An approach assigning probabilities to compatible families of time-ordered quantum alternatives. A decoherence functional measures interference between histories; when its relevant cross-terms vanish, coarse-grained probabilities add as in probability theory. Different incompatible families must not be combined as if they were one classical history space. The formalism does not require a fundamental external measurement apparatus.

An interpretation retaining universal unitary evolution of the quantum state, with definite measurement records described relative to outcome branches. Quantum decoherence explains the practical isolation and stable record basis of emergent branches. It does not insert a physical collapse selecting just one component. The emergence of branch structure and justification of Born rule weights involve additional interpretive arguments beyond writing the Schrödinger equation.

The need for mutually exclusive experimental arrangements or incompatible observables to exhibit different aspects of a quantum system. For example, path discrimination and interference contrast obey a tradeoff, while sharp Pauli operator $X$ and $Z$ measurements have incompatible eigenbases. Which-path information is an operational record, rather than a requirement that a conscious observer reads it.

An assemblage whose organisms are preserved where they lived. Local origin does not eliminate taphonomic bias: soft tissues can disappear and resistant parts can accumulate through time.

An assemblage containing remains transported from their original living sites. Hydraulic sorting can enrich durable parts or mix habitats; transport does not invariably disarticulate a carcass.

Return of previously swallowed material to the mouth for further chewing, followed by reswallowing. It helps expose plant material to microbial attack.

A base-excited oscillator has a moving support rather than a directly applied periodic force. If its damping depends on velocity relative to the support, $z=x-y$ obeys $m\ddot{z}+b\dot{z}+kz=-m\ddot{y}$. A model with damping against absolute velocity instead obeys $m\ddot{x}+b\dot{x}+kx=ky$ after subtracting its static equilibrium. These conventions give different transfer functions.

A force field specifies a force vector as a function of position and, when necessary, time, for a specified test body. It is a vector field. A time-independent conservative vector field has $\mathbf{F}=-\nabla V$, where $V$ is the body's potential energy; its work along a path equals the decrease in $V$. Position-dependent forces can have nonzero curl, so a scalar potential need not exist.

A force between molecules, including effects of electrostatics, polarizability and London dispersion forces. Hydrogen bonds describe a directional interaction pattern rather than an additional fundamental force to add independently to all those components.

A force density is force per unit volume, measured in $\mathrm{N}{\mathrm{m}}^{-3}$. Pressure gradients, the divergence of a viscous stress and gravitational body force have this dimension in a local momentum balance.

With anion radius $r_a$, cation radius $r_c$ and unlike-neighbour contact, the conventional cell gives $\eta =(\pi \sqrt{3}/4)(r_c^3+2r_a^3)/(r_c+r_a{)}^3$. At the eight-coordinate limiting contact ratio this is $\pi (3\sqrt{3}-4)/6$. The cation-contact tetrahedral-hole limit is a different geometric ideal.

Continuous withdrawal and consolidation of a connected carbon nanotube aerogel from a synthesis reactor. It differs from later spinning of a prepared liquid dispersion and from drawing a pre-grown substrate-supported forest.^[ref]

Suppose each independent turn makes the current player win with probability $p$, lose with probability $q$, or pass with probability $r$, with $p+q+r=1$ and $r<1$. The players have identical turn rules and alternate after a pass. Before the first pair of turns finishes, the starting player wins with probability $p+rq$. Passing twice returns to the initial state with probability $r^2$. First-step analysis therefore gives $P=p+rq+r^{2}P$, proving the formula. The geometric series of repeated restarts converges because $r<1$; infinite play has probability zero. For $r=1$, nobody ever wins, so that excluded case has no eventual winning probability summing to one.

The Robinson unification algorithm solves equations between first-order terms by finding a substitution making them identical. Repeatedly delete equal terms, decompose equal function symbols into argument equations, reject incompatible symbols, and substitute a variable by its proposed term. The occurs check rejects $x=f(x)$ for finite-term unification, since no finite term equals a proper term containing itself. For $f(x,a)=f(b,y)$ it returns $x\mapsto b$, $y\mapsto a$. Successful finite-term unification yields a most general unifier: every other unifier is obtained by further substitution. Implementations that omit the occurs check instead admit a different cyclic-term interpretation.

Four equally spaced lines can arise from three equivalent neighbours or from coincident couplings to different groups. Combining a doublet with a triplet of the same spacing gives intensities 1:3:3:1; line count alone does not identify a unique topology.

Splitting by two distinguishable spin-one-half neighbours with different resolved coupling magnitudes. The four line positions are displaced from the centre by every choice of plus or minus half of each coupling.

A first-order pattern combining a doublet splitting with a triplet splitting, commonly from one neighbour and a pair of equivalent neighbours.

For a symmetric one-dimensional finite square well, nondegenerate bound states alternate even and odd parity, starting with an even ground state. The $n$th state counted from zero has $n$ interior nodes. Outside the well, it decays with inverse length $\sqrt{2m(U-E_n)}/\hslash$. Matching both the wavefunction and its first derivative at a finite step distinguishes it from the infinite-well approximation.

For a symmetric attractive well of width $a$ and depth $V_0$, the first odd bound state obeys $-k\cot (ka/2)=q>0$, where $q$ is the inverse tail length. Its lowest branch has $\pi <ka<2\pi$. Since $k^2<2m{V}_0/{\hslash }^2$, existence requires $V_0>V_{\mathrm{odd}}$; it is also sufficient by continuity of the matching equation on that branch. At equality the exterior solution has zero decay rate and is not square-integrable. Thus a well at or below this threshold has only its even ground state.

If a normalized wavefunction has exterior tails $C{e}^{-q|x|}$ outside $|x|<a/2$, integration over both tails gives the title probability. For an even cosine interior $A\cos (kx)$, the remaining normalization contribution is $A^2[a/2+\sin (ka)/(2k)]$. Tail amplitudes defined at the edge rather than at the origin absorb the factor $e^{-qa/2}$, so their probability formula looks different despite describing the same state.