#
Complements of mathematics and calculus

A.Y. 2021/2022

Learning objectives

The course aims at: completing the students' knowledge in Mathematics, by studying some of the problems frequently encountered in Applied Sciences; providing the basic tools regarding the numerical simulation of mathematical problems of applicative interest, and the basic tools for an appropriate usage of Scientific Computing software.

Expected learning outcomes

The student will acquire a good knowledge of the mathematical foundations of linear algebra, of descriptive statistics and of numerical calculation; he/she will be able to frame some mathematical problems of applicative interest, and to correctly use the Scientific Calculation software to process data and simulate simple problems in the chemical field.

**Lesson period:**
Second semester

**Assessment methods:** Esame

**Assessment result:** voto verbalizzato in trentesimi

Course syllabus and organization

### Single session

Responsible

Lesson period

Second semester

The lectures are registered in synchronous mode.

All the material will be published on the site https://ariel.unimi.it/ and is organized as follows:

- detailed notes of the subject contents of each lesson in PDF format

- recording videos of the lecture and of the teacher's desktop

As far as practical computer experience is concerned, both the asynchronous mode is provided for explanations, and the synchronous mode in order to deepen and review the understanding of the subjects introduced during the asynchronous lessons.

The program of the course and the teaching material are unchanged.

The examination and evaluation procedures are unchanged.

The procedure for performing exams will be communicated on the site https://ariel.unimi.it/

All the material will be published on the site https://ariel.unimi.it/ and is organized as follows:

- detailed notes of the subject contents of each lesson in PDF format

- recording videos of the lecture and of the teacher's desktop

As far as practical computer experience is concerned, both the asynchronous mode is provided for explanations, and the synchronous mode in order to deepen and review the understanding of the subjects introduced during the asynchronous lessons.

The program of the course and the teaching material are unchanged.

The examination and evaluation procedures are unchanged.

The procedure for performing exams will be communicated on the site https://ariel.unimi.it/

**Course syllabus**

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.

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.

**Prerequisites for admission**

Numerical sets. Elementary functions. Sequences of real numbers. Differential and integral calculus for real functions in 1D and 2D. Ordinary differential equations. Vector and matrix algebra.

**Teaching methods**

Frontal lectures and tutorial exercises. Exercises and practical experiences in the computer room.

**Teaching Resources**

[Web site]: https://ariel.unimi.it/

Probability and Statistical Inference, 8th Edition, Robert V. Hogg, Elliot Tanis, 2010, Pearson

Elementary Numerical Analysis: An Algorithmic Approach Updated with MATLAB

S.D. Conte, Carl de Boor

SIAM, U.S., Classics in Applied Mathematics, 2018

Elementary linear algebra : applications version

Anton, HowardJohn Wiley and Sons

Probability and Statistical Inference, 8th Edition, Robert V. Hogg, Elliot Tanis, 2010, Pearson

Elementary Numerical Analysis: An Algorithmic Approach Updated with MATLAB

S.D. Conte, Carl de Boor

SIAM, U.S., Classics in Applied Mathematics, 2018

Elementary linear algebra : applications version

Anton, HowardJohn Wiley and Sons

**Assessment methods and Criteria**

The exam consists of: a written test, a computer test to be performed in Excel, a short oral test.

Students must complete all components of the final exam (written exam, computer test and oral exam) within a single exam session ("appello") of which there are six a year (January, February, April or May, June, July, September).

The evaluation of the written test allows a maximum mark of 24, with a minimum mark of 14 to be passed. The written test requires the solution of 6 exercise and the reply to 4 theoretical questions.

The evaluation of the computer test allows a maximum mark of 6, with a minimum mark of 3 to be passed. The evaluation of the oral test allows a maximum mark of 3 and it could either confirm or decrease the sum of the marks of the written and computer tests. The evaluation of the oral test could also determine the failure of the whole exam and the repetition of all tests in the future.

The oral test, of short duration, is based on the subject contents of all lessons: frontal lectures, tutorial exercises and practical experiences in the computer laboratory.

In the written and computer tests wrong answers don't give negative marks. The minimum marks in both the written and computer tests are required in order to be admitted to the oral test. At the discretion of the committee, the final score could be 30 with honors.

Students must complete all components of the final exam (written exam, computer test and oral exam) within a single exam session ("appello") of which there are six a year (January, February, April or May, June, July, September).

The evaluation of the written test allows a maximum mark of 24, with a minimum mark of 14 to be passed. The written test requires the solution of 6 exercise and the reply to 4 theoretical questions.

The evaluation of the computer test allows a maximum mark of 6, with a minimum mark of 3 to be passed. The evaluation of the oral test allows a maximum mark of 3 and it could either confirm or decrease the sum of the marks of the written and computer tests. The evaluation of the oral test could also determine the failure of the whole exam and the repetition of all tests in the future.

The oral test, of short duration, is based on the subject contents of all lessons: frontal lectures, tutorial exercises and practical experiences in the computer laboratory.

In the written and computer tests wrong answers don't give negative marks. The minimum marks in both the written and computer tests are required in order to be admitted to the oral test. At the discretion of the committee, the final score could be 30 with honors.

MAT/01 - MATHEMATICAL LOGIC - University credits: 0

MAT/02 - ALGEBRA - University credits: 0

MAT/03 - GEOMETRY - University credits: 0

MAT/04 - MATHEMATICS EDUCATION AND HISTORY OF MATHEMATICS - University credits: 0

MAT/05 - MATHEMATICAL ANALYSIS - University credits: 0

MAT/06 - PROBABILITY AND STATISTICS - University credits: 0

MAT/07 - MATHEMATICAL PHYSICS - University credits: 0

MAT/08 - NUMERICAL ANALYSIS - University credits: 0

MAT/09 - OPERATIONS RESEARCH - University credits: 0

MAT/02 - ALGEBRA - University credits: 0

MAT/03 - GEOMETRY - University credits: 0

MAT/04 - MATHEMATICS EDUCATION AND HISTORY OF MATHEMATICS - University credits: 0

MAT/05 - MATHEMATICAL ANALYSIS - University credits: 0

MAT/06 - PROBABILITY AND STATISTICS - University credits: 0

MAT/07 - MATHEMATICAL PHYSICS - University credits: 0

MAT/08 - NUMERICAL ANALYSIS - University credits: 0

MAT/09 - OPERATIONS RESEARCH - University credits: 0

Practicals: 16 hours

Laboratories: 16 hours

Lessons: 32 hours

Laboratories: 16 hours

Lessons: 32 hours

Shifts:

Professor:
Zampieri Elena

Corso A

Professor:
Fierro FrancescaCorso B

Professor:
Bressan NicolettaProfessor(s)