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 .

Abstract: Publication date: Available online 26 December 2016 Source:Expositiones Mathematicae Author(s): Celeste Damiani In this paper we introduce distinct approaches to loop braid groups, a generalization of braid groups, and unify all the definitions that have appeared so far in the literature, with a complete proof of the equivalence of these definitions. These groups have in fact been an object of interest in different domains of mathematics and mathematical physics, and have been called, in addition to loop braid groups, with several names such as of motion groups, groups of permutation-conjugacy automorphisms, braid-permutation groups, welded braid groups and untwisted ring groups. In parallel to this, we introduce an extension of these groups that appears to be a more natural generalization of braid groups from the topological point of view. Throughout the text we motivate the interest in studying loop braid groups and give references to some of their applications.

Abstract: Publication date: Available online 26 December 2016 Source:Expositiones Mathematicae Author(s): Jaume Llibre, Xiang Zhang In 1977 Lins Neto, de Melo and Pugh (Lins Neto et al., 1977)conjectured that the classical Liénard system x ̇ = y − F ( x ) , y ̇ = − x , with F ( x ) a real polynomial of degree n , has at most [ ( n − 1 ) / 2 ] limit cycles, where [ ⋅ ] denotes the integer part function. In this paper we summarize what is known and what is still open on this conjecture. For the known results on this conjecture we present a complete proof.

Abstract: Publication date: Available online 24 December 2016 Source:Expositiones Mathematicae Author(s): Anatoliy Manov, Viktor Zastavnyi Let α ∈ ( 0 , 1 ) , h ∈ R and let f α , h be an even function with the properties: f α , h ( x ) = 0 for x ∈ ( 1 , + ∞ ) , f α , h is linear over the intervals [ 0 , α ] and [ α , 1 ] , f α , h ( 0 ) = 1 , f α , h ( α ) = h , and f α , h ( 1 ) = 0 . In this paper we prove that f α , h is positive definite on R ⟺ m ( α ) ≤ h ≤ 1 − α , where m ( α ) = 0 if 1 / α ∉ N , and m ( α ) = − α if 1 / α ∈ N .

Authors:Sunil K. Chebolu; Dan McQuillan; Ján Mináč Abstract: Publication date: Available online 8 October 2016 Source:Expositiones Mathematicae Author(s): Sunil K. Chebolu, Dan McQuillan, Ján Mináč The year 2017 marks the 80th anniversary of Witt’s famous paper containing key results, including the Witt cancellation theorem, which form the foundation for the algebraic theory of quadratic forms. We pay homage to this paper by presenting a transparent and algebraic proof of the Witt cancellation theorem, which itself is based on a cancellation. We also present an overview of some recent spectacular work which is still building on Witt’s original creation of the algebraic theory of quadratic forms.

Authors:Benjamin Sambale Abstract: Publication date: Available online 7 October 2016 Source:Expositiones Mathematicae Author(s): Benjamin Sambale Let G be a permutation group on n < ∞ objects. Let f ( g ) be the number of fixed points of g ∈ G , and let { f ( g ) : 1 ≠ g ∈ G } = { f 1 , … , f r } . In this expository note we give a character-free proof of a theorem of Blichfeldt which asserts that the order of G divides ( n − f 1 ) … ( n − f r ) . We also discuss the sharpness of this bound.

Authors:R. Fioresi; F. Zanchetta Abstract: Publication date: Available online 7 October 2016 Source:Expositiones Mathematicae Author(s): R. Fioresi, F. Zanchetta In this paper we use the notion of Grothendieck topology to present a unified way to approach representability in supergeometry, which applies to both the differential and algebraic settings.

Authors:Nicolas Monod Abstract: Publication date: Available online 6 October 2016 Source:Expositiones Mathematicae Author(s): Nicolas Monod We propose elementary and explicit presentations of groups that have no amenable quotients and yet are SQ-universal. Examples include groups with a finite K ( π , 1 ) , no Kazhdan subgroups and no Haagerup quotients.

Authors:Alberto Navarro Abstract: Publication date: Available online 5 October 2016 Source:Expositiones Mathematicae Author(s): Alberto Navarro We prove that, for smooth quasi-projective varieties over a field, the K -theory K ( X ) of vector bundles is the universal cohomology theory where c 1 ( L ⊗ L ̄ ) = c 1 ( L ) + c 1 ( L ̄ ) − c 1 ( L ) c 1 ( L ̄ ) . Then, we show that Grothendieck’s Riemann–Rochh theorem is a direct consequence of this universal property, as well as the universal property of the graded K -theory G K • ( X ) ⊗ Q .

Authors:Dinh Trung Hoa; Mohammad Sal Moslehian; Cristian Conde; Pingping Zhang Abstract: Publication date: Available online 5 October 2016 Source:Expositiones Mathematicae Author(s): Dinh Trung Hoa, Mohammad Sal Moslehian, Cristian Conde, Pingping Zhang We extend an operator Pólya–Szegö type inequality involving the operator geometric mean to any arbitrary operator mean under some mild conditions. Utilizing the Mond–Pečarić method, we present some other related operator inequalities as well.

Authors:Tuomas Abstract: Publication date: Available online 2 October 2016 Source:Expositiones Mathematicae Author(s): Tuomas P. Hytönen This exposition presents a self-contained proof of the A 2 theorem, the quantitatively sharp norm inequality for singular integral operators in the weighted space L 2 ( w ) . The strategy of the proof is a streamlined version of the author’s original one, based on a probabilistic Dyadic Representation Theorem for singular integral operators. While more recent non-probabilistic approaches are also available now, the probabilistic method provides additional structural information, which has independent interest and other applications. The presentation emphasizes connections to the David–Journé T ( 1 ) theorem, whose proof is obtained as a byproduct. Only very basic Probability is used; in particular, the conditional probabilities of the original proof are completely avoided.

Authors:Nicholas Switala Abstract: Publication date: Available online 2 October 2016 Source:Expositiones Mathematicae Author(s): Nicholas Switala In this expository paper, we give a complete proof of van den Essen’s theorem that the de Rham cohomology spaces of a holonomic D -module are finite-dimensional in the case of a formal power series ring over a field of characteristic zero. This proof requires results from at least five of van den Essen’s papers as well as his unpublished thesis, and until now has not been available in a self-contained document.

Authors:Matt Clay Abstract: Publication date: Available online 1 October 2016 Source:Expositiones Mathematicae Author(s): Matt Clay We analyze the reduced ℓ 2 -homology groups of a finitely generated nonabelian free group, F . Specifically, the projection map onto the space of harmonic ℓ 2 -1-chains is explicitly described and a weak isomorphism from ( ℓ 2 ( F ) ) rk ( F ) − 1 to the space of harmonic ℓ 2 -1-chains is given.

Authors:Eivind Otto Hjelle; Alexander Schmeding Abstract: Publication date: Available online 3 August 2016 Source:Expositiones Mathematicae Author(s): Eivind Otto Hjelle, Alexander Schmeding In this article we study two “strong” topologies for spaces of smooth functions from a finite-dimensional manifold to a (possibly infinite-dimensional) manifold modelled on a locally convex space. Namely, we construct Whitney type topologies for these spaces and a certain refinement corresponding to Michor’s FD -topology. Then we establish the continuity of certain mappings between spaces of smooth mappings, e.g. the continuity of the joint composition map. As a first application we prove that the bisection group of an arbitrary Lie groupoid (with finite-dimensional base) is a topological group (with respect to these topologies). For the reader’s convenience the article includes also a proof of the folklore fact that the Whitney topologies defined via jet bundles coincide with the ones defined via local charts.

Authors:Barbara A. Shipman; Patrick D. Shipman; Stephen P. Shipman Abstract: Publication date: Available online 1 August 2016 Source:Expositiones Mathematicae Author(s): Barbara A. Shipman, Patrick D. Shipman, Stephen P. Shipman While conformal transformations of the plane preserve Laplace’s equation, Lorentz-conformal mappings preserve the wave equation. This property gives rise to a rich Lorentzian geometry in dimension 1 + 1 . In elucidating the geometry of the Lorentzian plane, this work provides a window into the field of pseudo-Riemannian geometry, where the Lorentzian plane is of fundamental importance. In the Lorentzian plane, curvilinear quadrilaterals and pairs of crossing curves are transformed under nonlinear Lorentz-conformal mappings via geometric constructions that can be expressed also by functional formulas. Classes of Lorentz-conformal maps are characterized by their symmetries under subgroups of the dihedral group of order eight, and unfoldings of non-invertible mappings into invertible ones are reflected in a change of the symmetry group. The questions are simple; but the answers are not obvious, yet have beautiful geometric, algebraic, and functional descriptions and proofs. This is due to the simple form of nonlinear Lorentz-conformal transformations in dimension 1 + 1 , provided by characteristic coordinates.

Authors:Stefan Abstract: Publication date: Available online 18 July 2016 Source:Expositiones Mathematicae Author(s): Stefan Barańczuk In this paper we consider certain dynamical local-global principle for Mordell–Weil type groups over number fields like S -units, abelian varieties with trivial ring of endomorphisms and odd algebraic K -theory groups.

Authors:Valentino Tosatti Abstract: Publication date: Available online 15 July 2016 Source:Expositiones Mathematicae Author(s): Valentino Tosatti We give an exposition of a theorem of Hirzebruch, Kodaira and Yau which proves the uniqueness of the Kähler structure of complex projective space, and of Yau’s resolution of the Severi Conjecture.

Authors:Saugata Basu; Laxmi Parida Abstract: Publication date: Available online 23 June 2016 Source:Expositiones Mathematicae Author(s): Saugata Basu, Laxmi Parida In this paper we study the relationship between a very classical algebraic object associated to a filtration of topological spaces, namely a spectral sequence introduced by Leray in the 1940’s, and a more recently invented object that has found many applications–namely, its persistent homology groups. We show the existence of a long exact sequence of groups linking these two objects and using it derive formulas expressing the dimensions of each individual groups of one object in terms of the dimensions of the groups in the other object. The main tool used to mediate between these objects is the notion of exact couples first introduced by Massey in 1952.

Abstract: Publication date: Available online 11 June 2016 Source:Expositiones Mathematicae Author(s): Paweł Wójcik In this paper we characterize the Birkhoff—James orthogonality for elements of K ( X ; Y ) . In this way we extend the Bhatia—Šemrl theorem. As an application, we consider the approximate orthogonality preserving property. Moreover, we give a new characterization of inner product spaces.

Authors:Aurélien Alvarez; Vincent Lafforgue Abstract: Publication date: Available online 11 June 2016 Source:Expositiones Mathematicae Author(s): Aurélien Alvarez, Vincent Lafforgue Nous donnons une démonstration élémentaire et relativement courte du fait suivant: tout groupe hyperbolique admet une action affine isométrique propre sur un espace ℓ p pour p suffisamment grand. Une première preuve de ce résultat a été donnée par Yu [22].

Authors:Alexander Rotkevich Abstract: Publication date: Available online 9 June 2016 Source:Expositiones Mathematicae Author(s): Alexander Rotkevich We give examples of Lebesgue spaces on which Cauchy and Cauchy–Leray–Fantappiè operators are not bounded.