A Pass to Variable Selection

Many regularized procedures produce sparse solution and therefore are sometimes used for variable selection in linear regression. It has been showed that regularized procedures are more stable than subset selection. Such procedures include LASSO, SCAD, and adaptive LASSO, to name just a few. However, their performance depends crucially on the tuning parameter selection. 

For the purpose of prediction, popular methods for the tuning parameter selection include Cp, cross-validation, and generalized cross-validation. For the purpose of variable selection, the most popular method for the tuning parameter selection is BIC. The selection consistencies of BIC for some regularized procedures have been shown. (Here the selection consistency means that the probability of selecting the data generating model is tending to one when the sample size goes to infinity, assuming that the data generating model is a subset of the full model.) 

Polignac's Conjecture with New Prime Number Theorem

In number theory, Polignac's conjecture was made by Alphonse de Polignac in 1849 and states: For any positive even number En, there are infinitely many prime gaps of size En. In other words: There are infinitely many cases of two consecutive prime numbers with difference En.

The conjecture has not yet been proven or disproven for a given value of En. In 2013 an important breakthrough was made by Zhang Yitang who proved that there are infinitely many prime gaps of size En for some value of En<70,000,000.

For En=6, it says there are infinitely many primes (p, p + 6). For En=4, it says there are infinitely many cousin primes (p, p + 4). For En=2, it is the twin prime conjecture that there are infinitely many twin primes (p, p + 2) as shown in Figure 1. For En=0, it is the new prime theorem.

Lie Group Methods for Eigenvalue Function

By considering a C∞ structure on the ordered non-increasing of elements of Rn, we show that it is a differentiable manifold. By using of Lie groups, we show that eigenvalue function is a submersion. This fact is used to prove some results. These results are applied to prove a few facts about spectral manifolds and spectral functions. Orthogonal matrices act on the real symmetric matrices as a Lie transformation group. This fact, also, is used to prove the results.

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Advances in Logic, Operations and Computational Mathematics

Journal of Applied & Computational Mathematics Volume 5, Issue 2 comprised of 7 research articles and 4 opinion articles and are focused on the innovation of polygon, Euler, linear and non-linear equations.

EL-Kholy et al., in their research article discussed about balanced folding over a polygon and Euler numbers. The study proved that for a balanced folding of a simply connected surface M, there is a subgroup of the group which is called all homeomorphisms of M that will acts 1- transitively on the 2-cells of M.

Gil et al., in their research have reported about the exponentially stabile non-linear, non-autonomous multi variable discrete systems. Based on the recent estimates on matrix equations, the findings suggest that a class of non-autonomous discrete-time systems is governed by semi-linear vector difference equations along with slowly varying linear parts.

Exploring Rack-bi algebra

Basic Lie theory relies heavily on the fundamental links between associative algebras, Lie algebras and groups. The study focuses on self-distributive multiplication on co algebras reveals that it leads to rack bi algebra, which is universal enveloping of the Lie algebra.

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Comprehension Process Overview

The thinking mode by images association is a reflection of the circulation of what we name "our awareness" in the set of sequences of memories corresponding to the perceptions of our senses: visual images, tactile sensations, sounds, etc., the complete set being stored in recoverable order of occurrence in various areas of the neocortex, each area corresponding to one of our senses. The sequences of events stored in the various neocortex areas are also mutually interconnected in such a way that we can easily access what we can remember of these sensory perceptions for any specific past event.

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Mathematica and LaTeX Integration

This document is about Mathex, a software package that allows embedding Mathematica macro expressions into a Latex document, combining together the power of the two systems. 

It works in this way: when you click F2 on your preferred editor, before of the TeX to PDF conversion, a pre-processor (written in the PERL language) call Mathematica, do the computation globally (i.e. treating the file as a single program) and replaces the macros in the final TeX. The macros should be inserted between the two conventional symbols <% ... %>, and can contain every valid Mathematica’s expression, or more expressions, separated by semicolon. As every expression and every graphics object get created every time on-the-fly, the process can be slow.

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Using Ancestral Information to Inform Analyses of Complex Data Sets

Over the last decade, improvements in sequencing technologies coupled with active development of association mapping methods have made it possible to link genotypes and quantitative traits in humans. Despite substantial progress in the ability to generate and analyze large data sets, however, genotype-phenotype associations are often difficult to find, even in studies with large numbers of individuals and genetic markers. This is due, in part, to the fact that effects of individual loci can be small and/or dependent on genetic variation at other loci or the environment. 

Tree-based mapping, which uses the evolutionary relatedness of sampled individuals to gain information during association mapping, has the potential to significantly improve our ability to detect loci impacting human traits. However, current tree-based methods are too computationally intensive and inflexible to be of practical use. Here, we compare tree-based methods with more classical approaches for association mapping and discuss how the limitations of these newer methods might be addressed. Ultimately, these advances have the potential to advance our understanding of the molecular mechanisms underlying complex diseases.