The Kelvin-Voigt model puts a spring and dashpot in parallel. Their common displacement makes their forces add. With the convention $e^{i\omega t}$, the complex stiffness is $k+i\omega \mu$, giving storage stiffness $k$ and loss stiffness $\omega \mu$. Under a force step, $x=(F_0/k)(1-e^{-t/\tau })$ with $\tau =\mu /k$. The series spring–dashpot is a Maxwell fluid analogue and has different relaxation and creep behavior.

An ideal dashpot resists relative motion with force proportional to velocity. Maintaining a displacement rate $\dot{x}$ requires force $\mu \dot{x}$ and dissipates power $\mu {\dot{x}}^2$ for $\mu >0$. Connecting it in parallel or series with a spring gives different constitutive laws.

This differential model separates normalized molecular orientation $B=A/\mathrm{tr}A$ from scalar stretch $\lambda$. With component-first velocity gradient $K$, it evolves by $D_{t}A=KA+A{K}^T-(A-I)/{\tau }_1$ and $D_t\lambda =\lambda B:K-(\lambda -1)/{\tau }_2$. Its relaxed state is $A=I$, $B=I/3$, $\lambda =1$. The displayed constant-relaxation stretch law is an idealization; variants can add saturation or nonlinear relaxation.

The Deborah number compares a material's viscoelastic relaxation time with the time scale of an imposed process. A small value allows substantial stress relaxation during that process; a large value can reveal stored elasticity. In small-amplitude oscillatory shear, one commonly uses $\mathrm{De}=\omega \tau$. Large frequency does not by itself invalidate a linear viscoelastic fluid approximation if the deformation amplitude remains sufficiently small.

The Weissenberg number measures deformation accumulated over a viscoelastic relaxation time, often as the product of that time and a shear rate or an extension rate. Its precise rate convention should be stated. Unlike a Deborah number based on externally imposed temporal variation, it can remain nonzero in a steady flow. A weak-flow constitutive expansion generally requires both slow temporal change and small deformation over the memory time.

The affine linear PTT fluid considered here uses a symmetric dimensionless tensor $A$ with $\stackrel{\nabla }{A}+(1+\alpha \mathrm{tr}A)A/\tau =2E$ and polymer stress $G_{0}A$. Here $\stackrel{\nabla }{A}$ is the upper-convected derivative, $G_0,\tau >0$ and the nonlinear relaxation parameter $\alpha$ is usually positive. Linearizing about rest gives the Maxwell fluid relaxation modulus $G(s)=G_0{e}^{-s/\tau }$. The PTT family also admits other relaxation functions and nonaffine derivatives; their nonlinear predictions need not equal those of this specified version.

Extend $f:[1,N]\cap \mathbb{Z}\to \mathbb{C}$ by zero, and assume $|f|\le 1$. In one dyadic endpoint convention, Type I sums are $\delta$-small if ${\sum }_{M\le m<2M}|{\sum }_{n\in I_m}f(mn)|\le \delta N$ for every dyadic $M\le N^{1/100}$ and every choice of integer intervals $I_m\subset [K,2K)$, $MK\le N$. Type II sums are $\delta$-small if $|{\sum }_{m\in [M,2M)}{\sum }_{n\in [K,2K)}{a}_m{b}_{n}f(mn)|\le \delta N$ for all dyadic $M,K\in [N^{1/100},N^{99/100}]$ and coefficients $|a_m|,|b_n|\le 1$. Empty tails are handled by zero extension.

Integrating every angle in the bond Fourier series of the Villain model imposes the Kronecker constraint ${\sum }_{\mu }[n_{x,\mu }-n_{x-\hat{\mu },\mu }]=0$. Thus its dual variables are divergence-free integer currents. On a planar lattice such currents are rotated gradients of integer heights on faces. Noncontractible current sectors must additionally be retained on a torus.

In two dimensions, the integer defects $q_x$ on dual faces are vortices of winding $2\pi q_x$. Here $G$ is the inverse positive lattice Laplacian with its boundary or zero-mode convention specified. At separations large compared with the lattice spacing, $G(r)=-(2\pi {)}^{-1}\log (r/a)+\text{constant}$, yielding the pair action $-2\pi \beta {\sum }_{i<j}{q}_i{q}_j\log (r_{ij}/a)$ and core self-energies. Like charges repel and opposite charges attract. With periodic boundaries the total charge is zero and the inverse is taken on the mean-zero subspace; harmonic winding sectors require their own factors.

For a thin disk in the harmonic vertical gravitational potential ${\Omega }_K^2{z}^2/2$, hydrostatic equilibrium and $p=c_s^2\rho$ with height-independent isothermal sound speed give the displayed Gaussian density. For perturbations with adiabatic index $\gamma$, the Brunt–Väisälä frequency is $N^2=(1-1/\gamma ){\Omega }_K^2{z}^2/H^2$. An isothermal background can therefore support buoyancy for adiabatic perturbations; isothermal perturbations instead have $\gamma =1$ and zero buoyancy frequency.

In a reflection-symmetric astrophysical disk, the vertical mass coordinate measures mass per unit area above the midplane. It satisfies $dm=\rho dz$, so ${\partial }_m={\rho }^{-1}{\partial }_z$ inside the material. If the total surface density is $\Sigma$, the upper face is $m=\Sigma /2$; the normalized coordinate $\zeta =2m/\Sigma$ ranges from zero to one. Integrals weighted by mass density become ordinary integrals in $m$, which is especially useful for radiative diffusion and the density-weighted mean kinematic viscosity.

For $n\ge 1$, a family of $2^{n-1}$ vertices in the Boolean hypercube has these minimum and maximum external vertex boundary sizes. The Harper theorem gives the minimum: in odd dimension take the lower half of the levels, and in even dimension add the middle-level sets containing coordinate one to all lower levels. The family of even Hamming weight attains the maximum, since every remaining vertex has an adjacent even-weight vertex.

For a specified set $S$ of vertices, a rooted separation is a pair $(A,B)$ with $A\cup B=V(G)$, $S\subseteq A$, and no edges between $A\setminus B$ and $B\setminus A$. Its order is $|A\cap B|$. It avoids a branch set $C$ when $A\cap C=\varnothing$. A graph with vertex connectivity at least $|S|$ has no rooted separation of order below $|S|$ avoiding a nonempty branch set: both sides of such a separation would be nonempty after deleting its separator.

A proper vector subspace of a vector space is one strictly smaller than the ambient space. In a finite-dimensional vector space, this is equivalent to having smaller dimension: a basis of the subspace can be extended to a basis of the ambient space, and strict inclusion requires at least one additional vector.

A connection is a linear map $\nabla :\Gamma (E)\to {\Omega }^1(B;E)$ satisfying $\nabla (fs)=df\otimes s+f\nabla s$. In a local frame it has the form $d+A$ for a matrix-valued one-form $A$.

A Clifford module bundle is a vector bundle over a Riemannian manifold with a fibrewise Clifford multiplication. A compatible connection on a vector bundle satisfies $[{\nabla }_X,c(\alpha )]=c({\nabla }_X\alpha )$ for the Levi-Civita connection on covectors. On a graded Clifford module bundle, Clifford multiplication exchanges the two grading summands. The associated Dirac operator is $D={\sum }_{i}c(e^i){\nabla }_{e_i}$. Its fibres are modules over the fibres of the Clifford algebra bundle.

A vector bundle over $S^n$ can be constructed by gluing two trivial bundles over its hemispheres with a transition map $\gamma :S^{n-1}\to GL(k,\mathbb{R})$. Extend this map constantly in an equatorial collar to obtain local vector bundle trivializations. A transition map into the special orthogonal group gives compatible fiber orientations and a fiber metric. Homotopies of transition maps give isomorphic bundles.

For a rank-$k$ smooth vector bundle, take the exterior power ${\Lambda }^r{E}_p$ in each fibre. A local frame gives a trivialization with fibre ${\Lambda }^r{\mathbb{R}}^k$, and a transition matrix $A$ acts by ${\Lambda }^{r}A$. Functoriality preserves the cocycle identities, producing a smooth vector bundle of rank $\left(\genfrac{}{}{0ex}{}{k}{r}\right)$, or the zero bundle when $r>k$. Its top exterior power is the determinant line bundle.

The vector bundle whose fibre at $p$ consists of linear maps $E_p\to E_p$. If a local frame changes by $e^{\prime }=eg$, the coefficient matrix of an endomorphism changes by $T^{\prime }=g^{-1}Tg$, giving smooth fibre-linear transitions. For a line bundle this bundle is canonically the trivial scalar bundle, because every fibre endomorphism is scalar multiplication. A connection on a vector bundle induces the endomorphism bundle connection.

A framing is a continuous ordered basis in every fiber, equivalently a trivialization of the vector bundle. A stable framing trivializes the bundle after adding trivial real summands, up to further stabilization and homotopy. For a smooth map it is the stable normal data that is framed, not necessarily the tangent bundle of either manifold.