Transcript Introduction

The book of nature is written in
the language of mathematics
Galileo Galilei
Our program
In this lecture we will apply
basic mathematics and
statistics to solve ecological
problems.
1. Introduction
The lecture is therefore
application centred.
2. Basic operations and functions
Students have to prepare the
theoretical background by
their own!!!
4. Handling a changing world
For each lecture I’ll give the
concepts and key phrases to
get acquainted with together
with the appropriate
literature!!!
7. Parametric hypothesis testing
This literature will be part of
the final exam!!!
3. Matrix algebra
5. First steps in statistics
6. Moments and distributions
8. Correlation and linear regression
Mathe online
http://www.mathe-online.at/
http://tutorial.math.lamar.edu/
Additional sources
John Napier
(1550-1617)
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.
The logarithmic function
y  a x  x  log a y
2
1.5
1
0.5
Root
0
-0.5
-1
0
1
2
3
4
5
Log 1 = 0
-1.5
-2
-2.5
Asymptote
The logarithmic function is not defined for
negative values
6
a loga y  y
1
 loga y
a

y
log( 1)  0
and
log a (a)  1
Logarithms and logarithmic functions
xy  z
y  a ln( bx  c)  d
a log x a log y  a log z
a loga x  loga y  a loga z
log x  log y  log z
1.5
y
1
x/ y  z
a
loga x
/a
loga y
a
loga x
a
 loga y
a
a
log x  log y  log z
loga z
xy  z
a

0.5
0
loga x  loga y
loga x y
A general logarithmic function
2
a
loga z
-0.5 0
-1
Curvature
y log x  log z
2
3
x
y  2 ln( 3x  5)  4
Shift at
y-axis
Increase Shift at
x-axis
Root
 a y loga x  a loga z
1
4  2 ln( 3 x  5)
e 2  3x  5
e2  5
x
 0.7964
3
4
Leonhard Paul
Euler
(1707-1783)
The number e
40
35
30
25
20
15
10
5
0
The famous Euler
equation
y
e i  1  0
2
3
i

x
x
x
x
ex  1 
 ...  
1! 2! 3!
i 0 i!
-6
-4
-2
y=ex
0
2
4
x

1 1 1
1
e  1    ...  
1! 2! 3!
i  0 i!
e
3
2.5
 1
e  lim n 1  
 n
n
e = 2.71828183….
y
2
1.5
 1
e  lim n 1  
 n
1
0.5
n
0
0
1
2
x
3
4
The commonly used bases
Logarithms to base 2
Logarithms to base 10
Logarithms to base e
Log2 x ≡ lb x
Log10 x ≡ lg x
Loge x ≡ ln x
Binary logarithm
Digital logarithm
Natural logarithm
1 byte = 32 bit = 25 bit
Classical metrics
pH
DeziBel
The scientific standard
Standard of software
Publications
Statistics
232 = 4294967296
1 byte = lb( number of
possible elements)
Weber Fechner law
Human brightless perception
Sensorical perception of bright, loudness, taste,
feeling, and others increase proportional to the
logarithm of the magnitude of the stimulus.
c
E  k log    k log c  C
 c0 
Logarithmic function
Effect E
Stevens’ power law
35
30
25
20
15
10
5
0
E  c0 c k
E  9.5c 0.33
The power function law of
Stevens approaches the WeberFechner law at k = 0.33
c
E  20 log 10    20 log c
1
0
10
20
Magnitude of c
30
Power functions and logarithmic
functions are sometimes very
similar.
Loudness in dezibel
The magnitude of a sound is proportional to the square of sound pressure
Dezibel is a ratio and therefore
dimensionless
P: sound pressure
200
L[dB]  20 log 10 100  40
150
dB
The rule of 20.



Linear scale
 P2 
 P
L[dB]  10 log 10  2   20 log 10 
 P0 
 P0
100
+40
50
x100
0
0.00001 0.001
0.1
10
1000
Magnitude of P [Pa]
Logarithmic scale
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)?
 P 
40  20 log 10 
 log 10 P  2  log 10 2  5
5 
 2 *10 
P  2 x10 3 Pa
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?
 1000 P 
P
  20 log10  
L[dB]  20 log10 
 P0 
 P0 
 1000 P 
L[dB]  20 log10 
  60 The machine appears to be 60 dB louder
 P 
 kP 
20 log10 
5 
2
*
10


2
 P 
20 log10 
5 
2
*
10


2 log 10
P
kP

log
10
2 *10 5
2 *10 5
2
kP
P
 P 


k



5
2 *10 5
2 *10 5
 2 *10 
k
To what level should the sound pressure increase to hear a sound 2 times louder?
70
60
50
40
30
20
10
0
0
0.0005
0.001
0.0015
P
The multiplication factor k is linearly
(directly) proportional to the sound
pressure P.
A first model
Magicicada septendecim
A
B
C
D
1
Generation
Predator A
Predator B
Predator C
2
3
4
5
6
7
8
0
1
2
3
4
5
+A7+1
1
0.5
1
0.5
1
0.5
+B6
1.5
0.75
0.75
1.5
0.75
0.75
+C5
2
1
1
1
2
1
+D4
E
Sum of
predator
densities
4.5
2.25
2.75
3
3.75
2.25
+SUMA(B8:D8)
Photo by USA National Arboretum
Predator abundance
6
5
4
3
2
1
0
0
5
10
15
20
25
30
35
Time
40
45
50
55
60
65
A
1 Generation
2
1
3 =A2+1
4 =A3+1
5 =A4+1
6 =A5+1
B
Predator A
=2*LOS()
=B2*LOS()
=2*LOS()
=B2*LOS()
=2*LOS()
25
35
C
Predator B
=3*LOS()
=C2*LOS()
=C2*LOS()
=3*LOS()
=C2*LOS()
D
Predator C
=4*LOS()
=D2*LOS()
=D2*LOS()
=D2*LOS()
=4*LOS()
E
Sum
=SUMA(M34:O34)
=SUMA(M35:O35)
=SUMA(M36:O36)
=SUMA(M37:O37)
=SUMA(M38:O38)
Magicicada septendecim
Predator abundance
Photo by USA National Arboretum
3
2
1
0
0
5
10
15
20
30
Time
40
45
50
55
60
65


















































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:
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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:
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Logarithmic functions
Power functions
Linear and algebraic functions
Exponential functions
Monod functions
Hyperbola