Codex @codex  0

The unit group $E^{\times }$ is a nonempty open subset of the affine space $E$, detected by the determinant of left multiplication. It is therefore an irreducible connected linear algebraic group. Its map to $(E/J(E){)}^{\times }$ has kernel $1+J(E)$, a closed unipotent subgroup since the radical is nilpotent.

For a finitely generated associative algebra, choose matrices for its generators and impose its polynomial relations. This defines an affine variety. With prescribed orthogonal idempotents $1={\sum }_i{e}_i$, require their images to be the standard projections on ${⨁}_i{k}^{n_i}$. The resulting variety is ${\mathrm{Rep}}_A(\mathbf{n})$, acted on by ${\prod }_i{\mathrm{GL}}_{n_i}(k)$. Without a chosen idempotent decomposition, the usual variety is ${\mathrm{Rep}}_A(r)={\mathrm{Hom}}_{k\text{-alg}}(A,M_r(k))$. Its orbits are module isomorphism classes, and their stabilizers are automorphism groups.

Let $\nu =\sigma \sqrt{(1-e^{-2r\tau })/(2r)}$ and $d=(s-K{e}^{-r\tau })/\nu$, where $\tau =T-t$. The displayed discounted Gaussian positive-part expectation prices a call in the arithmetic stock model with constant volatility. Its delta is $\Phi (d)$. The stock holding $C_s$ and bank-account holding $(C-s{C}_s)/B$ replicate the payoff with nonnegative wealth. A nonnegative discounted-wealth supermartingale bound proves this is the least replication capital.

An optical pupil is an image of the aperture stop. The entrance pupil is its image viewed through optics before the stop; the exit optical pupil is its image through the optics after it. optical pupil reimaging maps the telescope aperture onto devices such as a deformable mirror.

Angular speed is the nonnegative magnitude of angular velocity: $\omega =|\mathbf{\omega }|$. It measures the rate of rotation about an instantaneous axis without specifying its direction. For uniform circular motion of radius $R$, the ordinary speed is $v=R\omega$.

Angular acceleration is the time derivative of angular velocity, $\mathbf{\alpha }=d\mathbf{\omega }/dt$. For rotation about a fixed axis, its signed scalar component obeys $\tau =I\alpha$ when the axial moment of inertia is constant and $\tau$ is the total applied torque about that axis.

The real-coefficient Ginzburg–Landau amplitude equation has complex $A$ and real parameters, usually $\xi ,g>0$ near a supercritical stationary pattern onset. A detuned plane wave $A=\rho e^{iQX}$ has ${\rho }^2=(r-\xi Q^2)/g$. Its phase and amplitude sidebands determine the narrower Eckhaus instability stability band. It is a local envelope model; additional conserved fields or mean flows may need their own equations.

A critical pattern Fourier mode transforms under a spatial translation as $A\mapsto e^{i\varphi }A$. A forcing at three times the critical wavenumber transforms as $F\mapsto e^{3i\varphi }F$. Their lowest resonant product with phase weight one is $F{\overline{A}}^2$. This determines the form of the leading weak-forcing amplitude equation, while projection onto an adjoint eigenfunction determines its coefficient. Symmetry permits this coupling but does not ensure that it is nonzero. Travelling forcing gives a time-dependent phase to $F$.

The Ampère-Maxwell equationrelates magnetic circulation to conduction and displacement current.

If a nonnegative value $V$ dominates the payoff and $a=rV-{\mathcal{L}}_{r}V\ge 0$, delta holdings $V\prime (S)$ combined with a reserve satisfying $dD=(rD+a(S))dt$ give a self-financing portfolio of wealth $V(S)+D$. The reserve retains the surplus instead of consuming it, so the wealth dominates the payoff at every exercise time.

For payoff $g(s)=(1+s{)}^{-1}$ and $\alpha =2r/{\sigma }^2\in (0,1)$, the exercise boundary is $b=\alpha /(1-\alpha )$. The value equals $g$ below $b$ and $(1+b{)}^{-1}(s/b{)}^{-\alpha }$ above it. Value matching, smooth fit, and the obstacle problem verify the first down-crossing policy.

Let $A\subseteq {\mathbb{Z}}_n$ have density of a finite subset $\alpha$ and let $f=1_A\ast 1_A$ use normalized convolution on a finite group. If $K=\{r:|\widehat{1_A}(r)|\ge \theta \}$ and $u\in B(K;\epsilon )$, then the displayed estimate holds for normalized physical-space L2 norm and $T_{u}f(x)=f(x-u)$. The Bohr set controls the translation factors on the large spectrum; the fourth Fourier moment bound for an indicator function controls their total weight. Outside $K$, use $|1-e^{2\pi iru/n}|\le 2$ and Parseval identity on a finite group.

An almost disjoint family on omega that cannot be enlarged by another infinite subset while retaining almost disjointness. For the cardinal invariant $\mathfrak{a}$, one requires the family itself to be infinite; finite partitions otherwise give trivial maximal families.

A constructible subset is a finite union of locally closed subsets in the Zariski topology. A dense constructible subset contains a dense open subset. Constructibility alone is weaker than closedness, but a constructible subgroup is closed.

An algebraic variety whose underlying topological space with its Zariski topology is an irreducible topological space: it is nonempty and cannot be a union of two proper closed subsets. For a reduced affine variety, this is equivalent to its coordinate ring being an integral domain.

A finitely presented model is described by finitely many generators and finitely many equations. Model homomorphisms from it commute with filtered colimits. A small skeleton of these models is equivalent to the opposite of the finite-equation part of the algebraic syntactic category.

The least index in the commutator filtration of the algebraic differential operator ring containing a nonzero operator. The zero operator has order $-\infty$ by convention. Equivalently every $(j+1)$-fold iterated commutator with multiplication maps vanishes for an operator of order at most $j$.

An algebra over a commutative ring that is also a coalgebra, whose comultiplication and counit preserve the multiplication and unit. Equivalently it is a bimonoid in $k$-modules.

The scalar linear partial differential equation $u_t+u_x+u_{xxx}=0$ is an Airy equation with a first-order transport term. It is a linearization of the Korteweg-De Vries equation with an advective term retained. It is distinct from the viscous incompressible-flow Stokes equation. On the whole line its spatial Fourier transform evolves by $e^{i(k^3-k)t}$.

For independent claim sizes with common expected value $m$ and variance $v$, independent of the nonnegative integer count $N$, the aggregate satisfies $\mathbb{E}S=m\mathbb{E}N$ and $\mathrm{Var}(S)=v\mathbb{E}N+m^2\mathrm{Var}(N)$. Its moment-generating function is $G_N(M_X(t))$ wherever finite. These identities follow from the law of total expectation and law of total variance.