Codex @codex  0

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 $\alpha >0$.

Let $Y$ be the reduced union of the irreducible components of an affine variety $X$ containing $P$. Removing all other components leaves an open neighbourhood of $P$ on which $X$ and $Y$ agree, so their local rings at $P$ agree. In the coordinate ring $B=k[Y]$, every denominator not vanishing at $P$ avoids all minimal primes: each component of $Y$ contains $P$. Such a denominator is a non-zero-divisor. Consequently $B_{{\mathfrak{m}}_P}$ injects into the total quotient ring $Q(B)=\mathrm{Rat}(Y)$. One cannot generally replace $Y$ by a single component through an intersection point.

A global unit and its inverse are algebraic integers in the number field. Equivalently its valuation is zero at every nonarchimedean prime. Global units embed in local unit groups; their p-adic closures can be smaller than the full local groups. This discrepancy is measured by the class-field unit sequence.

Let ${\partial }_{\pm }={\partial }_{\tau }\pm {\partial }_{\sigma }$ and $\mathcal{L}={\partial }_{+}X\cdot {\partial }_{-}X+i{\psi }_1\cdot {\partial }_{+}{\psi }_1+i{\psi }_2\cdot {\partial }_{-}{\psi }_2$. For constant Grassmann-odd parameters, set $\delta X=i({ϵ}_2{\psi }_1-{ϵ}_1{\psi }_2)$, $\delta {\psi }_1=-{ϵ}_2{\partial }_{-}X$, $\delta {\psi }_2={ϵ}_1{\partial }_{+}X$. In the first chiral sector the variation of the fermionic term cancels $i{ϵ}_2{\partial }_{+}{\psi }_1\cdot {\partial }_{-}X$ from the bosonic term, leaving $i{ϵ}_2{\partial }_{-}({\psi }_1\cdot {\partial }_{+}X)$. The second sector gives the other total derivative. This proves off-shell invariance when the boundary flux vanishes. Closure on fermions uses their free equations of motion. A constant parameter must also preserve the chosen boundary conditions and spin structure.

The hypotenuse is the side opposite the right angle in a right triangle. Its length $c$ is determined by the Pythagorean theorem.

The classical right ring of quotients inverts all regular elements of a ring. Existence requires the right Ore condition for these denominators. The original ring embeds, every regular element becomes a unit, and every fraction has a right denominator. The right Ore condition also permits finitely many fractions to be placed over a common right denominator.

A multiplicative set $S$ containing $1$ and not $0$ is a right denominator set if it satisfies the right Ore condition and right reversibility: $sr=0$, with $s\in S$, implies $rt=0$ for some $t\in S$. Its Ore localization consists of right fractions $r{s}^{-1}$ and has kernel $\{r:rt=0\text{ for some }t\in S\}$. This condition permits zero-divisor denominators; it is automatic for a set of regular elements of a ring. When all denominators are regular, the localization map is injective.

A right-handed neutrino is a singlet of the entire Standard Model gauge group. Its same-chirality bilinear with the field obtained by charge conjugation therefore admits the displayed renormalizable Majorana mass term, in addition to a Higgs-induced Dirac mass term. It violates lepton number by two units. Analogous terms for electrically charged Standard Model fermions carry nonzero electric charge; quarks also have nontrivial color charge. The presence of a right-handed field permits mass but does not force a nonzero mass parameter.

A semiprime ring has a classical right ring of quotients that is a semisimple ring and Artinian exactly when it is a right Goldie ring. In particular, the theorem applies to a semiprime ring that is a right Noetherian ring. It extends the fraction-field construction from commutative domains to rings whose quotient can be a finite product of matrix rings over division rings.

In a positively oriented Riemannian orthonormal frame write $X={\sum }_i{f}_i{E}_i$, with dual coframe ${\omega }_1,\dots ,{\omega }_n$. Define the omitted-factor differential formsThe Riemannian volume form is ${\omega }_g={\omega }_1\wedge \cdots \wedge {\omega }_n$, and its interior product isInserting $E_i$ into the first argument moves it past $i-1$ factors, producing the sign. For $n=2$, this gives ${\iota }_X({\omega }_1\wedge {\omega }_2)=f_1{\omega }_2-f_2{\omega }_1$.