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A differentiable function has a $\beta$-Lipschitz gradient on a set $D$ if its gradient satisfies Lipschitz continuity there:All norms here are the Euclidean norm. The constant limits how fast the gradient changes; it is not a bound on the magnitude of the gradient. For $f(x)=(x_1^2+9x_2^2)/2$, the gradient is the linear map $x\mapsto \mathrm{diag}(1,9)x$. Its operator norm is $9$, so $\beta =9$ works globally and is sharp by taking $x-y=(0,1{)}^T$. The gradient itself is unbounded on the whole Euclidean space.

If $f$ has a continuous Hessian matrix on an open convex set $D$ and $\Vert {\nabla }^{2}f(z){\Vert }_2\le \beta$ throughout $D$, integrate that linear map along the line segment from $x$ to $y$:The operator norm bound proves the displayed Lipschitz continuity. Convexity of the domain keeps the entire line segment inside it. Bounding the Hessian matrix only at the endpoints is insufficient.

Stochastic gradient descent updates parameters using an unbiased or approximate gradient computed from one observation or a mini-batch rather than the complete data set.

For fixed prime $p$, a one-bounded function on ${\mathbb{F}}_p^n$ with $U^3$ norm at least $\delta$ has correlation bounded below in terms of $p$ and $\delta$ with a quadratic phase, with the usual nonclassical formulation in small characteristic.

A quadratic phase on ${\mathbb{F}}_p^n$ is a function $x\mapsto e_p(q(x))$, where $q$ is a quadratic form and $e_p(t)=e^{2\pi it/p}$. Its third multiplicative derivative is one, so its $U^3$ norm is one.

For functions indexed by $ϵ\in \{0,1{\}}^k$, the Gowers inner product iswhere $\mathcal{C}$ denotes complex conjugation.

The Fennoscandian Ice Sheet covered northern Europe during the last glacial period and produced ongoing uplift around the Baltic region after retreat.

The Laurentide Ice Sheet covered much of northern North America during the last glacial period and left a major postglacial-rebound signal after retreat.

Flexural isostasy models the lithosphere as an elastic plate that bends over a buoyant substrate. For a sinusoidal deflection of wavenumber $k$, bending contributes a restoring stress proportional to $B{k}^4$, where $B$ is the plate's bending stiffness.

Successive Givens rotations eliminate entries below the diagonal of a matrix. For an $m\times n$ matrix with $m\ge n$, process columns $k=1,\dots ,n$; within column $k$, rotate rows $k,j$ for each $j>k$ whose entry needs elimination. Apply each Givens row-pair update to columns $k,\dots ,n$. Both rows are zero in earlier columns, so these operations preserve earlier eliminations. No assumption of full column rank is needed to perform the eliminations.

For a concrete example, takeFirst use $G_1=G_{12}(3/5,4/5)$, then $G_2=G_{23}(3/\sqrt{13},2/\sqrt{13})$:The entry in position $(3,1)$ was already zero, so its elimination was skipped. The second Givens rotation combines the remaining pair $(3,2{)}^T$ in column two. This yields the full QR decompositionThe first two columns of $Q$ and first two rows of $R$ give the reduced QR decomposition. Direct multiplication verifies both $QR=A$ and $Q^{T}Q=I$. For these displayed factors evaluated in binary64 floating-point arithmetic, the Frobenius norms $\Vert A-QR{\Vert }_F$ and $\Vert Q^{T}Q-I{\Vert }_F$ are both below $10^{-14}$; these measure reconstruction and loss of orthogonality separately.

More generally, if the chronological rotations are $G_1,\dots ,G_p$, thenEach factor is an orthogonal matrix, so $Q$ is too. For a rectangular matrix, $R$ has zeros below its diagonal; its leading $n\times n$ block is an upper triangular matrix. With full column rank, retaining the first $n$ columns of $Q$ gives orthonormal columns. To impose a positive diagonal when necessary, flip matching columns of $Q$ and rows of $R$, preserving their product.

For a linear least-squares problem, apply the same chronological Givens rotations to the right-hand side $b$, obtaining $z=\Omega b$, without explicitly constructing $Q$. Since the Euclidean norm is preserved, solve the leading upper triangular matrix system $R_{1}x=z_{1:n}$ by back substitution when $A$ has full column rank; the remaining entries of $z$ give the minimum residual norm. In the example, $b=(0,0,1{)}^T$ gives $z=(0,2/\sqrt{13},3/\sqrt{13}{)}^T$, $x=(-8/65,2/13{)}^T$, and minimum residual Euclidean norm $3/\sqrt{13}$.

The locality is useful for a sparse matrix or an existing QR decomposition needing an extra row: only two rows change at each step, and the new row can be eliminated against the stored upper triangular matrix. For an $n\times n$ factor, this extra-row procedure takes $n$ rotations and $O(n^2)$ arithmetic operations. Locality does not guarantee preservation of sparsity: mixing rows can create nonzeros wherever either row had one. For a fully populated $m\times n$ input, the straightforward elimination uses ${\sum }_{k=1}^n(m-k)$ rotations, each touching $n-k+1$ columns, giving $O(m{n}^2)$ work. Storage layout, zero patterns and implementation affect speed; this count alone establishes no universal advantage over other QR decomposition algorithms.

The Euclidean norm preservation explains the appeal for numerical stability: exact rotations do not amplify a perturbation's length. Rounded coefficients and Givens row-pair updates still introduce roundoff error. Small reconstruction and orthogonality residuals check the computed factors, while the condition number of the original problem still controls how accurately a linear least-squares problem determines its solution.

Two phases coexist where their molar Gibbs free energies agree. Differentiating that equality along the coexistence curve relates its slope to the entropy and volume jumps.

For a reaction with stoichiometric changes ${\nu }_i$, varying the reaction extent at fixed temperature and pressure givesEquilibrium requires ${\sum }_i{\nu }_i{\mu }_i=0$.

A symmetric matrix obtained from a conservative five-point discretization of a diffusion equation has row centre $-s_i/h^2$ and radius at most $s_i/h^2$, where $h>0$ is the grid spacing and $s_i$ is the sum of its four nonnegative edge coefficients. Here the matrix represents the diffusion generator, with nonnegative off-diagonal entries; boundary elimination may reduce the row radius. If every coefficient is at most $a_{\max }>0$, then $s_i\le 4a_{\max }$, and the Gershgorin bounds for symmetric matrices place every eigenvalue in $[-8a_{\max }/h^2,0]$.

For the Forward Euler method, an eigenvalue $\lambda \le 0$ has amplification factor $1+\Delta t\lambda$. Requiring $|1+\Delta t\lambda |\le 1$ for every such eigenvalue gives the sufficient absolute stability condition $\Delta t/h^2\le 1/(4a_{\max })$. Symmetry makes this also prevent growth of the Euclidean norm. For $a_{\max }=2$ and $h=0.1$, the bound is $\lambda \in [-1600,0]$ and $\Delta t\le 0.00125$. At that step size the interval maps to amplification factors in $[-1,1]$. This illustrates why cheap Gershgorin disc bounds can select a stable time step before any eigenvalues are computed.

A coastal Kelvin wave is trapped within a Rossby deformation radius of a boundary. It propagates with the boundary on its right in the Northern Hemisphere and on its left in the Southern Hemisphere, and its linear shallow-water potential-vorticity anomaly vanishes.

An inertia-gravity wave is restored jointly by buoyancy or free-surface gravity and the Coriolis force.

For sufficiently small nonzero $\epsilon$,is an embedded antipodally invariant curve on the round sphere. It is not a great circle and hence not a geodesic, while the antipodal isometry exchanges its two complementary discs.

A non-equatorial latitude on the round sphere is preserved transitively by rotations about its axis but has nonzero geodesic curvature. Its two complementary spherical caps are not isometric.

If a closed oriented surface is cut along a curve into two isometric surfaces with boundary, their Euler characteristics and total Gaussian curvatures agree. Applying Gauss-Bonnet to both halves, whose induced boundary orientations are opposite, gives

If ambient isometries preserving a connected curve act transitively on it, then the magnitude of its geodesic curvature is constant. Continuity then makes the signed geodesic curvature either identically zero or of one fixed sign.

A coordinate $q_i$ is ignorable, or cyclic, when $\partial L/\partial q_i=0$. The Euler--Lagrange equation then gives ${\dot{p}}_i=0$, so its generalized momentum is conserved.

A generalized linear mixed model adds latent random effects to the linear predictor of a generalized linear model.