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Assume that a proposed statement fails for some natural number, and choose the least failing index using the well-ordering principle. Prove the necessary base cases and reduce the least failure to a smaller index that must satisfy the statement. If that smaller case implies the alleged failure also satisfies it, there is a contradiction. Reductions by a step larger than one require enough consecutive base cases to cover every possible residue class.

For a $C^1$ curve $u$, some $\delta >0$ must make the Weierstrass excess function nonnegative for every $x\in [a,b]$, every trial slope $q\in \mathbb{R}$, and all $y,p$ satisfying $|y-u(x)|<\delta$, $|p-u^{\prime }(x)|<\delta$. Thus the supporting-line property persists in a tube of nearby heights and reference slopes. The term “strengthened” here refers to this neighborhood requirement; strict excess for $q\ne p$ is a further condition.

At a strong local minimizer of a variational integral, almost every reference slope $u^{\prime }(x)$ is a point of convexity of $v\mapsto f(x,u(x),v)$. Equivalently its Weierstrass excess function is nonnegative for every trial slope. Uniformly small variations can have large derivatives, which is why this condition tests all trial slopes rather than only nearby ones. It is a necessary condition along the minimizing curve; the strengthened Weierstrass condition additionally controls a neighborhood of that curve and its slopes.

For $0<\delta <1$, choose the zero-trace piecewise affine curve with slope $\delta$ up to $1-\delta /4$ and slope $\delta -4$ thereafter. Periodicity makes $g^{\prime }$ equal to the same nonzero constant at both slopes, and $g^{\prime \prime }>0$ on small neighborhoods of both. For sufficiently small Lipschitz perturbations, local strict convexity bounds the energy difference below by $c\int {\varphi }^{\prime 2}$, since the common first-variation coefficient multiplies $\int {\varphi }^{\prime }=0$. This proves a strict weak minimum. The global supporting-tangent condition fails because its nonzero slope cannot stay below bounded $g$, so it is not a strong minimum. By contrast, tents of slopes $4n,-4n$ attain $g=-1$ everywhere and provide infinitely many distinct global minimizers.

Consider a smooth planar system $\dot{x}=y+P_3(x,y)+O(r^5)$, $\dot{y}=-x+Q_3(x,y)+O(r^5)$, where $P_3,Q_3$ are homogeneous cubic polynomials and $r^2=x^2+y^2$. The leading derivative of $r^2/2$ is the quartic polynomial $H_4=x{P}_3+y{Q}_3$. Let $\kappa$ be the angular average of $H_4$ on the unit circle. A homogeneous quartic correction $V_4$ solvinggives $\dot{V}=\kappa r^4+O(r^6)$ for $V=r^2/2+V_4$. The right side has zero angular average. Since $y{\partial }_x-x{\partial }_y=-{\partial }_{\theta }$, its nonconstant angular Fourier modes can be integrated individually to obtain a periodic quartic correction; only modes of orders two and four can occur after removing the constant average.

The correction is $O(r^4)$, so $V$ remains positive definite close to zero. If $\kappa <0$, its orbital derivative is strictly negative there, and the Second Lyapunov theorem proves asymptotic stability. If $\kappa >0$, apply the same argument in reversed time to obtain repulsion. If $\kappa =0$, this quartic calculation alone does not decide stability. This method converts the averaged damping of a weak focus into a pointwise sign for a Lyapunov function.

This real planar dynamical system has a saddle point at $(-1,0)$ and a repelling weak focus at $(1,0)$. The latter has purely imaginary linearized eigenvalues $\pm 2i$, so its linearization has closed elliptical orbits. Nonlinear terms nevertheless produce outward drift.

Set $X=2(x-1)$, $Y=y$, $\tau =2t$ and $r^2=X^2+Y^2$. The cubic expansion isThe linear rotation acts on scalar functions by $L=Y{\partial }_X-X{\partial }_Y$. For $H_2=r^2/2$, its cubic derivative is $X{Y}^2-3X^{2}Y/4-Y^3/2$. The homogeneous correction $H_3=7X^3/12+X{Y}^2/2+Y^3/3$ satisfies $L{H}_3$ equal to the negative of that derivative. The remaining quartic derivative of $H_2+H_3$ is $3X^2{Y}^2/2-X^{3}Y$. Withdirect differentiation givesfor all sufficiently small nonzero $r$. Meanwhile $H_2+H_3+H_4=r^2/2+O(r^3)>0$. In reversed time it is a strict Lyapunov function, proving asymptotic stability of the reversed system and hence repulsion forward in time. The angular speed stays close to $-1$, so the nearby nonzero trajectories spiral rather than approaching along a fixed direction. This explicit calculation illustrates why a centre in the linearization need not be a nonlinear centre.

Steam is water in the gas phase. Its state depends on temperature and pressure; a visible cloud above boiling water consists mainly of condensed droplets rather than isolated gas molecules.

Condensation of steam releases latent heat, allowing water vapour to transfer substantial energy. The distinction between saturated and superheated vapour matters when connecting phase equilibrium with heat transfer.

A historical quantitative test in which a standardized soap solution is added to a measured water sample until a specified persistent lather appears. A reagent blank accounts for the amount needed to form the lather itself.

For causal convolution on $t\ge 0$, the Laplace transform convolution theorem converts $x+\sin \ast x=F$ into $\bar{x}=[(p^2+1)/(p^2+2)]\bar{F}$. Since this multiplier is $1-1/(p^2+2)$, inversion gives the displayed resolvent kernel. The formula remains valid for locally integrable forcing on finite intervals: substitution and the Fubini theorem reduce verification to an identity between the two continuous sine kernels. Thus a global Laplace transform is a convenient derivation, rather than an essential growth restriction on every local forcing.

The derivative and convolution theorem for Laplace transforms give $[p+1/(p+2)]\bar{x}=1/p$. Partial fractions yield $\bar{x}=2/p-2/(p+1)-1/(p+1{)}^2$, so $x(t)=2-(2+t)e^{-t}$ tends to two. A direct check gives $x^{\prime }=(1+t)e^{-t}$ and $(e^{-2t}\ast x)(t)=1-(1+t)e^{-t}$. Introducing the memory variable $w=e^{-2t}\ast x$ gives the equivalent local system $x^{\prime }+w=1$, $w^{\prime }=x-2w$, $w(0)=0$, explaining why the transform denominator has a repeated pole at $p=-1$.

A vertex cover $C$ whose induced subgraph $G[C]$ is connected. The optimization problem minimizes $|C|$ subject to both conditions. A graph with edges in two distinct connected components of a graph has no connected vertex cover: covering each of these edges requires selected vertices in different components. Isolated vertices can be ignored when exactly one component contains all the edges. The edge-free case depends on whether the empty solution is admitted.

After a rapid switch to systolic compliance $C_s$, negligible valve resistance equalizes arterial and ventricular pressures. Neglecting runoff during this short stage conserves combined stored volume and gives the displayed pressure. Thereafter, with closed mitral valve, negligible venous pressure and constant $C_s$, the pressure decays as ${\hat{p}}_a{e}^{-t/[R_c(C_a+C_s)]}$. At firing duration $\Delta t_F$, the stroke volume is $V^{+}-V_0-C_s{\hat{p}}_a{e}^{-\Delta t_F/[R_c(C_a+C_s)]}$. Each approximation follows from an explicit valve or time-scale separation condition.

If an open aortic valve equilibrates pressure much faster than peripheral outflow, the arteries and ventricle share a common pressure while their combined blood volume is conserved during the short adjustment. Let arterial vascular compliance be $C_a$ and systolic ventricular elastance be $E_s$. Equating the initial and final arterial-plus-ventricular stored volumes gives the displayed pressure. Subsequent peripheral drainage sees the combined compliance $C_a+1/E_s$; during filling a closed aortic valve removes the ventricular contribution.

Compression of veins by skeletal muscles, with venous valves favouring forward flow.

Venesection at the elbow. The ancient instruction to open a hollow vein here does not mean opening the vena cava.

For a nonzero vector space with an infinite field or infinite basis, cardinality is the displayed maximum, since vectors are finite linear combinations of basis vectors. Over a finite field, infinite underlying size forces infinite dimension. Over a countably infinite field, every positive finite dimension is already countably infinite. The theory of infinite underlying spaces is therefore categorical in every infinite cardinal for a finite field, but only in uncountable cardinals for a countably infinite field.