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If a Lyapunov function is nonincreasing on a sublevel set, trajectories cannot cross its boundary outward.

A positive-definite quadratic form defines ellipsoidal sublevel sets and often turns a polynomial vector field into an exact factored orbital derivative.

An equilibrium has asymptotic stability when it is Lyapunov stable and every trajectory starting sufficiently nearby converges to it. Stability controls how far a small perturbation can wander; attraction supplies the eventual return. The definition does not require an exponential decay rate.

For $\dot{x}=-x^3$, the solution from $x_0$ is $x(t)=x_0/\sqrt{1+2x_0^{2}t}$. Its magnitude never increases and tends to zero, so the origin is asymptotically stable. The decay is algebraic rather than exponential, showing why exponential stability is a stronger condition. A center equilibrium provides a different comparison: neighboring trajectories stay close but circulate indefinitely instead of returning to the origin. These distinctions help interpret a measured response whose amplitude changes slowly or whose observation interval is too short to establish its eventual limit.

A positive-definite Lyapunov function with strictly negative orbital derivative away from the equilibrium proves asymptotic stability.

A positive-definite Lyapunov function with nonpositive orbital derivative proves Lyapunov stability of the equilibrium. Indeed, fix a sufficiently small ball $B_{\epsilon }$ in the function's domain and putContinuity and $V(0)=0$ give a ball $B_{\delta }$ on which $V<\alpha$. Since $V$ cannot increase along a trajectory, one starting in $B_{\delta }$ cannot cross the sphere $\Vert x\Vert =\epsilon$, on which $V\ge \alpha$. It therefore remains in $B_{\epsilon }$ for all forward time.

An equilibrium is Lyapunov stable if every neighbourhood contains a smaller neighbourhood whose forward trajectories remain in the original neighbourhood for all time.

Expansion reduces bolometric flux by one factor of $1+z$ from photon energy and one from arrival rate, in addition to inverse-area dilution.

The nonrelativistic limit takes characteristic speeds much smaller than $c$. Lorentz transformations then reduce to Galilean transformations at leading order.

The Lorentz group consists of real matrices preserving the Minkowski metric: ${\Lambda }^T\eta \Lambda =\eta$.

To first order around flat spacetime,and

In four dimensions, the trace-reversed perturbation isTrace reversal simplifies the linearized Einstein equations.

The linearized Riemann tensor defines an operator on two-forms byUnlike the metric perturbation itself, this operator is invariant under linearized gauge transformations.

For a differential equation $\dot{x}=f(x)$ with an equilibrium point $x_{\ast }$ and differentiable $f$, write $x=x_{\ast }+\eta$. The first-order Taylor expansion givesThe linearization discards the smaller remainder and uses the Jacobian matrix $Df(x_{\ast })$ to study small perturbations. Its eigenvalues predict exponential decay or growth when their real parts are nonzero.

For example, $\dot{x}=-x+x^3$ linearizes at zero to $\dot{\eta }=-\eta$, predicting local attraction. A zero eigenvalue makes the linear test inconclusive: $\dot{x}=-x^3$ and $\dot{x}=x^3$ have the same zero linearization but opposite stability behavior. Thus a linearization of a dynamical system supplies a local approximation with hypotheses, rather than a global replacement for the nonlinear dynamics.

An affine recurrence on a finite-dimensional vector space has the form $x_{n+1}=A{x}_n+b$, where $A$ is a fixed linear operator and $b$ a fixed vector. Induction givesThe sum is empty at $n=0$. If $I-A$ is invertible, the unique fixed point is $x_{\ast }=(I-A{)}^{-1}b$, and subtracting it reduces the recurrence to $x_n-x_{\ast }=A^n(x_0-x_{\ast })$. The sum formula remains valid when $I-A$ is singular, even when no fixed point exists. For example, $A=I$ gives the arithmetic progression $x_n=x_0+nb$. A stable affine recurrence additionally has spectral radius below one and converges to its fixed point.

For a sequence $(u_n)$, its ordinary generating function is the formal power seriesA constant-coefficient linear recurrence relation turns into an algebraic equation for $G$ after multiplication by powers of $x$ and summation; the initial values supply the correction polynomial. For example, if $u_0=a$, $u_1=b$ and $u_{n+2}=u_{n+1}+u_n$, thenHence $G$ is a rational function, interpreted as a formal power series at zero. No analytic convergence is needed. Variable coefficients can instead lead to operations such as differentiation, so the constant-coefficient hypothesis is part of this rational-function conclusion.

The simplex method solves a standard-form linear program by changing a simplex basis. A feasible basis expresses basic coordinates and the objective in terms of nonnegative nonbasic variables. An improving reduced cost selects a candidate entering variable, and the simplex ratio test chooses the largest step preserving nonnegative basic coordinates.

If this step is positive, the objective improves along an edge. Under degeneracy in linear programming, the step can be zero: the basis changes while the basic feasible solution and objective remain fixed. A negative reduced cost alone therefore does not promise strict improvement in that pivot. If the improving direction has no finite ratio limit, it gives an unbounded feasible ray.

When all reduced costs have the optimal sign, the simplex optimality criterion proves global optimality. Two-phase simplex supplies an initial feasible basis or an infeasibility certificate. A rule such as the Bland pivoting rule prevents cycling; arbitrary improving pivot choices need not terminate, and even terminating rules can require exponentially many steps on a Klee-Minty cube.

Linear programming duality pairs the standard maximization problemwith the minimization problemThe dual variables form nonnegative combinations of primal inequalities. If $x$ and $y$ are feasible, thenThis is weak duality, giving independently checkable bounds before any algorithm reaches an optimum.

For a feasible primal with finite optimal value, strong duality for finite-dimensional linear programs ensures attained primal and dual optima with equal values. Their objective difference decomposes into nonnegative products of variables and opposite slacks, yielding complementary slackness. The worked allocation in linear programming has primal point $(2,2)$ and dual multipliers $(2,1,0)$, both with value ten. Sign conventions change for minimization, equality rows and unrestricted variables; a dual formula belongs to its declared primal form.

Linear-fractional programming minimizes or maximizes a ratio of affine functions over a linear polyhedron, with the denominator required to be positive:For a threshold $t$, the sublevel condition is equivalent to the linear inequality $(c-td{)}^{T}x\le t\beta -\alpha$. Its sublevel sets are therefore convex, although the ratio need not be a convex function. The positivity restriction makes this multiplication preserve the inequality direction.^[ref]

The Charnes-Cooper transformation rescales by the denominator and introduces one extra variable to produce a linear program. Positive transformed scale gives an actual original feasible point; zero scale can encode a recession limit and must be interpreted separately when discussing attainment.

For a full-row-rank system $Ax=b$ with $m$ rows and variables $x\ge 0$, choose $m$ linearly independent columns as a simplex basis. Set all nonbasic variables to zero and solve $x_B=A_B^{-1}b$. If $x_B\ge 0$, the resulting vector is a basic feasible solution. Redundant equality rows can first be removed.

Equivalently, a feasible vector is basic when the columns indexed by its strictly positive coordinates are linearly independent. Those columns can be extended to a basis of the full column space because $A$ has row rank $m$. The extension may have zero-valued basic variables; this is degeneracy in linear programming.

For $x_1+x_2=1$ and $x_1,x_2\ge 0$, the two basic feasible solutions are $(1,0)$ and $(0,1)$. The feasible midpoint $(1/2,1/2)$ is not basic: its two positive columns are dependent. In this standard form, basic feasible solutions are exactly the extreme points of the feasible polyhedron, connecting basis algebra to the Fundamental theorem of linear programming.

For finite second moments and $\mathrm{Var}(X)>0$, the residual variance after affine projection onto one predictor $X$ is $\mathrm{Var}(Y)-\mathrm{Cov}(X,Y{)}^2/\mathrm{Var}(X)$, a covariance Schur complement.