Structure Of Multiplicative Group Research Articles

A penalized variable selection ensemble algorithm for high-dimensional group-structured data.

This paper presents a multi-algorithm fusion model (StackingGroup) based on the Stacking ensemble learning framework to address the variable selection problem in high-dimensional group structure data. The proposed algorithm takes into account the differences in data observation and training principles of different algorithms. It leverages the strengths of each model and incorporates Stacking ensemble learning with multiple group structure regularization methods. The main approach involves dividing the data set into K parts on average, using more than 10 algorithms as basic learning models, and selecting the base learner based on low correlation, strong prediction ability, and small model error. Finally, we selected the grSubset + grLasso, grLasso, and grSCAD algorithms as the base learners for the Stacking algorithm. The Lasso algorithm was used as the meta-learner to create a comprehensive algorithm called StackingGroup. This algorithm is designed to handle high-dimensional group structure data. Simulation experiments showed that the proposed method outperformed other R2, RMSE, and MAE prediction methods. Lastly, we applied the proposed algorithm to investigate the risk factors of low birth weight in infants and young children. The final results demonstrate that the proposed method achieves a mean absolute error (MAE) of 0.508 and a root mean square error (RMSE) of 0.668. The obtained values are smaller compared to those obtained from a single model, indicating that the proposed method surpasses other algorithms in terms of prediction accuracy.

On the Laxton group

In 1969, Laxton defined a multiplicative group structure on the set of rational sequences satisfying a fixed linear recurrence of degree two. He also defined some natural subgroups of the group, and determined the structures of their quotient groups. Nothing has been known about the structure of Laxton’s whole group and its interpretation. In this paper, we redefine his group in a natural way and determine the structure of the whole group, which clarifies Laxton’s results on the quotient groups. This definition makes it possible to use the group to show various properties of such sequences.

Algebraic representations of the weighted mean

We introduce a general framework for the algebraic representation of the weighted mean in the case in which priorities and values refer to a common aggregation domain, for instance in hierarchical multicriteria decision models of the AHP type. The general framework proposed is based on the semifield structure of open interval domains and provides a natural algebraic description of weighted mean aggregation, including the essential mechanism of normalization which transforms priorities into weights. Such description necessarily involves two operations, addition (abelian semigroup) and multiplication (abelian group), which generalize the role of addition and multiplication in P=(0,∞). In this sense, the semifield framework extends recent work by Cavallo, D'Apuzzo, and Squillante on the multiplicative group structure on the basis of the representation of pairwise comparison matrices and their associated priority vectors. We consider open interval domains S⊆R whose semifield structures are generated by bijections ϕ:S⊆R→P. Continuous (thus strictly monotonic) bijections play a central role. In such case, continuous strict triangular conorms/norms and uninorms emerge naturally in the representation of the semifield structure and, in their weighted version, they also provide the representation of the weighted mean, in both the arithmetic and geometric forms.

Multiplicative groups of division rings

Exactly 170 years ago, the construction of the real quaternion algebra by William Hamilton was announced in the Proceedings of the Royal Irish Academy. It became the first example of non-commutative division rings and a major turning point of algebra. To this day, the multiplicative group structure of quaternion algebras has not completely been understood. This article is a self-contained account of the recent developments on the multiplicative group structure of division rings.

Multiplicative groups of division rings

Exactly 170 years ago, the construction of the real quaternion algebra by William Hamilton was announced in the Proceedings of the Royal Irish Academy. It became the first example of non-commutative division rings and a major turning point of algebra. To this day, the multiplicative group structure of quaternion algebras have not completely been understood. This article is a long survey of the recent developments on the multiplicative group structure of division rings.

On the discrete mean value of the product of two Dirichlet L-functions

In Acta. Arith. 122(1), 51–56, 2006, Liu and the third author elaborated on the result of Louboutin to determine the coefficients of the exact formula for the discrete mean value \(\sum_{{\scriptstyle\chi\bmod q}\atop {\scriptstyle\chi(-1)=(-1)^{m}}}L(m,\chi)L(n,\bar{\chi})\) of the product of two Dirichlet L-functions, where m,n and χ are of the same parity. The method uses the Fourier series for the periodic Bernoulli polynomials and is rather computational. In this paper, we shall reveal the hidden algebraic structures and the intrinsic properties of the Bernoulli polynomials and of the Clausen functions to treat the more difficult case of m and χ being of opposite parity. The additive group structure of ℤ/qℤ is realized as the discrete Fourier transform while the multiplicative group structure of (ℤ/qℤ)× is realized as the subgroup of all even characters in the character group \(\widehat{(\mathbb{Z}/q\mathbb{Z})^{\times}}\) . The intrinsic property is the distribution property, which corresponds to the equally divided Riemann sum. This suggests the analogy between our result and continuous mean values in the form of definite integrals.

On the Bavrin Integro-Differential Operators and Their Applications to Solution of Functional Equations

We construct a multiplicative group structure in the class of integro-differential operators specific for a polydisk which was introduced by I. I. Bavrin. We indicate two applications of these operators to solution of functional equations.

Sex differences in self-reported delinquency and official records: A multiple-group structural modeling approach

There has been a large increase in research in the last 10 years or so on the nature, extent, and causes of female delinquency, especially on how patterns differ from those of male delinquency. Few research efforts, however, have systematically examined sex differences in the reliability and construct validity of the most common technique used in causal research-the self—report method. The present study addresses this issue by empirically examining male-female differences in self-report reliabilities and the effect of the prevalence and incidence of delinquent behavior on official contact (i.e., arrest, court referrals). Multiple-group covriance structure models are employed to test hypotheses of measurement and structural invariance. The major conclusion is that while self-report measurement parameters (e.g., reliabilities) may differ, the structural link between the prevalence and incidence of delinquent behavior and official contact is generally invariant with respect to sex.