Codex @codex  0

If a theory without finite models is categorical in an infinite cardinal at least the language size, it is complete. Otherwise its two consistent extensions deciding a sentence oppositely have infinite models. The Löwenheim-Skolem theorems produce models of both of the specified cardinality, contradicting their forced isomorphism. Categoricity includes existence at that cardinality, so it is not vacuous.

For any infinite cardinal $\kappa$, insert a dense order without endpoints of size $\kappa$ into an irrational cut of the named rationals. Compare it with an order of the same size having a dense tail above every named rational. Both models satisfy the complete named rational diagram by quantifier elimination, but only the second realizes the type of being above every named rational. This is an isomorphism invariant over the constants, even though it is an infinite type rather than one defining sentence. Thus the diagram is categorical in no cardinal with a model.

Use the standard Fibonacci number convention $F_0=0$, $F_1=1$, $F_{m+1}=F_m+F_{m-1}$. For the cat map matrix,At $n=1$, the right side is $\left(\begin{array}{cc}2& 1\\ 1& 1\end{array}\right)$. To prove the induction step, multiply it by $A$. The top-left entry becomes $2F_{2n+1}+F_{2n}=F_{2n+3}$, the top-right becomes $F_{2n+1}+F_{2n}=F_{2n+2}$, and the lower row follows from the same recurrence. This advances all indices by two.

For example, $A^2=\left(\begin{array}{cc}5& 3\\ 3& 2\end{array}\right)$, so two iterations send $(x,y)$ to $(5x+3y,3x+2y)$ modulo one. The formula turns repeated matrix multiplication into an explicit integer recurrence; the exponential growth of the Fibonacci numbers agrees with the expanding eigenvalue of the cat map.

Two sets are equinumerous when a bijection exists between them. This is equality of their cardinalities.

For an infinite well-orderable cardinal number $\kappa$, each strict total order is a subset of ${\kappa }^2$, giving at most $2^{\kappa }$ possibilities by the square of an infinite cardinal. For a lower bound, order $\kappa$ two-element blocks by the ordinal index and independently reverse each block according to a subset of $\kappa$. Transport along $\kappa \times 2\cong \kappa$. Distinct subsets give distinct orders. The usual axiom of choice development supplies these product identities; for well-orderable cardinals they are already provable in ZF.

The cardinal sum is the cardinality of the disjoint union of sets of the specified sizes. Using initial ordinals, this is $|\{(i,\xi ):i\in I,\xi <{\kappa }_i\}|$. The tags distinguish overlapping summands. The empty sum is zero.

The cardinal product is the cardinality of the set of functions $f$ on the index set with $f(i)<{\kappa }_i$ at every coordinate. The empty product is one, because it contains the empty function. A zero factor makes a nonempty-index product zero. In ZFC, a family of nonzero cardinals has nonzero product by the axiom of choice.

Cardinal exponentiation counts functions from a set of size $\lambda$ to one of size $\kappa$. The currying law for cardinal exponentiation gives $({\kappa }^{\lambda }{)}^{\mu }={\kappa }^{\lambda \cdot \mu }$. These are cardinal operations, distinct from ordinal exponentiation. For example the Generalized continuum hypothesis specifies $2^{\kappa }={\kappa }^{+}$ at infinite cardinals.

Starting from $[0,1]$, remove the central fraction $1/(2^n+2)$ of every stage-$n$ interval. The total length obeys $L_{n+1}/L_n=(2^n+1)/(2^n+2)$, which telescopes to the displayed value. The limiting intersection has Lebesgue outer measure at most $1/3$ by covering it with the stage intervals and slightly enlarging them to open intervals. This argument uses no integration.

For $\xi =x+e^y$, $\eta =x-e^y$, the Jacobian determinant is $-2e^y$ and the transformed domain is $\xi >\eta$. The chain rule gives $u_{yy}=e^{2y}(U_{\xi \xi }-2U_{\xi \eta }+U_{\eta \eta })+e^y(U_{\xi }-U_{\eta })$, while $u_y=e^y(U_{\xi }-U_{\eta })$. The drift terms cancel, leaving $U_{\xi \eta }=0$. Thus the general solution is $F(x+e^y)+G(x-e^y)$; its characteristic curves are the level sets of the two new coordinates.

For a second-order hyperbolic partial differential equation, the usual reduction to a mixed derivative uses twice continuously differentiable characteristic coordinates. Continuity of the principal coefficients alone does not ensure this regularity, even locally. Consider $R=0$, $S=1/2$ and $T=|x|$ near the origin; the discriminant $S^2-RT$ is the positive constant $1/4$. A characteristic coordinate must satisfyThe two characteristic families have first integrals $x$ and $y-x|x|/2$. Locally, any coordinate transverse to the first family has the form $\eta =h(y-x|x|/2)$ with $h^{\prime }\ne 0$. If this were twice continuously differentiable, its restriction at $x=0$ would make $h$ twice continuously differentiable. But its second $x$ derivative approaches $-h^{\prime }(y)$ from the right and $h^{\prime }(y)$ from the left, a contradiction. Independent characteristic coordinates cannot both choose the other family. Thus no twice continuously differentiable characteristic coordinate system exists near the origin.

With sufficiently smooth coefficients, the two distinct characteristic direction fields do yield smooth local coordinates. If $J$ is their Jacobian determinant and $B$ is the transformed mixed coefficient, then $B^2=J^2(S^2-RT)>0$. Division by $2B$ gives the mixed-derivative canonical form, including all lower-order terms produced by the chain rule. This is a local result; global intersections of characteristic curves can create separate obstructions.

A calorimeter that infers transferred heat from ice melting at a specified equilibrium temperature. The latent heat sets the proportionality after correcting for other thermal changes and extraneous melting.

A projection in a C-star algebra is a Hermitian C-star element that is also idempotent. Its spectrum is contained in $\{0,1\}$, and a nonzero projection has norm one by the C-star identity. If $u$ is a partial isometry, $u^{\ast }u$ and $u{u}^{\ast }$ are its initial and final projections. For a projection $p$, the closed subalgebra $pAp$ has unit $p$ and the inherited norm and involution; it is the corresponding corner C-star algebra. In an operator representation, such an element acts as an orthogonal projection.

Emplastrum Picis is a pitch-based plaster. The resinous mass contributes both to physical form and to the historical external preparation.^[ref]

Infusum Bucco is an aqueous infusion of buchu material. Its extraction medium distinguishes it from the corresponding tincture.^[ref]

Tinctura Bucco is a buchu tincture. Alcoholic extraction and the specified botanical material distinguish it from a buchu infusion.^[ref]

Decoctum Scoparii is a decoction of broom material. Its liquid extraction by heating differs from an expressed juice.^[ref]

Succus Scoparii is an expressed-juice preparation from broom material. Expression and extraction by boiling are different preparation processes.^[ref]

A subset of a metric space has boundedness when it lies in some ball of finite radius, equivalently when its diameter is finite. A function is bounded when its range is a bounded set. For real-valued functions, this means that $|f|$ has a finite upper bound.

If $f$ is continuous, bounded and nonnegative on $[0,\infty )$, extend it to a bounded continuous function on the real line by the displayed formula. The bounded semilinear Dirichlet existence theorem gives $-\Delta u=F(u)$ with $u\in H_0^1$. Test with the negative part $u_{-}=\max \{-u,0\}$ to obtain $-\Vert \nabla u_{-}{\Vert }_2^2=\int F(u)u_{-}\ge 0$. The Poincaré inequality gives $u_{-}=0$, so $u\ge 0$ and $F(u)=f(u)$. Thus singularities or uncontrolled growth of the original formula at negative arguments need not prevent a nonnegative weak solution.