A finite nonempty directed acyclic graph has a unique topological sort exactly when it has a Hamiltonian path. A Hamiltonian path forces the relative order of every consecutive pair and therefore of all vertices. Conversely, in a unique topological sort, every consecutive pair must have a forward edge: if such an edge were absent, swapping that adjacent pair would preserve all edge constraints and produce a second topological sort. The consecutive edges give a Hamiltonian path. Consequently one may compute any topological sort and test its consecutive edges, taking $O(n^2)$ time on an adjacency matrix representation.

An orientation of a topological manifold chooses a generator of every local homology group $H_n(M,M\setminus \{x\};\mathbb{Z})\cong \mathbb{Z}$, compatibly on coordinate neighbourhoods. Each choice is a local orientation of a manifold. Compatibility means the generators are locally restrictions of the same relative orientation class. It can be checked by choosing oriented coordinate balls and requiring agreement wherever their local classes overlap.

For a connected closed manifold, an orientation gives a unique fundamental class in $H_n(M;\mathbb{Z})$ whose restriction at each point is the chosen local generator. In an oriented triangulation this class is the sum of compatibly signed top simplices: interior faces cancel in opposite pairs, while the local coefficient in every top simplex is one. Reversing every local generator negates the fundamental class.

A reversing involution $R$ maps a trajectory $x(t)$ to another trajectory $Rx(-t)$. The chain rule gives the displayed condition. In contrast, an ordinary equivariance symmetry satisfies $F(Sx)=DS(x)F(x)$ and maps trajectories with time unchanged. Oddness $F(-x)=-F(x)$ expresses half-turn equivariance, not reversal by the half-turn. Damping usually breaks the mechanical candidate $R(x,y)=(x,-y)$.

A freely moving unstable particle of speed $v$ travels a laboratory distance $R$ in time $R/v$, accumulating proper time $R/(\gamma v)$. For rest-frame half-life $T$ and exponential decay, its survival probability is the displayed expression. At least half survive exactly when $\gamma vT\ge R$. The minimum speed is thereforeThe relevant flight length is $\gamma vT$, rather than $\gamma cT$ for a particle whose speed is appreciably less than $c$.

A computation-cost convention assigning constant cost to specified primitive operations. For exact unbounded integer arithmetic this can differ substantially from bit complexity: a multiplication of increasingly large values is one primitive operation but requires more bit operations. State which convention is used when estimating an algorithm's running time.

For every constant $c>0$, a deterministic multitape Turing machine with more than one tape and time complexity bounded by $T(n)$ has an equivalent machine with the same tape count and bound $T(n)/c+n+O(1)$, allowing a larger finite control and tape alphabet. Blocks of old symbols become single new symbols, and finite control implements blocks of old steps. The linear overhead accounts for initial input packing. If $T(n)/n\to \infty$, the overhead can be absorbed to obtain any fixed constant speed increase for sufficiently large inputs. This does not supply unlimited speed increases for operations on fixed hardware.^[ref]

A time complexity bounded by $2^{p(n)}$ for a polynomial $p$, where $n$ is the encoded input length. This usage includes bounds such as $2^{n^2}$; it need not mean only $2^{O(n)}$.

For isotropic linear elasticity with volumetric expansion coefficient $\alpha$, the stress-free thermal strain is $e^{\mathrm{th}}=\alpha (T-T_0)I/3$. Thus the thermal stress term is $-K\alpha (T-T_0)I$, where $K=\lambda +2\mu /3$ is the bulk modulus. A uniformly heated body with its full boundary fixed has zero displacement and the displayed compressive stress. For spatially varying temperature and zero body force, the displacement instead solves $\mu \Delta u+(\lambda +\mu )\nabla \mathrm{div}u=K\alpha \nabla T$ with its boundary values; the uniform formula is not automatic. A body force per volume $+K\alpha \nabla T$ cancels this thermal forcing in the equilibrium equation, but does not by itself cancel the stress or its boundary tractions.

Thermal strain is the strain that heating would produce without mechanical stress. In isotropic linear thermoelasticity it is a pure dilation: a volumetric expansion coefficient ${\alpha }_v$ corresponds to a linear expansion coefficient ${\alpha }_v/3$. The constitutive law acts on the mechanical strain $e-e^{\mathrm{th}}$. Spatially varying thermal strain need not be globally compatible with a stress-free displacement.

Thermal stress is stress generated when thermal expansion or contraction is constrained or incompatible. A uniformly heated isotropic body whose entire boundary is fixed has $e=0$ and $\tau =-K{\alpha }_v(T-T_0)I$. For a nonuniform temperature field, interior displacement generally remains nonzero even with a fixed outer boundary; equilibrium and compatibility must be solved together. Cancelling the divergence of a thermal stress term with a body force is not the same as making that stress vanish.

In a torsion-free divisible Abelian group, division by a nonzero integer is unique. The rational scalar operations are therefore a definitional expansion. Formulas reduce to Boolean combinations of rational linear equations: eliminate an existential variable in a conjunction by solving any equation with nonzero coefficient; if none fixes the variable, the finitely many nontrivial disequalities exclude only finitely many choices in an infinite group. Equations with zero coefficient reduce to parameter conditions.

Thus over a finite parameter set a complete one-variable type either specifies an element of its linear span, or excludes every element of that span. An uncountable rational vector space realizes the latter type because a finite-parameter span is countable. It is consequently an omega-saturated model in the additive language.

A temporal operator states how a proposition behaves across time in a temporal logic. In linear temporal logic, $X\varphi$ means that $\varphi$ holds at the next position, $F\varphi$ that it holds eventually, and $G\varphi$ that it holds at every later position, including the current one.

The binary strong until operator $\varphi U\psi$ requires a future position where $\psi$ holds and requires $\varphi$ at all earlier positions. Thus $G(\mathrm{request}\Rightarrow F\mathrm{response})$ expresses that every request eventually gets a response. The temporal operators distinguish such behavior from an ordinary Boolean condition evaluated at one instant. Branching-time logics additionally quantify over possible execution paths.

A temporal logic interpreted along infinite paths in a transition system. At position $i$, $X\alpha$ requires $\alpha$ at $i+1$, $F\alpha$ requires it at some $j\ge i$, and $G\alpha$ requires it at every $j\ge i$. Strong until and weak until express conditions that hold up to a later event. A system satisfies a specification when all relevant initial paths in a transition system satisfy it.

A branching temporal logic whose operators quantify over paths in a transition system from a state. In particular, $AG\Phi$, also written $\forall \square \Phi$, holds at $s$ when every infinite path in a transition system from $s$ satisfies $\Phi$ at every position, including its initial position.

Temperature measurement infers temperature from a reproducible physical response of a thermometer. Examples include a liquid column's thermal expansion, a sensor's electrical resistance and the thermal radiation emitted by a surface. Calibration connects the response to specified reference temperatures or a physical temperature scale.

A contact probe must exchange heat with the object, so its reading depends on time and thermal coupling. A small probe can respond rapidly, while a large probe can noticeably disturb a small sample. Heat conducted along the probe stem and electrical self-heating can shift the reading away from the object's undisturbed temperature.

A useful immersion experiment is to insert a probe progressively farther into a well-stirred bath and wait after each movement. A reading that changes with immersion depth reveals a heat-leak or equilibration problem. Repeating a calibration does not by itself remove such measurement-condition errors.

Potassium hydrogen tartrate is $\mathrm{KH}{\mathrm{C}}_4{\mathrm{H}}_4{\mathrm{O}}_6$, often written $\mathrm{K}{\mathrm{C}}_4{\mathrm{H}}_5{\mathrm{O}}_6$. It contains a singly charged hydrogen-tartrate ion rather than the fully deprotonated tartrate dianion.

Its remaining acidic hydrogen distinguishes its acid-base behavior from fully neutralized potassium tartrate. The familiar name cream of tartar refers to this compound.

A tartrate is an anion or salt derived from tartaric acid, a four-carbon dicarboxylic acid bearing two hydroxyl groups. The arrangement of the two stereogenic centres permits distinct stereoisomers.

Tartrate chemistry has a historical connection with chirality: salts of different handedness can have different crystal forms. Metal binding adds another distinction, as in potassium antimony tartrate, where the organic anion is part of a metal-containing chemical substance.

If a linear system $Ax=b$ has one solution $x_0$, every other solution differs from it by an element of the null space: $Ax=b$ is equivalent to $A(x-x_0)=0$. Its solution set is therefore the displayed affine subspace, of dimension equal to the nullity. If the system is inconsistent, the solution set is empty, rather than a translate of the kernel.