The gradient lemma

Lecture 19 : The Vanishing Lemma and Lemma 8 . 8

The Generalized XOR Lemma

The XOR Lemma states that a mapping is regular or balanced if and only if all the linear combinations of the component functions of the mapping are balanced Boolean functions. The main contribution of this paper is to extend the XOR Lemma to more general cases where a mapping may not be necessarily regular. The extended XOR Lemma has applications in the design of substitution boxes or S-boxes u...

3 the Switching Lemma

Today we show that PARITY is not in AC0. AC0 is a family of circuits with constant depth, polynomial size, and unbounded fan-in for the AND and OR gates. We establish this result through an application of the Switching Lemma. This result is the first use of randomization in its full power in complexity. Circuits were defined in previous lectures. In this lecture, we always assume that the circu...

The statin D-lemma

Correspondence to: Michael F. Holick; Email: [email protected] Yavuz et al. in 2009 reported that patients with hyperlipidemia who were in a prospective study evaluating rosuvastatin in hyperlipidemic patients made an unexpected observation. Blood concentrations of 25-hydroxyvitamin D [25(OH)D] and 1,25-dihydroxyvitamin D [1,25(OH)2D] increased after 8 weeks of treatment. In addition they observe...

The Farkas Lemma Revisited

The Farkas Lemma is extended to simultaneous linear operator and polyhedral sublinear operator inequalities by Boolean valued analysis. Introduction The Farkas Lemma, also known as the Farkas–Minkowski Lemma, plays a key role in linear programming and the relevant areas of optimization (cp. [1]). Some rather simple proof of the lemma is given in [2]. Using the technique of Boolean valued analys...