For $N\ge 2$, suppose a skew-Hermitian Lie algebra becomes all of $\mathfrak{u}(N)$ after adding the scalar identity direction. Scalar matrices commute, so $[\mathfrak{g},\mathfrak{g}]=[\mathfrak{u}(N),\mathfrak{u}(N)]=\mathfrak{su}(N)$ is contained in $\mathfrak{g}$. If $\mathfrak{g}$ contains any nonzero-trace generator, subtracting its traceless part supplies $iI$, and $\mathfrak{g}$ was already all of $\mathfrak{u}(N)$. This permits using identity-augmented commutator calculations without assuming an additional physical identity control.

Let $J$ be a symmetric unitary matrix and every skew-Hermitian control generator obey $x^{T}J+Jx=0$. The condition is closed under real linear combinations and commutators, and evolution preserves $U^{T}JU=J$. In a unitary basis where $J$ becomes the identity, the generators are real antisymmetric, so the Dynamical Lie algebra is contained in $\mathfrak{so}(N)$. The state quantity $|{\psi }^{T}J\psi |$ is invariant under the reachable evolution and insensitive to global phase, obstructing unrestricted pure-state controllability. This is a bilinear-form symmetry; it does not assert that $J$ commutes with the Hamiltonians.

Let $H_0=\mathrm{diag}(-{\omega }_1,0,-{\omega }_2)$ and $H_1=d_1(E_{12}+E_{21})+d_2(E_{23}+E_{32})$, with nonzero $d_j$, distinct positive ${\omega }_j$. The squared adjoint action of $-i{H}_0$ has different eigenvalues $-{\omega }_1^2,-{\omega }_2^2$ on the two adjacent skew-Hermitian couplings. Linear combinations of $-i{H}_1$ and its double commutator with $-i{H}_0$ isolate both couplings. Further commutators give their real antisymmetric partners, two traceless diagonal generators, and both $1$–$3$ generators, hence the special unitary Lie algebra $\mathfrak{su}(3)$. The drift has nonzero trace; subtracting its traceless part supplies $iI$, giving the Lie algebra of the unitary group $\mathfrak{u}(3)$ and unitary operator controllability under the usual freely timed real-control assumptions.

For a nonsingular skew-symmetric unitary $J$, if each skew-Hermitian control generator obeys $X^{T}J+JX=0$, then so do their real linear combinations and commutators. Differentiating $U^{T}JU$ shows that it stays equal to $J$. The reachable group is therefore contained in a unitary conjugate of the compact symplectic group. This is preservation of an alternating bilinear form, not commutation with $J$.

For $N\ge 2$, let $\mathfrak{g}$ be a Lie subalgebra of $\mathfrak{u}(N)$ with codimension one. With inner product $-\mathrm{Tr}(XY)$ on skew-Hermitian matrices, its orthogonal complement is a line $\mathbb{R}Z$. Invariance of the inner product shows that $[X,Z]$ lies in that line for every $X\in \mathfrak{g}$; skew-adjointness of the adjoint action makes $[X,Z]$ orthogonal to $Z$, hence zero. A nonscalar skew-Hermitian $Z$ has a centralizer of dimension at most $(N-1{)}^2+1<N^2-1$, by its orthogonal eigenspace decomposition, and cannot centralize all of $\mathfrak{g}$. Thus $Z$ is scalar and $\mathfrak{g}=\mathfrak{su}(N)$. This gives density operator controllability and all determinant-one gates, but not the central phase direction needed for unitary operator controllability.

A drift Quantum Hamiltonian is strongly regular when its energy levels are nondegenerate and distinct unordered transitions have distinct absolute nonzero Bohr frequencies. Sometimes regularity is required only on the graph of allowed control transitions. This lets polynomials in the squared drift adjoint action separate different coupling quadratures and permits frequency-selective resonant quantum control. This notion is unrelated to a strongly regular graph.

Let $H_0$ be diagonal and $H_1={\sum }_{j=1}^{N-1}{d}_j(E_{j,j+1}+E_{j+1,j})$, with every $d_j\ne 0$ and the adjacent absolute energy gaps distinct and nonzero. The squared drift adjoint action has distinct eigenvalues on these coupling components, so Lagrange interpolation isolates every adjacent symmetric coupling. One more commutator gives its antisymmetric partner. Their mutual brackets produce adjacent traceless diagonals, while brackets along the connected chain give all longer-range matrix units. Thus the special unitary Lie algebra is generated. Any nonzero-trace generator also supplies the central direction, giving all of $\mathfrak{u}(N)$. This is a sufficient condition in the finite-dimensional freely timed real-control model; connectivity alone does not remove frequency degeneracies.

For a rocket equation with exhaust mass rate $\dot{m}$ and incoming stationary mass rate $\dot{D}$, mass balance is $\dot{M}=\dot{D}-\dot{m}$. A short-interval momentum conservation calculation gives $M\dot{v}=u\dot{m}-v\dot{D}$. Constant time rates $\dot{D}=\alpha$, $\dot{m}=\beta$ give $M\dot{v}+\alpha v=u\beta$, unlike a constant accretion rate per distance, whose incoming mass rate is proportional to speed. For unequal rates and initial rest, $v=(u\beta /\alpha )[1-(M/M_0{)}^{\alpha /(\beta -\alpha )}]$.

Let $1<p<\infty$ and $q=p/(p-1)$. For a nonzero bounded linear functional $\ell$ on an Lp space, choose $f_0$ with $\ell (f_0)=1$ and a closest point $h$ in its closed kernel. Put $z=f_0-h$. The variational condition is $\int |z{|}^{p-2}\overline{z}k=0$ for every kernel element $k$. Since $f-\ell (f)z$ lies in the kernel, $\ell (f)=\int fg$ with $g=|z{|}^{p-2}\overline{z}/\Vert z{\Vert }_p^p\in L^q$. Hölder's inequality gives the upper norm bound; testing against $|g{|}^{q-2}\overline{g}$ gives equality. Thus the representation is an isometric linear isomorphism when the pairing is bilinear.

Closing an identity strand using compatible evaluation morphisms and coevaluation morphisms gives the categorical dimension in a pivotal or ribbon setting. For the standard two-dimensional quantum enveloping algebra of sl2 module, the weighted contraction has diagonal coefficients $q^{-1},q$, giving quantum dimension $q+q^{-1}=[2{]}_q$. The chosen duality and framing conventions matter.

A dual variable is a multiplier associated with a primal constraint in a dual linear program. In a minimization linear program with equality constraints $Ax=b$ and nonnegative primal variables, the Lagrangian $c^{T}x+y^T(b-Ax)$ yields the dual objective $b^{T}y$ and inequalities $A^{T}y\le c$; the equality-constraint multipliers $y$ are unrestricted in sign. For a transportation problem, row and column dual variables give potentials $u_i,v_j$ satisfying $u_i+v_j\le c_{ij}$. Complementary slackness makes occupied routes attain equality.

A linear code is self-dual when it equals its dual code under the standard dot product. For a length-$2k$ code with a rank-$k$ generator matrix $G$, this is equivalent to $G{G}^T=0$: orthogonality gives inclusion in the dual code, and equal dimensions give equality. Over ${\mathbb{F}}_2$, a generator matrix $(I\mid A)$ therefore generates a self-dual code exactly when $A{A}^T=I$.

On polynomials of degree at most $n$, the derivative-evaluation functionals $e_j(\varphi )={\varphi }^{(j)}(0)$ are dual to $p_j=x^j/j!$, since $e_i(p_j)={\delta }_{ij}$. Differentiation maps $p_j$ to $p_{j-1}$, with $p_0$ mapped to zero. Its matrix is a single nilpotent Jordan block with ones above the diagonal, and its dual has the transpose matrix.

A coordinate functional extracts one coefficient in a specified finite basis. It is an element of that basis's dual basis, rather than a coordinate rule independent of the basis. For the standard basis of $F^n$, the $i$th coordinate functional sends $(a_1,\dots ,a_n)$ to $a_i$.

For a family in the $n$-dimensional Boolean lattice, the loss of trace cardinality on deleting coordinate $i$ equals the number of pairs $S,S\cup \{i\}$ both in the family. If every direction has at least two such edges, downward coordinate compression preserves that property. The resulting down-set contains the empty set and all $n$ singletons. Each coordinate must also occur in a two-element member. Those pairs cover all $n$ coordinates and number at least $\lceil n/2\rceil$, giving the bound. For $n\ge 2$, the empty set, all singletons and a minimum collection of pairs covering every coordinate attain it.

Update the level and slope by ${\ell }_t=\alpha X_t+(1-\alpha )({\ell }_{t-1}+b_{t-1})$ and $b_t=\beta ({\ell }_t-{\ell }_{t-1})+(1-\beta )b_{t-1}$, with gains in $(0,1)$. The first update compares the new observation with the previous level extrapolated one step; the second smooths the latest level increment. The displayed forecast extrapolates that local slope. This is a double exponential smoothing method for a nonseasonal trending time series.

For $0<\alpha <1$, set $s_t^{(1)}=\alpha X_t+(1-\alpha )s_{t-1}^{(1)}$ and $s_t^{(2)}=\alpha s_t^{(1)}+(1-\alpha )s_{t-1}^{(2)}$. Use the displayed corrected level and slope, then forecast $a_t+h{b}_t$. For an exact linear trend after initialization transients, the first and second smooths have lags $(1-\alpha )/\alpha$ and twice that lag; their difference recovers the slope and the corrected level removes the lag. This is distinct from Holt's linear trend method.

Thermosolutal convection is double-diffusive convection driven by temperature and concentration gradients. Their buoyancy contributions can oppose one another, while unequal thermal diffusivity and solutal diffusivity allow steady, oscillatory and finite-amplitude states. For a layer heated from below with a stabilizing concentration gradient, thermal buoyancy is destabilizing and solutal buoyancy restoring.

The vector projection multiplies the scalar projection by the chosen unit vector, giving the orthogonal projection onto the linear span of a nonzero real Euclidean vector $\mathbf{c}$:For $\mathbf{a}=(3,4)$ and $\mathbf{c}=(2,0)$, this gives $\mathbf{p}=(6/4)(2,0)=(3,0)$ and $\mathbf{r}=(0,4)$; the dot product $\mathbf{r}\cdot \mathbf{c}=0$ checks the orthogonal projection residual. Replacing $\mathbf{c}$ by $(-2,0)$ gives the same $\mathbf{p}=(-6/4)(-2,0)=(3,0)$: reversing direction changes the scalar projection, but preserves the vector projection onto the line.

For a short proof of the nearest-point property, any vector $\mathbf{z}$ on this line satisfies $\Vert \mathbf{a}-\mathbf{z}{\Vert }^2=\Vert \mathbf{r}{\Vert }^2+\Vert \mathbf{p}-\mathbf{z}{\Vert }^2$ by the Pythagorean identity. The second term is uniquely minimized at $\mathbf{z}=\mathbf{p}$. This connects a directional measurement to the orthogonal projection used in approximation.^[ref]

The scalar projection of a real Euclidean vector $\mathbf{a}$ onto a nonzero Euclidean vector $\mathbf{c}$ is its signed component along the unit vector $\mathbf{c}/\Vert \mathbf{c}\Vert$:For $\mathbf{a}=(3,4)$, the scalar projection onto $(2,0)$ is $3$, while that onto $(-2,0)$ is $-3$. Reversing the measuring direction reverses the sign; the scalar projection is not necessarily a nonnegative length. For nonzero $\mathbf{a}$, it also equals $\Vert \mathbf{a}\Vert \cos \theta$, using the angle with $\mathbf{c}$ and its cosine. For the zero vector it is zero, even though that angle is undefined.