Authors:Etienne Couturier; Nicolas Jacquet Abstract: Publication date: Available online 15 June 2017 Source:Bulletin des Sciences Mathématiques Author(s): Etienne Couturier, Nicolas Jacquet A universal ordinary differential equation C ∞ of order 3 is constructed here. The equation is universal in the sense that any continuous function on a real segment can be approximated by a solution of this equation with an arbitrary accuracy in uniform norm.

Authors:Komla Domelevo; Stefanie Petermichl; Janine Wittwer Abstract: Publication date: Available online 29 May 2017 Source:Bulletin des Sciences Mathématiques Author(s): Komla Domelevo, Stefanie Petermichl, Janine Wittwer We show that the norm of the vector of Riesz transforms as operator in the weighted Lebesgue space L ω 2 is bounded by a constant multiple of the first power of the Poisson- A 2 characteristic of ω. The bound is free of dimension and optimal. Our argument requires an extension of Wittwer's linear estimate for martingale transforms to the vector valued setting with scalar weights, for which we indicate a proof. Extensions to L ω p for 1 < p < ∞ are discussed. Our arguments to exhibit sharpness at the critical exponent p = 2 require a martingale extrapolation theorem, for which we provide a proof. We also show that for n > 1 , the Poisson- A 2 class is properly included in the classical A 2 class.

Authors:Nikolaos Papageorgiou; Patrick Winkert Abstract: Publication date: Available online 26 May 2017 Source:Bulletin des Sciences Mathématiques Author(s): Nikolaos S. Papageorgiou, Patrick Winkert We consider a Dirichlet problem driven by the sum of a p-Laplacian and a Laplacian (known as a ( p , 2 ) -equation) and with a nonlinearity which exhibits asymmetric behavior as s → ± ∞ . More precisely, it is ( p − 1 ) -superlinear near +∞ (but without satisfying the Ambrosetti-Rabinowitz condition) and it is ( p − 1 ) -sublinear near −∞ and possibly resonant with respect to the principal eigenvalue of the p-Laplacian. Using variational tools along with Morse theory we prove a multiplicity theorem generating five nontrivial solutions (one is negative, two are positive, one is nodal and for the the fifth we do not have any information about its sign).

Authors:S.C. de Almeida; F.G.B. Brito; A.G. Colares Abstract: Publication date: Available online 22 May 2017 Source:Bulletin des Sciences Mathématiques Author(s): S.C. de Almeida, F.G.B. Brito, A.G. Colares In this paper we study the geometric properties of a couple of mutually orthogonal foliations with complementary dimensions. We recall that from Novikov's theorem, there is no foliation of S 3 by closed curves with integrable normal bundle. Nevertheless, for S 2 k + 1 , k ≥ 2 , Novikov's theorem is not applicable. In this paper we show that on odd-dimensional unit spheres there is no umbilical foliation with integrable normal bundle and divergence free mean curvature vector.

Authors:Indranil Biswas; Mahuya Datta Abstract: Publication date: Available online 15 May 2017 Source:Bulletin des Sciences Mathématiques Author(s): Indranil Biswas, Mahuya Datta A complex manifold or a symplectic manifold is automatically oriented. We investigate these structures in the context of non-orientable manifolds. Any smooth connected non-orientable manifold is equipped with a real line bundle of order two. Various structures which are defined only on oriented manifolds extend to non-orientable manifolds once they are twisted by this line bundle of order two. Our aim is to develop this theme.

Authors:Emilio Franco; Marcos Jardim; Simone Marchesi Abstract: Publication date: Available online 10 May 2017 Source:Bulletin des Sciences Mathématiques Author(s): Emilio Franco, Marcos Jardim, Simone Marchesi In the physicist's language, a brane in a hyperkähler manifold is a submanifold which is either complex or lagrangian with respect to three Kähler structures of the ambient manifold. By considering the fixed loci of certain involutions, we describe branes in Nakajima quiver varieties of all possible types. We then focus on the moduli space of framed torsion free sheaves on the projective plane, showing how the involutions considered act on sheaves, and proving the existence of branes in some cases.

Authors:Hélène Bommier-Hato; El Hassan Youssfi; Kehe Zhu Abstract: Publication date: Available online 8 May 2017 Source:Bulletin des Sciences Mathématiques Author(s): Hélène Bommier-Hato, El Hassan Youssfi, Kehe Zhu Sarason's Toeplitz product problem asks when the operator T u T v ‾ is bounded on various Hilbert spaces of analytic functions, where u and v are analytic. The problem is highly nontrivial for Toeplitz operators on the Hardy space and the Bergman space (even in the case of the unit disk). In this paper, we provide a complete solution to the problem for a class of Fock spaces on the complex plane. In particular, this generalizes an earlier result of Cho, Park, and Zhu.

Authors:Asma Azaiez; Hatem Zaag Abstract: Publication date: Available online 26 April 2017 Source:Bulletin des Sciences Mathématiques Author(s): Asma Azaiez, Hatem Zaag We consider a vector-valued blow-up solution with values in R m for the semilinear wave equation with power nonlinearity in one space dimension (this is a system of PDEs). We first characterize all the solutions of the associated stationary problem as an m-parameter family. Then, we show that the solution in self-similar variables approaches some particular stationary one in the energy norm, in the non-characteristic cases. Our analysis is not just a simple adaptation of the already handled real or complex case. In particular, there is a new structure of the set a stationary solutions.

Authors:Dongrui Wan; Wei Wang Abstract: Publication date: Available online 12 April 2017 Source:Bulletin des Sciences Mathématiques Author(s): Dongrui Wan, Wei Wang In this paper, we introduce the first-order differential operators d 0 and d 1 acting on the quaternionic version of differential forms on the flat quaternionic space H n . The behavior of d 0 , d 1 and △ = d 0 d 1 is very similar to ∂ , ∂ ‾ and ∂ ∂ ‾ in several complex variables. The quaternionic Monge-Ampère operator can be defined as ( △ u ) n and has a simple explicit expression. We define the notion of a closed positive current in the quaternionic case, and extend several results in complex pluripotential theory to the quaternionic case: define the Lelong number of a closed positive current, obtain the quaternionic version of Lelong-Jensen type formula, and generalize Bedford-Taylor theory, i.e., extend the definition of the quaternionic Monge-Ampère operator to locally bounded quaternionic plurisubharmonic functions and prove the corresponding convergence theorem.

Authors:Nikolaos S. Papageorgiou; Vicenţiu D. Rădulescu Abstract: Publication date: Available online 5 April 2017 Source:Bulletin des Sciences Mathématiques Author(s): Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu We consider a semilinear Robin problem driven by the Laplacian plus an indefinite and unbounded using a suitable version of the symmetric mountain pass theorem, we show that the problem has an infinity of nodal solutions whose energy level diverges to +∞.

Authors:Todd A. Oliynyk Abstract: Publication date: Available online 4 March 2017 Source:Bulletin des Sciences Mathématiques Author(s): Todd A. Oliynyk We demonstrate that a sufficiently smooth solution of the relativistic Euler equations that represents a dynamical compact liquid body, when expressed in Lagrangian coordinates, determines a solution to a system of non-linear wave equations with acoustic boundary conditions. Using this wave formulation, we prove that these solutions satisfy energy estimates without loss of derivatives. Importantly, our wave formulation does not require the liquid to be irrotational, and the energy estimates do not rely on divergence and curl type estimates employed in previous works.

Authors:Vianney Combet; Yvan Martel Abstract: Publication date: Available online 20 January 2017 Source:Bulletin des Sciences Mathématiques Author(s): Vianney Combet, Yvan Martel Let S be a minimal mass blow up solution of the critical generalized KdV equation as constructed in [25]. We prove both time and space sharp asymptotics for S close to the blow up time. Let Q be the unique ground state of (gKdV), satisfying Q ″ + Q 5 = Q . First, we show that there exist universal smooth profiles Q k ∈ S ( R ) (with Q 0 = Q ) and a constant c 0 ∈ R such that, fixing the blow up time at t = 0 and appropriate scaling and translation parameters, S satisfies, for any m ⩾ 0 , ∂ x m S ( t ) − ∑ k = 0 [ m / 2 ] 1 t 1 2 + m − 2 k Q k ( m − k ) ( ⋅ + 1 t t + c 0 ) → 0 in L 2 as t ↓ 0 . Second, we prove that, for 0 < t ≪ 1 , x ⩽ − 1 t − 1 , S ( t , x ) ∼ − 1 2 ‖ Q ‖ L 1 x − 3 / 2 , and related bounds for the derivatives of S ( t ) of any order. We also prove ∫ R S ( t , x ) d x = 0 .

Authors:Annamaria Canino; Luigi Montoro; Berardino Sciunzi; Marco Squassina Abstract: Publication date: Available online 20 January 2017 Source:Bulletin des Sciences Mathématiques Author(s): Annamaria Canino, Luigi Montoro, Berardino Sciunzi, Marco Squassina We investigate existence and uniqueness of solutions for a class of nonlinear nonlocal problems involving the fractional p-Laplacian operator and singular nonlinearities.

Authors:Han Abstract: Publication date: January 2017 Source:Bulletin des Sciences Mathématiques, Volume 141, Issue 1 Author(s): Qi Han In this paper, we study mainly the existence of multiple positive solutions for a quasilinear elliptic equation of the following form on R N , when N ≥ 2 , (0.1) − Δ N u + V ( x ) u N − 2 u = λ u r − 2 u + f ( x , u ) . Here, V ( x ) > 0 : R N → R is a suitable potential function, r ∈ ( 1 , N ) , f ( x , u ) is a continuous function of N-superlinear and subcritical exponential growth without having the Ambrosetti–Rabinowitz condition, while λ > 0 is a constant. A suitable Moser–Trudinger inequality and the compact embedding W V 1 , N ( R N ) ↪ L r ( R N ) are proved to study problem (0.1). Moreover, the compact embedding H V 1 ( R N ) ↪ L K t ( R N ) is also analyzed to investigate the existence of a positive ground state to the following nonlinear Schrödinger equation (0.2) − Δ u + V ( x ) u = K ( x ) g ( u ) with potentials vanishing at infinity in a measure-theoretic sense when N ≥ 3 .

Authors:Daniel Gonçalves; Danilo Royer Abstract: Publication date: Available online 2 December 2016 Source:Bulletin des Sciences Mathématiques Author(s): Daniel Gonçalves, Danilo Royer In this paper we further develop the theory of one-sided shift spaces over infinite alphabets, characterizing one-step shifts as edge shifts of ultragraphs and partially answering a conjecture regarding shifts of finite type (we show that there exists shifts of finite type that are not conjugate, via a conjugacy that is eventually finite periodic, to an edge shift of a graph ). We also show that there exists edge shifts of ultragraphs that are shifts of finite type, but are not conjugate to a full shift, a result that is not true for edge shifts of graphs. One of the key results needed in the proofs of our conclusions is the realization of a class of ultragraph C*-algebras as partial crossed products, a result of interest on its own.

Authors:David Kalaj Abstract: Publication date: Available online 17 November 2016 Source:Bulletin des Sciences Mathématiques Author(s): David Kalaj In this paper we extend Radó-Kneser-Choquet theorem for the mappings with weak homeomorphic Lipschitz boundary function and Dini's smooth boundary but without restriction on the convexity of the image domain, provided that the Jacobian satisfies a certain boundary condition. The proof is based on a recent extension of Radó-Kneser-Choquet theorem by Alessandrini and Nesi [1] and is used the approximation principle.

Authors:Pralay Chatterjee; Chandan Maity Abstract: Publication date: Available online 4 October 2016 Source:Bulletin des Sciences Mathématiques Author(s): Pralay Chatterjee, Chandan Maity In [1], the second de Rham cohomology groups of nilpotent orbits in all the complex simple Lie algebras are described. In this paper we consider non-compact non-complex exceptional Lie algebras, and compute the dimensions of the second cohomology groups for most of the nilpotent orbits. For the rest of cases of nilpotent orbits, which are not covered in the above computations, we obtain upper bounds for the dimensions of the second cohomology groups.

Authors:Sergio Albeverio; Iryna Garko; Muslem Ibragim; Grygoriy Torbin Abstract: Publication date: Available online 5 April 2016 Source:Bulletin des Sciences Mathématiques Author(s): Sergio Albeverio, Iryna Garko, Muslem Ibragim, Grygoriy Torbin In the present paper we study the dependence of fractal and metric properties of numbers which are non-normal resp. essentialy non-normal w.r.t. a chosen system of numeration. In particular, we solve open problems mentioned in [1] and prove that there exist expansions (the Q ⁎ -expansions or Q ⁎ -representations) for real numbers such that the corresponding sets of essentially non-normal numbers and even the whole set of non-normal numbers are of zero Hausdorff dimension. On the other hand, we show that in the same model of Q ⁎ -expansions it is possible to choose the matrix Q ⁎ in such a way that the corresponding set of essentially non-normal numbers is of full Lebesgue measure. Sufficient conditions for full dimensionality resp. zero dimensionality of the set of essentially non-normal numbers are also presented.