1. The book of nature is written in the language of mathematics Galileo Galilei

2. 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!!!

3. www.uni.torun.pl/~ulrichw Older scripts

4. Mathe online http://www.mathe-online.at/

5. http://tutorial.math.lamar.edu/

6. Additional sources

8. 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. Asymptote Root The logarithmic function The logarithmic function is not defined for negative values Log 1 = 0 John Napier (1550-1617)

9. Logarithms and logarithmic functions A general logarithmic function Shift at x-axis Shift at y-axis Increase Root Curvature

10. What is the logarithm of base 2 of 59049 if the the logarithm of 59049 of base 3 is 10?

11. The number e e = 2.71828183…. e y=e x Leonhard Paul Euler (1707-1783) The famous Euler equation

12. Logarithmic equations x = 0 x  -0.8 Mixed equations often do not have analytical solutions. Roots

13. The commonly used bases Logarithms to base 10Logarithms to base 2Logarithms to base e Log 10 x ≡ lg x Log 2 x ≡ lb x Log e x ≡ ln x Digital logarithm Binary logarithm Natural logarithm 1 byte = 32 bit = 2 5 bit 2 32 = 4294967296 1 byte = lb( number of possible elements) Classical metrics pH DeziBel The scientific standard Standard of software Publications Statistics

14. Weber Fechner law Sensorical perception of bright, loudness, taste, feeling, and others increase proportional to the logarithm of the magnitude of the stimulus. Logarithmic function The power function law of Stevens approaches the Weber- Fechner law at k = 0.33 Stevens’ power law Power functions and logarithmic functions are sometimes very similar. Human brightless perception

15. 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)? x100 +40 Logarithmic scale Linear scale The sound pressure is 100 times the threshold pressure.

16. How much louder do we hear a machine that increases its sound pressure by a factor of 1000? The machine appears to be 60 dB louder To what level should the sound pressure increase to hear a sound 2 times louder? The multiplication factor k is linearly (directly) proportional to the sound pressure P.

17. 10 ml of a solution of H 2 S has a pH of 5. What is the concentration of OH - after adding 100 ml HCN of pH 8. pH is the negative log 10 of H + concentration. What is the pH of 0.5mol*l -1 NaOH? The mass effect in physics, chemistry, biochemistry, and ecology The Arrhenius model assumes that reaction speed is directly proportional to the number of contacts an therefore the number of reactive atomes.

18. Living organisms are buffered systems Blood is a CO 2 – NaHCO 3 buffer at pH 7.5 What is the pH after injection of 100 ml 0.8mol*l -1 CH 3 COOH. Henderson Hasselbalch equation What is the pH of 0.2 mol l -1 C 2 H 5 COOH (pK = 4.75) and 0.1 mol l -1 NAOH?

19. Magicicada septendecim Photo by USA National Arboretum A first model

20. Magicicada septendecim Photo by USA National Arboretum

22. 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