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Pressure referred to an ideal vacuum, rather than to atmospheric pressure or another chosen reference. Gauge pressure instead subtracts atmospheric pressure.

A best affine state estimate minimizes mean squared error among estimates $a+Cy$ formed from the available observation vector $y$. With finite second moments and invertible observation covariance matrix, it isThis follows from the orthogonal projection of centered state components onto the linear span of centered observation components. For Jointly Gaussian state and observation it equals the conditional expectation; for general laws a nonlinear estimate can have lower mean squared error.

Process noise is the random contribution to a state transition in a state-space model. It represents uncertainty in how the state evolves. Measurement noise instead affects the observed value of the state. In the local-level state-space model $x_t=x_{t-1}+w_t$, $y_t=x_t+v_t$, the first disturbance changes the state itself; the second changes its measurement.

State prediction estimates a future state of a dynamical system using observations already available. A state-space model supplies the transition law. Unlike causal state filtering at the next time, the prediction has not yet incorporated that next observation; a Kalman prediction step therefore includes uncertainty from new process noise.

Causal state filtering estimates the current state of a dynamical system from observations up to the current time. Under quadratic loss, the conditional expectation is an optimal estimate. A Kalman filter obtains this estimate recursively for a linear Gaussian state-space model, without using future observations.

A composition of an integer $N>0$ is an ordered tuple of positive integers with sum $N$. Its length $k$ counts the parts. For example, $(3,2)$ and $(2,3)$ are different compositions of an integer, though they give the same integer partition. Stars and bars counts compositions of an integer with fixed length. ^[ref]

For a sufficiently smooth vector field at a hyperbolic equilibrium, the sums of generalized eigenspaces with negative and positive real parts are tangent to local invariant stable and unstable manifolds of the same dimensions. Their points converge to the equilibrium in forward and backward time respectively. Hyperbolicity excludes eigenvalues with zero real part; center manifolds require a separate theorem.

At a hyperbolic saddle, express a local stable manifold or unstable manifold as a graph over its tangent eigenspace. Substituting a power series for that graph into the vector field and equating its induced derivative to the other component gives successive coefficients. For $\dot{x}=x+x^2+2xy+3y^2$, $\dot{y}=-y+3x^2$, this gives $x=-y^2+y^3/2+O(y^4)$ on the stable graph and $y=x^2-x^3/2+O(x^4)$ on the unstable graph. These are local expansions, not globally invariant polynomial curves.

Starting at $N_1=1/b$ with $r>4$, the next two iterates are the maximum $M=r/(4b)$ and its image $L=4r^2/[b(r+4{)}^2]$. Since $1/b<L\le M$ and the squared-denominator population map decreases on $[L,M]$, its image there lies in $[L,M]$. Thus the displayed interval is invariant after the initial step, and both bounds are attained. The initial value itself is below $L$, so it must not be included in the lower-bound assertion.

A spreadsheet cell depends on itself along a nonempty directed path. For a separate formula-edit experiment, restore the initial inputs and replace only B1 by =A1+A2+C1. Now B1 $\to$ C1 $\to$ B1 is a directed cycle. Requesting D1 makes D1,C1,B1 active and then requests the still-active C1; the algorithm reports C1,B1,C1 rather than returning an old value. This witness follows references backwards, opposite to our dependency arrows; reversing it gives a directed cycle in the spreadsheet dependency graph.

D1 is not itself on that directed cycle, but it cannot receive a value from this evaluator because it needs C1. Independent requests for B2 and E1 still succeed. A topological order of the entire modified spreadsheet dependency graph is impossible: the two cyclic directed edges would require B1 before C1 and C1 before B1. For the initial inputs, the associated system of linear equations is even inconsistent: $B1=7+C1$ and $C1=B1+6$ imply $B1=B1+13$. Detecting a directed cycle is a structural check, not an attempt to solve these linear equations.

A real scalar field with the displayed potential is invariant under the Z2 symmetry $\varphi \mapsto -\varphi$, but a chosen broken-phase pure vacuum has nonzero vacuum expectation value of one sign. Expanding about $\varphi =v$ gives $V=\lambda v^2{h}^2+\lambda v{h}^3+\lambda h^4/4$. The excitation is massive, and no continuous broken generator requires a Goldstone boson. In a finite-volume theory with symmetry-preserving conditions, tunneling can produce a symmetric ground state. A broken pure vacuum is selected by taking the infinite-volume limit before removing an infinitesimal source, or by specifying appropriate boundary conditions. The two signs are related by the symmetry, although the individual pure vacua are not invariant under it.

After the GSO projection, the first massive open-string level has NS oscillator number $3/2$ and Ramond oscillator number $1$. Eight transverse directions give $8+64+\left(\genfrac{}{}{0ex}{}{8}{3}\right)=128$ NS states and $8\cdot 8+8\cdot 8=128$ R states. They form the massive Spin(9) representations $44+84$ and a gamma-traceless vector-spinor of dimension $9\cdot 16-16=128$. The identical chiral oscillator count in a closed string has the different mass normalization ${\alpha }^{\prime }{M}^2=4$.

Before the GSO projection, the Neveu–Schwarz sector has string oscillator levels $0,1/2,1,3/2$, with physical multiplicities $1,8,36,128$ in ten-dimensional light-cone gauge in string theory. The eight transverse worldsheet Majorana fermions supply eight single ${\psi }_{-1/2}$ states. Level one has eight ${\alpha }_{-1}$ states and $\left(\genfrac{}{}{0ex}{}{8}{2}\right)=28$ pairs of distinct ${\psi }_{-1/2}$ modes. Level $3/2$ has $8$ single ${\psi }_{-3/2}$ modes, $8\cdot 8$ mixed boson–fermion modes and $\left(\genfrac{}{}{0ex}{}{8}{3}\right)=56$ triples. Their squared masses are respectively $-1/(2{\alpha }^{\prime }),0,1/(2{\alpha }^{\prime }),1/{\alpha }^{\prime }$. In the Ramond sector, the Ramond zero-mode Clifford algebra gives a sixteen-dimensional ground space. At level one, ${\alpha }_{-1}^i$ and ${\psi }_{-1}^i$ each provide $8\cdot 16$ states, hence multiplicities $16,256$ at squared masses $0,1/{\alpha }^{\prime }$. No half-integer Ramond levels occur. The GSO projection keeps odd NS fermion number and one total Ramond chirality, giving $8$ bosons and $8$ fermions at zero mass, then $128$ of each at the first massive level.

The ferromagnetic Potts model on a finite graph assigns $q$ possible states to each graph vertex and rewards neighbouring equal states by the displayed weight, for $\beta \ge 0$. The interaction sum counts each unoriented edge once. The Edwards-Sokal coupling identifies its spin clusters with the random-cluster model at $p=1-e^{-\beta }$ and integer $q\ge 2$.

For a single linear filter followed by a rate nonlinearity and a Gaussian stimulus-history vector $s$ with covariance $C$, Gaussian integration by parts gives $\mathbb{E}[sg(k^{T}s)]\propto Ck$. Thus a spike-triggered average estimates $Ck$, rather than $k$ directly, when the stimulus is correlated. Covariance correction, with regularization if necessary, can recover the filter direction. For an even nonlinearity the proportionality factor can vanish; then a one-filter average is insufficient.

For the positive Laplace-Beltrami operator on the unit sphere $S^d$, with $d\ge 1$, the eigenvalues are $n(n+d-1)$ for integers $n\ge 0$. Their eigenspaces consist of restrictions of degree-$n$ harmonic polynomials and have dimensions $\left(\genfrac{}{}{0ex}{}{n+d}{d}\right)-\left(\genfrac{}{}{0ex}{}{n+d-2}{d}\right)$, with the second term zero for $n<2$. The Laplacian in polar coordinates gives the eigenvalue; Fischer decomposition and the Stone-Weierstrass theorem prove completeness.

A solid harmonic extends a spherical harmonic to a harmonic function in space with either the regular radial power $r^n$ or the irregular radial power $r^{-n-1}$. In three dimensions these are the two separated radial solutions of the Laplace equation at angular degree $n$.

For an angular spherical harmonic $Y$ of degree $n\ge 1$, the radial, tangential gradient and tangential curl families can be normalized as $\mathbf{P}=\hat{\mathbf{r}}Y$, $\mathbf{B}=r\nabla Y/\sqrt{n(n+1)}$ and $\mathbf{C}=-r\hat{\mathbf{r}}\times \nabla Y/\sqrt{n(n+1)}$. They depend only on direction. At $n=0$ only the radial family is defined by these formulas.

The angular monopole of a function on the sphere is its constant spherical-harmonic component, equal to its angular average. Subtracting it removes the mean temperature of a cosmic microwave background sky map without altering higher angular multipoles.

The angular dipole is the degree-one spherical harmonic component. A real angular dipole is a function $\mathbf{d}\cdot \hat{\mathbf{n}}$ for a fixed vector $\mathbf{d}$. Direction-dependent observer velocity and gravitational endpoint terms can therefore be discarded from higher-multipole calculations when they have this form.