### Some basic mathematics

```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
2. Basic operations and functions
3. Matrix algebra I
4. Matrix algebra II
5. Handling a changing world
The lecture is therefore
application centred.
6. The sum of infinities
7. Probabilities and distributions
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!!!
8. First steps in statistics
9. Moments and descriptive statistics
10. Important statistical distributions
11. Parametric hypothesis testing
12. Correlation and linear regression
13. Analysis of variance
14. Non-parametric testing
15. Cluster analysis
Older scripts
Modelling Biology
Modelling Biology
Basic Applications of Mathematics and
Statistics in the Biological Sciences
Basic Applications of Mathematics and
Statistics in the Biological Sciences
Part I: Mathematics
Script A
Introductory Course for Students of
Modelling Biology
Basic Applications of Mathematics and
Statistics in the Biological Sciences
Biology, Biotechnology and Environmental Protection
Part I: Mathematics
Werner Ulrich
Script B
Part II: Data Analysis and Statistics
Script A
Introductory Course for Students of
Biology, Biotechnology and Environmental Protection
Werner Ulrich
Introductory Course for Students of
Biology, Biotechnology and Environmental Protection
Werner Ulrich
UMK Toruń
2007
UMK Toruń
2007
www.uni.torun.pl/~ulrichw
UMK Torun
2007
Mathe online
http://www.mathe-online.at/
http://tutorial.math.lamar.edu/
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  loga 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
 log a y
a

y
log(1)  0
and
loga (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
a
log a x
/a
0.5
0
log a y
log a x  log a y
a
a
log a x
a
 log a y
log a z
a
log a z
-0.5 0
Curvature

2
3
x
y  2 ln(3x  5)  4
Shift at
y-axis
Increase Shift at
x-axis
xy  z
a
1
-1
log x  log y  log z
log a x y
A general logarithmic function
2
Root
 a y loga x  a loga z
y log x  log z
4  2 ln(3x  5)
e 2  3x  5
e2  5
x
 0.7964
3
4
ax  z  by
loga z  x  loga b y  y loga b
logb z  y  logb a x  x logb a
loga z  y loga b  logb z loga b
x  y loga b 
 x  x loga b logb a
y  x logb a 
1  loga b logb a
loga b 
What is the logarithm of base 2 of
59049 if the the logarithm of 59049 of
base 3 is 10?
x  10 log2 3  x  10*1.585  15.85
215.85  59049
1
logb a
loga ( x / y)  loga ( x)  loga ( y)   loga ( y / x)
loga ( y)
 c  loga ( y)  c loga ( x)  y  x c
loga ( x)
log2 3  1.585
log3 2  0.631
log2 3 * log3 2  1.585* 0.631 1
log 2
4
3
 0.415   log 2
3
4
Leonhard Paul
Euler
(1707-1783)
The number e
40
35
30
25
20
15
10
5
0
The famous Euler
equation
y
ei  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  limn 1  
 n
n
e = 2.71828183….
y
2
1.5
 1
e  limn 1  
 n
1
0.5
n
0
0
1
2
x
3
4
3
Logarithmic equations
ln(x  a)  b
2 x  ln(x  1)
ln(x  a)  b
2 x  ln(x  1)
e
ln( x  a )
e
e2 x  x  1
b
xa e  x e a
b
b
ln(ax)  ln(b)  c
2
1.5
1
x  -0.8
x=0
0.5
0
-2.5
-2
-1.5
-1
-0.5
-0.5
0
-1
Mixed equations often
do not have analytical
solutions.
ax  elnb c
x  elnbc / a
2.5
-1.5
Roots
y  ln(x  1)  1
y  ln(xe x 1 )  y  0
y  ln(x  1)  1  y  0
ln(x0  1)  1
ln(x0 e x0 1 )  0
x0 e x0 1  1
ln( x  1)
1
x
e
ln(x  1)
y
1 y  0
x
e
ln(x0  1)  e x0
x0  1  e
x0  1  e x0 1  1
powers are always  0
x0  1
no solut ionexist s
x0  e  1  2.718282.... 1
y  ln(xex1 )
y
0.5
1
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  c0c k
E  9.5c 0.33
The power function law of
Stevens approaches the WeberFechner law at k = 0.33
c
E  20 log10    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]  20log10 100  40
150
dB
The rule of 20.



Linear scale
 P2 
 P
L[dB]  10 log10  2   20 log10 
 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 log10 
 log10 P  2  log10 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 log10
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.
The mass effect in
physics, chemistry, biochemistry,
and ecology
Na   Cl   NaCl
[ Na  ][Cl  ]
K
[ NaCl]
pH is the negative log10 of H+ concentration.
The Arrhenius model assumes that
reaction speed is directly proportional to
the number of contacts an therefore the
number of reactive atomes.
[ H 3O  ][OH  ]
 1014
[ H 2O]
[H3O ][OH  ]  1014
pH   log10 ( H3O )
pH  pOH  14
10 ml of a solution of H2S has a pH of 5.
What is the concentration of OH- after adding 100 ml HCN of pH 8.
10* 5  100* 8  110pH  pH  7.(72)  pOH  6.(27)
What is the pH of 0.5mol*l-1 NaOH?
[ H 3O  ][OH  ]  1014  [ H 3O  ][m ol* l 1 ] * 0.5[m ol* l 1 ]  1014
[ H 3O  ] 
1
*1014  pH  (log10 2  14)  13.7
0.5
[HA]  [H2O]  [H3O ]  [ A ]  [H2O]
[ H 3O  ][ A ]
[ A ]
[ A ]

K
  log10 K   log10 [ H 3O ]  log10
 pK  pH  log10
[ HA]
[ HA]
[ HA]
Henderson Hasselbalch equation
What is the pH of 0.2 mol l-1 C2H5COOH (pK = 4.75) and 0.1 mol l-1 NAOH?
H3O  0.2CH3COO  0.1Na   0.1OH   0.1CH3COONa  0.1CH3COO  H2O
0.1
pH  4.75  log10 ( )  4.75
0.1
Living organisms are buffered systems
Blood is a CO2 – NaHCO3 buffer at pH 7.5
What is the pH after injection of 100 ml 0.8mol*l-1 CH3COOH.
H3O  NaCO3  NaHCO3  H2O
[ H 3O  ][NaCO3 ]
107.5 (1  0.1* 0.8)
7.5

 10  [ H 3O ] 
 pH  7.43
[ NaHCO3 ]
1.  0.1* 0.8
A first model
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)
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:
•
•
•
•
•
Greek alphabet
Logarithms, powers and roots: http://en.wikipedia.org/wiki/Logarithm
Logarithmic transformations and scales
Euler number (value, series and limes expression)
Prepare to the next lecture:
•
•
•
•
•
•
Logarithmic functions
Power functions
Linear and algebraic functions
Exponential functions
Monod functions
Hyperbola
```