arXiv subjects

Heinrich Hartmann

Publications and source records attributed to Heinrich Hartmann.

6 recordsLinked to original sources

arXiv2026

Discrete Fa\`a di Bruno via M\"obius Inversion

We approach discrete and differential Fa\`a di Bruno formulas from a M\"obius inversion angle. On the Boolean cube, Newton's discrete Taylor formula and the definition of iterated forward differences form a zeta--M\"obius dual pair, and composing two Taylor expansions and inverting once yields a closed discrete Fa\`a di Bruno formula at a fixed basepoint: for arbitrary maps $f, g$ between abelian groups, $$ \Delta(f \circ g;\,x;\,u_1,\dots,u_k) = \sum_{H \in \mathrm{Cov}(k)} \Delta(f;\,g(x);\,(\Delta(g;x;u_T))_{T\in H}), $$ where $\mathrm{Cov}(k)$ denotes the coverings of $[k]$ by nonempty subsets. Grouping repeated directions gives binomial versions on multi-index grids, and iterating gives formulas for $m$-fold composites, with integer covering coefficients governed by explicit cross and level recursions, a discrete analogue of the Constantine--Savits formulas. The relationship between coverings and partitions appearing in classical Fa\`a di Bruno formulas is exhibited in an algebraic setting. The discrete formulas are Taylor expansions over the function algebra of the Boolean cube, whose idempotent generators absorb overlapping products; in the differential analogue nilpotent generators annihilate overlaps and only partitions remain. We demonstrate how these algebraic identities can be lifted to the analytical setting of $C^n$ maps between Banach spaces, recovering the multivariate Fa\`a di Bruno formula of Constantine--Savits and extending it to composites of several maps. Boolean finite differences, binomial grid formulas, infinitesimal Taylor algebras, and Fr\'echet derivatives thus appear as four realizations of one M\"obius-dual Fa\`a di Bruno formula, connected by a flat family.

math.CO↗

arXiv2026

Fa\`a di Bruno is Taylor Composition

We approach Fa\`a di Bruno as a composition theorem for Taylor polynomials. For $C^k$ maps $\phi: E \to F$ and $\psi: F \to G$ between Banach spaces, let $T^k_\ast(\phi; x)$ denote the reduced Taylor polynomial of $\phi$ at $x$, obtained by removing the constant term. We show that $$T^k_\ast(\psi \circ \phi; x) = \pi_{\le k}\bigl(T^k_\ast(\psi; \phi(x)) \circ T^k_\ast(\phi; x)\bigr).$$ The proof is an elementary estimate of the Peano remainder and does not use partitions or combinatorial enumeration. Expanding this composition identity recovers the classical Fa\`a di Bruno formulas. Polarization gives the multivariate partition formula (L\'evy 2006), while coefficient extraction gives the multi-index formula (Constantine and Savits 1996). Our approach separates the functorial nature of Taylor approximation from the combinatorial bookkeeping of polarization and coefficient extraction. As an application, we give a general higher-order product rule.

math.GM↗

arXiv2020

Circllhist -- A Log-Linear Histogram Data Structure for IT Infrastructure Monitoring

The circllhist histogram is a fast and memory efficient data structure for summarizing large numbers of latency measurements. It is particularly suited for applications in IT infrastructure monitoring, and provides nano-second data insertion, full mergeability, accurate approximation of quantiles with a-priori bounds on the relative error. Open-source implementations are available for C/lua/python/Go/Java/JavaScript.

cs.DS↗

arXiv2015

Voting Behaviour and Power in Online Democracy: A Study of LiquidFeedback in Germany's Pirate Party

In recent years, political parties have adopted Online Delegative Democracy platforms such as LiquidFeedback to organise themselves and their political agendas via a grassroots approach. A common objection against the use of these platforms is the delegation system, where a user can delegate his vote to another user, giving rise to so-called super-voters, i.e. powerful users who receive many delegations. It has been asserted in the past that the presence of these super-voters undermines the democratic process, and therefore delegative democracy should be avoided. In this paper, we look at the emergence of super-voters in the largest delegative online democracy platform worldwide, operated by Germany's Pirate Party. We investigate the distribution of power within the party systematically, study whether super-voters exist, and explore the influence they have on the outcome of votings conducted online. While we find that the theoretical power of super-voters is indeed high, we also observe that they use their power wisely. Super-voters do not fully act on their power to change the outcome of votes, but they vote in favour of proposals with the majority of voters in many cases thereby exhibiting a stabilising effect on the system. We use these findings to present a novel class of power indices that considers observed voting biases and gives significantly better predictions than state-of-the-art measures.

cs.CY↗

arXiv2011

Period- and mirror-maps for the quartic K3

We study in detail mirror symmetry for the quartic K3 surface in P3 and the mirror family obtained by the orbifold construction. As explained by Aspinwall and Morrison, mirror symmetry for K3 surfaces can be entirely described in terms of Hodge structures. (1) We give an explicit computation of the Hodge structures and period maps for these families of K3 surfaces. (2) We identify a mirror map, i.e. an isomorphism between the complex and symplectic deformation parameters, and explicit isomorphisms between the Hodge structures at these points. (3) We show compatibility of our mirror map with the one defined by Morrison near the point of maximal unipotent monodromy. Our results rely on earlier work by Narumiyah-Shiga, Dolgachev and Nagura-Sugiyama.

math.AG↗

arXiv2010

Cusps of the Kähler moduli space and stability conditions on K3 surfaces

In [Ma1] S. Ma established a bijection between Fourier--Mukai partners of a K3 surface and cusps of the Kähler moduli space. The Kähler moduli space can be described as a quotient of Bridgeland's stability manifold. We study the relation between stability conditions $σ$ near to a cusp and the associated Fourier--Mukai partner Y in the following ways. (1) We compare the heart of $σ$ to the heart of coherent sheaves on Y. (2) We construct Y as moduli space of $σ$-stable objects. An appendix is devoted to the group of auto-equivalences of the derived category which respect the component $Stab^\dagger(X)$ of the stability manifold.

math.AG↗