An equilibrium point is asymptotically stable if it is Lyapunov stable and nearby solutions converge to it as time tends to infinity. See the definition.

An equivalence relation is a relation that behaves like equality: it is reflexive, symmetric and transitive.

Multiplication of natural numbers is the operation that sends two natural numbers $a$ and $b$ to their product $a \cdot b$. The product $a \cdot b$ is the sum of $b$ copies of $a$.

The linear span of a set of vectors is the set of all linear combinations of the vectors. It is the smallest subspace that contains the set.

The integral of a sum is the sum of the integrals. Constant factors can move out of the integral.

The complement of an or is the and of the complements, and the complement of an and is the or of the complements. Not (rain or snow) means no rain and no snow.

Sometimes the negation of the conclusion gives you more to work with. "If $n^2$ is even, then $n$ is even" is hard to start. "If $n$ is odd, then $n^2$ is odd" is easy, and it says the same thing.

A self-adjoint map is a pure stretch along perpendicular axes. The axes are the eigenvectors, and the stretch factors are the real eigenvalues. Every symmetric real matrix is diagonal in a suitable orthonormal basis.

The general solution of the homogeneous equation is $c_1 y_1 + c_2 y_2$ with constants. The idea is to let the constants vary with $x$. A smart extra condition makes the equations for $u_1'$ and $u_2'$ simple.

The characteristic polynomial contains the trace as its second coefficient, with a minus sign. For a $2 \times 2$ matrix, the full polynomial is $t^2 - (\text{trace})\,t + \text{determinant}$. So the trace and the determinant are part of the information of the characteristic polynomial.

Newton's identities convert between two ways of describing the roots of a polynomial: the sums of their powers, and the coefficients. Each identity gives the next quantity from the earlier ones.

A jump discontinuity is a point where both one-sided limits of a function exist but are different. The graph jumps from one height to another there. The node jump discontinuity gives the precise definition.

A primitive polynomial is a polynomial whose coefficients have no common factor except units. The definition of a primitive polynomial gives the condition.

Once you know the smallest solution, you get all the others by taking powers. Expand $(x_1 + y_1 \sqrt{d})^n$ and read the integer part and the coefficient of $\sqrt{d}$.

Insert a new element into a permutation written as cycles. It forms a new cycle of length $1$, or it goes into an existing cycle right after one of the $n$ old elements.

Differentiation respects sums and constant factors. So we can differentiate a sum term by term.

A norm on a real or complex vector space is a function that gives each vector a length. It satisfies the triangle inequality and absolute homogeneity, and only the zero vector has length zero.

Friction takes energy from the oscillating mass. With little friction, the mass still oscillates, but the swings get smaller. With much friction, the mass creeps back to rest without any swing.

A proof is a chain of small steps, and each step is certainly correct. It starts from things that you accept or already know, and it ends at the statement. After a correct proof, the statement is true in every case, not only in the cases that you tested.

Taking the orthogonal complement twice gives back the original subspace. The complement of the complement of a line is the same line.