A hypercube is the Cartesian product of $n$ equal-length closed intervals in Euclidean space. The centered cube $[-1,1{]}^n$ has $2^n$ vertices, each with coordinates in $\{-1,1\}$. Antipodal vertices determine $2^{n-1}$ body diagonals.

A geometric locus is the set of points satisfying a specified geometric or algebraic condition in an ambient space. For example, points at a fixed positive distance $r$ from $p$ form a circle in the plane and a sphere in three dimensions. In the complex plane, $|z-z_0|=r$ has Cartesian equation $(x-\mathrm{Re}{z}_0{)}^2+(y-\mathrm{Im}{z}_0{)}^2=r^2$, proving the circle description. A locus can also be a union, such as $x(y-1)=0$, which consists of two lines; domain restrictions in the original equation must be retained when simplifying it.

The Euclidean group $E(n)=O(n)⋉{\mathbb{R}}^n$ is the group of all isometries $x\mapsto Ax+b$ of Euclidean space, with $A$ an orthogonal matrix. Its orientation-preserving subgroup is $SE(n)=SO(n)⋉{\mathbb{R}}^n$, the semidirect product of the special orthogonal group with translations. The Euclidean motion of Euclidean three-space and the proper Euclidean motion of Euclidean three-space are the respective three-dimensional instances.

This surface consists of rays from a vertex $\mathbf{a}$ whose directions make a fixed angle $\beta$ with a unit axis vector $\hat{\mathbf{n}}$. The displayed equation follows from the dot product formula for that angle. For $0<\beta <\pi$, $\beta \ne \pi /2$, it gives one circular nappe; $\beta =\pi /2$ gives a plane, and $\beta =0,\pi$ gives a degenerate ray. Squaring the equation produces both nappes, so the unsquared sign distinguishes them. Generalized angular momentum in a radial monopole field confines particle directions to such a surface.

Figures are congruent when a rigid motion maps one onto the other, preserving all Euclidean distances.

Euclid's Elements is a thirteen-book mathematical work organized through definitions, postulates and proofs. Its topics include plane and solid Euclidean geometry, proportions and number theory. For example, Book I ends with the Pythagorean theorem and its converse, while Book IX includes a construction of even perfect numbers. Its enduring feature is the attempt to derive many geometric statements from a small explicit base of assumptions. Modern axiomatizations make additional assumptions about order, continuity and incidence explicit.^[ref]

In the Euclidean plane, precisely one line through a point outside a given line is parallel to it. This formulation is equivalent to Euclid's fifth postulate within the usual axioms of plane geometry.

A compass-and-straightedge construction uses an unmarked straightedge to draw a Euclidean line through two already constructed points, and a compass to draw a circle with a constructed centre and radius. New points arise from intersections of these lines and circles. The allowed operations matter: an accurate numerical drawing need not supply such a construction.

An ester consumed in the conductometric saponification assay, where hydrolysis under basic conditions produces ethanol and an acetate ion.

Ethyl acetoacetate is $\mathrm{C}{\mathrm{H}}_3\mathrm{COC}{\mathrm{H}}_2\mathrm{COOC}{\mathrm{H}}_2\mathrm{C}{\mathrm{H}}_3$, a beta-keto ester. Its central methylene group lies between two carbonyl-derived groups.

Delocalization stabilizes the corresponding enolate, making that carbon more acidic than an ordinary alkane carbon. This supports controlled carbon-carbon bond formation and subsequent transformations of the ester functionality.

For an absolutely irreducible real symmetry representation, an equivariant steady-state bifurcation with a simple transverse crossing of the scalar linear eigenvalue generically has an equilibrium branch for each axial isotropy type. Restricting the vector field to its one-dimensional fixed-point subspace reduces the existence problem to a scalar bifurcation equation. The lemma guarantees these branches; it does not exclude branches with smaller isotropy or higher-dimensional fixed-point subspaces.

At a nondegenerate equivariant Hopf bifurcation, each complex axial isotropy subgroup of the spatial group combined with temporal phase gives a branch of periodic solutions with that spatiotemporal symmetry. Assumptions include an isolated semisimple imaginary critical pair of the required representation type, transverse parameter crossing, and a nonzero cubic coefficient on the fixed-point plane. The statement guarantees existence and symmetry type, not stability or which parameter side contains every branch.

For the natural action of all signed coordinate permutations on ${\mathbb{R}}^3$, a cubic equivariant dynamical system has componentsReflecting $x_i$ alone reverses component $i$, so that component must be an odd function of $x_i$. Reflecting any other coordinate leaves component $i$ fixed, so it must be an even function of each other coordinate. Consequently the only cubic monomials are $x_i^3$ and $x_i{x}_j^2$. Permuting the other coordinates makes their coefficients equal; permuting all coordinates gives the same pair $a,b$ in every component. The same parity test excludes every quadratic term.

A branch with exactly $m\in \{1,2,3\}$ nonzero coordinates of common magnitude $r$ satisfiesThus its nonzero states require $r^2=-\mu /[a+(m-1)b]>0$, provided the denominator is nonzero. The three cases describe coordinate-axis, face-diagonal and body-diagonal states. If the denominator vanishes, the cubic truncation cannot determine that branch's amplitude; higher-order terms must be retained. Existence of a branch does not by itself determine its stability.

Let a translation symmetry with displacement $s\in {\mathbb{R}}^d$ act on nonzero complex Fourier modes byEach phase changes by $q_j\cdot s$. Therefore a real linear combination of locally unwrapped phases, ${\sum }_j{n}_j{\theta }_j$, is unchanged exactly when ${\sum }_j{n}_j{q}_j=0$. This is a kernel of a linear map calculation for the matrix whose columns are the wavevectors.

A globally defined expression on angular phases requires more care: $e^{i{\sum }_j{n}_j{\theta }_j}$ is independent of the representatives chosen modulo $2\pi$ precisely when every $n_j$ is an integer. The globally defined translation-invariant characters therefore satisfy both $n\in {\mathbb{Z}}^m$ and ${\sum }_j{n}_j{q}_j=0$. An arbitrary real vector in the matrix's kernel of a linear map gives a local unwrapped invariant, but need not give a single-valued function of the angular phases.

For example, take four planar wavevectors $q_1=(1,0)$, $q_2=(0,1)$, $q_3=(1,1)$ and $q_4=(1,-1)$. The two independent angular combinations are ${\theta }_3-{\theta }_1-{\theta }_2$ and ${\theta }_4-{\theta }_1+{\theta }_2$, each interpreted modulo $2\pi$. Translations can set the first two phases to zero; the other two then equal these combinations.

More generally, when four planar wavevectors span a common reciprocal lattice of rank two, their translation action has a closed two-dimensional torus as its image and the quotient of the four-phase torus is two-dimensional. Real rank two alone guarantees only two local independent combinations; without a commensurability condition, the translation image can be dense in a larger torus. At zero amplitude, a phase is undefined, so smooth centre manifold equations should first use real parts and imaginary parts of the complex amplitudes.

The strong equivalence principle extends universality of free fall to bodies with appreciable gravitational self-energy and requires local experiments, including gravitational ones, in a freely falling laboratory to be independent of the laboratory's location and velocity. This statement neglects external tidal variation across the laboratory; it does not remove Riemann curvature tensor. In general relativity, universal metric coupling, fixed local gravitational coupling and the absence of extra preferred-frame gravitational fields implement this principle.

Freely falling test particles with identical initial position and velocity follow the same trajectory independently of their material composition. The principle neglects appreciable self-gravity and nongravitational forces. The Einstein equivalence principle extends the local statement to nongravitational experiments; the strong equivalence principle also includes gravitational experiments and self-gravitating bodies.

The Einstein equivalence principle combines the weak equivalence principle with local Lorentz invariance and local position invariance of nongravitational experiments. In a sufficiently small freely falling laboratory, their results agree with special relativity independently of its location and velocity. The strong equivalence principle extends this claim to local gravitational experiments.

The prescription replaces partial derivatives in flat-spacetime matter equations by the Levi-Civita connection and the Minkowski metric by the curved metric tensor. It implements minimal gravitational coupling and reproduces flat-spacetime physics at a point in a local inertial frame. It is not a unique rule for rearranged higher-derivative equations: commuting partial derivatives before replacement can conceal a Riemann curvature tensor term afterwards. The derivative-order ambiguity in covariant Maxwell potential equations is an explicit example. The equivalence principle removes uniform gravitational acceleration locally but does not erase tidal curvature over a finite experiment.

Linear stability studies the leading evolution of a small perturbation around a specified solution. Near an equilibrium $x_{\ast }$ of $\dot{x}=f(x)$, write $x=x_{\ast }+\delta x$. Dropping terms quadratic and higher in the perturbation gives $\dot{\delta x}=J\delta x$, where $J$ is the Jacobian matrix $Df(x_{\ast })$.

An eigenvalue with positive real part gives a growing mode. If every eigenvalue has negative real part, sufficiently small perturbations decay and the smooth nonlinear equilibrium is locally asymptotically stable. Eigenvalues on the imaginary axis require additional analysis: nonlinear terms can change the outcome, and repeated neutral eigenvalues can have algebraic growth even in a linear system. The test also describes local behavior; a nearby stable equilibrium can coexist with a different remote attractor. State, parameter values and perturbation class are therefore part of a stability claim.

A meridian transit is a direction's crossing of the observer's celestial meridian. Upper meridian transit occurs at hour angle $0$ and gives the greatest geometric altitude in the fixed-direction model; lower meridian transit occurs at hour angle $12^{\mathrm{h}}$ and gives the least. Either crossing can lie below the astronomical horizon. ^[ref].