Moduli Theory and Classification Theory of Algebraic Varieties

This book provides an introduction to algebraic cycles on complex algebraic varieties, to the major conjectures relating them to cohomology, and even more precisely to Hodge structures on cohomology. The book is intended for both students and researchers, and not only presents a survey of the geometric methods developed in the last thirty years to understand the famous Bloch-Beilinson conjectures, but also examines recent work by the author. It focuses on two central objects: the diagonal of a variety—and the partial Bloch-Srinivas type decompositions it may have depending on the size of Chow groups—as well as its small diagonal, which is the right object to consider in order to understand the ring structure on Chow groups and cohomology. An exploration of a sampling of recent works by the author looks at the relation, conjectured in general by Bloch and Beilinson, between the coniveau of general complete intersections and their Chow groups and a very particular property satisfied by the Chow ring of K3 surfaces and conjecturally by hyper-Kähler manifolds. In particular, the book delves into arguments originating in Nori's work that have been further developed by others.

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ALGEBRAIC VARIETIES OVER FIELDS WITH DIFFERENTIATION

Mathematics of the USSR-Sbornik ◽

10.1070/sm1969v009n03abeh002054 ◽

1969 ◽

Vol 9 (3) ◽

pp. 389-413

Author(s):

Ju R Vaĭnberg

Keyword(s):

Algebraic Varieties

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Gradient boosting for linear mixed models

The International Journal of Biostatistics ◽

10.1515/ijb-2020-0136 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Colin Griesbach ◽

Benjamin Säfken ◽

Elisabeth Waldmann

Keyword(s):

Random Effects ◽

Mixed Models ◽

Selection Procedure ◽

Classification Theory ◽

Gradient Boosting ◽

Random Structure ◽

Boosting Algorithm ◽

The One ◽

Biased Estimates ◽

Selection Of

Abstract Gradient boosting from the field of statistical learning is widely known as a powerful framework for estimation and selection of predictor effects in various regression models by adapting concepts from classification theory. Current boosting approaches also offer methods accounting for random effects and thus enable prediction of mixed models for longitudinal and clustered data. However, these approaches include several flaws resulting in unbalanced effect selection with falsely induced shrinkage and a low convergence rate on the one hand and biased estimates of the random effects on the other hand. We therefore propose a new boosting algorithm which explicitly accounts for the random structure by excluding it from the selection procedure, properly correcting the random effects estimates and in addition providing likelihood-based estimation of the random effects variance structure. The new algorithm offers an organic and unbiased fitting approach, which is shown via simulations and data examples.

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