Complements of Mathematics and Calculus

Descriptive statistics: basic concepts and definitions. Numeric variables (discrete, continuous), non-numeric variables. Frequency distribution (absolute, relative, percentage, cumulative). Data grouping into classes: range, size, right/left open/close classes. Examples of Plots of frequency distribution: pie-charts, bar-charts, histograms. Position indices: mean, median, mode. Arithmetic, weighted and grouping data mean. Deviance, variance, standard quadratic deviation. Variation coefficient CV and CV%. Quadratic, geometric, harmonic mean. Calculation of the median, deciles, quartiles, percentiles for discrete data (finite case) and for grouping data into classes (continuous case). Basic concept of probability. Random variable, probability distribution, cumulative distribution function (discrete case). Expected value and variance of a discrete random variable. Continuous random variables. Uniform distribution: expected value and variance. Normal distribution: study of the properties of density of probability as a function of the parameters. Calculation of probability of data having normal distribution. Expected value and variance of a random variable Y=aX+b as a function of the expected value and variance of X. Linear regression, least square method in the discrete case. Covariance, coefficient of linear correlation. The calculation of least square methods based on second degree polynomials. Linear regression in the case of linearization of non-linear models.

The polynomial interpolation problem. Proof of the uniqueness theorem. Construction of the interpolation polynomil with Vandermonde method and with Lagrange method. Interpolation error. Runge counter-example. Interpolation vs extrapolation. Linear spline functions: definition, smoothness, construction using basis functions.

Numerical derivative: forward and backward approximation, mid-point formula, second derivative's approximation, Taylor series and error formulas.

Quadrature formulas for the approximation of definite integrals: rectangle, mid-point, trapezoidal and Cavalieri-Simpson rule. Newton-Cotes quadrature formulas: nodes, weights, open and closed formulas, degree of precision. Composite quadrature formulas; error's formulas. Computation of the weights of trapezoidal and Cavalieri-Simpson rules.

An introduction to the numerical approximation of ordinary differential equations. Explicit and implicit Euler's method: geometrical derivation, with Taylor expansion, using derivatives' approximation schemes, based on quadrature formulas. Crank-Nicolson and Heun methods. Definitions and examples of absolute stability and instability, definition of convergence.

Approximation of roots of functions: Bisection, Newton and chords' method. Order of convergence. Stop test. Global convergence theorem for the Newton method. Examples of fixed-point iterative methods.

Some recalls on vectors in the plane and in the space (R2 and R3): module, direction, sense; geometrical representation; notations; null vector; sum and product by a scalar (definition, properties, geometrical construction); unit vectors and basis vectors i,j; components and Cartesian representation; distance between two points.
Parallel and orthogonal vectors. Dot product. Norms of vectors. Linear combination of vectors. Linearly dependent and independent vectors. Orthogonality of the basis unit vectors i,j,k. Some generalization to the n-dimensional space.

Recalls on matrices in R(mxn): definition, equal matrices, sum and product by a real number, null matrix, identity matrix, opposite matrix, row and column vectors, diagonal and triangular matrices, transposed matrix, symmetric matrices. Product (matrix)x(vector), (matrix)x(matrix): definition, properties, examples. Norms of matrices: definition, properties, norm-1 and infinity-norm.
Product of diagonal, tridiagonal and triangular matrices. Determinant of a 2x2 and 3x3 matrix. Properties of determinants. Definition of inverse matrix. Linear systems Ax=b 2x2, 3x3, an overview of nxn systems. Homogeneous systems. Compatible (determined and undetermined) and incompatible (impossible) systems. Condition number of a square matrix. Condition number of a linear system in the particular case of perturbation of the vector of constant terms. Square submatrices, minor and rank of a matrix, augmented matrix [A|b]. Rouché-Capelli theorem (in the case A(nxn), without proof). Examples of determined, undetermined, impossible 2x2 and 3x3 systems. Degrees of freedom. Definition of eigenvalues and eigenvectors. Computation of eigenvalues and eigenvectors of 2x2 and 3x3 matrices. Norm-2 of a matrix and condition number in norm-2. Gauss elimination method. Cross product. Mixed product. Linear dependence and independence condition for n vectors in R^m. Recalls on: Parametric and Cartesian equation of a line in R^3, Equation of the plane, Parallel and orthogonal lines and planes.