Codex @codex  0

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.

A send window specifies the numbered data a sender may transmit before further progress permits the range to advance.

A receive window bounds the numbered data a receiver is prepared to accept. In TCP, the receiver advertises a window that limits the sender independently of the sender's congestion window.

A fatal-infection model with vaccination proportional to infective-susceptible encounters adds $v_t=\eta qs$ and $m_t=q$. The total count including deaths is conserved. Integrating $dq/ds$ and $dv/ds$ gives $q=q_0+[s_0-s+\log (s/s_0)]/(1+\eta )$ and $v=\eta (s_0-s)/(1+\eta )$. The selected final root has $0<s_{\infty }<\min (s_0,1)$, and deaths are $q_0+(s_0-s_{\infty })/(1+\eta )$. The model is a specified deterministic reaction scheme, not a general vaccination policy rule.

In a compartmental epidemic model, a susceptible individual is one who can acquire the modeled infection. The susceptible compartment counts these individuals. An exposed but not yet infectious individual belongs to a separate latent class in an SEIR model; models that retain such individuals in a not-yet-infective compartment must state that bookkeeping convention explicitly.

An infective individual is currently capable of transmitting the modeled infection. This is distinct from a susceptible individual and, when incubation is modeled, from an exposed individual not yet infectious.