Testing membership in lattices is of practical relevance, with applications to integer programming, error detection in lattice-based communication and cryptography. In this work, we initiate a systematic study of local testing for membership in lattices, complementing and building upon the extensive body of work on locally testable codes. In particular, we formally define the notion of local te...

We present quantum query complexity bounds for testing algebraic properties. For a set S and a binary operation on S, we consider the decision problem whether S is a semigroup or has an identity element. If S is a monoid, we want to decide whether S is a group. We present quantum algorithms for these problems that improve the best known classical complexity bounds. In particular, we give the fi...

We study the combinatorics of fully commutative elements in Coxeter groups of type H n for any n > 2. Using the results, we construct certain canonical bases for non-simply-laced generalized Temperley–Lieb algebras and show how to relate them to morphisms in the category of decorated tangles.

We address the attainability of the Hashin-Shtrikman bounds for multiphase composite materials. We demonstrate that the Hashin-Shtrikman bounds are not always attainable and give new restrictions on the attainable Hashin-Shtrikman bounds in terms of the conductivities and volume fractions of the constituent phases. New optimal microstructures are also constructed to attain the Hashin-Shtrikman ...

The function f : F2 → F2 is k-linear if it returns the sum (over F2) of exactly k coordinates of its input. We introduce strong lower bounds on the query complexity for testing whether a function is k-linear. We show that for any k ≤ n 2 , at least k−o(k) queries are required to test k-linearity, and we show that when k ≈ n 2 , this lower bound is nearly tight since 4 3 k+o(k) queries are suffi...