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After fixing the two cyclic angular momenta of a heavy symmetric top, its inclination has an effective potential $U(u)$ with $u=\cos \theta$. Write the reduced energy equation in the form $P(u)=2A(E-U(u))(1-u^2)$. A steady nonvertical inclination has $E=U(u_0)$ and $U^{\prime }(u_0)=0$, the latter coming from the reduced second-order Euler-Lagrange equation. Therefore $P(u_0)=P^{\prime }(u_0)=0$.

This proves the double-root condition without dividing by a velocity that vanishes in the very solution under study. A simple turning point has $P(u_0)=0$ but nonzero derivative, and is not a steady inclination. Exactly vertical spinning corresponds to $u=\pm 1$, where the inclination coordinate and its reduced denominator are singular; these endpoint configurations require a separate treatment and do not supply an interior double root.

On the real line, let $T(x)=\max (1-|x|/2,0)$. The Fourier transform of a triangular function gives $\hat{T}(\omega )=2{\sin }^2\omega /{\omega }^2$, with value two at zero. The Gaussian heat kernel has transform $e^{-t{\omega }^2}$. Hence $u=q_t\ast T$ solves the heat equation with initial profile $T$, and the inverse Fourier transform gives the displayed integral. Differentiating for positive time is justified by Gaussian decay. The uniform continuity of $T$ and the vanishing Gaussian mass outside any fixed neighborhood of zero prove uniform recovery of $T$ as $t\downarrow 0$. Convolution preserves total integral two, keeps the solution nonnegative, and smooths the corners for every positive time. Unlike the finite-interval problem in heat flow from a triangular initial temperature, this evolution has no endpoints through which heat can escape.

For thermal diffusivity $\kappa >0$ and thermal conductivity $k>0$, a semi-infinite conductor satisfies $u_t=\kappa u_{xx}$ for $x>0$. Take zero initial temperature and constant incoming heat flux $-k{u}_x(0,t)=Q$. Select the physical solution by boundedness at spatial infinity together with sufficient growth control for the time Laplace transform.

Writing the time transform as $U$ gives $U_{xx}=(p/\kappa )U$. The decaying branch is $U=A(p)e^{-x\sqrt{p/\kappa }}$, where $\mathrm{Re}\sqrt{p}>0$. The transformed flux condition gives $A=Q\sqrt{\kappa }/(k{p}^{3/2})$. Since $\mathcal{L}[2\sqrt{t}]=\sqrt{\pi }{p}^{-3/2}$, inversion gives the displayed surface temperature. More generally the full profile isDifferentiation verifies the heat equation and the applied flux; for every $x>0$ it tends to zero at initial time and decays at spatial infinity. Its surface temperature grows as $\sqrt{t}$, because the depth significantly heated by diffusion has the same scale.

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.

Historical formulation name; composition is identified by pharmacopoeial edition.

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 relation among quantities such as pressure, volume, temperature and amount of substance for a gas.

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.