We consider a perturbed version of the eigenvalue problem for the nonautonomous, degenerate Dirichlet p -Laplacian. The perturbation is indefinite. We prove an existence, nonexistence and multiplicity theorem which is global in the parameter λ , when λ ≥ λ ^ 1 a (where λ ^ 1 a is the principal eigenvalue of the differential operator).
Let A and B be two non-empty finite subsets of integers and n, k E Z(+). The sum of dilates of sets A { } and B can be defined: n & centerdot; A+k & centerdot;B = na+kb : a E A, b E B . Under certain assumptions, in 2017, Mistri [Notes Number Theory Discrete Math. 23 (2017), no. 4] proved that |A + k & centerdot; B| >= 4|A|-4, where A and B are non-empty finite subsets of integers and |k| >= 3. In this article, we consider the sumset p & centerdot;A+ k & centerdot;B, with k(> p) E Z(+), where p is an odd prime and (p, k) = 1, and prove that | p & centerdot; A +k & centerdot; B| >= 4|A| -4, where A and B are non-empty finite subsets of integers satisfying some properties. We also rectify the proof of Kaur and Singh [Comptes Rendus. Math & eacute;matique 362 (2024)] and generalize this extended inverse result for arbitrary values of k E Z(+)(>= 2).
We discuss various developments that occurred in the wake of Robin Hartshorne's seminar "Residues and duality". This seminar was based on notes written by Alexander Grothendieck in the summer of 1963, that have recently been published (R & eacute;sidus et dualit & eacute; - Pr & eacute;notes pour un "S & eacute;minaire Hartshorne" (R. Hartshorne, ed.), Documents Math & eacute;matiques, vol. 21, Soci & eacute;t & eacute; Math & eacute;matique de France, 2024).
We give a detailed account of the proof of a purity theorem for finite flat torsors due to Laurent Moret-Bailly. In addition, we show that a part of this result, namely the full faithfulness, holds also under weaker assumptions.
Grothendieck's seminar "Cohomologie & ell;-adique et Fonctions L (SGA 5)" was held at the IH & Eacute;S in the first semesters of 1965 and 1966, and published in 1977 in volume 589 of Lecture Notes in Mathematics, Springer-Verlag. In the first part of this text I review the chronology of this seminar. In the second part, I discuss several developments in its wake, concerning purity, the & ell;-adic formalism, the Lefschetz-Verdier formula, and the Grothendieck-Ogg-Shafarevich formula.
