User:Jmleonrojas/sandbox

Paper generation

• Paper generator
• http://www.plainenglish.co.uk/gobbledygook-generator.html

Tables

Euclid's algorithm example

(where ${\textstyle a,b\in \mathbb {Z} ^{+}}$ and ${\textstyle a>b}$):
${\textstyle a}$ is divided by ${\textstyle b}$, ${\textstyle a=bq_{0}+r_{0}}$, where ${\textstyle 0\leqslant r_{0}<b}$, then,

if ${\textstyle r_{0}=0}$, ${\textstyle \operatorname {gcd} (a,b)=b}$, END;

if ${\textstyle r_{0}\neq 0}$, ${\textstyle \operatorname {gcd} (a,b)=\operatorname {gcd} (b,r_{0})}$,

${\textstyle b}$ is divided by ${\textstyle r_{0}}$, ${\textstyle a=bq_{1}+r_{1}}$, where ${\textstyle 0\leqslant r_{1}<r_{0}}$, then,

if ${\textstyle r_{1}=0}$, ${\textstyle \operatorname {gcd} (b,r_{0})=r_{0}}$, END;

if ${\textstyle r_{1}\neq 0}$, ${\textstyle \operatorname {gcd} (b,r_{0})=\operatorname {gcd} (r_{0},r_{1})}$,

${\textstyle r_{0}}$ is divided by ${\textstyle r_{1}}$, ${\textstyle \ldots }$

so we get a strictly decreasing sequence of remainders ${\textstyle r_{0}>r_{1}>r_{2}>\ldots >r_{k-1}>r_{k}>r_{k+1}=0}$, and thus ${\textstyle \operatorname {gcd} (a,b)=\operatorname {gcd} (b,r_{0})=\operatorname {gcd} (r_{0},r_{1})=\operatorname {gcd} (r_{1},r_{2})=\cdots =\operatorname {gcd} (r_{k-2},r_{k-1})=\operatorname {gcd} (r_{k-1},r_{k})=r_{k}}$, END;

implementation
(given ${\textstyle a,b\in \mathbb {Z} ^{+}}$ and ${\textstyle a>b}$, it calculates the ${\textstyle \operatorname {gcd} (a,b)}$):

$${\begin{aligned}r_{0}&=a\\r_{1}&=b\\&\,\,\,\vdots \\r_{i+1}&=r_{i-1}-q_{i}r_{i}\quad \quad 0\leq r_{i+1}<r_{i}\\&\,\,\,\vdots \end{aligned}}$$

it stops when a null remainder ${\textstyle r_{k+1}=0}$ is reached, and then ${\textstyle \operatorname {gcd} (a,b)=r_{k}}$ (the last non zero remainder);

gcd(4947,1455) = 291

+----------+----------+--------+--------+
|          | q₀ = 3   | q₁ = 2 | q₂ = 2 |
+----------+----------+--------+--------+
| a = 4947 | b = 1455 | 582    | 291    |
+----------+----------+--------+--------+
| r₀ = 582 | r₁ = 291 | r₂ = 0 |        |
+----------+----------+--------+--------+

r₀ > r₁ > r₁₊₁ = 0 then: gcd(4947,1455) = gcd(4947,582) = gcd(582,291) = 291

Extended Euclidean algorithm example

implementation
(given ${\textstyle a,b\in \mathbb {Z} ^{+}}$ and ${\textstyle a>b}$, it calculates the ${\textstyle \operatorname {gcd} (a,b)}$ and two minimal Bézout coefficients ${\textstyle s}$ and ${\textstyle t}$):

$${\begin{aligned}r_{0}&=a&r_{1}&=b\\s_{0}&=1&s_{1}&=0\\t_{0}&=0&t_{1}&=1\\&\,\,\,\vdots &&\,\,\,\vdots \\r_{i+1}&=r_{i-1}-q_{i}r_{i}&0&\leq r_{i+1}<r_{i}&\\s_{i+1}&=s_{i-1}-q_{i}s_{i}\\t_{i+1}&=t_{i-1}-q_{i}t_{i}\\&\,\,\,\vdots \end{aligned}}$$

it stops when a null remainder ${\textstyle r_{k+1}=0}$ is reached, and then ${\textstyle \operatorname {gcd} (a,b)=r_{k}}$ (the last non zero remainder), ${\textstyle s=s_{k}}$ and ${\textstyle t=t_{k}}$, that is, ${\textstyle \operatorname {gcd} (a,b)=r_{k}=as_{k}+bt_{k}}$;

History

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Agenda de actividades

4 January World Braille Day 24 January International Day of Education 27 January International Holocaust Remembrance Day
Further Mathematics Academic Year 2019-2020 2nd semester Class (large group) meetings (48 h) and seminar/laboratory meetings (12 h)
Dates	Topics	Basic texts readings and study			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.
		Rosen 5th ed. Spain/USA	Rosen 7th ed. USA	Others	Rosen 5th ed. Spain/USA	Rosen 7th ed. USA	Others
Theme 1 Fundamentals (16 h LG and 5 h S/L)
Wed 29/1 Start date of classes	Symbolic logic, I: Propositional logic Propositional logic; The method of truth tables as a verification strategy.	§ 1.1.	§ 1.1; § 1.2.	WP+ (Logic).	§ 1.1 (8, 15, 30, 48, 42, 51-55, 59).	§ 1.1 (12, 21, 38); § 1.2 (12, 16, 19-23, 35).	WP+ (Logic).
Thu 30/1	Symbolic logic, I: Propositional logic Propositional equivalences; Formal derivation.	§ 1.2.	§ 1.3.	WP+ (Logic).	§ 1.2 (8, 10, 29, 51).	§ 1.3 (10, 12, 29, 57).	WP+ (Logic).
	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).
Group B Fri 31/1 Groups A and E Mon 3/2	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.	§ 1.1, 1.2; § 1.5.7; —.	§ 1.1, 1.2, 1.3; § 1.7.7; —.	Rosen 7th Global Edition: § 1.7 (normal forms); WP+ (Logic).	León-Rojas, J. M., KDEM DA/HD – 703-01.	León-Rojas, J. M., KDEM DA/HD – 703-01.	Rosen 7th Global Edition: § 1.7 (1, 2, 3, 4, 5, 6); WP+ (Logic); León-Rojas, J. M., Question selection no. 1 and 1st seminar/laboratory. Proofs and refutations, I.
Tue 4/2 World Cancer Day	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.	§ 1.5.7; — § 1.3.	§ 1.7.7; — § 1.4.	Rosen 7th Global Edition: § 1.7 (normal forms); WP+ (Logic).	— § 1.3 (8, 10, 22, 41, 48, 55); León-Rojas, J. M., KDEM DA/HD – 703-01.	— § 1.4 (8, 10, 24, 43, 52, 59); León-Rojas, J. M., KDEM DA/HD – 703-01.	Rosen 7th Global Edition: § 1.7 (normal forms) (1, 2, 3, 4, 5, 6); WP+ (Logic).
Wed 5/2	Symbolic logic, II: Predicate logic Reverse translation (from English to predicate logic).	§ 1.3.	§ 1.4.	WP+ (Logic).	§ 1.3 (8, 10, 22, 41, 48, 55).	§ 1.4 (8, 10, 24, 43, 52, 59).	WP+ (Logic).
Thu 6/2 International Day of Zero Tolerance for Female Genital Mutilation	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).	§ 1.4.	§ 1.5.	WP+ (Logic).	§ 1.4 (5, 8, 13, 18, 21, 28, 37, 44).	§ 1.5 (5, 8, 13, 18, 21, 28, 39, 48).	WP+ (Logic).
Group B Fri 7/2 Groups A and E Mon 10/2 World Pulses Day	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).	§ 1.5 (particularly, § 1.5.3, 1.5.6).	§ 1.6 (particularly, § 1.6.4, 1.6.7).	WP+ (Logic).	León-Rojas, J. M., KDEM DA/HD – 703-01	León-Rojas, J. M., KDEM DA/HD – 703-01	WP+ (Logic); León-Rojas, J. M., Question selection no. 2 and 2nd seminar/laboratory. Proofs and refutations, II.
Tue 11/2 International Day of Women and Girls in Science	Symbolic logic, III: Proofs Valid arguments and rules of inference.	§ 1.5.1, 1.5.2, 1.5.3, 1.5.4, 1.5.5, 1.5.6.	§ 1.6.	WP+ (Logic).	§ 1.5 (10, 12).	§ 1.6 (14, 16).	WP+ (Logic).
Wed 12/2	Symbolic logic, III: Proofs Introduction to proofs; Proof methods and strategy.	§ 1.5.7, 1.5.8, 1.5.9, 1.5.10; § 3.1.	§ 1.7; § 1.8.	WP+ (Logic).	§ 1.5 (20, 22, 30, 32, 35, 46, 58); § 3.1 (11, 13, 14, 19, 20, 27, 32, 38, 44, 49, 51).	§ 1.7 (14, 18, 24); § 1.8 (3, 7, 20, 23, 25, 26, 27, 42).	WP+ (Logic).
Thu 13/2 World Radio Day	Sets Sets; Set operations; Boolean algebra; partitions.	§ 1.6; § 1.7.	§ 2.1; § 2.2.	WP+ (Sets, relations and functions).	§ 1.6 (5, 6, 17, 24, 27, 30); § 1.7 (2, 10, 12, 20, 29, 34).	§ 2.1 (7, 8, 23, 32, 41, 46); § 2.2 (2, 14, 16, 26, 37, 42).	WP+ (Sets, relations and functions).
Group B Fri 14/2 Groups A and E Mon 17/2	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).	—	—	Antón and Casañ, 1987: § 3.2; Manzano and Huertas, 2006: § 4 (for Propositional logic), § 11 (for First-order logic); WP+ (Logic).	—	—	WP+ (Logic); León-Rojas, J. M., Question selection no. 3 and 3rd seminar/laboratory. Proofs and refutations, III.
Tue 18/2	Functions Correspondences, functions and mappings; injectivity, surjectivity and bijectivity; inverse function.	§ 1.8.	§ 2.3.	WP+ (Sets, relations and functions).	§ 1.8 (6, 8, 12, 13, 16, 17, 27, 29, 36, 45, 69).	§ 2.3 (6, 8, 12, 13, 20, 21, 35, 37, 44, 53, 77).	WP+ (Sets, relations and functions).
Wed 19/2	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]).	§ 7.1; § 7.3.	§ 9.1; § 9.3.	WP+ (Sets, relations and functions).	§ 7.1 (6, 8, 13, 20, 32, 34, 38); § 7.3 (10, 14, 26, 36).	§ 9.1 (6, 10, 15, 22, 34, 36, 40); § 9.3 (10, 14, 26, 36).	WP+ (Sets, relations and functions).
	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).
Thu 20/2 World Day of Social Justice	Relations, II Equivalence relations; Tolerance (or compatibility) relations; Partial, linear and strict preorders and orders; Hasse diagrams. Preference and indiference relations.	§ 7.5; § —; § 7.6; § —.	§ 9.5; § —; § 9.6; § —.	WP+ (Sets, relations and functions).	§ 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).	§ 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).	WP+ (Sets, relations and functions).
Group B Fri 21/2 International Mother Language Day Groups A and E Mon 24/2	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.	(§ 1.5.7, 1.5.8, 1.5.9, 1.5.10; § 3.1); § 3.3; § 3.3; § 3.4.	(§ 1.7; § 1.8); § 5.1; § 5.2; § 5.3.	G. Polya. How to solve it. Princeton, New Jersey (US-NJ), USA: Princeton University Press; WP+ (Logic).	§ 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.	(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).	G. Polya. How to solve it. Princeton, New Jersey (US-NJ), USA: Princeton University Press; WP+ (Logic); León-Rojas, J. M., Question selection no. 4 and 4th seminar/laboratory. Induction and recursion.
Tue 25/2	Relations, III Solving questions on relations.	—	—	WP+ (Sets, relations and functions).	—	—	WP+ (Sets, relations and functions). León-Rojas, J. M., Question selection no. 4 and 4th seminar/laboratory. Induction and recursion.
	Is there something greater than infinity? (Cardinality, I) Countable sets: $\mathbb {N}$, $\mathbb {Z}$ and $\mathbb {Q}$ are countable sets.	§ 3.2.5.	§ 2.5.1, 2.5.2.	WP+ (Cardinality).	§ 3.2.5 (31, 32, 34, 38);	§ 2.5 (1, 4, 16, 28).	WP+ (Cardinality).
Wed 26/2	Is there something greater than infinity? (Cardinality, II) $\mathbb {R}$ is an uncountable set; Computability; Cantor's Theorem and the Continuum Hypothesis.	§ 3.2.5; § 3.2.5: exercises 41, 42, 43; —.	§ 2.5.3; § 2.5.3 and exercises 37, 38, 39; § 2.5.3.	WP+ (Cardinality).	§ 3.2.5 (31, 32, 34, 38);	§ 2.5 (1, 4, 16, 28).	WP+ (Cardinality).
Thu 27/2	Algebraic structures, I Algebraic structures; Semigroups, monoids and groups;	—	—	Rosen 7th Global Edition: § 12.1; Rosen 7th Global Edition: § 12.2; WP+ (Algebraic structures).	—	—	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).
Group B Fri 28/2 Groups A and E Mon 2/3	Seminar/Laboratory No. 5: Cardinality and Algebraic Structures (Occasionally computer-assisted) hands-on word-problem-solving on issues related to: cardinality; algebraic structures.	—	—	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).	—	—	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.
Sun 1/3 Zero Discrimination Day
Tue 3/3 World Wildlife Day	Algebraic structures, II Homomorphisms; Rings, integral domains and fields.	—	—	Rosen 7th Global Edition: § 12.3; Rosen 7th Global Edition: § 12.4; WP+ (Algebraic structures).	—	—	Rosen 7th Global Edition: § 12.3 (4, 5, 7, 8); Rosen 7th Global Edition: § 12.4 (2, 3, 4, 5); WP+ (Algebraic structures).
Wed 4/3	Algebraic structures, III Solving some questions on algebraic structures.	—	—	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).	—	—	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).
Theme 2 Number theory (9 h LG and 3 h S/L) (1h LG Solving the mid-course preparatory exam)
Thu 5/3	Divisibility and modular arithmetic Divisibility; The division algorithm; Modular arithmetic.	§ 2.4.1, 2.4.2; § 2.4.4; § 2.4.6.	§ 4.1.1, 4.1.2; § 4.1.3; § 4.1.4, 4.1.5.	González Gutiérrez, 2004a, § 10.1, 10.4; González Gutiérrez, 2004d, § 13.1; WP+ (Number theory).	§ 2.4 (5, 6, 7); § 2.4 (10, 22, 34, 36); § 2.4 (38, 42, 44).	§ 4.1 (5, 6, 7); § 4.1 (10, 16, 18, 20); § 4.1 (26, 34, 36).	González Gutiérrez, 2004a, § 10.1, 10.4; González Gutiérrez, 2004d, § 13.1; WP+ (Number theory).
Group B Fri 6/3 Groups A and E Mon 9/3	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.	(Everything we have studied on the subject).	(Everything we have studied on the subject).	(Everything we have studied on the subject); WP+ (Number theory).	(Everything we have studied on the subject).	(Everything we have studied on the subject).	(Everything we have studied on the subject); WP+ (Number theory); León-Rojas, J. M., Question selection no. 6 and 6th seminar/laboratory. Divisibility, modular arithmetic, primes, GCD and congruences.
Sun 8/3 International Women's Day
Tue 10/3	Primes Prime numbers; The fundamental theorem of arithmetic.	§ 2.4.3.	§ 4.3.1, 4.3.2, 4.3.3, 4.3.4, 4.3.5; § 4.3.2.	González Gutiérrez, 2004b; WP+ (Number theory).	§ 2.4 (8, 12, 14, 15, 20, 24, 25, 26, 27).	§ 4.3 (2, 4, 6, 11, 18, 20, 21, 22, 23).	González Gutiérrez, 2004b; WP+ (Number theory).
Wed 11/3	Greatest common divisor (GCD) GCD and LCM; The Euclidean algorithm; Bézout's theorem and the extended Euclidean algorithm.	§ 2.4.5; § 2.5.5; § 2.6.2 and p. 180.	§ 4.3.6; § 4.3.7; § 4.3.8 and p. 273.	González Gutiérrez, 2004a, § 10.6, 10.7, 10.8; WP+ (Number theory).	§ 2.4 (17, 28); § 2.5 (21, 22); § 2.5 (2, 50).	§ 4.3 (15, 24); § 4.3 (33, 32); § 4.3 (40, 44).	González Gutiérrez, 2004a, § 10.6, 10.7, 10.8; WP+ (Number theory).
Thu 12/3	Solving congruences, I Linear congruences; The Chinese remainder theorem; Computer arithmetic with large integers.	§ 2.6.3; § 2.6.4; § 2.6.5.	§ 4.4.2; § 4.4.3; § 4.4.4.	WP+ (Number theory).	§ 2.6 (4, 5, 6, 7, 8, ...); § 2.6 (... ... ...); § 2.6 (... ... ...).	§ 4.4 (2, 5a, 6a, 5b, 6c, ...); § 4.4 (... ... ...); § 4.4 (... ... ...).	WP+ (Number theory).
Group B Fri 13/3 Groups A and E Mon 16/3	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.	—; § 2.6.8, 2.6.9, 2.6.10.	—; § 4.6.4, 4.6.5, 4.6.6, 4.6.7, 4.6.8.	González Gutiérrez, 2004c; WP+ (Number theory).	—; § 2.6 (46, 47, 45*).	—; § 4.6 (24, 27, 23*).	González Gutiérrez, 2004c; WP+ (Number theory); León-Rojas, J. M., Question selection no. 7 and 7th seminar/laboratory. Diophantine and congruence equations, I.
Tue 17/3	Solving congruences, II Fermat's little theorem and pseudoprimes; Euler's theorem and Wilson's theorem.	§ 2.6.6; p. 179.	§ 4.4.5 and § 4.4.6; p. 285.	WP+ (Number theory).	§ 2.6 (17, 28, 32, 34, 43, 44, 52, 56).	§ 4.4 (19, 38, 46, 48,41, 42, 58, 62).	WP+ (Number theory).
Wed 18/3	Divisibility rules Power residues and divisibility rules.	—	—	González Gutiérrez, 2004a, § 10.5; Sendra, Martín, Méndez and Ortiz, 2011; WP+ (Number theory).	—	—	González Gutiérrez, 2004a, § 10.5; Sendra, Martín, Méndez and Ortiz, 2011; WP+ (Number theory).
Thu 19/3	Diophantine equations Diophantine equations.	—	—	González Gutiérrez, 2004c; WP+ (Number theory).	—	—	González Gutiérrez, 2004c; WP+ (Number theory).
Group B Fri 20/3 International Francophonie Day International Day of Happiness Groups A and E Mon 23/3 World Meteorological Day	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.	(Everything we have studied on the subject).	(Everything we have studied on the subject).	(Everything we have studied on the subject).	(Everything we have studied on the subject).	(Everything we have studied on the subject).	(Everything we have studied on the subject); León-Rojas, J. M., Question selection no. 8 and 8th seminar/laboratory. Diophantine and congruence equations, II.
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
Sun 22/3 World Water Day
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	Applications of congruences, I Hashing functions (optional); Pseudorandom numbers (optional); Cryptography, I.	§ 2.4.7; § 2.4.7; § 2.6.7, 2.6.8, 2.6.9, 2.6.10.	§ 4.5.1; § 4.5.2; § 4.6.4, 4.6.5, 4.6.6, 4.6.7.	WP+ (Number theory).	§ 2.4 (48, 49); § 2.4 (50, 51, 52); § 2.6 (45, 46, 47).	§ 4.5 (2, 3); § 4.5 (6, 7, 8); § 4.6 (23, 24, 27).	WP+ (Number theory).
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	Applications of congruences, II Cryptography, II. RSA.	§ 2.6.7, 2.6.8, 2.6.9, 2.6.10.	§ 4.6.4, 4.6.5, 4.6.6, 4.6.7.	WP+ (Number theory).	§ 2.6 (45, 46, 47).	§ 4.6 (23, 24, 27).	WP+ (Number theory).
Thu 26/3	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).
Theme 3 Combinatorics (8 h LG and 3 h S/L)
Group B Fri 27/3 Groups A and E Mon 30/3	Seminar/Laboratory No. 9: Combinatorics, I (Occasionally computer-assisted) hands-on word-problem-solving on issues related to: combinatorics.	(Everything we have studied on the subject).	(Everything we have studied on the subject).	(Everything we have studied on the subject); Franco, Espinel and Almeida, 2008; Sedgewick, 1977; WP+ (Combinatorics).	(Everything we have studied on the subject).	(Everything we have studied on the subject).	(Everything we have studied on the subject); Franco, Espinel and Almeida, 2008; Sedgewick, 1977; WP+ (Combinatorics); León-Rojas, J. M., Question selection no. 9 and 9th seminar/laboratory. Combinatorics, I.
Tue 31/3	Combinatorics The basics of counting.	§ 4.1.	§ 6.1.	Franco, Espinel and Almeida, 2008, § 1, § 3.1, § 3.1.1; WP+ (Combinatorics).	§ 4.1 (6, 16, 22, 28, 33, 37, 41, 51, 52).	§ 6.1 (8, 16, 26, 32, 37, 41, 49, 67, 68).	Franco, Espinel and Almeida, 2008, § 1, § 3.1, § 3.1.1; WP+ (Combinatorics).
Wed 1/4	Combinatorics The pigeonhole principle.	§ 4.2.	§ 6.2.	Franco, Espinel and Almeida, 2008, § 1, § 3.1, § 3.1.1; WP+ (Combinatorics).	§ 4.2 (9, 10, 16, 17, 18, 20, 24, 25, 26, 36).	§ 6.2 (9, 10, 16, 17, 18, 20, 26, 27, 28, 40).	Franco, Espinel and Almeida, 2008, § 1, § 3.1, § 3.1.1; WP+ (Combinatorics).
Thu 2/4 World Autism Awareness Day	Combinatorics Permutations and combinations; Binomial coefficients and identities.	§ 4.3; § 4.4	§ 6.3; § 6.4.	Franco, Espinel and Almeida, 2008, § 4, § 5; WP+ (Combinatorics).	§ 4.3 (5, 12, 18, 22, 23, 35, 36, 37); § 4.4 (4, 8, 20, 22, 24, 33, 34).	§ 6.3 (5, 12, 18, 22, 23, 35, 36, 37); § 6.4 (4, 8, 20, 22, 24, 33, 34).	Franco, Espinel and Almeida, 2008, § 4, § 5; WP+ (Combinatorics).
	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.
Group B Fri 3/4 Lent Friday of Sorrows Groups A and E Mon 20/4	Seminar/Laboratory No. 10: Combinatorics, II (Occasionally computer-assisted) hands-on word-problem-solving on issues related to: combinatorics.	(Everything we have studied on the subject).	(Everything we have studied on the subject).	(Everything we have studied on the subject); Franco, Espinel and Almeida, 2008; WP+ (Combinatorics).	(Everything we have studied on the subject).	(Everything we have studied on the subject).	(Everything we have studied on the subject); Franco, Espinel and Almeida, 2008; WP+ (Combinatorics); León-Rojas, J. M., Question selection no. 10 and 10th seminar/laboratory. Combinatorics, II.
Sat 4/4 Lazarus Saturday International Day for Mine Awareness and Assistance in Mine Action (es)
Mon 6/4 Holy Week Holy Monday International Day of Sport for Development and Peace
Tue 7/4 Holy Week Holy Tuesday World Health Day
Wed 8/4 Holy Week Holy Wednesday
Thu 9/4 Holy Week Maundy Thursday
Fri 10/4 Holy Week Good Friday
Sun 12/4 Holy Week Resurrection Sunday International Day of Human Space Flight
Mon 13/4 Eastertide Easter Monday
Tue 14/4	Combinatorics Generalised permutations and combinations (variations, combinations and permutations, with repetition).	§ 4.5.1, 4.5.2, 4.5.3, 4.5.4.	§ 6.5.1, 6.5.2, 6.5.3, 6.5.4.	Franco, Espinel and Almeida, 2008, § 4.1.3, § 4.3, § 4.4; WP+ (Combinatorics).	§ 4.5 (10, 15, 16, 34, 56).	§ 6.5 (10, 15, 16, 34, 66).	Franco, Espinel and Almeida, 2008, § 4.1.3, § 4.3, § 4.4; WP+ (Combinatorics).
Wed 15/4	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.	§ 4.5.5 (~).	§ 6.5.5.	Franco, Espinel and Almeida, 2008, § 7.1; WP+ (Combinatorics).	§ 4.5.5 (22, 47, 50).	§ 6.5.5 (22, 47, 50).	Franco, Espinel and Almeida, 2008, § 7.1; WP+ (Combinatorics).
Thu 16/4	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.	§ 4.5.5 (~).	§ 6.5.5.	Franco, Espinel and Almeida, 2008, § 7.1; WP+ (Combinatorics).	§ 4.5.5 (22, 47, 50).	§ 6.5.5 (22, 47, 50).	Franco, Espinel and Almeida, 2008, § 7.1; WP+ (Combinatorics).
Group B Fri 17/4 Groups A and E Mon 4/5	Seminar/Laboratory No. 11: Combinatorics, III (Occasionally computer-assisted) hands-on word-problem-solving on issues related to: combinatorics.	(Everything we have studied on the subject).	(Everything we have studied on the subject).	(Everything we have studied on the subject); Franco, Espinel and Almeida, 2008; WP+ (Combinatorics).	(Everything we have studied on the subject).	(Everything we have studied on the subject).	(Everything we have studied on the subject); Franco, Espinel and Almeida, 2008; WP+ (Combinatorics); León-Rojas, J. M., Question selection no. 11 and 11st seminar/laboratory. Combinatorics, III.
Sun 19/4 UN Chinese Language Day
Tue 21/4 World Creativity and Innovation Day	Combinatorics Partitions of a set.			WP+ (Combinatorics);			WP+ (Combinatorics);
Wed 22/4 Earth Day	Combinatorics Additive decompositions of numbers.			WP+ (Combinatorics);			WP+ (Combinatorics);
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)
Theme 4 Difference equations (8 h LG and 2 h S/L) (1h LG Solving the end-course preparatory exam)
Group B Fri 24/4 International Day of Multilateralism and Diplomacy for Peace (ref) Groups A and E Mon 27/4	Seminar/Laboratory No. 12: Difference equations, I (Occasionally computer-assisted) hands-on word-problem-solving on issues related to: linear finite difference equations.	(Everything we have studied on the subject); § 6.3.	(Everything we have studied on the subject); § 8.3.	(Everything we have studied on the subject); Navas, Esteban and Quesada, 2009a; Navas, Esteban and Quesada, 2009b; WP+ (Finite difference equations (recurrence relations)).	(Everything we have studied on the subject); § 6.2 (45, 46, 47); § 6.3 (10, 11, 14, 15, 16).	(Everything we have studied on the subject); § 8.2 (45, 46, 47); § 8.3 (10, 11, 14, 15, 16).	(Everything we have studied on the subject); Navas, Esteban and Quesada, 2009a; Navas, Esteban and Quesada, 2009b; WP+ (Finite difference equations (recurrence relations)); León-Rojas, J. M., Question selection no. 12 and 12nd seminar/laboratory. Difference equations, I.
Sat 25/4 World Malaria Day International Delegate's Day (ref)
Sun 26/4 World Intellectual Property Day International Chernobyl Disaster Remembrance Day (ref)
Tue 28/4 World Day for Safety and Health at Work	Finite difference equations (recurrence relations) Linear finite difference equations, models and applications.	(§ 3.2.1, 3.2.2, 3.2.3, 3.2.4); § 6.1.	(§ 2.4); § 8.1.	Navas, Esteban and Quesada, 2009a, § 9.1, 9.2; Navas, Esteban and Quesada, 2009b; WP+ (Finite difference equations (recurrence relations)).	§ 6.1 (17, 22, 23, 25, 27, 36, 37, 42, 46).	§ 8.1 (1, 6, 7, 9, 11, 20, 21, 26, 30).	Navas, Esteban and Quesada, 2009a, § 9.1, 9.2; Navas, Esteban and Quesada, 2009b; WP+ (Finite difference equations (recurrence relations)).
Wed 29/4	Finite difference equations (recurrence relations) Homogeneous linear finite difference equations with constant coefficients, I.	§ 6.2.1, 6.2.2.	§ 8.2.1, 8.2.2.	Navas, Esteban and Quesada, 2009a, § 9.1, 9.2, 9.3; Navas, Esteban and Quesada, 2009b; WP+ (Finite difference equations (recurrence relations)).	§ 6.2 (2, 3, 4, 7, 8, 11, 12, 13, 14, 15, 17, 18).	§ 8.2 (2, 3, 4, 7, 8, 11, 12, 13, 14, 15, 17, 18).	Navas, Esteban and Quesada, 2009a, § 9.1, 9.2, 9.3; Navas, Esteban and Quesada, 2009b; WP+ (Finite difference equations (recurrence relations)).
Thu 30/4 International Jazz Day	Finite difference equations (recurrence relations) Homogeneous linear finite difference equations with constant coefficients, II.	§ 6.2.1, 6.2.2.	§ 8.2.1, 8.2.2.	Navas, Esteban and Quesada, 2009a, § 9.1, 9.2, 9.3; Navas, Esteban and Quesada, 2009b; WP+ (Finite difference equations (recurrence relations)).	§ 6.2 (2, 3, 4, 7, 8, 11, 12, 13, 14, 15, 17, 18).	§ 8.2 (2, 3, 4, 7, 8, 11, 12, 13, 14, 15, 17, 18).	Navas, Esteban and Quesada, 2009a, § 9.1, 9.2, 9.3; Navas, Esteban and Quesada, 2009b; WP+ (Finite difference equations (recurrence relations)).
Fri 1/5 International Workers' Day
Sat 2/5 World Tuna Day (es)
Sun 3/5 World Press Freedom Day
Tue 5/5 African World Heritage Day	Finite difference equations (recurrence relations) Non-homogeneous linear finite difference equation with constant coefficients, I.	§ 6.2.3.	§ 8.2.3.	Navas, Esteban and Quesada, 2009a, § 9.1, 9.2, 9.3 (esp. 9.3.2); Navas, Esteban and Quesada, 2009b; WP+ (Finite difference equations (recurrence relations)).	§ 6.2 (23, 24, 26, 31).	§ 8.2 (23, 24, 26, 31).	Navas, Esteban and Quesada, 2009a, § 9.1, 9.2, 9.3 (esp. 9.3.2); Navas, Esteban and Quesada, 2009b; WP+ (Finite difference equations (recurrence relations)).
Wed 6/5	Finite difference equations (recurrence relations) Non-homogeneous linear finite difference equation with constant coefficients, II.	§ 6.2.3.	§ 8.2.3.	Navas, Esteban and Quesada, 2009a, § 9.1, 9.2, 9.3 (esp. 9.3.2); Navas, Esteban and Quesada, 2009b; WP+ (Finite difference equations (recurrence relations)).	§ 6.2 (23, 24, 26, 31).	§ 8.2 (23, 24, 26, 31).	Navas, Esteban and Quesada, 2009a, § 9.1, 9.2, 9.3 (esp. 9.3.2); Navas, Esteban and Quesada, 2009b; WP+ (Finite difference equations (recurrence relations)).
Thu 7/5 Vesak Day (Held on the/a full moon day of May each year)	Finite difference equations (recurrence relations) Systems of linear finite difference equations, I.	—	—	Navas, Esteban and Quesada, 2009a, § 9.4; Navas, Esteban and Quesada, 2009b; WP+ (Finite difference equations (recurrence relations)).	—	—	Navas, Esteban and Quesada, 2009a, § 9.4; Navas, Esteban and Quesada, 2009b; WP+ (Finite difference equations (recurrence relations)).
	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.
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
Sat 9/5 Time of Remembrance and Reconciliation for Those Who Lost Their Lives during the Second World War World Migratory Bird Day (es)
Group B Mon 11/5 (please come and attend as far as possible) Groups A and E Mon 11/5	Seminario/Laboratorio N.º 13: Difference equations, II (Occasionally computer-assisted) hands-on word-problem-solving on issues related to: finite difference equations.	(Everything we have studied on the subject).	(Everything we have studied on the subject).	(Everything we have studied on the subject); Navas, Esteban and Quesada, 2009a; Navas, Esteban and Quesada, 2009b; WP+ (Finite difference equations (recurrence relations)).	(Everything we have studied on the subject).	(Everything we have studied on the subject).	(Everything we have studied on the subject); Navas, Esteban and Quesada, 2009a; Navas, Esteban and Quesada, 2009b; WP+ (Finite difference equations (recurrence relations)); León-Rojas, J. M., Question selection no. 13 and 13rd seminar/laboratory. Difference equations, II.
Tue 12/5	Finite difference equations (recurrence relations) Systems of linear finite difference equations, II.	—	—	Navas, Esteban and Quesada, 2009a, § 9.4; Navas, Esteban and Quesada, 2009b; WP+ (Finite difference equations (recurrence relations)).	—	—	Navas, Esteban and Quesada, 2009a, § 9.4; Navas, Esteban and Quesada, 2009b; WP+ (Finite difference equations (recurrence relations)).
Wed 13/5	Finite difference equations (recurrence relations) Systems of linear finite difference equations, III.	—	—	Navas, Esteban and Quesada, 2009a, § 9.4; Navas, Esteban and Quesada, 2009b; WP+ (Finite difference equations (recurrence relations)).	—	—	Navas, Esteban and Quesada, 2009a, § 9.4; Navas, Esteban and Quesada, 2009b; WP+ (Finite difference equations (recurrence relations)).
Thu 14/5 End of classes	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).
	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.
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
Mon 20/5 Start of June exam period
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
___ __/_	2019-2020 Final exam.
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Sat 6/7 End of June exam period
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Mon 22/6 Start of July exam period
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___ __/_	2019-2020 Resit final exam.
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Fri 10/7 End of July exam period
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Mon 20/7 End of term
... (See: International days currently observed by the United Nations).