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A closed superstring theory whose left and right Ramond sector projections have opposite ten-dimensional Majorana-Weyl chiralities. It has 32 real spacetime supercharges and is nonchiral. Its massless fields form type IIA supergravity; the RR gauge potentials have odd degrees, including a one-form and a three-form.

A closed superstring theory whose left and right Ramond projections have the same ten-dimensional chirality. It has 32 real supercharges and is chiral. Its low-energy fields form type IIB supergravity. Circle T-duality exchanges this theory with type IIA superstring theory.

The closed-string sector with NS sector excitations on both chiral sides. Its massless transverse tensor product is $8_v\otimes 8_v=35_v+28+1$, giving the graviton, Kalb–Ramond field and dilaton. These fields are common to the type II theories.

The closed-string sector with Ramond sector states on both chiral sides. The left and right spinor tensor product describes differential-form gauge fields. Opposite chiralities in type IIA superstring theory give the massless one-form and three-form, with transverse dimensions $8$ and $56$. Same chiralities instead give the type IIB form-field content.

A filling box model describes a turbulent plume rising through a confined fluid and spreading beneath its ceiling. Fluid entrainment transfers initially unmodified ambient fluid into the plume; the discharged fluid forms a descending filling-box first front. A maintained heat source can have negligible source volume flux while maintaining nonzero buoyancy flux. A finite injected fluid flux requires an equal exhaust if the container volume is fixed.

A buoyant plume with insufficient momentum for the corresponding pure plume balance. In the constant-entrainment top-hat model it has plume balance parameter $\Gamma >1$, unlike the momentum-excess forced plume. The equations accelerate the plume and drive the balance parameter toward one downstream. The definition depends on the chosen flux and entrainment convention; it is not merely a synonym for slow vertical speed.

An eddy diffusivity is a transport closure representing unresolved turbulence by a mean scalar flux proportional to a negative mean concentration gradient. A local estimate is $K\sim U\ell$, where $U$ and $\ell$ are turbulent speed and length scales. In a line plume with constant characteristic speed and width proportional to $z$, this gives $K\propto z$. Its coefficient is an independent closure parameter; fluid entrainment conservation alone does not determine it.

Mixing by turbulence is represented approximately by a diffusive flux down a mean concentration gradient. Its coefficient is the eddy diffusivity. A constant-coefficient model gives a $D{\nabla }^2\sigma$ term and a smoothing rate $D{k}^2$ for a Fourier mode. Effective diffusion can suppress slow gravitational concentration even when microscopic diffusion is negligible.

Product-to-sum formulas turn products of trigonometric functions into sums at combined arguments. The identity $2\sin A\sin B=\cos (A-B)-\cos (A+B)$ is especially useful for verifying orthogonality of sine Fourier series modes on an interval.

A sinusoid is a function $A\cos (\omega t+\varphi )$, where $A$ is its amplitude, $\omega$ its angular frequency and $\varphi$ its phase. For $\omega \ne 0$ its period is $2\pi /|\omega |$. A uniformly randomized phase can produce a strictly stationary process even though individual paths look periodic.

A closed polygonal tour through a finite point set, measured using Euclidean distance. Write $\mathrm{TS}(X)$ for the minimum length over cyclic visiting orders. A two-point tour traverses its segment twice; a one-point tour has length zero. The triangle inequality allows repeated visits to be shortcut when estimating this minimum.

A triangulation of a surface decomposes it into triangles meeting along full edges or vertices. On a closed surface, each edge is incident with two triangle faces, so $3F=2E$. Euler characteristic connects this local counting relation with the surface's topology.

The indiscrete topology on a set $X$ has only the empty set and the whole set as open sets. When $X$ has more than one point it is not a Hausdorff space. For example the quotient topology on $\mathbb{R}/\mathbb{Q}$ is indiscrete: a nonempty open saturated subset contains an interval and all its rational translates, which cover $\mathbb{R}$.

A family $(A_i)$ of subsets of a topological space is locally finite if every point has a neighbourhood meeting only finitely many of its members. On a compact space such a family has only finitely many nonempty members. For singleton subsets of a Riemann surface, this is the condition permitting prescribed poles of a meromorphic function without accumulation inside the surface.

The open sets are the empty set and subsets whose complement is countable. On an uncountable set, any two nonempty open sets intersect, so the space is not Hausdorff. On a countable underlying set this topology is discrete. The compact subsets of a cocountable space are finite, regardless of the cardinality of the underlying set.

Two quantum states are locally indistinguishable at a prescribed scale if all operators of norm at most one supported below that scale have nearly equal expectation values. This is a statement about restricted observables, not equality of the global states. An orthogonal pair can be locally indistinguishable on macroscopically large regions.

The local-pair condition for topological quantum order asks for an orthogonal partner state with local indistinguishability on supports of diameter at most a fixed fraction of the system diameter, with a fixed error smaller than one. This criterion captures a form of macroscopic information hidden from local observables; it is not a complete definition of every topological phase.

An innovation measures the part of the current observation not predicted from past observations. The linear innovation process uses orthogonal projection onto the past linear span. A conditional expectation predictor can use nonlinear information; the two predictors agree for a Gaussian process.

The difference $\Delta X_t=X_t-X_{t-1}$ can remove a unit-root stochastic trend. A seasonal difference operator instead subtracts an observation a full seasonal period earlier. Neither operation automatically removes a changing variance.

A process has period-$S$ second-order statistics when $E{X}_{t+S}=E{X}_t$ and $\mathrm{Cov}(X_{t+S},X_{s+S})=\mathrm{Cov}(X_t,X_s)$ for all times. Its variance can vary by season, so it need not be a weakly stationary process. A periodic covariance period need not be the fundamental period.