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Related Concept Videos

Radical Chain-Growth Polymerization: Chain Branching01:17

Radical Chain-Growth Polymerization: Chain Branching

The skeletal structure of polymers synthesized via radical polymerization is always branched. For example, the polymerization of ethylene by radical polymerization results in a low-density grade of polyethylene with a heavily branched skeletal structure. Here, the radical site abstracts hydrogen from the growing chain, and the radical site shifts from the end (a primary carbon center) to anywhere within the growing chain (a secondary carbon center). Consequently, the part of the chain from the...
The Chain Rule01:30

The Chain Rule

A system of interconnected gears provides a concrete physical interpretation of the Chain Rule in calculus. Consider three gears arranged in sequence, where the rotational speeds of the first, second, and third gears are represented by the variables x, z, and y, respectively. The first gear drives the second, and the second drives the third, so the motion of each gear depends on the one preceding it. This structure naturally leads to a two-stage variable relationship that can be analyzed using...
Structure of Alkanes02:23

Structure of Alkanes

The formation of carbon-carbon bonds leading to the creation of the carbon chain is the basis of organic chemistry. August Kekulé and Archibald Scott Couper independently developed this idea of carbon chain formation.
Hydrocarbons are the simplest organic compounds composed of carbons and hydrogens. Based on the bond order between carbons, the hydrocarbons are further classified into alkanes, alkenes, and alkynes. 
Alkanes are the simplest hydrocarbons with sp3 hybrid carbon atoms. These sp3...
Radical Chain-Growth Polymerization: Mechanism01:09

Radical Chain-Growth Polymerization: Mechanism

The radical chain-growth polymerization mechanism consists of three steps: initiation, propagation, and termination of polymerization. The polymerization initiates when a free radical generated from the radical initiator adds to the unsaturated bond in the monomer. The unpaired electron of the free radical and one π electron in the unsaturated bond creates a σ bond between the free radical and the monomer. As a result, the other π electron in the unsaturated bond converts this species into the...
Ziegler–Natta Chain-Growth Polymerization: Overview01:17

Ziegler–Natta Chain-Growth Polymerization: Overview

Ziegler–Natta polymerization is another form of addition or chain‐growth polymerization used for synthesizing linear polymers over branched polymers. The catalyst used for polymerization is the Ziegler–Natta catalyst, named after Karl Ziegler and Giulio Natta, who developed it in 1953. This catalyst is an organometallic complex of titanium tetrachloride and triethyl aluminum, with the active form of the catalyst being an alkyl titanium compound. Using the Ziegler–Natta catalyst, high molecular...
Chain Reactions01:29

Chain Reactions

Chain reactions involve highly reactive transient species, such as atoms or free radicals, as intermediates. These intermediates facilitate rapid reactions over an extended period. The process includes a series of steps: a reactive intermediate is consumed, reactants are converted to products, and the intermediate is regenerated. This cycle enables continuous repetition, amplifying the production of products with a small amount of intermediate. Chain reactions often utilize free radicals as...

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Related Experiment Video

Updated: Jun 20, 2026

Using Cholesky Decomposition to Explore Individual Differences in Longitudinal Relations between Reading Skills
06:52

Using Cholesky Decomposition to Explore Individual Differences in Longitudinal Relations between Reading Skills

Published on: September 17, 2019

Structural Learning of Chain Graphs via Decomposition.

Zongming Ma1, Xianchao Xie, Zhi Geng

  • 1School of Mathematical Sciences, LMAM, Peking University, Beijing 100871, China.

Journal of Machine Learning Research : JMLR
|September 18, 2009
PubMed
Summary
This summary is machine-generated.

We developed a computationally feasible method for learning chain graphs, a type of graphical model. This approach decomposes complex problems into smaller ones, proving effective for sparse graphs.

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Area of Science:

  • Statistics
  • Machine Learning
  • Graphical Models

Background:

  • Chain graphs are versatile graphical models encompassing Markov and Bayesian networks.
  • Understanding conditional independence structures is crucial for statistical modeling.

Purpose of the Study:

  • To propose a computationally feasible method for the structural learning of chain graphs.
  • To address the challenge of learning complex graphical models efficiently.

Main Methods:

  • Decomposing the chain graph learning problem into smaller subproblems on decomposed subgraphs.
  • Utilizing conditional independencies for decomposition without requiring complete subgraphs.
  • Developing algorithms for both skeleton recovery and arrow orientation.

Main Results:

  • The proposed method demonstrates competitive performance in simulations.
  • Effectiveness is particularly notable for sparse underlying graphs.
  • The decomposition strategy enhances computational feasibility.

Conclusions:

  • The presented method offers an efficient approach to structural learning for chain graphs.
  • This technique is valuable for analyzing complex conditional independence structures.
  • The decomposition approach is a promising direction for graphical model learning.