Surprenante beauté des mathématiques

Dans cet article, nous dissertons sur le côté esthétique des mathématiques. Selon nous, la beauté des mathématiques apparaît souvent au travers de raisonnements lumineux mais surprenants. Nous illustrons ce point au moyen de trois exemples concrets, simples et empruntés au mathématicien russe Lazebnik

Editorial

Em novembro de 2014, o professor Antonio Miguel nos fez o convite para que coordenássemos a organização de uma edição temática da Zetetiké. Inicialmente voltada para articulações entre Literatura, Educação e Matemática, a proposta foi modificada para se tornar mais abrangente e envolver não apenas a Literatura, mas o campo mais geral das Artes. Fixamos, então, as palavras Educação, Matemática e Artes para definir o tema da edição especial

SOLVING CONTEST PROBLEMS \ud VIA FORMAL PROGRAM VERIFICATION\ud

The interface between mathematics and computer science is many-sided. In particular, E.W. Dijkstra promoted a special “computer science” approach to mathematics problem solving. The approach combines a heuristic algorithm design and rigorous mathematical proof of algorithm correctness (in style of A. Hoare and R. Floyd). The paper sketches two problems of this kind in a form of tutorials for undergraduate students that are interested in different programming contests (like ACM International Collegiate Programming Contests). These tutorials took place at Novosibirsk State University in years 2005–2008. The paper also dioceses some direction for further research that emerge from the problems

Some interesting inequalities for a triangle

U radu su prezentirane neke zanimljive nejednakosti koje su vezane za trokut i točku unutar trokuta. Svakako najpoznatija takva nejednakost je čuvena Erdös -Mordellova nejednakost. Nejednakost je objavio P. Erdös 1935. godine u američkom časopisu American Mathematical Monthly, a iste godine je L. J. Mordell dao njen dokaz. Osim ove nejednakosti u radu se nalaze još neke zanimljive nejednakosti i njihovi dokazi.In this paper, we present some interesting inequalities for a triangle and a point inside the triangle. At all events, the most well-known of these inequalities is the famous Erdös-Mordell inequality. In 1935, Erdös published the inequality in the journal American Mathematical Monthly, and Mordell gave his proof in the same year. This inequality as well as some interesting inequalities and their proofs are presented in this paper

A powerful abelian square-free substitution over 4 letters

AbstractIn 1961, Paul Erdös posed the question whether abelian squares can be avoided in arbitrarily long words over a finite alphabet. An abelian square is a non-empty word uv, where u and v are permutations (anagrams) of each other. The case of the four letter alphabet Σ4={a,b,c,d} turned out to be the most challenging and remained open until 1992 when the author presented an abelian square-free (a-2-free) endomorphism g85 of Σ4∗. The size of this g85, i.e., |g85(abcd)|, is equal to 4×85 (uniform modulus). Until recently, all known methods for constructing arbitrarily long a-2-free words on Σ4 have been based on the structure of g85 and on the endomorphism g98 of Σ4∗ found in 2002.In this paper, a great many new a-2-free endomorphisms of Σ4∗ are reported. The sizes of these endomorphisms range from 4×102 to 4×115. Importantly, twelve of the new a-2-free endomorphisms, of modulus m=109, can be used to construct an a-2-free (commutatively functional) substitution σ109 of Σ4∗ with 12 image words for each letter.The properties of σ109 lead to a considerable improvement for the lower bound of the exponential growth of cn, i.e., of the number of a-2-free words over 4 letters of length n. It is obtained that cn>β−50βn with β=121/m≃1.02306. Originally, in 1998, Carpi established the exponential growth of cn by showing that cn>β−tβn with β=219/t=219/(853−85)≃1.000021, where t=853−85 is the modulus of the substitution that he constructs starting from g85

Mary Ellen Rudin: encontrando contraejemplos en topología

No estoy segura de que supiera que la derivada de sen x era cos x, pero podía probar todo tipo de teoremas sobre continuidad y diferenciabilidad, etcétera. Me tumbo en el sofá en la sala de estar con mi lápiz y papel, y pienso, y dibujo pequeñas imágenes, y pruebo esto y aquello... [...] Nunca me ha importado hacer matemáticas tumbada en el sofá en el medio de la sala de estar, con los niños pasando por encima de mí