arXiv subjects
Publications and source records attributed to Labib Haddad.
In 2004, Andrew Granville proved the following: A polynomial in variables $x,y,z,$ invariant under the map $\varphi: z \to -(x+y+z)$ is a polynomial in $b = z(x+y+z)$. We tell, curtly, its history and then give a variation on this theme.
On Cuesta-Conway numbers as an extension of Cantor's ordinals: A short introduction to surreal numbers. The class of Cuesta-Conway numbers, the surreal numbers, can be defined simply, starting from their normal forms (families of exponentials indexed by ordinals), as an extension of the reals and ordinals from which easily follow addition, multiplication, and total order relation. A construction of this class yields an increasing sequence of real-closed fields, preliminary, so to say. A complete proof is also given the well-known result that the class of surreal numbers is a totally ordered, commutative, real-closed field!
We give a notably simpler and shorter proof of H. B. Neumann's result which is stated, cursorly, like this. For any well-ordered subset, A, of a totally ordered semigroup, the set of products of any finite number of elements of A is itself well-ordered. Moreover, for each t, there are only a finite number of such products equal to t.
A discussion about conditions on the profiles of voters in order to avoid contradictory decisions that have been designed by Sen and others.
Comments about the paper by Elsholz, Fermat's last theorem implies Euclid's infinitude of primes, (2021), and simplification.
Transcendence criteria inspired by Kolberg's paper dated 1962. This is the second part of a note about Kolberg's proof that the values of the sums of a class of certain power series in x, for algebraic values of x, are transcendent. A method to find more such classes is presented and illustrated.
Transcendence criteria inspired by Kolberg's paper dated 1962. In his paper dated 1962, Kolberg states and proves a theorem on the transcendence of the values of the sums of a class of certain power series in x, for algebraic values of x. It builds on Lindemann's theorem and uses, en passant, a transcendence criterion for the values of certain rational fonctions. This last criterion is made explicit. We clarify certain delicate points of the proof and show how to extend its scope.
About Conway's surreal numbers: A letter to a friend (written in French). In memoriam John Horton Conway.
A translation from Spanish into French of a paper by N. Cuesta published in 1954. The paper deals mainly with partially, and totally, ordered sets. Two subjects are specially dealt with: Construction of new ordered sets starting from a family of those. Completion of ordered sets by tools akin to Dedekind cuts. Curiously enough, the so-called surreal numbers (later defined by Conway, in 1974) are already there, thirty years before.
We draw attention to a manuscript submitted to the HAL Open Archives by Francis Bessi\`ere, where he tries to base mathematics on a translative theory that could be shown consistant using only finitist methods, thus bypassing the impossibility shown by G\"odel for deductive theories, such as [ZF], for example.
A plea to open again the building site of finite simple groups in order to include finite simple hypergroups.
A mistake is pointed out in EGA I, Elements of algebraic geometry, concerning corollary (2.4.4) which is not correct. Quite likely, the hitch has already been noticed, and reported, but we know not who, nor where. New research is recommended to find out if the error has spread or not, in the book and, further, in the literature.
About the relation between a counterexample in the theory of numbers constructible by ruler and compass, due to Ian Stewart, and Alhazen's problem concerning the circular mirror.
A new tool is introduced, specially tailored for the study of order 2 bases of N. It is called the Erdos-Tuan algebra since it suits well their conjecture.
A short comment about a recent note by Kiss and Sandor on the maximum values of additive representation functions.
Full details are given for the definition and construction of the wreath product of two arbitrary Lie algebras, in the hope that it can lead to the definition of a suitable Lie group to be the wreath product of two given Lie groups. In the process, quite a few new notions are needed, and introduced. Such are, for example : Formal series with variables in a vector space and coefficients in some other vector space. Derivation of a formal series relative to another formal series. The Lie algebra of a vector space. Formal actions of Lie algebras over vector spaces. The basic formal action of a Lie algebra over itself (as a formal version of the analytic aspect of the infinitesimal operation law of a Lie groupuscule). More generally, the wreath product of two Lie algebras is defined, relative to a formal action of the second onto an arbitrary vector space. Main features are : A description of the triangular actions of wreath products over product vector spaces, and a Kaloujnine-Krasner type theorem : In essence, it says that all Lie extensions of a given Lie algebra by another Lie algebra are, indeed, subalgebras of their wreath product.
Formal actions of Lie algebras over vector spaces are introduced in a purely algebraic way, as a mimic of infinitesimal operations of Banach Lie algebras over Banach analytic manifolds. In analogy with the case of abstract groups, complete wreath products and triangular actions are then defined for Lie algebras acting "en cascade" over vector spaces. Finally, a Kaloujnine-Krasner type theorem for Lie algebra extensions is proved. ----- En mimant les lois d'opérations infinitésimales des algèbres de Lie sur les variété s analytiques banachiques, on introduit de manière purement algèbrique la notion d'action formelle d'une algèbre de Lie sur un espace vectoriel. Ensuite, par analogie avec le cas des groupes abstraits, et en faisant opérer les algèbres de Lie "en cascade", on définit produit d'entrelacement ("wreath product") et action triangulaire pour les algèbres de Lie. On démontre enfin un théorème du type Kaloujnine-Krasner pour les extensions d'algèbres de Lie.
A stroll taken around the landscape of Ramsey's Theory. One way first, "Down from infinite to finite", then, another way, "Up from disorder to order". An expose' made at the "Rencontres arithme'tique et combinatoire", Saint-Etienne, june 2006.