Jan 27, 2007

2007 week 05: Articles in Maths


Statistical Method Used Influences Results Of Observational Studies
A study comparing different statistical methods used to remove the effects of selection bias in observational studies finds that results may vary and caution may be warranted when interpreting findings of studies using certain methods, according to an article in the January 17 issue of JAMA.
"Randomized clinical trials cannot be undertaken in all situations in which evidence is needed to guide care. Well-designed observational studies are still needed to assess population effectiveness and to extend results to a general population setting. Our study serves as a cautionary note regarding their analysis and interpretation. First, propensity scores and propensity-based matching have the same limitations as multivariable risk adjustment model methods, and are no more likely to remove bias due to unmeasured confounding when strong selection bias exists. Second, instrumental variable analyses may remove both overt and hidden biases but are more suited to answer policy questions than to provide insight into a specific clinical question for a specific patient. Caution is advised regarding clinical protocols and policy statements for invasive care based on expected mortality benefits derived from traditional multivariable modeling and propensity score risk adjustment of observational studies," the researchers conclude.
Note: This story has been adapted from a news release issued by JAMA and Archives Journals.

Microlocal Analysis
In mathematical analysis, microlocal analysis is a term use to describe techniques developed from the 1950s onwards based on Fourier transforms related to the study of variable-coefficients-linear and nonlinear partial differential equations. This includes generalized functions, pseudo-differential operators, wave front sets, Fourier integral operators, and paradifferential operators.
The term microlocal implies localisation not just at a point, but in terms of cotangent space directions at a given point. This gains in importance on manifolds of dimension greater than one.
This entry was adapted from the Wikipedia article Microlocal analysis as of January 17, 2007.

The articles below contains text in the LaTex format. To view the article correctly, please visit the source link.

Fuzzy Subset
Fuzzy set theory is based on the idea that vague notions as "big", "near", "hold" can be modelled by "fuzzy subsets". A fuzzy subset of a set S is a map ... from S into the interval ... . More precisely, the interval ... is considered as a complete lattic ...

Eventual Property
Let be a set and a property on the elements of . Let be a net ( a directed set) in (that is, ). As each , either has or does not have property . We say that the net has property above if has property for all . Furthermore, we say that eventually has property if it has property above some .

Partial Ordering In A Topological Space
Let X be a T0 space. For any ... , we define a binary relation le as follows: ... Proposition . The binary relation just defined is a partial order. ... Clearly ... . Suppose next that ... and ... . If ... , then there is an open set A such that ... and . ...

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