A periodic orbit entering the local flow $\dot{x}=x^2+k^2$ at a fixed negative $\nu$ and leaving at positive $h$ has passage time $T_{\mathrm{loc}}=[\arctan (h/k)-\arctan (\nu /k)]/k\sim \pi /k$. The same coefficient holds if $-\nu /k\to \infty$. It is not uniform in the joint limit $\nu ,k\to 0$: if $\nu /k\to -C$, the leading coefficient is $\pi /2+\arctan C$. For entry at zero it is $\pi /2$, while entry at a fixed positive $\nu$ gives bounded passage time. These distinctions matter near a saddle-node separatrix-loop bifurcation.

At a planar saddle equilibrium with $m_{-}<0<m_{+}$, positive saddle quantity $m_{-}+m_{+}>0$ is equivalent to saddle index $\delta <1$. The local passage followed by the regular return has leading transverse map $s\mapsto C{s}^{\delta }$, $C>0$. Its derivative diverges as $s\downarrow 0$, so a nearby periodic orbit created by the saddle-loop splitting is repelling. This local mechanism agrees with the growth of the unstable periodic orbit born at a subcritical Hopf bifurcation in the quadratic Bogdanov–Takens unfolding.

If the first logarithmic perturbation ${\chi }_1$ is a centered linear functional of a Gaussian random field, the complex Gaussian exponential moment gives the displayed mean of the first Rytov approximation ${\psi }_R={\psi }_i{e}^{\mu {\chi }_1}$. This is the mean of the truncated logarithmic model. A nonzero mean of the omitted second logarithmic perturbation can change the true field mean at order ${\mu }^2$.

The generators $a,b,c,d$ obey $ab=qba$, $ac=qca$, $bd=qdb$, $cd=qdc$, $bc=cb$, $ad-da=(q-q^{-1})bc$ and $ad-qbc=1$. Matrix multiplication gives the coproduct, and the antipode matrix is $\left(\begin{array}{cc}d& -q^{-1}b\\ -qc& a\end{array}\right)$. For $q\ne 0,\pm 1$, its quotient with $c=0$, $a=u$, $d=u^{-1}$ and $b=v$ is the quantum Laurent-plane algebra $uv=qvu$, proving noncommutativity. The same quotient distinguishes $\Delta b$ from its flipped coproduct.

For the two-by-two RTT construction with the standard Yang-Baxter R-matrix, the displayed determinant commutes with the matrix generators and satisfies $\Delta D_q=D_q\otimes D_q$ and $\epsilon (D_q)=1$. Imposing $D_q=1$ gives the coordinate Hopf algebra of quantum SL2; adjoining its inverse instead gives the quantum general-linear coordinate algebra.

Given an invertible braiding $C$ on $V\otimes V$, the universal bialgebra coacting on $V$ for which $C$ is a comodule map is generated by the matrix coefficients $t_{ij}$. Its comultiplication is $\Delta t_{ij}={\sum }_k{t}_{ik}\otimes t_{kj}$. Colinearity gives quadratic relations. Writing $R=\tau C$ turns the relation into the familiar RTT construction with the appropriate ordered tensor factors. A Yang-Baxter R-matrix supplies a scalar coquasitriangular structure.

For a deterministic path $h$ in the Cameron-Martin space of Wiener measure, the law of $T_h\mathbf{B}$ is equivalent to the law of enhanced Brownian motion. The Cameron-Martin theorem gives equivalence for the underlying Brownian paths; the canonical enhancement of the translated path is exactly the rough-path translation, including its mixed area terms. Continuity of this translation transfers small-ball positivity at the identity to every control signature.

Let $P(z)={\prod }_{\ell }(z-r_{\ell })$ have distinct roots. Choose a disc around $r_j$ with radius $\delta$ smaller than every separation from $r_j$. On its boundary, $|P(z)|\ge \delta {\prod }_{\ell \ne j}(|r_j-r_{\ell }|-\delta )$. If a holomorphic perturbation $H$ satisfies the displayed inequality, Rouché's theorem shows that $P+H$ has exactly one root in that disc, counted with multiplicity. Discs confined to specified half-planes or quadrants give direct geometric root-location proofs without needing a global root-continuation argument.

For an indicator function of a subset of $[N]$ with subset density $\alpha$, a large Fourier coefficient of $1_A-\alpha 1_{[N]}$ gives a density increment on an arithmetic progression. The Dirichlet approximation theorem supplies a step $d\le \sqrt{N}$ on which the detected linear phase changes slowly. Split its residue classes into progressions of lengths comparable to ${\alpha }^2\sqrt{N}$. Replacing the phase by one constant on each cell preserves a fixed fraction of the Fourier correlation. Since the balanced function has total sum zero, positive discrepancies sum to half the total absolute discrepancy; one cell has the displayed increased density. One may take $c=1/(2304\pi )$ when $N$ is sufficiently large in terms of $\alpha$.

The rotations in three dimensions preserving a dodecahedron form a group of order $60$. There are $24$ nonidentity rotations about the six axes through opposite face centres, $20$ about the ten axes through opposite vertices, and $15$ half-turns about the fifteen axes through opposite edge midpoints. Its nonidentity conjugacy classes have sizes $12,12,20,15$: the centralizer of a rotation of order five or three consists of rotations about its oriented axis and has order five or three, respectively, while all edge half-turns are conjugate. A normal subgroup is a union of whole conjugacy classes containing the identity. No proper nontrivial sum of these class sizes plus one divides $60$, proving that it is a simple group. The centralizers of the fifteen involutions are five distinct Klein four-groups, each containing three involutions. Conjugation permutes these five subgroups nontrivially. The kernel is a normal subgroup, so simplicity makes this nontrivial action injective. Its sign map to a group of order two must be trivial, since simplicity would otherwise force an impossible injection of sixty elements into two. Thus its image lies in the alternating group $A_5$, and equality of orders makes the image all of $A_5$.

The orientation-preserving rotational symmetry group of a regular icosahedron has order sixty. Its conjugacy classes have sizes $1,15,20,12,12$: identity; half-turns about opposite-edge axes; rotations through $\pm 2\pi /3$ about opposite-face axes; and rotations through $\pm 2\pi /5$ and $\pm 4\pi /5$ about opposite-vertex axes, respectively. Conjugation carries an axis to its rotated axis and preserves the rotation angle. The transitive group action on edges, faces and vertices gives the indicated classes, with reversal of the axis identifying the two signs. A normal subgroup must contain whole conjugacy classes; none of the proper nonidentity sums of these class sizes is a divisor of sixty. Thus this group is a simple group.

For dimensionless Hermite index zero, the equatorial shallow-water dispersion relation factorizes as $(\omega +k)({\omega }^2-k\omega -1)=0$. The root $\omega =-k$ is generally spurious: for $v=C{e}^{-y^2/2}$, the original equations require ${\delta }^{\prime }-y\delta =i(2k-1/k)v$, where $\delta =u-\eta$. Multiplication by $e^{-y^2/2}$ and integration over the real line shows that decay at both ends requires $C=0$ unless $k^2=1/2$. At that exception the root already belongs to the physical quadratic; $u=\eta =i\omega yv$ is regular and no extra wave branch is created. The assertion concerns nonzero $k$ and angular frequency.

In a mixed-characteristic discretely valued field, a primitive $p^a$th root of unity reduces to one. For every $j$ prime to $p$, the ratio $({\zeta }^j-1)/(\zeta -1)$ is a unit, so all factors of ${\Phi }_{p^a}(1)=p$ have the same valuation. Taking valuations gives the formula. Because the left side is a positive integer for a normalized valuation, the existence of such a root forces $p^{a-1}(p-1)\mid v(p)$, bounding the possible $p$-power torsion.

Multiplicity records how many times an object occurs when repetitions carry mathematical information.

A root of a polynomial lying on the complex unit circle. For a numerical amplification polynomial, a simple such root gives an undamped oscillatory mode. A repeated one supplies polynomial growth in the step index and fails the root condition for a multistep method.

Two real polynomials with only real roots of a polynomial interlace when their ordered roots alternate, counting multiplicity of a root. For equal polynomial degrees, one ordering is the displayed chain; the opposite ordering is also permitted. Consecutive degrees give the analogous chain with one extra root at each end for the higher-degree polynomial. Strict interlacing uses strict inequalities. Weak interlacing permits common roots and follows as a limit of strict interlacing.

In this ideal rocket equation model, rocket dry mass is the mass remaining after all usable rocket propellant has been expelled: structure, engines and payload. It is the final mass in the worked example; if rocket propellant remains at shutdown, the final mass is larger than the rocket dry mass.

Let $G$ be the event of membership in the first of two groups and $A$ the event of selection. Write $p=P(G)$, $a=P(G\mid A)$ and $q=P(A)>0$, with $0<p,a<1$. Bayes' theorem gives the conditional probabilities $r_1=P(A\mid G)=aq/p$ and $r_0=P(A\mid G^c)=(1-a)q/(1-p)$. Their risk ratio is the displayed expression: the unknown overall selection probability cancels. Equivalently, it is the ratio of the group-membership odds after selection to those before selection.

When both selection probabilities lie strictly between zero and one, the selection odds ratio is $[r_1/(1-r_1)]/[r_0/(1-r_0)]$ and generally depends on $q$. Thus compositions alone do not identify that odds ratio or provide sample sizes for a standard error. When $a<p$, the first group's observed selection rate is lower; this comparison alone does not establish a causal effect of group membership.

A decision rule is admissible if no other rule has risk function no greater at every parameter and strictly smaller at one parameter. Parameter estimation is one decision problem; its admissible rules are admissible estimators.

For positive wealth and increasing twice differentiable utility function, this dimensionless coefficient measures sensitivity to proportional wealth risk. If it is constant, integration gives $U^{\prime }(W)=c{W}^{-\alpha }$ with $c>0$, and hence $U(W)=c{W}^{1-\alpha }/(1-\alpha )+d$ for $\alpha \ne 1$ or $c\log W+d$ for $\alpha =1$. Positive, zero and negative coefficients correspond respectively to risk aversion, risk neutrality and risk seeking. The usual CRRA utility subclass takes $\alpha >0$.