Weak vs Strong Dependency - Mathematics Stack Exchange In probability, weak dependence of random variables is a generalization of independence that is weaker than the concept of a martingale A (time) sequence of random variables is weakly dependent if distinct portions of the sequence have a covariance that asymptotically decreases to 0 as the blocks are further separated in time
What exactly does linear dependence and linear independence imply . . . A broader perspective on linear dependence is the theory of relations in group theory Roughly speaking, a relation is some equation satisfied by the elements of a group, e g $(ab)^{-1}=b^{-1}a^{-1}$; relations basically amount to declaring how group elements depend on each other
Is there such a thing as quadratic independence (and higher . . . There is also a general concept of dependence relations which includes linear dependence and algebraic dependence as special cases Haven't searched online but you can find it in N Jacobson, Basic Algebra vol II, section 3 6
vectors - Is there any difference between linear dependence . . . Given another vector which is in the span of these vectors, it is "coplanar" with them (in the same plane) So being coplanar does mean linear dependence (to the basis of a given plane) Colinear is the same idea but more general, the dependence doesn't have to be in a plane, it can be a hyperplane etc
partial differential equations - Continuous dependence on parameters . . . "Linear PDEs" is already a very large class, but if you restrict to a well-understood class of problems with established estimates then you can often get as-smooth-as-possible dependence on the coefficients Here's a very rough sketch of how to proceed for a simple example:
linear algebra - Row operations do not change the dependency . . . Dependence between two columns is a very special case, as you point out, with one column being a scalar multiple of the other The general case would be a linear combination of columns equal to a zero column without all the coeffcients being zeros $\endgroup$