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Sharp R-8680

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Comments to date: 6. Page 1 of 1. Average Rating:
BigJules 11:33pm on Thursday, October 7th, 2010 
After 1 month of horror and pain with my iPod earbuds, I was desperate for a nice set of ear phones for office, recreation, and home chore use. Would certainly purchase again, especially at price from this merchant. Sound is overall exceptionally high quality, and the ear buds fit comfortably.
ForumPosting1 4:15pm on Friday, September 3rd, 2010 
the unit are very small and easy to carry and look high tech too. I like it as compare to the bulky headphone I had used before. For the price. I bought these to go with my Sansa Fuze to replace the included earbuds. It is definitely a worthwhile upgrade.
rocafella 9:03am on Tuesday, August 24th, 2010 
Great Bass and overall volume, nice fit, durable Slightly canned sounding Clear sound, excellent bass, extremely comfortable. Same as all earbuds – cord noise. No case (not a big deal)
loopy67 6:42pm on Monday, July 19th, 2010 
Great for the price. The 5 star rating if more so for the price, eco-friendly packaging than audio quality. Sennheiser small ear buds Excellent Service from the vendor - package arrived very quickly and is exactly what I needed.
harveyfrey 6:01pm on Monday, June 14th, 2010 
Great little earphones I have previously purchased these for myself when I wanted a cheap pair to go to the gym. Excellent Great buy! Sound quality is great and does the job perfectly. Used it for my phone; does have a 2.5mm jack. Would recommend!
David 10:33am on Monday, June 14th, 2010 
Sounds great Newegg or the manufacturer should have noted that one channel is longer than the other. Annoying. Nice bass and also hi-fidelity to the original soundtrack ; The wires are of great quality,.

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

MARYLAND METRICS Technical Data Chart Pages included in this file Cold drawn flats EN 10277-1 / h11 Flat bars heat resistant DIN 1017 Cold drawn flats EN 10277-1 / h11 Key steel cold drawn Flat and Square DIN 6880 / h9 Half rounds cold rolled Cold rolled and bright annealed Surface finish accord. to 2R/2H* Tolerance h11 Equal Angles Dimension accord. EN 10056 Hot rolled, internal radius round (r) Execution accord. to EN 10088-3 Mill specific tolerances Equal Angles Dimensions accord. to EN 10056 Internal radii laser welded (r) Execution accord. to EN 10088-3 Mill specific tolerances Equal Angles cold rolled Dimensions accord. to DIN 1022 Internal radius sharp (r) Execution accord. to EN 10088-3 Equal Angles cold drawn Dimension accord. to DIN 59370 All radii sharp (r) Execution accord. to EN 10088-3 Equal Angles polished Dimensions accord. to EN 10056 Execution Unequal Angles Dimension accord. EN 10056 Hot rolled, internal radius round (r) Execution accord. to EN 10088-3 Unequal Angles Dimensions accord. to EN 10056 Internal radius welded (r) Execution accord. to EN 10088-3 Unequal Angles cold rolled Dimensions accord. to DIN 1022 Internal radius sharp (r) Execution accord. to EN 10088-3 Unequal Angles cold drawn Dimensions accord. to DIN 59370 Unequal Angles polished Dimension accord. to EN 10056 Execution T-Sections Dimensions accord. to EN 10055 Internal radii round (r) Execution accord. to EN 10088-3 T-Sections Dimensions accord. to EN 10055 Internal radii laser welded (r) Execution accord. to EN 10088-3 Statical properties T Sections cold drawn Dimensions accord. to EN 10055 All radii sharp Execution accord. to EN 10088-3 Unequal T-Sections Dimensions accord. to EN 10055 Internal radii laser welded (r) Execution accord. to EN 10088-3 Equal Angles polished Dimensions accord. to EN 10056 Execution Unequal Angles Dimension accord. EN 10056 Hot rolled, internal radius round (r) Execution accord. to EN 10088-3 Unequal Angles Dimensions accord. to EN 10056 Internal radius welded (r) Execution accord. to EN 10088-3 Unequal Angles cold rolled Dimensions accord. to DIN 1022 Internal radius sharp (r) Execution accord. to EN 10088-3 Unequal Angles cold drawn Dimensions accord. to DIN 59370 All radii sharp (r) Execution accord. to EN 10088-3 Unequal Angles polished Dimension accord. to EN 10056 Execution Channels UNP Dimension accord. to DIN 1026, hot rolled Execution accord. to EN 10088-3 Statical values Channels UAP Channels with parallel flanges, hot rolled Internal radius rounded (r) Execution according to EN 10088-3 Channels UAP Laser Channels with parallel flanges Execution according to EN 10088-3 Statical values Unequal T-Sections Dimensions accord. to EN 10055 Internal radii laser welded (r) Execution accord. to EN 10088-3 IPE Beams Dimensions accord. to EN 1025 Internal radii laser welded (r) Execution accord. to EN 10088-3 Statical properties HEA - Beams Dimensions accord. to EN 1025 Internal radii laser welded (r) Execution accord. to EN 10088-3 Statical properties HEB - Beams Dimensions accord. to EN 1025 Internal radii laser welded (r) Execution accord. to EN 10088-3 Statical properties Special Beams Internal radii laser welded (r) Execution accord. to EN 10088-3 Statical properties British Universal Beams (UB) in accordance with BS 4-1: 1993, Internal radii laser welded Statical properties Mill specific tolerances Z Sections Dimensions accord. to EN 10055 Internal radii laser welded (r) Execution accord. to EN 10088-3 HEA - Beams Dimensions accord. to EN 1025 Internal radii laser welded (r) Execution accord. to EN 10088-3 Statical properties HEB - Beams Dimensions accord. to EN 1025 Internal radii laser welded (r) Execution accord. to EN 10088-3 Statical properties Special Beams Internal radii laser welded (r) Execution accord. to EN 10088-3 Statical properties British Universal Beams (UB) in accordance with BS 4-1: 1993 Statical properties Mill specific tolerances Dimension Tolerance TechnicalData Chart ISO tolerances for shafts

Available from: MARYLAND METRICS P.O. Box 261 Owings Mills, MD 21117 USA ph: (410)358-3130 (800)638-1830 fx: (410)358-3142 (800)872-9329 web: http://mdmetric.com Click here for a request for quotation form: http://mdmetric.com/rfq.htm email: sales@mdmetric.com

Cold drawn flats

EN 10277-1 / h11

a (mm) 100 120

0,10 0,14 0,19 0,24 0,13 0,19 0,26 0,32 0,16 0,24 0,32 0,40 0,29 0,38 0,48 0,36 0,48 0,60 0,38 0,51 0,64 0,43 0,58 0,72 0,48 0,64 0,80 0,60 0,80 1,00 0,72 0,96 1,20 0,84 1,12 1,40 0,96 1,28 1,60 1,08 1,44 1,80 1,20 1,60 2,00 1,44 1,92 2,40 2,24 2,80 2,56 3,20 3,60 4,00 4,80
6 0,38 0,48 0,57 0,72 0,77 0,86 0,96 1,20 1,44 1,68 1,92 2,16 2,40 2,88 3,36 3,84 4,32 4,80 5,76

t (mm) 20

0,64 0,76 0,96 1,02 1,15 1,44 1,60 1,92 2,24 2,56 2,72 3,20 3,84 4,48 5,12 5,76 6,40 7,68
0,96 1,20 1,28 1,44 1,60 2,00 2,40 2,80 3,20 3,60 4,00 4,80 5,60 6,40 7,20 8,00 9,60
1,44 1,54 1,73 1,92 2,40 2,88 3,36 3,84 4,32 4,80 5,76 6,56 7,68 8,64 9,60
2,16 2,40 3,00 3,60 4,20 4,80 5,40 6,00 7,20 8,40 9,60 10,8 12,0
4,00 4,80 5,60 6,40 7,20 8,00 9,60 11,2 12,8 14,4 16,0
6,00 7,00 8,00 9,00 10,0 12,0 14,0 16,0 18,0 20,0
8,40 9,60 10,8 12,0 14,4 16,8 19,2 21,6 24,0
11,2 12,6 14,0 16,8 19,6 22,4 25,2 28,0
14,4 16,0 19,2 24,0 25,6 28,8 32,0
18,0 21,6 25,2 28,8 32,4 36,0

24,0 28,0 32,0 36,0 40,0

36,0 38,4 43,2 48,0

Grade: 1.4404 /AISI 316L

Note: many of these products are not normally stocked in the USA, so please allow sufficient lead time for delivery.

Flat bars heat resistant

DIN 1017
a (mm) 5 0,80 1,00 1,20 1,40 1,60 1,80 2,00 2,40 2,80 3,20 3,60 3,93 4.32 4,80 5,10 5,60 5,85 6,40 7,20 8,00 10,0 12,0,96 1,20 1,44 1,68 1,92 2,16 2,40 2,88 3,36 3,84 4,32 4,71 5,18 5,76 6,12 6,60 7,07 7,68 8,64 9,60 12,0 14,1,44 1,60 1,92 2,24 2,56 2,72 3,20 3,84 4,48 5,12 5,76 6,28 6,91 7,68 8,16 8,80 9,42 10,2 11,5 12,8 16,0 19,2 t (mm) 10 1,60 2,00 2,40 2,80 3,20 3,60 4,00 4,80 5,60 6,40 7,20 7,85 8,64 9,60 10,2 11,0 11,8 12,8 14,4 16,0 20,0 24,1,92 2,40 2,88 3,36 3,84 4,32 4,80 5,76 6,56 7,68 8,64 9,42 10,4 11,3 12,3 13,2 14,1 15,4 17,3 19,2 24,0 28,8
15 2,40 3,00 3,60 4,20 4,80 5,40 6,00 7,20 8,40 9,60 10,8 11,8 13,9 14.1 15,3 16,5 17,7 19,2 21,6 24,0 30,0 36,0 20
4,80 5,60 6,40 7,20 8,00 9,60 11,2 12,8 14,4 15,7 17,2 18.8 20,4 22.0 23,6 25,5 28,8 32,0 40,0 48,0
Grades: 1.4828 and 1.4841

a (mm) 100

3 0,24 0,36 0,48 0,60
4 0,32 0,48 0,64 0,80 0,96 1,28
5 0,40 0,60 0,80 1,00 1,20 1,60 2,00 2,40
6 0,48 0,72 0,96 1,20 1,44 1,92 2,40 2,88 3,84 4,32 4,80

8 0,96 1,44 1,60 1,92 2,56 3,20 3,84 5,12 5,76 6,40
t (mm) 10 1,20 1,60 2,00 2,40 3,20 4,00 4,80 6,40 7,20 8,00
1,92 2,40 2,88 3,84 4,80 5,76 7,68 8,64 9,60
2,40 3,00 3,60 4,80 6,00 7,20 9,60
4,00 4,80 6,40 8,00 9,60 12,80

Grade: 1.4305 /AISI 303

a (mm) 0,0,07 0,13 0,12 0,20

Key steel cold drawn

Flat and Square DIN 6880 / h9 t (mm) 11
0,28 0,31 0,38 0,44 0,47 0,63 0,57 0,75 0,94 0,66 0,99 1,26 0.99 1,56 1,26 1,88 1,55 2,42 2,75 2,20 3,52 4,52 5,65 6,91 8,83 Grade: AISI 316

Half rounds cold rolled

Cold rolled and bright annealed Surface finish accord. to 2R/2H* Tolerance h11, Grade 1.4404

[ kg/m ]

* 32 * 40

3.18 4.95

* Cold drawn sections, surface finish accord. to 2H.
Flat half rounds cold rolled

20 * 30

0.67 1.71

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Equal Angles

Dimension accord. EN 10056 Hot rolled, internal radius round (r) Execution accord. to EN 10088-3: 1D
0.45 0.65 0.61 2.56 4.60 2.12 2.76 4.00 5.15 2.36 3.07 5.13 7.09 5.00 4.58 6.24 7.05 8.70 5.03 6.84 8.69 6.42 10.3 5.82 6.89 7.95 10.0 11.1 7.37 11.8 8.33 9.65 10.9

[ kg/m ] 9.28 10.8 12.2

Tolerances accord. to EN 10056
Dimension 15 - 20 mm 30 - 45 mm 50 mm 60 - 100 mm

Tolerances +/- 1.0 mm +/- 1.5 mm +/- 2.0 mm +/- 3.0 mm

90 mm 100 mm

+/- 0.4 mm +/- 0.6 mm
15 - 20 mm 30 mm 40 mm 45 - 60 mm 65 - 80 mm 90 - 100 mm
2 mm 4 mm 4.5 mm 6.5 mm 8.5 mm 10 mm

Grades

1.4301 / 1.4307 / 1.4401 / 1.4404 / 1.4571

Mill specific tolerances

Dimension h b s t Grades mm mm mm < 7 mm 7 mm 10 mm > 10 mm Tolerance +/- 1.0 mm +/- 1.5 mm +/- 0.5 m +/- 0.5 mm + 0/- 1.0 mm + 0/- 1.0 mm + 0/- 1.5 mm

Techni calD at C har a t

I t er SO ol ances f orshaf s t
m i l m eters / m m li over i cl di g n u n h9 h11

0 --------------120

example: 20h9 = 20.00/19.948

0 --------400

tol eran ce i m i n cron s / m i crom etres
Werkstoff number 1.4301 1.4305 1.4307 1.4401 1.4404 1.4571 1.4828 1.4841
AISI Nearest Fit 304L 316 316L 316Ti 309 314
European Steel Designation X5CrNi18-10 X8CrNiS18-9 X2CrNi18-9 X5CrNiMo17-12-2 X2CrNiMo17-12-2 X6CrNiMoTi17-12-2 X15CrNiSi20-12 X15CrNiSi25-21
Dimensions accord. to EN 10056 Internal radii laser welded (r) Execution accord. to EN 10088-3: 1D
[ kg/m ] 10.3 13.5 16.6 11.2 14.8 18.1 21.9 23.6 12.2 16.1 20.0 23.8 25.7 13.0 17.2 21.3 25.4 13.9 18.7 22.9 27.3 29.8 14.9 19.7 24.5 28.8 32.0 17.0 22.5 28.0 33.4 36.1 41.4 18.9 25.1 31.2 37.2 40.2 46.2

Dimension

Tolerance

+/- 0.5 mm

Equal Angles cold rolled
Dimensions accord. to DIN 1022 Internal radius sharp (r) Execution accord. to EN 10088-3: 2H

a 20 25

[ kg/m ] 0.89 1.13 1.37 1.79 2.43 3.00 3.80 4.51 5.47

Toleranzen

40 mm 60 mm mm mm mm mm 60 mm
+/- 0.4 mm +/- 0.5 mm +/- 0.3 mm +/- 0.4 mm +/- 0.5 mm +/- 0.6 mm r 1.5 mm

Equal Angles cold drawn

Dimension accord. to DIN 59370 All radii sharp (r) Execution accord. to EN 10088-3: 2H

a 60 60

s 3 2.8
[ kg/m ] 0.29 0.45 0.65 0.61 0.89 0.93 1.13 1.47 1.37 1.79 2.20 2.11 1.85 2.43 3.00 3.40 3.80 4.51 5.47 7.17
Tolerances accord. to DIN 59370
15 mm 60 mm 4 mm 8 mm 15 mm 30 mm 60 mm
+/- 0.15 mm +/- 0.20 mm +/- 0.10 mm +/- 0.15 mm r 0.5 mm r 0.8 mm r 1.0 mm

Equal Angles polished

Dimensions accord. to EN 10056 Execution: G 220 /G 240 /G 320
Tolerances accord. To EN 10056
Dimension 15 - 20 mm 30 - 45 mm 50 mm 60 mm 60 mm 15 - 20 mm 30 mm 40 mm 45 - 60 mm
Tolerance +/- 1.0 mm +/- 1.5 mm +/- 2.0 mm +/- 3.0 mm +/- 0.4 mm 2 mm 4 mm 4.5 mm 6.5 mm

Unequal Angles

[ kg/m ] 0.65 0.89 1.01 1.13 1.47 1.80 2.11 1.37 1.79 2.20 2.59 1.62 2.10 2.60 3.07 1.73 2.27 2.80 3.31 1.85 2.43 3.00 3.54 2.09 2.75 3.40 4.02 3.40 4.03 4.62 5.25 3.80 4.51 5.82 4.40 5.26 6.05 6.82 7.60

b 75 75

[ kg/m ] 4.60 5.47 6.33 7.17 7.99 4.80 5.71 6.61 7.46 8.31 9.15 5.00 5.95 6.89 7.81 8.71 9.60 5.47 6.33 7.17 7.96 8.73 6.67 7.71 8.77 9.73 10.70 6.91 9.09 7.66 8.86 11.20

[ kg/m ] 6.91 9.09 11.20 7.66 8.85 10.05 11.09 12.30 8.10 9.42 10.70 11.95 13.10
Tolerances accord. to EN 10056-2
50 mm 60 - 100 mm 90 mm 100 mm 20 - 40 mm 45 - 60 mm 65 - 80 mm 90 - 100 mm
+/- 1.0 mm +/- 2.0 mm +/- 0.4 mm +/- 0.6 mm 4 mm 6.5 mm 8.5 mm 10 mm
Dimensions accord. to EN 10056 Internal radius welded (r) Execution accord. to EN 10088-3: 1D

b 100 100

s 10 12
[ kg/m ] 9.45 12.2 14.9 9.05 11.8 14.6 9.55 12.6 15.5 13.8 16.8 15.5 19.0 14.7 18.1 21.6 16.6 20.1 24.2 18.5 22.5 27.4

+/- 0.5 mm +/- 0.5 mm

Unequal Angles cold rolled

b 20 20

[ kg/m ] 1.13 1.47 1.37 1.79
2.27 2.79 2.98 4.51 3.40 3.80 4.51 5.47 7.17 6.91 9.09 11.2

mm mm mm

+/- 0.5 mm +/- 0.3 mm r 1 mm
Unequal Angles cold drawn
Dimensions accord. to DIN 59370 All radii sharp (r) Execution accord. to EN 10088-3: 2H
2.27 2.79 2.98 4.51 3.40 3.80 4.51
Tolerances accord. To DIN 59370

mm mm mm mm

+/- 0.20 mm +/- 0.15 mm r 0.8 mm r 1.0 mm

Unequal Angles polished

Dimension accord. to EN 10056 Execution: G 220 /G 240 /G 320

T-Sections

Dimensions accord. to EN 10055 Internal radii round (r) Execution accord. to EN 10088-3: 1D

b 30 35

s=t 3 3.4
[ kg/m ] 0.88 1.15 1.13 1.30 1.47 1.37 1.79 2.11
Tolerances accord. to EN 10055

h b s/t

35 mm 35 mm

+/- 1.0 mm +/- 1.0 mm

Dimensions accord. to EN 10055 Internal radii laser welded (r) Execution accord. to EN 10088-3: 1D

b 140 140

s=t 13 15
[ kg/m ] 2.10 1.85 2.40 3.00 3.40 2.30 3.10 3.80 4.50 5.90 2.80 3.70 4.60 5.45 7.15 7.50 7.40 9.70 12.20 12.40 15.10 18.40 25.10 21.60 27.80 31.80

mm mm 5 mm

+/- 0.5 mm +/- 0.5 mm +/- 0.5 mm

Statical properties

G kg/m

Ix cm4

Wx cm3

Iy cm4

Wy cm3

T Sections cold drawn

Dimensions accord. to EN 10055 All radii sharp Execution accord. to EN 10088-3: 2H

b 50 50

s=t 5 6
[ kg/m ] 1.37 1.79 2.11 2.42 3.00 3.40 3.80 4.51
mm mm mm internal external
+/- 0.2 mm +/- 0.2 mm +/- 0.15mm 1 mm 0.5 mm

Unequal T-Sections

Dimension mm mm 4 mm
Tolerance +/- 0.5 mm +/- 0.5 mm +/- 0.5 mm

a 50 60

Dimension 15 - 20 mm 30 - 45 mm 50 mm 60 mm 60 mm 15 - 20 mm 30 mm 40 mm 45 - 60 mm Tolerance +/- 1.0 mm +/- 1.5 mm +/- 2.0 mm +/- 3.0 mm +/- 0.4 mm 2 mm 4 mm 4.5 mm 6.5 mm

a s r Grades

Dimension a/b s 50 mm 60 - 100 mm 90 mm 100 mm 20 - 40 mm 45 - 60 mm 65 - 80 mm 90 - 100 mm Tolerance +/- 1.0 mm +/- 2.0 mm +/- 0.4 mm +/- 0.6 mm 4 mm 6.5 mm 8.5 mm 10 mm

r Grades

Dimension a s Grades mm mm Tolerance +/- 0.5 mm +/- 0.5 mm
[ kg/m ] 1.13 1.47 1.37 1.79 1.49 2.27 2.79 2.98 4.51 3.40 3.80 4.51 5.47 7.17 6.91 9.09 11.2

Dimension a s r Grades mm mm mm Tolerance +/- 0.5 mm +/- 0.3 mm r 1 mm
Dimensions accord. to DIN 59370 All radii sharp (r) Execution accord. to EN 10088-3: 2H a 60 b s 5 6
[ kg/m ] 1.13 1.47 1.37 1.79 1.49 2.27 2.79 2.98 4.51 3.40 3.80 4.51
Dimension a s r Grades mm mm mm mm Tolerance +/- 0.20 mm +/- 0.15 mm r 0.8 mm r 1.0 mm

b 40 40

Channels UNP
Dimension accord. to DIN 1026, hot rolled Execution accord. to EN 10088-3: 1D

[ kg/m ] 0.86

3.5 4.5 7

1.37 1.74 4.27

1.78 2.30 4.80

3.86 5.60

4.37 5.10
Tolerances accord. to EN 10279 tab.1

+/+/+/+/-

1.5 2.0 1.5 2.0

3 7.5 mm

3.5 10.0 mm > 10.0 mm
+0*/- 0.5 mm +0*/- 1.0 mm
* Plus tolerance is limited by weight per meter

Statical values

UNP 50

UNP 65

UNP 80

UNP 100

UNP 120

UNP 140

UNP 160

Channels UAP

Channels with parallel flanges, hot rolled Internal radius rounded (r) Execution according to EN 10088-3: 1D
[ kg/m ] 2.30 3.60 4.17 4.37 5.90 7.05 7.65 8.90 10.90
Tolerances accord. to EN 10279 tab. 2
+/- 2.0 mm +/- 1.5 mm +/- 0.5 mm
1.4301 / 1.4307 / 1.4401 / 1.4404
Laser Channels with parallel flanges Execution according to EN 10088-3: 1D
s 6.6.0 7.6 6.6 7.7.8 7.8 7.5 8.8 8.8 10
t 8.8.5 9.6 9.6 10.10.8 10.8 10.5 11.8 11.8 10
[ kg/m ] 4.90 5.90 7.05 7.60 8.65 6.10 7.65 8.90 10.60 13.40 9.20 10.90 13.70 9.90 11.80 16.20 12.90 14.90 16.80 17.90 13.80 18.10 18.80 14.80 19.40 21.20 22.00 16.60 22.00 25.30 18.70 24.60 30.40

h 280 300

b 95 100
s 9.0 9.5 9.0 10.0 10.0 10.0
t 12.5 13.0 13.5 14.0 15.0 16.0
[ kg/m ] 29.40 33.20 34.40 37.90 41.80 46.20
mm mm mm mm 7 - 10 mm 9 - 16 mm

+/+/+/+/+/+/-

2.0 3.0 1.5 2.0 0.5 1.0

mm mm mm mm mm mm

UAP 80 *

UAP 100*

UAP 120*

UAP 130*

UAP 140*

UAP 150

UAP 160*

UAP 175

UAP 180*

UAP 200*

UAP 220*

UAP 240*

UAP 250

UAP 260*

UAP 280*

UAP 300*

* Sizes accord. to DIN 1026
Dimensions accord. to EN 10055 Internal radii laser welded (r) Execution accord. to EN 10088-3: 1D h 60 b s=t 5.8.5 10

h s r t

[ kg/m ] 1.79 2.76 3.64 4.66 3.20 6.20 9.00 12.80

h b s/t Grades Dimension mm mm 4 mm Tolerance +/- 0.5 mm +/- 0.5 mm +/- 0.5 mm
h mm 60 b mm t mm 5.8.s mm 5.8.G kg/m 1.79 2.80 3.40 4.75 3.20 6.33 9.60 12.80 F mm1640 Ix cm4 0.62 1.50 2.91 5.11 5.19 8.86 21.1 43.2 Wx cm3 0.41 0.79 1.27 1.90 1.70 2.89 5.51 9.35 Iy cm4 2.14 5.23 9.93 17.2 7.22 30.0 71.0 144.4 Wy cm3 1.07 2.09 3.31 4.42 2.41 7.49 14.2 24.1

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IPE Beams

Dimensions accord. to EN 1025 Internal radii laser welded (r) Execution accord. to EN 10088-3: 1D
Abbrev. IPE 80 IPE 100 IPE 120 IPE 140 IPE 160 IPE 180 IPE 200

IPE 220 IPE 240

IPE 270 IPE 300

270 300

s 3.8 4.1 4.4 4.7 5.0 5.3 5.6 5.9
t 5.2 5.7 6.3 6.9 7.4 8.0
[ kg/m ] 6.00 8.10 10.40 12.90 15.80 18.80

IPE 330 IPE 360

330 360

IPE 400

6.2 6.6 7.1 7.5 8.0 8.6

8.5 9.2 9.8 10.2

22.40 25.20

29.40 35.20

10.7 11.5

41.40 47.30

12.7 13.5

56.00 63.60

mm mm 3.8 8.6 mm < 6.5 mm 6.5 mm
+/- 1.0 mm +/- 0.5 mm + 0.4/- 0.7 mm + 0.8/- 0.4 mm + 0.5/- 1.0 mm

Abbrev.

IPE 100

IPE 120

IPE 140

IPE 160

IPE 180

IPE 200

IPE 220

IPE 240

IPE 270

IPE 300

IPE 330

IPE 360

HEA - Beams
Dimensions accord. to EN 1025 Internal radii laser welded (r) Execution accord. to EN 10088-3: 1D
HEA HEA HEA HEA HEA HEA HEA 220

HEA 240 HEA 260

HEA 280 HEA 300

270 290

280 300

5.6 6.5 7.0 7.5 7.5 8.0

8.9.5 10
16.7 19.9 24.7 30.4 35.5 42.3

12.5 13

48.6 58.5

63.6 72.9

+/- 1.0 mm +/- 0.5 mm

6.5 mm

8 13.5 mm

+ 0/- 1.0 mm

HEA 100

HEA 120

HEA 140

HEA 160

HEA 180

HEA 200

HEA 220

HEA 240

HEA 260

HEA 280

HEA 300

80.18260

HEB - Beams
Abbrev. HEB 100 HEB 120 HEB 140 HEB 160 HEB 180 HEB 200 HEB 220 HEB 240 HEB 260 HEB 280 HEB 300

s 6 6.8 8.9.10 10.5 11

t 17.19
[ kg/m ] 20.4 26.7 33.7 42.6 51.2 61.3 71.5 83.2 93.117
mm mm mm mm < 7 mm 7 mm 10 mm > 10 mm
+/- 1.0 mm +/- 1.5 mm +/- 0.5 mm +/- 1.0 mm +/- 0.5 mm + 0/- 1.0 mm + 0/- 1.0 mm + 0/- 1.5 mm

HEB 100

HEB 120

HEB 140

HEB 160

HEB 180

HEB 200

HEB 220

HEB 240

10600 11260

HEB 260

11800 14920

HEB 280

13100 19270

HEB 300

14900 25170

Special Beams
Internal radii laser welded (r) Execution accord. to EN 10088-3: 1D

Abrev. H 80

H H H H

150 160

H 240 H 250

t 8 12.15 16

[ kg/m ] 14.3 16.6 18.2 31.6 35.2 31.9 26.3 32.9 49.0 39.8 71.1 62.1 97.0

mm mm mm mm mm 12 mm

1.0 1.5 0.5 1.0 0.5 1.0
British Universal Beams (UB)
in accordance with BS 4-1: 1993, Internal radii laser welded (r)
Abbrev. 127 x 76 x x 89 x x 102 x x 102 x x 133 x x 133 x x 102 x x 102 x x 102 x x 146 x x 146 x x 146 x x 102 x x 102 x x 102 x 33
UB UB UB UB UB UB UB UB UB UB UB UB UB UB UB UB 305 x 127 x 37
h 127 152.4 177.8 203.2 203.2 206.257.2 260.4 251.259.6 305.1 308.7 312.7
b 76 88.7 101.2 101.8 133.2 133.9 101.6 101.9 102.2 146.1 146.4 147.3 101.6 101.8 102.4
s 4 4.5 4.8 5.4 5.7 6.4 5.7 6.0 6.3 6.0 6.3 7.2 5.8 6.0 6.6
t 7.6 7.7 7.9 9.3 7.8 9.6 6.8 8.8.6 10.9 12.7 7.0 8.8 10.8
[ kg/m ] 13.0 16.0 19.0 23.1 25.1 30.0 22.0 25.2 28.3 31.1 37.0 43.0 24.8 28.2 32.8

UB UB UB UB UB UB UB UB

356 356

x x x x x x x x

127 171
304.1 307.2 311.0 303.4 306.6 310.4 349.0 353.4
123.3 124.3 125.165.7 166.9 125.4 126.0
7.1 8.0 9.6.7 7.9 6.0 6.6
10.7 12.1 14.0 10.2 11.8 13.7 8.5 10.7
37.0 41.9 48.1 40.3 46.1 54.0 33.1 39.1

UB UB UB UB

356 406

x x x x

171 140

351.358 363.4

171.1 171.5 172.2 173.2

7 7.4 8.1 9.1

9.7 11.15.7

45.0 51.0 57.0 67.1

UB 406 x 140 x 46 UB 406 x 178 x 54

398.0 403.2

141.8 142.2

6.4 6.8

8.6 11.2

39.0 46.0

UB 406 x 178 x 60

$EEUHY

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NJP FP FP FP FP

FP FP FP

8%[[

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mm mm mm < 7 mm 7 mm 10 mm > 10 mm
+/- 1.0 mm +/- 1.5 mm +/- 0.5 m +/- 0.5 mm + 0/- 1.0 mm + 0/- 1.0 mm + 0/- 1.5 mm

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Z Sections
Abrev. Z 40 Z 50 Z 60 Z 80 Z 100 Z 120 Z 140 Z 160

h 140 160

b 65 70

s 4.6.8 8.5

t 5 5.11
[ kg/m ] 4.30 5.30 6.30 8.60 11.3 14.3 18.0 21.6

h b s t

mm mm 4.5 8.5 mm 5.0 11.0 mm

1.0 0.5 0.5 0.5

HEA HEA HEA HEA HEA HEA HEA HEA HEA HEA HEA
5.6 6.5 7.0 7.5 7.5 8.0 8.5

8.9.12.13.5

16.7 19.9 24.7 30.4 35.5 42.3 48.6 58.5 63.6 72.9 80.4
Dimension h b s t Grades mm mm 6.5 mm 8 13.5 mm Tolerance +/- 1.0 mm +/- 0.5 mm +/- 0.5 mm + 0/- 1.0 mm
Abbrev. HEA 100 HEA 120 HEA 140 HEA 160 HEA 180 HEA 200 HEA 220 HEA 240 HEA 260 HEA 280 HEA 300 h mm 290 b mm 300 s mm 5.6 6.5 7.0 7.5 7.5 8.0 8.5 t mm 8.9.12.13.5 G kg/m 16.70 19.90 24.70 30.40 35.50 42.30 48.60 58.50 63.60 72.90 F mm9730 Ix cm13670 Wx cm3 72.1260 Iy cmWy cm3 26.8 38.5 55.6 76.340 421
Dimensions accord. to EN 1025 Internal radii laser welded (r) Execution accord. to EN 10088-3: 1D Abbrev. HEB 100 HEB 120 HEB 140 HEB 160 HEB 180 HEB 200 HEB 220 HEB 240 HEB 260 HEB 280 HEB 300 h 300 b 300 s 6 6.8 8.9.10 10.t 17.19
Dimension h b s t Grades mm mm mm mm < 7 mm 7 mm 10 mm > 10 mm Tolerance +/- 1.0 mm +/- 1.5 mm +/- 0.5 mm +/- 1.0 mm +/- 0.5 mm + 0/- 1.0 mm + 0/- 1.0 mm + 0/- 1.5 mm
Abbrev. HEB 100 HEB 120 HEB 140 HEB 160 HEB 180 HEB 200 HEB 220 HEB 240 HEB 260 HEB 280 HEB 300 h mm 300 b mm 300 s mm 6 6.8 8.9.10 10.t mm 17.19 G kg/m 20.4 26.7 33.7 42.6 51.2 61.3 71.5 83.2 93.117 F mmIx cmWx cm3 89.1680 Iy cmWy cm3 33.5 52.9 78.571

Kurzz. H 100

[ kg/m ] 16.6

+/+/+/+/+/+/1.0 1.5 0.5 1.0 0.5 1.0 mm mm mm mm mm mm

b t h r s

in accordance with BS 4-1: 1993, Internal radii laser welded (r)
UB UB UB UB UB UB UB UB UB UB UB UB UB UB UB UB UB UB UB UB UB UB UB UB UB
Abbrev. 127 x 76 x x 89 x x 102 x x 102 x x 133 x x 133 x x 102 x x 102 x x 102 x x 146 x x 146 x x 146 x x 102 x x 102 x x 102 x x 127 x x 127 x x 127 x x 165 x x 165 x x 165 x x 127 x x 127 x x 171 x x 171 x 51
h 127 152.4 177.8 203.2 203.2 206.257.2 260.4 251.259.6 305.1 308.7 312.7 304.1 307.2 311.0 303.4 306.6 310.4 349.0 353.4 351.358 363.4 398.0 403.2 402.6 406.4
b 76 88.7 101.2 101.8 133.2 133.9 101.6 101.9 102.2 146.1 146.4 147.3 101.6 101.8 102.4 123.3 124.3 125.165.7 166.9 125.4 126.0 171.1 171.5 172.2 173.2 141.8 142.2 177.7 177.9
s 4 4.5 4.8 5.4 5.7 6.4 5.7 6.0 6.3 6.0 6.3 7.2 5.8 6.0 6.6 7.1 8.0 9.6.7 7.9 6.0 6.7.4 8.1 9.1 6.4 6.8 7.7 7.9
t 7.6 7.7 7.9 9.3 7.8 9.6 6.8 8.8.6 10.9 12.7 7.0 8.8 10.8 10.7 12.1 14.0 10.2 11.8 13.7 8.5 10.7 9.7 11.15.7 8.6 11.2 10.9 12.8
[ kg/m ] 13.0 16.0 19.0 23.1 25.1 30.0 22.0 25.2 28.3 31.1 37.0 43.0 24.8 28.2 32.8 37.0 41.9 48.1 40.3 46.1 54.0 33.1 39.1 45.0 51.0 57.0 67.1 39.0 46.0 54.1 60.1
UB 356 x 171 x 57 UB 356 x 171 x 67 UB 406 x 140 x 39 UB 406 x 140 x 46 UB 406 x 178 x 54 UB 406 x 178 x 60
Abbrev. UB 127 x 76 x 13 UB 152 x 89 x 16 G A Iy cm834 Wy Wpl,y cm3 cm3 iy cm Avz cm2 Iz cm4 55.7 89.8 Wz cm3 Wpl,z cm3 iz cm Ss mm IT cm4 Iwx10-3 cm6 1.98 4.69 9.85 15.37 29.33 37.34 18.16 22.92 27.89 65.88 85.61 103.1 27.18 34.79 44.04 71.94 84.32 164.1 194.4 233.6 80.97 236.4 285.2 329.2 410.9 kg/m cm2 13.0 16.52 16.0 20.32 074.6 84.2 5.353 6.427 109.5 123.3 6.407 8.178 14.67 22.58 1.837 28.1 2.796 20.24 31.18 2.102 28.8 3.546
UB 178 x 102 x 19 19.0 24.152.5 171.3 7.476 9.852 136.7 27.02 41.59 2.374 29.5 4.416 UB 203 x 102 x 23 23.1 29.207.2 234.1 8.462 12.38 163.9 32.19 49.75 2.361 32.9 7.022 UB 203 x 133 x 25 25.1 31.230.3 257.7 8.556 12.82 307.6 46.19 70.94 3.102 30.2 6.095 UB 203 x 133 x 30 30.0 38.280 314.4 8.705 14.58 384.7 57.45 88.22 3.173 34.5 10.43 16.7 148.7 29.18 46.01 2.154 31.7 6.557 UB 254 x 102 x 22 22.0 28.223.10.07 15.62 119.3 23.49 37.27 2.064 28.2 4.348 UB 254 x 102 x 25 25.2 32.265.5 305.5 10.32 UB 254 x 102 x 28 28.3 36.307.6 352.8 10.54 17.79 178.5 34.94 54.85 2.224 35.2 9.657 UB 254 x 146 x 31 31.1 39.351.1 393.1 10.55 16.37 447.5 61.26 94.13 3.358 32.1 8.677 UB 254 x 146 x 37 37.0 47.432.6 483.2 10.83 17.59 570.6 77.96 119.4 3.478 UB 254 x 146 x 43 43.0 54.504.1 566.3 10.93 20.15.37 677.4 91.97 141.1 3.517 41.5 23.97

UB 305 x 102 x 25 24.8 31.292.1 342.0 11.87 18.850 122.9 24.20 38.81 1.972 28.7 4.977 UB 305 x 102 x 28 28.2 35.347.6 402.9 12.23 19.827 155.4 30.53 48.45 2.081 32.5 7.511 UB 305 x 102 x 33 32.8 41.415.8 480.8 12.47 22.063 194.1 37.91 60.04 2.154 37.1 12.29 UB 305 x 127 x 37 37.0 47.470.2 538.4 12.32 23.416 335.4 54.40 85.27 2.667 38.9 14.95 UB 305 x 127 x 42 41.9 53.533.6 613.5 12.39 26.442 388.8 62.55 98.41 2.698 42.6 21.42 UB 305 x 165 x 40 40.3 51.560.5 623.1 12.87 20.09 764.4 92.65 141.7 3.859 36.83 14.74 UB 305 x 165 x 46 46.1 58.645.12.98 22.53 895.7 108.1 165.5 3.905 40.73 22.20 UB 305 x 165 x 54 54.0 68.753.6 846.1 13.04 26.127.4 195.6 3.932 45.73 34.90 UB 356 x 127 x 33 33.1 42.472.7 542.9 13.99 23.057 280.2 44.69 70.29 2.579 35.0 8.972 UB 356 x 171 x 45 45.0 57.686.7 774.6 14.51 26.79 811.1 94.81 146.6 3.761 38.35 16.16 UB 356 x 171 x 51 51.0 64.796.4 896.0 14.76 28.66 968.3 112.92 174.2 3.862 42.35 24.00 UB 356 x 171 x 57 57.0 72.14.87 31.128.71 198.8 3.908 46.05 33.59 UB 356 x 171 x 67 67.1 85.15.09 35.157.3.992 52.45 55.90
UB 305 x 127 x 48 48.1 61.615.7 710.7 12.50 29.902 461.0 73.59 116.1 2.744 47.4 32.18 101.22
UB 356 x 127 x 39 39.1 49.575.6 658.5 14.30 25.694 357.8 56.80 89.05 2.681 40.0 15.15 104.74
UB 406 x 140 x 39 39.0 49.628.6 723.7 15.87 27.569 409.8 57.80 90.85 2.873 35.6 10.99 154.92 UB 406 x 140 x 46 46.0 58.778.0 887.6 16.35 29.834 538.1 75.68 118.1 3.029 41.2 19.07 206.19 UB 406 x 178 x 54 54.1 68.16.48 33.114.94 178.3 3.848 41.45 23.50 34.135.3.965 45.45 33.465.2 531.7 607.1 UB 406 x 178 x 60 60.1 76.16.8 UB 406 x 178 x 74 74.2 94.17
UB 406 x 178 x 67 67.1 85.16.87 38.152.68 236.6 3.995 49.35 46.40 41.172.4.044 53.45 63.10

Technical Data Chart

ISO tolerances for shafts
millimeters / mm over including h9 h11
tolerance in microns / micrometres

Markovian semigroups, time translation operator and a new development of Hida product between Hida distributions on sharp time elds

Sergio ALBEVERIO

Minoru W. YOSHIDA

\dag er

May 31, 2007
Abstract A new denition of product between the elements of Hida distributions dened on the sharp time free eld is given. By this procedure the space of Hida distributions has the structure of ring. This newly dened product is dierent to the well known Stransform. It is much more complicated, but might have more fruitful strucrures than Stransform.

Preliminaries

Throughout this note, we denote by $d\in N$, where is the set of natural numbers, the space-time dimension, and we understand that $d-1$ is the space dimension and 1 is the dimension of time. Correspondingly, we use the notations
$x\equiv(t,\tilde{x})\in Rx\mathbb{R}^{d-1}$
Let $S(R^{d})$ (resp. $S(R^{d-1})$ ) be the Schwartz space of rapidly decreasing test functions on the dimensional Euclidean space (resp. $d-1$ dimensional Euclidean space ), equipped with the usual topology by which it is a Fr\echet nuclear space. ) be the topological dual space of $S(R^{d})$ (resp. $S(W^{-1})$ ). Let $S(R^{d})$ (resp. In order to simplify the notations, in the equel, by the symbol $D$ we denote both and $d-1$. In each discussion we exactly explain the dimension (space-time or space) of the eld on which we are working. Now, suppose that on a complete probability space we are given an isonormal Gaussian process $B^{D}=\{B^{D}(h), h\in L^{2}(R^{D} ; \lambda^{D})\}$ , where denotes the Lebesgue measure on (cf., e.g., [HKPS], $[SiSi]$ , [AY1,2] and references therein). Precisely, is a centered Gaussian family of random variables such that

$\mathbb{R}^{d}$

$R^{d-1}$

$S(\mathbb{R}^{d-1})$

$(\zeta l,\mathcal{F},P)$

$\lambda^{D}$

$\mathbb{R}^{D}$

$B^{D}$

$E[B^{D}(h)B^{D}( 9)|=\int_{R^{p}}$
(x) $g(x)\lambda^{D}(\ )$ ,
$g\in L^{2}(R^{D} ; \lambda^{D})$

We write

$B_{\omega}^{D}($
$)= \int_{R^{D}}h(y)\dot{B}_{w}^{D}(y)dy$

$\omega\in\zeta$

Inst. Angewandte MathemaLik, Universit\"at Bonn, Wegelerstr. 6, D-53115 Bonn (Germany), SFB611; $BiBoS$ ; CERFIM, Locarno; Acc. ArchI USI, MendrisIo; MathemaSica, $Unive\iota 8iR$ di ltento -mail wyoshIdaGipcku.kan8ai-u.ac.jp $fax+816$ 63303770. Kansai Univ., Depl. Mathematics, 564-8680 Yamate-Tyou 3-&35 Suita Osaka(Japan)
$lel\ovalbox{\tt\small REJECT} ura$ $I\epsilon l$

$\uparrow e$

Namely, is the Gaussian white noise on in the framework of Hida calculus (cf., e.g., [HKPS], $[SiSi]$ ). By the framework of the calculus on the abstract Wiener spaces, $\dot{B}(y)dy$ is written by (cf., [Nu], [AFY]). As far as the discussions in section 1, each analysis given here is not singular (cf. (1.6) and (1.7)), and we may use the notation of the calculus on the abstract Wiener spaces, but to make the symbol clear and for the discussions in section 2 (going into the more singular discussions) we prefer to use the notations of white noise analysis. We are considering a massive scalar eld and suppose that we are given a mass $m>0$. Let and resp. be the , resp. $d-1$ , dimensional Laplace operator, and dene the pseudo dierential operators and as follows:

$\dot{B}_{w}^{D}(\cdot)$

$W_{w}^{D}(dy)$

$\Delta_{d}$

$\Delta_{d-1}$

$L_{-r}1$

$H_{-\pi}\iota$
$L_{-\#}=(-\Delta_{d}+m^{2})^{-\int}$

(1.2) (1.3)

$H_{-:}=(-\Delta_{d-1}+m^{2})^{-i}$
same symbols as and we also denote the integral kernels of the corresponding pseudo dierential operators, i.e., the Fourier inver se transforms of the corresponding symbols of the pseudo dierential operators. By making use of stochastic integral expressions, we dene two extremely important random elds the Nelsons Euclidean free eld, and $h$ , the sharp time free eld, as follows:

By the

$L_{-:}$

$H_{-\tau}1$

$\phi_{N_{i}}$

$d\geq 2$

$\phi_{N}(\cdot)\equiv\int_{R^{d}}L_{-\}}(x-\cdot)\dot{B}^{d}(x)dx$
$(\cdot)\equiv\prime_{R^{d-1}}H_{-}$
$(\tilde{x}-\cdot)\dot{B}^{d-1}(\tilde{x})$
These denitions of and resp. seems formal, but they are rigorously dened $S(W)$ and resP. $S(R^{d-1})$ valued random variables through a limiting Procedure as (cf. $[AY1,2]$ ), more precisly it has been shown that

$\phi_{N}$

$\phi_{0}$
$P(\phi_{N}(\cdot)\in B_{d}^{a,b})=1$

$a^{l},$

such that
$\min(1, \frac{2a}{d})+\frac{2}{d}>1,$

$b>d$

$P(h\in B_{d-1}^{a,b})=1$
$\min(1, \frac{2a}{d-1})+\frac{1}{d-1}>1,$

$b>d-1$

$(1.7)$

Here for each

$S^{J}(R^{D})$
$D>0$ , the Hilbert spaces , is dened by

$a,$ $b,$

$B_{d}^{a_{:}b}$
, which is a linear subspace of

$B_{d}^{a,b}$

$B_{D}^{a,b}=\{(|x|^{2}+1)^{A}4(-\Delta_{D}+1)^{+\S}f:f\in L^{2}(R^{D} ; \lambda^{D})\}$
where $x\in R^{D}$ and is given by

$<u|v>$

$\lambda$
denotes the Lebesgue measure on IR, the scalar product of
$\int_{R^{D}}\{(-\Delta_{D}+1)^{3}((1+|x|^{2})^{-\eta}bu(x))\}$
$\{(-\Delta_{D}+1)^{g}((1+|x|^{2})^{-*}v(x))\}dx$

$u,$ $v\in B_{D}^{a,b}$

eld on IR4 (cf. (1.6)).
Let be the probability measure on $S(R^{d-1}arrow R)$ which is the probability law (cf. (1.7)), and of the sharp time free eld on be the probability which is the probability law of the Nelsons Euclidean free measure on

$\mu_{0}$

$(\Omega,\mathcal{F}, P)$

$\mu_{N}$

$S(R^{d}arrow \mathbb{R})$

We denote

$\phi_{0}(\varphi)\equiv<\phi_{0},$ $\varphi>\equiv\int_{\mathbb{R}^{d-1}}(H_{-4\iota\varphi})(\vec{x})\dot{B}^{d-1}(\vec{x})d\vec{x}$
$\phi_{0}(\varphi_{1})\cdots\phi_{0}(\varphi_{n})$
$= \int_{X^{k(i-1)}}H_{-:}\varphi_{1}(\tilde{x}_{1})\cdots H_{-4A}\varphi_{1}(\vec{x}_{k}):\dot{B}^{d-1}(\tilde{x}_{1})\cdots\dot{B}^{d-1}(\tilde{x}_{k})$

x&\sim l

$\dot{B}^{d-1}$
$d \tilde{x}_{k}\in\bigcap_{q\geq 1}L^{q}(\mu_{0})$
$\varphi,\varphi_{j}\in S(R^{d-1}arrow \mathbb{R})$

$j=1,$ $\cdots k$ ,

$k\in N$

(1.10)

where (1.10) is the k-th multiple stochastic integral with respect to the Gaussian white noise. on ; is nothing more than an element of the n-th Wiener Since, ; , it also adomits an expression by means of the Hermite polynomial chaos of $j=1,$ , (cf., e.g., [AY1,2] and references therein). of

$L^{2}(\mu_{0})$

$\phi_{0}(\varphi_{j}),$

$\cdots$

Remark 1. From the view point of the notational rigorousness, are the distribution valued random variables on the probability space (S), the notation such as :
$h( \varphi_{1})\cdots\phi_{0}(\varphi_{n}):\in\bigcap_{q\succeq 1}L^{q}(\mu_{0})$

$\mathcal{F},$

and $P$ ), hence
is incorrect. However in the above and in the sequel, sinoe there is no aimbiguity, for the simplicity of the notations we use the notations (with an obvious and interpretation) to indicate the measurable functions $X$ and resp. on the measure $(S(R^{d}),\mu_{N},\mathcal{B}(S(M)))$ such that spaces and resp.
$(S(R^{d-1}),\mu 0,\mathcal{B}(S(\mathbb{R}^{d-1})))$

$P(\{\omega :

\phi_{0}(\omega)\in A\})=$

$(\{\phi :

X(\phi)\in A\})$
$A\in \mathcal{B}(S(R^{d-1}))$
\phi_{N}(\omega)\in A\})=\mu_{N}(\{\phi :

Y(\phi)\in A\})$

$A\in \mathcal{B}(S(\mathbb{R}^{d}))$

respectively, where

$B(S)$
denotes the Borel -eld of the topological space

$\sigma$

$H\#\equiv(-\Delta_{d-1}+m^{2})\}$.

and dene the operator

(\varphi 1).

$d\Gamma(H_{1,\tau})$

(for the notations cf. Remark 1.)

$(\varphi_{n})$

$(H)(:h(\varphi_{1})\cdots\phi_{0}(\varphi_{n}):)=:$

$.+$ ;

$(H\varphi_{1})h(\varphi_{2})\cdots$

$:+\cdots$

$h(\varphi_{n-1})h(H_{8}\varphi_{k})$
We state the weU known fundamental structures on the $bee$ eld with the space time dimension as follows:
Proposition 1.1 T operator with the natural domain an on essentially self adjoint non negative operator, and \it is a generator of the Markoman } , ($i.e$. satisfying the properties of positivity se migroup, denoted by and hypercontractive contraction $(q\in[0_{i}\infty]))$ , moreover

$d\Gamma(H_{i})$

$\dot{u}$
$T_{\ell}\equiv e^{-td\Gamma(H)}$

$pre\delta en\dot{n}ng$

$L^{q}(\mu_{0})$

$T_{t}i\epsilon$

The operator on the sharp time he eld in such a way th at Nelson Euclidean free eld

$T_{t}\phi_{0}(\varphi)$

identied

with the

$\int_{S(R^{d-1}arrow R)}$
$T_{t_{1}}((T_{t_{2}}<\cdot,\varphi_{2}>s,s)(\cdot)<\cdot,$ $\varphi_{1}>S,S)(\phi)\mu o(d\phi)$
$E^{\mu N}[<\phi,\varphi_{1}\cross\delta_{\{t_{1}\}}(\cdot)><\phi,\varphi_{2}x\delta_{\{t_{1}+t_{2}\}}(\cdot)>]$
The operator on the sharp time ee eld lation gropup on the Wightman ftee eld, and

$e^{-itd\Gamma(H}*$

$e^{itd\Gamma(H)}\phi_{0}(\varphi)e^{-:td\Gamma(H)}$

is the time trans(1.14)

are the eld operators on the jfhee eld with the space time dimension.

Denition of Hida product

$d\in N$

$(d\geq 2)$

be a given space time dimension, and and be the corresponding sharp time free eld and Nelsons Euclidean free eld dened by (1.5) and (1.4) respectively, and respectively. and be the probability laws of and , let For real

$\gamma$

$H_{-\gamma}\equiv(-\Delta_{d-1}+m^{2})^{-\gamma}$

For $r\in N$, let

$\Lambda_{r,d-1}\in C_{0}^{\infty}(R^{d-1}arrow \mathbb{R}_{\vdash})$
be a given function such that
$0\leq\Lambda_{r,d-1}(\tilde{x})\leq 1(\vec{x}\in\Psi^{-1})$
$\Lambda_{r,d-1}\equiv 1(|xarrow|\leq r)$
$\Lambda_{r,d-1}\equiv 0(|\tilde{x}|\geq r+1)$
for $p\in N$ dene a Hida distribution
$\bigcap_{q\geq 1}L^{q}(\mu 0)$
$<:\phi_{0}^{2p}:,\Lambda_{r,d-1}>on$

as fonows:

the space the test functions,
$<:\phi_{0}^{2p}:,\Lambda_{r,d-1}>$
$\equiv\int_{(R^{d-1})^{2p}}\{\int_{R^{d-1}},arrow\}$
$\dot{B}^{d-1}(\vec{x}_{1})\cdots\dot{B}^{d-1}(i_{2p}):d\vec{x}_{1}\cdots d\vec{x}_{2p}$
here, all the way of using notations follow the rule given by Remark 1. For $d=2$ $(d-1=1)$ we know that
$<: \phi_{0}^{2p}:,\Lambda_{r,1}>\in\bigcap_{q\geq 1}L^{q}(\mu_{0})$
But our main interest is concentrated on the case whered $=4$ $(d-1=3)$ , and in :, not a random variable any more, but a Hida distribution. this case In the sequel, if there is no indication of the dimension in each discussion, then we should understand that the consideration is canied out on $d=4,$ $d-1=3$. :, Let us dene a new multiplication between two Hida distributions $A_{r.3}>$. We denote this new multiplication procedure as Hida product. :, and It produces one Hida distribution from another two, and the resulting distributions are dierent from the ones derived through the well known S-transform and others. Hence, equipping this multiplication, the space of Hida distribution has the structure of ring.

$<:\phi_{0}^{2p}$

$\Lambda_{r,3}>is$

$\Lambda_{r.3}>$

We have to stress that the Hida distributions generated through this new production procedure are much more complicated than the ones given by the known multiplication procedures, but they are much more fruitful, in fact by these distributions (operators) we may dene the non-trivial interactions on the 4-dimensional space time quantum eld.
be Denition. (Hida produ ) For the space time dimension $d=4$ , let : $j\Lambda_{r,S}>be$ sharp time free eld with $d-1=3$. Let the Hida distribution the
$the2<:\phi_{0^{p}\prime}r_{:.\Lambda_{r,3}>\cross<:\phi_{0}^{2p}:,\Lambda_{r,3}>isdefinedasadistributiononthespaceofthe}^{thWickwerofthesharptimeheefie1ddefinedby(2.3).TheHidapr\sigma duct}po_{\mathcal{H}}$

test functions

$n_{q\geq 1}L^{q}(\mu_{0})$

as follows:

$\Lambda_{r,3}>x<:\phi_{0}^{2p}:,$

$\Lambda_{r,3}>$

$<:\phi_{0}^{2p}:,\Lambda_{r,3}>\cross^{\mathcal{H}}<:\phi_{0}^{2p}:,$ $\Lambda_{r,S}>$
$=<:\phi_{\mathfrak{v}}^{2p}:,$

(all the terms that

are not Hida distributions),

explicitly

$<:\phi_{0}^{2p}:,\Lambda_{r,3>}x^{\mathcal{H}}<:\phi_{0}^{2p}.,$ $\Lambda_{r,3}>$
$\equiv\int_{(R^{S})^{4p}}\{\int_{R^{g}}\iota\sim\}\{\sim y^{\simarrow\}}$
$\dot{B}^{3}(\tilde{x}_{1})\cdots\dot{B}^{3}(\vec{x}_{2p})\dot{B}^{3}(x_{1}^{\tilde{\prime}})\cdots\dot{B}^{3}(x_{2p}^{\tilde{\prime}}):d\tilde{x}_{1}\cdots d\tilde{x}_{2p}dx_{1}^{\tilde{\prime}}\cdots dx_{2p}^{\tilde{\prime}}$
+16 $\int_{(R^{8})^{p-2}}[\int_{R^{\theta}xR\}\{\wedgearrow$
$xH_{-\#}(\tilde{y}-y)dyarrow dy^{\tilde{\prime}]}\sim$
$\dot{B}^{3}(\tilde{x}_{1})\cdots\dot{B}^{3}(\tilde{x}_{2p-1})\dot{B}^{3}(x_{1}^{\tilde{\prime}})\cdots\dot{B}^{3}(x_{2p-1}^{\tilde{\prime}})$
$\cross d\tilde{x}_{1}\cdots d\tilde{x}_{2p-1}dx_{1}^{\vec{\prime}}\cdots\tilde{M}_{2p-1}$
i) To get a production between two. if we use Remark 2. the S-transform, then we may only have the rst term of (2.4) and do not have the second term of (2.4).
$<:\phi_{0}^{2p}:,\Lambda_{r.S}>$
For the 6-th power the corresponding Hida product, , involves much complicated terms. In particular, it denoted by possesses the following important term having a hexagonal form:
$(<:\phi_{0}^{2p} :, \Lambda_{r,3}>)_{i}^{6}$
$(<:\phi_{0}^{2p} :, \Lambda_{r,3}>)_{H}^{6}$
$\int_{(R)}x2p-12[\int_{(R^{\theta})^{6}}\{\Lambda_{r,3}(y_{1}^{arrow})\prod_{k=1}^{2p-2}H_{-41}(y_{1}^{arrow}-\tilde{x}_{1,k})\}$
$xH_{-;}(y_{1}^{\sim}-y_{2}^{\sim})\{\Lambda_{r,3}(y_{2}^{\sim})\prod_{k=1}^{2p-2}H_{-4A}(y_{2}^{\sim}-\tilde{x}_{2,k})\}$
$xH_{-\}}(y_{2}^{\sim}-y_{3}^{\vee})\{A,,3(y_{3}^{\vee})\prod_{k=1}^{2p-2}H_{-:}(y_{3}^{\sim}-\vec{x}_{3,k})\}x\cdots$
$xH_{-;}(y_{6}^{arrow}-y_{1}^{\vee})\{\Lambda_{r,3}(y_{6}^{\sim})\prod_{k=1}^{2p-2}H_{-A,4}$

$y_{6}^{arrow}$

,k)}dy\tilde l.

$dy_{6}^{arrow]}$

$\dot{B}^{3}(\tilde{x}_{1,1})\cdots\dot{B}^{3}(\tilde{x}_{6,2p-2})$
$d\vec{x}_{1,1}\cdots d\vec{x}_{6,2p-2}$
By using the Hida product we can dene the power series of Hida distributions, in we can set
$(e^{-\lambda<:\phi_{0}^{lp}:,\Lambda_{r,\theta>}})_{H} \equiv\sum_{n=0}^{\infty}\frac{1}{n!}(-\lambda<:\phi_{0}^{2p}., \Lambda_{r,3}>)_{\mathcal{H}}^{n}$

particular

$\lambda\in C$
By modifying the usual multiplication procedures of the random Remark 3. variables on the sharp time eld to the Hida product, we can dene the operator (cf.

(1.14))

$(e^{:t<:\phi_{0}^{2p}:,\Lambda_{r.\}>})_{\mathcal{H}}e^{itd\Gamma(H)}\phi_{0}(\varphi)e^{-itd\Gamma(H)}$
, which has the structure of the ring. with the domain [GrotS] debes the operators which are dened by making use of the S-transform, and [AGW] constructs the operators through the comvolution of pseudo dierential opetators with generalized white noises, and they howd that these operators do not include the non trivial interactions. $[NaMu]$ introduces an anther important approach of the construction of the eld by choosing special test functions that are not. For $d=4$ , every existing known result on the eld operator with non trivial , is a statement that the eld operator can not be really interaction, in particular an operator but a form on a Hilbert space and henoe it does not admit an operation of productions.
$\bigcap_{q>1}L^{q}(\mu_{0})$

$\Phi_{4}^{4}$

References
[AFY] Albeverio, S., Ferrario, B., Yoshida, $M$ I.W.: On the essential self-adjointness of Wick powers of relativistic elds and of elds unitary equivalent to random elds. Acta Applicande Mathematicae, 80, 309-334 (2004).
[AGW] Albeverio, S., Gortschalk, H., Wu, J.-L.: Convoluted generalized white noise, Schwinger functions and their analytic continuation to Wightman functions. Rev. Math. Phys. 8 (1996), 763-817. [AY1]
S. Albeverio, M.W. Yoshida: Multiple Stochastic Integral Construction of non-Gaussian Reection Positive Generalized Random Fields, SFB 611 PrePrint, 241, 2006.
Albeverio, S., Yoshida, M. W.: $H-C^{1}$ maps and elliptic SPDEs with polynomial and exponential perturbations of Nelsons Euclidean free eld. J. thnctional Analysis, 196, 265-322 (2002). L. Gross: Logarithmic Sobolev inequalities and contractive properties of semi, Springer-Verlag, Berlin, 1993. groups, in Lecture Notes in Math ematics

$15\theta S$

[GrotS] Grothaus, M., Streit, L.: Construction of relativistic quantum elds in the framework of white noise analysis. J. Math. Phys. 40 (1999), 5387-5405.
[HKPS] Hida, T., Kuo, H.-K., Pottho, J., Streit, L.: White Nois : An Innite $D;men8iond$ Calculus. Kluwer Academic Publishers, Dordrecht, 1993.

$[NaMu]$

Comm. Math. Phys. 46 (1976), no. 2, 119-134.

$[SiSi]$

Nagamachi, S., Mugibayashi, N.:
Hyperfunction quantum eld theory.
Malliavin calculus and related topics. Springer-Verlag, New $York/Heidelberg/Berlin$ , 1995.

Nualart, D.: T

Si Si: Poisson noise, innite symmetric group an stochastic integrals based. in The Fifth L\evy Seminar, Dec., 2006. on:

$\dot{B}(t)^{2}:$

Simon, B.: The Euclidean (Quantum) Field Theory, Princeton Univ. Press, Princeton, NJ., 1974.

$P(\Phi)_{2}$

 

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