4 January
World Braille Day
24 January
International Day of Education
27 January
International Holocaust Remembrance Day
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Further Mathematics
Academic Year 2019-2020
2nd semester
Class (large group) meetings (48 h) and seminar/laboratory meetings (12 h)
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Dates
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Topics
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Basic texts readings and study
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Following is a selection of training exercises, essentially instrumental. Do not forget those that are worked in class and seminar/lab meetings. The concreteness of the examples implies in no case a course content cut. It is strongly recommended to solve exercises and other questions, the more the better. There is a more than enough bibliography to which to consult.
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Rosen 5th ed. Spain/USA
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Rosen 7th ed. USA
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Others
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Rosen 5th ed. Spain/USA
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Rosen 7th ed. USA
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Others
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Theme 1
Fundamentals
(16 h LG and 5 h S/L)
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Wed
29/1
Start date of classes
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Symbolic logic, I: Propositional logic
- Propositional logic;
- The method of truth tables as a verification strategy.
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- § 1.1 (8, 15, 30, 48, 42, 51-55, 59).
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- § 1.1 (12, 21, 38);
- § 1.2 (12, 16, 19-23, 35).
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Thu
30/1
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Symbolic logic, I: Propositional logic
- Propositional equivalences;
- Formal derivation.
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University project Discrete and numerical mathematics (optional out-of-class activity): Beginning date of the academic component of the project in the 2nd semester of the academic year 2018-2019. You should read its descriptive web page, Wikipedia:School and university projects/Discrete and numerical mathematics. Once you have read that web page, and if you are interested in the project and only if you have queries or need help to do what you have been told (on that web page) to do or want to help your colleagues to do it or want to share questions, concerns or suggestions about the project, you could attend at 4:00 p.m., to Room C8 (meeting will finish at no later than 5:30 p.m.). (Bring a computer if you need help). (This meeting will be in Spanish).
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Group B
Fri
31/1
Groups A and E
Mon
3/2
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Seminar/Laboratory No. 1: Proofs and refutations, I
- (Occasionally computer-assisted) hands-on word-problem-solving on issues related to formal and informal arguments, essentially using:
- truth tables,
- reductio ad absurdum, or
- normal forms.
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- § 1.1, 1.2, 1.3;
- § 1.7.7;
- —.
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- Rosen 7th Global Edition: § 1.7 (normal forms);
- WP+ (Logic).
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Tue
4/2
World Cancer Day
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Symbolic logic, I: Propositional logic
- Reductio ad absurdum (proof by contradiction);
- Normal forms.
Symbolic logic, II: Predicate logic
- Predicates, variables, quantifiers, negation of quantifiers and logical equivalences.
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- Rosen 7th Global Edition: § 1.7 (normal forms);
- WP+ (Logic).
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- Rosen 7th Global Edition: § 1.7 (normal forms) (1, 2, 3, 4, 5, 6);
- WP+ (Logic).
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Wed
5/2
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Symbolic logic, II: Predicate logic
- Reverse translation (from English to predicate logic).
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- § 1.3 (8, 10, 22, 41, 48, 55).
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- § 1.4 (8, 10, 24, 43, 52, 59).
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Thu
6/2
International Day of Zero Tolerance for Female Genital Mutilation
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Symbolic logic, II: Predicate logic
- Nested quantifiers; order of quantifiers; negating nested quantifiers;
- Translation from predicate logic to English;
- Reverse translation (from English to predicate logic).
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- § 1.4 (5, 8, 13, 18, 21, 28, 37, 44).
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- § 1.5 (5, 8, 13, 18, 21, 28, 39, 48).
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Group B
Fri
7/2
Groups A and E
Mon
10/2
World Pulses Day
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Seminar/Laboratory No. 2: Proofs and refutations, II
- (Occasionally computer-assisted) hands-on word-problem-solving on issues related to formal and informal arguments, essentially using:
- natural deduction calculus (Gentzen-style).
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- § 1.5 (particularly, § 1.5.3, 1.5.6).
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- § 1.6 (particularly, § 1.6.4, 1.6.7).
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Tue
11/2
International Day of Women and Girls in Science
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Symbolic logic, III: Proofs
- Valid arguments and rules of inference.
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- § 1.5.1, 1.5.2, 1.5.3, 1.5.4, 1.5.5, 1.5.6.
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Wed
12/2
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Symbolic logic, III: Proofs
- Introduction to proofs;
- Proof methods and strategy.
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- § 1.5.7, 1.5.8, 1.5.9, 1.5.10;
- § 3.1.
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- § 1.5 (20, 22, 30, 32, 35, 46, 58);
- § 3.1 (11, 13, 14, 19, 20, 27, 32, 38, 44, 49, 51).
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- § 1.7 (14, 18, 24);
- § 1.8 (3, 7, 20, 23, 25, 26, 27, 42).
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Thu
13/2
World Radio Day
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Sets
- Sets;
- Set operations; Boolean algebra; partitions.
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- § 1.6 (5, 6, 17, 24, 27, 30);
- § 1.7 (2, 10, 12, 20, 29, 34).
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- § 2.1 (7, 8, 23, 32, 41, 46);
- § 2.2 (2, 14, 16, 26, 37, 42).
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Group B
Fri
14/2
Groups A and E
Mon
17/2
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Seminar/Laboratory No. 3: Proofs and refutations, III
- (Occasionally computer-assisted) hands-on word-problem-solving on issues related to formal and informal arguments, essentially using:
- Beth and Hintikka semantic tableaux (Smullyan&Jeffrey-style).
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—
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—
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- Antón and Casañ, 1987: § 3.2;
- Manzano and Huertas, 2006: § 4 (for Propositional logic), § 11 (for First-order logic);
- WP+ (Logic).
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—
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—
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Tue
18/2
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Functions
- Correspondences, functions and mappings; injectivity, surjectivity and bijectivity; inverse function.
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- § 1.8 (6, 8, 12, 13, 16, 17, 27, 29, 36, 45, 69).
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- § 2.3 (6, 8, 12, 13, 20, 21, 35, 37, 44, 53, 77).
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Wed
19/2
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Relations, I
- Relations and their properties (mainly: reflexivity, irreflexivity, symmetry, asymmetry, antisymmetry, transitivity, intransitivity and connexity);
- Representing relations (mainly using: correspondences, sets, cartesian diagrams, binary matrices and directed graphs [digraphs]).
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- § 7.1 (6, 8, 13, 20, 32, 34, 38);
- § 7.3 (10, 14, 26, 36).
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- § 9.1 (6, 10, 15, 22, 34, 36, 40);
- § 9.3 (10, 14, 26, 36).
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University project Discrete and numerical mathematics (optional out-of-class activity): (First checkpoint). Due date for having joined the English-language Wikipedia, if not yet, and for having chosen the articles of which you become responsible (follow the indications on the project page and on the contributions page).
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Thu
20/2
World Day of Social Justice
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Relations, II
- Equivalence relations;
- Tolerance (or compatibility) relations;
- Partial, linear and strict preorders and orders; Hasse diagrams.
- Preference and indiference relations.
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- § 7.5 (3, _, 7, 8, 10, 18, 26, 29, 31, 46, 48);
- § 7.6 (2, 3, 4, 5, 10, 13, 16, 28, 32, 36, 49, 51, 56, 59).
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- § 9.5 (3, 8, 11, 12, 16, 24, 36, 41, 43, 60, 62);
- § 9.6 (8, 9, 10, 11, 16, 19, 22, 34, 38, 42, 55, 57, 62, 67).
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Group B
Fri
21/2
International Mother Language Day
Groups A and E
Mon
24/2
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Seminar/Laboratory No. 4: Induction and recursion
- (Occasionally computer-assisted) hands-on word-problem-solving on issues related to:
- mathematical induction;
- strong induction and well order;
- structural induction.
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- (§ 1.5.7, 1.5.8, 1.5.9, 1.5.10; § 3.1);
- § 3.3;
- § 3.3;
- § 3.4.
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- (§ 1.7; § 1.8);
- § 5.1;
- § 5.2;
- § 5.3.
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- G. Polya. How to solve it. Princeton, New Jersey (US-NJ), USA: Princeton University Press;
- WP+ (Logic).
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- § 3.3 (8, 10, 6, 14, 11, 20, 25, 50, 38, 60);
- § 3.3 (_, 34, _);
- § 3.4 (__, __, __);
- León-Rojas, J. M., KDEM DA/HD – 703-01.
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- (León-Rojas, J. M., KDEM DA/HD – 703-01);
- § 5.1 (4, 6, 10, 18, 25, 32, 45, 50, 54, 62); § 5.2 (4, 10, 26).
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Tue
25/2
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Relations, III
- Solving questions on relations.
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—
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—
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—
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—
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Is there something greater than infinity? (Cardinality, I)
- Countable sets:
, and are countable sets.
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- § 3.2.5 (31, 32, 34, 38);
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Wed
26/2
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Is there something greater than infinity? (Cardinality, II)
is an uncountable set;
- Computability;
- Cantor's Theorem and the Continuum Hypothesis.
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- § 3.2.5;
- § 3.2.5: exercises 41, 42, 43;
- —.
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- § 2.5.3;
- § 2.5.3 and exercises 37, 38, 39;
- § 2.5.3.
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- § 3.2.5 (31, 32, 34, 38);
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Thu
27/2
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Algebraic structures, I
- Algebraic structures;
- Semigroups, monoids and groups;
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—
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—
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—
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—
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- Rosen 7th Global Edition: § 12.1 (1, 2);
- Rosen 7th Global Edition: § 12.2 (2, 4, 5, 8, 12, 17, 18, 19, 20, 26, 31, 36, 40);
- WP+ (Algebraic structures).
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Group B
Fri
28/2
Groups A and E
Mon
2/3
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Seminar/Laboratory No. 5: Cardinality and Algebraic Structures
- (Occasionally computer-assisted) hands-on word-problem-solving on issues related to:
- cardinality;
- algebraic structures.
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—
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—
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- Rosen 7th Global Edition: § 12.1;
- Rosen 7th Global Edition: § 12.2;
- Rosen 7th Global Edition: § 12.3;
- Rosen 7th Global Edition: § 12.4;
- WP+ (Algebraic structures).
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—
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—
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- Rosen 7th Global Edition: § 12.1 (1, 2);
- Rosen 7th Global Edition: § 12.2 (2, 4, 5, 8, 12, 17, 18, 19, 20, 26, 31, 36, 40);
- Rosen 7th Global Edition: § 12.3 (4, 5, 7, 8);
- Rosen 7th Global Edition: § 12.4 (2, 3, 4, 5);
- WP+ (Algebraic structures);
- León-Rojas, J. M., Question selection no. 5 and 5th seminar/laboratory. Cardinality and Algebraic structures.
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Sun
1/3
Zero Discrimination Day
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Tue
3/3
World Wildlife Day
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Algebraic structures, II
- Homomorphisms;
- Rings, integral domains and fields.
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—
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Wed
4/3
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Algebraic structures, III
- Solving some questions on algebraic structures.
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—
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—
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- Rosen 7th Global Edition: § 12.1;
- Rosen 7th Global Edition: § 12.2;
- Rosen 7th Global Edition: § 12.3;
- Rosen 7th Global Edition: § 12.4;
- WP+ (Algebraic structures).
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—
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—
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- Rosen 7th Global Edition: § 12.1 (1, 2);
- Rosen 7th Global Edition: § 12.2 (2, 4, 5, 8, 12, 17, 18, 19, 20, 26, 31, 36, 40);
- Rosen 7th Global Edition: § 12.3 (4, 5, 7, 8);
- Rosen 7th Global Edition: § 12.4 (2, 3, 4, 5);
- WP+ (Algebraic structures).
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Theme 2
Number theory
(9 h LG and 3 h S/L)
(1h LG Solving the mid-course preparatory exam)
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Thu
5/3
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Divisibility and modular arithmetic
- Divisibility;
- The division algorithm;
- Modular arithmetic.
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- § 2.4.1, 2.4.2;
- § 2.4.4;
- § 2.4.6.
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- § 4.1.1, 4.1.2;
- § 4.1.3;
- § 4.1.4, 4.1.5.
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- § 2.4 (5, 6, 7);
- § 2.4 (10, 22, 34, 36);
- § 2.4 (38, 42, 44).
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- § 4.1 (5, 6, 7);
- § 4.1 (10, 16, 18, 20);
- § 4.1 (26, 34, 36).
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Group B
Fri
6/3
Groups A and E
Mon
9/3
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Seminar/Laboratory No. 6: Divisibility, modular arithmetic, primes, GCD and congruences
- (Occasionally computer-assisted) hands-on word-problem-solving on issues related to:
- divisibility;
- modular arithmetic;
- primes;
- GCD;
- congruences.
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- (Everything we have studied on the subject).
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- (Everything we have studied on the subject).
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- (Everything we have studied on the subject).
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- (Everything we have studied on the subject).
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Sun
8/3
International Women's Day
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Tue
10/3
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Primes
- Prime numbers;
- The fundamental theorem of arithmetic.
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- § 4.3.1, 4.3.2, 4.3.3, 4.3.4, 4.3.5;
- § 4.3.2.
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- § 2.4 (8, 12, 14, 15, 20, 24, 25, 26, 27).
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- § 4.3 (2, 4, 6, 11, 18, 20, 21, 22, 23).
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Wed
11/3
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Greatest common divisor (GCD)
- GCD and LCM;
- The Euclidean algorithm;
- Bézout's theorem and the extended Euclidean algorithm.
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- § 2.4.5;
- § 2.5.5;
- § 2.6.2 and p. 180.
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- § 4.3.6;
- § 4.3.7;
- § 4.3.8 and p. 273.
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- § 2.4 (17, 28);
- § 2.5 (21, 22);
- § 2.5 (2, 50).
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- § 4.3 (15, 24);
- § 4.3 (33, 32);
- § 4.3 (40, 44).
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Thu
12/3
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Solving congruences, I
- Linear congruences;
- The Chinese remainder theorem;
- Computer arithmetic with large integers.
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- § 2.6.3;
- § 2.6.4;
- § 2.6.5.
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- § 4.4.2;
- § 4.4.3;
- § 4.4.4.
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- § 2.6 (4, 5, 6, 7, 8, ...);
- § 2.6 (... ... ...);
- § 2.6 (... ... ...).
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- § 4.4 (2, 5a, 6a, 5b, 6c, ...);
- § 4.4 (... ... ...);
- § 4.4 (... ... ...).
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Group B
Fri
13/3
Groups A and E
Mon
16/3
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Seminar/Laboratory No. 7: Diophantine and congruence equations, I
- (Occasionally computer-assisted) hands-on word-problem-solving on issues related to:
- diophantine equations;
- congruence equations.
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- —;
- § 2.6.8, 2.6.9, 2.6.10.
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- —;
- § 4.6.4, 4.6.5, 4.6.6, 4.6.7, 4.6.8.
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Tue
17/3
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Solving congruences, II
- Fermat's little theorem and pseudoprimes; Euler's theorem and Wilson's theorem.
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- § 4.4.5 and § 4.4.6; p. 285.
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- § 2.6 (17, 28, 32, 34, 43, 44, 52, 56).
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- § 4.4 (19, 38, 46, 48,41, 42, 58, 62).
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Wed
18/3
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Divisibility rules
- Power residues and divisibility rules.
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Thu
19/3
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Diophantine equations
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Group B
Fri
20/3
International Francophonie Day
International Day of Happiness
Groups A and E
Mon
23/3
World Meteorological Day
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Seminar/Laboratory No. 8: Diophantine and congruence equations, II
- (Occasionally computer-assisted) hands-on word-problem-solving on issues related to:
- diophantine equations;
- congruence equations;
- applications of congruences.
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- (Everything we have studied on the subject).
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- (Everything we have studied on the subject).
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- (Everything we have studied on the subject).
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- (Everything we have studied on the subject).
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- (Everything we have studied on the subject).
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Sat
21/3
World Poetry Day
International Day for the Elimination of Racial Discrimination
International Nowruz Day (es)
World Down Syndrome Day
International Day of Forests
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Sun
22/3
World Water Day
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Tue
24/3
International Day for the Right to the Truth Concerning Gross Human Rights Violations and for the Dignity of Victims (es)
World Tuberculosis Day
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Applications of congruences, I
- Hashing functions (optional);
- Pseudorandom numbers (optional);
- Cryptography, I.
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- § 2.4.7;
- § 2.4.7;
- § 2.6.7, 2.6.8, 2.6.9, 2.6.10.
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- § 4.5.1;
- § 4.5.2;
- § 4.6.4, 4.6.5, 4.6.6, 4.6.7.
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- § 2.4 (48, 49);
- § 2.4 (50, 51, 52);
- § 2.6 (45, 46, 47).
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- § 4.5 (2, 3);
- § 4.5 (6, 7, 8);
- § 4.6 (23, 24, 27).
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Wed
25/3
Annunciation
International Day of Solidarity with Detained and Missing Staff Members
International Day of Remembrance of the Victims of Slavery and the Transatlantic Slave Trade
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Applications of congruences, II
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- § 2.6.7, 2.6.8, 2.6.9, 2.6.10.
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- § 4.6.4, 4.6.5, 4.6.6, 4.6.7.
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Thu
26/3
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Exam review: large-group class dedicated to share ideas and solutions in a whole-class discussion about the mid-course preparatory exam (done as homework).
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Theme 3
Combinatorics
(8 h LG and 3 h S/L)
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Group B
Fri
27/3
Groups A and E
Mon
30/3
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Seminar/Laboratory No. 9: Combinatorics, I
- (Occasionally computer-assisted) hands-on word-problem-solving on issues related to:
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- (Everything we have studied on the subject).
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- (Everything we have studied on the subject).
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- (Everything we have studied on the subject).
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- (Everything we have studied on the subject).
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Tue
31/3
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Combinatorics
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- § 4.1 (6, 16, 22, 28, 33, 37, 41, 51, 52).
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- § 6.1 (8, 16, 26, 32, 37, 41, 49, 67, 68).
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Wed
1/4
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Combinatorics
- The pigeonhole principle.
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- § 4.2 (9, 10, 16, 17, 18, 20, 24, 25, 26, 36).
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- § 6.2 (9, 10, 16, 17, 18, 20, 26, 27, 28, 40).
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Thu
2/4
World Autism Awareness Day
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Combinatorics
- Permutations and combinations;
- Binomial coefficients and identities.
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- § 4.3 (5, 12, 18, 22, 23, 35, 36, 37);
- § 4.4 (4, 8, 20, 22, 24, 33, 34).
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- § 6.3 (5, 12, 18, 22, 23, 35, 36, 37);
- § 6.4 (4, 8, 20, 22, 24, 33, 34).
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University project Discrete and numerical mathematics (optional out-of-class activity): (Second checkpoint). You should have continually been working in your contributions, publishing each update, along with the corresponding themes to which they belong are worked in class, and linking each new major contribution on the contributions page of the project. Furthermore, you must publish, also on an ongoing basis, in your logbook (sandbox), the part of your self-report that deals with what you have developed so far.
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Group B
Fri
3/4
Lent
Friday of Sorrows
Groups A and E
Mon
20/4
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Seminar/Laboratory No. 10: Combinatorics, II
- (Occasionally computer-assisted) hands-on word-problem-solving on issues related to:
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- (Everything we have studied on the subject).
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- (Everything we have studied on the subject).
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- (Everything we have studied on the subject).
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- (Everything we have studied on the subject).
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Sat
4/4
Lazarus Saturday
International Day for Mine Awareness and Assistance in Mine Action (es)
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Mon
6/4
Holy Week
Holy Monday
International Day of Sport for Development and Peace
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Tue
7/4
Holy Week
Holy Tuesday
World Health Day
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Wed
8/4
Holy Week
Holy Wednesday
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Thu
9/4
Holy Week
Maundy Thursday
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Fri
10/4
Holy Week
Good Friday
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Sun
12/4
Holy Week
Resurrection Sunday
International Day of Human Space Flight
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Mon
13/4
Eastertide
Easter Monday
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Tue
14/4
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Combinatorics
- Generalised permutations and combinations (variations, combinations and permutations, with repetition).
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- § 4.5.1, 4.5.2, 4.5.3, 4.5.4.
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- § 6.5.1, 6.5.2, 6.5.3, 6.5.4.
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- § 4.5 (10, 15, 16, 34, 56).
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- § 6.5 (10, 15, 16, 34, 66).
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Wed
15/4
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Combinatorics
- Distributing objects to boxes when the order of objects in each box does not matter and both the objects and the boxes may be distinguishable or not.
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Thu
16/4
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Combinatorics
- Distributing objects to boxes when the order of objects in each box matters and both the objects and the boxes may be distinguishable or not.
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Group B
Fri
17/4
Groups A and E
Mon
4/5
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Seminar/Laboratory No. 11: Combinatorics, III
- (Occasionally computer-assisted) hands-on word-problem-solving on issues related to:
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- (Everything we have studied on the subject).
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- (Everything we have studied on the subject).
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- (Everything we have studied on the subject).
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- (Everything we have studied on the subject).
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Sun
19/4
UN Chinese Language Day
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Tue
21/4
World Creativity and Innovation Day
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Combinatorics
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Wed
22/4
Earth Day
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Combinatorics
- Additive decompositions of numbers.
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Thu
23/4
Saint George's Day
World Book and Copyright Day
UN English Language Day
UN Spanish Language Day
International Girls in ICT Day (es)
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Theme 4
Difference equations
(8 h LG and 2 h S/L)
(1h LG Solving the end-course preparatory exam)
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Group B
Fri
24/4
International Day of Multilateralism and Diplomacy for Peace (ref)
Groups A and E
Mon
27/4
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Seminar/Laboratory No. 12: Difference equations, I
- (Occasionally computer-assisted) hands-on word-problem-solving on issues related to:
- linear finite difference equations.
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- (Everything we have studied on the subject);
- § 6.3.
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- (Everything we have studied on the subject);
- § 8.3.
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- (Everything we have studied on the subject);
- § 6.2 (45, 46, 47);
- § 6.3 (10, 11, 14, 15, 16).
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- (Everything we have studied on the subject);
- § 8.2 (45, 46, 47);
- § 8.3 (10, 11, 14, 15, 16).
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Sat
25/4
World Malaria Day
International Delegate's Day (ref)
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Sun
26/4
World Intellectual Property Day
International Chernobyl Disaster Remembrance Day (ref)
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Tue
28/4
World Day for Safety and Health at Work
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Finite difference equations (recurrence relations)
- Linear finite difference equations, models and applications.
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- (§ 3.2.1, 3.2.2, 3.2.3, 3.2.4);
- § 6.1.
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- § 6.1 (17, 22, 23, 25, 27, 36, 37, 42, 46).
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- § 8.1 (1, 6, 7, 9, 11, 20, 21, 26, 30).
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Wed
29/4
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Finite difference equations (recurrence relations)
- Homogeneous linear finite difference equations with constant coefficients, I.
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- § 6.2 (2, 3, 4, 7, 8, 11, 12, 13, 14, 15, 17, 18).
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- § 8.2 (2, 3, 4, 7, 8, 11, 12, 13, 14, 15, 17, 18).
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Thu
30/4
International Jazz Day
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Finite difference equations (recurrence relations)
- Homogeneous linear finite difference equations with constant coefficients, II.
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- § 6.2 (2, 3, 4, 7, 8, 11, 12, 13, 14, 15, 17, 18).
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- § 8.2 (2, 3, 4, 7, 8, 11, 12, 13, 14, 15, 17, 18).
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Fri
1/5
International Workers' Day
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Sat
2/5
World Tuna Day (es)
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Sun
3/5
World Press Freedom Day
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Tue
5/5
African World Heritage Day
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Finite difference equations (recurrence relations)
- Non-homogeneous linear finite difference equation with constant coefficients, I.
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Wed
6/5
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Finite difference equations (recurrence relations)
- Non-homogeneous linear finite difference equation with constant coefficients, II.
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Thu
7/5
Vesak Day
(Held on the/a full moon day of May each year)
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Finite difference equations (recurrence relations)
- Systems of linear finite difference equations, I.
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University project Discrete and numerical mathematics (optional out-of-class activity): (Third and last checkpoint). You should have continually been working in your contributions, publishing each update, along with the corresponding themes to which they belong are worked in class, and linking each new major contribution on the contributions page of the project. Furthermore, you must publish, also on an ongoing basis, in your logbook (sandbox), the part of your self-report that deals with what you have developed so far (in this case all you have done). Starting from now until the ending date, you can review all what you have done and you can correct minor errors and complete other small details.
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Fri
8/5
Academic celebration, Cáceres School of Technology (es)
Time of Remembrance and Reconciliation for Those Who Lost Their Lives during the Second World War
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Sat
9/5
Time of Remembrance and Reconciliation for Those Who Lost Their Lives during the Second World War
World Migratory Bird Day (es)
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Group B
Mon
11/5
(please come and attend as far as possible)
Groups A and E
Mon
11/5
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Seminario/Laboratorio N.º 13: Difference equations, II
- (Occasionally computer-assisted) hands-on word-problem-solving on issues related to:
- finite difference equations.
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- (Everything we have studied on the subject).
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- (Everything we have studied on the subject).
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- (Everything we have studied on the subject).
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- (Everything we have studied on the subject).
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Tue
12/5
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Finite difference equations (recurrence relations)
- Systems of linear finite difference equations, II.
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Wed
13/5
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Finite difference equations (recurrence relations)
- Systems of linear finite difference equations, III.
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Thu
14/5
End of classes
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Exam review: large-group class dedicated to share ideas and solutions in a whole-class discussion about the end-course preparatory exam (done as homework).
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University project Discrete and numerical mathematics (optional out-of-class activity): Ending date of the academic component in the 2nd semester of the academic year 2019-2020.
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15 May
International Day of Families
16 May
International Day of Living Together in Peace (es)
International Day of Light (es)
17 May
World Telecommunication and Information Society Day
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Mon
20/5
Start of June exam period
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20 May
World Bee Day
21 May
World Day for Cultural Diversity for Dialogue and Development
International Tea Day (UN) (ref)
22 May
International Day for Biological Diversity
23 May
International Day to End Obstetric Fistula (es)
29 May
International Day of United Nations Peacekeepers
31 May
World No Tobacco Day
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___ __/_
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2019-2020 Final exam.
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...
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Sat
6/7
End of June exam period
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...
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Mon
22/6
Start of July exam period
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...
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___ __/_
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2019-2020 Resit final exam.
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...
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Fri
10/7
End of July exam period
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...
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Mon
20/7
End of term
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(See: International days currently observed by the United Nations).
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