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A momentum eigenstate is an eigenstate of the momentum operator. In one dimension it is a plane wave $e^{ikx}$ with momentum eigenvalue $\hslash k$.

The nearly circular orbit equationhas radial angular frequency $1-3m^2/h^2$ to leading order. The perihelion advances by $6\pi m^2/h^2$ per orbit.

The Catalan number $C_n$ counts Dyck paths with semilength of a Dyck path $n$. For a nonnegative integer $n$,The count $C_0=1$ includes the empty path. Of the $\left(\genfrac{}{}{0ex}{}{6}{3}\right)=20$ paths with three up steps and three down steps, exactly five stay at nonnegative height. Their shapes and balanced parentheses encodings appear below.^[ref]

The useful connection is a shared decomposition. A Dyck path splits uniquely at its first-return decomposition of a Dyck path; a binary tree splits at its root into left and right subtrees; a binary parenthesization splits at its last operation. Each removes one outer unit and leaves an ordered pair of smaller objects. The resulting Catalan recurrence lets one count several different structures with the same calculation. A count of five does not mean the structures are physically identical: a bijection supplies the precise translation between them.

A polynomial moment functional evaluates a polynomial through a weighted integral, such as $p\mapsto \int p(t)w(t)dt$; finitely many moments determine it on a bounded-degree polynomial space.

If $q_1,\dots ,q_n\in V^{\ast }$ are linearly independent, thenmaps $V$ onto $F^n$. Otherwise a nonzero functional annihilating its proper image would give a nontrivial linear relation among the $q_j$.

For a vector subspace $U\le V$,In finite dimension, $\dim U^{\circ }=\dim V-\dim U$.

An annihilator of a vector subspace collects the linear functionals that ignore every input from that vector subspace. For $U=\mathrm{span}\{(1,2)\}\subseteq {\mathbb{R}}^2$, a linear functional $\phi (x,y)=ax+by$ vanishes on $U$ exactly when $a+2b=0$. Hence $U^{\circ }$ is the one-dimensional span of $(x,y)\mapsto 2x-y$. The zero level set in the linear functional diagram is precisely $U$.

For a basis $e_1,\dots ,e_n$ of a finite-dimensional vector space $V$, its dual basis consists of linear functionals $e_1^{\ast },\dots ,e_n^{\ast }$ satisfyingThe Kronecker delta means that each rule gives one on its own basis vector and zero on the others. These rules extract coordinates:Applying $e_i^{\ast }$ to a linear combination removes every term except the $i$th. Consequently the dual basis is a basis of the dual space, and every linear functional has expansion $\phi ={\sum }_i\phi (e_i)e_i^{\ast }$. ^[ref].

Coordinate extraction works for a nonorthogonal basis too. With $b_1=(1,0)$ and $b_2=(1,1)$, the dual basis isThe checks $b_1^{\ast }(b_1)=1$, $b_1^{\ast }(b_2)=0$, $b_2^{\ast }(b_1)=0$, and $b_2^{\ast }(b_2)=1$ fix the two rules. For $v=(3,1)$ they give coordinates $(2,1)$, so $v=2b_1+b_2$. The usual dot products with $b_1,b_2$ instead give $3,4$ and do not extract these coordinates: simple projection formulas depend on an orthonormal basis.

The same idea explains polynomial interpolation. On polynomials of polynomial degree at most $n$, evaluation at distinct interpolation nodes $x_0,\dots ,x_n$ gives the dual basis to the Lagrange cardinal polynomials, because ${\ell }_i(x_j)={\delta }_{ij}$. Samples are coordinates in this basis, so the reconstruction $p={\sum }_{i}p(x_i){\ell }_i$ is a dual-basis expansion. Differentiation weights from nodal evaluations and factorial monomial basis for polynomial differentiation apply the same principle to different measurements.

A linear functional is a linear map from a vector space $V$ to its scalar field $F$:Its output is one scalar, rather than an arbitrary vector in another space. In ${\mathbb{R}}^2$, $\phi (x,y)=2x-y$ gives $\phi (3,1)=5$, $\phi (1,2)=0$, and $\phi (4,3)=5$: adding the two inputs adds their measured values.

The level sets $\phi (v)=c$ visualize what the measurement retains and discards. For this example they are parallel straight lines $y=2x-c$. Moving parallel to $(1,2)$ changes the input but leaves $\phi$ unchanged. The zero level set is its kernel of a linear map, while the other level sets are translates of that vector subspace.

Inputs need not be geometric arrows. On the vector space of polynomials of polynomial degree at most two, evaluating at zero is a linear functional, as is taking the definite integral from zero to one. For $p(t)=a+bt+c{t}^2$, they give $p(0)=a$ and ${\int }_0^{1}p(t)dt=a+b/2+c/3$. Thus polynomial evaluation and integration are two different measurements of the same polynomial. Coordinate extraction is another measurement, developed under dual basis.

A linear functional is defined by linearity, not by continuity. In finite-dimensional real or complex normed vector spaces it is automatically continuous; in infinite dimension it need not be. The continuous dual space adds this requirement when small changes in the input should give small changes in the output.

A covector is an element of the dual space, hence a linear functional on a vector space. In finite-dimensional coordinates, it acts as a row multiplied by a coordinate vector column. The row $\left(\begin{array}{cc}2& -1\end{array}\right)$ acts on $\left(\begin{array}{c}3\\ 1\end{array}\right)$ to give five. The terminology emphasizes its role as a scalar-valued measurement and its behavior under a change of basis.

For example, the total derivative of $f(x,y)=x^2+y^2$ at $(1,2)$ sends a small displacement $h$ to the linear change $df(h)=2h_x+4h_y$. This covector describes the first-order response. The gradient $\nabla f=(2,4)$ represents the same rule through the chosen Euclidean inner product. In differential geometry, the cotangent space collects these scalar-valued measurements of tangent vectors.

A bilinear form $B$ on $V$ is invariant exactly when $v\mapsto B(v,-)$ is a $G$-homomorphism $V\to V^{\ast }$. On an irreducible representation, Schur's lemma makes a nonzero such form nondegenerate and unique up to scale.

For complex representations $V,W$ of a finite group,Under the $G\times G$ action $(g,h)\alpha =\sigma (h)\alpha \rho (g^{-1})$, its character is

The fixed points of $F^n$ areThus $F^n$ has $2^n-1$ fixed points.

If $x=0.b_1{b}_2{b}_3\dots$ in base two, thenBinary cylinders make transitivity and density of periodic points transparent: concatenate prescribed finite blocks for transitivity and repeat a finite block for a periodic point.

If $A\subseteq {\mathbb{F}}_p^n$ and $|A+A|\le K|A|$, then $A$ is contained in a vector subspace $H$ satisfying

If finite sets satisfy $|A+B|\le K|A|$, then the Plünnecke-Ruzsa inequality bounds iterated sumsets and difference sets by

The nonlinear Euler--Lagrange equations couple both angular accelerations and contain centrifugal terms quadratic in angular velocity.

The kinetic energy contains the coupling $m{l}_1{l}_2\cos ({\theta }_1-{\theta }_2){\dot{\theta }}_1{\dot{\theta }}_2$, while gravity supplies one cosine potential for each mass height.

For a smooth projective curve of genus $g$ and a canonical divisor $K$,

This is the full linear system of divisors, obtained by using every global section of the associated line bundle. For a Cartier divisor on an integral projective variety, the complete linear system $|D|$ consists of its effective representatives under linear equivalence of Cartier divisors. A basis of $H^0(X,{\mathcal{O}}_X(D))$ defines a rational map to projective space.

For a nonzero rational function $f$ on a smooth projective curve $C$, its principal divisor isIt records the zeros of $f$ with positive multiplicity and its poles with negative multiplicity. Every principal divisor has degree zero.