An analogue of Artin’s conjecture for multiplicative subgroups of the rationals

Given ${\Gamma \subset \mathbb{Q}^*}$ a multiplicative subgroup and ${m \in \mathbb{N}^+}$ , assuming the Generalized Riemann Hypothesis, we determine an asymptotic formula for the number of primes p ≤ x for which ind _p Γ = m, where ind _p Γ = (p ? 1)/|Γ _p | and Γ _p is the reduction of Γ modulo p. This problem is a generalization of some earlier works by Cangelmi–Pappalardi, Lenstra, Moree, Murata, Wagstaff, and probably others. We prove, on GRH, that the primes with this property have a density and, in the case when Γ contains only positive numbers, we give an explicit expression for it in terms of an Euler product. We conclude with some numerical computations.     

An extension of Barta’s Theorem and geometric applications

We prove an extension of a theorem of Barta and we give some geometric applications. We extend Cheng’s lower eigenvalue estimates of normal geodesic balls. We generalize Cheng-Li-Yau eigenvalue estimates of minimal submanifolds of the space forms. We show that the spectrum of the Nadirashvili bounded minimal surfaces in have positive lower bounds. We prove a stability theorem for minimal hypersurfaces of the Euclidean space, giving a converse statement of a result of Schoen. Finally we prove generalization of a result of Kazdan–Kramer about existence of solutions of certain quasi-linear elliptic equations. Bessa and Montenegro were partially supported by CNPq Grant.     

Liouville numbers and Schanuel’s Conjecture

In this paper, using an argument of P. Erd?s, K. Alniaçik, and É. Saias, we extend earlier results on Liouville numbers, due to P. Erd?s, G.J. Rieger, W. Schwarz, K. Alniaçik, É. Saias, E.B. Burger. We also produce new results of algebraic independence related with Liouville numbers and Schanuel’s Conjecture, in the framework of ${G_\delta}$ -subsets.     

An extension of Wilf’s conjecture to affine semigroups

Let \(\mathcal {C}\subset \mathbb {Q}^p_+\) be a rational cone. An affine semigroup \(S\subset \mathcal {C}\) is a \(\mathcal {C}\)-semigroup whenever \((\mathcal {C}\setminus S)\cap \mathbb {N}^p\) has only a finite number of elements. In this work, we study the tree of \(\mathcal {C}\)-semigroups, give a method to generate it and study the \(\mathcal {C}\)-semigroups with minimal embedding dimension. We extend Wilf’s conjecture for numerical semigroups to \(\mathcal {C}\)-semigroups and give some families of \(\mathcal {C}\)-semigroups fulfilling the extended conjecture. Other conjectures formulated for numerical semigroups are also studied for \(\mathcal {C}\)-semigroups.     

An analogue of Beurling’s theorem for the Jacobi transform

In this paper, we prove Beurling＇s theorem for the Jacobi transform, from which we derive some other versions of uncertainty principles.     

Baker’s conjecture and Eremenko’s conjecture for functions with negative zeros

We introduce a new technique that allows us to make progress on two long standing conjectures in transcendental dynamics: Baker's conjecture that a transcendental entire function of order less than 1/2 has no unbounded Fatou components, and Eremenko's conjecture that all the components of the escaping set of an entire function are unbounded. We show that both conjectures hold for many transcendental entire functions whose zeros all lie on the negative real axis, in particular those of order less than 1/2. Our proofs use a classical distortion theorem based on contraction of the hyperbolic metric, together with new results which show that the images of certain curves must wind many times round the origin.     

An analog of shemetkov’s conjecture for fischer classes of finite groups

We describe some methods for constructing Fischer classes of finite groups by means of the operators defined by given properties of Hall π-subgroups. It is in particular proved that, for a Fischer class $\mathfrak{F}$ and a set of primes π, the class of all finite π-soluble $C_\pi \mathfrak{F}$ -groups, i.e., of all groups whose Hall π-subgroups belong to $\mathfrak{F}$ , is a Fischer class.     

An extension of Matkowski’s and Wardowski’s fixed point theorems with applications to functional equations

Aequationes mathematicae - In this paper, we prove a fixed point theorem for a system of maps on the finite product of metric spaces. Our result generalizes the result of Matkowski (Bull Acad Pol...     

Serre’s conjecture and its consequences

After some generalities about the absolute Galois group of \mathbb Q\mathbb Q, we present the historical context in which Serre made his modularity conjecture. This was recently proved by Wintenberger and the author ([22], [23]), with an input of Kisin ([24]). The focus of these notes is on the applications of the conjecture. Some of the applications are based on the methods used in the proof.     

Graph norms and Sidorenko’s conjecture

Let H and G be two finite graphs. Define h _H (G) to be the number of homomorphisms from H to G. The function h _H (·) extends in a natural way to a function from the set of symmetric matrices to ℝ such that for A _G , the adjacency matrix of a graph G, we have h _H (A _G ) = h _H (G). Let m be the number of edges of H. It is easy to see that when H is the cycle of length 2n, then h _H (·)^1/m is the 2n-th Schatten-von Neumann norm. We investigate a question of Lovász that asks for a characterization of graphs H for which the function h _H (·)^1/m is a norm.     

Boyle’s conjecture and perfect localizations

ABSTRACT

In this article we study the behavior of left QI-rings under perfect localizations. We show that a perfect localization of a left QI-ring is a left QI-ring. We prove that Boyle’s conjecture is true for left QI-rings with finite Gabriel dimension such that every hereditary torsion theory in the Gabriel filtration is perfect. As corollary, we get that Boyle’s conjecture is true for left QI-rings which satisfy the restricted left socle condition, a result proved by Faith in [6 Faith, C. (1976). On hereditary rings and Boyle’s conjecture. Arch. Math. 27(1):113–119.[Crossref] , [Google Scholar]].     

Old and new applications of Katona’s circle

The present paper is to honour Gyula Katona, my teacher on the occasion of his 80th birthday. Its main content is threefold, new proofs of old results (e.g. the De Bonis, Katona, Swanepoel Theorem on butterfly-free families), new results obtained via the Katona Circle (e.g. for the sum of the sizes of non-empty cross-intersecting families), the solution for the Katona Circle of some notoriously difficult problems (e.g. Erdős Matching Conjecture).     

Algorithmic complexity of Greenberg’s conjecture

Archiv der Mathematik - Let k be a totally real number field and p a prime. We show that the “complexity” of Greenberg’s conjecture ( $$\lambda = \mu = 0$$ ) is governed (under...     

Generalizations of Graham’s pebbling conjecture

We investigate generalizations of pebbling numbers and of Graham’s pebbling conjecture that $\pi \left(G\square H\right)\le \pi \left(G\right)\pi \left(H\right)$, where $\pi \left(G\right)$ is the pebbling number of the graph $G$. We develop new machinery to attack the conjecture, which is now twenty years old. We show that certain conjectures imply others that initially appear stronger. We also find counterexamples that shows that Sjöstrand’s theorem on cover pebbling does not apply if we allow the cost of transferring a pebble from one vertex to an adjacent vertex to depend on the weight of the edge and we describe an alternate pebbling number for which Graham’s conjecture is demonstrably false.     

A proof of snevily’s conjecture

We prove Snevily’s conjecture, which states that for any positive integer k and any two k-element subsets {a ₁, …, a _k } and {b ₁, …, b _k } of a finite abelian group of odd order there exists a permutation π ∈ S _k such that all sums a _i + b _π(i) (i ∈ [1, k]) are pairwise distinct.