On scrimshaw precursors: a 13th-century carved and engraved sperm whale tooth

In der einschlägigen Forschung zur Walfängervolkskunst des "Scrimshaw" - den seit den 1820er Jahren populär gewordenen Seemannsarbeiten speziell auf und aus Pottwal- und Walroßzahn, Barten und Walknochen - werden künstlerisch bearbeitete Gegenstände anderer Kulturen aus denselben Materialien als "Scrimshaw"-Vorläufer bezeichnet. Aus dem europäischen Mittelalter sind zwar zahlreiche Beispiele von Schnitzereien aus Walroßzahn und Walknochen bekannt, nicht jedoch aus Pottwalzahn, dem beinah stereotypen Werkstoff der "Scrimshander" des 19. und 20. Jahrhunderts. Der "Königsspiegel", ein um 1250/60 in Norwegen geschriebener pädagogischer Text, erwähnt gleichwohl Schnitzereien aus Pottwalzahn. Auch aus dem 16. und 17. Jahrhundert sind textliche Hinweise auf skandinavisches Kunstgewerbe aus den Zähnen dieses Meeressäugers bekannt. Im Anschluß an die Präsentation dieser Quellen wird ein konkretes Beispiel vorgestellt: Es handelt sich um ein beschnitztes und graviertes Salbenhorn aus Pottwalzahn, das sich in der Sammlung christlicher Kunst und Kultur des Museums der Universität Bergen, Norwegen, lnv. MA 437, befindet. Aus dem massiven Dentin wurde ein Greif herausgeschnitzt, der Ausguß wurde wie der Kopf eines gotischen Wasserspeiers gestaltet. Auf einer glatten Seitenfläche wurde später ein kleines Tondo mit Pflanzen- und Vogelmotiv eingraviert. Es trägt eine Inschrift, die vom Bergenser Museum als griechisch, hier aber als russisch identifiziert wurde, deren Sinn aber unklar bleibt. Anlass, die vom Museum vorgenommene Datierung des Zahns in das 13. Jahrhundert zu revidieren, besteht allerdings nicht

On the volume functional of compact manifolds with boundary with constant scalar curvature

We study the volume functional on the space of constant scalar curvature metrics with a prescribed boundary metric. We derive a sufficient and necessary condition for a metric to be a critical point, and show that the only domains in space forms, on which the standard metrics are critical points, are geodesic balls. In the zero scalar curvature case, assuming the boundary can be isometrically embedded in the Euclidean space as a compact strictly convex hypersurface, we show that the volume of a critical point is always no less than the Euclidean volume bounded by the isometric embedding of the boundary, and the two volumes are equal if and only if the critical point is isometric to a standard Euclidean ball. We also derive a second variation formula and apply it to show that, on Euclidean balls and ''small'' hyperbolic and spherical balls in dimensions 3 to 5, the standard space form metrics are indeed saddle points for the volume functional

Q2Q_2-free families in the Boolean lattice

For a family $\mathcal{F}$ of subsets of [n]=\{1, 2, ..., n} ordered by inclusion, and a partially ordered set P, we say that $\mathcal{F}$ is P-free if it does not contain a subposet isomorphic to P. Let $ex(n, P)$ be the largest size of a P-free family of subsets of [n]. Let $Q_2$ be the poset with distinct elements a, b, c, d, a<b, c<d; i.e., the 2-dimensional Boolean lattice. We show that $2N -o(N) \leq ex(n, Q_2)\leq 2.283261N +o(N),$ where $N = \binom{n}{\lfloor n/2 \rfloor}$. We also prove that the largest $Q_2$-free family of subsets of [n] having at most three different sizes has at most 2.20711N members.Comment: 18 pages, 2 figure

On Bernoulli Decompositions for Random Variables, Concentration Bounds, and Spectral Localization

As was noted already by A. N. Kolmogorov, any random variable has a Bernoulli component. This observation provides a tool for the extension of results which are known for Bernoulli random variables to arbitrary distributions. Two applications are provided here: i. an anti-concentration bound for a class of functions of independent random variables, where probabilistic bounds are extracted from combinatorial results, and ii. a proof, based on the Bernoulli case, of spectral localization for random Schroedinger operators with arbitrary probability distributions for the single site coupling constants. For a general random variable, the Bernoulli component may be defined so that its conditional variance is uniformly positive. The natural maximization problem is an optimal transport question which is also addressed here

On spectral minimal partitions: the case of the sphere

We consider spectral minimal partitions. Continuing work of the the present authors about problems for planar domains, [23], we focus on the sphere and obtain a sharp result for 3-partitions which is related to questions from harmonic analysis, in particular to a conjecture of Bishop

Efficient binary fuzzy measure representation and Choquet integral learning

The Choquet integral (ChI), a parametric function for information aggregation, is parameterized by the fuzzy measure (FM), which has 2N real-valued variables for N inputs. However, the ChI incurs huge storage and computational burden due to its exponential complexity relative to N and, as a result, its calculation, storage, and learning becomes intractable for even modest sizes (e.g., N = 15). Inspired by empirical observations in multi-sensor fusion and the more general need to mitigate the storage, computational, and learning limitations, we previously explored the binary ChI (BChI) relative to the binary fuzzy measure (BFM). The BChI is a natural _t for many applications and can be used to approximate others. Previously, we investigated different properties of the BChI and we provided an initial representation. In this article, we propose a new efficient learning algorithm for the BChI, called EBChI, by utilizing the BFM properties that add at most one variable per training instance. Furthermore, we provide an efficient representation of the BFM (EBFM) scheme that further reduces the number of variables required for storage and computation, thus enabling the use of the BChI for \big N". Finally, we conduct experiments on synthetic data that demonstrate the efficiency of our proposed techniques