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Stefan Neuwirth

Publications and source records attributed to Stefan Neuwirth.

30 records · Page 2Linked to original sources
arXiv2020

Lorenzen's reshaping of Krull's Fundamentalsatz for integral domains (1938--1953)

Krull's Fundamentalsatz, the generalisation of the main theorem of elementary number theory to integral domains, is the starting point of Lorenzen's career in mathematics. This article traces a conceptual history of Lorenzen's successive reformulations of the Fundamentalsatz on the basis of excerpts of his articles. An edition of the extant correspondence of Lorenzen with Hasse, Krull, and Aubert provides a better understanding of the context of these investigations.

math.HO↗
arXiv2020

Lorenzen's proof of consistency for elementary number theory [with an edition and translation of "Ein halbordnungstheoretischer Widerspruchsfreiheitsbeweis'']

We present a manuscript of Paul Lorenzen that provides a proof of consistency for elementary number theory as an application of the construction of the free countably complete pseudocomplemented semilattice over a preordered set. This manuscript rests in the Oskar-Becker-Nachlass at the Philosophisches Archiv of Universit{ä}t Konstanz, file OB 5-3b-5. It has probably been written between March and May 1944. We also compare this proof to Gentzen's and Novikov's, and provide a translation of the manuscript.

math.HO↗
arXiv2018

Lattice-ordered groups generated by an ordered group and regular systems of ideals

Unbounded entailment relations, introduced by Paul Lorenzen (1951), are a slight variant of a notion which plays a fundamental rôle in logic (see Scott 1974) and in algebra (see Lombardi and Quitté 2015). We call systems of ideals their single-conclusion counterpart. If they preserve the order of a commutative ordered monoid G and are equivariant w.r.t. its law, we call them equivariant systems of ideals for G: they describe all morphisms from G to meet-semilattice-ordered monoids generated by (the image of) G. Taking an article by Lorenzen (1953) as a starting point, we also describe all morphisms from a commutative ordered group G to lattice-ordered groups generated by G through unbounded entailment relations that preserve its order, are equivariant, and satisfy a regularity property invented by Lorenzen (1950); we call them regular entailment relations. In particular, the free lattice-ordered group generated by G is described through the finest regular entailment relation for G, and we provide an explicit description for it; it is order-reflecting if and only if the morphism is injective, so that the Lorenzen-Clifford-Dieudonné theorem fits into our framework. Lorenzen's research in algebra starts as an inquiry into the system of Dedekind ideals for the divisibility group of an integral domain R, and specifically into Wolfgang Krull's "Fundamentalsatz" that R may be represented as an intersection of valuation rings if and only if R is integrally closed: his constructive substitute for this representation is the regularisation of the system of Dedekind ideals, i.e. the lattice-ordered group generated by it when one proceeds as if its elements are comparable.

math.LO↗
arXiv2017

Roger Apéry: humour at the service of a free and original thinking on constructive mathematics

This article is an invitation to read a famous text of Roger Ap{é}ry, "Math{é}matique constructive", published in the book "Penser les math{é}matiques: s{é}minaire de philosophie et math{é}matiques de l'{É}cole normale sup{é}rieure (J. Dieudonn{é}, M. Loi, R. Thom)" [arXiv:1705.05581]. It also proposes a commented bibliography for constructive mathematics and a technical note on measure theory.

math.HO↗
arXiv2012

Transfer of Fourier multipliers into Schur multipliers and sumsets in a discrete group

We inspect the relationship between relative Fourier multipliers on noncommutative Lebesgue-Orlicz spaces of a discrete group and relative Toeplitz-Schur multipliers on Schatten-von-Neumann-Orlicz classes. Four applications are given: lacunary sets; unconditional Schauder bases for the subspace of a Lebesgue space determined by a given spectrum, that is, by a subset of the group; the norm of the Hilbert transform and the Riesz projection on Schatten-von-Neumann classes with exponent a power of 2; the norm of Toeplitz Schur multipliers on Schatten-von-Neumann classes with exponent less than 1.

math.FA↗
arXiv2007

The maximum modulus of a trigonometric trinomial

Let Lambda be a set of three integers and let C_Lambda be the space of 2pi-periodic functions with spectrum in Lambda endowed with the maximum modulus norm. We isolate the maximum modulus points x of trigonometric trinomials T in C_Lambda and prove that x is unique unless |T| has an axis of symmetry. This permits to compute the exposed and the extreme points of the unit ball of C_Lambda, to describe how the maximum modulus of T varies with respect to the arguments of its Fourier coefficients and to compute the norm of unimodular relative Fourier multipliers on C_Lambda. We obtain in particular the Sidon constant of Lambda.

math.FA↗
arXiv2004

Cycles and 1-unconditional matrices

We characterize the 1-unconditional subsequences of the canonical basis (e_rc) of elementary matrices in the Schatten-von-Neumann class S^p . The set I of couples (r,c) must be the set of edges of a bipartite graph without cycles of even length 4<=l<=p if p is an even integer, and without cycles at all if p is a positive real number that is not an even integer. In the latter case, I is even a Varopoulos set of V-interpolation of constant 1. We also study the metric unconditional approximation property for the space S^p_I spanned by (e_rc)_{(r,c)\in I} in S^p .

math.FA↗
arXiv2001

Metric unconditionality and Fourier analysis

We investigate several aspects of almost 1-unconditionality. We characterize the metric unconditional approximation property UMAP in terms of ``block unconditionality''. Then we focus on translation invariant subspaces $L^p_E(T)$ and $C_E(T)$ of functions on the circle and express block unconditionality as arithmetical conditions on $E$. Our work shows that the spaces $L^p_E(T)$, $p$ an even integer, have a singular behaviour from the almost isometric point of view: property UMAP does not interpolate between spaces $L^p_E(T)$ and $L^{p+2}_E(T)$. These arithmetical conditions are used to construct counterexamples for several natural questions and to investigate the maximal density of such sets $E$. We also prove that if $E=\{n_k\}_{k\ge1}$ with $|n_{k+1}/n_k|\to\infty$, then $C_E(T)$ has UMAP and we get a sharp estimate of the Sidon constant of Hadamard sets. Finally, we investigate the relationship of metric unconditionality and probability theory.

math.FA↗
arXiv2001

The size of bipartite graphs with girth eight

Reiman produced a quadratic inequality for the size of bipartite graphs of girth six. We get its counterpart for girth eight, a cubic inequality. It is optimal in as far as it admits the algebraic structure of generalized quadrangles as case of equality. This enables us to obtain the optimal estimate e ~ v^(4/3) for balanced bipartite graphs. We also get an optimal estimate for very unbalanced graphs.

math.CO↗
arXiv2001

Lacunary matrices

We study unconditional subsequences of the canonical basis e_rc of elementary matrices in the Schatten class S^p. They form the matrix counterpart to Rudin's Lambda(p) sets of integers in Fourier analysis. In the case of p an even integer, we find a sufficient condition in terms of trails on a bipartite graph. We also establish an optimal density condition and present a random construction of bipartite graphs. As a byproduct, we get a new proof for a theorem of Erdos on circuits in graphs.

math.FA↗
arXiv2001

The Sidon constant of sets with three elements

We solve an elementary extremal problem on trigonometric polynomials and obtain the exact value of the Sidon constant for sets with three elements {n0, n1, n2}: it is \sec(π/ 2n) for n = max(n1-n0, n2-n0, n2-n1) / gcd(n1-n0, n2-n0).

math.CA↗