A purely imaginary root of $\lambda =-e^{-\lambda \tau }$ has frequency one and first appears at $\tau =\pi /2$. Roots cross with positive real derivative $1/(1+{\tau }_c^2)$; all roots lie in the left half-plane for smaller positive delay, since a hypothetical root $a+ib$ with $a\ge 0$ would require both $|b|\tau \ge \pi /2$ and $|b|\le 1$. A small-amplitude expansion $u=1+a\cos (\omega t)+a^2[\cos (2\omega t)/5+\sin (2\omega t)/10]+\cdots$ yields $\omega =1-3a^2/20+\cdots$ and $\tau =\pi /2+(3\pi -2)a^2/40+\cdots$, verifying a nondegenerate Hopf bifurcation. The limiting period is $2\pi =4{\tau }_c$.

For a twice continuously differentiable deformation map, each row of its cofactor matrix has zero divergence in reference coordinates. In three dimensions write $(\mathrm{cof}Dy{)}_{iA}=\frac{1}{2}{\epsilon }_{ijk}{\epsilon }_{ABC}{y}_{j,B}{y}_{k,C}$. Applying ${\partial }_A$ gives terms in which a symmetric pair of second partial derivatives contracts with an antisymmetric Levi-Civita symbol, so each term vanishes. This identity is the cancellation that makes the Piola-Kirchhoff stress tensor transform force balance correctly between configurations.

Two constant deformation gradients are rank-one connected when their difference is an outer product $a\otimes N$. For a nonzero jump its matrix rank is one; equality $A=B$ is the trivial compatible case. The vector $N$ specifies a reference interface normal, and $a$ is the jump amplitude. The Hadamard jump condition identifies this algebraic condition with continuity of an affine deformation across a planar interface. The normal can be normalized by rescaling $a$ without changing the product.

In the usual first-order Hilbert system, logical implication discharge requires the appropriate generalization restriction: a variable generalized in the derivation must not be free in the assumption being discharged. Equivalently use a proof system whose generalization rule already forbids variables free in open assumptions. One can first discharge a temporary assumption and then generalize if the quantified variable is not free in the remaining assumptions. Ignoring this side condition would allow deriving universal conclusions from a single arbitrary instance as an open hypothesis.

Conditional modus ponens derives $A\to C$ from $A\to B$ and $A\to (B\to C)$. It is the step which translates a modus ponens rule inference when discharging a common assumption. In the case-elimination Hilbert calculus, the third scheme first gives $A\to (\neg B\to C)$; the second scheme combines the two conditionals to give $A\to C$.

In ZF, an injection $h:\omega \to X$ yields an injective nonsurjective self-map by shifting $h(n)$ to $h(n+1)$ and fixing the complement. Conversely, if an injection $g:X\to X$ omits $a$, the natural-number recursion theorem defines $h(n)=g^n(a)$. Any repetition would, after cancellation, put $a$ in the image of $g$. Thus $h$ is injective. The argument uses no axiom of choice, even when some infinite sets are Dedekind-finite sets.

A database query has the same result when its ambient domain is extended while all input database relations are unchanged.

A relational schema is in Boyce-Codd normal form if the determinant of every non-trivial functional dependency is a superkey.

A relational schema is in fourth normal form if the determinant of every non-trivial multivalued dependency is a superkey. This implies Boyce-Codd normal form because every functional dependency implies a multivalued dependency.

A decomposition of a relational schema whose component relational projections always reconstruct the original database relation by natural join, without extra database tuples.

A decomposition in which the database integrity constraints can be enforced using constraints on individual component database relations, without computing their relational join.

Let $F(\zeta ,t)$ map the unit disk conformally to a planar Darcy flow region, with $F(0,t)=0$ at a point source of flux $2\pi Q$. With zero velocity potential on the boundary, the logarithmic source potential becomes $\varphi =Q\log |\zeta |$. Differentiating the inverse map at fixed physical position gives ${\varphi }_t=-Q\mathrm{Re}(F_t/(\zeta F_{\zeta }))$, while $|\nabla \varphi {|}^2=Q^2/|\zeta F_{\zeta }{|}^2$. The boundary law ${\varphi }_t+|\nabla \varphi {|}^2=0$ therefore gives the displayed equation, for nonzero $Q$.

The vector $\zeta F_{\zeta }d\theta$ is outward normal times arclength on $\zeta =e^{i\theta }$. Consequently the equation gives the area growth law $d|D|/dt=2\pi Q$. For a circular map $F=R(t)\zeta$, it reduces to $R\dot{R}=Q$, and hence $R(t{)}^2=R(0{)}^2+2Qt$.

On a $d$-dimensional vector space, a cyclic vector $v$ gives a basis $v,Tv,\dots ,T^{d-1}v$. Write $T^{d}v=-{\sum }_{j<d}{c}_j{T}^{j}v$ and set $f(t)=t^d+{\sum }_{j<d}{c}_j{t}^j$. Then $f(T)v=0$. Since polynomials in $T$ commute with $T$, $f(T)$ kills every basis vector, so $f(T)=0$. No nonzero polynomial of degree less than $d$ can kill $v$, by linear independence. Hence the minimal polynomial is $f$. The matrix in this basis is $C(f)$, and the characteristic polynomial of a companion matrix is $f$ as well.

The CRC generator polynomial is the agreed divisor $G(x)$ used in a cyclic redundancy check. In the basic binary convention it is a monic polynomial of polynomial degree $r$ with $G(0)=1$. A nonzero error polynomial $E$ goes undetected exactly when $G\mid E$.

This algebra gives useful detection guarantees. A single changed bit has $E=x^j$ and cannot be divisible by $G$ when $r\ge 1$. More generally, suppose the first and last changed bits span at most $r$ consecutive positions. Write $E=x^{j}B$ with $B(0)=1$ and $\mathrm{deg}B<r$. Since $G$ and $x$ are coprime polynomials, $G\mid E$ would imply $G\mid B$, contradicting the polynomial degree bound. Every such error pattern is therefore detected.

If $x+1$ divides $G$, every error changing an odd number of bits is also detected: its error polynomial satisfies $E(1)=1$, whereas every multiple of $G$ has value zero at $1$.

If an algebra acts on a vector space only by scalar multiples of the identity, every vector generates at most a one-dimensional submodule. A higher-dimensional such module is therefore not cyclic. For example upper triangular $n$-by-$n$ matrices act by the first diagonal entry on their first-row ideal. For $n>1$ this ideal is a noncyclic submodule of the cyclic regular module. Thus a submodule of a cyclic module need not be cyclic, even over a finite-dimensional algebra.

Along the antipodal two-cycle of $F(x,y)=(y,a(x-x^3/3)-y)$, the Jacobian matrix is $B=\left(\begin{array}{cc}0& 1\\ -2a& -1\end{array}\right)$ at both points. Its two-step periodic-orbit multipliers are squares of the roots of ${\lambda }^2+\lambda +2a=0$. The Jury criterion makes the cycle attracting precisely in the displayed interval, and repelling for $a>1/2$. This stable family is distinct from the always unstable nonzero-sum cycles.

The nonzero-sum two-cycles of the cubic swap map exist for $a>3$. Their two-step characteristic polynomial has trace $T$ and determinant $D$ as displayed. Since $1-T+D=-2(a-3)(a+1)<0$, one periodic-orbit multiplier exceeds one, proving instability throughout. The other crosses minus one at $a=(1+\sqrt{29})/2$; the cycles are saddles before that threshold and repelling afterwards. At $a=3$ they meet the fixed branches in an unstable subcritical-flip branch.

The alphabets assigned to participants determine their shared CSP events. An event in both assigned alphabets requires both participants; one assigned only to a single participant advances that participant. Replicated composition synchronizes all participants whose assigned alphabets contain the event.

A state with no outgoing CSP internal action. A CSP stable failure records a refusal only after such a state has been reached.

The first visible CSP event resolves the choice; the environment can select an event offered by only one branch. CSP internal actions can advance a branch without selecting it.