Over a field of characteristic zero, for an antisymmetric $2n\times 2n$ matrix, the Pfaffian is $\mathrm{Pf}(A)=(2^{n}n!{)}^{-1}{\sum }_{\sigma \in S_{2n}}\mathrm{sgn}(\sigma ){\prod }_{j=1}^n{A}_{\sigma (2j-1),\sigma (2j)}$. Equivalently, it is the coefficient of the top exterior power of the associated alternating form, with the usual factorial normalization. Its square is the determinant. In size four it is $a_{12}{a}_{34}-a_{13}{a}_{24}+a_{14}{a}_{23}$, so its vanishing characterizes the simple nonzero bivectors of the Klein quadric.

For a square matrix with row vectors $r_i$, the absolute determinant is at most the product of the Euclidean norms of its rows. If the rows are dependent the determinant is zero. Otherwise Gram-Schmidt process leaves the determinant unchanged and replaces each row by its component orthogonal to its predecessors. The resulting orthogonal rows have lengths no greater than the original rows, and their volume is their product. This proves the inequality; equality holds for mutually orthogonal rows.

For an invertible square matrix $A$ and $Ax=b$, replace column $i$ by $b$ to form $A_i$. Multilinearity of the determinant and $b={\sum }_j{x}_j{A}_{\cdot j}$ give $\det A_i=x_i\det A$: all other summands have repeated columns and vanish. Division gives the formula. With integer coefficients this supplies explicit rational solutions and, together with the Hadamard determinant inequality, bounds their numerators and denominators.

For matrices $A\in {\mathbb{R}}^{r\times m}$ and $B\in {\mathbb{R}}^{m\times r}$ with $r\le m$,where the indices of $S$ are taken in increasing order. To prove it, write the $j$th column of $AB$ as ${\sum }_q{B}_{qj}{A}_{:,q}$ and expand its determinant by multilinearity. Terms with a repeated $q$ vanish by alternation. For each remaining index set $S=\{s_1<\cdots <s_r\}$, collect all permutations $q_j=s_{\sigma (j)}$. The factor from $A$ is $\mathrm{sgn}(\sigma )\det A_{:,S}$; the sum of the products $\mathrm{sgn}(\sigma ){\prod }_j{B}_{s_{\sigma (j)},j}$ is $\det B_{S,:}$. This proves the formula. Applying it to selected rows and columns proves that a product of compatible totally nonnegative matrices is totally nonnegative.

Let $L$ be the triangular matrix with $L_{ij}=1$ for $j\le i$ and zero otherwise. Then $(L{L}^T{)}_{ij}=\min (i,j)$ and $\det L=1$, so the determinant of $L{L}^T$ is one. More generally, for $0=t_0<t_1<\cdots <t_n$, the matrix with entries $\min (t_i,t_j)$ factors as $L\mathrm{diag}(t_k-t_{k-1})L^T$ and has determinant ${\prod }_k(t_k-t_{k-1})$. This is the finite-dimensional structure behind the Brownian covariance kernel.

For a weighted acyclic directed graph, let $M_{ij}$ be the sum of path weights from $A_j$ to $B_i$. Expanding $\det M$ gives a signed sum of path families. At the first intersection in a fixed topological order, swap the two path tails. The operation preserves weight and reverses the endpoint permutation's sign, so intersecting families cancel. Only vertex-disjoint families remain. If planarity forces these families to join identically indexed endpoints, their sum has positive sign. Applied to northeast paths with horizontal weight $x_r$ at height $r$, this gives the skew Jacobi–Trudi identity.

Response surface methodology fits local regression approximations to a quantitative response as experimental settings change. A first-order model estimates a local gradient; augmenting the design permits quadratic regression, curvature assessment and local optimization. Extrapolating a fitted surface outside the experimentally supported region requires caution.

A contrast is a linear combination ${\sum }_i{c}_i{\mu }_i$ of group means with ${\sum }_i{c}_i=0$. It compares responses without depending on a common additive level. Treatment contrasts and factorial contrasts specify comparisons in a design of experiments.

A Latin square of order $q$ is a $q\times q$ array of $q$ symbols, each occurring once in each row and column. Used as a row-column design, its symbols label treatments. Centered treatment indicators are orthogonal to both row and column indicators, because every row-treatment and column-treatment pair occurs once. This balance separates additive effects from two nuisance directions.

Optimal experimental design chooses observation locations and allocations to optimize a specified criterion for a specified statistical model. Criteria can control the determinant of an estimator's covariance matrix or its worst prediction variance. Optimality depends on the model and the permitted design class, including whether allocations are integer or approximate.

In three space dimensions, rescale $A_i(x)$ to $\lambda A_i(\lambda x)$ and $\Phi (x)$ to $\Phi (\lambda x)$. The gauge curvature then scales with two derivatives, and the adjoint covariant derivative of the Higgs field with one. Magnetic, Higgs-gradient and potential energies give the displayed formula. Stationarity requires $V_B=V_D+3V_U$, which permits nonzero energies. In the Bogomolny-Prasad-Sommerfield monopole limit $V_U=0$, the Bogomolny equations give $V_B=V_D$.

For a smooth gauge field on ${\mathbb{R}}^3$ with finite positive magnetic energy, dilate its connection one-form by $A_i^{(s)}(x)=s{A}_i(sx)$. Its gauge curvature scales as $F_{ij}^{(s)}(x)=s^2{F}_{ij}(sx)$, so the energy scales as $s^4{s}^{-3}E=sE$. A static stationary configuration must have zero derivative at $s=1$, forcing $E=0$ and hence $F=0$. This Derrick scaling argument assumes the dilation preserves the boundary conditions and that boundary terms vanish. It does not exclude four-dimensional Yang-Mills instantons, whose Euclidean action is scale invariant.

For a smooth finite-energy static planar Skyrme model field, Derrick scaling gives zero quartic energy, hence $[R_1,R_2]=0$ pointwise. The quartic first variation then vanishes, leaving the sigma-model equation ${\partial }_i{R}_i=0$. The right-current Maurer-Cartan equation gives ${\partial }_1{R}_2-{\partial }_2{R}_1=[R_1,R_2]=0$. Each current component is therefore harmonic on the plane and square-integrable. The mean-value estimate on expanding disks makes each component zero, so $U$ is constant. This excludes smooth finite-energy planar static solutions of this potential-free SU(2) model, without misapplying the same conclusion to a pure curved-target sigma model.

Every derivative has the intermediate-value property, even when the derivative is discontinuous.

The right and left one-sided derivatives are the limits of the difference quotient restricted respectively to positive and negative increments:At an interior point, differentiability is equivalent to existence and equality of these two finite limits. For the absolute value function, $f_{+}^{\prime }(0)=1$ and $f_{-}^{\prime }(0)=-1$, so it is continuous at zero but not differentiable there.

If a continuous real function on an interval has the displayed limit zero at every interior point, it is affine. On any compact subinterval subtract its endpoint chord. If the difference were positive somewhere, adding $\epsilon (t-a)(t-b)$ for sufficiently small positive $\epsilon$ would retain a positive interior maximum. At a maximum every symmetric second difference is nonpositive, but its normalized limit would be $2\epsilon >0$. Thus the function lies below its chord; applying the same argument to its negative gives equality. Real and imaginary parts give the complex-valued version. Existence of ordinary second derivatives is unnecessary.

For a continuous function with affine formulas $At+B$ to the left of $c$ and $A^{\prime }t+B^{\prime }$ to its right, the quotient $[F(c+h)-2F(c)+F(c-h)]/h$ equals $A^{\prime }-A$ for positive $h$. If it tends to zero, the slopes agree, and continuity at $c$ makes the intercepts agree as well. This detects slope jumps that ordinary pointwise derivatives away from the corner cannot detect.

For a continuous function with affine formulas $At+B$ to the left of $c$ and $A^{\prime }t+B^{\prime }$ to its right, the quotient $[F(c+h)-2F(c)+F(c-h)]/h$ equals $A^{\prime }-A$ for positive $h$. If it tends to zero, the slopes agree, and continuity at $c$ makes the intercepts agree as well. This detects slope jumps that ordinary pointwise derivatives away from the corner cannot detect.

The symmetric second derivative is the displayed limit when it exists. For a twice continuously differentiable function it equals the ordinary second derivative by Taylor expansion; the definition itself does not require ordinary differentiability. A continuous function with vanishing symmetric second derivative throughout an interval must be affine, by vanishing symmetric second derivative forces an affine function. For a twice-integrated Fourier series, sinc-squared weights express this quotient, connecting the operation to Riemann summation of a convergent trigonometric series.

For a differentiable function, a local linear approximation replaces nearby values by its value plus its derivative times the input change. The error divided by that change tends to zero. It uses the first-order Taylor polynomial; the statement concerns a limit near the specified point, not a guaranteed approximation far from it.