Computer Society Press Research Articles

Talagrand Inequality at Second Order and Application to Boolean Analysis

This note is concerned with an extension, at second order, of an inequality on the discrete cube $$C_n=\{-\,1,1\}^n$$ (equipped with the uniform measure) due to Talagrand (Ann Probab 22:1576–1587, 1994). As an application, the main result of this note is a theorem in the spirit of a famous result from Kahn et al. (cf. Proceedings of 29th Annual Symposium on Foundations of Computer Science, vol 62. Computer Society Press, Washington, pp 68–80, 1988) concerning the influence of Boolean functions. The notion of the influence of a couple of coordinates $$(i,j)\in \{1,\ldots ,n\}^2$$ is introduced in Sect. 2, and the following alternative is obtained: For any Boolean function $$f\,:\, C_n\rightarrow \{0,1\}$$, either there exists a coordinate with influence at least of order $$(1/n)^{1/(1+\eta )}$$, with $$\, 0<\eta <1$$ (independent of f and n), or there exists a couple of coordinates $$(i,j)\in \{1,\ldots ,n\}^2$$ with $$i\ne j$$, with influence at least of order $$(\log n/n)^2$$. In Sect. 4, it is shown that this extension of Talagrand inequality can also be obtained, with minor modifications, for the standard Gaussian measure $$\gamma _n$$ on $${\mathbb {R}}^n$$; the obtained inequality can be of independent interest. The arguments rely on interpolation methods by semigroup together with hypercontractive estimates. At the end of the article, some related open questions are presented.

Notes on Contributors

<i>Notes on Contributors</i>

Re-inventing the introductory computer graphics course: providing tools for a wider audience

Traditionally, the introductory computer graphics course in computer science has focused on fundamental algorithms and techniques for creating images and animations. This was reflected in ACM/IEEE Curriculum 91 (Tucker AB, Barnes BH. Computing curricula 1991: Report of the ACM/IEEE/CS Curriculum Task Force. New York: Silver Spring, MD: ACM Press/IEEE Computer Society Press, 1991). Computer graphics and similar subjects are expected to play a larger role in undergraduate computer science in the future (Cunninghams. SIGCSE Bulletin 1998;30(4):4a–7a), and this is being discussed in the ACM/IEEE Curriculum 2001 project. In the last few years the approach to teaching this course has changed to take advantage of more powerful graphics tools. This paper describes an approach to the introductory computer graphics course that increases its value as a tool for the student in the sciences, mathematics, or engineering as well as providing a sound introduction to the subject for the computer science student. This approach is compatible with the recommendations of the recent Graphics and Visualization Education Workshop (Reports of the Graphics and Visualization Education 99 workshop are published online at www.eg.org/WorkingGroups/GVE/GVE99 and www.education.siggraph.org/conferences/GVE99 and will appear in Computer Graphics and Computer Graphics Forum) while focusing on serving an expanded audience.