 (1995) 141-150.
~ ~e11k0v5k~, H. N1jme1jer/ 5y5tem5 C0ntr01 L e t ~
[5] 5. ~e11k0v5k~, Numef1c~ ~90f1~m ~ r ~ e n0n5m0~h 5m~1~10n 0f t r 1 a n 9 ~ 5y~em5 Ky6ernet1ka 32 (1996~ t0 appea~ [~ J.M. C0r0n and L. P r ~ Ad~n9 an 1nte9rat0r ~ r the ~a61112at10n p ~ e m , 5y5~m5 C0ntr01 L e ~ 17 (1991) 89-104. [7] A. ~ 0 r 1 , N0n11near C0n~01 5y5~m~ An 1n~0duct10n (5pf1n9e~ He~d6e~, 1989). [8] 8. Jaku6c2yk and W. Re5p0nde~ 0n f1neaf12af10n0f c0m~1 5y~em~ 8u~ Acad 5eL P0~na~e 5er 5c~ Mat~ 28 (1980) 517-522. [~ V.1. K0~60~ C0~Ha611~y and ~a61~1~ 0fcea~n n 0 ~ e ~ 5y~em, D1fferen~1a1~nye Uravnen~a 9 (1973) 614-619. [1~ H. N ~ m e ~ and A~. van der 5cha~ N0n11near Dynam1ca1 C0mr01 5y~em5 (5pf1n9e~ New Y0rk, 1990). [11] W. Re5p0ndeL 0 n dec0mp051t10n 0f n0nf1near c0mrd 5y~ tem5, 5y~em5 C0n~01 L~L 1 (1982) 301-30~ [12] W. Re5p0ndek, 6106~ a5pec~ 0f 1~eaf12m10~ e4u1v~ence ~ p01yn0m1~1 ~m~5 and dec0mp05~0n 0f n0n11near c0ntr01 5y~em~ ~: M. ~ and M. Ha2ew~kd, ed5., A~e6r~c

[13] [1~

[1~ [1~ [1~
and 6e0m~r1c M~h0d5 ~ N0n11near C0ntr01 7he0ry (R~d~, D 0 ~ c ~ , 1986) 257-283. W. Re5p0ndek, pf1v~e c0mmun~at10n, 5e~em6~ 1992. C. ~m~e~ H. N ~ m e ~ ~nd J. 75mh~ N0n5m0~h 5ta61112a6111~ and ~ed6ack 1~ear12at10n 0f ~ r ~ e - t 1 m e n 0 ~ e ~ ~y~em~ Mem0mndum N0. 1190, De~. A p ~ d M ~ h e m ~ 5 , Un~. 7 w e ~ 7he N~hef1and5; t0 appear 1n: 1 n ~ a t. ~ R06u~ N0n11near C0nt~L H. 5u~mann, 0~1~ 0f ~m111e5 0f vect0r f1e1d5 and 1nte9m6111ty 0f ~f16ut10n~ 7ran~ Am. Math. 50~ 180 (1973) 171-188. H.J. 5u~man~ L1e 6mck~5 and ~ca1 c0mr01~6111ty: a 5uff1c~ c0n~f10n ~ r ~ ~ p ~ 5y5tem, 51AM ~ C0ntr01 0pt1m. 21 (1983) 686-713. J. 7~nm~ A ~ c ~ ~ a ~ 1 ~ n ~ r 1me~0nne~ed 5y~em~ 5y5tem5 C0n~01 Let~ 18 (1992) 429-434. J. 7 ~ , P a ~ 1 ~ - ~ e 9106a~ 1 ~ n ~ r 9 e n ~ tf1a n 9 d ~ 5y5tem5, 5y5~m5 C0n~01 L ~ 24 (1995) 139-145. J. 751n1a~ V ~ 0 n 5 0f 50~a9~5 ~ p ~ t0 5tate 5ta6111~ c0nd1t~n and 0 ~ p ~ ~ed6ack ~06~ ~a61112at10n,m appea~