Statistical inference based on robust low-rank data matrix approximation

The singular value decomposition is widely used to approximate data matrices with lower rank matrices. Feng and He [Ann. Appl. Stat. 3 (2009) 1634-1654] developed tests on dimensionality of the mean structure of a data matrix based on the singular value decomposition. However, the first singular values and vectors can be driven by a small number of outlying measurements. In this paper, we consider a robust alternative that moderates the effect of outliers in low-rank approximations. Under the assumption of random row effects, we provide the asymptotic representations of the robust low-rank approximation. These representations may be used in testing the adequacy of a low-rank approximation. We use oligonucleotide gene microarray data to demonstrate how robust singular value decomposition compares with the its traditional counterparts. Examples show that the robust methods often lead to a more meaningful assessment of the dimensionality of gene intensity data matrices.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1186 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Two Particle States in a Box and the ss-Matrix in Multi-Channel Scattering

Using a quantum mechanical model, the exact energy eigenstates for two-particle two-channel scattering are studied in a cubic box with periodic boundary conditions. A relation between the exact energy eigenvalue in the box and the two-channel $S$-matrix elements in the continuum is obtained. This result can be viewed as a generalization of the well-known L\"uscher's formula which establishes a similar relation in elastic scattering.Comment: 4 pages, typeset with ws-ijmpa.cls. Talk presented at International Conference on QCD and Hadronic Physics, June 16-20, 2005, Beijing, China. One reference adde

Professor Chen Ping Yang's early significant contributions to mathematical physics

In the 60's Professor Chen Ping Yang with Professor Chen Ning Yang published several seminal papers on the study of Bethe's hypothesis for various problems of physics. The works on the lattice gas model, critical behaviour in liquid-gas transition, the one-dimensional (1D) Heisenberg spin chain, and the thermodynamics of 1D delta-function interacting bosons are significantly important and influential in the fields of mathematical physics and statistical mechanics. In particular, the work on the 1D Heisenberg spin chain led to subsequent developments in many problems using Bethe's hypothesis. The method which Yang and Yang proposed to treat the thermodynamics of the 1D system of bosons with a delta-function interaction leads to significant applications in a wide range of problems in quantum statistical mechanics. The Yang and Yang thermodynamics has found beautiful experimental verifications in recent years.Comment: 5 pages + 3 figure

On correspondences between toric singularities and (p,q) webs

We study four-dimensional N=1 gauge theories which arise from D3-brane probes of toric Calabi-Yau threefolds. There are some standing paradoxes in the literature regarding relations among (p,q)-webs, toric diagrams and various phases of the gauge theories, we resolve them by proposing and carefully distinguishing between two kinds of (p,q)-webs: toric and quiver (p,q)-webs. The former has a one to one correspondence with the toric diagram while the latter can correspond to multiple gauge theories. The key reason for this ambiguity is that a given quiver (p,q)-web can not capture non-chiral matter fields in the gauge theory. To support our claim we analyse families of theories emerging from partial resolution of Abelian orbifolds using the Inverse Algorithm of hep-th/0003085 as well as (p,q)-web techniques. We present complex inter-relations among these theories by Higgsing, blowups and brane splittings. We also point out subtleties involved in the ordering of legs in the (p,q) diagram

Z-D Brane Box Models and Non-Chiral Dihedral Quivers

Generalising ideas of an earlier work \cite{Bo-Han}, we address the problem of constructing Brane Box Models of what we call the Z-D Type from a new point of view, so as to establish the complete correspondence between these brane setups and orbifold singularities of the non-Abelian G generated by Z_k and D_d under certain group-theoretic constraints to which we refer as the BBM conditions. Moreover, we present a new class of ${\cal N}=1$ quiver theories of the ordinary dihedral group d_k as well as the ordinary exceptionals E_{6,7,8} which have non-chiral matter content and discuss issues related to brane setups thereof