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Maths

[edit]

The following example uses the bar notation to calculate 0.012 × 0.85 = 0.0102:

* This step makes the mantissa between 0 and 1, so that its antilog (10mantissa) can be looked up.

When X has standard deviation ? = 0.25, the distribution of Y is weakly skewed. Using formulas for the log-normal distribution, we find:

Indeed, the median is about one third on the way from mean to mode.

When X has a larger standard deviation, ? = 1, the distribution of Y is strongly skewed. Now

Here, Pearson's rule of thumb fails.

Ratio of volumes of a cone, sphere and cylinder of the same radius and height

[edit]
A cone, sphere and cylinder of radius r and height h.

The above formulas can be used to show that the volumes of a cone, sphere and cylinder of the same radius and height are in the ratio 1 : 2 : 3, as follows.

Let the radius be r and the height be h (which is 2r for the sphere).

The discovery of the 2 : 3 ratio of the volumes of the sphere and cylinder is credited to Archimedes.[1]

Ratio of surface areas of a sphere and cylinder of the same radius and height

[edit]

The above formulas can be used to show that the volumes of a sphere and cylinder of the same radius and height are in the ratio 2 : 3, as follows.

Let the radius be r and the height be h (which is 2r for the sphere).

The discovery of this ratio is credited to Archimedes.[2]


Wikitext tables

[edit]
4:1:1       4:2:0       4:2:2       4:4:4
Y'CrCb                                
                               
= = = =
Y'                                
                               
+ + + +
1 2 3 4  J = 4 1 2 3 4  J = 4 1 2 3 4  J = 4 1 2 3 4  J = 4
(Cr, Cb) 1  a = 1 1 2  a = 2 1 2  a = 2 1 2 3 4  a = 4
1  b = 1      b = 0 1 2  b = 2 1 2 3 4  b = 4
¼ horizontal resolution,
full vertical resolution
½ horizontal resolution,
½ vertical resolution
½ horizontal resolution,
full vertical resolution
full horizontal resolution,
full vertical resolution
Phase transitions of matter ()
To
From
Solid Liquid Gas Plasma
Solid
Melting Sublimation
Liquid Freezing
Vaporization
Gas Deposition Condensation
Ionization
Plasma Recombination


Patients with bowel cancer
(as confirmed on endoscopy)
Total population (pop.)
= 2030
Condition positive Condition negative Prevalence
= (TP + FN) / pop.
= (20 + 10) / 2030
1.48%
Accuracy (ACC)
= (TP + TN) / pop.
= (20 + 1820) / 2030
90.64%
Fecal
occult
blood

screen
test
outcome
Test
outcome
positive
True positive (TP)
= 20
(2030 × 1.48% × 67%)
False positive (FP)
= 180
(2030 × (100% − 1.48%) × (100% − 91%))
Positive predictive value (PPV), precision
= TP / (TP + FP)
= 20 / (20 + 180)
= 10%
False discovery rate (FDR)
= FP / (TP + FP)
= 180 / (20 + 180)
= 90.0%
Test
outcome
negative
False negative (FN)
= 10
(2030 × 1.48% × (100% − 67%))
True negative (TN)
= 1820
(2030 × (100% − 1.48%) × 91%)
False omission rate (FOR)
= FN / (FN + TN)
= 10 / (10 + 1820)
0.55%
Negative predictive value (NPV)
= TN / (FN + TN)
= 1820 / (10 + 1820)
99.45%
True positive rate (TPR), recall, sensitivity
= TP / (TP + FN)
= 20 / (20 + 10)
66.7%
False positive rate (FPR), fall-out, probability of false alarm
= FP / (FP + TN)
= 180 / (180 + 1820)
= 9.0%
Positive likelihood ratio (LR+)
= TPR/FPR
= (20 / 30) / (180 / 2000)
7.41
Diagnostic odds ratio (DOR)
= LR+/LR−
20.2
F1 score
= 2 × precision × recall/precision + recall
0.174
False negative rate (FNR), miss rate
= FN / (TP + FN)
= 10 / (20 + 10)
33.3%
Specificity, selectivity, true negative rate (TNR)
= TN / (FP + TN)
= 1820 / (180 + 1820)
= 91%
Negative likelihood ratio (LR−)
= FNR/TNR
= (10 / 30) / (1820 / 2000)
0.366
Condition
(as determined by "Gold standard")
Condition Positive Condition Negative
Test
Outcome
Test
Outcome
Positive
True Positive False Positive
(Type I error)
Positive predictive value =
Σ True Positive
Σ Test Outcome Positive
Test
Outcome
Negative
False Negative
(Type II error)
True Negative Negative predictive value =
Σ True Negative
Σ Test Outcome Negative
Sensitivity =
Σ True Positive
Σ Condition Positive
Specificity =
Σ True Negative
Σ Condition Negative


LEGEND No food restrictions Food sale/trade restrictions / ran tests
No cases
  
  
Suspected cases
  
  
Known cases
  
  
Deaths
  
  
Exposure to the outbreak start from April 21, 2011 with permanent or temporary stay in Germany, consumption of a food acquired in Germany or close contact with a HUS case.[3]
Number of cases by country until 31 May 2011[4]
Country Deaths Confirmed cases Suspected cases
 Germany 17[5] 450 1 200
 United Kingdom 0 3 -
 Sweden 1 41 -
 Denmark 0 14 26
 Netherlands 0 2 -
 France 0 0 3[6]
 Spain 0 1[7] -
 Austria 0 - -
 Norway 0 - -
 Switzerland 0 2 -
 Finland 0 - 1
 Poland 0 1 -
 United States 0 3 -
Centripetal acceleration for some values of radius and magnitude of velocity
|v|


  r
1 m/s
3.6 km/h
2.2 mph
2 m/s
7.2 km/h
4.5 mph
5 m/s
18 km/h
11 mph
10 m/s
36 km/h
22 mph
20 m/s
72 km/h
45 mph
50 m/s
180 km/h
110 mph
100 m/s
360 km/h
220 mph
Slow walk Bicycle City car Aerobatics
10 cm
3.9 in
Laboratory
centrifuge
10 m/s²
1.0 g
40 m/s²
4.1 g
250 m/s²
25 g
1.0 km/s²
100 g
4.0 km/s²
410 g
25 km/s²
2500 g
100 km/s²
10000 g
20 cm
7.9 in
5.0 m/s²
0.51 g
20 m/s²
2.0 g
130 m/s²
13 g
500 m/s²
51 g
2.0 km/s²
200 g
13 km/s²
1300 g
50 km/s²
5100 g
50 cm
1.6 ft
2.0 m/s²
0.20 g
8.0 m/s²
0.82 g
50 m/s²
5.1 g
200 m/s²
20 g
800 m/s²
82 g
5.0 km/s²
510 g
20 km/s²
2000 g
1 m
3.3 ft
Playground
carousel
1.0 m/s²
0.10 g
4.0 m/s²
0.41 g
25 m/s²
2.5 g
100 m/s²
10 g
400 m/s²
41 g
2.5 km/s²
250 g
10 km/s²
1000 g
2 m
6.6 ft
500 mm/s²
0.051 g
2.0 m/s²
0.20 g
13 m/s²
1.3 g
50 m/s²
5.1 g
200 m/s²
20 g
1.3 km/s²
130 g
5.0 km/s²
510 g
5 m
16 ft
200 mm/s²
0.020 g
800 mm/s²
0.082 g
5.0 m/s²
0.51 g
20 m/s²
2.0 g
80 m/s²
8.2 g
500 m/s²
51 g
2.0 km/s²
200 g
10 m
33 ft
Roller-coaster
vertical loop
100 mm/s²
0.010 g
400 mm/s²
0.041 g
2.5 m/s²
0.25 g
10 m/s²
1.0 g
40 m/s²
4.1 g
250 m/s²
25 g
1.0 km/s²
100 g
20 m
66 ft
50 mm/s²
0.0051 g
200 mm/s²
0.020 g
1.3 m/s²
0.13 g
5.0 m/s²
0.51 g
20 m/s²
2 g
130 m/s²
13 g
500 m/s²
51 g
50 m
160 ft
20 mm/s²
0.0020 g
80 mm/s²
0.0082 g
500 mm/s²
0.051 g
2.0 m/s²
0.20 g
8.0 m/s²
0.82 g
50 m/s²
5.1 g
200 m/s²
20 g
100 m
330 ft
Freeway
on-ramp
10 mm/s²
0.0010 g
40 mm/s²
0.0041 g
250 mm/s²
0.025 g
1.0 m/s²
0.10 g
4.0 m/s²
0.41 g
25 m/s²
2.5 g
100 m/s²
10 g
200 m
660 ft
5.0 mm/s²
0.00051 g
20 mm/s²
0.0020 g
130 m/s²
0.013 g
500 mm/s²
0.051 g
2.0 m/s²
0.20 g
13 m/s²
1.3 g
50 m/s²
5.1 g
500 m
1600 ft
2.0 mm/s²
0.00020 g
8.0 mm/s²
0.00082 g
50 mm/s²
0.0051 g
200 mm/s²
0.020 g
800 mm/s²
0.082 g
5.0 m/s²
0.51 g
20 m/s²
2.0 g
1 km
3300 ft
High-speed
railway
1.0 mm/s²
0.00010 g
4.0 mm/s²
0.00041 g
25 mm/s²
0.0025 g
100 mm/s²
0.010 g
400 mm/s²
0.041 g
2.5 m/s²
0.25 g
10 m/s²
1.0 g
secure channel insecure channel
Alice Charles Bob
constructs 4 packets, each containing one bit of her message and a valid MAC
SerialBitMAC
11234
20890
30456
41678
adds 4 chaff packets with inverted bits and invalid MAC, shown in italics (chaffing)
SerialBitMAC
10321
11234
20890
21987
30456
31543
40765
41678
discards packets with invalid MAC to recover the message (winnowing)

In this example, Alice wishes to send the message "1001" to Bob. For simplicity, assume that all even MAC are valid and odd ones are invalid.
Ending Percentage
0 7.5
1 0.3
2 0.3
3 0.8
4 0.3
5 28.6
6 0.3
7 0.4
8 1.0
9 60.7
NarratorWife      
|          
  Father — Stepdaughter
      |
Narrator

Family tree showing how
the narrator of the song
is his own grandfather.

Moon Sun
At perigee
(nearest)
At apogee
(farthest)
At perihelion
(nearest)
At aphelion
(farthest)
Mean radius, r 1,737.10 kilometres
(1,079.38 miles)
696,000 kilometres
(432,000 miles)
Distance, d 363,104 km
(225,622 mi)
405,696 km
(252,088 mi)
147,098,070 km
(91,402,500 mi)
152,097,700 km
(94,509,100 mi)
Angular diameter,
2×arctan(r / d)
32' 54"
(0.5482°)
29' 26"
(0.4907°)
32' 32"
(0.5422°)
31' 28"
(0.5244°)
Apparent size
to scale
Rank in
descending order
1st 4th 2nd 3rd
Celestial body Angular diameter Relative size (10 pixels per arcsecond)
Sun 31.6′ – 32.7′ 28.7 – 29.7 times the maximum value for Venus (orange bar below)
Moon 29.3′ – 34.1′ 26.6 – 31.0 times the maximum value for Venus (orange bar below)
Venus 9.565″ – 66.012″

Jupiter 29.800″ – 50.115″

Saturn 14.991″ – 20.790″

Mars 3.492″ – 25.113″

Mercury 4.535″ – 13.019″

Uranus 3.340″ – 4.084″

Neptune 2.179″ – 2.373″

Ceres 0.330″ – 0.840″

Vesta 0.20" – 0.64"

Pluto 0.063″ – 0.115″

R Doradus 0.052″ – 0.062″

Betelgeuse 0.049″ – 0.060″

Eris 0.034" – 0.089″

Alphard 0.00909″
Alpha Centauri A 0.007″
Canopus 0.006″
Sirius 0.005936″
Altair 0.003″
Deneb 0.002″
Proxima Centauri 0.001″
Visible to
typical
human eye
Apparent
magnitude
Brightness
relative
to Vega
Number of stars
brighter than
apparent magnitude[8]
Yes -1 250% 1
0 100% 4
1 40% 15
2 16% 48
3 6.3% 171
4 2.5% 513
5 1.0% 1 602
6 0.40% 4 800
No 7 0.16% 14 000
8 0.063% 42 000
9 0.025% 121 000
10 0.010% 340 000
Apparent
magnitude
Brightness
relative to
magnitude 0
Example Apparent
magnitude
Brightness
relative to
magnitude 0
Example Apparent
magnitude
Brightness
relative to
magnitude 0
Example
-27 6.3×1010 Sun -7 630 SN 1006 supernova 13 6.3×10-6 3C 273 quasar
-26 2.5×1010 -6 250 International Space Station (max) 14 2.5×10-6 Pluto (max)
-25 1.0×1010 -5 100 Venus (max) 15 1.0×10-6
-24 4.0×109 -4 40 16 4.0×10-7 Charon (max)
-23 1.6×109 -3 16 Jupiter (max) 17 1.6×10-7
-22 6.3×108 -2 6.3 Mercury (max) 18 6.3×10-8
-21 2.5×108 -1 2.5 Sirius 19 2.5×10-8
-20 1.0×108 0 1.0 Vega 20 1.0×10-8
-19 4.0×107 1 0.40 Antares 21 4.0×10-9 Callirrhoe (small satellite of Jupiter)
-18 1.6×107 2 0.16 Polaris 22 1.6×10-9
-17 6.3×106 3 0.063 Andromeda galaxy 23 6.3×10-10
-16 2.5×106 4 0.025 Acubens 24 2.5×10-10
-15 1.0×106 5 0.010 Vesta asteroid (max) 25 1.0×10-10 Fenrir (small satellite of Saturn)
-14 4.0×105 6 4.0×10-3 typical limit of naked eye 26 4.0×10-11
-13 1.6×105 Full moon 7 1.6×10-3 Ceres (max) 27 1.6×10-11 visible light limit of 8m ground-based telescopes
-12 6.3×104 8 6.3×10-4 Neptune (max) 28 6.3×10-12
-11 2.5×104 9 2.5×10-4 29 2.5×10-12
-10 1.0×104 10 1.0×10-4 typical limit of 7x50 binoculars 30 1.0×10-12
-9 4.0×103 Iridium flare 11 4.0×10-5 31 4.0×10-13
-8 1.6×103 12 1.6×10-5 32 1.6×10-13 visible light limit of Hubble Space Telescope
Status Configuration Name Year built Deadweight tonnage IMO ship identification number Photograph
Retired Passenger (upper deck)
Vehicle (lower deck)[9]
Pulau Labuan 1971 139 7038408
In service Passenger (upper deck)
Vehicle (lower deck)
Pulau Rawa 1975 262 7343736
In service Passenger (upper deck)
Vehicle (lower deck)
Pulau Talang Talang 1975 262 7343748
In service Passenger (upper deck)
Vehicle (lower deck)
Pulau Undan 1975 262 7343724
In service Vehicle (both decks) Pulau Rimau 1980 100 7911076
In service Passenger (upper deck)
Vehicle (lower deck)
Pulau Angsa 1981 100 8010491
In service Vehicle (both decks) Pulau Kapas 1981 280 8101082
In service Vehicle (both decks) Pulau Payar 2002 440 9254393
In service Vehicle (both decks) Pulau Pinang 2002 440 9275244
Age
group
Estimated July 2010 population in millions[10]
Africa Asia Europe Latin America
and the
Caribbean
Northern
America
Oceania
0–4 155.3 155.3
 
360.4 360.4
 
39.67 39.67
 
53.83 53.83
 
23.54 23.54
 
3.079 3.079
 
5–9 136.2 136.2
 
354.5 354.5
 
37.01 37.01
 
55.52 55.52
 
22.66 22.66
 
2.875 2.875
 
10–14 120.2 120.2
 
364.8 364.8
 
37.32 37.32
 
55.12 55.12
 
21.72 21.72
 
2.844 2.844
 
15–19 108.1 108.1
 
374.6 374.6
 
42.54 42.54
 
54.11 54.11
 
23.93 23.93
 
2.829 2.829
 
20–24 97.21 97.21
 
379.4 379.4
 
51.28 51.28
 
52.08 52.08
 
24.00 24
 
2.838 2.838
 
25–29 83.89 83.89
 
343.8 343.8
 
53.33 53.33
 
49.47 49.47
 
24.42 24.42
 
2.721 2.721
 
30–34 69.28 69.28
 
310.5 310.5
 
52.96 52.96
 
45.52 45.52
 
22.17 22.17
 
2.497 2.497
 
35–39 55.60 55.6
 
315.4 315.4
 
52.88 52.88
 
41.34 41.34
 
22.71 22.71
 
2.594 2.594
 
40–44 45.15 45.15
 
297.5 297.5
 
53.46 53.46
 
37.64 37.64
 
23.03 23.03
 
2.382 2.382
 
45–49 37.82 37.82
 
248.4 248.4
 
55.15 55.15
 
34.33 34.33
 
25.74 25.74
 
2.362 2.362
 
50–54 31.68 31.68
 
212.2 212.2
 
53.08 53.08
 
28.68 28.68
 
24.91 24.91
 
2.128 2.128
 
55–59 25.68 25.68
 
188.5 188.5
 
48.30 48.3
 
23.72 23.72
 
21.72 21.72
 
1.862 1.862
 
60–64 20.11 20.11
 
135.1 135.1
 
41.80 41.8
 
18.07 18.07
 
18.62 18.62
 
1.676 1.676
 
65–69 14.78 14.78
 
100.0 100
 
31.57 31.57
 
13.88 13.88
 
13.72 13.72
 
1.227 1.227
 
70–74 10.43 10.43
 
78.13 78.13
 
32.85 32.85
 
10.66 10.66
 
10.25 10.25
 
0.939 0.939
 
75–79 6.367 6.367
 
53.71 53.71
 
24.09 24.09
 
7.564 7.564
 
8.223 8.223
 
0.702 0.702
 
80–84 3.100 3.1
 
29.75 29.75
 
17.89 17.89
 
4.870 4.87
 
6.755 6.755
 
0.564 0.564
 
85–89 1.036 1.036
 
12.70 12.7
 
9.446 9.446
 
2.460 2.46
 
4.201 4.201
 
0.318 0.318
 
90–94 0.229 0.229
 
3.882 3.882
 
2.689 2.689
 
0.921 0.921
 
1.714 1.714
 
0.120 0.12
 
95–99 0.030 0.03
 
0.775 0.775
 
0.795 0.795
 
0.253 0.253
 
0.425 0.425
 
0.031 0.031
 
100+ 0.003 0.003
 
0.090 0.09
 
0.089 0.089
 
0.044 0.044
 
0.063 0.063
 
0.004 0.004
 

References

[edit]
  1. ^ Rorres, Chris. "Tomb of Archimedes: Sources". Courant Institute of Mathematical Sciences. Retrieved 2007-01-02.
  2. ^ Rorres, Chris. "Tomb of Archimedes: Sources". Courant Institute of Mathematical Sciences. Retrieved 2007-01-02.
  3. ^ Case definition for HUS-cases associated with the outbreak in Germany
  4. ^ Noticia 20minutos.es
  5. ^ La OMS dice que la bacteria letal de Alemania pertenece a una cepa desconocida de 'E. coli'
  6. ^ Infected cucumbers: three suspicious cases in France. lemonde.fr. Retrieved on June 1st, 2011.
  7. ^ Hospitalizan a un hombre en San Sebastián por infección de 'E.coli'
  8. ^ "Magnitude". National Solar Observatory—Sacramento Peak. Archived from the original on 2008-02-06. Retrieved 2006-08-23.
  9. ^ "Off-limit zone held waiting folk: Captain", New Straits Times, 1 Mar 1989
  10. ^ File 1A: Total population (both sexes combined) by five-year age group, major area, region and country, annually for 1950-2010 (thousands)