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This note offers an elementary proof of the Siegel-Walfisz theorem for primes in arithmetic progressions.
The purpose of this article is to formulate a number of probabilistic hidden-variable theorems, to provide proofs in some cases, and counterexamples to some conjectured relationships. The first theorem is the fundamental one. It asserts the…
A simple theory of the covariant derivatives, deformed derivatives and relative covariant derivatives of multivector and multiform fields is presented using algebraic and analytical tools developed in previous papers.
The classical propositional assumption-based model is extended to incorporate probabilities for the assumptions. Then it is placed into the framework of evidence theory. Several authors like Laskey, Lehner (1989) and Provan (1990) already…
I expound here in a more detailed way a proof of an important Serini's theorem, which I have already sketched in a previous Note. Two related questions are briefly discussed.
A minimal presentation of the cohomology ring of the flag manifold $GL_n/B$ was given in [A. Borel, 1953]. This presentation was extended by [E. Akyildiz-A. Lascoux-P. Pragacz, 1992] to a non-minimal one for all Schubert varieties. Work of…
We develop a new approach to prove multiplier theorems in various geometric settings. The main idea is to use martingale transforms and a Gundy-Varopoulos representation for multipliers defined via a suitable extension procedure. Along the…
A very simple and short proof of the polynomial matrix spectral factorization theorem (on the unit circle as well as on the real line) is presented, which relies on elementary complex analysis and linear algebra.
We prove the Mumford-Tate conjecture for those abelian varieties over number fields, whose simple factors of their adjoint Mumford-Tate groups have over $\dbR$ certain (products of) non-compact factors. In particular, we prove this…
We construct new rational approximants of Euler's constant that improve those of Aptekarev et al. (2007) and Rivoal (2009). The approximants are given in terms of certain (mixed type) multiple orthogonal polynomials associated with the…
We show that the Bayesian star paradox, first proved mathematically by Steel and Matsen for a specific class of prior distributions, occurs in a wider context including less regular, possibly discontinuous, prior distributions.
Representations of primes by simple quadratic forms, such as $\pm a^2\pm qb^2$, is a subject that goes back to Fermat, Lagrange, Legendre, Euler, Gauss and many others. We are interested in a comprehensive list of such results, for $q\le…
We present an elementary proof of the fundamental theorem of algebra, following Cauchy's version but avoiding his use of circular functions. It is written in the same spirit as Littlewood's proof of 1941, but reduces it to more elementary…
Conditions for Bayesian posterior robustness have been examined in recent literature. However, many of the proofs seem to be long and complicated. In this paper, we first summarize some basic lemmas that have been applied implicitly or…
We state an improved version of the conjecture of Langlands and Rapoport, and we prove the conjecture for a large class of Shimura varieties. In particular, we obtain the first proof of the (original) conjecture for Shimura varieties of…
We give a simplified proof of analyticity of pluripolar multifunctions
By creating a new method, the author proved the well-known world's baffling problems Goldbach conjecture, twin primes conjecture, the Proposition (C) and the Proposition $n^2+1$.
We present a short and completely elementary proof for a double sum studied by Brent and Osburn in arXiv:1309.2795v2.
These are some notes on the two Milnor conjectures and their proofs (due to Voevodsky, Orlov-Vishik-Voevodsky, and Morel).
Final version, appeared in C.R. Acad. Sci. Paris 339 (2004), no4, 257--259.