Programmes - Faculty of Sciences, first year of the undergraduate course

1. DEGREE COURSES IN MATHEMATICS, PHYSICS, INFORMATION TECHNOLOGY AND RELATED SUBJECTS

1.a. Mathematics Prerequisites

Number sets and arithmetic:

  • Numerical fractions: operations and inequalities
  • Decimal representation; in which cases can a fraction be expressed in decimal form?
  • Relative rational numbers: properties of operations; product cancellation law. How is the for the product justified?
  • Inequalities and their fundamental properties; absolute values; approximate numerical calculations.
  • Intuitive notion of real number. Arithmetic mean and geometric mean of two positive numbers.
  • Division with remainder between natural integers (a precise statement that expresses the meaning of the division with remainder of a by b….)
  • Divisibility, greatest common divisor, least common multiple. Euclid's algorithm for calculating the greatest common divisor Prime numbers. There are infinitely many prime numbers: how can you prove it? Decomposition of an integer into prime factors. (A precise statement, without proof)

Algebra:

  • Elements of literal calculus: use of letters, use of parentheses
  • Polynomials (notion of degree of a monomial, of a polynomial). Algebraic operations between polynomials. Algebraic fractions
  • Division with remainder between polynomials (a precise statement that expresses the meaning of the division between two polynomials). Divisibility of a polynomial by x-a.
  • Polynomials as functions and the identity theorem of polynomials (A precise statement, even without proof).

Geometry:

  • Knowledge of the meaning of the terms: axiom (postulate), theorem, lemma, corollary, hypothesis, thesis….
  • Elements of plane geometry: incidence, perpendicularity, parallelism.
  • The parallel postulate · Convex figures, convex polygons
  • Geometric transformations of the plane and their composition (symmetries with respect to a straight line and with respect to a point; translations and rotations; homotheties and similarities)
  • Properties of plane figures, particularly in relation to symmetries
  • The theorems of Thales, Euclid and Pythagoras
  • The parallelogram; vectors and operations on them
  • Segmental and angular properties of the circle (chords, secants, tangents, inscribed angle)
  • Measurement of angles; sum of the internal and external angles of a convex polygon
  • Biunique correspondence between real numbers and the points of the straight line.
  • The Cartesian plane: representation of straight lines, circles; of the parabola, of the ellipse, of the hyperbola (appropriately taking the axes)
  • Geometry of space: incidence, perpendicularity, parallelism. Angle between straight line and plane. Dihedral and trihedral.
  • Convex polyhedra. Euler's formula. Regular polyhedra.
  • The sphere, the cone, the cylinder.

The language of sets, equations and inequalities:

  • Elementary language of sets
  • Relations (in particular: equivalence and order)
  • Applications (functions). Injective, surjective, bijective applications
  • Elements of combinatorics: given the finite sets A and B, number of applications of A in B (dispositions with repetitions…), number of injective applications of A in B (simple arrangements…). Number of subsets of k elements, in a set of n elements (combinations…).
  • Equations and inequalities. Equations (and inequalities) deduced from an assigned equation (or inequality). Equations and inequalities equivalent to each other.
  • Linear systems of two equations and two unknowns, and their interpretations in the Cartesian plane.
  • nth root (in the set of positive real numbers)
  • Second degree equations; relations between the coefficients and the roots
  • Graph of a second degree trinomial

Sequences, elementary functions:

  • Succession; arithmetic and geometric progressions
  • Limit of a sequence; sum of a geometric series.
  • Powers with relative rational exponents (and positive base!)
  • Exponential functions and logarithmic functions; their graphic representations. Decimal logarithm and its relationship with the decimal representation of numbers.
  • Length of a circle and arc length of a circle
  • Measurement of angles in radians.
  • Definition of cosine, sine, tangent and first properties
  • Criteria of congruence of triangles and related trigonometric problems: theorem of sines and Carnot's theorem. Graphs of circular functions. Addition theorem for circular functions. Definition of arccosine, arcsine, arctangent functions and their graphs.
  • Area of a flat region (defined, for example, with grid lines, as is done with graph paper...)
  • Area of polygons and equidecomposability. Area of the circle.
  • Volume of a solid. Cavalieri's principle.
  • Volume of the cylinder, the cone, the pyramid, the sphere.
  • Areas and volumes of similar figures.

Complements:

A - Elements of Mathematical Analysis
Knowledge of some elements of mathematical analysis is useful not so much for access to university courses in mathematics as for courses in which mathematics is used right from the start (in particular, courses in Physics) Knowledge of:

  • limits, continuity for functions of one variable
  • derivative of a function; rules of derivation
  • increasing and decreasing functions; maxima and minima: convexity and concavity; inflections
  • definite integral and its first properties
  • primitive of a function; the fundamental theorem of integral calculus.
     
  • Calculation of areas and volumes

Please Note: The use of small programmable calculators is also useful for a deeper understanding of the elements of Mathematical Analysis: it is very instructive to use them for calculating the roots of equations, definite integrals, etc. …..

B - Elements of linear algebra
With applications to physics, operations research, etc…..

C - Elements of probability calculus and statistics
These are very important topics in themselves, and are particularly useful to anyone who studies  an experimental science.

1.b.Physics prerequisites

  • Kinematics. Newton's laws. Elementary rigid body mechanics. Principles of conservation of energy, linear momentum and angular momentum. Collisions. Law of universal gravitation and Kepler's laws. Mechanical oscillators.
  • Fundamentals of fluid mechanics, Bernoulli's theorem. Wave phenomena and basics of acoustics. Thermology. Gas laws. The first two principles of thermodynamics. thermodynamic cycles. Kinetic theory of gases. Geometric optics: reflection and refraction of light. Wave properties of light: interference and diffraction. Electrostatics. Electric currents. Magnetostatics. electromagnetic induction. Oscillating circuits and electromagnetic waves.
  • fundamentals of special relativity.
  • Crisis of classical physics: blackbody, photoelectric effect, Bohr’s atom, Compton effect.

2. DEGREE COURSES IN CHEMISTRY, PHARMACEUTICAL CHEMISTRY AND TECHNOLOGY, GEOLOGICAL SCIENCES AND RELATED SUBJECTS

2. a. Chemistry prerequisites

  • States of aggregation of matter. Homogeneous and heterogeneous systems. State transitions. Separation and purification of substances. Elements and compounds. chemical reactions. Weight relationships in chemical reactions. Properties of gases, liquids and solids. Properties of solutions. Ionic solutions. Electrolysis. Electronic structure of atoms. Chemical bonding and molecular structures. Periodic system of elements.
  • Thermochemistry. Elements of chemical kinetics. Chemical equilibrium. Acid-base reactions, redox reactions, formation reactions of coordination compounds. Descriptive inorganic chemistry: hydrogen, halogens, oxygen, sulphur, nitrogen, phosphorus, alkali and alkaline-earth metals. Carbon chemistry: isomerism and stereoisomerism of organic molecules. Hydrocarbons. Functional groups and nomenclatures. Fundamental types of organic reactions: addition, substitution and elimination reactions.

2.b. Mathematics Prerequisites

Number sets and arithmetic:

  • Numerical fractions: operations and inequalities
  • Decimal representation; in which cases can a fraction be expressed in decimal form?
  • Relative rational numbers: properties of operations; product cancellation law.
  • Inequalities and their fundamental properties; absolute values; approximate numerical calculations.
  • Intuitive notion of real number. Arithmetic mean and geometric mean of two positive numbers.
  • Division with remainder between natural integers. Greatest common divisor, least common multiple.

Prime numbers. Decomposition of an integer into prime factors.

Algebra:

  • Elements of literal calculus: use of letters, use of parentheses
  • Polynomials (notion of degree of a monomial, of a polynomial). Sum and product between polynomials.  Division with remainder between polynomials. Divisibility of a polynomial by x-a.
  • Polynomials as functions and the identity theorem of polynomials.

Geometry:

  • Knowledge of the meaning of the terms: axiom (postulate), theorem, lemma, corollary, hypothesis, thesis….
  • Elements of plane geometry: incidence, perpendicularity, parallelism.
  • The parallel postulate · Convex figures, convex polygons
  • Geometric transformations (symmetries translations and rotations; homotheties and similarities)
  • Properties of plane figures, particularly in relation to symmetries
  • The theorems of Thales, Euclid and Pythagoras
  • Vectors and operations on them
  • Biunique correspondence between real numbers and points on the line
  • The Cartesian plane: representation of straight lines, circles; of the parabola, of the ellipse, of the hyperbola  
  • Geometry of space: incidence, perpendicularity, parallelism. Angle between straight line and plane.  
  • The sphere, the cone, the cylinder.

The language of sets, equations and inequalities:

  • Elementary language of sets
  • Relations (in particular: equivalence and order)
  • Applications (functions). Injective, surjective, bijective applications
  •  Elements of combinatorics: given the finite sets A and B, number of applications of A in B (dispositions with repetitions…), number of injective applications of A in B (simple arrangements…). Number of subsets of k elements, in a set of n elements (combinations…).
  • Equations and inequalities.
  • Linear systems of two equations and two unknowns, and their interpretations in the Cartesian plane.
  • nth root (in the set of positive real numbers)
  • Second degree equations; relations between the coefficients and the roots
  • Graph of a second degree trinomial
  • Sequences, elementary functions. Arithmetic and geometric progressions; sum of a geometric series.
  • Exponential functions and logarithmic functions; their graphic representations. Decimal logarithm and its relationship with the decimal representation of numbers.
  • Length of a circle and arc length of a circle
  • Measurement of angles in radians.
  •  Definition of cosine, sine, tangent and first properties
  • Criteria of congruence of triangles and related trigonometric problems: theorem of sines and Carnot's theorem. Graphs of circular functions. Addition theorem for circular functions. Definition of arccosine, arcsine, arctangent functions and their graphs.
  • Area of polygons and equidecomposability. Area of the circle.
  • Volume of a solid. Cavalieri's principle.
  • Volume of the cylinder, the cone, the pyramid, the sphere.
  • Areas and volumes of similar figures.

2. c. Physics prerequisites

  • Kinematics. Newton's laws. Elementary rigid body mechanics. Principles of conservation of energy, linear momentum and angular momentum. Collisions. Law of universal gravitation and Kepler's laws. Mechanical oscillators.
  • Fundamentals of fluid mechanics, Bernoulli's theorem. Wave phenomena and basics of acoustics. Thermology. Gas laws. The first two principles of thermodynamics. thermodynamic cycles. Kinetic theory of gases. Geometric optics: reflection and refraction of light. Wave properties of light: interference and diffraction. Electrostatics. Electric currents. Magnetostatics. electromagnetic induction. Oscillating circuits and electromagnetic waves.
  • Introduction to special relativity.
  • Crisis of classical physics: blackbody, photoelectric effect, Bohr’s atom, Compton effect.

3. DEGREE COURSES IN BIOLOGY, NATURAL AND ENVIRONMENTAL SCIENCES AND RELATED SUBJECTS

3.a. Biology prerequisites

Chemistry of molecules of biological interest:

  • chemical structure of nucleotides and amino acids
  • covalent bonds and hydrogen bonds
  • enantiomers and their biological significance
  • structure of DNA and RNA
  • proteins: primary, secondary, tertiary structures
  • protein phosphorylation and its biological significance as intracellular signal
  • Adenosine-tri-phosphate (ATP): chemical structure and biological significance
  • Enzymes and enzyme-substrate interactions
  • Allosteric modifications
  • Mechanism by which enzymes catalyze biological reaction        

Classical genetics and molecular genetics:

  • Mendelian inheritance, sex chromosomes, Hardy-Weinberg law
  • crossing over
  • mitochondrial genome inheritance
  • genetic code
  • transfer RNA and ribosomal RNA
  • control of DNA transcription in prokaryotes (operon)
  • control of DNA transcription in eukaryotes (enhancer)
  • processing of messenger RNA (splicing)
  • protein synthesis
  • main classes of proteins: enzymes, structural proteins, factors that regulate gene expression
  • structure of chromatin and chromosomes

Cell biology:

  • Structure of the eukaryotic cell membrane
  • Intracellular organelles: Golgi apparatus, endoplasmic reticulum, lysosome, their role in protein synthesis and degradation
  • Structure of the nucleus: membrane, euchromatin heterochromatin and nucleolus
  • mitochondria, cellular respiration and ATP production
  • The cytoskeleton: microtubules and associated proteins
  • cell replication     

Genetic engineering and molecular techniques:

  • Restriction Enzymes
  • Plasmids and Insertion of Exogenous Genes into Prokaryotic Cells
  • Electrophoresis on DNA gel
  • Marking of DNA, RNA and Proteins with Radioactive Tracers
  • DNA Sequencing       

Biology of organs and systems:

  • Structure of the myocyte and molecular aspects of muscle contraction
  • Aerobic and anaerobic musculature
  • Principles of anatomy and functioning of the heart and the circulatory system
  • Hemoglobin and oxygen transport
  • Insulin and other hormones encoded by genes, fundamentals of their biological function
  • Steroid-type hormones, fundamentals of their biological function
  • fundamentals of the immune system. Antibodies, humoral and cell-mediated immune response
  • fundamentals of anatomy of the nervous system: cortex, cerebellum, subcortical structures, spinal cord, peripheral nervous system.
  • fundamentals of anatomy of the sense organs: structure of the eye and ear
  • Structure of the neuron: axon, dendrites and synapses.
  • Neuronal communication: neurotransmitters and their mechanism of action
  • Action potential and signal coding as frequency of action potentials

Behavioral biology:

  • Signals for intraspecific communication
  • Innate component of signals used for intraspecific communication
  • Dominance and structure of social species
  • Influence of environment in learning social behaviors (e.g. parental care)
  • Song learning in birds
  • Language learning in humans
  • Learning by association
  • Learning by imitation      

Molecular evolution and evolution of organisms:

  • The concept of gene mutation
  • Positive and negative selection
  • Neutral mutations and their use as a "molecular clock"
  • Population genetics: bottleneck, founder effect and genetic drift
  • Sexual and asexual reproduction
  • Darwinian fitness
  • Mechanisms of speciation
  • Evolution of vertebrates and of the human species

3.b. Mathematics Prerequisites

Number sets and arithmetic:

  • Numerical fractions: operations and inequalities
  • Decimal representation; in which cases can a fraction be expressed in decimal form?
  • Relative rational numbers: properties of operations; product cancellation law.
  • Inequalities and their fundamental properties; absolute values; approximate numerical calculations.
  • Intuitive notion of real number. Arithmetic mean and geometric mean of two positive numbers.
  • Division with remainder between natural integers.  Greatest common divisor, least common multiple.
  • Prime numbers.  Decomposition of an integer into prime factors.

Algebra:

  • Elements of literal calculus: use of letters, use of parentheses
  • Polynomials (notion of degree of a monomial, of a polynomial). Sum and product of  polynomials.
  • Division with remainder between polynomials. Divisibility of a polynomial by x-a.
  • Polynomials as functions and the identity theorem of polynomials

Geometry:

  • Knowledge of the meaning of the terms: axiom (postulate), theorem, lemma, corollary, hypothesis, thesis….
  • Elements of plane geometry: incidence, perpendicularity, parallelism.
  • Geometric transformations of the plane and their composition (symmetries with respect to a straight line and with respect to a point; translations and rotations; homotheties and similarities)
  • Properties of plane figures, particularly in relation to symmetries
  • The theorems of Thales, Euclid and Pythagoras
  • Vectors and operations on them
  • Biunique correspondence between real numbers and the points of the straight line.
  • The Cartesian plane: representation of straight lines, circles; of the parabola, of the ellipse, of the hyperbola  
  • Geometry of space: incidence, perpendicularity, parallelism. Angle between straight line and plane.
  •  The sphere, the cone, the cylinder.

The language of sets, equations and inequalities:

  • Elementary language of sets
  • Relations (in particular: equivalence and order)
  • Applications (functions). Injective, surjective, bijective applications
  •  Elements of combinatorics: given the finite sets A and B, number of applications of A in B (dispositions with repetitions…), number of injective applications of A in B (simple arrangements…). Number of subsets of k elements, in a set of n elements (combinations…).
  • Equations and inequalities.
  • Linear systems of two equations and two unknowns, and their interpretations in the Cartesian plane.
  • nth root (in the set of positive real numbers)
  • Second degree equations; relations between the coefficients and the roots
  • Graph of a second degree trinomial
  • Sequences, elementary functions:
  • Arithmetic and geometric progressions; sum of a geometric series.
  • Exponential functions and logarithmic functions; their graphic representations. Decimal logarithm and its relationship with the decimal representation of numbers.
  • Length of a circle and  arc length of a circle
  • Measurement of angles in radians.
  • Definition of cosine, sine, tangent and first properties
  • Criteria of congruence of triangles and related trigonometric problems: theorem of sines and Carnot's theorem. Graphs of circular functions. Addition theorem for circular functions. Definition of arccosine, arcsine, arctangent functions and their graphs.
  • Area of polygons and equidecomposability. Area of the circle.
  • Volume of a solid. Cavalieri's principle.
  • Volume of the cylinder, the cone, the pyramid, the sphere.
  • Areas and volumes of similar figures.

3.c. Physics prerequisites

  • Kinematics. Newton's laws. Elementary rigid body mechanics. Principles of conservation of energy, linear momentum and angular momentum. Collisions. Law of universal gravitation and Kepler's laws. Mechanical oscillators.
  • Fundamentals of fluid mechanics, Bernoulli's theorem. Wave phenomena and basics of acoustics. Thermology. Gas laws. The first two principles of thermodynamics. thermodynamic cycles. Kinetic theory of gases. Geometric optics: reflection and refraction of light. Wave properties of light: interference and diffraction. Electrostatics. Electric currents. Magnetostatics. electromagnetic induction. Oscillating circuits and electromagnetic waves.
  • Introduction to special relativity.
  • Crisis of classical physics: blackbody, photoelectric effect, Bohr’s atom, Compton effect.