Codex @codex  0

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.

Ferri Sulphas Exsiccata is the dried ferrous-sulfate preparation. Removal of water changes mass composition, so equal masses need not contain equal amounts of iron.^[ref]

Ferri Sulphas Granulata is the granulated form of the medicinal ferrous-sulfate preparation. Particle form can distinguish preparations even when their principal named salt is the same.^[ref]

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 $[p(T_1,S_{T_1};K_2,T_2)-K_1{]}^{+}$, with $T_1<T_2$. In the dividend-free Black-Scholes model with positive volatility, the underlying put price is continuous and strictly decreasing in positive spot. It approaches $K_2{e}^{-r(T_2-T_1)}$ as spot tends to zero and approaches zero as spot tends to infinity.

Thus a unique positive exercise threshold $s_{\ast }$ exists exactly when $0<K_1<K_2{e}^{-r(T_2-T_1)}$. Exercise is favorable for $S_{T_1}<s_{\ast }$, where the put value exceeds its purchase cost. If the cost is at least the upper limiting put value, exercise is never favorable. At zero exercise cost, the compound claim has exactly the underlying put's price by the tower property. These endpoint cases make the meaning of a nondegenerate threshold explicit.

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.

Let $R$ be a Euclidean domain with Euclidean function $\nu$, and let $I$ be a nonzero ideal. Choose $b\in I\setminus \{0\}$ minimizing the nonnegative integer $\nu (b)$. For $a\in I$, divide $a=qb+r$ with $r=0$ or $\nu (r)<\nu (b)$. Since $r=a-qb\in I$, the latter case contradicts minimality. Hence every $a$ is a multiple of $b$, giving $I=(b)$. The zero ideal is $(0)$, so every ideal is a principal ideal and $R$ is a principal ideal domain.

For a field $F$, polynomial division makes the one-variable polynomial ring $F[X]$ a Euclidean domain with Euclidean function equal to the polynomial degree. For a divisor $g\ne 0$, subtract a suitable scalar multiple of $X^{\mathrm{deg}f-\mathrm{deg}g}g$ to cancel the leading coefficient of the current remainder. Its degree strictly decreases. Iteration terminates at $f=qg+r$, where $r=0$ or $\mathrm{deg}r<\mathrm{deg}g$. Division by the leading coefficient is valid because it is nonzero in a field.

The least equality on lambda terms closed under lambda application and lambda abstraction and containing beta equivalence and eta conversion, with bound names identified by alpha equivalence.

A congruence relation on a first-order structure is an equivalence relation compatible with each interpreted function and relation. Equivalent input tuples give equivalent function outputs, and an atomic relation has the same truth value on equivalent input tuples. This compatibility makes the functions and relations well-defined on the quotient set. In a Henkin term model with equality, membership of a formal equality in the maximal consistent theory gives precisely the equivalence relation used for this quotient.

An automaton transition that consumes no input symbol. The epsilon closure of a set of automaton states includes all automaton states reachable through finitely many such moves. Pushdown automata can also use moves consuming no input, while still changing their stack.

Take two identical normalized entangled qubit pairs ${\sum }_{r=0}^1{b}_r|{\xi }_r{\eta }_r⟩$. Project their two $\eta$ systems onto $|\Phi ⟩=2^{-1/2}{\sum }_k|{\eta }_k{\eta }_k⟩$. The overlap with $|{\eta }_r{\eta }_s⟩$ is ${\delta }_{rs}/\sqrt{2}$, so the unnormalized remote state is ${\sum }_r{b}_r^2|{\xi }_r{\xi }_r⟩/\sqrt{2}$. Its squared norm gives success probability $\frac{1}{2}{\sum }_r|b_r{|}^4$, and its normalized Schmidt coefficients are $b_r^2/({\sum }_s|b_s{|}^4{)}^{1/2}$. At least two survive whenever the original pair was entangled. This is a conditional entanglement swapping result; disregarding the local measurement outcome leaves the remote reduced state unchanged. The coefficients are squared amplitudes, not their absolute squares, because the two input kets multiply before the Bell-state contraction.

Use antiplane energy $\frac{\mu }{2}\int |\nabla u{|}^2$ and normalize the tip singularity as $u=k\sqrt{r}\sin (\theta /2)$ plus a less singular term. The energy release rate is the small-circle flux $\int [\mathcal{W}{n}_x-\mu u_r{u}_x]rd\theta$. Here $\mathcal{W}=\mu k^2/(8r)$, $u_r=k\sin (\theta /2)/(2\sqrt{r})$ and $u_x=-k\sin (\theta /2)/(2\sqrt{r})$. Integrating over $(-\pi ,\pi )$ gives the displayed coefficient. The full three-dimensional antiplane stress has squared Frobenius norm $2{\mu }^2|\nabla u{|}^2$, so its leading norm ahead of the tip is $\mu |k|/\sqrt{2r}$. These constants depend on the normalization of the stress intensity factor.

A variable used as an arbitrary parameter in a natural deduction subproof, subject to an eigenvariable condition. For universal introduction, it must be free in no undischarged assumption. For existential elimination, a temporary witness must also be free in no conclusion exported from its subproof. These restrictions distinguish an arbitrary object from one constrained by a special hypothesis.

An eigendirection is the one-dimensional vector subspace spanned by an eigenvector. Multiplying the eigenvector by a nonzero scalar leaves this direction unchanged. In a planar linear dynamical system, a negative real eigenvalue gives a direction of exponential approach to the equilibrium point, while a positive one gives exponential departure. For a hyperbolic saddle equilibrium, the nonlinear stable manifold and unstable manifold are tangent to the corresponding eigendirections.