The book of nature is written in the language of mathematics
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Transcript of The book of nature is written in the language of mathematics
The book of nature is written in
the language of mathematicsGalileo Galilei
1. Introduction
2. Basic operations and functions
3. Matrix algebra
4. Handling a changing world
5. First steps in statistics
6. Moments and distributions
7. Parametric hypothesis testing
8. Correlation and linear regression
Our program
In this lecture we will apply basic mathematics and statistics to solve ecological problems.
The lecture is therefore application centred.
Students have to prepare the theoretical background by their own!!!
For each lecture I’ll give the concepts and key phrases to get acquainted with together with the appropriate literature!!!
This literature will be part of the final exam!!!
Mathe onlinehttp://www.mathe-online.at/
http://tutorial.math.lamar.edu/
Additional sources
Logarithms and logarithmic functions
A logarithm is that number with which we have to take another number (the
base) to the power to get a third number.
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
0 1 2 3 4 5 6
Asymptote
Root
The logarithmic function
The logarithmic function is not defined for negative values
Log 1 = 0
1)(log
0)1log(
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Logarithms and logarithmic functions
dcbxay )ln(A general logarithmic function
-1
-0.5
0
0.5
1
1.5
2
0 1 2 3 4
y
x
4)53ln(2 xyShift at x-axis
Shift at y-axis
Increase
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53
)53ln(24
2
2
ex
xe
xRoot
Curvature
00.5
11.5
22.5
3
0 1 2 3 4
y
x
The number e
0
32
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ix
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0 !1...
!31
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i ie
n
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11lim
e = 2.71828183….
n
n ne
11lim
05
10152025303540
-6 -4 -2 0 2 4
y
x
e
y=ex
Leonhard Paul Euler (1707-1783)
01ieThe famous Euler equation
The commonly used bases
Logarithms to base 10Logarithms to base 2 Logarithms to base e
Log10 x ≡ lg xLog2 x ≡ lb x Loge x ≡ ln x
Digital logarithmBinary logarithm Natural logarithm
1 byte = 32 bit = 25 bit
232 = 4294967296
1 byte = lb( number of possible elements)
Classical metrics pHDeziBel
The scientific standardStandard of softwarePublicationsStatistics
Weber Fechner law
Sensorical perception of bright, loudness, taste, feeling, and others increase proportional to the
logarithm of the magnitude of the stimulus.
CckcckE
loglog
0
Logarithmic function
05
101520253035
0 10 20 30
Effec
t E
Magnitude of c
kccE 0The power function law of
Stevens approaches the Weber-Fechner law at k = 0.33
Stevens’ power law
33.05.9 cE
ccE log201
log20 10
Power functions and logarithmic functions are sometimes very
similar.
Human brightless perception
0102
0
2
10 log20log10][PP
PPdBL
Loudness in dezibel
Dezibel is a ratio and therefore dimensionless
P: sound pressure
The rule of 20.
The magnitude of a sound is proportional to the square of sound pressure
The threshold of hearing is at 2x10-5 Pascal. This is by definition 0 dB. What is the sound pressure at normal talking (40 dB)?
PaxP
PP
3
1010510
102
52log2log10*2
log2040
0
50
100
150
200
0.00001 0.001 0.1 10 1000
dB
Magnitude of P [Pa]
x100
+40
40100log20][ 10 dBL
Logarithmic scale
Line
ar s
cale
The sound pressure is 100 times the threshold pressure.
How much louder do we hear a machine that increases its sound pressure by a factor of 1000?
601000log20][
log201000log20][
10
010
010
PPdBL
PP
PPdBL
The machine appears to be 60 dB louder
To what level should the sound pressure increase to hear a sound 2 times louder?
510
510
10*2log20
10*2log20
2P
kP
55
2
5
510510
10*210*210*2
10*2log
10*2log2
PkkPP
kPP 010203040506070
0 0.0005 0.001 0.0015
k
PThe multiplication factor k is linearly (directly) proportional to the sound
pressure P.
0
1
2
3
4
5
6
0 5 10 15 20 25 30 35 40 45 50 55 60 65
Time
Pre
dato
r ab
unda
nce
Magicicada septendecimPhoto by USA National Arboretum
A B C D E
1 Generation Predator A Predator B Predator CSum of predator densities
2 0 1 1.5 2 4.53 1 0.5 0.75 1 2.254 2 1 0.75 1 2.755 3 0.5 1.5 1 36 4 1 0.75 2 3.757 5 0.5 0.75 1 2.258 +A7+1 +B6 +C5 +D4 +SUMA(B8:D8)
A first model
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Time
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A B C D E1 Generation Predator A Predator B Predator C Sum2 1 =2*LOS() =3*LOS() =4*LOS() =SUMA(M34:O34)3 =A2+1 =B2*LOS() =C2*LOS() =D2*LOS() =SUMA(M35:O35)4 =A3+1 =2*LOS() =C2*LOS() =D2*LOS() =SUMA(M36:O36)5 =A4+1 =B2*LOS() =3*LOS() =D2*LOS() =SUMA(M37:O37)6 =A5+1 =2*LOS() =C2*LOS() =4*LOS() =SUMA(M38:O38)
Magicicada septendecimPhoto by USA National Arboretum
Alpha Beta Gamma Delta Epsilon Zeta Eta
Theta Jota Kappa Lambda My Ny Xi Omikron Pi Rho
Sigma Tau Ypsilon Phi Chi Psi Omega
Home work and literature
Refresh:
• Greek alphabet• Logarithms, powers and roots: http://en.wikipedia.org/wiki/Logarithm• Logarithmic transformations and scales• Euler number (value, series and limes expression)• Radioactive decay
Prepare to the next lecture:
• Logarithmic functions • Power functions• Linear and algebraic functions• Exponential functions• Monod functions• Hyperbola