Cross Product Bialgebras - Part I

The subject of this article are cross product bialgebras without co-cycles. We establish a theory characterizing cross product bialgebras universally in terms of projections and injections. Especially all known types of biproduct, double cross product and bicross product bialgebras can be described by this theory. Furthermore the theory provides new families of (co-cycle free) cross product bialgebras. Besides the universal characterization we find an equivalent (co-)modular description of certain types of cross product bialgebras in terms of so-called Hopf data. With the help of Hopf data construction we recover again all known cross product bialgebras as well as new and more general types of cross product bialgebras. We are working in the general setting of braided monoidal categories which allows us to apply our results in particular to the braided category of Hopf bimodules over a Hopf algebra. Majid's double biproduct is seen to be a twisting of a certain tensor product bialgebra in this category. This resembles the case of the Drinfel'd double which can be constructed as a twist of a specific cross product.Comment: 33pages, t-angles.sty file needed (in xxx.lanl). Various Examples added, to be published in Journal of Algebr

Algorithm selection on data streams

We explore the possibilities of meta-learning on data streams, in particular algorithm selection. In a first experiment we calculate the characteristics of a small sample of a data stream, and try to predict which classifier performs best on the entire stream. This yields promising results and interesting patterns. In a second experiment, we build a meta-classifier that predicts, based on measurable data characteristics in a window of the data stream, the best classifier for the next window. The results show that this meta-algorithm is very competitive with state of the art ensembles, such as OzaBag, OzaBoost and Leveraged Bagging. The results of all experiments are made publicly available in an online experiment database, for the purpose of verifiability, reproducibility and generalizability

Rational minimax approximation via adaptive barycentric representations

Computing rational minimax approximations can be very challenging when there are singularities on or near the interval of approximation - precisely the case where rational functions outperform polynomials by a landslide. We show that far more robust algorithms than previously available can be developed by making use of rational barycentric representations whose support points are chosen in an adaptive fashion as the approximant is computed. Three variants of this barycentric strategy are all shown to be powerful: (1) a classical Remez algorithm, (2) a "AAA-Lawson" method of iteratively reweighted least-squares, and (3) a differential correction algorithm. Our preferred combination, implemented in the Chebfun MINIMAX code, is to use (2) in an initial phase and then switch to (1) for generically quadratic convergence. By such methods we can calculate approximations up to type (80, 80) of $|x|$ on $[-1, 1]$ in standard 16-digit floating point arithmetic, a problem for which Varga, Ruttan, and Carpenter required 200-digit extended precision.Comment: 29 pages, 11 figure

Towards Meta-learning over Data Streams

Modern society produces vast streams of data. Many stream mining algorithms have been developed to capture general trends in these streams, and make predictions for future observations, but relatively little is known about which algorithms perform particularly well on which kinds of data. Moreover, it is possible that the characteristics of the data change over time, and thus that a different algorithm should be recommended at various points in time. Figure 1 illustrates this. As such, we are dealing with the Algorithm Selection Problem [9] in a data stream setting. Based on measurable meta-features from a window of observations from a data stream, a meta-algorithm is built that predicts the best classifier for the next window. Our results show that this meta-algorithm is competitive with state-of-the art data streaming ensembles, such as OzaBag [6], OzaBoost [6] and Leveraged Bagging [3]

On the evaluation of matrix elements in partially projected wave functions

We generalize the Gutzwiller approximation scheme to the calculation of nontrivial matrix elements between the ground state and excited states. In our scheme, the normalization of the Gutzwiller wave function relative to a partially projected wave function with a single non projected site (the reservoir site) plays a key role. For the Gutzwiller projected Fermi sea, we evaluate the relative normalization both analytically and by variational Monte-Carlo (VMC). We also report VMC results for projected superconducting states that show novel oscillations in the hole density near the reservoir site

In-Plane Spectral Weight Shift of Charge Carriers in YBa2Cu3O6.9YBa_2Cu_3O_{6.9}

The temperature dependent redistribution of the spectral weight of the $CuO_2$ plane derived conduction band of the $YBa_2Cu_3O_{6.9}$ high temperature superconductor (T_c = 92.7 K) was studied with wide-band (from 0.01 to 5.6 eV) spectroscopic ellipsometry. A superconductivity - induced transfer of the spectral weight involving a high energy scale in excess of 1 eV was observed. Correspondingly, the charge carrier spectral weight was shown to decrease in the superconducting state. The ellipsometric data also provide detailed information about the evolution of the optical self-energy in the normal and superconducting states