The in-plane tensile force per unit transverse length in a thin elastic sheet produces the linear transverse restoring force $T{\eta }_{xx}$ per unit area. A spatially uniform displacement has zero curvature and hence no elastic-sheet tension force. This explains why infinite elastic-sheet tension blocks fixed nonzero-wavenumber acoustic membrane waves but does not make the uniform mode rigid. The sheet's inertia, by contrast, acts also on uniform oscillations.

Two diametrically opposite rotating blades with equal tangential loads have zero total rotating force. Their leading compact acoustic dipoles cancel, but their retarded positions and moving-surface retarded Jacobians leave an acoustic quadrupole. For line load $2Fr/a^2$ and angular speed $\Omega$, the first radiating density is $2F{\Omega }^{2}a{\sin }^2\theta \sin (2\Omega {\tau }_0)/(3\pi c_0^{4}R)$. The twofold geometry gives frequency $2\Omega$, while projection into the rotor plane gives the amplitude directivity ${\sin }^2\theta$.

For harmonic convention $e^{i\omega t-ikx}$, choose $\mathrm{Re}\gamma >0$ by continuation from $\mathrm{Im}\omega <0$ and impose vertical factors $e^{-\gamma y}$ above the interface and $e^{\gamma y}$ below it. At positive real frequency, $\gamma =i\sqrt{k_0^2-k^2}$ for propagating Fourier components and is positive real for evanescent components. This analytic prescription fixes the radiation branch and how the Fourier contour passes poles.

This is the total number of vertices in graph components with orders in the specified half-open interval. It differs from the number of those graph components. Write $N_{\ge k}$ for the total above the lower endpoint. Then $N_{[k,\ell )}=N_{\ge k}-N_{\ge \ell }$. Disjoint size bands count disjoint sets of vertices at a common time, even though their mass may move between bands as an Achlioptas process adds edges.

For the two-choice Achlioptas process, fix $a>0$ and $K\ge 1$. Conditional on any current graph with $N_{\ge K}\ge an$, put $W$ equal to the vertices in its large graph components. There are at most $n/K$ such initial graph components. If after $s$ more steps no graph component contains $an/3$ vertices of $W$, greedily assign final graph components to two groups so that each meets at least $|W|/3\ge an/3$ vertices of $W$. This induces a set partition of the initial large graph components, of which there are at most $2^{n/K}$ possibilities.

For any fixed such split $W=A\cup B$, a uniform candidate edge joins $A$ to $B$ with probability at least $2a^2/9$. If both candidates do so, the selected edge cannot avoid the crossing. Thus the probability that the split survives all $s$ steps is at most $(1-a^4/81{)}^s$. The deliberately weaker constant also covers distinct candidate sampling for sufficiently large $n$. In the absent-edge version, as long as the split has survived, every $A$--$B$ edge is absent, so conditioning on absence can only increase this crossing probability. A union bound with $s=\lceil A(a)n/K\rceil$ and $A(a)=81(\log 2+2)/a^4$ bounds failure by $e^{-2n/K}$. For fixed $K$, a further union bound makes the conclusion simultaneous over all starting steps $m\le 3n$. This proof is independent of how the rule chooses between the two candidates.

Fix positive $\alpha$, $D\ge 2$ and $B>0$. Suppose a size band $[k,Dk)$ of an Achlioptas process contains at least $\alpha n$ vertices at time $m$. For every fixed $k$, at least $\beta n$ of those vertices remain in unchanged graph components throughout the next $T=\lceil Bn/k\rceil$ steps, with high probability, where one may take $\beta =\alpha e^{-16BD}/4$. This is a deliberately nonoptimal positive constant.

For independent uniform candidate edges, an initial graph component with $w<Dk$ vertices is untouched if none of the $2T$ candidate edges meets it. A uniform edge hits it with probability at most $3w/n$. For sufficiently large $n$, $(1-3w/n{)}^{2T}\ge e^{-16BD}$, so the expected value of the untouched vertex mass $Y$ is at least $\alpha e^{-16BD}n$. Replacing one row of two candidate edges can change $Y$ by at most $8Dk$, because only the initial graph components hit by an old or new endpoint can change status. The McDiarmid inequality therefore makes $Y\ge \alpha e^{-16BD}n/2$ except on an event of probability $e^{-cn/k}$, for a positive constant $c$ depending only on $\alpha ,B,D$. An untouched graph component stays unchanged whichever offered edge is selected.

For candidates restricted to absent edges, generate each by an independent uniform proposal stream and reject already present edges. During any interval of $O(n)$ steps with $O(n)$ present edges, each proposal has rejection probability $O(1/n)$ conditional on the past. The number of rejections is at most $O(\log n)$ except on an event of probability smaller than any inverse power of $n$: if there were $r$ rejections among $2T+r$ proposals, a union bound over their positions bounds this by $\left(\genfrac{}{}{0ex}{}{2T+r}{r}\right)(C/n{)}^r$. The first $2T$ proposals are independent and give the previous untouched-mass estimate. Extra proposals can touch at most $2r$ additional initial graph components, losing only $O(Dk\log n)=o(n)$ vertices for fixed $k$. This leaves the stated $\beta n$ bound. A union bound makes these estimates simultaneous over starting times $m\le 3n$. They hold through the entire interval, so deterministic or random stopping times within it are allowed.

A fixed-choice Achlioptas process cannot have its largest component of a graph jump from $o(n)$ to a fixed positive fraction of $n$ within $o(n)$ steps, with high probability. The proof combines forced merging of large components and persistence of unsampled graph components. An assumed jump forces positive vertex mass in each band $[k,Dk)$ slightly before the jump: insufficient smaller components can feed the final large component, while enough components above $Dk$ would merge too early. Persistence carries a fixed positive amount of each band to a common time. Taking sufficiently many disjoint bands with $k=1,D,D^2,\dots$ then counts more than $n$ vertices. The number of offered candidates is fixed throughout this argument. Riordan and Warnke establish this obstruction and broader continuity results in their original research paper.

Weibull distributions with a common shape $k>0$ form both an accelerated life family and a proportional hazards family. Their hazard functions are $h_z(t)=k{\lambda }_z{t}^{k-1}$, whose ratio is ${\lambda }_z/{\lambda }_0$. Their time multiplier is $({\lambda }_0/{\lambda }_z{)}^{1/k}$. In the representation $\log T_z=a+bz+cX$ with $c>0$ and density $e^{x-e^x}$ for $X$, the parameters are $k=1/c$ and ${\lambda }_z=e^{-(a+bz)/c}$, giving time multiplier $e^{bz}$ and hazard multiplier $e^{-bz/c}$.

For $\log T_z=bz-X$, with $e^X$ unit exponential distribution and $b=\log 2$, the survivor functions are $S_z(t)=1-e^{-2^z/t}$ for $t>0$. These are time-scaled Fréchet distributions, so they form an accelerated life family. However their hazard ratio is $2/(e^{1/t}+1)$, tending to zero as $t\downarrow 0$ and to one as $t\to \infty$. It is not constant. Thus a positive-scale assumption matters when deriving a proportional hazards family from the usual log-location-scale model.

For any finite set of couplings in $z_N(t)={\prod }_k\cos (2g_{k}t)$, the simultaneous Dirichlet approximation theorem supplies arbitrarily late near-alignment of all phases, or an exact common period. Thus finite bath coherence does not tend to zero as time tends to infinity. This remains true for almost every realization of continuous random couplings; an ensemble mean can decay while each realization has recurrences.

Independent random couplings suppress typical coherence as the bath size increases. This conclusion concerns an ensemble or a specified large-bath limit, and must be distinguished from a pointwise long-time limit for one finite realization. Ensemble spin-bath coherence supplies exact mean and mean-square formulas for a uniform coupling distribution, while short-time Gaussian spin-bath decoherence gives the initial decay scale.

Consider a zero-sum game in which a hider selects one of $n$ locations and a searcher inspects locations until finding it. Inspection costs are $c_i>0$, paid also at the discovered location; the hider maximizes total cost and the searcher minimizes it. Put $c={\sum }_i{c}_i$. The value of a zero-sum game isThe hider uses probabilities $c_i/c$. Every complete search order then has the same expected payoff: summing each cost multiplied by the probability that its location or a later one hides the object gives $({\sum }_i{c}_i^2+{\sum }_{i<j}{c}_i{c}_j)/c=v$.

For the searcher's matching guarantee, inspect location $i$ first with probability $c_i/c$, then inspect the remaining locations in a uniformly random order. For distinct $i,j$, the probability that $i$ precedes $j$ isIf the object is at $j$, expected total cost is therefore $c_j+{\sum }_{i\ne j}{c}_i\mathrm{Pr}(i\text{ precedes }j)=v$. Both guarantees hold for every opponent choice, proving optimality for any finite number of locations, including the three-location case.

The induced worldvolume metric is the pullback of the target metric tensor by a brane embedding. For a nondegenerate timelike embedding, $\sqrt{-\det h}$ is its invariant worldvolume volume density. The $p=1$ specialization is the induced worldsheet metric.

An auxiliary worldvolume metric is an independent metric introduced in a brane action, distinct from the induced worldvolume metric. Its algebraic equation can eliminate it. For the Polyakov action of a string, it is determined only up to a Weyl transformation.

Under a superpotential critical-point shift, $m\mapsto M=m+gs$ and the linear coefficient vanishes, while $M^2=\Delta$ is unchanged. At either supersymmetric root $v$, the complex fermion mass parameter is $W^{\prime \prime }(v)=\pm \sqrt{\Delta }$. The physical scalar and fermion masses are both $\sqrt{|\Delta |}$, independent of the additive superpotential constant. A repeated critical point has a massless multiplet and a quartic leading scalar potential.

For $0<\epsilon <2/(3\sqrt{3})$, the well-mixed ventilation temperature balance has one stable upward-flow temperature and two downward-flow temperatures. The heat-removal curve on the latter interval is $\sqrt{K_A}\theta \sqrt{W-G_T\theta }$, maximal at $2W/(3G_T)$. Its lower-temperature root is stable and its higher-temperature root unstable. The two meet at a saddle-node bifurcation. The unstable root separates the basins of the stable wind-dominated and buoyancy-dominated states.

For a normal variety with affine coordinate ring $A$ and a Weil divisor $D$, the fractional ideal of functions whose principal Weil divisor is at least $D$ isFor an effective prime divisor this is its height-one prime ideal. If the divisor is Cartier locally, this ideal is locally principal. A nonprincipal height-one prime ideal in a normal local ring can therefore detect a non-Cartier divisor.

Two Weil divisors are linearly equivalent when their difference is a principal Weil divisor. This is the equivalence relation defining the divisor class group, and applies to normal varieties of any dimension, not just curves.

For $D={\sum }_P{a}_{P}P$, its support is the union of the prime divisors $P$ with $a_P\ne 0$. Negative as well as positive coefficients contribute. A point outside the support has a neighbourhood on which the divisor is zero.

For Fréchet spaces, the two continuity conditions for a linear map are equivalent. Original continuity makes every target continuous linear functional compose to a source continuous linear functional, proving weak continuity. Conversely, weak continuity makes these compositions continuous in the original source topology. They separate target points, so limits in the graph of a linear operator remain in the graph. The closed graph theorem for Fréchet spaces finishes the proof.

Without the hypotheses, equal continuous dual spaces do not imply equal topologies. The identity from an infinite-dimensional Hilbert space with its weak topology to its norm topology is weak-to-weak continuous but not continuous in those original topologies.