# Faculty of Sciences, I year undergraduate course

1. DEGREE COURSES IN MATHEMATICS, PHYSICS AND INFORMATION TECHNOLOGY AND SIMILAR

1.a. Mathematics prerequisites

Numerical 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; cancellation law of product. How is the <rule of signs> 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 division with remainder of a by b ...)
· Divisibility, greatest common divisor, least common multiple. Euclid's algorithm for the calculation of the greatest common divisor
· Prime numbers. There is an infinite number of prime numbers: how can we demonstrate this?
· Breakdown of an integer into prime factors. (A precise statement, without proof)

Algebra:

· Elements of literal calculation: use of letters, use of brackets
· 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 for polynomials (A precise statement, even without proof).

Geometry:

· Know the meaning of the terms: axiom (postulate), theorem, lemma, corollary, hypothesis, thesis ... ..
· Elements of plane geometry: incidence, perpendicularity, parallelism.
· The postulate of parallels
· 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, homoteties and similarities)
· Properties of plane shapes, 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, arc of a given angle)
· Measurement of angles; sum of internal angles and of external angles of a convex polygon
· Bijective correspondence between real numbers and points on the line.
· The Cartesian plane: representation of the lines, of the circles; of the parabola, of the ellipse and of the hyperbola (taking the axes appropriately)
· Geometry of space: incidence, perpendicularity and parallelism. Angle between line and plane. Dihedrons and trihedrons.
· 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
· Some elements of combinatorial calculus: given the finite sets A and B, the number of applications of A in B (dispositions with repetitions ...), number of injective applications of A in B (simple dispositions ...). 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 in two equations and two variables, 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
· Graphic of a second degree trinomial

Successions, Elementary Functions:
· Successions; arithmetic and geometric progressions
· Limit of a succession; 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 relation to the decimal representation of numbers.
· Length of the circle and an arc of a circle
· Measurement of angles in radians.
· Definition of the cosine, sine and tangent and their prime properties
· Triangle congruency criteria and related trigonometric problems:
sine theorem and Carnot theorem. Graphs of circular functions. Addition theorem for circular functions. Definition of the functions arc cosine, arc sine, arc tangent, and their graphs.
· Area of a plane region (defined, for example, with division into squares, as in practice with graph paper ... ..)
· Area of polygons and equidecomposibility.  Area of circle.
· Volume of a solid. Cavalieri's principle.
· Volume of the cylinder, of the cone and of the sphere
· Areas and volumes of similar figures.

Complements:

A - Elements of Mathematical Analysis

The 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 from the beginning (in particular, courses in Physics)

It is necessary to know:

· Limits, continuity of functions of a variable
· Derivative of a function; derivation rules
· Increasing and decreasing functions; maxima and minima: convexity and concavity; inflections
· Definite integral and its prime properties
· Primitive function; the fundamental theorem of integral calculus.
· Calculation of areas and volumes

N.B. Also for the purposes of a deeper understanding of the elements of Mathematical Analysis, it is useful to use small programmable calculators: it is very instructive to use them for the calculation of roots of equations, of definite integrals, etc.

B - Elements of linear algebra

With applications to physics, operational research etc.

C - Elements of probability calculus and statistics

These are very important themes in themselves, and can be particularly useful for those who undertake the study of an experimental science.

1.b. Physics prerequisites

Kinematics. Newton's laws. Elementary mechanics of rigid body. Principles of conservation of energy, linear impulse and angular momentum. Collisions. Law of universal gravitation and Kepler's laws. Mechanical oscillators.

Basics of fluid mechanics, Bernoulli's theorem. Wave phenomena and notions of acoustics.

Thermology. Laws of gases. 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.

Elements of special relativity.

Crisis of classical physics: black body, photoelectric effect, Bohr's atom, Compton effect.

2. DEGREE COURSES IN CHEMISTRY, CHEMISTRY AND PHARMACEUTICAL TECHNOLOGY, GEOLOGY AND SIMILAR

2. a. Chemistry Prerequisites

States of matter aggregation. Homogeneous and heterogeneous systems. Phase transitions. Separation and purification of substances. Elements and compounds. Chemical reactions. Ponderal relationships in chemical reactions. Properties of gases, liquids and solids. Properties of solutions. Ionic solutions. Electrolysis. Electronic structure of atoms. Chemical bond and molecular structures. Periodic system of elements. Thermochemistry. Elements of chemical kinetics. Chemical balance. Acid-base reactions, oxidation-reduction reactions, formation reactions of co-ordination compounds. Descriptive inorganic chemistry: hydrogen, halogens, oxygen, sulphur, nitrogen, phosphorus, alkaline and alkaline earth metals. Chemistry of carbon: isomerism and stereoisomerism of organic molecules. Hydrocarbons. Functional groups and nomenclatures. Basic types of organic reactions: addition, substitution and elimination reactions.

2.b. Mathematics prerequisites:
Numerical sets and arithmetic:
Numerical fractions: operations and inequalities.
Decimal representation; in which cases can a fraction be expressed in decimal form?
Rational and relative numbers: properties of operations; cancellation law
of the product.
Inequalities and their fundamental properties; absolute values; approximated 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. Breakdown of an integer into prime factors.

Algebra:
Elements of literal calculus: use of letters, use of brackets.
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 principle of the identity 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 (symmetries, translations and rotations, homoteties and
similitudes). Properties of plane shapes, particularly in relation to
symmetries.
The theorems of Thales, Euclid and Pythagoras.
Vectors and operations on them.
Bijective correspondence between real numbers and points on the line.
The Cartesian plane: representation of lines, circles, parabola,
ellipse and hyperbola.

Geometry of space: incidence, perpendicularity, parallelism. Angle between 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 relations).

Applications (functions). Injective, surjective, bijective applications.

Some elements of combinatorial calculus: given the finite sets A and B, the number of applications of A in B (dispositions with repetitions ), number of injective applications of A in B (simple dispositions). Number of subsets of k elements, in a set of n elements (combinations).

Equations and inequalities:
Linear systems in two equations and two variables, and their interpretations on the
Cartesian plane.
Nth root (in the set of positive real numbers).
Second-degree equations; relations between coefficients and roots. Graph of
a second degree trinomial.

Successions, elementary functions. Arithmetic and geometric progressions; sum of a geometric series.
Exponential functions and logarithmic functions; their graphic representations.
Decimal logarithm and its relation to the decimal representation of numbers.

Length of the circle and of an arc of a circle. Measurement of angles in radians.
Definition of the cosine, sine and tangent and their prime properties.
Congruency of triangles criteria and related trigonometric problems: sine theorem and Carnot theorem. Graphs of circular functions. Addition theorem for circular functions. Definition of the arccosine, arcsine and arctangent functions and their graphs.
Area of polygons and equidecomposibility. Area of circle.
Volume of a solid. Cavalieri's principle. Volume of the cylinder, of the cone and of the sphere.
Areas and volumes of similar figures.

2. c. Physics prerequisites

Kinematics. Newton's laws. Elementary mechanics of the rigid body. Principles of conservation of energy, linear impulse and angular momentum. Collisions. Law of universal gravitation and Kepler's laws. Mechanical oscillators.

Basics of fluid mechanics, Bernoulli's theorem. Wave phenomena and elements of acoustics.

Thermology. Laws of gases. 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.

Elements of special relativity.

Crisis of classical physics: black body, photoelectric effect, Bohr's atom, Compton effect.

3. DEGREE COURSES IN BIOLOGY, NATURAL AND ENVIRONMENTAL SCIENCES AND SIMILAR

3.a. Biology prerequisites

Chemistry of molecules with biological interest:
chemical structure of nucleotides and amino acids
covalent bonds and hydrogen bonds
enantiomers and their biological significance
DNA and RNA structure
proteins: primary, secondary and tertiary structures
phosphorylation of proteins and its biological significance as an intracellular signal
Adenosine-tri-phosphate (ATP): chemical structure and biological significance
Enzymes and enzyme-substrate interactions
Allosteric changes
Mechanism of catalysis of biological reactions by enzymes

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

Cell biology:
Structure of the eukaryotic cell membrane
Intracellular organelles: Golgi apparatus, endoplasmic reticulum, lysosome, their role synthesis and degradation of proteins
Structure of the nucleus: membrane, euchromatin, heterochromatin and nucleolus
mitochondria, cellular respiration and ATP production
The cytoskeleton: microtubules and proteins associated with them
cellular 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 by means of radioactive tracers
DNA sequencing

Biology of organs and systems
Myocyte structure and molecular aspects of muscle contraction
Aerobic and anaerobic musculature
Principles of anatomy and functioning of the heart and the circulatory system
Haemoglobin and oxygen transport
Insulin and other gene encoded hormones; notes on their biological function
Steroid hormones; notes on their biological function
Notes on the immune system. Antibodies, humoral and cell-mediated immune response
Elements of the anatomy of the nervous system: cortex, cerebellum, subcortical structures, spinal cord, peripheral nervous system.
Elements of anatomy of the sense organs: structure of the eye and of the ear
Structure of the neuron: axon, dendrites and synapses.
Neuronal communication: neurotransmitters and their mechanism of action
The action potential and the coding of the signal as a frequency of action potentials

Biology of behaviour
Signals for intraspecific communication
The innate component of the signals used for intraspecific communication
Dominance and structure of social species
Influence of the environment in learning social behaviours (e.g. parental care)
Learning of singing in birds
Learning of language in the human species
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 "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

Numerical 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; cancellation law of product.
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.  Breakdown of an integer into prime factors.

Algebra:
Elements of literal calculation: use of letters, use of brackets
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 for 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 (symmetries, translations and rotations; homoteties and similarities)
Properties of plane shapes, particularly in relation to symmetries
The theorems of Thales, Euclid and Pythagoras
Vectors and operations on them
Bijective correspondence between real numbers and points on the line.
The Cartesian plane: representation of lines, circles, parabola, ellipse and hyperbola.
Geometry of space: incidence, perpendicularity and parallelism. Angle between 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
Some elements of combinatorial calculus: given the finite sets A and B, the number of applications of A in B (dispositions with repetitions), the number of injective applications of A in B (simple dispositions). Number of subsets of k elements, in a set of n elements (combinations).

Equations and Inequalities.
Linear systems in two equations and two variables, 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. Graphic of a second degree trinomial
Successions, Elementary Functions. Arithmetic and geometric progressions; sum of a geometric series.
Exponential functions and logarithmic functions; their graphic representations.
Decimal logarithm and its relation to the decimal representation of numbers.
Length of the circle and of an arc of a circle. Measurement of angles in radians.
Definition of the cosine, sine and tangent and their prime properties
Triangle congruency criteria and related trigonometric problems:
sine theorem and Carnot theorem. Graphs of circular functions. Addition theorem for circular functions. Definition of the functions arccosine, arcsine, arctangent, and their graphs.
Area of polygons and equidecomposibility.  Area of circle.
Volume of a solid. Cavalieri's principle. Volume of the cylinder, of the cone and of the sphere
Areas and volumes of similar figures.

3.c.Physics prerequisites

Kinematics. Newton's laws. Elementary mechanics of rigid body. Principles of conservation of energy, linear impulse and angular momentum. Collisions. Law of universal gravitation and Kepler's laws. Mechanical oscillators.
Basics of fluid mechanics, Bernoulli's theorem. Wave phenomena and notions of acoustics.
Thermology. Laws of gases. 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.
Elements of special relativity.
Crisis of classical physics: black body, photoelectric effect, Bohr's atom, Compton effect.