We evaluate multiple polylogarithm values at sixth roots of unity up to weight six, i.e. of the form G(a1, . . . , aw; 1) where the indices ai are equal to zero or a sixth root of unity, with a1 6= 1. For w ≤ 6, we construct bases of the linear spaces generated by the real and imaginary parts of G(a1, . . . , aw; 1) and obtain a table for expressing them as linear combinations of the elements o...

when s1, . . . , sk are positive integers and z a complex number in the unit disk. For k = 1, this is the classical polylogarithm Lis (z). These multiple polylogarithms can be defined also in terms of iterated Chen integrals and satisfy shuffle relations. Multiple polylogarithms in several variables are defined for si ≥ 1 and |zi | < 1(1 ≤ i ≤ k) by Li(s1,...,sk)(z1, . . . , zk) = ∑ n1>n2>···>n...

We introduce a q-analog of the multiple harmonic series commonly referred to as multiple zeta values. The multiple q-zeta values satisfy a q-stuffle multiplication rule analogous to the stuffle multiplication rule arising from the series representation of ordinary multiple zeta values. Additionally, multiple q-zeta values can be viewed as special values of the multiple q-polylogarithm, which ad...

We introduce a q-analog of the multiple harmonic series commonly referred to as multiple zeta values. The multiple q-zeta values satisfy a q-stuffle multiplication rule analogous to the stuffle multiplication rule arising from the series representation of ordinary multiple zeta values. Additionally, multiple q-zeta values can be viewed as special values of the multiple q-polylogarithm, which ad...

We examine the power of Boolean functions with low L 1 norms in several settings. In large part of the recent literature, the degree of a polynomial which represents a Boolean function in some way was chosen to be the measure of the complexity of the Boolean function (see, e. have high degree, but small L 1 norms. So, in conjunction with communication complexity, instead of the degree, the L 1 ...

We examine the power of Boolean functions with low L 1 norms in several settings. In large part of the recent literature, the degree of a polynomial which represents a Boolean function in some way was chosen to be the measure of the complexity of the Boolean function (see, e. have high degree, but small L 1 norms. So, in conjunction with communication complexity, instead of the degree, the L 1 ...