In this article, we consider the ergodic optimization of top Lyapunov exponent. We prove that there is a unique maximising measure exponent for typical matrix cocyles. By using results obtain, in any nonuniquely minimal dynamical system, Lyapunov-irregular points are

If A is a primitive matrix, then there is a smallest power of A (its fully indecomposable exponent) that is fully indecomposable, and a smallest power of A (its strict fully indecomposable exponent) starting from which all powers are fully indecomposable. We obtain bounds on these two exponents.

We prove that Alekhnovich’s algorithm can be used for row reduction of skew polynomial matrices. This yields an O(ln log(n)) decoding algorithm for l-Interleaved Gabidulin codes of length n, where ω is the matrix multiplication exponent, improving in the exponent of n compared to previous results.

We introduce a relaxation of the notion of tensor rank, called s-rank, and show that upper bounds on the s-rank of the matrix multiplication tensor imply upper bounds on the ordinary rank. In particular, if the “s-rank exponent of matrix multiplication” equals 2, then ω = 2. This connection between the s-rank exponent and the ordinary exponent enables us to significantly generalize the group-th...

We derive a new renormalization group to calculate a non-trivial critical exponent of the divergent correlation length which gives a universality classification of essential singularities in infinite-order phase transitions. This method resolves the problem of a vanishing scaling matrix in infinite-order phase transitions. The exponent is obtained from the maximal eigenvalue of a scaling matrix...

We use a benchtop NMR technique to monitor entangled polymer melt dynamics over 5 decades in time covering regimes II-IV of the tube model. We confirm the familiar molar mass scaling exponents of the regime transitions, but we observe a mass-dependent time scaling exponent describing segmental fluctuations in the constrained-Rouse regime II up to high molecular weights. Local chain motions are ...