A modal operator forms a new formula from a formula, such as $A\mapsto \square A$. In provability logic, it represents an assertion that a proof of $A$ exists, rather than an assertion that $A$ is true. A box is not interchangeable with an unboxed formula.

Provability logic studies modal principles valid when $\square A$ is interpreted by an arithmetical provability predicate. The distribution axiom corresponds to formalized modus ponens rule, necessitation corresponds to verification of actual proofs, and the Löb axiom is the Internal form of Löb's theorem. The standard system capturing these principles is Gödel-Löb logic.

The necessitation rule permits boxing a theorem in a modal proof system. It does not permit boxing an arbitrary temporary assumption. Under an arithmetic interpretation, it uses the external first of the derivability conditions: an actual proof of $A$ supplies a proof that $A$ has a proof. This distinction matters when proving the Internal form of Löb's theorem.

Positive introspection says that a boxed assertion can itself be boxed. For an arithmetical provability predicate it is the third of the Hilbert-Bernays-Löb derivability conditions: provability of $A$ implies provability of the assertion that $A$ is provable. It follows from formalized Sigma-1 completeness when provability assertions are Sigma-1. It is not the reflection principle $\square A\to A$.

An occurrence of a propositional variable in a modal formula is modalized if it lies within the scope of a box operator. A variable occurs modalized in a formula when all its occurrences do. For example $p$ is modalized in $\square p\to q$, while $q$ is not. Fixed-point and uniqueness theorems for Gödel-Löb logic impose this syntactic condition on the variable being solved for.

A modal formula is built from propositional variables and logical constants using Boolean connectives and modal operators, such as the box operator. In provability logic, arithmetic realization replaces propositional letters by arithmetic sentences and interprets a box as the corresponding arithmetic provability assertion. Modal validity, theoremhood and truth under a particular arithmetic realization are distinct notions.

The normal distribution axiom says that a boxed implication and its boxed antecedent give the boxed consequent. Together with the necessitation rule, it permits boxed versions of propositional deductions. Under an arithmetic interpretation it is the internal modus-ponens condition among the Hilbert-Bernays-Löb derivability conditions. It does not permit dropping a box to assert the enclosed sentence.

Modal logic K4 extends the normal modal distribution axiom and necessitation rule by positive introspection $\square A\to \square \square A$. Its Kripke semantics uses transitive accessibility relations. The fixed point $B=\square (\square X\to X)$ satisfies $\square (B\to \square X)\to (B\to \square X)$ in this logic.

Uniform substitution in modal logic replaces each occurrence of a propositional variable by the same modal formula. Normal modal formal proof systems with axiom schemes are closed under this operation: substituted axioms remain instances, modus ponens is preserved, and necessitation commutes with substitution. Fresh auxiliary letters permit substituting a derived fixed point into fixed-point uniqueness theorems without changing other parameters.

A function continuous across an interface can have different one-sided normal derivatives. For instance, $u(x,y)=y$ when $y<0$ and $u(x,y)=0$ when $y\ge 0$ is continuous, satisfies $u_{xx}+\mathrm{sign}(y)u_{yy}=0$ on each open side, and has zero tangential second derivative on the interface. Its lower $y$-derivative is one and its upper $y$-derivative is zero. Thus the normal derivative does not exist there as a two-sided derivative.

More generally, for functions smooth on each side with one-sided first derivative traces, global $C^1$ regularity would force both traces to equal the derivative on the interface. Unequal traces therefore preclude a globally classical $C^2$ solution as well. A condition on the function value alone is not a condition on its normal derivative; a specified transmission law or an appropriate weak formulation is additional information.

Use signature $(+---)$ and future-pointing timelike vectors $U,V$. Their time components exceed their spatial magnitudes, so the same holds for their sum by the Euclidean triangle inequality. In the rest frame of $U$, write $U=(a,0)$, $a=\sqrt{g(U,U)}>0$, and let $b=\sqrt{g(V,V)}>0$. Then $g(U,V)=a{V}^0\ge ab$. Expanding the squared norm of $U+V$ proves the displayed inequality. Equality requires $V$ to have zero spatial component in this frame, equivalently $V$ is a positive scalar multiple of $U$. The shared future time orientation is essential.

This explicit numerical method is a two-stage Runge-Kutta method and a one-step method. For a sufficiently smooth ordinary differential equation, the chain rule gives $y^{\prime \prime }=f_x+f_{y}f$, and Taylor's theorem shows that its local truncation error is $O(h^3)$. Uniform Lipschitz continuity of $f$ in its solution argument gives an increment-function Lipschitz constant at most $L(1+hL/2)$. The Lipschitz one-step error bound then proves second-order numerical convergence on a fixed time interval. This method uses an intermediate prediction from $y_n$; the explicit midpoint linear multistep method instead requires a separate previous value $y_{n-1}$.

The Eustachian tube connects the middle ear to the nasopharynx, enabling pressure equalization, mucosal clearance and protection from nasopharyngeal contents.

The tympanic cavity is the air-containing middle-ear space housing the auditory ossicles. It communicates with the nasopharynx through the auditory tube. The cavity, its lining and the structures suspended within it have different mechanical and developmental roles.

The tubotympanic recess is an embryonic extension of the first pharyngeal pouch contributing to the auditory tube and middle-ear cavity. Its endodermal origin differs from the ectodermal contribution to the external ear passage.

The tensor tympani muscle is a middle-ear muscle attaching to the malleus. Its contraction alters the mechanics of the tympanic membrane and ossicular chain. Its innervation is associated with the mandibular division of the trigeminal nerve, rather than the facial supply to stapedius.

Substrate replication competes with saturating enzyme clearance in this fast-slow model, where $K>\lambda >0$ and $ϵ\ll 1$. The physical strip $u\ge 0$, $0\le v\le 1$ is invariant. If $0<\eta <\lambda /K$, the origin is attracting and a positive saddle exists at $u_e=\lambda /\eta -K$, $v_e=1-\eta K/\lambda$. Eliminating the fast complex fraction gives $u^{\prime }\simeq \eta u(u-u_e)/(u+K)$, separating extinction from runaway. For initially zero complexes, the stable-manifold threshold is $u_{\ast }=u_e+ϵ(1-\eta )v_e+O({ϵ}^2)$, away from singular parameter limits. To derive the correction, write the stable manifold as $u=u_e+ϵ\varphi (v)$; its invariance equation gives ${\varphi }^{\prime }=-(1-\eta )$ and $\varphi (v_e)=0$. A Dulac factor $1/u$ has strictly negative weighted divergence, excluding a periodic alternative to the two basins. Saturated clearance and autocatalytic growth can therefore create an unstable concentration threshold rather than a stable nonzero balance.

Let $X$ be a nonempty compact metric space and let $f:X\to X$ satisfy $d(f(x),f(y))<d(x,y)$ whenever $x\ne y$. Then $f$ has exactly one fixed point. The distance condition implies continuity, and $x\mapsto d(f(x),x)$ attains a minimum by compactness. If a minimizer $p$ were not a fixed point, then $d(f^2(p),f(p))<d(f(p),p)$ would contradict minimality. Two distinct fixed points would contradict the same strict distance inequality directly. A uniform Lipschitz constant below $1$ is unnecessary; compactness replaces the quantitative hypothesis of the contraction mapping theorem.

Given a metric space $(X,d)$ and $a\in X$, define a new metric by $d_a(x,x)=0$ and $d_a(x,y)=d(x,a)+d(a,y)$ for distinct points. It is the distance obtained by joining every point to a central vertex with its original distance to $a$ as edge length. Symmetry and positive definiteness are immediate. With three distinct points the triangle inequality follows from the extra nonnegative term $2d(z,a)$ in the route through an intermediate point; repeated-point cases are immediate.

Every $b\ne a$ is isolated, since $B_{d_a}(b,d(b,a))=\{b\}$. At the centre, however, $B_{d_a}(a,r)=B_d(a,r)$. Thus a map out of this space is continuous exactly when it is continuous at $a$ in the original metric. The closure of $B$ is $B$ together with $a$ if $a\notin B$ and $d(a,B)=0$; otherwise it is simply $B$. This construction isolates every point except the centre while preserving exactly its original neighbourhood filter.

For an exponential integral with phase $\varphi$, a constant-phase contour holds the imaginary part of $\varphi$ fixed. Along it, decreasing real part of $\varphi$ gives steepest descent, while increasing real part gives steepest ascent. At a cubic stationary point, six rays alternate between descent and ascent.