Density and duality theorems for regular Gabor frames

We investigate Gabor frames on locally compact abelian groups with time-frequency shifts along non-separable, closed subgroups of the phase space. Density theorems in Gabor analysis state necessary conditions for a Gabor system to be a frame or a Riesz basis, formulated only in terms of the index subgroup. In the classical results the subgroup is assumed to be discrete. We prove density theorems for general closed subgroups of the phase space, where the necessary conditions are given in terms of the "size" of the subgroup. From these density results we are able to extend the classical Wexler-Raz biorthogonal relations and the duality principle in Gabor analysis to Gabor systems with time-frequency shifts along non-separable, closed subgroups of the phase space. Even in the euclidean setting, our results are new

Molecular subgroups of adult medulloblastoma: a long-term single-institution study

Background Recent transcriptomic approaches have demonstrated that there are at least 4 distinct subgroups in medulloblastoma (MB); however, survival studies of molecular subgroups in adult MB have been inconclusive because of small sample sizes. The aim of this study is to investigate the molecular subgroups in adult MB and identify their clinical and prognostic implications in a large, single-institution cohort. Methods We determined gene expression profiles for 13 primary adult MBs. Bioinformatics tools were used to establish distinct molecular subgroups based on the most informative genes in the dataset. Immunohistochemistry with subgroup-specific antibodies was then used for validation within an independent cohort of 201 formalin-fixed MB tumors, in conjunction with a systematic analysis of clinical and histological characteristics. Results Three distinct molecular variants of adult MB were identified: the SHH, WNT, and group 4 subgroups. Validation of these subgroups in the 201-tumor cohort by immunohistochemistry identified significant differences in subgroup-specific demographics, histology, and metastatic status. The SHH subgroup accounted for the majority of the tumors (62%), followed by the group 4 subgroup (28%) and the WNT subgroup (10%). Group 4 tumors had significantly worse progression-free and overall survival compared with tumors of the other molecular subtypes. Conclusions We have identified 3 subgroups of adult MB, characterized by distinct expression profiles, clinical features, pathological features, and prognosis. Clinical variables incorporated with molecular subgroup are more significantly informative for predicting adult patient outcome

Credibility of subgroup analyses by socioeconomic status in public health intervention evaluations:An underappreciated problem?

There is increasing interest amongst researchers and policy makers in identifying the effect of public health interventions on health inequalities by socioeconomic status (SES). This issue is typically addressed in evaluation studies through subgroup analyses, where researchers test whether the effect of an intervention differs according to the socioeconomic status of participants. The credibility of such analyses is therefore crucial when making judgements about how an intervention is likely to affect health inequalities, although this issue appears to be rarely considered within public health. The aim of this study was therefore to assess the credibility of subgroup analyses in published evaluations of public health interventions. An established set of 10 credibility criteria for subgroup analyses was applied to a purposively sampled set of 21 evaluation studies, the majority of which focussed on healthy eating interventions, which reported differential intervention effects by SES. While the majority of these studies were found to be otherwise of relatively high quality methodologically, only 8 of the 21 studies met at least 6 of the 10 credibility criteria for subgroup analysis. These findings suggest that the credibility of subgroup analyses conducted within evaluations of public health interventions’ impact on health inequalities may be an underappreciated problem. Keywords: Health inequalities, Health inequities, Equity and public health interventions, Policy impact by socioeconomic statu

SUBIC: A Supervised Bi-Clustering Approach for Precision Medicine

Traditional medicine typically applies one-size-fits-all treatment for the entire patient population whereas precision medicine develops tailored treatment schemes for different patient subgroups. The fact that some factors may be more significant for a specific patient subgroup motivates clinicians and medical researchers to develop new approaches to subgroup detection and analysis, which is an effective strategy to personalize treatment. In this study, we propose a novel patient subgroup detection method, called Supervised Biclustring (SUBIC) using convex optimization and apply our approach to detect patient subgroups and prioritize risk factors for hypertension (HTN) in a vulnerable demographic subgroup (African-American). Our approach not only finds patient subgroups with guidance of a clinically relevant target variable but also identifies and prioritizes risk factors by pursuing sparsity of the input variables and encouraging similarity among the input variables and between the input and target variable

A concave pairwise fusion approach to subgroup analysis

An important step in developing individualized treatment strategies is to correctly identify subgroups of a heterogeneous population, so that specific treatment can be given to each subgroup. In this paper, we consider the situation with samples drawn from a population consisting of subgroups with different means, along with certain covariates. We propose a penalized approach for subgroup analysis based on a regression model, in which heterogeneity is driven by unobserved latent factors and thus can be represented by using subject-specific intercepts. We apply concave penalty functions to pairwise differences of the intercepts. This procedure automatically divides the observations into subgroups. We develop an alternating direction method of multipliers algorithm with concave penalties to implement the proposed approach and demonstrate its convergence. We also establish the theoretical properties of our proposed estimator and determine the order requirement of the minimal difference of signals between groups in order to recover them. These results provide a sound basis for making statistical inference in subgroup analysis. Our proposed method is further illustrated by simulation studies and analysis of the Cleveland heart disease dataset

Semi-Invariant Terms for Gauged Non-Linear Sigma-Models

We determine all the terms that are gauge-invariant up to a total spacetime derivative ("semi-invariant terms") for gauged non-linear sigma models. Assuming that the isotropy subgroup $H$ of the gauge group is compact or semi-simple, we show that (non-trivial) such terms exist only in odd dimensions and are equivalent to the familiar Chern-Simons terms for the subgroup $H$. Various applications are mentioned, including one to the gauging of the Wess-Zumino-Witten terms in even spacetime dimensions. Our approach is based on the analysis of the descent equation associated with semi-invariant terms.Comment: section 6 expande

Polar orbitopes

We study polar orbitopes, i.e. convex hulls of orbits of a polar representation of a compact Lie group. The face structure is studied by means of the gradient momentum map and it is shown that every face is exposed and is again a polar orbitope. Up to conjugation the faces are completely determined by the momentum polytope. There is a tight relation with parabolic subgroups: the set of extreme points of a face is the closed orbit of a parabolic subgroup of G and for any parabolic subgroup the closed orbit is of this form.Comment: 24 pages. To appear on Communications in Analysis and Geometr