A smooth projective curve of gonality three has a basepoint-free pencil of degree three. For a nonhyperelliptic curve, a fibre of this pencil spans a line in its canonical map: Riemann-Roch theorem gives $h^0(K_C-D)=g-2$. The union of these lines is a scroll of degree $g-2$ in ${\mathbb{P}}^{g-1}$. Its smooth ruled model may map to a cone in low genus.

For canonical global four-dimensional N=1 supersymmetry, define $F^i=\overline{W_i}$, $H_{ij}=W_{ij}$ and $G_{ia}=g^a{\bar{\varphi }}_j(T^a{)}^j{}_i$ at a stationary vacuum. Gauge invariance of the superpotential gives $G^{T}F=0$, while stationarity of $V={\sum }_i|W_i{|}^2+\frac{1}{2}{\sum }_a(D^a{)}^2$ gives $HF-GD=0$. Therefore the canonically normalized fermion mass matrix, after a harmless gaugino phase choice, obeysIf supersymmetry is broken, this vector is nonzero and specifies a massless Goldstino. The matrix with off-diagonal blocks $-G$ instead acts on $(F,D)$; it is related to the canonical matrix by a change of normalization, not by equality of canonical entries. Additional zero modes can coexist with the Goldstino.

A right Goldie ring satisfies the right-sided conditions in the definition of a Goldie ring. Every right Noetherian ring does: its right annihilators are right ideals, and its right regular module has finite uniform dimension.

Phosphatidylcholine, abbreviated PC, is a glycerophospholipid with a phosphate-choline head. Its zwitterionic head illustrates why a hydrophilic region need not have a nonzero net electric charge. See Lodish et al.: membrane lipids.

A Cauchy hypersurface is an achronal hypersurface met exactly once by every inextendible timelike curve. Its topology records the global spatial topology of a globally hyperbolic spacetime.

In the two-class globally coupled underdominant metapopulation, the rectangle $3/4\le x\le 1$, $0\le y\le 1/4$ is forward invariant under the displayed bound. On its inner edges, selection points inward with magnitude $3s/32$ while outward migration is at most $(3/4)\sigma (1-q)$ or $(3/4)\sigma q$; the outer edges also point inward. Thus prescribed initial state $(1,0)$ cannot approach global allele fixation when $0<q<1$. Maximizing $f(u)/u=s(1-u)(2u-1)$ gives $s/8$ at $u=3/4$, making the bound optimal within this particular symmetric worst-case rectangle argument, not a universal sharp migration threshold.

With equally many initially pure habitats of each allele, symmetry preserves $y=1-x$. The heterogeneous equilibrium exists for $\sigma <s/2$. Its full-system eigenvalues are $-s+3\sigma$ in the mean-frequency direction and $-s+2\sigma$ in the antisymmetric direction, so it attracts general nearby perturbations only for $\sigma <s/3$. Exact symmetry can preserve both alleles beyond this robust range; at stronger migration it approaches the unstable uniform midpoint.

For a conserved charge $Q$ and invariant vacuum, its global symmetry commutes with the S-matrix. Matrix elements between charge eigenstates give the displayed selection rule. Constant electromagnetic phases cancel at every QED vertex, and nonzero physical amplitudes conserve total charge. Local gauge invariance strengthens this global rule to the Ward-Takahashi identity.

For the whole-line linear complex Ginzburg-Landau equation with real positive diffusion $\gamma$ and $\mu (x)={\mu }_0-\nu {ϵ}^2{x}^2$, removing drift and scaling $\xi =\sqrt{ϵ}(\nu /\gamma {)}^{1/4}x$ gives the Hermite differential equation. Decay at both infinities quantizes the spectral parameter to $2j+1$. The strongest mode is $j=0$, giving global instability iff ${\mu }_0>U^2/(4\gamma )+ϵ\sqrt{\gamma \nu }$. The confinement correction vanishes in the slow-profile limit, recovering the homogeneous absolute wave-packet instability threshold.

For a differentiable approximation $\eta$, put $d={\eta }^{\prime }-f(t,\eta )$. If $f$ is Lipschitz continuous with constant $L$ on the relevant region, the exact solution's error satisfies the displayed bound by Gronwall's inequality. Thus a bound on the residual over the whole interval supplies an a posteriori global error estimate. A small estimated defect at a single step does not furnish this bound by itself.

If a one-step map is Lipschitz with factor at most $1+L{h}_n$ and its exact-start defect is at most $C{h}_n^{p+1}$, iteration of the error recurrence gives the displayed global bound on a time interval of length $T$. Since $\sum h_n\le T$, the defect sum is at most $T(\max h_n{)}^p$. Local tolerance control alone is not a global accuracy certificate: propagation, the number of steps, stiffness, roundoff and nonlinear-solve errors also enter. The discrete Gronwall inequality organizes this distinction.

Left matrix multiplication by a Givens rotation replaces rows $i,j$ of a matrix byBoth right-hand sides use the old entries. An in-place algorithm must save the first result temporarily, update the second entry, and only then overwrite the first. The reference BLAS DROT implementation follows this order with its temporary DTEMP.^[ref]

For elimination in column $k$, construct $(c,s)$ from $(A_{ik},A_{jk})$. This sets the lower component to zero in exact arithmetic. In floating-point arithmetic the computed expression can leave a small remainder; after computing the other updated entries, explicitly store $A_{jk}^{\prime }=0$ and $A_{ik}^{\prime }=r$. Previously eliminated columns stay zero if both participating rows are already zero there. This is why the ordering in QR decomposition by Givens rotations matters.

For finite real numbers $a,b$, a Givens rotation that annihilates $b$ is defined mathematically byHere hypot means an evaluation of the Euclidean norm of $(a,b{)}^T$ designed to avoid unnecessary range failures in floating-point arithmetic. The zero case selects the identity matrix, avoiding division by zero. If $a=0,b\ne 0$, then $c=0,s=\mathrm{sign}(b)$; if $b=0,a\ne 0$, then $c=\mathrm{sign}(a),s=0$. Thus the retained component is nonnegative, including when $a<0$. In exact arithmetic, $ca+sb=r$ and $-sa+cb=0$.

Computing the square root of a*a + b*b directly can fail even when the final norm is representable. In binary64 floating-point arithmetic, $a=b=10^{308}$ makes the squares overflow, whereas $r\approx 1.4142\times 10^{308}$ is finite. With $a=b=10^{-300}$ the squares round to zero, although $r\approx 1.4142\times 10^{-300}$ is representable. A range-safe hypot avoids these intermediate failures.

The simple divisions $a/r,b/r$ still need care when $r$ is a subnormal number: rounding $r$ can cause large relative errors. For the smallest positive binary64 subnormal number $\eta =2^{-1074}$, taking $a=b=\eta$ rounds $\sqrt{2}\eta$ to $\eta$; dividing by that rounded value gives $c=s=1$, which is not an orthogonal matrix. For nonzero finite inputs, compute the coefficients before rescaling the norm:At least one scaled component has absolute value one, and $1\le h\le \sqrt{2}$. This construction keeps the coefficients meaningful even when rounding $r$ is severe. It cannot make an unrepresentably large $r$ finite, preserve an arbitrarily tiny $s$ against underflow, or repair inputs already rounded to zero. The row updates themselves can also overflow if their exact results exceed the available range.

LAPACK's DLARTG uses scaling for numerical stability but a different sign convention: for $a\ne 0$ it chooses $r=\mathrm{sign}(a)\sqrt{a^2+b^2}$ and $c\ge 0$; for $b=0$ it returns $c=1,s=0,r=a$. Its block still has the lower-left minus sign. Do not combine its returned coefficients with an assumed nonnegative $r$.^[ref]

For a prime-power alphabet size $q$, a linear $[n,k,\ge d]$ code exists if $V_q(n-1,d-2)<q^{n-k}$. Choose parity-check columns successively, avoiding combinations of at most $d-2$ earlier columns. Fewer than $q^{n-k}$ vectors are forbidden, so such a choice is possible. No set of at most $d-1$ columns is dependent, and the resulting kernel has dimension at least $k$.

// Target: probability-and-statistics.bigb

For $0\le \delta \le 1-1/q$, take logarithms of the Gilbert–Varshamov bound and apply the Hamming ball volume exponent. The Varshamov bound for linear codes gives the same achievable lower bound within linear codes for prime-power $q$.

// Target: algebra.bigb

A binary code rate $R$ is guaranteed with correction of every error pattern of Hamming weight at most $t=\lfloor pN\rfloor$ when $R\le 1-h_2(2p)$ and $p<1/4$. Choose minimum Hamming distance $2t+1$. The Gilbert–Varshamov bound and entropy bound for a Hamming ball give at least $2^{N(1-h_2(2p))}$ codewords. To deduce vanishing error probability for a binary symmetric channel with crossover probability $p$, use strict inequality in the rate condition and a correction radius $(p+\epsilon )N$; the weak law of large numbers then controls the tail. Correcting exactly the mean number of errors does not by itself ensure reliable transmission.

// Target: algebra.bigb

A positive diagonal matrix $D=\mathrm{diag}(d_1,\dots ,d_n)$, with $d_i>0$, is an invertible matrix and changes the coordinate scales through the similarity transformation $B=D^{-1}AD$. Its diagonal entries are still $a_{ii}$, while $b_{ij}=a_{ij}{d}_j/d_i$. Thus the Gershgorin discs have the same centres but different radii. Matrix similarity preserves the characteristic polynomial and eigenvalues, so the original and scaled disk unions are both valid bounds and may be intersected. Suitable scaling can compensate for badly unbalanced off-diagonal entries; it need not shrink every radius.^[ref]

For example,The original Gershgorin discs are $|z-2|\le 100$ and $|z-3|\le 0.01$. Their union is simply the huge first disk; the small second disk contains no eigenvalue. After scaling, both radii are $1$, giving $|z-2|\le 1$ or $|z-3|\le 1$. The two Gershgorin discs overlap, so their individual eigenvalue counts are not certified. The scaled matrix is a real symmetric matrix, however, so matrix similarity now proves that the original matrix has real eigenvalues in $[1,4]$ and is an invertible matrix. Directly, both matrices have characteristic polynomial ${\lambda }^2-5\lambda +5$, henceThe scaling changes the available bounds, while these eigenvalues stay fixed.

For a real symmetric matrix, or more generally a Hermitian matrix, the finite-dimensional spectral theorem makes all eigenvalues real. Intersecting its Gershgorin discs with the real axis givesThe union preserves gaps that the single outer interval loses. If every $a_{ii}-r_i>0$, all eigenvalues are positive, so the matrix is a positive-definite matrix. This is the positive-diagonal strictly diagonally dominant matrix criterion. For the three-by-three example, $1.7\le \lambda \le 9.4$ gives a positive-definite matrix certificate and bounds its spectral condition number of a positive-definite matrix by $9.4/1.7\approx 5.529412$; the computed value is about $4.544364$.

Strictness matters. The symmetric matrix $\left(\begin{array}{cc}1& -1\\ -1& 1\end{array}\right)$ has positive diagonal and is a diagonally dominant matrix, but its eigenvalues are $0$ and $2$: it is singular. For a general complex matrix, the condition $\mathrm{Re}{a}_{ii}>r_i$ only puts the eigenvalues in the open right half-plane. The Hermitian matrix assumption is what converts that statement into positive definiteness.

Let $C$ be a connected component of the union of the row Gershgorin discs of an $n\times n$ matrix. If $C$ comprises $k$ of the indexed Gershgorin discs, then it contains exactly $k$ eigenvalues, counted with algebraic multiplicity. Coincident disks still count separately, once for each row. In particular, a Gershgorin disc disjoint from all the others contains one simple eigenvalue. More generally, any union of $k$ indexed Gershgorin discs disjoint from the remaining $n-k$ contains exactly $k$ eigenvalues, whether or not that union is connected.^[ref]

The separation is essential: overlapping Gershgorin discs do not individually promise one eigenvalue each. Discs that touch at a boundary point belong to the same connected component. For example, the matrix $\left(\begin{array}{ccc}2& 1& 0\\ 0& 2& 0\\ 0& 0& 8\end{array}\right)$ has Gershgorin discs of radii $1,0,0$ with centres $2,2,8$. The component around $2$ comprises two indexed disks and contains the eigenvalue $2$ with algebraic multiplicity two; the isolated point at $8$ contains the remaining eigenvalue.

For intuition, continuously turn on the off-diagonal entries using the diagonal matrix $E=\mathrm{diag}(a_{11},\dots ,a_{nn})$ and $A(t)=E+t(A-E)$, $0\le t\le 1$. At $t=0$ the eigenvalues are the disk centres; at every later time the Gershgorin discs stay inside their final union. The continuity of the eigenvalues as a collection means that no eigenvalue can cross a gap between final components. The count gives a useful certificate that an eigenvalue cluster is complete.^[ref]

For an initially resting shallow water layer whose height rises linearly from $-h$ to $h$ across $(-L,L)$, let $R_d=\sqrt{gH}/|f|$ and $\alpha =L/R_d$. Conserved linear shallow-water potential vorticity gives $R_d^2{\eta }_s^{\prime \prime }-{\eta }_s=-{\eta }_0$. Boundedness, oddness, and continuity of height and slope give ${\eta }_s/h=x/L-e^{-\alpha }\sinh (x/R_d)/\alpha$ inside the ramp, and ${\eta }_s/h=\mathrm{sgn}(x)[1-(\sinh \alpha /\alpha )e^{-|x|/R_d}]$ outside. The balanced velocity is $u_s=0$, $v_s=g{\eta }_s^{\prime }/f$. Narrow ramps spread over the Rossby deformation radius; wide ramps retain their initial height except for narrow matching layers.