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The book of nature is written in the language of mathematics. Galileo Galilei. 1. Introduction 2. Basic operations and functions 3. Matrix algebra I 4. Matrix algebra II 5. Handling a changing world 6. The sum of infinities 7. Probabilities and distributions - PowerPoint PPT Presentation
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The book of nature is written in
the language of mathematicsGalileo Galilei
1. Introduction
2. Basic operations and functions
3. Matrix algebra I
4. Matrix algebra II
5. Handling a changing world
6. The sum of infinities
7. Probabilities and distributions
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
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!!!
www.uni.torun.pl/~ulrichw
Modelling Biology
Basic Applications of Mathematics and Statistics in the Biological Sciences
Part I: Mathematics
Script A
Introductory Course for Students of
Biology, Biotechnology and Environmental Protection
Werner Ulrich
UMK Toruń 2007
Modelling Biology
Basic Applications of Mathematics and Statistics in the Biological Sciences
Part I: Mathematics
Script B
Introductory Course for Students of
Biology, Biotechnology and Environmental Protection
Werner Ulrich
UMK Toruń 2007
Modelling Biology
Basic Applications of Mathematics and Statistics in the Biological Sciences
Part II: Data Analysis and Statistics
Script A
Introductory Course for Students of
Biology, Biotechnology and Environmental Protection
Werner Ulrich
UMK Torun 2007
Older scripts
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(
aand
a
ya
ya
yxay
y
y
ax
a
a
1
log
log
log
John Napier(1550-1617)
zyxaa
aaa
zxy
zyx
zyx
aaa
logloglog
logloglog
logloglog
zyxaa
aaaaa
zyx
zyx
zyxyx
aaa
aaaaa
logloglog
/
/
logloglog
logloglogloglog
zxy
aaa
zxzxyyx
y
aaa
loglog
logloglog
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
7964.03
5
53
)53ln(24
2
2
ex
xe
xRoot
Curvature
What is the logarithm of base 2 of 59049 if the the logarithm of 59049 of
base 3 is 10?
590492
85.15585.1*103log10
85.15
2
xx
1631.0*585.12log*3log631.02log585.13log
32
3
2
bzbyz
axayz
bybxz
bza
abaa
bx
bb
ay
aa
yx
loglogloglog
logloglog
logloglog
ab
ab
abxxaxybyx
ba
ba
bab
a
log1log
loglog1
loglogloglog
caa
a
a
aaaa
xyxcycxy
xyyxyx
)(log)(log)(log)(log
)/(log)(log)(log)/(log
43log415.0
34log 22
00.5
11.5
22.5
3
0 1 2 3 4
y
x
The number e
0
32
!...
!3!2!11
i
ix
ixxxxe
0 !
1...!3
1!2
1!1
11i i
e
n
n ne
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
Logarithmic equations
bax )ln(
aexeax
ee
bax
bb
bax
)ln(
)ln(
)1ln(2 xx
1
)1ln(22
xe
xxx
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
-2.5 -2 -1.5 -1 -0.5 0 0.5 1
x = 0x -0.8
cbax )ln()ln(
aex
eaxcb
cb
/ln
ln
Mixed equations often do not have analytical solutions.
1)1ln( xy
1....718282.211
1)1ln(01)1ln(
0
0
0
exex
xyxy
)ln( 1 xxey
111
1
0)ln(
0)ln(
0
10
10
10
1
0
0
0
xex
ex
ex
yxey
x
x
x
x
1)1ln( xexy
existssolution no0 always are powers
)1ln(
01)1ln(
00
x
x
ex
yexy
Roots
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.
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.
pH is the negative log10 of H+ concentration.
)27.(6)72.(71108*1005*10 pOHpHpH
)(log 310 OHpH
14 pOHpH
What is the pH of 0.5mol*l-1 NaOH?
7.13)142(log10*5.0
1][
10]*[5.0*]*][[10]][[
1014
3
14113
143
pHOH
lmollmolOHOHOH
143 10]][[ OHOH
The mass effect inphysics, chemistry, biochemistry,
and ecology
NaClClNa
KNaClClNa
][]][[
The Arrhenius model assumes that reaction speed is directly proportional to the number of contacts an therefore the number of reactive atomes.
14
2
3 10][
]][[
OHOHOH
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.
OHNaHCONaCOOH 2333
43.78.0*1.0.1
)8.0*1.01(10][10][
]][[ 5.7
35.7
3
33
pHOHNaHCO
NaCOOH
][][][][][ 232 OHAOHOHHA
][][log
][][log][loglog
][]][[
1010310103
HAApHpK
HAAOHK
HAAOHK
Henderson Hasselbalch equation
What is the pH of 0.2 mol l-1 C2H5COOH (pK = 4.75) and 0.1 mol l-1 NAOH?
75.4)1.01.0(log75.4 10 pH
OHCOOCHCOONaCHOHNaCOOCHOH 23333 1.01.01.01.02.0
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
0
1
2
3
0 5 10 15 20 25 30 35 40 45 50 55 60 65
Time
Pre
dato
r abu
ndan
ce
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
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