Meta-Analyzing Multiple Omics Data With Robust Variable Selection

High-throughput omics data are becoming more and more popular in various areas of science. Given that many publicly available datasets address the same questions, researchers have applied meta-analysis to synthesize multiple datasets to achieve more reliable results for model estimation and prediction. Due to the high dimensionality of omics data, it is also desirable to incorporate variable selection into meta-analysis. Existing meta-analyzing variable selection methods are often sensitive to the presence of outliers, and may lead to missed detections of relevant covariates, especially for lasso-type penalties. In this paper, we develop a robust variable selection algorithm for meta-analyzing high-dimensional datasets based on logistic regression. We first search an outlier-free subset from each dataset by borrowing information across the datasets with repeatedly use of the least trimmed squared estimates for the logistic model and together with a hierarchical bi-level variable selection technique. We then refine a reweighting step to further improve the efficiency after obtaining a reliable non-outlier subset. Simulation studies and real data analysis show that our new method can provide more reliable results than the existing meta-analysis methods in the presence of outliers.

Functional identities on upper triangular matrix rings

Let R be a subset of a unital ring Q such that 0 ∈ R. Let us fix an element t ∈ Q. If R is a (t; d)-free subset of Q, then Tn(R) is a (t′; d)-free subset of Tn(Q), where t′ ∈ Tn(Q), $\begin{array}{} t_{ll}' \end{array} $ = t, l = 1, 2, …, n, for any n ∈ N.

THE MAXIMUM SIZE OF -SUM-FREE SETS IN CYCLIC GROUPS

A subset$A$of a finite abelian group$G$is called$(k,l)$-sum-free if the sum of$k$(not necessarily distinct) elements of$A$never equals the sum of$l$(not necessarily distinct) elements of $A$. We find an explicit formula for the maximum size of a$(k,l)$-sum-free subset in$G$for all$k$and$l$in the case when$G$is cyclic by proving that it suffices to consider$(k,l)$-sum-free intervals in subgroups of $G$. This simplifies and extends earlier results by Hamidoune and Plagne [‘A new critical pair theorem applied to sum-free sets in abelian groups’,Comment. Math. Helv. 79(1) (2004), 183–207] and Bajnok [‘On the maximum size of a$(k,l)$-sum-free subset of an abelian group’,Int. J. Number Theory 5(6) (2009), 953–971].

STRONGLY NP-HARDNESS OF SQUARE-FREE SUBSET PRODUCT

We are given 3N integers; find N of them whose product is square free.

The number of additive triples in subsets of abelian groups

A set of elements of a finite abelian group is called sum-free if it contains no Schur triple, i.e., no triple of elementsx,y,zwithx+y=z. The study of how large the largest sum-free subset of a given abelian group is had started more than thirty years before it was finally resolved by Green and Ruzsa a decade ago. We address the following more general question. Suppose that a setAof elements of an abelian groupGhas cardinalitya. How many Schur triples mustAcontain? Moreover, which sets ofaelements ofGhave the smallest number of Schur triples? In this paper, we answer these questions for various groupsGand ranges ofa.

The Sharp Threshold for Maximum-Size Sum-Free Subsets in Even-Order Abelian Groups

We study sum-free sets in sparse random subsets of even-order abelian groups. In particular, we determine the sharp threshold for the following property: the largest such set is contained in some maximum-size sum-free subset of the group. This theorem extends recent work of Balogh, Morris and Samotij, who resolved the caseG= ℤ2n, and who obtained a weaker threshold (up to a constant factor) in general.