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For a formal power series over a field of characteristic zero, the displayed coefficientwise operation chooses constant coefficient zero. Its formal derivative is the original formal power series: differentiating the term with exponent $n+1$ cancels its denominator. Conversely, formally integrating a formal derivative removes the original constant term. These are algebraic identities; analytic convergence is not required.

For a symmetric gap $h(x)=\sqrt{1+x^2}$ on $[-1,1]$, unit lower-wall speed and equal zero gauge pressures at the ends, the flux average for a translating lubrication gap gives $Q=\pi /(4\sqrt{2})$. Integrating $p_x=6(h-2Q)/h^3$ gives the displayed odd pressure. It has a positive maximum at $x=-\sqrt{{\pi }^2/8-1}$ and a negative minimum at the opposite point. The converging and diverging portions of a symmetric gap therefore have opposite pressure lobes even though the geometry itself is even: the directed wall motion breaks reflection symmetry of the flow.

If a normal function on an ordinal begins at a countable value and preserves countability from $\Phi (\xi )$ to $\Phi (\xi +1)$, transfinite induction shows it maps all countable ordinals to countable ordinals. At countable limits this uses countable unions of countable sets, hence axiom of countable choice in the usual proof. Iteration from zero either stabilizes at a countable fixed point or has a countable limit fixed by continuity. Applying this to ${\omega }^{\alpha }$ gives the countable epsilon number ${ϵ}_0$.

Precision comparisons using Fisher information must specify what one observation records. A Bernoulli distribution with success probability $\theta$ gives information $I_{\mathrm{Bern}}=1/[\theta (1-\theta )]$ per trial. A geometric distribution on $1,2,\dots$ records the number of trials until the first success. Its log-likelihood for one waiting time $X$ is $\log \theta +(X-1)\log (1-\theta )$, soAt equal numbers of observations, geometric waiting times therefore provide $1/\theta$ times as much information, and the asymptotic variance of their maximum-likelihood estimator is $\theta$ times the Bernoulli variance. But each waiting time uses an expected $1/\theta$ trials. Information per expected underlying trial is consequently $\theta I_{\mathrm{geom}}=I_{\mathrm{Bern}}$. This separates an observation count from the cost of collecting the observations.

If $X$ has an exponential distribution with rate $\lambda >0$, a full observation has Fisher information $1/{\lambda }^2$. Recording only $Y=1_{\{X>T\}}$ for a fixed $T>0$ instead gives a Bernoulli distribution with parameter $r=e^{-\lambda T}$. The chain rule for its score function givesTo see the strict inequality, write $e^s-1=s+s^2/2+s^3/6+\cdots$. For $0<s\le 1$ the first two terms already exceed $s^2$. For $s\ge 1$, the function $e^s-1-s^2$ is positive at one, its derivative $e^s-2s$ is positive there, and that derivative is increasing for $s\ge 1$. Thus every positive threshold discards information. For independent observations, both information quantities multiply by the sample size. The Cramér-Rao bound and asymptotic normality of a maximum likelihood estimator explain the corresponding loss of attainable precision.

For $n$ independent exponential lifetimes with rate $\lambda$, recording all lifetimes gives Fisher information $n/{\lambda }^2$. Recording only the count alive at a fixed time $T>0$ gives a binomial distribution with success probability $s=e^{-\lambda T}$. Its information in $s$ is $n/[s(1-s)]$, and the change of parameter $ds/d\lambda =-Ts$ gives the displayed information.

The retained fraction is $R(z)=z^2/(e^z-1)$, $z=\lambda T$. It is strictly below one, so every single snapshot loses information. The inequality follows, for instance, from $h(z)=e^z-1-z^2$: its derivative is $e^z-2z>0$ for positive $z$, because that derivative has its minimum at $z=\log 2$ with value $2-2\log 2>0$. Also $R(z)$ tends to zero as $z\downarrow 0$ and as $z\to \infty$.

Differentiating $R$ shows that the best snapshot time solves $2(1-e^{-z})=z$. The difference of the two sides is strictly concave, starts at zero with positive derivative, and eventually becomes negative, so it has exactly one positive root, approximately $1.59362$. The largest retained fraction is about $0.64761$. Thus even an optimally timed snapshot keeps less than two thirds of the full lifetime information. This is a local design result: choosing that time requires a nominal rate or a preceding estimate.

A first-order term containing no variables. Constant symbols are ground terms; applying a function symbol to ground terms produces another ground term.

A closed term contains no variables, so its value in a first-order structure does not depend on an assignment. In a language containing only relations and the constant zero, zero is the only closed term. A term-generated model has every element denoted by such a term.

After replacing functions by graph predicates and constants by singleton predicates, $E_t(z)$ says that $z$ is the value of a term $t$. For a unary function graph $P$ and constant predicate $Q$, use $E_x(z)=(z=x)$, $E_c(z)=Q(z)$ and $E_{f(t)}(z)=\exists y(E_t(y)\wedge P(y,z))$. Fresh witnesses avoid variable capture. Totality and uniqueness axioms make each value formula have exactly one witness under every assignment.

A first-order structure and an assignment determine the value of every term recursively: a variable has its assigned value, a constant has its interpreted value, and a compound function term is evaluated by applying the interpreted function to the recursively evaluated arguments. This operation supplies the input values for the satisfaction clauses of atomic formulas.

A family of resolution calculi for first-order Horn clauses. A selection function chooses either the positive head atom alone or a nonempty set of body atoms. Permitted inferences respect those choices; fair saturation is refutationally complete. Function symbols need not be excluded, so completeness does not by itself guarantee termination. ^[ref]

Assume in addition that the probability space is a metric space, continuous integrable functions are dense in $L^1$, and the diameters of its partition cells tend to zero at each point. For a continuous $g$, the oscillation average tends to zero because every point in $M_k(x)$ is within its diameter of $x$. For $h=f-g$, the upper limit of the oscillation of $f$ is bounded by the partition maximal function of $h$ plus $|h(x)|$. The first-crossing maximal inequality for refining partitions and Chebyshev's inequality bound the measure where this exceeds $2\epsilon$ by $2\Vert h{\Vert }_1/\epsilon$. Approximation proves the displayed almost everywhere assertion. Refinement supplies the maximal estimate; shrinking diameter supplies the approximation argument. Without shrinking diameter, a constant partition generally converges only to its cell averages.

A finite axiomatization of a class of structures is a finite set of first-order sentences whose first-order models are exactly that class. The existence of such a set is finite axiomatizability.

A finite abelian group is a cyclic group if and only if it has at most one subgroup of each order dividing its order. A cyclic group $C_n$ has exactly one subgroup of order $d$ for each $d\mid n$, generated by the $n/d$th power of a generator.

Conversely, the structure theorem for finitely generated modules over a principal ideal domain expresses $A$ as $C_{d_1}\oplus \cdots \oplus C_{d_s}$ with $2\le d_1\mid \cdots \mid d_s$. If $s\ge 2$, choose a prime number $p\mid d_1$. The first and second summands then provide distinct subgroups of order $p$, contradicting the hypothesis. Hence $s\le 1$, so $A$ is cyclic, including the trivial group when $s=0$. This criterion combines with the root bound for a polynomial to prove that a finite subgroup of a field multiplicative group is cyclic.

Historical formulation name; composition is identified by pharmacopoeial edition.

Historical formulation name; composition is identified by pharmacopoeial edition.

If $B$ satisfies the universal theory of $A$, every existential sentence true in $B$ is true in $A$, since its falsehood in $A$ would supply a universal negation. Apply the existential-theory embedding criterion to $B$ and the complete theory of $A$ to obtain $A^{\ast }$. Conversely universal truth descends to substructures. The elementarily equivalent model need not be the original structure itself.

An object, or a term denoting an object, that satisfies the predicate asserted by an existential quantification. In a natural deduction use of the existential quantifier elimination rule, a fresh eigenvariable represents an otherwise unspecified witness inside a subproof. The eigenvariable condition prevents it from occurring freely in the exported conclusion or the other undischarged assumptions; violating this restriction is witness escape.

This compound European contingent claim gives the right to buy a later-expiring put option at time $T_1$ for $K_1$. Its payoff is the positive part of the put value minus this exercise cost. Under the Black-Scholes model the put value decreases strictly with the stock spot, so a nondegenerate positive exercise cost gives a unique upper spot threshold for exercise. With zero exercise cost, the compound claim has exactly the underlying put's price by the tower property.

An exchange option permits delivery of one asset in exchange for another at maturity, producing the positive-part payoff $(S_1(T)-S_2(T){)}^{+}$. In a complete finite market its price is the discounted expected value under the unique risk-neutral measure. For a two-period joint binomial model, evaluating this payoff at all four terminal paths and weighting by their risk-neutral path probabilities gives its price directly.