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The humeral head is the rounded proximal articular region of the humerus that meets the glenoid cavity. Its large curved surface relative to the socket permits substantial shoulder motion. Stability consequently relies on capsule, labrum and muscular control as well as bone geometry.

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.

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.

Space heating supplies energy to an occupied region to maintain or raise its temperature. The thermal balance includes heat transfer through walls, exchange with ventilating air, stored internal energy and internal sources. A simple lumped thermal model writes $C\dot{T}=P-G(T-T_{\mathrm{out}})$ when all losses can be approximated by one effective thermal conductance.

At steady temperature, supplied power balances the ongoing losses; during warm-up, some also increases stored internal energy. Hot-water heating, steam heating and hot-air heating differ in how energy is carried to the room. A heat pump can deliver heat partly extracted from outside, whereas an electrical resistance heater converts supplied electrical work locally. Comparing room temperature with input power after a small controlled step tests both the effective loss coefficient and the thermal time constant of the simplified model.

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.

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.