Authors:Maria Pia Gualdani; Nicola Zamponi Pages: 509 - 538 Abstract: Publication date: August 2017 Source:Bulletin des Sciences Mathématiques, Volume 141, Issue 6 Author(s): Maria Pia Gualdani, Nicola Zamponi In this paper we prove new constructive coercivity estimates and convergence to equilibrium for a spatially non-homogeneous system of Landau equations with moderately soft potentials. We show that the nonlinear collision operator conserves each species' mass, total momentum, total energy and that the Boltzmann entropy is nonincreasing along solutions of the system. The entropy decay vanishes if and only if the Boltzmann distributions of the single species are Maxwellians with the same momentum and energy. A linearization of the collision operator is computed, which has the same conservation properties as its nonlinear counterpart. We show that the linearized system dissipates a quadratic entropy, and prove existence of spectral gap and exponential decay of the solution towards the global equilibrium. As a consequence, convergence of smooth solutions of the nonlinear problem toward the unique global equilibrium is shown, provided the initial data are sufficiently close to the equilibrium. Our proof is based on new spectral gap estimates and uses a strategy similar to [11] based on an hypocoercivity method developed by Mouhot and Neumann in [27].

Authors:Edward L. Green; Sibylle Schroll Pages: 539 - 572 Abstract: Publication date: August 2017 Source:Bulletin des Sciences Mathématiques, Volume 141, Issue 6 Author(s): Edward L. Green, Sibylle Schroll In this paper we introduce a generalization of a Brauer graph algebra which we call a Brauer configuration algebra. As with Brauer graphs and Brauer graph algebras, to each Brauer configuration, there is an associated Brauer configuration algebra. We show that Brauer configuration algebras are finite dimensional symmetric algebras. After studying and analysing structural properties of Brauer configurations and Brauer configuration algebras, we show that a Brauer configuration algebra is multiserial; that is, its Jacobson radical is a sum of uniserial modules whose pairwise intersection is either zero or a simple module. The paper ends with a detailed study of the relationship between radical cubed zero Brauer configuration algebras, symmetric matrices with non-negative integer entries, finite graphs and associated symmetric radical cubed zero algebras.

Abstract: Publication date: February 2018 Source:Bulletin des Sciences Mathématiques, Volume 142 Author(s): Omar Anza Hafsa, Jean-Philippe Mandallena We study relaxation of nonconvex integrals of the calculus of variations in the setting of Cheeger–Sobolev spaces when the integrand does not have polynomial growth and can take infinite values.

Abstract: Publication date: Available online 5 December 2017 Source:Bulletin des Sciences Mathématiques Author(s): A.J. Parameswaran, Mihai Tibăr We find natural and convenient conditions which allow us to produce classes of genuine real map germs with Milnor tube fibration, either with Thom regularity or without it.

Abstract: Publication date: Available online 13 October 2017 Source:Bulletin des Sciences Mathématiques Author(s): Jesús Suárez de la Fuente We characterize the subspaces X of ℓ 1 satisfying Grothendieck's theorem in terms of extension of nonnegative quadratic forms q : X ⟶ R to the whole ℓ 1 .

Abstract: Publication date: Available online 6 October 2017 Source:Bulletin des Sciences Mathématiques Author(s): Giovanni Bazzoni, Juan Carlos Marrero We present some examples of locally conformal symplectic structures of the first kind on compact nilmanifolds which do not admit Vaisman metrics. One of these examples does not admit locally conformal Kähler metrics and all the structures come from left-invariant locally conformal symplectic structures on the corresponding nilpotent Lie groups. Under certain topological restrictions related with the compactness of the canonical foliation, we prove a structure theorem for locally conformal symplectic manifolds of the first kind. In the non compact case, we show that they are the product of a real line with a compact contact manifold and, in the compact case, we obtain that they are mapping tori of compact contact manifolds by strict contactomorphisms. Motivated by the aforementioned examples, we also study left-invariant locally conformal symplectic structures on Lie groups. In particular, we obtain a complete description of these structures (with non-zero Lee form) on connected simply connected nilpotent Lie groups in terms of locally conformal symplectic extensions and symplectic double extensions of symplectic nilpotent Lie groups. In order to obtain this description, we study locally conformal symplectic structures of the first kind on Lie algebras.

Abstract: Publication date: Available online 21 September 2017 Source:Bulletin des Sciences Mathématiques Author(s): Sarbeswar Pal Let X be a smooth projective curve over complex numbers of arbitrary genus. In this paper we show that the locus of non-very stable bundles of rank 2 and determinant isomorphic to a fixed line bundle over X of degree 1 is a closed subvariety of codimension 1. We also give an explicit description of this closed subvariety for the genus 2 case.

Abstract: Publication date: Available online 19 September 2017 Source:Bulletin des Sciences Mathématiques Author(s): Olgur Celikbas, Kei-ichiro Iima, Arash Sadeghi, Ryo Takahashi A celebrated conjecture of Auslander and Reiten claims that a finitely generated module M that has no extensions with M ⊕ Λ over an Artin algebra Λ must be projective. This conjecture is widely open in general, even for modules over commutative Noetherian local rings. Over such rings, we prove that a large class of ideals satisfy the extension condition proposed in the aforementioned conjecture of Auslander and Reiten. Along the way we obtain a new characterization of regularity in terms of the injective dimensions of certain ideals.

Abstract: Publication date: Available online 14 September 2017 Source:Bulletin des Sciences Mathématiques Author(s): Karin Baur, Eleonore Faber, Sira Gratz, Khrystyna Serhiyenko, Gordana Todorov We study mutations of Conway-Coxeter friezes which are compatible with mutations of cluster-tilting objects in the associated cluster category of Dynkin type A. More precisely, we provide a formula, relying solely on the shape of the frieze, describing how each individual entry in the frieze changes under cluster mutation. We observe how the frieze can be divided into four distinct regions, relative to the entry at which we want to mutate, where any two entries in the same region obey the same mutation rule. Moreover, we provide a combinatorial formula for the number of submodules of a string module, and with that a simple way to compute the frieze associated to a fixed cluster-tilting object in a cluster category of Dynkin type A in the sense of Caldero and Chapoton.

Abstract: Publication date: Available online 12 September 2017 Source:Bulletin des Sciences Mathématiques Author(s): Michael Anshelevich, John D. Williams In the setting of distributions taking values in a C ⁎ -algebra B , we define generalized Jacobi parameters and study distributions they generate. These include numerous known examples and one new family, of B -valued free binomial distributions, for which we are able to compute free convolution powers. Moreover, we develop a convenient combinatorial method for calculating the joint distributions of B -free random variables with Jacobi parameters, utilizing two-color non-crossing partitions. This leads to several new explicit examples of free convolution computations in the operator-valued setting. Additionally, we obtain a counting algorithm for the number of two-color non-crossing pairings of relative finite depth, using only free probabilistic techniques. Finally, we show that the class of distributions with Jacobi parameters is not closed under free convolution.

Abstract: Publication date: Available online 6 September 2017 Source:Bulletin des Sciences Mathématiques Author(s): D. Pellegrino, J. Santos, D. Serrano-Rodríguez, E. Teixeira We prove a nonlinear regularity principle in sequence spaces which produces universal estimates for special series defined therein. Some consequences are obtained and, in particular, we establish new inclusion theorems for multiple summing operators. Of independent interest, we settle all Grothendieck's type ( ℓ 1 , ℓ 2 ) theorems for multiple summing multilinear operators. We further employ the new regularity principle to solve the classification problem concerning all pairs of admissible exponents in the anisotropic Hardy–Littlewood inequalities.

Authors:Trivedi Abstract: Publication date: Available online 1 August 2017 Source:Bulletin des Sciences Mathématiques Author(s): V. Trivedi Here we consider the set of bundles { V n } n ∈ N associated to the plane trinomial curves k [ x , y , z ] / ( h ) . We prove that the Frobenius semistability behaviour of the reduction mod p of V n is a function of the congruence class of p modulo 2 λ h (an integer invariant associated to h). As one of the consequences of this, we prove that if V n is semistable in char 0 then its reduction mod p is strongly semistable, for p in a Zariski dense set of primes. Moreover, for any given finitely many such semistable bundles V n , there is a common Zariski dense set of such primes.

Authors:Aline Bonami; Justin Feuto; Sandrine Grellier; Luong Dang Ky Abstract: Publication date: Available online 29 July 2017 Source:Bulletin des Sciences Mathématiques Author(s): Aline Bonami, Justin Feuto, Sandrine Grellier, Luong Dang Ky We give an atomic decomposition of closed forms on R n , the coefficients of which belong to some Hardy space of Musielak–Orlicz type. These spaces are natural generalizations of weighted Hardy–Orlicz spaces, when the Orlicz function depends on the space variable. One of them, called H log , appears naturally when considering products of functions in the Hardy space H 1 and in BMO. As a main consequence of the atomic decomposition, we obtain a weak factorization of closed forms whose coefficients are in H log . Namely, a closed form in H log is the infinite sum of the wedge product between an exact form in the Hardy space H 1 and an exact form in BMO. The converse result, which generalizes the classical div–curl lemma, is a consequence of [4]. As a corollary, we prove that the real-valued H log space can be weakly factorized.

Authors:M. Hitrik; K. Pravda-Starov; J. Viola Abstract: Publication date: Available online 29 July 2017 Source:Bulletin des Sciences Mathématiques Author(s): M. Hitrik, K. Pravda-Starov, J. Viola We study accretive quadratic operators with zero singular spaces. These degenerate non-selfadjoint differential operators are known to be hypoelliptic and to generate contraction semigroups which are smoothing in the Schwartz space for any positive time. In this work, we study the short-time asymptotics of the regularizing effect induced by these semigroups. We show that these short-time asymptotics of the regularizing effect depend on the directions of the phase space, and that this dependence can be nicely understood through the structure of the singular space. As a byproduct of these results, we derive sharp subelliptic estimates for accretive quadratic operators with zero singular spaces pointing out that the loss of derivatives with respect to the elliptic case also depends on the phase space directions according to the structure of the singular space. Some applications of these results are then given to the study of degenerate hypoelliptic Ornstein-Uhlenbeck operators and degenerate hypoelliptic Fokker-Planck operators.

Authors:JinRong Wang; Michal Fečkan; Yong Zhou Abstract: Publication date: Available online 29 July 2017 Source:Bulletin des Sciences Mathématiques Author(s): JinRong Wang, Michal Fečkan, Yong Zhou In this paper, we discuss a new class of fractional order differential switched systems with coupled nonlocal initial and impulsive conditions in R n . We firstly derive a solution formula for this system. Secondly, we utilize three well-known fixed point methods to present the existence results. Moreover, we use Schauder topological degree theory to show a new existence result for resonant case: Landesman-Lazer conditions. Finally, we introduce the concepts of Ulam's type stability and present new stability results in the space of fractional version piecewise continuous functions.

Authors:I. Area; M. Foupouagnigni; E. Godoy; Y. Guemo Tefo Abstract: Publication date: Available online 29 July 2017 Source:Bulletin des Sciences Mathématiques Author(s): I. Area, M. Foupouagnigni, E. Godoy, Y. Guemo Tefo In this work explicit representations of the moments of certain hypergeometric continuous and discrete weight functions are obtained. An algorithm for constructing a sequence of bivariate orthonormal polynomials by using these moments is presented. Finally, the linearization problem between orthonormal polynomials is explicitly solved.

Authors:Iván Area; Mohammad Masjed-Jamei Abstract: Publication date: Available online 29 July 2017 Source:Bulletin des Sciences Mathématiques Author(s): Iván Area, Mohammad Masjed-Jamei By using a generalization of Sturm-Liouville problems in q-difference spaces, a class of symmetric q-orthogonal polynomials with four free parameters is introduced. The standard properties of these polynomials, such as a second order q-difference equation, the explicit form of the polynomials in terms of basic hypergeometric series, a three term recurrence relation and a general orthogonality relation are presented. Some particular examples are then studied in detail.

Authors:Sandra Pott; Andrei Stoica Abstract: Publication date: Available online 8 July 2017 Source:Bulletin des Sciences Mathématiques Author(s): Sandra Pott, Andrei Stoica For a matrix A 2 weight W on R p , we introduce a new notion of W-Calderón-Zygmund matrix kernels, following earlier work in [11]. We state and prove a T1 theorem for such operators and give a representation theorem in terms of dyadic W-Haar shifts and paraproducts, in the spirit of [7]. Finally, by means of a Bellman function argument, we give sharp bounds for such operators in terms of bounds for weighted matrix martingale transforms and paraproducts.

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.