Decoctum Cinchonae Flavae is a decoction of yellow cinchona bark. Heating during extraction distinguishes it from the corresponding infusion.^[ref]

Infusum Cinchonae Flavae is an infusion of yellow cinchona bark. It differs in preparation process from the corresponding decoction.^[ref]

Extractum Cinchonae Flavae Liquidum is a liquid extract of yellow cinchona bark. Its concentration and extraction distinguish it from an infusion of the same source material.^[ref]

Tinctura Cinchonae Flavae is a tincture of yellow cinchona bark. The specified bark and solvent extraction together define the historical article.^[ref]

Tinctura Cinchonae Composita is a compound cinchona tincture. Additional ingredients distinguish it from a tincture of one specified bark material alone.^[ref]

When $\alpha =\kappa =\eta =1$, the instability threshold of a spatial mode with squared wave number $s$ is $\beta =s+2+1/s$. On a rectangle $L_1=1$, $L_2=1/2$, modes varying in both coordinates have $s\ge 5{\pi }^2$. Since the threshold increases for $s>1$, the first such mode is $(n,m)=(1,1)$, at ${\beta }_{crit}=5{\pi }^2+2+(5{\pi }^2{)}^{-1}$. At equality it is neutral; strict instability begins above this value.

An unbiased coefficient estimator with known variance $v$ gives a conservative level-$\alpha$ test of a zero coefficient by rejecting when its absolute value is at least $\sqrt{v/\alpha }$. Chebyshev's inequality bounds the null rejection probability by $\alpha$. This test needs the variance assumptions but no Gaussian error distribution; a normal-reference test requires additional distributional justification.

For $0<a<b$, the displayed polynomial minimizes ${\max }_{t\in [a,b]}|p(t)|$ among real polynomials of degree at most $k$ with $p(0)=1$. Its $k+1$ alternating extremal values prove the assertion: a strictly smaller competitor would make the difference change sign in each consecutive extremal interval and also vanish at zero outside $[a,b]$, giving too many roots. The normalizing factor makes either the monic or the usual Chebyshev polynomial convention give the same ratio. For $\kappa =b/a$ and $q=(\sqrt{\kappa }-1)/(\sqrt{\kappa }+1)$, its maximum is $2q^k/(1+q^{2k})$. This yields an error bound for polynomial acceleration of Richardson iteration.

The positive characteristic of a ring which is an integral domain is a prime number. The characteristic cannot be one because $1\ne 0$. If its least positive value were $n=ab$ with $1<a,b<n$, minimality would make $a1$ and $b1$ nonzero, while $(a1)(b1)=n1=0$ would give zero divisors.

The analogous result holds for a nonzero commutative nonunital ring without zero divisors, using the characteristic of a rng. If $n=ab$ were composite and minimal, choose $r$ with $ar\ne 0$. For every $s$, $(ar)(bs)=n(rs)=0$, so cancellation forces $bs=0$ for every $s$, contradicting minimality of $n$.

In a finite unital ring $R$, some positive multiple of $1$ is zero, because two of the finitely many successive multiples coincide. If $n=\mathrm{char}R$, the permutation $r\mapsto r+1$ has every cycle of a permutation of length exactly $n$: returning to $r$ after $k$ steps means $k1=0$, whose least positive solution is $n$. These disjoint cycles partition the finite set $R$, so $n$ divides its cardinality. The zero ring has cardinality and characteristic both one.

The radial transport equation has projected characteristic curves $(x,y)=e^s(x_0,y_0)$, which remain on positive rays. Data on $x+y=1$ reach precisely the half-plane $x+y>0$, since $x+y=e^s$. A datum $u(a,1-a)=h(a)$ therefore gives $u(x,y)=(x+y{)}^{k}h(x/(x+y))$ in that half-plane. The opposite half-plane is not determined by these data. Extending a characteristic line through the origin would join distinct flow orbits across a point where the projected vector field vanishes.

Expurgation discards messages with unusually high conditional decoding error. If a codebook with $2M$ equally likely messages has average error at most $ϵ$, at least $M$ messages have individual error at most $2ϵ$, by Markov's inequality. Retaining those messages and restricting the decoder's candidates gives a codebook with maximal error at most $2ϵ$, provided removal of candidates cannot spoil a previously successful decoding. This is the case for bounded-distance decoding with errors on ambiguity, and for a decoder that retains its former decisions whenever the decoded message remains.

For a sequence of block lengths $n$, halving a book changes its logarithmic rate by $1/n$, which tends to zero. Thus a vanishing average-error bound from a random coding construction can be converted into a vanishing maximal decoding error probability at the same limiting rate. The probabilistic construction first supplies a deterministic book with sufficiently small average error; expurgation is a subsequent deterministic step.

Random coding constructs a channel code by sampling its codewords from a specified probability distribution and bounding the expected decoding error over that ensemble. If the expected average error is at most $ϵ$, at least one deterministic codebook has average error at most $ϵ$; otherwise every book would exceed the expectation bound. This is the probabilistic method applied to communication codes.

For independently uniform binary codewords, a fixed competing word falls in any received word's radius-$r$ Hamming ball with probability $2^{-n}{\sum }_{j=0}^r\left(\genfrac{}{}{0ex}{}{n}{j}\right)$. The union bound over competing messages, together with a bound on excessive channel noise, yields the basic random-coding error estimate. Expurgation of a codebook can then convert average-error control to maximal decoding error probability with negligible loss of asymptotic rate.

Historical formulation name; composition is identified by pharmacopoeial edition.

The attachment region of a tapeworm.

A reproductive unit of the body of many adult tapeworms.

A bladder-like larval form with an invaginated scolex, occurring in the life cycles of some Taenia.

Taenia is a genus of tapeworms whose adult and larval stages use different hosts. Adults possess a scolex and reproductive proglottids, while tissue larvae vary among species. Taenia solium and Taenia multiceps illustrate different larval forms.

The broad fish tapeworm, historically described under different generic names.

Echinococcus is a genus of tapeworms with intestinal adults in a definitive host and tissue larvae in an intermediate host. The larval form produces hydatid cysts; different species have different patterns of cyst growth.^[ref]