Repeatedly apply the public RSA cryptosystem encryption map to a ciphertext. A return to that ciphertext reveals its preceding iterate as the plaintext. A nontrivial greatest common divisor between an iterate-minus-start and the modulus instead enables integer factorization. Eventual return alone supplies no useful running-time bound.

Recover a plaintext from an RSA cryptosystem ciphertext and the public key, without a supplied private key. Integer factorization of the modulus suffices; the existence of an efficient general reduction in the reverse direction is not known.

The time derivative of a vector in inertial axes equals its component derivative in rotating axes plus the change of the rotating basis itself. Each basis vector obeys ${\dot{\mathbf{e}}}_j=\mathbf{\Omega }\times {\mathbf{e}}_j$, proving the displayed identity by differentiating $\mathbf{a}={\sum }_j{a}_j{\mathbf{e}}_j$. Applying it twice to position produces the Coriolis acceleration $2\mathbf{\Omega }\times \mathbf{u}$ and the centripetal term $\mathbf{\Omega }\times (\mathbf{\Omega }\times \mathbf{x})$. A time-dependent rotation adds $\dot{\mathbf{\Omega }}\times \mathbf{x}$; constant angular velocity removes that last term.

For self-adjoint operators $A,B$ and a normalized quantum state in their relevant product operator domains, put $u=A\psi$, $v=B\psi$, $z=⟨u,v⟩$. With the inner product linear in its second entry, $⟨C⟩=2\mathrm{Im}z$. The Cauchy-Schwarz inequality givesEquality holds exactly when $u,v$ are linearly dependent and $\mathrm{Re}z=0$. This includes either zero vector; if both vectors are nonzero it is equivalent to $B\psi =i\lambda A\psi$ for real nonzero $\lambda$. Applying this inequality to the centred observables $A-⟨A⟩I$ and $B-⟨B⟩I$ gives the Robertson uncertainty principle.

Expanding the squared velocity $v_G+\omega \times r$ and integrating over mass removes the mixed term because the centre-of-mass first moment is zero. The remaining rotational integral is half the scalar product of angular velocity and angular momentum about the centre of mass. Principal axes diagonalize that rotational quadratic form.

Consider an axially symmetric rigid body rotating about a fixed pivot, with transverse and axial principal moments of inertia $A$ and $C$ and orientation-dependent potential energy $Mgh\cos \theta$. Let the inclination $\alpha$ be nonvertical, $0<\alpha <\pi$, and let $n$ be the axial angular velocity of a symmetric top. Conservation of the two cyclic angular momenta makes both $n$ and the steady precession rate $\Omega$ constant. The tilt Euler-Lagrange equation, after division by $\sin \alpha$, givesWhen $\cos \alpha \ne 0$, a real precession rate exists exactly when the quadratic discriminant obeys $C^2{n}^2-4AMgh\cos \alpha \ge 0$. At horizontal inclination, the equation is linear: if $Cn\ne 0$, then $\Omega =Mgh/(Cn)$. If $Cn=0$ and $Mgh\ne 0$, there is no steady horizontal solution. Vertical inclinations require the original undivided equation instead. The moments of inertia and lever arm $h$ must all refer to the same fixed pivot, rather than mixing central and pivot values.

For $H(u)=ru{e}^{-u}$ and $L=\log r$, a nonzero second-iterate fixed point satisfies $u+H(u)=2L$. A genuine pair has the displayed form, and its periodic-orbit multiplier is $(1-L{)}^2-d^2$. At $r=9$, the pair is exactly $\log 3,3\log 3$, with periodic-orbit multiplier $(1-\log 3)(1-3\log 3)\simeq 0.2264$. For large $r$ its upper point is asymptotic to $2\log r$ and the pair periodic-orbit multiplier to $1-2\log r$, so eventual instability follows despite the small derivative at the upper point.

Proliferating inflammatory tissue that extends over a surface, particularly synovial tissue in rheumatoid arthritis. At a joint it can contribute to cartilage and bone damage through an active tissue process.

Pressure within the pleural cavity; it is distinct from alveolar pressure.

Air pressure inside a pulmonary alveolus.

Distending pressure difference between alveolar pressure and pleural pressure.

Failure of the respiratory system to maintain adequate gas exchange for the body’s needs. Insufficient oxygenation and inadequate carbon dioxide elimination describe different physiological components.

A disorder of the airways involving variable respiratory symptoms and variable airflow limitation. Airway inflammation and altered responsiveness connect episodic narrowing with changes that may persist between attacks.

At a simple pole, write $f(z)=c/(z-a)+h(z)$ with $h$ holomorphic near $a$ and $c$ its residue. On the arc $z=a+ϵe^{it}$, the singular term integrates exactly to $ic(\beta -\alpha )$. The remainder has absolute integral at most $ϵ(\beta -\alpha )\sup |h|$, which tends to zero. Reversing orientation reverses the sign. In particular a clockwise upper semicircle contributes $-i\pi c$ in the shrinking limit.

A higher-order pole need not have such a limit: $(z-a{)}^{-2}$ has residue zero but integral $2/ϵ$ on a positively oriented upper semicircle. Thus the simple-pole assumption controls all the potentially divergent terms of the principal part of a Laurent series.

Committing a completed instruction's architectural effects in program order. An instruction with no register result still needs completion status before instruction retirement can pass it.

The proximal tubule is the early tubular segment after the renal corpuscle. It reabsorbs a large fraction of filtered water and solutes and participates in secretion. Its transport workload makes it a major site where cellular metabolism is coupled to whole-body fluid regulation.

A disorder predominantly involving the renal glomeruli. Changes in filtration-barrier integrity can produce protein or blood in urine, while the effect on total filtration varies with the process and extent.

Inflammation involving kidney tissue. More specific anatomical terms identify whether glomeruli, tubules or interstitium are predominantly affected.

Severe loss of renal function such that normal clearance and regulation cannot be maintained adequately. The condition can develop acutely or chronically and is not synonymous with every small increase in serum creatinine.