A complete theory in a countable first-order language is omega-stable if over every countable set of parameters it has only countably many complete one-types. The theory of a free bijection is an example: a new point is either a specified integer translate of a parameter or lies in a new orbit. It is omega-stable but not omega-categorical.

A Herbrand interpretation equal to the least Herbrand model of the positive reduct obtained relative to that interpretation.

Ground the rules, delete every rule having a negative body atom true in the candidate $I$, and remove all remaining negative premises. The result is a positive program whose least Herbrand model is compared with $I$.

A consequence holds skeptically if it is true in every admitted model, in particular every stable model when using stable model semantics. Exhibiting one supporting stable model establishes only existence of support, not skeptical entailment.

SQL NULL marks a missing or unknown value. Ordinary comparisons involving SQL NULL, including NULL = NULL, produce the truth value unknown. To test a scalar value for absence, use IS NULL; to compare while treating two missing values alike, use IS NOT DISTINCT FROM.^[ref]

This three-valued logic changes query reasoning. 5 NOT IN (2, NULL) is unknown: five differs from two, but the missing value might be five. A relational selection retains true conditions, so unknown does not pass the filter.^[ref]

Database aggregation has operation-specific rules. On values 2, NULL, 4, COUNT(*) counts three rows, COUNT(value) counts two non-null values and AVG(value) is three. With no input rows, COUNT returns zero while AVG and SUM return SQL NULL. Therefore a blanket rule that all database aggregation ignores SQL NULL is insufficient.^[ref]

For a polynomial $f\in K[x]$ of degree $n$, its splitting field $L$ has $[L:K]\mid n!$, including in positive characteristic when the polynomial is inseparable. Prove this by induction on the degree. If $f$ is irreducible, adjoining a root gives degree $n$ and leaves a polynomial of degree $n-1$ to split, so the tower law for field extensions gives the divisibility. If $f=gh$ with positive degrees $r,s<n$, split $g$ first and then $h$ over the resulting field. The degree divides $r!s!$, which divides $(r+s)!$. This separate reducible case avoids the invalid step of assuming that the degree of an arbitrarily chosen irreducible factor must divide $n$.

In ordinary bottom-up splaying, locate the selected node by a binary search tree search and repeatedly move it toward the root. If its parent is the root, perform a zig step. Otherwise use a zig-zig step when the node and its parent occupy the same left/right directions, or a zig-zag step when their directions differ. Each tree rotation preserves the inorder traversal and hence the represented key order. The process ends with the selected key at the root. Unsuccessful access policies can instead move the last visited node to the root; arguments about successful calls should say the requested key exists.

Given a polar axis, the equatorial plane passes through the sphere's centre and is perpendicular to that axis. In spherical coordinates with polar angle $\theta$, it is the plane $\theta =\pi /2$. For motion in a spherically symmetric gravitational field, conservation of angular momentum permits rotation of axes so that a nonradial orbit lies in such a plane. This is a choice of orientation, rather than an additional physical restriction on a spherically symmetric problem.

For a pulse in which only infectives diffuse, the travelling-wave equations imply $v_{-}=\eta (s^{\ast }-s_{-})/(1+\eta )$ and $m_{-}=\log (s^{\ast }/s_{-})/(1+\eta )$. Integrating total biological population balance with vanishing infective derivative at both infinities gives the displayed final-size relation. Its nontrivial root requires $0<s_{-}<1<s^{\ast }$. Thus deaths are exactly $(s^{\ast }-s_{-})/(1+\eta )$, while live density behind the wave is $(s_{-}+\eta s^{\ast })/(1+\eta )$. The root is independent of $\eta$ in this particular pulse model.

Suppose two spanning trees have the same first $n$ edges in nondecreasing weight order. If the first tree's next edge $e$ is no heavier than the second tree's next edge, it can be inserted into the second tree while preserving the common prefix and without increasing any sorted edge weight. If $e$ already occurs there, it ties for the next position. Otherwise its fundamental cycle contains an edge $f$ outside the prefix, since the prefix together with $e$ is acyclic. Every such $f$ is at least as heavy as $e$. Exchanging $f$ for $e$ preserves the spanning tree. Replacing a multiset entry by a no-larger entry cannot increase any order statistic. Tied edges can be ordered to preserve the prescribed prefix.

A spanning tree with greatest total edge weight. Apply Kruskal's algorithm in decreasing weight order, or negate every edge weight and compute a minimum spanning tree. Every graph path within a maximum spanning tree solves the widest path problem between its endpoints.

The continuous linear functional $f_a(x)=\sum a_n{x}_n$ on the real space of sequences converging to zero corresponds to $a\in {\ell }_1$ and has operator norm $\sum |a_n|$. Equality $f_a(x)=\Vert f_a\Vert$ on the closed unit ball forces $x_n=\mathrm{sgn}(a_n)$ on the support of $a$, since every term $|a_n|-a_n{x}_n$ is nonnegative. Such an $x$ can tend to zero only if that support is finite. Conversely, the finite sign sequence attains equality for every finitely supported sequence $a$.

The closed linear span of these vectors is all of $c_0$. Otherwise the Hahn-Banach theorem gives a nonzero continuous functional annihilating them, represented by $a\in {\ell }_1$. At the first index $m$ where $a_m\ne 0$, divide ${\sum }_j{a}_j{t}_n^j=0$ by $t_n^m$. The remaining tail has magnitude at most $t_n{\sum }_{j>m}|a_j|\to 0$, forcing $a_m=0$, a contradiction. This is an elementary coefficient-isolation argument for density by annihilators.

The longest nondecreasing prefix of a specified segment. For a nonempty segment $u[a..b)$, its endpoint is the first $c$ with $a<c<b$ and $u[c-1]>u[c]$, or $b$ if no such descent exists. Scanning adjacent pairs takes time proportional to the returned run length. An empty segment has an empty maximal initial run.

The Hamming number stream merges its shared multiples by two, three and five. To emit its first $n$ values, each of the fixed number of mapped streams and merge nodes advances at most $n+O(1)$ positions. With lazy evaluation that shares each evaluated list cell, this takes $\Theta (n)$ comparisons and fixed-factor multiplications. The statement counts arithmetic operations, rather than bits: the growing exact integers make their bit-operation costs larger. Recomputing independent copies of the recursively defined stream loses this sharing argument.

The assertion follows using only Sylow theorems. Write the order instead as $p{q}^2$ to separate the prime of multiplicity one. If $p<q$, the number of Sylow $q$-subgroups divides $p$ and is one modulo $q$, so one such subgroup is normal. If $p>q$ and the Sylow $p$-subgroup is not normal, its number must be $q^2$, forcing $p\mid (q-1)(q+1)$. Since $p>q$, this forces $p=q+1$ and hence $p=3,q=2$. In that order-twelve exception, four order-three subgroups account for eight nonidentity elements. The four remaining elements contain every order-four Sylow subgroup, so that subgroup is unique and normal. In every case a normal Sylow subgroup and its quotient have prime or prime-square order, hence are abelian groups. They provide a series with abelian factors and prove solubility. Orders with the squared exponent lowered to one or zero follow by the same simpler counting.

Sodium carbonate decahydrate is $\mathrm{N}{\mathrm{a}}_2\mathrm{C}{\mathrm{O}}_3\cdot 10{\mathrm{H}}_2\mathrm{O}$, a hydrated crystalline form of sodium carbonate. Its formula contains ten waters of crystallization per salt formula unit.

Loss of this water changes the measured mass and solid phase, so it cannot be replaced gram-for-gram by anhydrous sodium carbonate in a stoichiometric calculation. This is a clear example of why hydration belongs in a compound specification.

The bounded angular function has ordinary derivatives $D_{j}u={\delta }_{1j}/r-x_1{x}_j/r^3$ away from zero. Their magnitudes are at most $2/r$, whose $p$-th power is integrable on the unit ball exactly below the dimension threshold in this estimate. To verify the weak derivative across zero, integrate by parts outside a ball of radius $\epsilon$ against a compactly supported smooth test. The boundary term has magnitude at most $C{\epsilon }^{n-1}$, tending to zero for $n>1$, while the derivative volume term converges by integrability. This proves the stated Sobolev membership with those weak derivatives. The dimensional condition distinguishes it from the one-dimensional sign function, whose jump produces a Dirac derivative.

The duodenum is the first small-intestinal segment after the stomach. Pancreatic and biliary secretions enter its region to support digestion and modification of gastric contents. Its anatomy connects digestive secretions from accessory organs with material arriving through the pylorus.

The ileum is the distal part of the small intestine before entry into the large intestine. Its specialized transport includes recovery of bile-associated material and absorption of vitamin B12 under the appropriate carrier conditions. It also contains prominent gut-associated lymphoid structures.