The same underlying module as an associative algebra, with multiplication $a{\cdot }_{\mathrm{op}}b=ba$. Right multiplications identify $H^{\mathrm{op}}$ with ${\mathrm{End}}_H(H)$ for the left regular module.

A finite-dimensional unital associative algebra over a field admitting a linear functional $\epsilon$ whose pairing $(a,b)\mapsto \epsilon (ab)$ is nondegenerate. The dual numbers with functional extracting the nilpotent coefficient are an example; semisimplicity is unnecessary.

The free associative algebra has a basis of finite words in its generators, including the empty word as identity. Multiplication concatenates words without commutativity relations. It is the path algebra of the quiver with r loops. With one generator it is a polynomial algebra.

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.

For the arithmetic stock model with constant volatility, let $\sigma >0$, $\tau =T-t>0$, $a=K{e}^{-r\tau }$, andThus $\nu =\sigma \sqrt{(1-e^{-2r\tau })/(2r)}$ for $r\ne 0$, and $\nu =\sigma \sqrt{\tau }$ for $r=0$. With $\Phi ,\varphi$ the standard normal distribution function and standard normal density, the European call option price isIndeed, the discounted terminal price is conditionally Gaussian with mean $s$ and variance ${\nu }^2$. Integrating its positive part above the discounted strike price gives this expression. At maturity it reduces continuously to $(s-K{)}^{+}$.

Differentiation gives option delta $C_s=\Phi (d)$. Holding $C_s$ stock units and $(C-s{C}_s)/B$ bank-account units gives a self-financing portfolio replicating the financial payoff, under the usual stochastic-integrability conditions. Its nonnegative wealth is admissible, and the discounted-wealth supermartingale bound shows that this is the least replication capital among nonnegative-wealth strategies.

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 $j$.