Variance function of boolean additive convolution

Additive Genetic Variance

Decompositions of the Free Additive Convolution

We introduce and study a new type of convolution of probability measures called the orthogonal convolution, which is related to the monotone convolution. Using this convolution, we derive alternating decompositions of the free additive convolution μ` ν of compactly supported probability measures in free probability. These decompositions are directly related to alternating decompositions of the ...

Parsing Unary Boolean Grammars Using Online Convolution

Consider context-free grammars generating strings over a one-letter alphabet. For the membership problem for such grammars, stated as “Given a grammar G and a string an, determine whether an is generated by G”, only a näıve O(|G| ·n2)-time algorithm is known. This paper develops a new algorithm solving this problem, which is based upon fast multiplication of integers, works in time |G| · n log ...

Lower bound for monotone Boolean convolution

Any monotone Boolean circuit computing the n-dimensional Boolean convolution requires at least n2 and-gates. This matches the obvious upper bound. The previous best bound for this problem was Ω(n4/3), obtained by Norbert Blum in 1981. More generally, exact bounds are given for all semi-disjoint bilinear forms.

Additive Autocorrelation of Resilient Boolean Functions

In this paper, we introduce a new notion called the dual function for studying Boolean functions. First, we discuss general properties of the dual function that are related to resiliency and additive autocorrelation. Second, we look at preferred functions which are Boolean functions with the lowest 3-valued spectrum. We prove that if a balanced preferred function has a dual function which is al...