We prove that for any 4 points in the plane that belong to 2 parallel lines, there is no linear dependence between the associated time-frequency translates of any nontrivial Schwartz function. If mild Diophantine properties are satisfied, we also prove linear independence in the category of L2(R) functions.

Proof. We will prove the general case of conditional dependence measure since the other case follows trivially as a special case when Z = ∅. The kernel-free property of the dependence measures is used to prove the result. The proof essentially uses change of variables formulas for transformation of random variables. From Theorem 1, we have

We give a new proof of the fact that the vanishing of generalized Wronskians implies linear dependence of formal power series in several variables. Our results are also valid for quotients of germs of analytic functions.

We present a quantitative analysis of recent data on the kinetics of ATP hydrolysis, which has presented a puzzle regarding the load dependence of the Michaelis constant. Within the framework of coarse grained two-state ratchet models, our analysis not only explains the puzzling data, but provides a modified Michaelis law, which could be useful as a guide for future experiments.

Permalloy antidot arrays with different square hole sizes 120