This article in French, with a large English introduction, is a survey about applications of bi-quantization theory in Lie theory. We focus on a conjecture of M. Duflo. Most of the applications are coming from our article with Alberto Cattaneo [13] and some extensions are relating discussions with my student [9]. The end of the article is completely new. We prove that the conjecture E = 1 impli...

The Hall ratio of a graph $G$ is the maximum value $v(H) / \alpha(H)$ taken over all non-null subgraphs $H$ $G$. For any graph, lower-bound on its fractional chromatic number. In this note, we present various constructions graphs whose number grows much faster than their ratio. This refutes conjecture Harris.

An old conjecture of Sierpiński asserts that for every integer k > 2, there is a number m for which the equation φ(x) = m has exactly k solutions. Here φ is Euler’s totient function. In 1961, Schinzel deduced this conjecture from his Hypothesis H. The purpose of this paper is to present an unconditional proof of Sierpiński’s conjecture. The proof uses many results from sieve theory, in particul...

Let M be a closed symplectic manifold and suppose M → P → B is a Hamiltonian fibration. Lalonde and McDuff raised the question whether one always has H∗(P ;Q) = H∗(M ;Q) ⊗ H∗(B;Q) as vector spaces. This is known as the c–splitting conjecture. They showed, that this indeed holds whenever the base is a sphere. Using their theorem we will prove the c–splitting conjecture for arbitrary base B and f...

The generalized k-connectivity κk(G) of a graphG, which was introduced by Chartrand et al.(1984) is a generalization of the concept of vertex connectivity. Let G and H be nontrivial connected graphs. Recently, Li et al. gave a lower bound for the generalized 3-connectivity of the Cartesian product graph G H and proposed a conjecture for the case that H is 3-connected. In this paper, we give two...

This paper investigates what is the Hausdorff distance between the set of Euler curves of a Lipschitz continuous differential inclusion and the set of Euler curves for the corresponding convexified differential inclusion. It is known that this distance can be estimated by O( √ h), where h is the Euler discretization step. It has been conjectured that, in fact, an estimation O(h) holds. The pape...

Using Bailey’s $$_{10}\phi _9$$ transformation formula, we prove a family of q-congruences modulo the square cyclotomic polynomial, which were previously observed by author and Zudilin (J Math Anal Appl 475:1636–1646, 2019). As an application, confirm conjecture in (Electron Res Arch 28:1031–1036, 2020). This also partially reproves special case Swisher’s (H.3) conjecture.

The Atiyah conjecture for a discrete group G states that the L-Betti numbers of a finite CW-complex with fundamental group G are integers if G is torsion-free, and in general that they are rational numbers with denominators determined by the finite subgroups of G. Here we establish conditions under which the Atiyah conjecture for a torsion-free group G implies the Atiyah conjecture for every fi...

For example, K. Künnemann [Ku] proved that if X is a projective space, then the conjecture is true. Here we fix a notation. We say a Hermitian line bundle (H, k) on X is arithmetically ample if (1) H is f -ample, (2) the Chern form c1(H∞, k∞) is positive definite on the infinite fiber X∞, and (3) there is a positive integer m0 such that, for any integer m ≥ m0, H(X, H) is generated by the set {...

چکیده the issue of validity and non-validity of absolute conjecture, although frequently studied and researched by great scholars, is still in need of further research, since the studies and researches carried out so far are typically based on the presuppositions that have been regarded as indisputable and needless of study and research; whereas, in fact, they are not indisputable and need furt...