Abstract: Publication date: Available online 10 January 2018 Source:Expositiones Mathematicae Author(s): Felipe Cano, Nuria Corral, Rogério Mol In this survey paper, we take the viewpoint of polar invariants to the local and global study of non-dicritical holomorphic foliation in dimension two and their invariant curves. It appears a characterization of second type foliations and generalized curve foliations as well as a description of the G S V -index in terms of polar curves. We also interpret the proofs concerning the Poincaré problem with polar invariants.

Abstract: Publication date: Available online 10 January 2018 Source:Expositiones Mathematicae Author(s): Andrei K. Lerner, Fedor Nazarov This article is divided into two parts. In the first part we present a general theory of the dyadic lattices. In the second part we show several applications of this theory to harmonic analysis: a decomposition of an arbitrary measurable function in terms of its local mean oscillations, and a pointwise bound of Calderón–Zygmund operators by sparse operators.

Abstract: Publication date: Available online 10 January 2018 Source:Expositiones Mathematicae Author(s): Rajendra Bhatia, Tanvi Jain, Yongdo Lim The metric d ( A , B ) = tr A + tr B − 2 tr ( A 1 ∕ 2 B A 1 ∕ 2 ) 1 ∕ 2 1 ∕ 2 on the manifold of n × n positive definite matrices arises in various optimisation problems, in quantum information and in the theory of optimal transport. It is also related to Riemannian geometry. In the first part of this paper we study this metric from the perspective of matrix analysis, simplifying and unifying various proofs. Then we develop a theory of a mean of two, and a barycentre of several, positive definite matrices with respect to this metric. We explain some recent work on a fixed point iteration for computing this Wasserstein barycentre. Our emphasis is on ideas natural to matrix analysis.

Abstract: Publication date: Available online 2 January 2018 Source:Expositiones Mathematicae Author(s): Peter Crooks This expository article is an introduction to the adjoint orbits of complex semisimple groups, primarily in the algebro-geometric and Lie-theoretic contexts, and with a pronounced emphasis on the properties of semisimple and nilpotent orbits. It is intended to build a foundation for more specialized settings in which adjoint orbits feature prominently (ex. hyperkähler geometry, Landau-Ginzburg models, and the theory of symplectic singularities). Also included are a few arguments and observations that, to the author’s knowledge, have not yet appeared in the research literature.

Abstract: Publication date: Available online 1 January 2018 Source:Expositiones Mathematicae Author(s): Martín Argerami The matricial range of the 2 × 2 matrix E 21 (i.e., the 2 × 2 unilateral shift) is described very simply: it consists of all matrices with numerical radius at most 1 ∕ 2 . The known proofs of this simple statement, however, are far from trivial and they depend on subtle results on dilations. We offer here a brief introduction to the matricial range and a recap of those two proofs, following independent work of Arveson and Ando in the early 1970s.

Abstract: Publication date: Available online 14 November 2017 Source:Expositiones Mathematicae Author(s): Takis Konstantopoulos, Linglong Yuan, Michael A. Zazanis This paper presents a stochastic approach to theorems concerning the behavior of iterations of the Bernstein operator B n taking a continuous function f ∈ C [ 0 , 1 ] to a degree- n polynomial when the number of iterations k tends to infinity and n is kept fixed or when n tends to infinity as well. In the first instance, the underlying stochastic process is the so-called Wright–Fisher model, whereas, in the second instance, the underlying stochastic process is the Wright–Fisher diffusion. Both processes are probably the most basic ones in mathematical genetics. By using Markov chain theory and stochastic compositions, we explain probabilistically a theorem due to Kelisky and Rivlin, and by using stochastic calculus we compute a formula for the application of B n a number of times k = k ( n ) to a polynomial f when k ( n ) ∕ n tends to a constant.

Abstract: Publication date: Available online 10 November 2017 Source:Expositiones Mathematicae Author(s): Philip S. Hirschhorn We give a complete and careful proof of Quillen’s theorem on the existence of the standard model category structure on the category of topological spaces. We do not assume any familiarity with model categories.

Abstract: Publication date: Available online 10 November 2017 Source:Expositiones Mathematicae Author(s): Serge Dubuc We consider the class of self-affine functions. Firstly, we characterize all nowhere differentiable self-affine continuous functions. Secondly, given a self-affine continuous function ϕ , we investigate its Hölder properties. We find its best uniform Hölder exponent and when ϕ is C 1 , we find the best uniform Hölder exponent of ϕ ′ . Thirdly, we show that the Hölder cut of ϕ takes the same value almost everywhere for the Lebesgue measure. This last result is a consequence of the Borel strong law of large numbers.

Abstract: Publication date: Available online 1 September 2017 Source:Expositiones Mathematicae Author(s): Alessandra Bertapelle, Cristian D. González-Avilés We present a detailed and elementary construction of the inverse perfection of a scheme and discuss some of its main properties. We also establish a number of auxiliary results (for example, on inverse limits of schemes) which do not seem to appear in the literature.

Abstract: Publication date: Available online 30 August 2017 Source:Expositiones Mathematicae Author(s): Ravi P. Agarwal, Janusz Brzdęk, Jacek Chudziak Composite type functional equations in several variables play a significant role in various branches of mathematics and they have several interesting applications. Therefore their stability properties are of interest. The aim of this paper is to present a survey of some results and methods concerning stability of several of such equations; especially those somehow connected to the well known Gołąb-Schinzel equation.

Abstract: Publication date: Available online 30 August 2017 Source:Expositiones Mathematicae Author(s): Karim Johannes Becher, Thomas Unger Pfister’s Local-Global Principle states that a quadratic form over a (formally) real field is weakly hyperbolic (i.e. represents a torsion element in the Witt ring) if and only if its total signature is zero. This result extends naturally to the setting of central simple algebras with involution. The present article provides a new proof of this result and extends it to the case of signatures at preorderings. Furthermore the quantitative relation between nilpotence and torsion is explored for quadratic forms as well as for central simple algebras with involution.

Abstract: Publication date: Available online 30 August 2017 Source:Expositiones Mathematicae Author(s): S. Hou, J. Xiao This note geometrically generalizes the main theorem in H. Shahgholian, A characterization of the sphere in terms of single-layer potentials, Proc. Amer. Math. Soc. 115 (1992) 1167-1168, from R 3 to R n ≥ 2 via the Minkowski functional of a convex body with the origin in its interior.

Abstract: Publication date: Available online 30 August 2017 Source:Expositiones Mathematicae Author(s): Umberto Zannier The present article offers an overview of the basic theory of heights of algebraic points on varieties, illustrating some applications, mainly of diophantine type. The exposition is informal and with little detail, modelled on three lectures delivered by the author at the Fields Institute in Toronto in February 2017.

Abstract: Publication date: Available online 30 August 2017 Source:Expositiones Mathematicae Author(s): Martin Brandenburg Alexander L. Rosenberg has constructed a spectrum for abelian categories which is able to reconstruct a quasi-separated scheme from its category of quasi-coherent sheaves. In this note we present a detailed proof of this result which is due to Ofer Gabber. Moreover, we determine the automorphism class group of the category of quasi-coherent sheaves.

Abstract: Publication date: Available online 30 August 2017 Source:Expositiones Mathematicae Author(s): S.J. Bhatt, S.H. Kulkarni The Gelfand-Mazur Theorem, a very basic theorem in the theory of Banach algebras states that: (Real version) Every real normed division algebra is isomorphic to the algebra of all real numbers R , the complex numbers C or the quarternions H ; (Complex version) Every complex normed division algebra is isometrically isomorphic to C . This theorem has undergone a large number of generalizations. We present a survey of these generalizations and also discuss some closely related unsettled issues.

Abstract: Publication date: Available online 30 August 2017 Source:Expositiones Mathematicae Author(s): Carlo Morosi, Livio Pizzocchero We consider the inequalities of Gagliardo–Nirenberg and Sobolev in R d , formulated in terms of the Laplacian Δ and of the fractional powers D n ≔ − Δ n with real n ⩾ 0 ; we review known facts and present novel, complementary results in this area. After illustrating the equivalence between these two inequalities and the relations between the corresponding sharp constants and maximizers, we focus the attention on the ℒ 2 case where, for all sufficiently regular f : R d → C , the norm ‖ D j f ‖ ℒ r is bounded in terms of ‖ f ‖ ℒ 2 and ‖ D n f ‖ ℒ 2 , for 1 ∕ r = 1 ∕ 2 − ( ϑ n − j ) ∕ d , and suitable values of j , n , ϑ (with j , n possibly noninteger). In the special cases ϑ = 1 and ϑ = j ∕ n + d ∕ 2 n (i.e., r = + ∞ ), related to previous results of Lieb and Ilyin, the sharp constants and the maximizers can be found explicitly; we point out that the maximizers can be expressed in terms of hypergeometric, Fox and Meijer functions. For the general ℒ 2 case, we present two kinds of upper bounds on the sharp constants: the first kind is suggested by the literature, the second one is an alternative proposal of ours, often more precise than the first one. We also derive two kinds of lower bounds. Combining all the available upper and lower bounds, the sharp constants are confined to quite narrow intervals. Several examples are given, including the numerical values of the previously mentioned bounds.

Abstract: Publication date: Available online 17 May 2017 Source:Expositiones Mathematicae Author(s): Vugar E. Ismailov We consider the approximation of a continuous function, defined on a compact set of the d -dimensional Euclidean space, by sums of two ridge functions. We obtain a necessary and sufficient condition for such a sum to be a best approximation. The result resembles the classical Chebyshev equioscillation theorem for polynomial approximation.

Abstract: Publication date: Available online 17 May 2017 Source:Expositiones Mathematicae Author(s): Jean-Pierre Kahane, Eric Saias Let f be a function from N ∗ to C not identically 0 . We call it C M O if f ( a b ) = f ( a ) f ( b ) for all a and b and ∑ n = 1 + ∞ f ( n ) = 0 . We give properties and examples of C M O functions. A basic example goes back to Euler, namely f ( p ) = − 1 ∕ p for every prime number p . We study how far from this example the C M O character is kept. The zeroes of Dirichlet series are implied in this study, as well as in other examples. The relation between C M O and the generalized Riemann hypothesis is pointed out at the end of the article.

Abstract: Publication date: Available online 17 May 2017 Source:Expositiones Mathematicae Author(s): Avner Kiro, Mikhail Sodin For a class of functions γ analytic in the angle { s : arg ( s ) < α 0 } with π 2 < α 0 < π , we describe the asymptotic behaviour of the entire function E ( z ) = ∑ n ≥ 0 z n γ ( n + 1 ) and of the analytic function K ( z ) = 1 2 π i ∫ c − i ∞ c + i ∞ z − s γ ( s ) d s that solves the moment problem ∫ 0 ∞ t n K ( t ) d t = γ ( n + 1 ) , n ≥ 0 .

Abstract: Publication date: Available online 8 May 2017 Source:Expositiones Mathematicae Author(s): Alexandru Chirvasitu, Souleiman Omar Hoche, Paweł Kasprzak The lattice of subgroups of a group is the subject of numerous results revolving around the central theme of decomposing the group into ”chunks” (subquotients) that can then be compared to one another in various ways. Examples of results in this class would be the Noether isomorphism theorems, Zassenhaus’ butterfly lemma, the Schreier refinement theorem, the Dedekind modularity law, and last but not least the Jordan-Hölder theorem. We discuss analogues of the above-mentioned results in the context of locally compact quantum groups and linearly reductive quantum groups. The nature of the two cases is different: the former is operator algebraic and the latter Hopf algebraic, hence the corresponding two-part organization of our study. Our intention is that the analytic portion be accessible to the algebraist and vice versa. The upshot is that in the locally compact case one often needs further assumptions (integrability, compactness, discreteness). In the linearly reductive case on the other hand, the quantum versions of the results hold without further assumptions. Moreover the case of compact / discrete quantum groups is usually covered by both the linearly reductive and the locally compact framework, thus providing a bridge between the two.

Abstract: Publication date: Available online 18 March 2017 Source:Expositiones Mathematicae Author(s): Pete L. Clark We give an exposition of a recent result of M. Kapovich on the cardinal Krull dimension of the ring of holomorphic functions on a connected C -manifold. By reducing to the one-dimensional case we give a stronger lower bound for Stein manifolds.

Abstract: Publication date: Available online 5 January 2017 Source:Expositiones Mathematicae Author(s): Caroline Lassueur, Jacques Thévenaz It is well-known that a finite group possesses a universal central extension if and only if it is a perfect group. Similarly, given a prime number p , we show that a finite group possesses a universal p ′ -central extension if and only if the p ′ -part of its abelianization is trivial. This question arises naturally when working with group representations over a field of characteristic p .