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The cumulative function of the finite positive spectral measure of a stationary time series is nondecreasing and right-continuous, with total mass equal to the variance. For an absolutely continuous measure, $F(\lambda )={\int }_{-\pi }^{\lambda }f(u)du$, where $f$ is the time-series spectral density. Under angular-frequency normalization, ${\gamma }_k={\int }_{-\pi }^{\pi }{e}^{ik\lambda }dF(\lambda )$ and, when the covariance series is absolutely summable, $f(\lambda )=(2\pi {)}^{-1}{\sum }_{k\in \mathbb{Z}}{\gamma }_k{e}^{-ik\lambda }$. Atoms describe persistent periodic components and cannot be represented by an ordinary density.

For uncorrelated zero-mean unit-variance amplitudes $A,B$, the process $A\cos ({\omega }_{0}t)+B\sin ({\omega }_{0}t)$ has covariance $\cos ({\omega }_{0}k)$ and hence the displayed two-atom spectral measure of a stationary time series. Independence or Gaussian amplitudes are unnecessary for weak stationarity. A phase shift need not preserve the whole joint amplitude distribution, so strict stationarity cannot be inferred from these second moments alone.

For $L=d^2/d{x}^2$ on $(0,\pi )$ with zero endpoints, the zero-parameter Green function is $-x_{<}(\pi -x_{>})/\pi$. Its eigenfunctions are $\sqrt{2/\pi }\sin (nx)$ with eigenvalues $-n^2$. At $x=\xi =\pi /2$, the kernel is $-\pi /4$ and its expansion contains precisely the odd reciprocal squares, giving the displayed identity.

For two real centered time series $X_t,Y_t$, require joint weak stationarity: all auto- and cross-covariances depend only on the time lag. Individual weak stationarity alone does not impose this condition on their joint behavior. With the convention ${\gamma }_{XY}(h)=\mathbb{E}[X_t{Y}_{t+h}]$, define the cross-spectrum byAssume absolute summability of the auto- and cross-covariance sequences, so the displayed Fourier series and the two individual spectral densities are continuous. Reality and joint weak stationarity giveConsequently the spectral matrix $F(\lambda )=\left(\begin{array}{cc}{f}_{XX}& f_{XY}\\ f_{YX}& f_{YY}\end{array}\right)$ is a Hermitian matrix. It is positive semidefinite, because its quadratic forms are nonnegative spectral densities of complex linear combinations of the jointly stationary series. Its nonnegative determinant therefore yields $|f_{XY}(\lambda ){|}^2\le f_{XX}(\lambda )f_{YY}(\lambda )$.

The imaginary part records directional lag information. For white noise $X_t$ of variance $v$ and $Y_t=X_{t-1}$, only ${\gamma }_{XY}(1)=v$ is nonzero. Hence $f_{XY}(\lambda )=v{e}^{-i\lambda }/(2\pi )$, although both individual spectral densities equal $v/(2\pi )$. The cross-spectrum is generally complex, rather than real and even like a real series' own spectral density.

Under angular-frequency normalization, the displayed time-series spectral density gives ${\gamma }_0={\pi }^2$ and ${\gamma }_k=2(1-(-1{)}^k)/k^2$ for nonzero integer $k$. This follows from the even integral $2{\int }_0^{\pi }(\pi -\lambda )\cos (k\lambda )d\lambda$. All nonzero even-lag covariances vanish while the odd-lag covariances are positive.

Volumetric heat capacity is heat capacity per unit volume, with units $\mathrm{J}{\mathrm{m}}^{-3}{\mathrm{K}}^{-1}$. For homogeneous material of mass density $\rho =m/V$, it equals $\rho c$, using specific heat capacity $c$ for the same constraint. It measures the heat required per small temperature increase for a chosen occupied volume. In a material model using specific heat capacity at constant pressure, thermal diffusivity is $\alpha =k/(\rho c_p)$, where $k$ is thermal conductivity.

Near the identity, the matrix logarithm of a unitary matrix is skew-Hermitian. Restrict the logarithm's operator norm below $\pi /n$; determinant one then forces its trace to be zero, since $\det U=\exp (\mathrm{tr}\log U)$ and no nonzero multiple of $2\pi i$ fits this bound. The matrix exponential gives the inverse chart on a ball in the displayed real Lie algebra, of dimension $n^2-1$. Left translation supplies compatible charts on the whole special unitary group.

Project a based map $\alpha :S^r\to SO(s+1)$ to $S^s$. Since $r<s$, the resulting homotopy class vanishes. Lift a based null-homotopy by the special orthogonal sphere fibration. Its endpoint lies in the fiber $SO(s)$. If the lift moves its base point, right-multiply at time $t$ by the inverse of its base-point value, which lies in the fiber. This keeps the projected homotopy unchanged and makes the lift based. Thus $\alpha$ is based-homotopic to a map into $SO(s)$, proving the asserted surjectivity.

The scalar center has order $\gcd (n,q-1)$. For $n>1$ the quotient is simple except at $(n,q)=(2,2),(2,3)$. Its faithful primitive projective-point action, abelian fixed-center transvections, elementary-matrix generation and perfectness give the Iwasawa simplicity lemma proof.

A determinant-one matrix over ${\mathbb{F}}_2$ permutes the three nonzero vectors of ${\mathbb{F}}_2^2$. This group action is faithful because fixing two basis vectors forces the matrix to be the identity. There are $(4-1)(4-2)=6$ invertible matrices, all of determinant one since the only nonzero field element is one, so the action identifies the group with the symmetric group $S_3$. Reduction of integer determinant-one matrices modulo two is onto: $\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$ and $\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$ induce two distinct transpositions, which generate $S_3$. Its kernel consists of matrices congruent to the identity modulo two, the level-two principal congruence subgroup.

Over a field of odd characteristic, an involutory matrix has minimal polynomial dividing $(X-1)(X+1)$, so it is diagonalizable with eigenvalues in $\{1,-1\}$. In dimension two, determinant one requires either both eigenvalues one or both minus one. Hence the only element of order two in $S{L}_2$ is $-I$; the identity has order one.

Modulational instability is growth of long-wavelength perturbations of a uniform or periodic wave through coupling of its modulation to nonlinear dynamics. It can destabilize an exact plane wave even when the unperturbed amplitude remains constant. Linearizing the amplitude and phase produces coupled sidebands rather than independent phase shifts.

For the focusing nonlinear Schrodinger equationthe background $\psi =A{e}^{2i{A}^{2}t}$ has real amplitude $A>0$. Substitute $\psi =(A+u+iv)e^{2i{A}^{2}t}$ with real small $u,v$. The linearized equations areFor a Fourier mode proportional to $e^{\sigma t+iqx}$,Thus $0<|q|<2A$ has an exponentially growing branch. The maximum growth rate is $2A^2$, attained at $q^2=2A^2$. The zero mode is neutral, and larger modulation wavenumbers have oscillatory linear eigenvalues. These are predictions of the linearized problem; they do not determine the eventual nonlinear pattern. A finite domain must admit a wavenumber inside the unstable band.

The coefficients and evolution variable depend on the model: fibre modulational instability uses propagation distance and retarded time, while the Benjamin-Feir instability gives a related long-wave criterion for the complex Ginzburg–Landau equation.

For a symmetric sparse matrix, its matrix graph has one vertex per unknown and an edge $i-j$ whenever the off-diagonal entry $a_{ij}$ is structurally nonzero. Simultaneous row and column permutation relabels this graph. Nonsymmetric patterns are instead represented by directed or bipartite graphs when row and column roles must be distinguished.

Fill-in consists of new nonzero positions created by Gaussian elimination, absent from the original sparse matrix in the same ordering. For a Cholesky decomposition $PA{P}^T=C{C}^T$, compare $C$ with the lower triangle of $PA{P}^T$, not with the unpermuted input. Each new undirected edge of the elimination graph contributes one structural position in $C$. The sparse elimination of a four-leaf star gives six such positions when the center is eliminated first, and none when the leaves are eliminated first.

Symbolic factorization predicts possible nonzeros from the pattern alone. An initially zero $a_{ij}$ becomes nonzero in a single scalar step if both $a_{ik}$ and $a_{kj}$ are nonzero; however, at later steps several contributions can cancel. For an exact example, letEliminate indices $1,2$ in that order. The first Schur complement, in order $(2,3,4)$, isThe second pivot element updates the $(3,4)$ entry to $-1/3-(-1/2)(1)/(3/2)=0$. The remaining block is $\mathrm{diag}(7/3,7/3)$, so all four pivot elements are positive and $H$ is a symmetric positive-definite matrix. The elimination graph retains the structural edge $3-4$, but this exact choice of coefficients makes the numerical entry vanish. A small change to $H_{34}=H_{43}$ generally removes the cancellation.

Such an exact zero is different from discarding a small value computed with floating-point arithmetic. The symbolic factorization deliberately reserves space for possible entries without depending on special numerical identities. A perfect elimination ordering creates no structural fill-in at all; an ordering with cancellations need not have that property.

At each step choose a remaining vertex of minimum degree in the current elimination graph. This heuristic reduces the size of the immediate Schur complement update and often reduces fill-in, but does not guarantee a globally optimal ordering. Its degrees must reflect earlier fill, rather than just the original graph.

A vertex separator divides the remaining unknowns into disconnected subdomains. Nested dissection recursively eliminates each subdomain before its separator, keeping interaction localized until late in the factorization. For a regular two-dimensional mesh with $N$ unknowns, separators have size $O(\sqrt{N})$; the resulting direct factorization uses $O(N^{3/2})$ work and $O(N\log N)$ storage under the usual geometric separator assumptions.

For a symmetric factorization in a fixed ordering, the elimination tree assigns the displayed parent to a column of the lower-triangular factor when that set is nonempty. A column with no later entry is a root. The tree records dependencies of descendant updates on later pivots, supports symbolic factor allocation, and exposes parallel subtrees. It describes the factorization ordering, not merely a spanning tree of the original matrix graph.

Symbolic factorization computes possible nonzero positions and update dependencies from a sparse matrix pattern and ordering before calculating numerical factor values. The elimination graph predicts fill and the elimination tree organizes dependencies. Numerical pivoting or exceptional cancellation can make the computed factor pattern differ from this structural prediction.

Associate an undirected graph with a symmetric matrix's off-diagonal nonzero pattern. Eliminating vertex $i$ updates the remaining matrix by $a_{jk}\mapsto a_{jk}-a_{ji}{a}_{ik}/a_{ii}$. Only pairs of neighbors of $i$ can acquire fill-in. If the graph is a tree, remove leaves successively: each eliminated vertex has at most one remaining neighbor, so only a diagonal entry can change. Every Schur complement of a positive-definite matrix is positive definite, so no zero pivot forces a different ordering. Leaf removal therefore constructs a perfect elimination ordering and a fill-free factorization.

For a nontrivial strongly connected directed graph, a coherent cyclic order of a digraph and the Dilworth theorem produce a spanning circuit family with at most its independence number circuits. Put a largest consecutive stable block first and duplicate its vertices as sinks. Use forward edges and edges into the sink copies to form an acyclic auxiliary graph. Its reachability order has width equal to the block size. Expand a chain partition to paths, then exchange intersecting path suffixes until the permutation of source-to-sink labels has as many cycles as possible. Paths in the same permutation cycle are disjoint, so identifying copied endpoints produces simple directed circuits. A windmill of directed triangles sharing one vertex shows that the bound is sharp.