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The area postrema is a specialized vascular region near the Caudal fourth ventricle, involved in detecting circulating signals and contributing to emetic responses. Its close relation to vagal sensory circuits connects blood-borne information with coordinated behavior.

The funiculus separans is a narrow ridge in the Caudal floor of the fourth ventricle, separating the vagal trigone from the area postrema. It is a surface landmark, not a complete boundary between all underlying neural pathways.

The use of resolution on ground clauses. A finite refutation ends in the empty ground clause and establishes unsatisfiability of the chosen ground instances. Herbrand's theorem explains the completeness of searching for such refutations for universal equality-free first-order clauses.

A rule that combines first-order clauses by unifying complementary first-order literals and retaining the other literals under the resulting first-order substitution. Completeness requires the appropriate calculus and search conditions.

For a finite graph of maximum degree $\Delta \ge 1$, a random independent edge coloring and the Lovász local lemma prove existence of a coloring with no repetitively colored four-edge path when $k\ge 8{\Delta }^{3/2}$, and no repetitively colored six-edge path when $k\ge 5{\Delta }^{5/3}$.

A fixed edge lies in at most $\ell {\Delta }^{\ell -1}$ unoriented paths of length $\ell$. Hence a four-edge event has a closed dependency neighbourhood of size at most $16{\Delta }^3$ and probability $k^{-2}$; the symmetric local-lemma product is at most $16e{\Delta }^3/k^2$. The corresponding six-edge quantities are $36{\Delta }^5$, $k^{-3}$ and $36e{\Delta }^5/k^3$. The stated constants make both products less than one.

Both lengths can be excluded simultaneously with $k\ge C{\Delta }^{5/3}$ for an absolute constant $C$. Apply the asymmetric Lovász local lemma with parameters $x_4=2/k^2$, $x_6=2/k^3$. A path of length $\ell \in \{4,6\}$ meets at most $4\ell {\Delta }^3$ four-edge paths and $6\ell {\Delta }^5$ six-edge paths. The sums of neighbour parameters are therefore bounded by constants times $C^{-2}{\Delta }^{-1/3}+C^{-3}$. Choosing $C$ sufficiently large makes both products in the local-lemma inequalities at least one half, establishing the claim.

Suppose a reference polynomial $q$ has nonzero alternating errors of magnitude $|h|$, and another point $\eta$ has error magnitude $M>|h|$. Insert $\eta$ and delete a reference point so that the retained errors, in spatial order, still alternate. Solving the new alternation equations gives the displayed strict increase in reference height, with $w_j={\prod }_{k\ne j}(z_j-z_k{)}^{-1}$ on the new ordered set. To prove this, apply the order-$(n+1)$ divided difference to the old and new degree-at-most-$n$ polynomials. Both vanish, so the new signed height is the sum of the old errors times $w_j$, divided by ${\sum }_j(-1{)}^j{w}_j$. All these signed weight products have one sign. Its absolute value is therefore the positive weighted average of one value $M$ and $n+1$ values $|h|$.

For an interior $\eta$, replace the endpoint of its containing reference interval with the same error sign. If $\eta$ is exterior, replace the nearest endpoint when the signs agree; if they disagree, insert $\eta$ and drop the farthest endpoint. These rules preserve ordered alternation. The nearest point with the same sign need not be an adjacent point on an irregular grid, and an unqualified nearest-same-sign replacement does not ensure the strict increase.

For positive-precondition STRIPS with a positive conjunctive goal, remove every delete effect and use the exact shortest relaxed plan cost. Every original plan still succeeds in the relaxed model because facts can only accumulate, so its relaxed optimum is an admissible heuristic. The cost of an arbitrary relaxed plan is an upper bound on the relaxed optimum and need not be an admissible heuristic for the original problem.

Map original states to simpler abstract states, include every projected original transition at no greater cost, and map original goals to abstract goals. An original goal-reaching graph path then projects to an abstract goal-reaching walk, so the exact abstract distance is an admissible heuristic. A pattern database stores such distances for repeated lookup.

A natural join combines two database relations by equating every identically named attribute and retaining one copy of each common attribute. If people(id,name) contains (7,Ada) and scores(id,mark) contains (7,90), their natural join contains (7,Ada,90). With no common names it becomes a Cartesian product. Matching several common columns requires all their equalities, so adding a same-named column can silently change the result. An explicit join condition is often clearer when schemas evolve. In SQL, duplicates and null values follow the implementation's join semantics.^[ref]

If $Z$ is a Banach space and $X$ is a norm-closed reflexive linear subspace of $Z^{\ast }$, evaluation gives the displayed surjection. Restriction $R:Z^{\ast \ast }\to X^{\ast }$ is weak-star continuous and maps the closed unit ball onto the target ball by the Hahn-Banach theorem. The Goldstine theorem then makes $T(B_Z)$ weak-star dense in $B_{X^{\ast }}$, where $T=R{J}_Z$. Reflexivity of $X$ makes this target weak-star topology equal to its weak topology. Convexity and the Hahn-Banach separation theorem convert weak density of this ball image to norm density.

To pass from density to surjectivity, approximate a desired $f$ by $T{z}_1$ with $\Vert z_1\Vert \le \Vert f\Vert$ and residual at most $\Vert f\Vert /2$. Repeat on each residual. The chosen vectors have summable norms, with sum at most $2\Vert f\Vert$, and the residuals tend to zero geometrically. Completeness of $Z$ produces $z={\sum }_n{z}_n$ with $Tz=f$. Using any smaller residual ratio proves the infimum lifting norm equals $\Vert f\Vert$, although that infimum need not be attained. This argument proves onto-ness, rather than merely density of the evaluation functionals.

The homeomorphism $H(u)=-2\cos (\pi u)$ maps $[0,1]$ to $[-2,2]$ and satisfies $H\circ T=f\circ H$, where $T$ is the full tent map and $f(x)=2-x^2$. This transfers dense periodic points, topological mixing and sensitivity to the quadratic interval map. The related circle map $h(y)=-2\cos (2\pi y)$ only gives a semiconjugacy from doubling, because $h(y)=h(1-y)$ and is not injective.

An autocatalytic species promotes its own growth while consuming a substrate that promotes that growth. Its reaction Jacobian has the displayed signs after choosing the activator first. This differs from the activator-inhibitor sign pattern $\left(\begin{array}{cc}+& -\\ +& -\end{array}\right)$, where the activator stimulates its inhibitor. Appropriate magnitudes, diffusivities and admissible spatial modes are needed for a Turing instability; signs alone are insufficient.

For $0<k\le 1$, seek a rational function $r(x)=ax/(1+bx)$ that has $r(0)=0$, $r(1)=1$ and $r^{\prime }(1)=k$, matching the value and first derivative of $x^k$ at one. The last two conditions give $a=1+b$ and $a/(1+b{)}^2=k$, henceFor $x\in [0,1]$ its denominator is positive, so it has no pole there. The usual degree-$(1,1)$ Padé approximant about one is instead $[(1-k)+(1+k)x]/[(1+k)+(1-k)x]$, matching the next Taylor series coefficient while generally failing the zero-endpoint condition.

The limiting case $k=0$ needs care: for each fixed $x>0$, the constrained expression tends to one, while its value at $x=0$ remains zero for every $k>0$. Cancelling $x/x$ gives the constant extension one only after abandoning the condition $r(0)=0$. Thus the three original constraints cannot all hold in this family at $k=0$.

Let $U=aT+bI$ on a nonzero finite-dimensional vector space over a field, with $a\ne 0$. An invariant block for $T$ is also invariant for $U$. If $v,Tv,\dots ,T^{d-1}v$ is its cyclic vector basis, then $v,Uv,\dots ,U^{d-1}v$ is another basis: the binomial theorem gives a triangular change-of-basis matrix with diagonal $1,a,\dots ,a^{d-1}$. The monic polynomial $g(t)=a^{d}f((t-b)/a)$ annihilates this block, and its degree is $d$, so it is the block's minimal polynomial. In the new cyclic basis the block is $C(g)$.

The substitution preserves divisibility of the monic invariant factors of a linear operator. Apply it to every block to obtain the rational canonical form of $U$. If $a=0$, then $U=bI$ is a scalar matrix instead: all invariant factors are $t-b$, one for each dimension. This exceptional case cannot be obtained by dividing by $a$.

Let an invertible matrix $T$ have monic invariant factors of a linear operator $f_j$ of degrees $d_j$. Their constant coefficients are nonzero, since their product is the characteristic polynomial and $\det T\ne 0$. On a cyclic block, $f(T)=0$ implies $T^{-d}f(T)=0$, so $f^{\vee }(T^{-1})=0$, where $f^{\vee }(t)=t^{d}f(1/t)/f(0)$ is monic of degree $d$. The vector $v$ remains cyclic for $T^{-1}$: multiplying $v,T^{-1}v,\dots ,T^{-(d-1)}v$ by the invertible map $T^{d-1}$ gives the reversed original cyclic basis. Thus this polynomial is the minimal polynomial of the inverse block.

Normalized reciprocation preserves divisibility because $(fg{)}^{\vee }=f^{\vee }{g}^{\vee }$ when both constant coefficients are nonzero. Consequently the rational canonical form of $T^{-1}$ is the direct sum of the companion matrices $C(f_j^{\vee })$. The normalization by $f_j(0)$ is essential to make each reciprocal polynomial monic.

For a nonzero finite-dimensional cyclic linear operator, identify its module with $K[X]/(d)$, where $d$ is the monic minimal polynomial. Its kernel consists of classes of $h$ with $d\mid Xh$. If $d(0)\ne 0$, coprimality gives $d\mid h$, so the kernel is zero. If $d=Xe$, it consists precisely of the multiples of $e$ modulo $Xe$, a one-dimensional space. Thus the multiplicity of zero as a root of $d$ does not increase the nullity. The same companion block reads off invertibility from the constant term and gives the characteristic and minimal polynomials directly, while explicitly listing eigenvalues may require factoring $d$.

If each successive inertial frame moves at the same signed velocity $u$, with $|u|<c$, relative to the preceding frame along a common axis, the rapidity adds. The final velocity relative to the first frame is therefore the displayed expression. When $u=c/N$ and $N\ge 2$, the limit is $c\tanh 1$, since $N\mathrm{artanh}(1/N)\to 1$.

For finite-dimensional linear maps $T:U\to V$ and $S:V\to W$, the restriction of $T$ to $\ker (ST)$ maps onto $\mathrm{im}T\cap \ker S$ and has kernel $\ker T$. The rank-nullity theorem on this restriction proves the title identity. Since the intersection has dimension at most $\dim \ker S$, it implies $n(ST)\le n(T)+n(S)$. Equivalently, $r(ST)\ge r(T)+r(S)-\dim V$, the rank inequality for a composition. Also $\mathrm{im}(ST)=S(\mathrm{im}T)$ directly gives the upper bounds $r(ST)\le \min (r(S),r(T))$.

Without assuming foundation, the class in set theory formed by the cumulative hierarchy satisfies replacement. Relativize a functional logical formula to that definable class in set theory and use ambient replacement to collect its range. Every output belongs to the hierarchy class in set theory, and the rank-bounding result places the range set itself in that class in set theory.

Retain each of $N$ objects independently with probability $r$, independently of the nonnegative integer-valued random variable $N$. The retained count $Y$ has the displayed conditional binomial distribution. The probability generating function of an independent random sum gives $G_Y(s)=G_N(1-r+rs)$. If $N$ has finite variance, the law of total expectation and law of total variance giveThe first term is the average conditional variance of the independent retention decisions; the second is the variance in their conditional expected value.