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The unit repeats an element and multiplication selects the two outer diagonal entries. Both unit laws hold directly, and the associativity law selects the same two diagonal entries of an eight-entry array. Its algebras for a monad are precisely rectangular bands, including the empty algebra.

For a uniform disc of radius $a$ and mass $M$, annuli of radius $r$ and thickness $dr$ have mass $(M/\pi a^2)2\pi rdr$. Their axial moment of inertia sums to $I={\int }_0^a{r}^{2}dm=M{a}^2/2$.

A uniform planar annulus of outer radius $a$, inner radius $b<a$ and mass $M$ has surface density $M/[\pi (a^2-b^2)]$. Integrating $r^2$ over its mass gives the axial moment of inertia $I=M(a^2+b^2)/2$. Subtracting the inner disc from a complete disc gives the same result, provided their surface densities match.

// Target: electromagnetism.bigb

If a fixed-axis wheel of radius $R$ drives a fixed-axis roller of radius $S$ without slip, their angular-speed magnitudes satisfy $\sigma =(R/S)\omega$. Their kinetic energy is $\frac{1}{2}[I+J(R/S{)}^2]{\omega }^2$, so the displayed reflected moment of inertia governs the wheel coordinate. A roller resisting torque of magnitude $\gamma {\sigma }^2$ consumes power $\gamma {\sigma }^3$ and therefore reflects to wheel torque $\gamma (R/S{)}^3{\omega }^2$. The squared ratio belongs to inertia; the cubed ratio belongs to this quadratic resisting torque.

For a uniform solid cylinder of radius $a$, length $L$ and mass density $\rho$, a coaxial shell carries mass $dm=2\pi \rho Lrdr$. The axial moment of inertia is $I={\int }_0^a{r}^{2}dm=\pi \rho L{a}^4/2=m{a}^2/2$. The axial moment of inertia is independent of the length once mass and radius are fixed. A thin cylindrical shell instead has $I=m{a}^2$, demonstrating why the uniform solid assumption matters.

The centroid of a uniform solid hemisphere of radius $a$ lies $3a/8$ from its flat face. About a diameter of the underlying sphere parallel to that face, its moment of inertia is $2m{a}^2/5$. The parallel axis theorem therefore gives $I_G=2m{a}^2/5-m(3a/8{)}^2=83m{a}^2/320$ about the parallel centroidal axis. A thin hemispherical shell has a different centroid and inertia.

For a toroidally compactified heterotic string, the torus metric, antisymmetric tensor and gauge Wilson lines determine a positive $d$-plane in a charge space of signature $(d,d+16)$, modulo lattice identifications. The continuous dimension is $d(d+16)$; the coupling is an additional modulus. The three-torus case has fifty-seven shape and Wilson-line parameters.

For stable compact Riemann surfaces with $n$ distinct labelled marked points, where $2g-2+n>0$, the displayed complex dimension counts independent conformal shapes and positions after conformal automorphisms are removed. On the sphere it is $n-3$, since three points fix a Möbius transformation. On a torus it is $1+n-1=n$: one shape modulus and $n$ positions, less one translation. For genus $g\ge 2$, the unmarked shape dimension is $3g-3$ and continuous conformal automorphisms vanish. The shape dimension also follows from the Riemann-Roch theorem applied to holomorphic quadratic differentials.

The socle of a module is the sum of all its simple submodules, equivalently its largest semisimple module.

A left $R$-module consists of an abelian group $(M,+)$ and scalar multiplication $R\times M\to M$, written $(r,m)\mapsto rm$, where $R$ is a ring with identity element $1$. For every $r,s\in R$ and $m,n\in M$, requireThe module elements can be added to one another and multiplied by scalars; a product of two module elements is not part of the definition. These conventions agree with ^[ref]. For a noncommutative ring, the side matters: a right module instead has $(mr)s=m(rs)$. Over a commutative ring, the two conventions agree.

Over a field $k$, these are exactly the vector space axioms. A nonzero $a\in k$ has a multiplicative inverse, so $am=0$ implies $m=a^{-1}(am)=0$. Over the integers, $2$ has no multiplicative inverse: $2x=1$ has no solution in $\mathbb{Z}$. In the module $\mathbb{Z}/6\mathbb{Z}$, division can also lose uniqueness: $2\overline{1}=2\overline{4}=\overline{2}$. A ring element which is a unit still acts invertibly on every module; the difficulty lies with scalars which are not units.

For a module $M$ over a commutative ring $A$, its character module is the additive-group dual into $D=\mathbb{Q}/\mathbb{Z}$, with action $(a\varphi )(m)=\varphi (am)$. The group $D$ is an injective cogenerator of abelian groups. The natural adjunction ${\mathrm{Hom}}_A(N,M^{\ast })\cong {\mathrm{Hom}}_{\mathbb{Z}}(N{\otimes }_{A}M,D)$ sends $h$ to the balanced pairing $(n,m)\mapsto h(n)(m)$.

A topological abelian group $A$ with a continuous action of a topological group $G$ by group automorphisms. Continuity means joint continuity of $G\times A\to A$. For a discrete coefficient group this is equivalent to every element of $A$ having an open stabilizer.

For a finite-dimensional algebra, the Auslander–Reiten quiver has one vertex for each isomorphism class of finite-dimensional indecomposable modules, and its arrows record irreducible module homomorphisms. A homomorphism is irreducible when it is neither a split injection nor a split surjection and every factorization forces the first map to split injectively or the second to split surjectively.

A finite-dimensional algebra has finite representation type when there are only finitely many isomorphism classes of finite-dimensional indecomposable modules. This counts module types, not the dimensions or the number of direct sums formed from them. For a block of a group algebra, finite representation type is equivalent to a cyclic defect group of a block.

For $P$ cyclic of order $q=p^n$ in characteristic $p$, its group algebra is $k[t]/(t^q)$. The nilpotent Jordan normal form classifies its indecomposable finite-dimensional modules as $V_i$, $1\le i\le q$. A module homomorphism $V_i\to V_j$ is determined by an element annihilated by $t^i$, so its dimension is $\min (i,j)$.

Here $\mathcal{O}$ is a complete discrete valuation ring with fraction field $K$ of characteristic zero and residue field $k$ of characteristic $p$. The system is splitting for the finite group and its subgroups when their simple representations over $K$ and $k$ are absolutely irreducible. It permits ordinary and modular representation theory to be compared by reduction of lattices modulo the maximal ideal.

A p-local module here is a finite direct sum of induced representations from normalizers $N_G(Q)$ of nontrivial p-subgroups $Q$. This usage includes induction from $G$ itself when $G$ normalizes such a subgroup. Green correspondence and induction on vertex order express every finite module, or lattice over a complete valuation ring, as a difference of p-local modules after adjoining projective modules.

The modular lambda function is invariant under the projective level-two principal congruence subgroup. That group acts freely, and its quotient is the level-two modular curve. Adding its three modular cusps gives a sphere, and $\lambda$ has cusp values $0,1,\infty$, with simple orders in exponential cusp coordinates. Therefore it identifies the quotient with the sphere minus those three points. Since the complex upper half-plane is simply connected, this is a universal covering map, with the projective level-two group as its deck transformation group.

Choose $4a_j+6b_j=k-12j$ with nonnegative exponents, for every $j$ with $k-12j\ge 0$ and $k-12j\ne 2$. The Eisenstein series and the modular discriminant give modular forms with integer coefficients and distinct first powers of $q$. The valence formula for the modular group shows that these form a basis. Successively subtracting their leading terms proves that integral coefficients up to the greatest such $j$ force all coefficients to be integral. This is more precise than merely spanning by arbitrary monomials in $E_4,E_6$.

A weight-$2k$ modular form descends as an invariant tensor differential. At an elliptic point of order $e$, a local invariant coordinate is $w=u^e$, so its descended coefficient has a pole of order at most $\lfloor k(e-1)/e\rfloor$. At infinity, $d\tau =dq/(2\pi iq)$ gives a pole of order at most $k$. Thus $D=k[\infty ]+\lfloor k/2\rfloor [i]+\lfloor 2k/3\rfloor [\rho ]$ bounds the poles. Conversely these bounds give holomorphic pullbacks. The Riemann-Roch theorem on the compactified modular curve proves finite-dimensionality; since $X(1)$ is the Riemann sphere, the dimension is $\max (0,1-k+\lfloor k/2\rfloor +\lfloor 2k/3\rfloor )$.