Every injection from an ordinal into a set $X$ has domain strictly below the Hartogs number $h(X)$. Therefore all partial injections with ordinal domain and range in $X$ form a set of graphs contained in $\mathcal{P}(h(X)\times X)$. Under extension, the union of a chain remains a partial injection of this kind. Zorn's lemma then supplies a maximal one, and extending at an unused range element shows that it must be a bijection onto $X$. This gives the well-ordering theorem with the set-sizedness of the poset made explicit.

The semimembranosus is a medial hamstring muscle contributing to hip extension, knee flexion and medial rotation of the flexed leg. Its tendon also gives an expansion associated with the oblique popliteal ligament. One muscle can therefore contribute to both active motion and passive support.

Gum arabic is a gum obtained from certain acacia relatives, used for its dispersion and stabilizing properties. It is a complex biological material rather than one molecular species.

Mistura Guaiaci is a guaiacum-resin mixture. The dispersed resin and aqueous vehicle have different physical properties.^[ref]

For a nonempty inclusion chain of group operations on subsets of a fixed set, the union is a group operation on the union of their carriers. Any finite list of carrier elements lies in one chain member, so its multiplication and associativity agree there. Nested identities coincide because the only idempotent of a group is its identity; inverses remain valid in larger members. Thus the union supplies the chain upper bound required by Zorn's lemma.

Let $x$ have two nonadjacent neighbours $y,z$ in a finite graph $G$, with $G-\{y,z\}$ connected. Then $\chi (G)\le \Delta (G)$. Give $y,z$ one common colour. In a spanning tree of the remaining graph rooted at $x$, order vertices from larger to smaller distance from the root and colour greedily in that order. Every vertex before $x$ has an uncoloured parent, so at most $\Delta (G)-1$ colours are forbidden there. At $x$, the two initial neighbours use the same colour, so again at most $\Delta (G)-1$ distinct colours are forbidden. This constructs a proper colouring from a palette of $\Delta (G)$ colours. It is a useful local step in Brooks' theorem.

For a linearly independent ordered family in a real inner product space, require each unit vector $w_j$ to lie in the span of the first $j$ inputs and be orthogonal to all earlier inputs. That orthogonal part of the successive span is one-dimensional. Subtracting the earlier orthonormal components gives its nonzero generator $r_j$. Every possible unit choice is therefore $\pm r_j/\Vert r_j\Vert$. Since $⟨r_j,u_j⟩=\Vert r_j{\Vert }^2>0$, the positive normalization fixes the sign uniquely. This distinguishes uniqueness of the normalized ordered construction from the many possible orthonormal bases of the whole space.

A matrix and its transpose have the same characteristic polynomial, allowing two independent Gershgorin eigenvalue counting arguments for one spectrum. If isolated row discs certify simple eigenvalues near one group of centres and isolated column discs certify the remaining ones, pairwise separation of all the certified small discs establishes a simple spectrum even if neither full family has pairwise disjoint discs.

For example, let $n=2m$, the diagonal centres be $1,\dots ,n$, and all adjacent off-diagonal entries have magnitude strictly below $1/4$. Add a corner entry $a_{n1}=c$. The row radii are below $1/4,1/2,\dots ,1/2,|c|+1/4$; the column radii reverse the two endpoint bounds. If $|c|\le m-3/4$, rows isolate centres $1,\dots ,m$ and columns isolate centres $m+1,\dots ,n$. The nearest potentially enlarged-disc contact has centre separation $m$, while the radius sum is strictly less than $|c|+3/4\le m$. Each of the $n$ certified small discs contains one simple eigenvalue. For real entries these eigenvalues are real, since a nonreal conjugate pair would put two roots in the same conjugation-invariant isolated disc.

This is a sufficient uniform certificate. Overlap beyond this bound does not prove repeated eigenvalues; individual matrices can have simple real spectra well beyond a Gershgorin disc separation bound.

Give the geometric realization of an abstract simplicial complex the quotient topology obtained by gluing the disjoint union of its standard simplexes along their labelled faces. A subset $U$ is open exactly when $U\cap |\sigma |$ is open in every closed geometric simplex $|\sigma |$. This is the weak realization topology. It is the topology of its CW complex structure; the word weak here refers to the simplexwise test, rather than to the weak topology of a topological vector space. A map out of the realization is continuous exactly when its restriction to every simplex is continuous.

Represent a point by finitely supported barycentric coordinates ${\lambda }_v\ge 0$ with ${\sum }_v{\lambda }_v=1$ and support a simplex. Restricting the displayed Euclidean metric on these coordinates gives a metric realization. The coordinate-sum is finite because both points have finite support. For a finite simplicial complex, this agrees with the weak topology of a simplicial realization: the coordinate map from the finite quotient realization is a continuous bijection from a compact space to a Hausdorff space, hence a homeomorphism. For an infinite complex the two topologies can differ, as the countable-star examples demonstrate.

Suppose reflection in a plane is an isometry of a smooth surface, and its intersection with the plane is a regular curve. That curve is a geodesic when parametrized by arclength. At any intersection point, reflection fixes both the point and every tangent vector to the intersection. It carries the geodesic with these initial data to one with the same data. Local uniqueness of the geodesic equations forces it to remain in the fixed plane. This is the one-dimensional instance of fixed-point sets of Riemannian isometries being totally geodesic.

For a band matrix, elimination without pivoting retains bounded lower and upper bandwidths in its triangular factors. A tridiagonal system therefore needs only linear work and storage, as in the Thomas algorithm. Pivoting can change the precise fill pattern and must be included in the bandwidth and cost analysis when required. The inverse need not be banded, so solving through the factors preserves sparsity that explicit inversion would lose.

A gas law relates equilibrium variables such as pressure, volume, temperature and amount of substance. The ideal gas law $pV=nRT=N{k}_{B}T$ provides a useful dilute-gas model. Holding temperature and particle number fixed predicts $p\propto 1/V$; holding pressure and particle number fixed predicts $V\propto T$ with absolute temperature.

A piston and cylinder provide a direct comparison: vary one controlled variable slowly enough for the gas to equilibrate, then record the remaining pair. The constraint is part of the law being tested; rapid compression can heat the gas and need not follow an isothermal curve. At larger density, real gas effects require a more suitable equation of state. A gas law describes possible equilibrium states, while the first law of thermodynamics additionally determines the heat and work along a specified process.

A historical physical name for a gas or vapour, emphasizing its mobility and appreciable compressibility. It does not imply an exactly ideal gas or prevent a vapour from undergoing condensation.

A galvanometer using oppositely directed, nearly equal magnetic dipole moments to reduce the net restoring torque of the approximately uniform terrestrial field. It still needs a weak restoring mechanism for a reproducible deflection; it does not shield the needles from that field.

For fixed mean turn length and fixed conductor volume, a coil of $N$ turns has conductor length proportional to $N$ and cross-sectional area proportional to $1/N$. Its electrical resistance is therefore $R_{\mathrm{coil}}=a{N}^2$. With source electromotive force $\mathcal{E}$ and internal resistance $R_s$, suppose the small-deflection driving torque is proportional to $NI=N\mathcal{E}/(R_s+a{N}^2)$. Its derivative has the sign of $R_s-a{N}^2$, so the maximum occurs at $R_{\mathrm{coil}}=R_s$. Geometry, damping and finite response time can alter the practical optimization; this is a conditional winding model, not a universal rule for all galvanometers.

A current-measuring coil and compass arranged so that the coil's central magnetic field is perpendicular to the horizontal terrestrial field. With unchanged coil geometry and terrestrial field, the electric current ratio is the ratio of the tangents of the needle deflections. A small needle and controlled alignment make the central-field approximation useful.

The set of database attributes determined by $X$ under given functional dependencies. Start from $X$ and repeatedly add the right-hand side of each functional dependency whose left-hand side is already included.

A full ternary tree whose leaves all have the same depth. At depth $j$ there are $3^j$ vertices, so a tree with $h+1$ levels has $N_h={\sum }_{j=0}^h{3}^j=(3^{h+1}-1)/2$ vertices and $3^h$ leaves, by the finite geometric series formula. For a rose tree representation, recursively flattening children and joining their lists with ordinary list concatenation copies each label once per ancestor. The work is therefore $\Theta ((h+1)3^h)=\Theta (N_h\log N_h)$ when the complete output is demanded. Passing an already constructed suffix through the recursion instead emits each label once, giving $\Theta (N_h)$ work.

Let $f$ be a Borel measurable function in L1 space on $\mathbb{R}$, and let $a,b$ be measurable real functions of $x$ with $b(x)>0$ and $W={\int }_{\mathbb{R}}|a(x)|/b(x)dx<\infty$. The joint integrand $G(x,y)=a(x)f(b(x)y)$ is measurable, and Tonelli's theorem followed by Lebesgue substitution on an interval givesFor each fixed $x$, substitute $z=b(x)y$ in the inner $y$ integral. The absolute Jacobian is $1/b(x)$, leaving $|a(x)|\Vert f{\Vert }_1/b(x)$. Integrating in $x$ proves the formula and absolute integrability. Fubini's theorem then permits both orders of signed integration, givingThe one-dimensional sections are integrable for almost every value of the other variable; the theorem does not assert integrability of every section.