Paysages Mathématiques

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Une citation mathématique chaque jour & quelques actualités mathématiques. A mathematical quote every day & some mathematical news. Site : https://paysmaths.net #mathématiques #mathematics #maths #math

Citation de János Kollár : "En mathématiques, vous n'avez pas besoin de matériel coûteux ni de centaines d'assistants. Vous êtes seul face à l'inconnu, réussissant ou échouant de votre propre chef."
Quote from János Kollár : "In mathematics, you do not need expensive equipment or hundreds of assistants. You are up against the unknown alone, succeeding or failing on your own wit."
Comprehensive presentation of the "Theorem of the Day", starting with a statement of this theorem. Euler’s Product Formula for Sine : For any complex number z, sin (τ/2)z = (τ/2)z(1 − z^2/1^2) (1 − z^2/2^2) (1 − z^2/3^2)· · · .
Portrait photographique de George Sarton, et une citation : "Retirez les développements mathématiques de l'histoire de la science, et vous supprimez le squelette qui soutenait et maintenait ensemble tout le reste."
Photographic portrait of George Sarton, and a quote : "Take the mathematical developments out of the history of science, and you suppress the skeleton which supported and kept together all the rest."
Comprehensive presentation of the "Theorem of the Day", starting with a statement of this theorem. The Classification of Archimedean 4-Polytopes : The Archimedean polytopes in four dimensions are: 1. the polytopes of Boole Stott–Wythoff type; 2. the prismatic polytopes; 3. Cartesian products of regular polygons; 4. Thorold Gosset’s snub-24-cell (1900); 5. the Grand Antiprism (Conway–Guy, 1965.
Photographic portrait of Shakuntala Devi, and a quote : "Stability in an unstable world – that is what maths is all about. That is why I say maths is the pulse of our lives ; the thread that keeps us together ; the thread that never breaks."
Portrait photographique de Shakuntala Devi, et une citation : "De la stabilité dans un monde instable – c'est ce que sont les maths. C'est pourquoi je dis que les maths sont le pouls de nos vies ; le fil qui nous maintient ensemble ; le fil qui ne se brise jamais."
Portrait of Pierre-Simon Laplace, and a quote : "The most important questions of life [are], for the most part, only problems of probability."
Portrait photographique de Pierre-Simon Laplace, et une citation : "[Les] questions les plus importantes de la vie [ne sont], pour la plupart, que des problèmes de probabilité."
Comprehensive presentation of the "Theorem of the Day", starting with a statement of this theorem. Sperner’s Lemma : Suppose the interior of a triangle is triangulated (that is, divided up internally into small triangles). The vertices of the triangle are coloured red, green and blue, respectively. All other vertices, where lines meet inside or around the outside edges of the triangle, are also coloured red, green or blue with the restriction that no edge of the main triangle contains all three colours. Let TC be the number of small triangles whose vertices are coloured red, green and blue in clockwise order; and let TA be the number of small triangles whose vertices are coloured red, green and blue in anticlockwise order. Then |TC − TA| = 1. In particular, the total number of small red-green-blue triangles must be odd, and is certainly neverzero.
Photographic portrait of David Hilbert, and a quote : "Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts. For with all the variety of mathematical knowledge, we are still clearly conscious of the similarity of the logical devices, the relationship of the ideas in mathematics as a whole and the numerous analogies in its different departments."
Portrait photographique de David Hilbert, et une citation : "Selon moi, la Science mathématique est un entier indivisible, un organisme dont la force vitale a pour condition l'indissolubilité de ses parties. En effet, quelle que soit la diversité des matières de notre Science dans ses détails, nous n'en sommes pas moins frappés de l'équivalence des procédés logiques, de la parenté des idées dans l'ensemble de la Science ainsi que des nombreuses analogies dans ses différents domaines."
Comprehensive presentation of the "Theorem of the Day", starting with a statement of this theorem. The Quadratic Formula : The quadratic equation ax^2 + 2bx + c = 0, with complex values of a, b and c, is solved by the two, possibly equal, x values x = (−b ± √(b^2 − ac)) / a.
Portrait photographique de Gian-Carlo Rota, et une citation : "Les faits mathématiques sont aussi utiles que les faits de tout autre science. Aussi abscons qu'ils puissent paraître à première vue, ils finissent tôt ou tard par trouver des applications pratiques. [...] Les faits mathématiques d'aujourd'hui sont le tremplin de la science de demain."
Photographic portrait of Gian-Carlo Rota, and a quote : "The facts of mathematics are as useful as the facts of any other science. No matter how abstruse they may first seem, sooner or later they find their way back to practical applications. [...] The facts of today's mathematics are the springboard for the science of tomorrow."
Comprehensive presentation of the "Theorem of the Day", starting with a statement of this theorem. Bregman’s Theorem : Let A be an n × n (0, 1)-matrix, with non-zero row sums r_i, i = 1, . . . , n. Then the permanent of A satisfies the inequality, per(A) ≤ ∏_(i=1)^n ((r_i)!)^(1/ri).
Photographic portrait of Jacques Hadamard, and a quote : "Good mathematicians when they make [errors], which is not infrequent, soon perceive and correct them. As for me (and mine is the case of many mathematicians), I make many more of them than my students do; only I always correct them so that no trace of them remains in the final result."
Portrait photographique de Jacques Hadamard, et une citation : "Les bons mathématiciens, quand ils font [des erreurs], ce qui n'est pas rare, s'en aperçoivent rapidement et les corrigent. Quant à moi (et mon cas est celui de beaucoup de mathématiciens), j'en fais beaucoup plus que mes élèves ; seulement, je les corrige toujours de sorte qu'il n'en reste aucune trace dans le résultat final."
Comprehensive presentation of the "Theorem of the Day", starting with a statement of this theorem. The Design of the Century : There exists a 2-chromatic Steiner system S (2, 4, 100) in which every block contains three points of one colour and one point of the other colour.
Photographic portrait of Benjamin Peirce, and a quote : "Mathematics is not the discoverer of laws, for it is not induction; neither is it the framer of theories, for it is not hypothesis; but it is the judge over both, and it is the arbiter to which each must refer its claims; and neither law can rule nor theory explain without the sanction of mathematics."
Portrait photographique de Benjamin Peirce, et une citation : "La mathématique ne découvre pas des lois, car elle n'est pas l'induction ; elle n'élabore pas non plus des théories, car elle n'est pas l'hypothèse ; mais elle est le juge de l'une et de l'autre, et elle est l'arbitre auquel chacune doit soumettre ses affirmations ; et aucune loi ne peut régir ni aucune théorie expliquer sans la sanction de la mathématique."
Comprehensive presentation of the "Theorem of the Day", starting with a statement of this theorem. The McIver–Neumann Half-n Bound : Let Ω be a set of order n, n≠3, and let G be a permutation group acting on Ω. Then G may be generated by ⌊ n/2 ⌋ elements.
Portrait photographique de Charles Proteus Steinmetz, et une citation : "Il y a près de cent ans, les grands mathématiciens du XIXe siècle, les Allemands Gauss et Riemann, le Russe Lobatschewsky, le Hongrois Bolyai et d'autres, ont étudié les fondements de la géométrie, ce qui les a conduits à développer des systèmes différents de la géométrie euclidienne et basés sur différents ensembles d'axiomes. Par conséquent, lorsque finalement, dans la théorie de la relativité, la physique a dépassé le cadre de la géométrie euclidienne, les mathématiques des nouvelles caractéristiques de l'espace étaient déjà pleinement développées."
Photographic portrait of Charles Proteus Steinmetz, and a quote : "Nearly a hundred years ago the great mathematicians of the nineteenth century, the Germans Gauss and Riemann, the Russian Lobatschewsky, the Hungarian Bolyai and others, investigated the foundations of geometry, which led them to the development of systems different from the Euclidean and based on different sets of axioms. Therefore, when finally, in the relativity theory, physics advanced beyond the range of Euclidean geometry, the mathematics of the new space characteristics was already fully developed."
Comprehensive presentation of the "Theorem of the Day", starting with a statement of this theorem. The Skolem–Noether Theorem : Let R, S be finite dimensional algebras, R simple and S central simple. If f, g : R → S are homomorphisms then there is an element s S such that, for all r R, ∈ ∈ g(r) = s^(−1) f (r)s.
Quote from Arthur Lefevre : "Formal thought [...] is the means of all exact knowledge; and a correct understanding of the main formal sciences, Logic and Mathematics, is the proper and only safe foundation for a scientific education."
"La pensée formelle [...] est le moyen de toute connaissance exacte ; et une compréhension correcte des principales sciences formelles, la logique et les mathématiques, est le fondement approprié, et le seul sûr, d'une éducation scientifique." – Arthur Lefevre #citation #mathématiques #maths #math
Comprehensive presentation of the "Theorem of the Day", starting with a statement of this theorem. The Albert–Brauer–Hasse–Noether Main Theorem : Every finite-dimensional noncommutative division algebra over a number field is a cyclic algebra.
Photographic portrait of Alain Connes, and a quote : "Mathematical activity requires daily practice. If you stop for too long, the know-how erodes. The habits are lost. Fortunately, this is only temporary."
Portrait photographique d'Alain Connes, et une citation : "L’activité mathématique exige une pratique quotidienne. Si l’on s’interrompt trop longtemps, le savoir-faire s’érode. Les automatismes se perdent. C’est heureusement transitoire."
Comprehensive presentation of the "Theorem of the Day", starting with a statement of this theorem. The Hardy–Ramanujan Asymptotic Partition Formula : For n a positive integer, let p(n) denote the number of unordered partitions of n, that is, unordered sequences of positive integers which sum to n; then the value of p(n) is given asymptotically by p(n) ˜ (1/4n √3)e^(τ √(n/6)).
Portrait photographique de William Thurston, et une citation : "Dans les mathématiques, il ne s'agit pas de nombres, d'équations, de calculs ou d'algorithmes : il s'agit de comprendre."
Photographic portrait of William Thurston, and a quote : "Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding."
Comprehensive presentation of the "Theorem of the Day", starting with a statement of this theorem. Khinchin’s Continued Fraction Theorem : There is a constant K such that, for almost all real numbers x, if x has continued fraction expansion x = [a0; a1, a2, . . .] then the geometric means of the sets {a1, a2, . . . , an}, n ≥ 1, converge to K as n → ∞.
Citation de Mariana Cook : "La vérité est l'autorité ultime en mathématiques. Un théorème doit être prouvé pour être vrai. Souvent, après une décennie de travail, une preuve ne fera qu'une page. Elle sera magnifique dans sa simplicité."
Quote from Mariana Cook : "Truth is the ultimate authority in mathematics. A theorem must be proven to be true. Often, after a decade of work, a proof will only be a page long. It will be beautiful in its simplicity."
Comprehensive presentation of the "Theorem of the Day", starting with a statement of this theorem. Minkowski’s Convex Body Theorem : Let L(B) = {Bx | x Z^n} be the integer lattice whose ∈ points are all integer-weighted sums of the n linearly independent basis vectors forming B, an n × n matrix over R. Let S be a convex subset of R^n, closed under negation, whose volume exceeds 2^n| det(B)|. Then S contains a nonzero point of L(B).
Portrait photographique de Paul Erdős, et une citation : "Toute activité humaine, bonne ou mauvaise, à l'exception des mathématiques, doit prendre fin."
Photographic portrait of Paul Erdős, and a quote : "Every human activity, good or bad, except mathematics, must come to an end."
Comprehensive presentation of the "Theorem of the Day", starting with a statement of this theorem. Euler’s Product Formula for ζ(s) : For any complex number s having real part Re(s) > 1, ζ(s) = 1 + 1/2^s + 1/3^s + 1/ 4^s + 1/5^s + . . . = (1 − 1/p1^s)^(−1) (1 − 1/p2^s)^(−1) (1 − 1/ p3^s)^(−1)· · · where p1, p2, p3, . . . are the prime numbers.
Photographic portrait of Stanislaw Ulam, and a quote : "It is not so much whether a theorem is useful that matters, but how elegant it is."
Portrait photographique de Stanislaw Ulam, et une citation : "Ce n'est pas tant l'utilité d'un théorème qui compte, mais son élégance."
Comprehensive presentation of the "Theorem of the Day", starting with a statement of this theorem. Nevanlinna’s Five-Value Theorem : If f and g are meromorphic functions of one complex variable and f and g have the same inverse images (ignoring multiplicities) on five distinct values then they are identically equal.
Portrait photographique de Henri Poincaré, et une citation : "Deviner avant de démontrer ! Ai-je besoin de rappeler que c'est ainsi que se sont faites toutes les découvertes importantes !"
Photographic portrait of Henri Poincaré, and a quote : "Guessing before proving! Need I remind you that it is so that all important discoveries have been made?"
Comprehensive presentation of the "Theorem of the Day", starting with a statement of this theorem. A Generalised Hlawka Inequality : Let x1, x2, . . . , xn be a sequence of vectors in an inner product space (e.g. R^m) with norm ‖ · ‖. Denote by d_k the sum of all norms of k-sums of these vectors, i.e. d_k = ∑_(X {1, . . . , n}, |X| = k) ‖∑_(i X) xi ‖. Then, for n ≥ 3, ⊆ ∈ d_2 + . . . + d_(n−1) ≤ (2^(n−2) − 1) (d_1 + d_n).
Photographic portrait of Godfrey Harold Hardy, and a quote : "If intellectual curiosity, professional pride, and ambition are the dominant incentives to research, then assuredly no one has a fairer chance of gratifying them than a mathematician."
Portrait photographique de Godfrey Harold Hardy, et une citation : "Si la curiosité intellectuelle, la fierté professionnelle et l'ambition sont les principales motivations de la recherche, alors il est certain que personne n'a plus de chance de les satisfaire qu'un mathématicien."
Comprehensive presentation of the "Theorem of the Day", starting with a statement of this theorem. L’Hospital’s Rule : Let f (x) and g(x) be real-valued functions of a real variable x. Let a ∈ R and suppose that, in some open interval containing a, f (x) and g(x) are differentiable except possibly at x = a, with g′(x) being nowhere zero. Suppose, moreover that lim_(x→a) f = lim_(x→a) g = 0 or |lim_(x→a) f | = | lim_(x→a) g| = ∞. Then lim_(x→a) f ′(x)/g′(x), provided this limit exists, satisfies lim_(x→a) (f (x) / g(x)) = lim_(x→a) (f ′(x) / g′(x))
Portrait of Eric Temple Bell, and a quote : "Today mathematical invention (discovery, if you prefer) is going forward more vigorously than ever. The only thing, apparently, that can stop its progress is a general collapse of what we have been pleased to call civilization. If that comes, mathematics may go underground for centuries, as it did after the decline of Babylon; but if history repeats itself, as it is said to do, we may count on the spring bursting forth again, fresher and clearer than ever, long after we and all our stupidities shall have been forgotten."
Portrait d'Eric Temple Bell, et une citation : "Aujourd'hui, l'invention mathématique (la découverte, si vous préférez) progresse plus vigoureusement que jamais. La seule chose, apparemment, qui puisse arrêter ses progrès, est un effondrement général de ce que nous avons eu le plaisir d'appeler civilisation. Si cela se produit, les mathématiques risquent d'entrer dans la clandestinité pendant des siècles, comme ce fut le cas après le déclin de Babylone ; mais si, comme on le dit, l'histoire se répète, nous pouvons être sûr que la source rejaillira, plus fraîche et plus claire que jamais, longtemps après que nous et toutes nos idioties auront été oubliés."
Comprehensive presentation of the "Theorem of the Day", starting with a statement of this theorem. De Morgan’s Laws : If B, a set containing at least two elements, and equipped with the operations +, × and ′ (complement), is a Boolean algebra, then, for any x and y in B, (x + y)′ = x′ × y′, and (x × y)′ = x′ + y′.
Photographic portrait of Vladimir Voevodsky, and a quote : "Constructive logic is so much more interesting than classical logic and it is so much closer to the true behavior of intelligible things in the world : in classical logic, because of double negation – double negation equals identity –, there is no difference between positive and negative statements so positive and negative statements are indistinguishable [...]. But when you get an option of doing interesting things without this identification, you suddenly understand that you can actually do interesting mathematics in a world of logic where not every statement is a negative statement, where there is an actual distinction between positive and negative things. I think this is much closer to the truth."
Portrait photographique de Vladimir Voevodsky, et une citation : "La logique constructive est beaucoup plus intéressante que la logique classique et elle est beaucoup plus proche du véritable comportement des choses intelligibles dans le monde : dans la logique classique, en raison de la double négation – la double négation est égale à l'identité –, il n'y a pas de différence entre les énoncés positifs et négatifs, de sorte que les énoncés positifs et négatifs sont indiscernables [...]. Mais lorsqu'on a la possibilité de faire des choses intéressantes sans cette identification, on comprend soudain qu'il est possible de faire des mathématiques intéressantes dans un monde logique où chaque énoncé n'est pas un énoncé négatif, où il y a une distinction réelle entre les choses positives et les choses négatives. Je pense que c'est beaucoup plus proche de la vérité."
Comprehensive presentation of the "Theorem of the Day", starting with a statement of this theorem. The Transversal Matroid Theorem : Let E be a finite set and let A = {Ai | i = 1, . . . , t} be a family of subsets of E. Let T denote the collection of partial transversals of A, i.e. those subsets X of E having the property that each element of X may choose a distinct member of A to which it belongs (more precisely: those subsets X = {x1, . . . , xk}, 0 ≤ k ≤ t, for which there is a subset {y1, . . . , yk} of {1, . . . , t} satisfying xi ∈ Ayi, i = 1, . . . , k). Then T forms the collection of independent sets of a matroid.
Photographic portrait of Bertrand Russell, and a quote : "Symbolism is useful because it makes things difficult. (This is not true of the advanced parts of mathematics, but only of the beginnings.) What we wish to know is, what can be deduced from what. Now, in the beginnings, everything is self-evident ; and it is very hard to see whether one self-evident proposition follows from another or not. [...] Hence we invent some new and difficult symbolism, in which nothing seems obvious. Then we set up certain rules for operating on the symbols, and the whole thing becomes mechanical."
Portrait photographique de Bertrand Russell, et une citation : "Le symbolisme est utile parce qu'il rend les choses difficiles (ceci n'est pas vrai pour les parties avancées des mathématiques, mais seulement pour les débuts). Ce que nous voulons savoir, c'est ce qui peut être déduit de quoi. Or, au début, tout est évident ; et il est très difficile de voir si une proposition évidente découle d'une autre ou non. [...] Nous inventons donc un symbolisme nouveau et difficile, dans lequel rien ne semble évident. Ensuite, nous établissons certaines règles pour opérer sur les symboles, et tout devient mécanique."
Comprehensive presentation of the "Theorem of the Day", starting with a statement of this theorem. The Rule of Sarrus : In the projective plane of order 3 let the line at infinity consist of points R, C, P and M, assigned the values 0, 0, +1 and −1, respectively. Suppose that the other nine points are assigned the entries of a 3 × 3 matrix A in such a way that the rows of the matrix are assigned to the three parallel lines meeting at R, the columns to the parallel lines meeting at C, and the main diagonal to a line through P. Then the determinant of A may be evaluated as: det A = ∑_(lines L) ∏_(x on L) (value of point x).