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For a holomorphic function on the unit disk fixing zero, $g(z)=f(z)/z$ has a removable singularity at zero and a bounded holomorphic extension. A bounded holomorphic function evaluated along planar Brownian motion is a bounded stopped martingale before domain exit. Stop at the first exit from a concentric radius-$r$ disk with $|z|<r<1$. Optional stopping gives $g(z)={\mathbb{E}}_{z}g(Z_{T_r})$, hence $|g(z)|\le 1/r$. Letting $r$ increase to one proves the displayed bound.

If $|g(z_0)|=1$ at a nonzero point, put $c=g(z_0)$. The same expectation identity forces $\mathrm{Re}(\overline{c}g(Z_{T_r}))=1$ almost surely. The exit density is strictly positive on every boundary arc by the Poisson kernel for the unit disk, and $g$ is continuous there, so $g=c$ on that circle. The expectation identity for every starting point inside it then gives $g=c$ throughout that smaller disk. Letting $r$ increase to one shows $f(z)=cz$ on the full disk, with $|c|=1$. This derives both the inequality and its rigidity using Brownian harmonic measure.

A process scheduling policy that can replace a running, still-runnable task with another task. Time-sliced round-robin scheduling uses a hardware timer interrupt to regain control when a scheduling quantum expires.

The time at which all work in a scheduling instance has finished, equivalently the largest completion time measured from a common start. For divisible job sizes $t_i\ge 0$ and positive machine speeds $s_j$, when each machine's completion time is its total assigned size divided by its speed, the minimum makespan is ${\sum }_i{t}_i/{\sum }_j{s}_j$. The lower bound follows from total processing capacity; assigning the fraction $s_j/{\sum }_h{s}_h$ of each job to machine $j$ attains it.

For single-machine jobs all available at time zero, order by nondecreasing deadline. Keep a tentative on-time list. Whenever appending a job makes its completion exceed its deadline, delete a job of largest processing time from that list and place it on a rejected list. Append all rejected jobs at the end. This minimizes the number of late jobs: at the first missed deadline at least one job of that prefix must be rejected, and rejecting a longest one frees at least as much processing time as any alternative. The exchange argument applies recursively to the remaining prefixes. The algorithm needs no ordering of the rejected jobs to determine the minimum count.

A calcium salt contains calcium ions, usually $\mathrm{C}{\mathrm{a}}^{2+}$, together with charge-balancing anions. Calcium chloride and calcium carbonate are familiar examples with markedly different water solubilities.

Writing the correct formula requires total charge balance. The two-charged cation can also coordinate several oxygen-containing groups, connecting calcium-salt chemistry with minerals, solution complexation, and biological binding sites.

Potassium carbonate is the ionic compound ${\mathrm{K}}_2\mathrm{C}{\mathrm{O}}_3$. In water, carbonate participates in proton-transfer equilibria and gives a basic solution.

Its formula is fixed by charge balance between ${\mathrm{K}}^{+}$ and $\mathrm{C}{\mathrm{O}}_3^{2-}$. Carbonate and bicarbonate are distinct protonation states, so potassium carbonate and potassium bicarbonate do not have the same acid-base capacity per mole.

Magnesium carbonate has ideal composition $\mathrm{MgC}{\mathrm{O}}_3$; natural and commercial materials can also include hydrated or basic carbonate forms. Magnesite is the anhydrous carbonate mineral.

Carbonate reacts with acid to release carbon dioxide, while the magnesium remains in solution or another product. A material's precise phase and composition must therefore be specified rather than inferred solely from its common label.

Sodium sulfate is $\mathrm{N}{\mathrm{a}}_2\mathrm{S}{\mathrm{O}}_4$, occurring in anhydrous and hydrated solid forms. Two sodium ions balance each sulfate dianion.

A hydrated crystal includes chemically specified water in its formula unit. This changes the relation between measured mass and amount of sodium sulfate, and can alter the temperature-dependent phase behavior of the material.

A potassium salt contains potassium ions, usually ${\mathrm{K}}^{+}$, balanced by anions. Examples include potassium chloride, potassium sulfate, and potassium carbonate.

The shared cation does not give these compounds identical solubility or acid-base behavior. Those properties also depend on the anion, crystal structure, hydration, and the solution conditions.

Salivary secretion produces and modifies the fluid entering the mouth from salivary glands. Secretory cells supply components that are further modified during passage through ducts. Changes in flow rate can therefore change composition as well as total fluid volume.

The submandibular gland is a major salivary gland associated with the region beneath the mandible and the floor of the mouth. Its duct passes forward to open near the tongue. A ductal blockage therefore affects a defined drainage route rather than all oral secretions equally.

The sublingual gland is a major salivary gland beneath the oral lining beside the tongue. Numerous small ducts open into the floor of the mouth. Its location distinguishes it from the deeper and more posterior submandibular gland.

The parotid gland is a major salivary gland near the side of the face, anterior and inferior to the ear. Its duct crosses the cheek to reach the mouth. Branches of the facial nerve traverse the gland, an important spatial relation distinct from its secretory innervation.

This scalar quasilinear partial differential equation results by imposing $h=Q/u$ in a frictional river momentum equation and measuring $X$ upstream. It retains a relaxation toward $u_0=Q^{1/3}$ and carries information at speed $c(u)=Q/(F^2{u}^2)-u$. Near equilibrium the relaxation time is $\epsilon F^2{u}_0/3$. A slowly moving downstream boundary can generate an attached boundary layer when $c(u)$ exceeds its speed throughout the relaxation. The model retains momentum dynamics only: exactly constant flux in both original Saint-Venant equations would instead force time-independent depth.

An attack exploiting a known relation between plaintexts encrypted using one textbook RSA cryptosystem key. For public exponent three and a known nonzero additive difference, two cubic equations can yield a linear relation recovering a plaintext. A small unknown difference can be enumerated.

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.