"Antisemitismus?" : ein Nachwort zum Artikel von Volkmar Frobenius

Ludwig Holländer

Io. Frobenius pio lectori S.D. Diui Hilaij Pictauoru[m] episcopi lucubrationes /

Operum diui Hilarii Pictauorum episcopi, tomus secundus, con portada, paginación y signaturas propias.

Ioannes Frobenius studiosis quandoquidem nobis animus est, antiquis autoribus excudendis rem iuuare literaria[m], damus nu[n]c vobis Panegyricos ... /

Mode of access: Internet.

On Projective Plane of Order 13 with a Frobenius Group of Order 39 as a Collineation Group

One of the most outstanding problems in combinatorial mathematics and geometry is the problem of existence of finite projective planes whose order is not a prime power.

Improved image set classification via joint sparse approximated nearest subspaces

Existing multi-model approaches for image set classification extract local models by clustering each image set individually only once, with fixed clusters used for matching with other image sets. However, this may result in the two closest clusters to represent different characteristics of an object, due to different undesirable environmental conditions (such as variations in illumination and pose). To address this problem, we propose to constrain the clustering of each query image set by forcing the clusters to have resemblance to the clusters in the gallery image sets. We first define a Frobenius norm distance between subspaces over Grassmann manifolds based on reconstruction error. We then extract local linear subspaces from a gallery image set via sparse representation. For each local linear subspace, we adaptively construct the corresponding closest subspace from the samples of a probe image set by joint sparse representation. We show that by minimising the sparse representation reconstruction error, we approach the nearest point on a Grassmann manifold. Experiments on Honda, ETH-80 and Cambridge-Gesture datasets show that the proposed method consistently outperforms several other recent techniques, such as Affine Hull based Image Set Distance (AHISD), Sparse Approximated Nearest Points (SANP) and Manifold Discriminant Analysis (MDA).

Transverse plane wave analysis of short elliptical chamber mufflers: An analytical approach

Short elliptical chamber mufflers are used often in the modern day automotive exhaust systems. The acoustic analysis of such short chamber mufflers is facilitated by considering a transverse plane wave propagation model along the major axis up to the low frequency limit. The one dimensional differential equation governing the transverse plane wave propagation in such short chambers is solved using the segmentation approaches which are inherently numerical schemes, wherein the transfer matrix relating the upstream state variables to the downstream variables is obtained. Analytical solution of the transverse plane wave model used to analyze such short chambers has not been reported in the literature so far. This present work is thus an attempt to fill up this lacuna, whereby Frobenius solution of the differential equation governing the transverse plane wave propagation is obtained. By taking a sufficient number of terms of the infinite series, an approximate analytical solution so obtained shows good convergence up to about 1300 Hz and also covers most of the range of muffler dimensions used in practice. The transmission loss (TL) performance of the muffler configurations computed by this analytical approach agrees excellently with that computed by the Matrizant approach used earlier by the authors, thereby offering a faster and more elegant alternate method to analyze short elliptical muffler configurations. (C) 2010 Elsevier Ltd. All rights reserved.

Zero-sum risk-sensitive stochastic games on a countable state space

Infinite horizon discounted-cost and ergodic-cost risk-sensitive zero-sum stochastic games for controlled Markov chains with countably many states are analyzed. Upper and lower values for these games are established. The existence of value and saddle-point equilibria in the class of Markov strategies is proved for the discounted-cost game. The existence of value and saddle-point equilibria in the class of stationary strategies is proved under the uniform ergodicity condition for the ergodic-cost game. The value of the ergodic-cost game happens to be the product of the inverse of the risk-sensitivity factor and the logarithm of the common Perron-Frobenius eigenvalue of the associated controlled nonlinear kernels. (C) 2013 Elsevier B.V. All rights reserved.

Asynchronous Gossip for Averaging and Spectral Ranking

We consider two variants of the classical gossip algorithm. The first variant is a version of asynchronous stochastic approximation. We highlight a fundamental difficulty associated with the classical asynchronous gossip scheme, viz., that it may not converge to a desired average, and suggest an alternative scheme based on reinforcement learning that has guaranteed convergence to the desired average. We then discuss a potential application to a wireless network setting with simultaneous link activation constraints. The second variant is a gossip algorithm for distributed computation of the Perron-Frobenius eigenvector of a nonnegative matrix. While the first variant draws upon a reinforcement learning algorithm for an average cost controlled Markov decision problem, the second variant draws upon a reinforcement learning algorithm for risk-sensitive control. We then discuss potential applications of the second variant to ranking schemes, reputation networks, and principal component analysis.

Some theorems in asymptotic analysis

Basing ourselves on the analysis of magnitude of order, we strictly prove fundamental lemmas for asymptotic integral, including the cases of infinite region. Then a general formula for asymptotic expansion of integrals is given. Finally, we derive a sufficient condition for an ordinary differential equation to possess a solution of the Frobenius series type at finite irregular singularities or branching points.

Topics in randomized numerical linear algebra

This thesis studies three classes of randomized numerical linear algebra algorithms, namely: (i) randomized matrix sparsification algorithms, (ii) low-rank approximation algorithms that use randomized unitary transformations, and (iii) low-rank approximation algorithms for positive-semidefinite (PSD) matrices.

Randomized matrix sparsification algorithms set randomly chosen entries of the input matrix to zero. When the approximant is substituted for the original matrix in computations, its sparsity allows one to employ faster sparsity-exploiting algorithms. This thesis contributes bounds on the approximation error of nonuniform randomized sparsification schemes, measured in the spectral norm and two NP-hard norms that are of interest in computational graph theory and subset selection applications.

Low-rank approximations based on randomized unitary transformations have several desirable properties: they have low communication costs, are amenable to parallel implementation, and exploit the existence of fast transform algorithms. This thesis investigates the tradeoff between the accuracy and cost of generating such approximations. State-of-the-art spectral and Frobenius-norm error bounds are provided.

The last class of algorithms considered are SPSD "sketching" algorithms. Such sketches can be computed faster than approximations based on projecting onto mixtures of the columns of the matrix. The performance of several such sketching schemes is empirically evaluated using a suite of canonical matrices drawn from machine learning and data analysis applications, and a framework is developed for establishing theoretical error bounds.

In addition to studying these algorithms, this thesis extends the Matrix Laplace Transform framework to derive Chernoff and Bernstein inequalities that apply to all the eigenvalues of certain classes of random matrices. These inequalities are used to investigate the behavior of the singular values of a matrix under random sampling, and to derive convergence rates for each individual eigenvalue of a sample covariance matrix.

On the set of eigenvalues of a class of equimodular matrices

The structure of the set ϐ(A) of all eigenvalues of all complex matrices (elementwise) equimodular with a given n x n non-negative matrix A is studied. The problem was suggested by O. Taussky and some aspects have been studied by R. S. Varga and B.W. Levinger.

If every matrix equimodular with A is non-singular, then A is called regular. A new proof of the P. Camion-A.J. Hoffman characterization of regular matrices is given.

The set ϐ(A) consists of m ≤ n closed annuli centered at the origin. Each gap, ɤ, in this set can be associated with a class of regular matrices with a (unique) permutation, π(ɤ). The association depends on both the combinatorial structure of A and the size of the a_ii. Let A be associated with the set of r permutations, π₁, π₂,…, π_r, where each gap in ϐ(A) is associated with one of the π_k. Then r ≤ n, even when the complement of ϐ(A) has n+1 components. Further, if π(ɤ) is the identity, the real boundary points of ɤ are eigenvalues of real matrices equimodular with A. In particular, if A is essentially diagonally dominant, every real boundary point of ϐ(A) is an eigenvalues of a real matrix equimodular with A.

Several conjectures based on these results are made which if verified would constitute an extension of the Perron-Frobenius Theorem, and an algebraic method is introduced which unites the study of regular matrices with that of ϐ(A).

Matrices doblemente estocásticas y mayorización

El proyecto tiene como objetivo el estudio de las propiedades más importantes de las matrices doblemente estocásticas y algunas aplicaciones. Se comienza analizando algunas propiedades espectrales de las matrices no negativas de las que aquellas son un caso particular y se demuestra, en particular, el Teorema de Perron-Frobenius. Posteriormente se discute en detalle la relación entre las matrices doblemente estocásticas y la mayorización de vectores reales y el importante teorema de Birkhoff. El proyecto finaliza desarrollando algunas aplicaciones de este tipo de matrices.

一类Koblitz椭圆曲线的快速点乘

考虑一类特征3的Koblitz椭圆曲线的快速点乘算法.在这类曲线上适合建立低带宽的、可证明安全的密码体制.结果显示,利用这类曲线的复乘性质,使用模约减和Frobenius展开技巧,这类曲线上存在一种不带预计算的快速点乘算法,其运算速度是通常的重复加倍-点加算法的6倍.该算法的快速优化原理与有限域算术优化和椭圆曲线点的坐标表示的选取无关.

Robust Tensor Analysis With L1-Norm

Tensor analysis plays an important role in modern image and vision computing problems. Most of the existing tensor analysis approaches are based on the Frobenius norm, which makes them sensitive to outliers. In this paper, we propose L1-norm-based tensor analysis (TPCA-L1), which is robust to outliers. Experimental results upon face and other datasets demonstrate the advantages of the proposed approach.