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Forthe Fourier convention $\hat{u}(\theta )={\sum }_m{e}^{-im\theta }{u}_m$ givesprovided the denominator does not vanish.

The velocity of a surface curve is tangent. If its acceleration is normal, thenso every affinely parametrized geodesic has constant speed.

The free product $A\ast B$ is the amalgamated free product over the trivial group. Its nonidentity elements have unique reduced alternating normal forms.

The alternating harmonic series ${\sum }_{n\ge 1}(-1{)}^{n+1}/n$ converges. Its even partial sums increase, its odd partial sums decrease, and the two limits agree because their difference is the next term. In particular, every even partial sum is a lower bound and every odd partial sum is an upper bound for the limit.

This alternates the signs of the terms of the harmonic series, which itself diverges.

An Alfvén number is the ratio of a flow speed to the corresponding Alfvén speed. For a poloidal flow parallel to a poloidal magnetic field, $A^2={\mu }_0\rho u_p^2/B_p^2$.

This group has seven conjugacy classes: the identity; two classes of five nonidentity translations, distinguished by square class of the translation parameter; and, for each of the four $r\in Q\setminus \{1\}$, one class $\{(r,b):b\in {\mathbb{F}}_{11}\}$ of size eleven.

A bounded operator $T$ on a real or complex Hilbert space is a normal operator when $T^{\ast }T=T{T}^{\ast }$. Equivalently,The forward direction follows by taking inner products. Conversely the norm equality makes $⟨(T^{\ast }T-T{T}^{\ast })x,x⟩=0$ for every $x$, so the polarization identity forces the self-adjoint difference to vanish.

Every Hermitian operator and every unitary operator is normal. The complex diagonal matrix $\mathrm{diag}(2,i)$ is normal but neither self-adjoint nor unitary. In contrast, the unilateral shift $S{e}_n=e_{n+1}$ on the l2 sequence space has $S^{\ast }S=I$ and $S{S}^{\ast }=I-P_{e_1}$, so it is an isometry but not normal. For unbounded operators, normality additionally involves closedness and matching operator domains; the everywhere-defined bounded criterion above cannot be applied while ignoring them.

For a subspace $U$ of a finite-dimensional inner-product space,This follows directly by moving $T$ across the inner product and using $(U^{\perp }{)}^{\perp }=U$.

A bounded operator $T:H\to H$ on a real or complex Hilbert space is Hermitian, or self-adjoint, when $T=T^{\ast }$. Equivalently,Its directional values $q_T(x)=⟨Tx,x⟩$ are real, since this identity makes $q_T(x)$ equal its complex conjugate. Reality allows either sign; a positive operator additionally requires nonnegative values in every direction.

The diagram compares Hermitian quadratic forms along paths of unit vectors. The Hermitian matrix $D=\mathrm{diag}(1,-1)$ takes the values $1$ and $-1$ on the standard basis vectors; $P=\mathrm{diag}(1,0)$ takes the values $1$ and $0$; andsatisfies $H{e}_1=e_1$ and $H{e}_3=3e_3$. These eigenvectors form an orthonormal basis, and every $x=a{e}_1+b{e}_3$ has $q_H(x)=|a{|}^2+3|b{|}^2$. Thus $H$ is a positive operator, with eigenvalues $1,3$ and operator norm $3$.

Three conditions describe different geometry. A Hermitian operator satisfies $T=T^{\ast }$; a normal operator satisfies $T^{\ast }T=T{T}^{\ast }$; a unitary operator satisfies $T^{\ast }T=T{T}^{\ast }=I$, where $I$ is the identity operator. Every Hermitian operator and every unitary operator is a normal operator. Concrete matrices separate the conditions:

The spectrum of a bounded Hermitian operator is real. In finite dimension, the spectral theorem supplies an orthonormal basis of eigenvectors, turning its action into independent real scalings. This makes the geometry easy to interpret: each eigenvalue records a signed response along an orthogonal direction. An arbitrary bounded Hermitian operator in infinite dimension need not have any eigenvectors; the general spectral theorem uses a projection-valued measure. MIT's lecture 21 develops the real spectrum and the orthogonality of eigenspaces.^[ref]

A historical connection is the study of forms with conjugate variables. In his 1901 account of Charles Hermite's work, Camille Jordan describes these forms and emphasizes their role in reducing forms with real or complex coefficients. In today's notation, $x^{\ast }Hx$ is the diagonal value of a Hermitian form; the conjugate variables make a real-valued expression possible even with complex coefficients.^[ref]

For a densely defined operator outside this bounded setting, the displayed pairing identity alone defines a symmetric operator. A self-adjoint operator must also have the same operator domain as its adjoint operator.

For $H=p^2/(2m)-1/|q|=-|E|$, collision continuation between the two turning points gives $I=\sqrt{2m/|E|}$.

When a Hamiltonian parameter varies slowly compared with the orbital period and no separatrix is crossed, the action changes only at higher adiabatic order.

The action is an adiabatic invariant under these assumptions; other slowly varying systems can have different adiabatic invariants.

For two homogeneous inviscid fluids, specify displacement amplitudes along one fixed interface normal. At a real positive frequency, their ratios equal those of the corresponding normal velocities. For propagating plane waves with incidence angles ${\theta }_j$ from the normal, letContinuity of normal velocity and pressure givesThe condition $|r_n|=|t_n|$ therefore becomes $|Z_{1,n}-Z_{2,n}|=2Z_{1,n}$. Positive impedances make this equivalent to $Z_{2,n}=3Z_{1,n}$, giving $r_n=-1/2$, $t_n=1/2$.

At normal incidence this is simply ${\rho }_2{c}_2=3{\rho }_1{c}_1$. At oblique incidence, Snell's law also requires $\sin {\theta }_1/c_1=\sin {\theta }_2/c_2$, so both angular factors enter. Equal normal displacement amplitudes then correspond to reflected and transmitted power fractions$1/4$ and$3/4$, because transmitted power has an impedance factor. Full displacement magnitudes along each ray use different cosine factors; the measured component must be stated.

For a P-wave incident at angle $\theta$ from a liquid $(\rho ,c,\lambda )$ onto $({\rho }^{\prime },c^{\prime },{\lambda }^{\prime })$, with transmitted angle ${\theta }^{\prime }$ obeying Snell law for elastic and acoustic waves,No reflection is equivalent to ${\rho }^{\prime }{c}^{\prime }\cos \theta =\rho c\cos {\theta }^{\prime }$.

For a line bundle $L$ on an abelian variety $A$ and translations $T_x(a)=a+x$, the Theorem of the square gives an isomorphism of line bundlesConsequently $x\mapsto T_x^{\ast }L\otimes L^{-1}$ is a group homomorphism from $A$ to its dual abelian variety. These are isomorphism classes: no canonical trivialization is supplied without normalization data. On an elliptic curve, translating a degree-one divisor changes its degree-zero difference class additively, illustrating the identity. The Theorem of the Cube controls the additional compatibility when the translation points themselves vary.^[ref]

An invariant differential on a group scheme is a differential one-form preserved by every left translation. For a smooth group scheme $G$ over a field $k$, translation identifiesThus a covector at the identity extends uniquely to a left-invariant form. On the additive algebraic group the invariant form is $dx$; on the multiplicative algebraic group it is $dx/x$, since $d(ax)/(ax)=dx/x$. An abelian variety of dimension $g$ therefore has $g$ linearly independent invariant one-forms, providing the local linear data behind its global group geometry.

A local form of the Mumford rigidity lemma says: if $X$ is a complete integral variety over an algebraically closed field, and a morphism $f:X\times Y\to Z$ sends one fibre $X\times \{y_0\}$ to a point, then on $X\times U$ for some open neighbourhood $U$ of $y_0$, it factors through the projection to $U$. To see the mechanism, choose an affine neighbourhood of that image point. Properness of the projection supplies $U$ whose whole product maps into that neighbourhood. Regular functions on the complete integral $X$ are constant, forcing the factorization there. Pointed global versions add further hypotheses; the local conclusion alone should not be silently replaced by a global one.^[ref]

The root condition for a multistep method says that every root of a polynomial $\xi$ of its first characteristic polynomial $\rho$ satisfies $|\xi |\le 1$, and each root with $|\xi |=1$ is simple. For constant coefficients this is equivalent to zero-stability.

A root of multiplicity $m$ produces recurrence modes $n^j{\xi }^n$ for $0\le j<m$. If $|\xi |<1$, the exponential decay dominates every such polynomial factor, so repeated interior roots are allowed. At $|\xi |=1$, any nonconstant polynomial factor grows, explaining the simplicity requirement. A root outside the unit disk gives exponential growth even when simple. Thus $\rho =(\zeta -1)(\zeta -1/2{)}^2$ satisfies the condition, while $(\zeta -1{)}^2$ does not.

The seven consecutive zeros followed by one force degree at least eight. Degree eight is attained by $s_{n+8}=s_n$, with initial state $10000000$ and feedback polynomial $x^8+1$.

Shell crossing occurs when the Lagrangian-to-Eulerian map ceases to be one-to-one. Its Jacobian determinant vanishes, the pressureless single-stream density formally diverges, and several streams subsequently occupy the same position.

For an irrotational displacement $\mathbf{s}=-{\nabla }_q\Phi$, the deformation tensor is the symmetric Hessian $d_{ij}=-{\partial }_i{\partial }_j\Phi$. Its real eigenvalues determine the principal collapse directions.