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  • Coefficients and higher order derivatives of cyclotomic polynomials: Old and new
    Expos. Math. (IF 0.491) Pub Date : 2020-03-16
    Andrés Herrera-Poyatos; Pieter Moree

    The nth cyclotomic polynomial Φn(x) is the minimal polynomial of an nth primitive root of unity. Its coefficients are the subject of intensive study and some formulas are known for them. Here we are interested in formulas which are valid for all natural numbers n. In these a host of famous number theoretical objects such as Bernoulli numbers, Stirling numbers of both kinds and Ramanujan sums make their

  • On a spectral theorem of Weyl
    Expos. Math. (IF 0.491) Pub Date : 2020-03-14
    Nigel Higson; Qijun Tan

    We give a new proof of a theorem of Weyl on the continuous part of the spectrum of Sturm–Liouville operators on the half-line with asymptotically constant coefficients. Earlier arguments, due to Weyl and Kodaira, depended on particular features of Green’s functions for linear ordinary differential operators. We use a concept of asymptotic containment of C∗-algebra representations that has geometric

  • Extinction times of an inhomogeneous Feller diffusion process: A PDE approach
    Expos. Math. (IF 0.491) Pub Date : 2020-03-09
    Florian Lavigne; Lionel Roques

    We focus on the distribution of the extinction times of a population whose size N(t) follows a Feller diffusion process with inhomogeneous growth term r(t). Obtaining a precise description of the extinction times and of their dependence with respect to r(t) has important applications in adaptive biology, for understanding “evolutionary rescue” phenomena. A formula for the distribution of the extinction

  • Quantum Spacetime, Quantum Geometry and Planck scales
    Expos. Math. (IF 0.491) Pub Date : 2020-02-14
    Sergio Doplicher

    The Principles of Quantum Mechanics and of Classical General Relativity indicate that Spacetime in the small (Planck scale) ought to be described by a noncommutative C* Algebra, implementing spacetime uncertainty relations. A model C* algebra of Quantum Spacetime and its Quantum Geometry is described. Interacting Quantum Field Theory on such a background is discussed, with open problems and recent

  • Uniquely separable extensions
    Expos. Math. (IF 0.491) Pub Date : 2020-02-10
    Lars Kadison

    The separability tensor element of a separable extension of noncommutative rings is an idempotent when viewed in the correct endomorphism ring; so one speaks of a separability idempotent, as one usually does for separable algebras. It is proven that this idempotent is full if and only if the H-depth is 1 (H-separable extension). Similarly, a split extension has a bimodule projection; this idempotent

  • The emergence of time
    Expos. Math. (IF 0.491) Pub Date : 2020-02-07
    Roberto Longo

    Classically, one could imagine a completely static space, thus without time. As is known, this picture is unconceivable in quantum physics due to vacuum fluctuations. The fundamental difference between the two frameworks is that classical physics is commutative (simultaneous observables) while quantum physics is intrinsically noncommutative (Heisenberg uncertainty relations). In this sense, we may

  • A hydrodynamic exercise in free probability: Setting up free Euler equations
    Expos. Math. (IF 0.491) Pub Date : 2020-02-06
    Dan-Virgil Voiculescu

    For the free probability analogue of Euclidean space endowed with the Gaussian measure we apply the approach of Arnold to derive Euler equations for a Lie algebra of non-commutative vector fields which preserve a certain trace. We extend the equations to vector fields satisfying non-commutative smoothness requirements. We introduce a cyclic vorticity and show that it satisfies vorticity equations and

  • Cotangent bundles for “matrix algebras converge to the sphere”
    Expos. Math. (IF 0.491) Pub Date : 2020-02-06
    Marc A. Rieffel

    In the high-energy quantum-physics literature one finds statements such as “matrix algebras converge to the sphere”. Earlier I provided a general setting for understanding such statements, in which the matrix algebras are viewed as compact quantum metric spaces, and convergence is with respect to a quantum Gromov–Hausdorff-type distance. More recently I have dealt with corresponding statements in the

  • A note on commutators in algebras of unbounded operators
    Expos. Math. (IF 0.491) Pub Date : 2020-02-06
    Richard V. Kadison; Zhe Liu; Andreas Thom

    We show that the identity is the sum of two commutators in the algebra of all operators affiliated with a von Neumann algebra of type II1, settling a question, in the negative, that had puzzled a number of us.

  • Hutchinson without Blaschke: An alternative way to fractals
    Expos. Math. (IF 0.491) Pub Date : 2020-01-16
    Mihály Bessenyei; Evelin Pénzes

    The original approach of Hutchinson to fractals considers the defining equation as a fixed point problem, and then applies the Banach Contraction Principle. To do this, the Blaschke Completeness Theorem is essential. Avoiding Blaschke’s result, this note presents an alternative way to fractals via the Kuratowski noncompactness measure. Moreover, our technique extends the existence part of Hutchinson’s

  • On extreme contractions between real Banach spaces
    Expos. Math. (IF 0.491) Pub Date : 2020-01-16
    Debmalya Sain; Kallol Paul; Arpita Mal

    We completely characterize extreme contractions between two-dimensional strictly convex and smooth real Banach spaces, perhaps for the very first time. In order to obtain the desired characterization, we introduce the notions of (weakly) compatible point pair (CPP) and μ-compatible point pair (μ-CPP) in the geometry of Banach spaces. As a concrete application of our abstract results, we describe all

  • An extension of a q-deformed Heisenberg algebra and its Lie polynomials
    Expos. Math. (IF 0.491) Pub Date : 2020-01-15
    Rafael Reno S. Cantuba; Mark Anthony C. Merciales

    Let F be a field, and fix a q∈F. The q-deformed Heisenberg algebra H(q) is the unital associative algebra over F with generators A, B and a relation which asserts that AB−qBA is the multiplicative identity in H(q). We extend H(q) into an algebra R(q) defined by generators A, B and a relation which asserts that AB−qBA is central in R(q). We identify all elements of R(q) that are Lie polynomials in A

  • A species approach to Rota’s twelvefold way
    Expos. Math. (IF 0.491) Pub Date : 2019-12-19
    Anders Claesson

    An introduction to Joyal’s theory of combinatorial species is given and through it an alternative view of Rota’s twelvefold way emerges.

  • The automorphism group of the universal Coxeter group
    Expos. Math. (IF 0.491) Pub Date : 2019-12-04
    Olga Varghese

    We study fixed point properties of the automorphism group of the universal Coxeter group Aut(Wn). In particular, we prove that whenever Aut(Wn) acts by isometries on complete d-dimensional CAT(0) space with d<⌊n2⌋, then it must fix a point. We also prove that Aut(Wn) does not have Kazhdan’s property (T). Further, strong restrictions are obtained on homomorphisms of Aut(Wn) to groups that do not contain

  • Automorphism group actions in complex analysis
    Expos. Math. (IF 0.491) Pub Date : 2019-12-04
    Steven G. Krantz

    In this paper we describe the subject of automorphism groups of domains in complex space. This has been an active area of research for fifty years or more, and continues to be dynamic and developing today. We discuss noncompact automorphism groups, the Bun Wong/Rosay theorem, the Greene/Krantz conjecture, semicontinuity of automorphism groups, the method of scaling, and other current topics. Contributions

  • A note on crossed products
    Expos. Math. (IF 0.491) Pub Date : 2019-12-04
    Wojciech Chojnacki

    We give an exposition of two fundamental results of the theory of crossed products. One of these states that every regular representation of a reduced crossed product is faithful whenever the underlying Hilbert space representation of the C∗-algebra that together with an automorphism group gives rise to the crossed product is faithful. The other result states that a full and a reduced crossed products

  • All classifiable Kirchberg algebras are C∗-algebras of ample groupoids
    Expos. Math. (IF 0.491) Pub Date : 2019-07-17
    Lisa Orloff Clark; James Fletcher; Astrid an Huef

    In this note we show that every Kirchberg algebra in the UCT class is the C∗-algebra of a Hausdorff, ample, amenable and locally contracting groupoid. The non-unital case follows from Spielberg’s graph-based models for Kirchberg algebras. Our contribution is the unital case and our proof builds on Spielberg’s construction.

  • Distances on the moduli space of complex projective structures
    Expos. Math. (IF 0.491) Pub Date : 2019-06-22
    Gianluca Faraco

    Let S be a closed and oriented surface of genus g at least 2. In this (mostly expository) article, the object of study is the space P(S) of marked isomorphism classes of projective structures on S. We show that P(S), endowed with the canonical complex structure, carries exotic hermitian structures that extend the classical ones on the Teichmüller space T(S) of S. We shall notice also that the Kobayashi

  • Equidistribution of orbits of isometries on compact Riemannian manifolds
    Expos. Math. (IF 0.491) Pub Date : 2019-06-20
    Adrián Gordillo-Merino; José Navarro

    Let S be a finitely generated semigroup of isometries of a compact Riemannian manifold. We present a simple argument to prove that, if there exists a point x whose orbit Sx is dense, then the orbit of any point is equidistributed.

  • Which domains have two-sided supporting unit spheres at every boundary point?
    Expos. Math. (IF 0.491) Pub Date : 2019-06-20
    Marta Lewicka; Yuval Peres

    We prove the quantitative equivalence of two important geometrical conditions, pertaining to the regularity of a domain Ω⊂RN. These are: (i) the uniform two-sided supporting sphere condition, and (ii) the Lipschitz continuity of the outward unit normal vector. In particular, the answer to the question posed in our title is: “Those domains whose unit normal is well defined and has Lipschitz constant

  • Dickson’s Lemma, Higman’s Theorem and Beyond: A survey of some basic results in order theory
    Expos. Math. (IF 0.491) Pub Date : 2019-06-20
    Erhard Aichinger; Florian Aichinger

    We provide proofs for the fact that certain orders have no infinite descending chains and no infinite antichains.

  • Categorical notions of fibration
    Expos. Math. (IF 0.491) Pub Date : 2019-06-14
    Fosco Loregian; Emily Riehl

    Fibrations over a category B, introduced to category theory by Grothendieck, encode pseudo-functors Bop⇝Cat, while the special case of discrete fibrations encodes presheaves Bop→Set. A two-sided discrete variation encodes functors Bop×A→Set, which are also known as profunctors from A to B. By work of Street, all of these fibration notions can be defined internally to an arbitrary 2-category or bicategory

  • Differential structure associated to axiomatic Sobolev spaces
    Expos. Math. (IF 0.491) Pub Date : 2019-06-13
    Nicola Gigli; Enrico Pasqualetto

    The aim of this note is to explain in which sense an axiomatic Sobolev space over a general metric measure space (à la Gol’dshtein–Troyanov) induces – under suitable locality assumptions – a first-order differential structure.

  • The Bateman–Horn conjecture: Heuristic, history, and applications
    Expos. Math. (IF 0.491) Pub Date : 2019-06-13
    Soren Laing Aletheia-Zomlefer; Lenny Fukshansky; Stephan Ramon Garcia

    The Bateman–Horn conjecture is a far-reaching statement about the distribution of the prime numbers. It implies many known results, such as the prime number theorem and the Green–Tao theorem, along with many famous conjectures, such the twin prime conjecture and Landau’s conjecture. We discuss the Bateman–Horn conjecture, its applications, and its origins.

  • Some remarks on Birkhoff–James orthogonality of linear operators
    Expos. Math. (IF 0.491) Pub Date : 2019-06-11
    Debmalya Sain; Arpita Mal; Kallol Paul

    We study Birkhoff–James orthogonality of compact (bounded) linear operators between Hilbert spaces and Banach spaces. Applying the notion of semi-inner-products in normed linear spaces and some related geometric ideas, we generalize and improve some of the recent results in this context. In particular, we obtain a characterization of Euclidean spaces and also prove that it is possible to retrieve the

  • Continued fractions of arithmetic sequences of quadratics
    Expos. Math. (IF 0.491) Pub Date : 2019-06-11
    Menny Aka

    Let x be a quadratic irrational and let P be the set of prime numbers. We show the existence of an infinite set S⊂P such that the statistics of the period of the continued fraction expansions along the sequence px:p∈S approach the ‘normal’ statistics given by the Gauss–Kuzmin measure. Under the generalized Riemann hypothesis, we prove that there exist full density subsets S⊂P and T⊂N satisfying the

  • Lelong–Poincaré formula in symplectic and almost complex geometry
    Expos. Math. (IF 0.491) Pub Date : 2019-05-20
    Emmanuel Mazzilli; Alexandre Sukhov

    In this paper, we present two applications of the theory of singular connections developed by Harvey and Lawson (1993). The first one is a version of the Lelong–Poincaré formula with estimates for sections of vector bundles over an almost complex manifold. The second one is a convergence theorem for divisors associated to a general family of symplectic submanifolds constructed by Donaldson (1996) (the

  • Dynamics, points and places at infinity, and the inversion of polynomial self-maps of R2
    Expos. Math. (IF 0.491) Pub Date : 2019-05-15
    Luis Fernando Mello; Frederico Xavier

    In this partly expository paper we give two applications of ideas from dynamical systems to the study of the injectivity properties of a polynomial local diffeomorphism F=(F1,F2):R2→R2 (by the work of Pinchuk, these maps need not be globally injective). I) The Jacobian conjecture claims that all polynomial local biholomorphisms G=(G1,G2):ℂ2→ℂ2 must be injective. By the Abhyankar–Moh theory in algebraic

  • A new first-principles approach for the catenary
    Expos. Math. (IF 0.491) Pub Date : 2019-05-08
    George Victor McIlvaine

    This paper reports on findings relating to catenaries since the publication in Expositiones Mathematicae of Denzler and Hinz’s pioneering 1999 paper, Catenaria Vera – the True Catenary. New governing differential equations and explicit solutions are derived for the catenary in positive and negative radial potentials with physical constants incorporated in the derivations. In keeping with precedent

  • Stoïlow’s theorem revisited
    Expos. Math. (IF 0.491) Pub Date : 2019-05-06
    Rami Luisto; Pekka Pankka

    Stoïlow’s theorem from 1928 states that a continuous, open, and light map between surfaces is a discrete map with a discrete branch set. This result implies that such maps between orientable surfaces are locally modeled by power maps z↦zk and admit a holomorphic factorization. The purpose of this expository article is to give a proof of this classical theorem having readers in mind that are interested

  • Entropy numbers of finite-dimensional embeddings
    Expos. Math. (IF 0.491) Pub Date : 2019-04-04
    Marta Kossaczká; Jan Vybíral

    Entropy numbers and covering numbers of sets and operators are well known geometric notions, which found many applications in various fields of mathematics, statistics, and computer science. Their values for finite-dimensional embeddings id:ℓpn→ℓqn, 0

  • The unit theorem for finite-dimensional algebras
    Expos. Math. (IF 0.491) Pub Date : 2019-03-22
    Hendrik W. Lenstra

    The unit theorem that forms the subject of the present article, is a theorem from algebra that has a combinatorial flavour, and that originated in fact from algebraic combinatorics. Beyond a proof, we also address applications, one of which is a proof of the normal basis theorem from Galois theory.

  • On the ℓ1 non-embedding in the James Tree Space
    Expos. Math. (IF 0.491) Pub Date : 2019-02-14
    Ioakeim Ampatzoglou

    James Tree Space (JT), introduced by R. James in James (1974), is the first Banach space constructed having non-separable conjugate and not containing ℓ1. James actually proved that every infinite dimensional subspace of JT contains a Hilbert space, which implies the ℓ1 non-embedding. In this expository article, we present a direct proof of the ℓ1 non-embedding, using Rosenthal’s ℓ1-Theorem (Rosenthal

  • Parallelisability of 3-manifolds via surgery
    Expos. Math. (IF 0.491) Pub Date : 2019-02-14
    Sebastian Durst; Hansjörg Geiges; Jesús Gonzalo Pérez; Marc Kegel

    We present two proofs that all closed, orientable 3-manifolds are parallelisable. Both are based on the Lickorish–Wallace surgery presentation; one proof uses a refinement of this presentation due to Kaplan and some basic contact geometry. This complements a recent paper by Benedetti–Lisca.

  • Closed geodesics on geodesic spaces of curvature <∞
    Expos. Math. (IF 0.491) Pub Date : 2018-12-18
    Pedro Ontaneda; Cristina Salviano

    We show how to use classical methods to prove that a compact, connected, non-contractible geodesic space of curvature <∞ has at least one closed geodesic.

  • Dissipative extension theory for linear relations
    Expos. Math. (IF 0.491) Pub Date : 2018-11-05
    Josué I. Rios-Cangas; Luis O. Silva

    This work is devoted to dissipative extension theory for dissipative linear relations. We give a self-consistent theory of extensions by generalizing the theory on symmetric extensions of symmetric operators. Several results on the properties of dissipative relations are proven. Finally, we deal with the spectral properties of dissipative extensions of dissipative relations and provide results concerning

  • Sturm’s theorem on zeros of linear combinations of eigenfunctions
    Expos. Math. (IF 0.491) Pub Date : 2018-10-26
    Pierre Bérard; Bernard Helffer

    Motivated by recent questions about the extension of Courant’s nodal domain theorem, we revisit a theorem published by C. Sturm in 1836, which deals with zeros of linear combination of eigenfunctions of Sturm–Liouville problems. Although well known in the nineteenth century, this theorem seems to have been ignored or forgotten by some of the specialists in spectral theory since the second half of the

  • Geometry of the Sasakura bundle
    Expos. Math. (IF 0.491) Pub Date : 2018-10-25
    Cristian Anghel

    The Sasakura bundle is a relatively recent appearance in the world of remarkable vector bundles on projective spaces. In fact, it is connected with some surfaces in P4 which were missed in early classification papers. The aim of this review is to present various, scattered in the literature, aspects concerning the geometry of this bundle. The last part will be devoted to the place of this bundle in

  • The Euler and Springer numbers as moment sequences
    Expos. Math. (IF 0.491) Pub Date : 2018-08-09
    Alan D. Sokal

    I study the sequences of Euler and Springer numbers from the point of view of the classical moment problem.

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