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The book of nature is written in the language of mathematics

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

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The book of nature is written in the language of mathematics

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  1. Thebook of natureiswritteninthelanguage 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 stepsin statistics 9. Moments and descriptivestatistics 10. Important statistical distributions 11. Parametric hypothesis testing 12. Correlation and linear regression   13. Analysis of variance 14. Non-parametric testing 15. Clusteranalysis 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. Older scripts www.uni.torun.pl/~ulrichw

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

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

  6. Additionalsources

  7. Logarithms and logarithmicfunctions John Napier (1550-1617) A logarithmisthatnumberwithwhich we have to takeanothernumber (thebase) to thepower to get a third number. The logarithmic function Root Log 1 = 0 Asymptote Thelogarithmicfunctionis not defined for negativevalues

  8. Logarithms and logarithmicfunctions A general logarithmicfunction Curvature Shift aty-axis Increase Shift atx-axis Root

  9. Whatisthelogarithm of base 2 of 59049 ifthethelogarithm of 59049 of base 3 is 10?

  10. Leonhard Paul Euler (1707-1783) Thenumber e Thefamous Euler equation y=ex e • e = 2.71828183….

  11. Logarithmicequations x  -0.8 x = 0 Mixedequationsoften do not haveanalyticalsolutions. Roots

  12. Thecommonlyusedbases Logarithms to base 2 Logarithms to base 10 Logarithms to base e Log10 x ≡ lg x Logex ≡ ln x Log2 x ≡ lb x Digital logarithm Natural logarithm Binarylogarithm The scientific standard Standard of software Publications Statistics Classicalmetrics pH DeziBel 1 byte = 32 bit =25 bit 232 = 4294967296 1 byte = lb( number of possibleelements)

  13. Weber Fechner law Humanbrightlessperception Sensoricalperception of bright, loudness, taste, feeling, and othersincreaseproportional to thelogarithm of themagnitude of thestimulus. Logarithmicfunction Stevens’ power law Thepowerfunction law of Stevens approachesthe Weber-Fechner law at k = 0.33 Power functions and logarithmicfunctionsaresometimesverysimilar.

  14. Loudnessindezibel Themagnitude of a soundisproportional to thesquare of soundpressure Therule of 20. Linearscale +40 Dezibelis a ratio and thereforedimensionless P: soundpressure x100 Logarithmicscale 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)? Thesoundpressureis 100 timesthethresholdpressure.

  15. 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.

  16. The mass effectin physics, chemistry, biochemistry, and ecology pHisthenegative log10 of H+concentration. The Arrhenius model assumesthatreactionspeedisdirectlyproportional to thenumber of contacts an thereforethenumber of reactiveatomes. 10 ml of a solution of H2S has a pH of 5. Whatistheconcentration of OH- afteradding 100 ml HCN of pH 8. WhatisthepH of 0.5mol*l-1NaOH?

  17. Henderson Hasselbalchequation WhatisthepH of 0.2 mol l-1 C2H5COOH (pK = 4.75) and 0.1 mol l-1 NAOH? Livingorganismsarebuffered systems Bloodis a CO2 – NaHCO3bufferatpH 7.5 WhatisthepHafterinjection of 100 ml 0.8mol*l-1 CH3COOH.

  18. A first model Magicicadaseptendecim Photo by USA National Arboretum

  19. Magicicadaseptendecim Photo by USA National Arboretum

  20. Home work and literature • Refresh: • Greek alphabet • Logarithms, powers and roots: http://en.wikipedia.org/wiki/Logarithm • Logarithmictransformations and scales • Euler number (value, series and limes expression) • Radioactivedecay • Prepare to thenextlecture: • Logarithmicfunctions • Power functions • Linear and algebraicfunctions • Exponentialfunctions • Monodfunctions • Hyperbola

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