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This first-order-in-time field theory has an annihilation-field expansion $\Psi =\int a_{\mathbf{p}}{e}^{-i{\mathbf{p}}^{2}t/(2m)+i\mathbf{p}\cdot \mathbf{x}}{d}^{3}p/(2\pi {)}^3$. Its adjoint creates particles; an antiparticle creation term is not required by a relativistic mass shell or Lorentz-invariant microcausality. It is often an effective low-energy sector of a relativistic charged theory after the antiparticle sector is omitted. This does not mean that antimatter cannot be treated nonrelativistically: one may introduce a separate nonrelativistic field for that species.

For real scalar fields with quartic potential energy density $V_4=\lambda ({\varphi }_1^4+{\varphi }_2^4)+2\mu {\varphi }_1^2{\varphi }_2^2$, nonnegativity in every direction is equivalent to the displayed conditions. Necessity follows on an axis and on ${\varphi }_1={\varphi }_2$. Sufficiency follows by writing $V_4=\lambda ({\varphi }_1^2-{\varphi }_2^2{)}^2+2(\lambda +\mu ){\varphi }_1^2{\varphi }_2^2$. With a positive quadratic mass term, equality cases still give a stable isolated classical vacuum at the origin. The quartic alone is strictly positive away from the origin precisely when $\lambda >0$ and $\mu >-\lambda$. Rotation symmetry additionally requires $\mu =\lambda$.

For a neutral formula unit, let ${\nu }_i$ count ions of type $i$ with electric charge $z_{i}e$, where $e$ is the elementary charge. The stoichiometry must satisfyFor ${\mathrm{Na}}^{+}$ and ${\mathrm{Cl}}^{-}$, the smallest positive counts are $1:1$. For ${\mathrm{Ca}}^{2+}$ and ${\mathrm{Cl}}^{-}$, they are $1:2$, giving calcium chloride $\mathrm{CaC}{\mathrm{l}}_2$: $1(2)+2(-1)=0$. The formula counts ions, not the number of nearest neighbours. A counterion is the oppositely charged partner needed to balance a charged species; its identity can change solubility and other properties without changing the identity of that species.^[ref]

A salt of potassium cations and permanganate anions, supplying permanganate in a redox titration.

This cubic smooth function has one isolated degenerate critical point at the origin, with three ascending and three descending sectors. Adding $-tx$ with $t>0$ replaces it by two nondegenerate saddle points at $(\pm \sqrt{t/3},0)$. The Hessians there have opposite diagonal entries $6x$ and $-6x$. It illustrates why the number of smooth critical points can be smaller than that required for a Morse function.

The maximum amount of CPU execution allotted in one turn by round-robin scheduling. A task can finish before consuming its scheduling quantum. When other tasks remain runnable, a shorter scheduling quantum provides more frequent turns at the cost of more frequent context switches.

A weak geometric $p$-rough path is a continuous path in the free step-N nilpotent Lie group, with $N=\lfloor p\rfloor$, whose increments have finite $p$-variation for the Carnot-Carathéodory distance. Its multiplicative increments are ${\mathbf{X}}_{s,t}={\mathbf{X}}_s^{-1}\otimes {\mathbf{X}}_t$. They obey the Chen identity and the algebraic symmetries of path signatures.

A geometric $p$-rough path is a limit, in the rough path metric, of path signatures of sufficiently regular paths. It is a weak geometric p-rough path. At the same exponent these definitions need not coincide; weak geometric paths can be approximated after slightly weakening the variation exponent. The Brownian enhancement is geometric for every $p>2$.

This inhomogeneous variation metric compares the tensor increments level by level. A common alternative is the Hölder metric ${\rho }_{\alpha }={\max }_k{\sup }_{s<t}|X_{s,t}^{(k)}-Y_{s,t}^{(k)}|/|t-s{|}^{k\alpha }$. On a finite interval convergence in this metric implies $p$-variation convergence whenever $p\alpha \ge 1$. Both topologies control the iterated integrals needed by the universal limit theorem.

The path signature collects the ordered iterated Riemann-Stieltjes integrals of a continuous path of bounded variation. Its truncation belongs to the free step-N nilpotent Lie group. It multiplies under concatenation by the Chen identity, is independent of orientation-preserving reparametrisation of the whole path, and records more than its endpoint displacement.

The Stratonovich enhancement of Brownian motion is a geometric $p$-rough path for every $p>2$. Its second level consists of Lévy area and the prescribed symmetric tensor. Its increments satisfy the Chen identity and have finite Hölder rough-path norms at every exponent below $1/2$.

A rough differential equation interprets a differential equation using the higher-order data of its driving rough path. For $2<p<3$, its local expansion includes $V_i(Y_s)X_{s,t}^i+D{V}_j(Y_s)V_i(Y_s){\mathbb{X}}_{s,t}^{ij}$, with a remainder controlled at order greater than one. For a smooth driver it is the usual controlled ordinary differential equation; for enhanced Brownian motion it is the corresponding Stratonovich equation.

A control is a continuous nonnegative function on ordered pairs of times, zero on the diagonal and superadditive as displayed. It measures the size of increments of a rough path, typically through $|X_{s,t}^{(k)}|\le \omega (s,t{)}^{k/p}$. Continuous finite variation gives suitable controls, which let local estimates be stated without tying them to a particular time parametrization.

Fourier transform microwave spectroscopy excites gaseous molecules with a pulse of electromagnetic radiation, then records the emitted time-dependent electric field. A Fourier transform extracts transition frequencies from the decaying signal. In a cavity implementation, two facing mirrors form a resonator; tuned excitation, receiver electronics and digitization replace the molecule-counting readout of molecular beam electric resonance. The pulsed radiation prepares quantum coherence between rotational quantum states; it does not set the whole gas spinning at one classical angular velocity.

For documented laboratory context, Evolution of Microwave Spectroscopy at the National Bureau of Standards (NBS) and the National Institute of Standards and Technology (NIST) by F. J. Lovas and others (§5, printed pp. 280–282) describes the NIST cavity instrument, phase-locked sources, mirror tuning and detector-mixer modifications. That account explains one instrument family; it does not establish the electronics of every later commercial spectrometer.

In a rotating reference frame, this apparent force acts away from the rotation axis. The corresponding centrifugal acceleration is the force per unit mass.

A mass $m$ in a rotating reference frame with varying angular velocity experiences this apparent force. Dividing it by $m$ gives the Euler acceleration.

For a uniformly rotating incompressible fluid of constant mass density, pressure and conservative body forces combine as the gradient of this scalar. The centrifugal term has a negative sign. Pressure enforces incompressible flow, whereas the velocity-dependent Coriolis force remains explicit and cannot be included in a scalar position potential. Adding a spatially constant function of time to $\psi$ does not change the motion.

These roots are taken in an algebraic closure, not merely in ${\mathbb{F}}_q$. If the characteristic of a field is $p$ and $n=p^{a}m$ with $p\nmid m$, then $X^n-1=(X^m-1{)}^{p^a}$; hence there are $m$ distinct roots. They form a cyclic group by the finite multiplicative subgroup of a field is cyclic theorem. In particular they have order $n$ when $\gcd (n,q)=1$.

Let $S$ be a finite multiset of nonzero complex numbers such that taking the $m^2$th powers permutes it for every positive integer $m$. Taking $m=2$ and iterating the resulting permutation shows that each $z\in S$ obeys $z^{4^r}=z$ for some $r\ge 1$, hence is a root of unity. Let $d$ be a common multiple of their orders. The $d^2$th powers are all one, and the assumed permutation then forces every original element to be one. For a unitary matrix, the spectral theorem for normal operators further forces the matrix itself to be the identity.

A square root of unity is an element $x$ of a ring with identity satisfying $x^2=1$. In a field of characteristic different from two, the only possibilities are $1$ and $-1$: the difference of squares identity $(x-1)(x+1)=0$ has no other roots because a field has no zero divisors. In a general ring more solutions may occur; modulo eight the four residue classes $1,3,5,7$ all have square one.