Definition of Mathematical Areas
For the Methods in Theoretical Biology Project
Introduction
This document defines the mathematical areas that will be used in the database of the Methods in Theoretical Biology project. The definition is based on the "Basic Methods in Theoretical Biology" document (view as HTML, pdf), in which the most often used mathematical techniques in theoretical biology are described. The definition of the areas is based on the mathematical subjects that are needed to study a particular type of theory. In this way, study books of that subject can be used as background reading for a particular area. In each area, basic, medium and advanced exercises and examples can be found. It is advisable to be comfortable with at least most of the material in a given difficulty level to advance to the next. The notion of difficulty is based on the judgement of the author and the editorial board.
Area Index
The following mathematical areas have been defined:- Methodology
- Mathematical concepts
- Calculus
- Linear algebra
- Probability
- Numerical methods
- Processes
- Systems
- Model based statistics
- Mixture
Methodology
The study of method and procedure. A summary of the required knowledge for this area can be found in chapter 1 of the methods document.
Associated mathematical keywords
| allometric function | Arrhenius | auxiliary theory |
| Buckingham theorem | coherence | consistency |
| deductive logic | design experimental | dimension |
| empirical cycle | extensive variable | experimental biology |
| falsification | inductive logic | intensive variable |
| module | organization scale | Pascal |
| predicate logic | propositional logic | support |
| testability | theoretical biology | usefulness |
| verification |
Recommended literature
No free books found so far, please mail the Executing editor if you have suggestions.
Mathematical Concepts
Basic concepts that are used in mathematics. A summary of the required knowledge for this area can be found in sections 2.1 to 2.3 of the methods document.
Associated mathematical keywords
|
set Venn diagram complement partition union intersection subset disjoint Boolean operator de Morgan rule interval |
interval closed open interval half-open interval unbounded interval factorial over operator image domain injective operator surjective operator |
bijective operator scalar real number absolute value integer rational number complex number real part imaginary part conjugate |
Recommended literature
Basic Concepts of Mathematics, Elias Zakon Available for self study without chargeThis book helps the student complete the transition from purely manipulative to rigorous mathematics, using extensive explanations of basic concepts.
Calculus
The branch of mathematics concerned with the differentiation and integration of functions. A summary of the required knowledge for this area can be found in section 2.4 of the methods document.
Associated mathematical keywords
|
function range independent variable dependent variable parameter monotonous function even function odd function homogeneous function polynomial function degree rational function algebraic function composition function inner function outer function transcendental function base implicit definition inverse function sine cosecant cosine secant |
tangent cotangent limit right-hand limit left-hand limit continuous function vertical tangent line asymptote Order order sequence term Fibonacci sequence recursive definition convergence series geometric series harmonic series divergence Riemann zeta function binomial series differentiable function implicit differentiation second derivative |
inflection point L'Hopital rule Maclaurin series Taylor series tangent line tangent parabola approximation partial derivative gradient directional derivative implicit partial differentiation Hessian matrix chain rule intermediate variable tangent plane critical point minimum value maximum value local maximum local minimum global minimum global maximum determinant saddle point |
constraint Lagrange multiplier antiderivative indefinite integration integrand definite integration average value logarithmic function exponential function gamma function beta function Leibniz theorem partial integration Jacobian determinant Jacobian matrix algebra theorem root determinant isocline circle ellipse hyperbola sphere |
Recommended literature
Elementary Calculus, an Approach Using Infinitesimals, H. Jerome Keisler On-line edition available without chargeContaining a more extensive, intuitive discussion of concepts.
The Calculus Bible, Prof. G. S. Gill Available for reading without charge.
A formal approach to undergraduate area calculus.
Linear Algebra
The part of algebra that deals with the theory of linear equations and linear transformation. A summary of the required knowledge for this area can be found in section 2.5 of the methods document.Associated mathematical keywords
|
matrix element size matrix matrix product direct product trace square matrix diagonal matrix identity matrix zero matrix |
triangular matrix vector idempotent transposition symmetric matrix determinant singular matrix nonsingular matrix linearly dependent rank |
full rank inverse matrix non-homogeneous linear system generalized inverse matrix eigen value eigen vector bilinear form quadratic form positive definite negative definite |
inner product length vector norm of vector Pythagoras theorem outer product cosine orthogonal orthonormal rotation matrix |
Recommended literature
Linear Algebra, J. Hefferson, Available for reading without chargeAn undergraduate textbook on linear algebra meant to challenge students without being overwhelming. It introduces many interesting applications.
Introduction to Matrix Algebra, A.K. Kaw, Available for reading without charge
A contemporary, applied approach to linear algebra, suitable for self study. Uses examples to illustrate every concept.
Elementary Linear Algebra, K.R. Matthews Available for reading without charge
A more formal and concise approach to linear algabra for undergraduate students.
Probability
The branch of mathematics concerned with the analysis of the likelihood of events occurring. A summary of the required knowledge for this area can be found in section 2.6 of the methods document.Associated mathematical keywords
|
random variable
discrete random variable continuous random variable sample frequency distribution probability distribution probability density function distribution function survivor function expectation |
mean variance moment central moment standard deviation variation coefficient stochastically dependent covariance correlation coefficient |
Poisson distribution binomial distribution Bernoulli distribution multinomial distribution geometric distribution negative binomial distribution homogeneous distribution beta distribution exponential distribution |
gamma distribution normal distribution central limit theorem Chi-square distribution conditional probability truncated distribution marginal distribution mixture convolution integral |
Recommended literature
Basic Probability, C. M. Grinstead, J. L. Snell, available under GNU General Public License.This Book introduces probability without too much formal mathematics, in a "somewhat leisurely style" using well - written introductions and examples.
Numerical Methods
Methods that are needed to find numerical values for quantities that cannot be obtained analytically. A summary of the required knowledge for this area can be found in section 2.7 of the methods document.Associated mathematical keywords
numerical integrationstiff differential equation
numerical differentiation
Newton Raphson iteration
simplex
Recommended literature
Numerical Recipes, by Numerical Recipes Software available for reading onlineBook accompanying the "Numerical Recipes" software. Offers practical advice on how to employ numerical techniques, and explains those used in the library.
Processes
Models for any series of changes, especially natural ones. A summary of the required knowledge for this area can be found in section 3.1 of the methods document.Associated mathematical keywords
|
transient state
steady state discrete time process continuous time process map deterministic process |
stochastic process correlation function chain Markov branching process Markov process process Markov |
birth and death process Yule process point process counting process interval process orderly process |
renewal process intensity ergodic process hazard rate central limit theorem |
Recommended literature
No free books found so far, please mail the Executing editorif you have suggestions.Systems
Sytems are an assemblage of inter-related elements comprising a unified whole, mathematical systems theory has its specific way of dealing with systems. A summary of the required knowledge for this area can be found in section 3.2 of the methods document.
Associated mathematical keywords
|
state variable input differential equation output initial value boundary value feedback |
bifurcation memory point attractor cyclic attractor chaos strange attractor neutral stability |
local stability global stability continuation normal form Floquet multiplier bifurcation diagram operating diagram |
Recommended literature
No free books found so far, please mail theExecuting editorif you have suggestions.Model Based Statistics
The branch of mathematics concerned with estimation of model parameters and testing hypotheses about their values. A summary of the required knowledge for this area can be found in Chapter 4 of the methods document.Associated mathematical keywords
|
transformation measurement scale measurement scale measurement unit measurement significant figure floating point random error systematic error |
precision accuracy absolute error relative error interpolation spline function knot Bezier curve control point |
null hypothesis alternative hypothesis significance level type 1 error type 2 error power test Neyman-Pearson theorem likelihood function maximum likelihood estimate |
maximum likelihood estimator bias confidence interval deviance regression model unidentifiable parameter random generator seed |