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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.

If a surface lies outside a sphere centered at the origin, its minimum distance to the sphere is its minimum radial distance minus the sphere radius. For each surface point, the triangle inequality gives that lower bound, and the radial point on the sphere attains it. Thus minimizing the squared norm on the surface suffices.

The decreasing production function and increasing linear loss have one positive constant equilibrium $u_{\ast }$. It scales as ${\delta }^{-1/(n+1)}$ for small loss and ${\delta }^{-1}$ for large loss. With $a=n{u}_{\ast }^{n-1}/(1+u_{\ast }^n{)}^2>0$, the linear characteristic equation is $\sigma +\delta +a{e}^{-\sigma d}=0$. It has no real nonnegative root, and zero is never a root. Thus loss of stability from the zero-delay stable state must occur through a nonzero imaginary conjugate pair.

If $a>\delta$, the first crossing has $\omega =\sqrt{a^2-{\delta }^2}$ and $d_c=\arccos (-\delta /a)/\omega$. Differentiation of the characteristic equation gives $\mathrm{Re}{\sigma }^{\prime }(d_c)={\omega }^2/[(1+\delta d_c{)}^2+{\omega }^2{d}_c^2]>0$, proving transversality. If $a\le \delta$, the magnitude inequality $|\sigma +\delta |=a|e^{-\sigma d}|$ rules out a root in the closed right half-plane. The model needs an initial history on the whole delay interval. The delay represents a production/maturation lag rather than instantaneous production controlled by the current cell count.

Transient blood-forming endothelium that generates haematopoietic cells through an endothelial-to-haematopoietic transition.

The principal axes of an ellipse are the two perpendicular symmetry axes through its center of an ellipse. For a symmetric positive-definite matrix $M$, its orthonormal eigenvectors give those directions and its positive eigenvalues ${\lambda }_1,{\lambda }_2$ give semiaxis lengths ${\lambda }_1^{-1/2},{\lambda }_2^{-1/2}$. This follows by orthogonal diagonalization of the defining quadratic form. In ${\xi }^2/A^2+{\upsilon }^2/B^2=1$, the axes are already horizontal and vertical, with semiaxes $|A|,|B|$ and full lengths $2|A|,2|B|$. Equal semiaxes give a circle, for which every perpendicular pair through the center can serve.

The electric potential difference between the ground-contact points of the two feet. A ground current can make it nonzero.

The polarization of an elastic wave in an isotropic solid specifies the direction of its displacement field amplitude relative to its wavevector. The longitudinal wave component is parallel to the wavevector; the transverse wave component is perpendicular. For a real unit propagation vector $n$, $a_L=(a\cdot n)n$ and $a_T=a-a_L$. An evanescent constituent has a complex spatial derivative vector, so the corresponding amplitude ratio can be imaginary. The longitudinal and transverse constituents of a Rayleigh wave combine into ellipse particle motion at the free surface.

A shear wave in an isotropic elastic material has displacement perpendicular to its wavevector, with zero divergence for a plane wave. The Navier-Cauchy equation with inertia then reduces to $\rho {\mathbf{u}}_{tt}=\mu \Delta \mathbf{u}$, giving the displayed wave speed. At a boundary it can be generated by mode conversion from a compressional wave, even when the incident displacement is purely longitudinal.

For an isotropic elastic layer $|y|<h$ between two identical elastic half-spaces, an antiplane displacement $w=f(y)e^{i(kx-\omega t)}$ has $m^2={\omega }^2/c_{s1}^2-k^2>0$ in the layer and exterior decay rate ${\ell }^2=k^2-{\omega }^2/c_{s2}^2>0$. Hence a trapped mode has phase speed between the layer and cladding shear speeds. Continuity of displacement and shear traction gives the first displayed equation for even modes $f\propto \cos (my)$ and the second for odd modes $f\propto \sin (my)$.

In a rigid-cladding limit with nonzero decay rate, the interfaces become displacement nodes. Even transverse wavenumbers are $(n+1/2)\pi /h$, $n\ge 0$, and odd ones are $n\pi /h$, $n\ge 1$. The minimum even-mode dispersion branch is ${\omega }^2=c_{s1}^2[k^2+{\pi }^2/(4h^2)]$. When the cladding speed is finite, exterior decay remains a separate requirement; satisfying this lower cutoff alone is insufficient.

In an isotropic half-space $y<0$ with zero displacement at $y=0$, an incident longitudinal wave directed along $(\cos \alpha ,\sin \alpha )$ reflects into both longitudinal and shear waves. Let $c_p^2=(\lambda +2\mu )/\rho$ and $c_s^2=\mu /\rho$. Horizontal phase matching gives $\sin \beta =(c_s/c_p)\cos \alpha$ for a reflected shear wavevector proportional to $(\sin \beta ,-\cos \beta )$ and polarization $(\cos \beta ,\sin \beta )$. The two zero-displacement equations at the boundary give the displayed coefficients. Since $c_s<c_p$ in a stable isotropic solid, both reflected branches are propagating for every genuine incident angle $0<\alpha <\pi$, with no critical-angle obstruction for this incident-wave type. Changing to a stress-free boundary changes the amplitude equations and coefficients.

For a homogeneous isotropic plate of thickness $h$, Young's modulus $E$ and Poisson's ratio $\nu$, the bending stiffness is $D=E{h}^3/[12(1-{\nu }^2)]$. Its quadratic bending energy isIt follows by integrating the strain energy density $\frac{1}{2}{\tau }_{ij}{e}_{ij}$ through the thickness: linear bending strains are proportional to $z$, and ${\int }_{-h/2}^{h/2}{z}^{2}dz=h^3/12$. Taking a first variation and integrating by parts twice gives $D{\Delta }^{2}w=0$ without applied transverse load. Clamped boundary variations have both zero value and zero normal derivative.

A sheet-like elastic component of a vessel wall, especially prominent in the tunica media of elastic arteries.