In a partially fixed two-colour edge coloring of a complete graph, complete all unassigned edges independently and fairly. For a fixed clique with $r$ unassigned edges, its conditional probability of becoming monochromatic is zero if both colours already occur; $2^{-r}$ if a colour is fixed and no conflicting colour occurs; and $2^{1-r}$ if none is fixed. Sum these conditional probabilities over the cliques to obtain a computable conditional expectation. Fix each next edge to a colour that does not increase this conditional expectation, yielding a deterministic minimization construction.

A static liquid meniscus rising on a perfectly wetting vertical wall has dimensional height $z_0=\sqrt{2}{\ell }_c$, where ${\ell }_c=\sqrt{\sigma /(\rho g)}$ is the capillary length. In capillary-length coordinates, write the interface $x=h(z)$. The Young–Laplace equation gives $h_{zz}/(1+h_z^2{)}^{3/2}=z$ and the first integral $h_z/\sqrt{1+h_z^2}=z^2/2-1$. The integration constant follows from the horizontal far bath, where $z\to 0$ and $h_z\to -\infty$. Zero contact angle means $h=h_z=0$ at the wall, so $z_0=\sqrt{2}$. Expanding and integrating there gives $h\sim (z-z_0{)}^2/\sqrt{2}$, the parabolic shape used to match an entrained film.

Represent each generated sequence position by an immutable head and a lazily memoized successor, and give each iterator its own pointer into the shared chain. Computing a successor invokes the privately owned generator once; lagging copies reuse that same successor. Consequently a mutable programming closure generating Fibonacci numbers can support interleaved independent cursor positions. Sharing only the generator and a copied current value would instead consume the generator's mutable state twice. This construction can retain all cells between the fastest and slowest live cursors; it does not guarantee constant memory usage.

The inferior olive is a medullary nuclear complex providing climbing-fiber input to the cerebellum. Its connections participate in cerebellar timing and plasticity. A teaching-related role in one task should be distinguished from a claim that every olivary signal encodes the same error variable.

The nucleus ambiguus is a medullary neuronal group contributing branchial motor pathways to pharyngeal and laryngeal muscles and containing cardiac vagal populations. Different cell groups have different targets. The anatomical label therefore spans more than one physiological role.

The dorsal motor nucleus of the vagus contains preganglionic parasympathetic neurons whose axons enter the vagus nerve. Many outputs regulate thoracic and abdominal organs. It is distinct from sensory vagal nuclei and from the branchial motor populations of the nucleus ambiguus.

The nucleus of the solitary tract is another name for the solitary nucleus. It receives visceral sensory input and contributes to distributed reflex circuits, including those regulating cardiorespiratory and digestive functions.

The spinal trigeminal nucleus is an elongated brainstem sensory complex involved particularly in pain and temperature information from the face and associated territories. Its Caudal extension and incoming tract distinguish it from a single compact relay at the entry of the trigeminal nerve.

The hypoglossal nucleus contains motor neurons whose axons travel through the hypoglossal nerve to most muscles of the tongue. It lies near the floor of the fourth ventricle, beneath the hypoglossal trigone. Surface position and motor connections describe different aspects of the same nucleus.

Nucleus tractus solitarius is the Latin-derived name commonly used for the solitary nucleus. The name refers to a central sensory complex associated with the solitary tract, rather than to a separate peripheral ganglion.

The solitary nucleus is a medullary sensory complex receiving visceral and taste-related inputs through several cranial nerves. Its connections contribute to autonomic and behavioral responses. Nucleus tractus solitarius and nucleus of the solitary tract are alternate names for this complex.

These historical papers use the First and Second Examination divisions printed in the original volume. They are not mapped to the years or stages of the modern medicine course of the University of Oxford.

Exclusion of the entire carcass from food use. This differs from condemnation of an isolated diseased organ; the nature and distribution of disease, and the applicable jurisdiction and date, matter.^[ref]

A sequence of real measurable functions converges in measure to $f$ when the displayed condition holds for every positive threshold. It controls the measure of the exceptional set at each index, rather than requiring eventual closeness on each sample path. On a probability space it is exactly convergence in probability. On a finite measure space, almost-everywhere convergence implies convergence in measure by continuity of measure on the decreasing tail events. The converse need not hold for the full sequence, but convergence in measure always admits an almost-everywhere convergent subsequence.

For a classical solution of $u_t-\Delta u+cu=0$ with $c\ge 0$, the positive part of its maximum is bounded by the corresponding maximum on the initial and lateral parabolic boundary. When $c>0$, a positive maximum in space at an interior or final time gives $u_t\ge 0$, $\Delta u\le 0$, and $u_t-\Delta u+cu>0$, a contradiction. For $c=0$, a small strictly increasing exponential time perturbation gives the weak inequality by a limiting argument. Applied to both signs of the difference of two solutions, the principle proves uniqueness for fixed initial and boundary data.

A nonconstant holomorphic function cannot have a positive interior local minimum of its modulus. Wherever it is nonzero, its reciprocal is holomorphic; a positive modulus minimum would give a modulus maximum of the reciprocal, contradicting the maximum modulus principle. Thus if a holomorphic function on a neighborhood of a closed disk has modulus at the center strictly smaller than its minimum boundary modulus, it has a zero inside the disk. Otherwise the Cauchy estimate applied to its reciprocal would give the opposite inequality.

For finite random variables $X,Y$, let $P_e$ be the error probability of maximum a posteriori decoding of $X$ from $Y$. Then $P_e\le H(X\mid Y)/2$ when conditional entropy is measured in bits. At each positive-probability output $y$, the optimum success probability is the largest posterior atom ${\Pi }_y$, so its conditional error is $1-{\Pi }_y$. Apply the entropy lower bound from the largest atom to that posterior to get $2(1-{\Pi }_y)\le H(X\mid Y=y)$, then average over $Y$. An arbitrary decoder need not satisfy this bound; it uses the optimum conditional decision rule.

For an undirected finite graph, its forest polytope is the matroid independence polytope of its graphic matroid. It has the exact inequality descriptionsHere $\kappa (V,A)$ counts all connected components of a graph, including isolated vertices, and $E(S)$ denotes the edges internal to $S$. A maximal forest has one fewer edge than vertices in each component, giving the rank $|V|-\kappa (V,A)$. The matroid independence polytope theorem therefore proves the first description. It implies the second since the rank of $E(S)$ is at most $|S|-1$. Conversely, split any $A$ into its vertex components $S_j$; nonnegativity gives $x(A)\le {\sum }_{j}x(E(S_j))\le {\sum }_j(|S_j|-1)=|V|-\kappa (V,A)$. No spanning requirement is imposed, so an optimum with negative edge weights can leave vertices isolated.

Given an initial value $a$ and a specified successor rule on a set, there is a unique function on the natural numbers with that initial value and successor rule. Finite compatible approximations give existence; induction gives agreement of any two approximations and uniqueness. Recursive definitions of addition and multiplication apply this theorem with the other argument fixed.

For a continuously differentiable energy on the connected set of positive-determinant matrices, material frame indifference differentiates to the displayed covariance of the first Piola-Kirchhoff stress tensor. Conversely that covariance makes $W(QF)-W(F)$ independent of $F$. Call the constant $c(Q)$. Composition gives $c(Q_1{Q}_2)=c(Q_1)+c(Q_2)$. Continuity and compactness of the rotation group make $c$ bounded, while $c(Q^n)=nc(Q)$ forces every $c(Q)$ to vanish. Thus no additive rotation-dependent energy ambiguity remains. Homogeneity means the energy has no explicit dependence on the material point; it is separate from this observer-independence property.