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A shape function is a local basis function associated with a degree of freedom of a finite element. For nodal degrees of freedom it is one at its own node and zero at the other nodes, allowing local interpolation from the nodal values. For a bilinear square finite element, the four products of linear coordinate functions provide the nodal shape functions.

An isoparametric finite element uses the same reference shape functions to interpolate its geometry and its unknown field. With reference functions $N_i$ and physical nodes ${\mathbf{x}}_i$, the geometry map is $F={\sum }_i{\mathbf{x}}_i{N}_i$ and the physical functions are $N_i\circ F^{-1}$. The map must be one-to-one and have a nonvanishing Jacobian determinant. For bilinear quadrilaterals, compatible common edges have affine geometry maps and identical linear field traces, giving a continuous conforming finite element space. On a general quadrilateral the pulled-back functions need not be polynomials in physical coordinates.

For a rectangular Poisson equation discretization with commuting negative-definite directional matrices $X,Y$, two alternating implicit directional solves give the displayed iteration matrix. If their shared eigenvectors have eigenvalues $-{\xi }_p,-{\eta }_q$ with ${\xi }_p,{\eta }_q>0$, the iteration eigenvalues are $(\alpha -{\xi }_p)(\alpha -{\eta }_q)/[(\alpha +{\xi }_p)(\alpha +{\eta }_q)]$. For $\alpha >0$ their moduli are less than one, proving convergence. At $\alpha =0$ the composed iteration is the identity and the forcing terms cancel, so arbitrary initial errors persist. Strict positivity of the parameter is therefore essential for this splitting.

Normalize the closed SIR model by its initial susceptible count, writing $u(0)=1$, $v(0)=v_0>0$, $w(0)=0$ and $\tau =\gamma t$. Then $u+v+w=1+v_0$, and the SIR susceptible-recovered identity becomes $u=e^{-{\mathcal{R}}_{0}w}$. HenceThe positive root of the quadratic approximation isThe displayed small-seed formula requires ${\mathcal{R}}_0>1$, ${\mathcal{R}}_0{w}_{\ast }\ll 1$ and ${\mathcal{R}}_0^2{v}_0\ll ({\mathcal{R}}_0-1{)}^2$. In particular, a regime is $0<{\mathcal{R}}_0-1\ll 1$ with a much smaller positive seed. Setting the seed exactly to zero produces the disease-free trajectory instead of an outbreak.

Compute adjacent Fibonacci numbers together, starting at $(0,1)$ and applying $(a,b)\mapsto (b,a+b)$. The shared pair avoids repeating the same recursive subproblem and gives $n$ additions for $F_n$. This counts arithmetic operations; exact integers grow with $n$.

For positive integers $m,n$, the Fibonacci numbers satisfy $F_m\mid F_{mn}$. The Fibonacci addition formula gives $F_{mn+m}=F_{m-1}{F}_{mn}+F_m{F}_{mn+1}$, so mathematical induction on $n$ proves the assertion. This explains why every fifth Fibonacci number is divisible by $F_5=5$.

The Cesàro means of the Fourier series of a continuous periodic function converge uniformly to that function. The Fejér kernel is nonnegative with total mass one after normalization, and its mass outside every fixed neighborhood of zero tends to zero. Uniform continuity controls the near contribution, while the vanishing tail controls the rest. For an $L^1$ function the same positive-kernel argument gives an $L^1$ contraction, which is useful even when pointwise behavior is unavailable.

Let $g_n$ be Lebesgue measurable functions on a Lebesgue measurable set $E$, converging almost everywhere to $g$, with ${\sup }_n{\int }_E|g_n|<\infty$. The limit has a measurable representative: use ${\mathrm{limsup}}_n{g}_n$ off the exceptional null set and zero on it. The completeness of Lebesgue measure makes any alternative values on that null set measurable as well. Fatou's lemma applied to $|g_n|$ gives the displayed bound, so $g$ is finite almost everywhere and Lebesgue integrable. A uniform L1 norm bound alone does not imply convergence in L1 space; the strict inequality in Fatou lemma supplies a shrinking-spike example.

If $Y_n$ are random variables bounded above by an integrable function $Z$, with all relevant expectations well defined, Fatou's lemma applied to the nonnegative variables $Z-Y_n$ gives ${\mathrm{limsup}}_n\mathbb{E}{Y}_n\le \mathbb{E}{\mathrm{limsup}}_n{Y}_n$. In particular, bounded indicator random variables that tend almost surely to zero have probabilities tending to zero.

The membranous component of superficial perineal fascia.

In an alternating-access model a protein binding site faces either side of a membrane; both empty and substrate-bound carriers can switch orientation. Mass-action binding on each side and conformational transition rates give a four-state carrier cycle coupled to two free-substrate pools. Conservation counts all four carrier states as one protein each and counts bound substrate with the free-substrate pools. Equal source and sink fluxes preserve the total substrate while maintaining throughput.

The complexity class of formal languages decidable in deterministic exponential time. It contains NP: enumerate the finitely many choices of a polynomial-length nondeterministic computation and simulate each branch with a polynomial time cutoff. There are at most $b^{p(n)}$ branches for fixed branching bound $b$ and polynomial $p$.

For fixed real $x$, the base $1+x/n$ is positive for sufficiently large $n$. Applying Taylor's theorem to the natural logarithm at one gives $n\log (1+x/n)=x+O(n^{-1})$. Continuity of the exponential function then proves the displayed limit. In particular $(1-z/n{)}^n\to e^{-z}$ for fixed $z\ge 0$, a basic source of exponential distribution limits in probability theory.

For a real symmetric positive-definite matrix $A$ and nonzero linear-system residual $r=b-Ax$, minimize $f(x+\alpha r)$ for $f(x)=x^{T}Ax/2-b^{T}x$. Differentiating with respect to $\alpha$ gives $-r^{T}r+\alpha r^{T}Ar=0$, yielding the displayed positive step. Positive definiteness makes it the unique minimum along the line. Applying the same quotient to a nonsymmetric or indefinite matrix is a formal iteration, but this minimization proof and its convergence consequences no longer apply.

For the angular endpoint conditions $\theta (0)=\theta (1)=0$, take $U=(EI/L){\int }_0^1[{\theta }^{\prime 2}/2-{\pi }^2\lambda (1-\cos \theta )]d\xi$. The Poincaré inequality and $1-\cos t\le t^2/2$ give $U\ge 0$ for $0\le \lambda \le 1$. At $\lambda =1$ a nonzero continuous angle still has strictly positive energy because $1-\cos t<t^2/2$ for $t\ne 0$. Thus the straight state remains the unique global minimum at the critical parameter even though the quadratic variation vanishes on the first sine mode.

Along that mode the energy is ${\pi }^{2}EI{A}^4/(64L)+O(A^6)$ at criticality. For $\lambda >1$ a sufficiently small first-mode variation lowers energy, so the straight state loses local minimality. Near the first threshold, expansion of the Euler strut equation gives $A_0(A_0^2-8{\lambda }_1)=0$ when $\lambda =1+ϵ{\lambda }_1$ and $\theta \sim \sqrt{ϵ}{A}_0\sin \pi \xi$. The resulting supercritical pitchfork bifurcation is stabilized by the same positive quartic term. A zero second variation is therefore an inconclusive quadratic test, not proof of failure of minimization.

For a smooth solution of $y^{\prime }=f(x,y)$, Taylor's theorem gives the displayed local truncation error when an Euler method step starts at an exact solution value. Thus the one-step defect is $O(h^2)$. Dividing it by $h$ gives a normalized local truncation error of $O(h)$; these are different conventions for reporting the same residual. Also $y^{\prime \prime }=f_x+f_{y}f$ along the solution by the chain rule. A global $O(h)$ error conclusion additionally requires the usual regularity and stability control over successive steps.

Specifying endpoint tangent angles of an inextensible rod does not by itself specify its endpoint positions. A planar Euler elastica satisfies $x^{\prime }=\cos \theta$, $y^{\prime }=\sin \theta$, so equal endpoint heights require the displayed integral to vanish. For constant compressive resultant $(-P,0)$, the angular equation is $EI{\theta }^{\prime \prime }+P\sin \theta =0$, giving $P\Delta y=-EI[{\theta }^{\prime }(L)-{\theta }^{\prime }(0)]$. Thus a nontrivial branch containing an odd number of pendulum half-periods cannot have equal endpoint heights and the same zero endpoint angles with zero transverse resultant. At small amplitude the tangent-only eigenfunctions are $\sin (n\pi s/L)$, with $P_n=EI(n\pi /L{)}^2$; equal endpoint heights retain even $n$ and give the first fixed-clamp load $4{\pi }^{2}EI/L^2$. Other clamp modes can require a nonzero transverse reaction. Axial shortening is also needed when the reference endpoint separation equals the rod length.

For an orthogonal matrix $Q$, the identity $Q^{T}Q=I$ gives $\Vert Qx{\Vert }_2^2=x^T{Q}^{T}Qx=x^{T}x$. Thus multiplication by $Q$ preserves the Euclidean norm. Applying this on both input and output sides proves $\Vert UA{V}^T{\Vert }_2=\Vert A{\Vert }_2$ for orthogonal matrices $U,V$ and the Euclidean operator norm. The analogous invariance of the Frobenius norm follows by summing column lengths or by a trace calculation.

Let $r>0$, let $S^{n-1}$ be the unit sphere in Euclidean space, and let $M$ be a nonempty subset of ${\mathbb{R}}^n$. For each $x\in M$,The reverse triangle inequality gives the lower bound, and the radial point $rx/\Vert x\Vert$ attains it when $x\ne 0$; when $x=0$, every point of the sphere is at distance $r$. Taking the infimum givesIf a surface $M$ lies outside the sphere, meaning $\Vert x\Vert \ge r$ for all $x\in M$, this becomes the title formula ${\inf }_{x\in M}\Vert x\Vert -r$. Minimizing the squared norm is then equivalent when a minimizer exists. The word minimum requires attainment: it follows, for example, when $M$ is compact. In ${\mathbb{R}}^3$, the surface $M=\{(s,t,0):s>r+1\}$ has distance $1$ from the sphere, but no point of $M$ attains that distance.

An erythroblast is a developing red-cell precursor. Its successive changes include hemoglobin accumulation and, in mammals, eventual loss of the nucleus before the mature circulating state. Cell identity and developmental stage must therefore be specified when interpreting a blood-forming population.