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An inertia tensor can be isotropic even when the body is not a sphere. An example is a uniform thin disc of radius $2a$ and mass $M$, with a thin rod of length $\sqrt{3}a$ and mass $2M$ attached normally at its center. The combined center of mass is one third of the rod length from the disc. Standard disc and rod moments and the parallel axis theorem give all three principal moments equal to $2M{a}^2$, so $J=2M{a}^{2}I$. The angular momentum is therefore a fixed scalar times the angular velocity. In torque-free motion, conservation of the inertial angular-momentum vector makes the angular-velocity vector constant for every initial spin. Rotational symmetry of the shape is not required; equality of its second moments is enough.

For linear interpolation at zero and $h>0$, an error $\delta$ in the second sampled value, with the first exact, changes the linear interpolant by $\Delta p(x)=\delta x/h$. On $[0,h]$, its uniform norm is exactly $|\delta |$, whereas the derivative change is the constant $\delta /h$. With independent absolute bounds $|{\delta }_0|,|{\delta }_1|\le \delta$ on both samples, the value change remains at most $\delta$ and the derivative change is at most $2\delta /h$, attained for opposite endpoint errors.

Thus bounded absolute sample noise can remain bounded under value interpolation while becoming unbounded under numerical differentiation as $h\to 0$. Fixed nonzero sample noise also prevents exact convergence of the value approximation; stability against perturbations and convergence to exact data are different assertions. For a second-sample error $aϵ(h)$ with fixed $a>0$, value convergence requires $ϵ(h)\to 0$, while derivative convergence requires $ϵ(h)=o(h)$ in little o notation, in addition to vanishing truncation error.

Equal volumes of ideal gases at the same temperature and pressure contain equal amounts of substance. Named after Amedeo Avogadro. From $pV=nRT$, $V/n=RT/p$ is common to them.

A hyperelastic material is homogeneous when its strain energy density $W(F)$ has no explicit dependence on the reference configuration position. Its first Piola-Kirchhoff stress tensor is $T=DW(F)$. Homogeneity of the material does not require the deformation gradient to be spatially constant.

For a material isotropy and material frame indifference strain energy density $W(F)=h(I_1,I_2,I_3)$, take $B=F{F}^T$, $I_1=\mathrm{tr}B$, $I_2=((\mathrm{tr}B{)}^2-\mathrm{tr}{B}^2)/2$, $I_3=\det B=J^2$, $J=\det F>0$. The chain rule and $d(\det F)=J{F}^{-T}:dF$ give $DW=2h_{1}F+2h_2(I_{1}I-B)F+2I_3{h}_3{F}^{-T}$. Multiplying by $F^T/J$ yields the displayed Cauchy stress tensor. Material isotropy allows an identity term and a quadratic term in $B$, as well as the term proportional to $B$.

An unconstrained natural state is a stress-free homogeneous deformation gradient. For $J>0$, its equivalent conditions are $DW(F)=0$ and $J^{-1}DW(F)F^T=0$. An interior differentiable minimum of the strain energy density is a natural state, although a stationary state need not minimize the energy. Constraints require allowing their reaction stress; the unconstrained equivalence must then be modified. Under strict Baker-Ericksen inequalities, an isotropic natural state is a dilation times a proper rotation.

On positive-determinant deformation gradients, the first Piola-Kirchhoff stress is $\mu (F-J^{-1}{F}^{-T})$. Its zero set is $SO(3)$: multiplication by $F^T$ gives $F{F}^T=J^{-1}I$, and determinants give $J^5=1$. These are also the global minimizers, since $|F{|}^2\ge 3J^{2/3}$ and $3J^{2/3}/2+J^{-1}$ has its unique minimum $5/2$ at $J=1$. The minimum energy is $5\mu /2$ rather than zero; an additive constant changes no stress.

A hyperelastic energy well is a set of deformation gradients minimizing the stored-energy function at a fixed temperature. Material frame indifference makes each well invariant under left multiplication by proper rotations. A phase-changing crystal can have a single austenitic rotational well at high temperature and several martensitic wells below a transition. Continuous deformations cannot freely join arbitrary wells: planar interfaces require a rank-one connection between the selected gradients.

The four-dimensional hypercube contains a graph subdivision of $K_{3,3}$, although no hypercube contains an ordinary $K_{3,3}$ subgraph: two distinct cube vertices have at most two common neighbors. One subdivision uses branch sets $\{0000,0011,1100\}$ and $\{0001,0010,0100\}$. Six connections are direct; the remaining paths are $0011\to 0111\to 0110\to 0100$, $1100\to 1101\to 0101\to 0001$, and $1100\to 1110\to 1010\to 0010$. Their internal vertices are distinct. Appending zero coordinates embeds the construction in every higher-dimensional cube, proving nonplanarity.

A group homomorphism from ${\mathbb{Z}}_m$ to the additive cyclic group ${\mathbb{Z}}_n$ is determined by the image $a$ of $1$, subject to $ma\equiv 0(\mathrm{mod}n)$. Conversely every such $a$ gives a well-defined map $k\mapsto ka$. If $d=\gcd (m,n)$, the condition is equivalent to $n/d$ dividing $a$. There are exactly $d$ such residues modulo $n$, proving the formula. More generally, homomorphisms from ${\mathbb{Z}}_m$ to any group correspond to target elements whose order of a group element divides $m$.

Take precisely the propositional tautologies as axioms and supply no inference rules. A one-line proof then establishes exactly the valid propositional formulas, so this calculus is sound and complete for theoremhood. Its axiom set is decidable by finite truth tables. This construction does not by itself give strong completeness for consequence from arbitrary assumptions.

One presentation of classical propositional logic uses the schemes $A\to (B\to A)$, $(A\to (B\to C))\to ((A\to (\neg B\to C))\to (A\to C))$, and $(A\to B)\to (A\to (\neg B\to C))$, together with modus ponens rule and the case rule from $A\to B$ and $\neg A\to B$ to $B$. Its deduction theorem for propositional logic follows by induction on logical derivations: conditional modus ponens handles a modus ponens rule line, and the second scheme handles a case-rule line.

On $\mathbb{C}[x]$, equip polynomials with the inner productThe linear operatorpreserves the space $V_n$ of polynomials of polynomial degree at most $n$. Integration by parts, whose boundary terms vanish by Gaussian decay, givesThus $T$ equals its adjoint on an inner product space. If a nonzero polynomial of degree $n$ is an eigenvector, its leading coefficient forces its eigenvalue to be $-2(n+1)$.

One can construct all these eigenvectors without solving a differential equation. The one-dimensional space $L_n=V_n\cap V_{n-1}^{\perp }$ is invariant: for $u\in V_{n-1}$ and $v\in L_n$, the displayed identity gives $⟨u,Tv⟩=⟨Tu,v⟩=0$. Its unit vector of positive real leading coefficient is therefore an eigenvector of degree $n$. These choices are exactly the Gram-Schmidt process applied to $1,x,x^2,\dots$, producing normalized Hermite polynomials. Their first three members areEvery polynomial is a finite linear combination of these orthonormal eigenvectors.

For the Hermite polynomials in the Rodrigues normalization $H_n=(-1{)}^n{e}^{t^2}{D}^n{e}^{-t^2}$, differentiating gives $H_{n+1}=2t{H}_n-H_n^{\prime }$ with $H_0=1$. Induction givesInsert this into the exponential generating series and reindex $n=m+2j$. The result factors into ${\sum }_m(2tz{)}^m/m!$ times ${\sum }_j(-z^2{)}^j/j!$, proving the displayed identity. Equivalently Taylor expansion of $e^{-(t-z{)}^2}$ in $z$ gives the same formula. This generating function keeps the normalization distinct from the Probabilists' Hermite polynomial convention.

A first-order structure whose domain is the Herbrand universe, with constants denoting themselves and function symbols acting by syntactic term formation. Its predicates are specified by assigning truth values to ground atoms. With first-order equality, ordinary equality of syntactic terms imposes additional restrictions; more general term models may require quotienting by a congruence.

Equivalently, the predicate meanings of a Herbrand interpretation correspond to a subset of its Herbrand base.

The set of all ground atoms formed using a Herbrand universe and the signature's predicates.

For $n\in {\mathbb{Z}}_3$, the equation $x^3+3x+y^3+3y=n$ has a solution in the p-adic integers exactly for the displayed residue classes. The polynomial $F(t)=t^3+3t$ has values $0,4,5$ modulo nine, whose pairwise sums are $0,1,4,5,8$. For sufficiency choose initial pairs $(0,0),(-1,-1),(1,0),(-1,0),(1,1)$ for those five classes respectively, fix the second coordinate, and lift the first. Its derivative is $3(x^2+1)$, always of 3-adic norm $1/3$, while its second Taylor coefficient is $3x$, of norm below one. The residual is divisible by nine, so either the strict or the Boundary case of Hensel's lemma with a small quadratic coefficient applies.

For viscous fluid inside a circular fluid free boundary between parallel plates of separation $h$, the depth-averaged Hele-Shaw flow satisfies $\bar{u}=-K\nabla p$, $K=h^2/(12\mu )$. With volume injection rate $Q$, ${\bar{u}}_r=Q/(2\pi hr)$ and $R^2=R_0^2+Qt/(\pi h)$. Neglecting surface tension and exterior viscosity, pressure is constant at the interface. Perturb its radius by $a_n\cos n\theta$ and solve the interior Laplace equation with a regular pressure term proportional to $r^n\cos n\theta$. The linearized pressure boundary condition gives $\delta p(R)=a_{n}Q/(2\pi hKR)$, hence velocity perturbation $-n{a}_{n}Q/(2\pi h{R}^2)$. The displaced base velocity adds another $-a_{n}Q/(2\pi h{R}^2)$, proving the displayed growth law. Injection stabilizes the circular blob; suction destabilizes it. An exact radial solution can therefore exist while giving a fragile experimental prediction. These signs concern viscous fluid inside the blob, rather than the opposite geometry of a low-viscosity injected bubble.

If $X\subseteq \mathbb{R}$ is not compact, there is an unbounded continuous function $X\to \mathbb{R}$. By the Heine-Borel theorem, either $X$ is unbounded, when $f(x)=x$ works, or it is not closed. In the latter case choose $a\in \overline{X}\setminus X$; the continuous function $f(x)=|x-a{|}^{-1}$ is unbounded because points of $X$ approach the missing limit point $a$. This proof uses the specific compactness characterization of real subspaces; boundedness of a general metric space does not imply compactness.