Universidad Pública de Navarra - Nafarroako Univertsitate Publikoa

Module/Subject matter

Module: Quantitative methods.

Subject matter: Mathematics.

Up

Contents

Diagonalization of square matrices: eigenvectors and eigenvalues, characteristic polynomial, diagonalizable matrices. Quadratic forms: polynomial and matrix expression, symmetrical matrices Diagonalization. Convergence and geometric series: convergence of successions, geometric series, criteria of convergence of series, sum of series. Introduction to dynamic models: differential equations, difference equations.

Up

Descriptors

Intermediate Differential Calculus, Integral Calculus and Linear Algebra.

Up

General proficiencies

• CG03: Oral and written communication in their mother tongue.
• CG05: Computer skills relevant to the field of study.
• CG07: The capacity to solve problems.
• CG09: The capacity to work as part of a team.
• CG16: Working in high-pressure environments.
• CG17: A capacity for self-reliant learning.

Up

Specific proficiencies

• CE03: To discern relevant information and data in microeconomic and macroeconomic data which a non-professional would be unable to recognise.
• CE04: To perform economic analysis applying professional criteria, preferably criteria based on the use of technical instruments.
• The students should get used to working at a proficiency level with the most fundamental techniques of Calculus, namely: real numbers, basic binary operations, maps and real-valued functions, single equations, systems of equations, and so on.
• The students should improve their ability to state, pose questions related to, mathematically interpret and finally solve, problems arising in Economics, as well as to criticize, compare, discuss, evaluate and draw conclusions from the results obtained.
• The students should be familiar with the scope and different possible uses of the contents of the subject matter, not only as a tool for the better understanding of several other complementary subjects, but also (and not less important), as relevant devices to be used for Decision-Making in typical situations coming from Economics, Management and/or Business Administration studies.

Up

Learning outcomes

Up

Methodology

The methodology used throughout this subject matter aims to give a positive answer to the following challenges (among others):

• Lead the student to be responsible of his/her own process of learning and acquiring knowledge.
• Establish a preference where the practical items are more important than the theoretical ones. To do so, the lecturer will furnish the basically tools to solve problems, but not disregarding the most important theoretical facts, that will allow the students to know why things are as they are.
• Enable and recommend the systematic use of computer devices and the access to the Internet.

To do so, the lecturer(s) will develop the scheduled activities in the following way:

1. Theoretical classes (that will include exercises) will be taught in the lecture room by the lecturer(s) in charge of this subject matter. Along these lectures, the lecturer(s) will introduce and develop the most important facts, highlighting them by means of the inclusion of a wide list of examples and practical cases to be analyzed. Therefore, the applications are understood as a base and ground on which the theory leans, and not only as a mere list of exercises. The exposition made by the lecturer(s) will show the main lines and aspects of the subject matter. The complementary use of the bibliography, access to the Internet, or use of the tutorials and consulting hours to complete, improve and reinforce the items shown in the lecture room, is a task to be done by the students.
2. Practical and problem-solving sessions, in which the lecturer(s) will state and help the students to solve (in the lecture room) typical problems related to the subject matter. To do so, the lecturer(s) will select, when available, "case studies", that is: key problems that perhaps are not trivial or straightforward, and have some special difficulty. These problems should show in a clear way how the most important theoretical facts interact to get a solution, output or result. These kinds of lectures will require an active (and compulsory) participation of the students that should have tried to solve (in advance) the problems stated. Having this in mind, it is very important here that the students work in teams in which they will comment, discuss, analyze, get solutions and draw conclusions from the problems they solve.
3. The personal work of the student is crucial here, since the subject is exigent and forces the student to update, at the same speed in which it is taught, the amount of matter shown in the lecture room. To cope with this, the student must develop techniques of self-training. To control all this process, the interaction student-lecturer through tutorials and consulting hours is compulsory.
4. The student must get used to working in teams as a complement to his/her individual process of learning. Having this in mind, the lecturer(s) should also provide the students with sessions of "group-tutorials", in which a team of students will expose the achievements obtained (working in a team) on schemes and models to be analyzed and discussed.
5. Systematic "control tests" should be scheduled by the lecturer(s) to be done in the lecture room by the students in a period of, say, once a fortnight, to have a clear idea about how the students are understanding and using the main concepts and ideas. This is a way to control the "continuity" in the way in which the students interact with the subject matter. Notice that to assimilate the main ideas of Mathematics, a process of continuous learning is required.

Up

Languages

English.

Up

Evaluation

Up

Agenda

1. ALGEBRA

1.1 Complex numbers.

1.2 Matrix diagonalization.

1.3 Quadratic forms.

2. DYNAMICAL SYSTEMS (GENERAL THEORY)

2.1 Preliminary concepts on dynamical systems.

2.2 Differential equations.

2.3 Systems of linear differential equations.

3. DISCRETE DYNAMICAL SYSTEMS

3.1 Sequences of real numbers.

3.2 Series of real numbers.

3.3 Difference equations: the linear case.

3.4 Systems of linear difference equations.

Up

Bibliography

Access the bibliography that your professor has requested from the Library.

By means of the tool "MiAulario", the students will have access to schemes, lists of exercises, links to complementary material, etc., related to the subject matter.

We do not recommend any particular textbook on this subject. There are many possible books on Differential Calculus, Integral Calculus, Linear Algebra, Differential Equations, Dynamical Systems, Difference Equations and related items, even from a point of view of Economics or reportedly addressed to students of Economics and/or Business Administration, that can be found in any "average-size" universitary library.

Perhaps the reference "Mathematical models in the Social, Management and Life Sciences" , by D.N. Burghes and A. D. Wood (Ellis Horwood. Chichester, UK. 1984) is an excellent reference to find "case-studies" related to the main concepts to be developed throughout the semester in this subject matter.

For the sake of completeness, we include below a list (not exhaustive, and by no means the only possible one) of texts in Spanish that can be used in the preparation of some lectures.

LIST (texts in Spanish) FOLLOWS:

• Balbás, A. y otros (1998) "Análisis matemático para la Economía II". A C Madrid.
• Chiang, A. C. (1987) "Métodos fundamentales de economía matemática". McGraw-Hill. Madrid.
• Fernández Pérez, G. y otros (2003) "Ecuaciones diferenciales y en diferencias: sistemas dinámicos". Thomson.
• Lomelí, H. y Rumbos, B. (2003) "Métodos dinámicos en economía". Thomson.
• Martínez Estudillo, F. J. (2005) “Introducción a las Matemáticas para la Economía” Desclée de Brouwer.
• Takahashi, T. (1991) "Ecuaciones en diferencias con aplicaciones". Grupo Editorial Iberoamericana.

Up

Location

Location details

Up