A stamen is a flower's male reproductive organ, typically an anther supported by a stamen filament. The anther produces and releases pollen.^[ref]

A network circulation is a flow with zero net supply at every vertex. A feasible residual network circulation changes one feasible flow into another with the same prescribed supplies. It decomposes into directed cycles, giving the negative-cycle optimality test for minimum-cost flow.

Catastrophic cancellation occurs when subtraction of nearly equal approximations removes their matching leading digits and exposes earlier roundoff error. The subtraction can itself be exact on the stored operands: the information may already have been lost while obtaining them. Goldberg distinguishes this from cancellation of accurately known quantities, which can be harmless.^[ref]

For $x>0$, consider the single real-valued functionThe rationalization of a square-root difference proves equality in exact arithmetic. The two evaluations use the same input $x$, but the direct version first forms a value near one and then subtracts one. The rearranged version keeps the small input in its numerator and divides by a denominator near two.

For binary64 x = 1e-16, the rounded sum 1.0 + x is one, so sqrt(1.0 + x) - 1.0 is zero. The rationalized evaluation returns approximately $5\times 10^{-17}$. A 100-digit Decimal calculation using the exact stored input gives relative errors of one and about $2.50\times 10^{-17}$ respectively. At x = 1e-12, the direct result is about $5.0004445029\times 10^{-13}$, with relative error $8.89\times 10^{-5}$; the rearranged result is about $4.99999999999875\times 10^{-13}$, with relative error $1.21\times 10^{-17}$.

The plot uses Decimal.from_float(x) to preserve the binary input exactly, computes $x/(\sqrt{1+x}+1)$ at 100 decimal digits, and verifies the reference at 160 digits. Exact conversion of a stored binary value is documented by Python.^[ref] Each error is $|\hat{f}-f|/|f|$, with the computed result converted exactly to Decimal before comparison. Hence the curves measure evaluation error for one common input, rather than a mixture of input conversion error and evaluation error. The plotted interval $10^{-16}\le x\le 10^{-1}$ is far above the subnormal number range.

The original problem is well conditioned with respect to relative changes in $x$. Using the derivative $f^{\prime }(x)=1/(2\sqrt{1+x})$ and rationalization of a square-root difference, its relative condition number isIt tends to one as $x$ tends to zero. The direct evaluation instead introduces the intermediate difference $s-1$ with $s=\sqrt{1+x}$; an error of order unit roundoff in $s$ is large relative to $f(x)\approx x/2$. This is a failure of the chosen algorithm's accuracy despite a small condition number for the same real-valued function. The rationalized evaluation avoids that intermediate catastrophic cancellation, retaining relative accuracy near the unit roundoff scale throughout the plotted interval. Recognizing this structure can improve a calculation far more than merely printing extra digits.

A subnormal number is a nonzero representable value smaller in magnitude than the least positive normalized value ${\beta }^{e_{\min }}$. Keep the exponent scale fixed at $e_{\min }$ and allow a zero leading digit: the positive values are $k{\beta }^{e_{\min }-p+1}$ for integers $1\le k<{\beta }^{p-1}$. This fills the gap toward zero with constant absolute spacing, at the cost of fewer significant digits. Goldberg describes this gradual approach to zero and why it matters for differences of small numbers.^[ref]

For binary64 the least positive normalized value is $2^{-1022}$, and subnormal numbers are spaced by $2^{-1074}$. With nearest rounding and gradual underflow enabled, an exact result $z$ with $|z|<2^{-1022}$ has absolute rounding error at most $2^{-1075}$. A uniform relative bound by unit roundoff fails here: $z=2^{-1075}$ rounds to zero under the even-last-digit tie rule, giving relative error one. Machine epsilon measures resolution near one; it does not measure the distance from zero to the first positive value.

For floating-point arithmetic with precision $p$ in radix $\beta$ and rounding to nearest, unit roundoff is the half-gap scale $u={\beta }^{1-p}/2$. If a nonzero exact real number $z$ lies in the normal finite range, its nearest rounded value satisfiesIndeed, the absolute distance to the nearest value is at most ${\Delta }_e/2$, and $|z|\ge {\beta }^e$ gives the bound. For a correctly rounded addition, subtraction, multiplication or division, apply this estimate to the exact result of that one operation on the stored operands. The bound does not include earlier roundoff error or automatically bound a whole algorithm. Results outside the finite range and tiny subnormal numbers require separate treatment. Rounding in a directed mode also changes the bound.^[ref]

Here machine epsilon means the gap from one to the next larger value in normalized floating-point arithmetic:It is a relative resolution scale near one, not the smallest positive representable value and not a fixed absolute error for all magnitudes. In binary64, $p=53$ giveswhere $u$ is unit roundoff for rounding to nearest. Some authors instead call $u$ their machine epsilon; MIT 18.335J uses this half-gap convention. Always check which definition an error bound uses.^[ref]

The gap below one is smaller: the preceding binary64 value is $1-2^{-53}$, whereas the next value above one is $1+2^{-52}$. Under nearest-even rounding, 1.0 + 2.0**-53 is exactly 1.0: the exact sum is a tie and one has the even final binary digit. By contrast, 1.0 + 2.0**-52 is the next value. Thus searching for the smallest increment that changes one by trial addition also depends on the rounding convention. At $2^{53}$ the next larger binary64 value is $2^{53}+2$, illustrating why a small absolute increment can disappear at large magnitude.

The predator-prey first integral $H=cX-d\log X+bY-a\log Y$ changes by the displayed amount when $h$ prey are removed. At a fixed prey threshold this increment is constant; sufficiently high thresholds reduce oscillation energy until the threshold is no longer crossed. At a fixed predator threshold, the increment depends on the prey abundance and on the crossing direction. A rising-predator balanced-quota cycle has negative return multiplier of magnitude greater than one; its leading small-oscillation approximation is marginal rather than a robust attractor.

Automorphisms $\sigma (z)={\zeta }_{n}z$ and $\tau (z)=1/z$ generate a dihedral group of order $2n$. The invariant $w=z^n+z^{-n}$ has rational map degree $2n$, so $[\mathbb{C}(z):\mathbb{C}(w)]=2n$. The fixed-field theorem for a finite automorphism group gives the same degree over its fixed field, proving equality. For odd prime $n$, the nontrivial proper subgroups are the rotation group and the $n$ reflection groups. Their fixed fields are $\mathbb{C}(z^n)$ and $\mathbb{C}(z+{\zeta }_n^{-j}/z)$ for reflections ${\sigma }^j\tau$, respectively.

For potential energy of two-layer entrainment with constant $B=g^{\prime }h$, the fixed-energy turbulent mixed layer assumption gives $u_U^{2}h=C_K^2{\Omega }^2{R}^2{h}_0$. If $u_I=C_I{u}_U$ and the interfacial power model for entrainment applies with fixed coefficients, integration gives the displayed depth law with $A=5C_W{c}_D{C}_I^3{C}_K^3{\Omega }^2{R}^3/(B{h}_0)$. The relation applies only before the layer reaches the tank bottom. Its interfacial Richardson number $B{d}_I/(h{u}_I^2)$ stays constant for fixed interface thickness, whereas it decreases like $h^{-1}$ in the constant-speed entrainment model.

The four commuting independent stabilizer generators define a two-dimensional stabilizer code. The identity and fifteen single-site Pauli errors have sixteen distinct four-bit error syndromes, so their two-dimensional images form an orthogonal direct sum of the physical space. This proves nondegenerate single-site correction and the perfect quantum error-correcting code property. The operators $X^{\otimes 5}$ and $Z^{\otimes 5}$ commute with the stabilizer and anticommute with each other. Multiplying $X^{\otimes 5}$ by $XZZXI$ gives, up to phase, $IYYIX$, a nontrivial weight-three logical operator, proving distance exactly three.

For $V=1-2M/r^2+Q^2/r^4$, an orthonormal-frame curvature calculation gives $(V^{\prime \prime }{)}^2+6(V^{\prime }/r{)}^2+12((1-V)/r^2{)}^2$. Expansion gives the displayed invariant, diverging at zero radius. It is finite at every positive root of V. Thus horizon coordinate singularities are distinct from the central curvature singularity.

When an experiment's configuration is observed, inference conditions on the experiment actually performed rather than averaging over its possible configurations. Conditioning on an ancillary statistic is a central application; freedom from a nuisance parameter must still be checked in the resulting conditional distribution.

For first-order logic, every sentence true in every model of a theory is derivable from that theory. Equivalently, every consistent theory has a model.

The fragment $L^k$ permits at most $k$ distinct variable names, but allows bound names to be reused. It can contain formulas of arbitrarily large quantifier depth. On a linear order, alternating two variable names can express an increasing chain of any prescribed finite length.

Let $A$ be a commutative algebra over a field $k$, with a homomorphism $A\to k$ restricting to the identity on $k$, and let $I$ be its kernel. If $A$ is generated by finitely many elements, subtracting their images in $k$ gives finitely many generators in $I$. Every polynomial in those generators, modulo $I^2$, is a linear combination of their classes. Thus $I/I^2$, the algebraic cotangent space at this point, is finite-dimensional. For $A=k+xk[x,y]$ and $I=xk[x,y]$, the classes $x{y}^j$, $j\ge 0$, form an infinite basis modulo $I^2=x^{2}k[x,y]$. This is a subalgebra of a polynomial ring which is not a finitely generated algebra.

Consider the advection equation $u_t+a{u}_x=0$ with constant speed $a>0$, no sources, and periodic boundary conditions on $[0,1]$. Use $N$ equal cells of width $h=1/N$, time step $k>0$, and indices modulo $N$. The exact solution is the periodic translation $u(x,t)=u_0(x-at)$. At a face, the Riemann problem transports the left state to the right, so the two-state upwind numerical flux is $F(L,R)=aL$, with $F(U,U)=aU$.^[ref]

The finite volume method with this numerical flux and an explicit Euler method step givesHere $U_j^n$ represents a cell average; the recurrence also occurs in the upwind finite difference scheme. Impose the specific CFL condition $0<\nu \le 1$. Each updated cell average is then a convex combination of two old ones. This proves preservation of their initial minimum and maximum, including nonnegativity for nonnegative data, and contraction of the maximum norm of the difference of two solutions on this periodic grid. These statements concern this constant-speed scalar update with this time step.

For an original synthetic example, take $N=8$, $a=1$, $h=1/8$, $k=1/16$, hence Courant number $\nu =1/2$. Choose a function constant on each successive cell, with the following values, so these are exact initial cell averages. One step givesFor example, cell zero receives from cell seven across the periodic face: $U_0^1=(0+1)/2=1/2$. All eight face transfers cancel in the total by shared face flux cancellation. More generally this periodic recurrence conserves the total algebraically for any $\nu$, even though the CFL condition is needed for the convex combination argument. With the same initial data and $\nu =3/2$, one step gives $U_2^1=-1$ and $U_4^1=5/2$, despite preserving the total.

After four steps, $t=1/4$, the exact translated profile has moved two cells to the right. Repeated averaging instead gives the explicit binomial coefficient formulawith all indices modulo eight. The diagram compares these computed cell averages with the exact translated cell averages. Their integrals agree, but the numerical jumps spread across cells. The finite volume method is useful for jump discontinuities because it retains the integral balance even at a jump; conservation alone does not guarantee a sharp profile. At $\nu =1$ this particular update is an exact one-cell shift, whereas for $0<\nu <1$ it mixes neighboring states. No accuracy conclusion at a jump follows just from the conserved total.

In a finite volume method, orient every face numerical flux to the right. For cells $1,\dots ,N$, write the discrete total as $M^n={\sum }_{j=1}^N{h}_j{U}_j^n$. If $u$ is a mass per unit length, this total approximates mass; for a different conserved field, it measures that field's integral. Multiplying the update by the cell width gives an amount balance, so adjacent cells contributeThe interior face contributes $-k{F}_{j+1/2}^n+k{F}_{j+1/2}^n=0$. Summing all cells givesWith periodic boundary conditions, the two boundary labels denote the same physical face and must use the same numerical flux; then $M^{n+1}=M^n$. On an open interval the total changes by the boundary transfers. Unequal widths require the weighted total $\sum h_j{U}_j$, not the unweighted sum of cell averages.

The diagram uses two unit-width cells, $k=1/4$, cell averages $(2,1)$ and right-oriented numerical fluxes $(1,3,2)$ at their three faces. The updated cell averages are $(3/2,5/4)$: the total changes from $3$ to $11/4$, exactly $k(1-2)=-1/4$. The shared interior transfer $k\cdot 3=3/4$ cancels. If the receiving cell instead used $3+\epsilon$ for that same face, the total would acquire the spurious term $k\epsilon$. Conservation is therefore an algebraic property of sharing the transfer, even when the numerical flux is a poor approximation. In floating-point arithmetic, independently rounded cell updates may still leave small roundoff errors in the computed total.

The Jacobian matrix of a deformation map carries an infinitesimal reference line element to its current image. Its determinant $J=\det F$ is the local volume ratio. An orientation-preserving deformation has $J>0$, and incompressibility imposes $J=1$.

A stretch tensor isolates local changes of material length from the rotation contained in a deformation gradient. For an invertible orientation-preserving $F$, the polar decomposition in continuum mechanics iswhere $R$ is a rotation matrix and $U,V$ are positive-definite symmetric matrices. The right stretch tensor $U$ acts on directions in the reference configuration; the left stretch tensor $V$ acts on directions in the current configuration.

The eigenvalues of either stretch tensor are the principal stretches. A value two means a material line in the corresponding principal direction doubles its length, while a value one means unchanged length. For a pure rotation, both stretch tensors are $I$, even though $F$ need not equal $I$. This makes them useful when an experiment includes substantial specimen rotation that should not be mistaken for strain.

Simple shear displaces one coordinate direction in proportion to a transverse coordinate while preserving volume. Its deformation gradient has a single off-diagonal component $\gamma$. Simple shear flow describes the corresponding velocity field rather than the finite deformation.