Looking at MacLane's thesis on proof theory in the light of combinatory logic

Erwin Engeler

Let $R \cup B$ be a set of $n$ points in $\mathbb{R}^2$, and let $k \in 1..n$. Our goal is to compute a line that "best" separates the "red" points $R$ from the "blue" points $B$ with at most $k$ outliers. We present an efficient semi-online dynamic data structure that can maintain whether such a separator exists. Furthermore, we present efficient exact and approximation algorithms that compute a linear separator that is guaranteed to misclassify at most $k$, points and minimizes the distance to the farthest outlier. Our exact algorithm runs in $O(nk + n \log n)$ time, and our $(1+\varepsilon)$-approximation algorithm runs in $O(\varepsilon^{-1/2}((n + k^2) \log n))$ time. Based on our $(1+\varepsilon)$-approximation algorithm we then also obtain a semi-online data structure to maintain such a separator efficiently.

Erwin Glazenburg Frank Staals Marc van Kreveld

Given two sets $R$ and $B$ of $n$ points in the plane, we present efficient algorithms to find a two-line linear classifier that best separates the "red" points in $R$ from the "blue" points in $B$ and is robust to outliers. More precisely, we find a region $\mathcal{W}_B$ bounded by two lines, so either a halfplane, strip, wedge, or double wedge, containing (most of) the blue points $B$, and few red points. Our running times vary between optimal $O(n\log n)$ and around $O(n^3)$, depending on the type of region $\mathcal{W}_B$ and whether we wish to minimize only red outliers, only blue outliers, or both.

Erwin Glazenburg Thijs van der Horst Tom Peters Bettina Speckmann Frank Staals

We report a neutron scattering study of the spin correlations for the spin 5/2, two-dimensional antiferromagnet Rb_2MnF_4 in an external magnetic field. Choosing fields near the system's bicritical point, we tune the effective anisotropy in the spin interaction to zero, constructing an ideal S=5/2 Heisenberg system. The correlation length and structure factor amplitude are closely described by the semiclassical theory of Cuccoli et al. over a broad temperature range but show no indication of approaching the low-temperature renormalized classical regime of the quantum non-linear sigma model.

R. L. Leheny R. J. Christianson R. J. Birgeneau R. W. Erwin

We consider d-dimensional random surface models which for d=1 are the standard (tied-down) random walks (considered as a random ``string''). In higher dimensions, the one-dimensional (discrete) time parameter of the random walk is replaced by the d-dimensional lattice \Z^d, or a finite subset of it. The random surface is represented by real-valued random variables \phi_i, where i is in \Z^d. A class of natural generalizations of the standard random walk are gradient models whose laws are (formally) expressed as P(d\phi) = 1/Z \exp[-\sum_{|i-j|=1}V(\phi_i-\phi_j)] \prod_i d\phi_i, V:\R -> R^+ convex, and with some growth conditions. Such surfaces have been introduced in theoretical physics as (simplified) models for random interfaces separating different phases. Of particular interest are localization-delocalization phenomena, for instance for a surface interacting with a wall by attracting or repulsive interactions, or both together. Another example are so-called heteropolymers which have a noise-induced interaction. Recently, there had been developments of new probabilistic tools for such problems. Among them are: o Random walk representations of Helffer-Sj\"ostrand type, o Multiscale analysis, o Connections with random trapping problems and large deviations We give a survey of some of these developments.

Erwin Bolthausen

We present results from a preliminary analysis of a recently-completed, multicolor imaging survey of nearby, early-type barred galaxies in the field, carried out with the WIYN telescope and supplemented with archival HST images. This forms a reasonably complete sample of nearby, bright, barred S0 and Sa galaxies in the field. The excellent seeing provided by WIYN allows us to examine the galaxies for central features such as circumnuclear rings and secondary bars; we find some evidence for dust lanes within secondary bars. The most striking results is the high frequency of double bars: our analysis suggests that at least $\sim 20%$ of barred S0-Sa galaxies possess secondary bars}. We also find an excellent candidate triply barred galaxy: NGC 2681.

Peter Erwin Linda S. Sparke

This pre-print contains the abstracts of seminars (including key references) presented at the ESI workshop on mathematical problems in quantum gravity held during July and August of 1996. Contributors include A. Ashtekar, J. Baez, F. Barbero, A. Barvinsky, F. Embacher, R. Gambini, D. Giulini, J. Halliwell, T. Jacobson, R. Loll, D. Marolf, K. Meissner, R. Myers, J. Pullin, M. Reisenberger, C. Rovelli, T. Strobl and T. Thiemann. While these contributions cover most of the talks given during the workshop, there were also a few additional speakers whose contributions were not received in time.

Peter Aichelburg Abhay Ashtekar

We describe in greater detail the exactly solvable microscopic model we have developed for analyzing the strain-mediated interaction of vacancy lines in a pseudomorphic adsorbate system (Phys. Rev. Lett., to appear). The model is applied to Ga/Si(112) by extracting values for the microscopic parameters from total-energy calculations. The results, which are in good agreement with experimental observations, reveal an unexpectedly complex interplay between compressive and tensile strain within the mixed Ga-Si surface layer.

S. C. Erwin A. A. Baski L. J. Whitman R. E. Rudd

Interstitials and vacancies in the Abrikosov phase of clean Type II superconductors are line imperfections, which cannot extend across macroscopic equilibrated samples at low temperatures. We argue that the entropy associated with line wandering nevertheless can cause these defects to proliferate at a sharp transition which will exist if this occurs below the temperature at which the crystal actually melts. Vortices are both entangled and crystalline in the resulting ``supersolid'' phase, which in a dual ``boson'' analog system is closely related to a two-dimensional quantum crystal of He$^4$ with interstitials or vacancies in its ground state. The supersolid {\it must} occur for $B\gg B_\times$, where $B_\times$ is the decoupling field above which vortices begin to behave two-dimensionally. Numerical calculations show that interstitials, rather than vacancies, are the preferred defect for $B\gg \phi_0/\lambda_\perp^2$, and allow us to estimate whether proliferation also occurs for $B\,\lot\,B_\times$.The implications of the supersolid phase for transport measurements, dislocation configurations and neutron diffraction are discussed.

Erwin Frey David R. Nelson Daniel S. Fisher

We establish a lower bound on the noncoherent capacity pre-log of a temporally correlated Rayleigh block-fading single-input multiple-output (SIMO) channel. Our result holds for arbitrary rank Q of the channel correlation matrix, arbitrary block-length L > Q, and arbitrary number of receive antennas R, and includes the result in Morgenshtern et al. (2010) as a special case. It is well known that the capacity pre-log for this channel in the single-input single-output (SISO) case is given by 1-Q/L, where Q/L is the penalty incurred by channel uncertainty. Our result reveals that this penalty can be reduced to 1/L by adding only one receive antenna, provided that L \geq 2Q - 1 and the channel correlation matrix satisfies mild technical conditions. The main technical tool used to prove our result is Hironaka's celebrated theorem on resolution of singularities in algebraic geometry.

Erwin Riegler Veniamin I. Morgenshtern Giuseppe Durisi Shaowei Lin Bernd Sturmfels Helmut Bölcskei