The Hopf normal form is a local normal form of a dynamical system on the critical two-dimensional centre manifold. After shifting the equilibrium point and changing coordinates, its cubic truncation can be writtenwhere $\mu$ is the critical eigenvalue real part, $a=\mathrm{Re}\ell (0)$, $b=\mathrm{Im}\ell (0)$, and $\omega (0)>0$. Smooth parameter dependence of the cubic coefficients and higher-order terms are omitted in this truncation. The real coefficient $a$ controls radial saturation; the imaginary coefficient $b$ changes the angular frequency with amplitude. This finite-order reduction, rather than literal equality to a cubic equation everywhere, is the meaning of the Hopf normal form for a general system.^[ref]

In the cubic truncation, $r_{\ast }^2=-\mu /a$ whenever $\mu a<0$. Substituting into the angular equation gives $\Omega =\omega (\mu )-b\mu /a$. For the full smooth system, the leading normal form radius satisfies $r_{\ast }^2=-\mu /a+o(|\mu |)$ and the period tends to $2\pi /\omega (0)$. The exact circle and exactly constant angular frequency of the special model $a=-1$, $b=0$, $\omega (\mu )=\omega$ are conveniences of that model. In the original coordinates the small periodic orbit need not be a circle. Radial scaling can normalize a nonzero $a$ to $-1$ or $+1$, so $\sqrt{|\mu |}$ is a normalized leading amplitude, not a universal physical radius.

The convolution inverse $S$ of the identity on a bialgebra: $m(S\otimes 1)\Delta =j\epsilon =m(1\otimes S)\Delta$. Existence makes the bialgebra a Hopf algebra; $S$ reverses multiplication.

The full ${\mathrm{Hom}}_k(M,N)$ carries the action $(hf)(u)=\sum h_{(1)}f(S(h_{(2)})u)$, making it the internal hom right adjoint to $-\otimes M$. No finite-dimensionality or inverse antipode is needed.

For a left module over a Hopf algebra, a vector $v$ is invariant when $hv=\epsilon (h)v$. In the internal hom for modules over a Hopf algebra, invariant maps are exactly the R-module homomorphisms for the acting algebra.

The based suspension collapses both ends and the basepoint cylinder of $X\times I$ to one basepoint. The loop space and reduced suspension are adjoint in based homotopy theory. For a connected based CW complex, the James reduced product models the loop space of its reduced suspension.

Study homotopy after killing torsion, especially for simply connected spaces of finite type. The rational Whitehead theorem detects rational homotopy equivalences by rational homology, while Sullivan minimal models encode rational homotopy generators and cohomological relations.

A nonempty space is acyclic with the chosen coefficients if its reduced homology vanishes in every degree. Equivalently it has the homology of a point. It need not be contractible.

A positive-degree class whose reduced homology diagonal vanishes. In a torsion-free homology Hopf algebra, powers of an even primitive class have binomial diagonals. Dualizing that diagonal produces the multiplication of a divided power algebra.

Homogeneity of degree $k$ means that multiplying all arguments by a common positive factor $a$ multiplies the output by $a^k$. A homogeneous function is the function possessing this property. This is different from degree-one positive homogeneity; investment values with constant relative risk aversion utility usually have degree $1-R$.

A continuous bijection that is an open map is a homeomorphism. If $g=f^{-1}$ and $U$ is open in the original domain, then $g^{-1}(U)=f(U)$ is open by the open-map property. This proves continuity of the inverse directly.

A holomorphic primitive of $f$ is a holomorphic function $F$ with $F^{\prime }=f$. A holomorphic $f$ on a domain has such a primitive exactly when its integrals around all closed piecewise smooth curves vanish. Integrating from a fixed starting point then constructs a path-independent primitive.

On an oriented Euclidean metric four-space, the Hodge star operator on two-forms is an orthogonal involution. Its two eigenspaces each have dimension three. A self-dual two-form has eigenvalue $+1$, while an anti-self-dual two-form has eigenvalue $-1$. The splitting uses the metric-normalized Riemannian volume form; an arbitrarily rescaled volume in the defining wedge identity would rescale the operator itself.

For a sample term $\max \{0,1-a_{i}x\}$, the active hinge set contains indices with $1-a_{i}x>0$. At strict negative margins the hinge is inactive, while zero margins allow fractional hinge-loss optimality weights. Away from zero margins, the active set stays fixed on a sufficiently small candidate solution neighborhood.

For $\mathcal{H}f(x)={\pi }^{-1}\mathrm{pv}\int f(y)/(x-y)dy$ and $\hat{f}(\xi )=\int e^{-ix\xi }f(x)dx$, the Fourier transform multiplier is $-i\mathrm{sgn}\xi$. The Heaviside function transform and the double-transform reflection identity give the sign. Plancherel theorem proves the $L^2$ isometry, and multiplication by $i\xi$ proves commutation with differentiation. The map need not preserve the Schwartz space.

For a Schwartz function $f$, the Hilbert transform satisfies $\mathcal{H}f(x)=(\int f)/(\pi x)+o(|x{|}^{-1})$ at both infinities. Splitting the inverse Fourier transform at the sign multiplier's jump and integrating each half-line by parts extracts the zero-frequency value $\hat{f}(0)$; the remainder vanishes by the Riemann-Lebesgue lemma. A nonzero zeroth moment therefore obstructs Schwartz decay.

A linear map $V:\mathcal{H}\to \mathcal{K}$ is an isometry when it preserves inner products, equivalently $V^{†}V=I$. It need not be surjective. It maps an orthonormal basis to an orthonormal family, and $\rho \mapsto V\rho V^{†}$ preserves the nonzero eigenvalues and the Von Neumann entropy of a density operator.

For a generator of a group $g$ of order $M$ and $x=g^y$, the function $f(a,b)=g^a{x}^{-b}$ hides the subgroup $\{(yb,b):b\in {\mathbb{Z}}_M\}$ of ${\mathbb{Z}}_M^2$. Each fiber is an affine coset $\{(yb+k,b):b\in {\mathbb{Z}}_M\}$. Measuring the function register gives its normalized coset state. Applying the positive-exponent quantum Fourier transform to both input registers gives amplitudeThe finite geometric series vanishes off $c_2=-y{c}_1$ and equals $M$ on that line. Thus each allowed output pair has probability $1/M$, independently of the coset offset. If $\gcd (c_1,M)=1$, a modular inverse recovers $y=-c_2{c}_1^{-1}$.

For $S=a\mathrm{tr}(M^2)/2+b\mathrm{tr}(M^4)/4$ with $a>0$, the Wick theorem givesThere are eight adjacent external-attachment contractions with one free matrix-index sum and four opposite attachments with no free sum. The second tensor structure is needed at finite $N$; omitting it is an additional large-$N$ approximation, not merely restricting to connected Feynman diagrams.

Imposing vanishing off-diagonal entries with auxiliary integrations and an off-diagonal Grassmann Gaussian integral gives a determinant of the commutator map on the off-diagonal matrix units. Its eigenvalues are ${\lambda }_i-{\lambda }_j$, $i\ne j$. Up to a constant sign, their product is the squared Vandermonde determinant. It contributes $-2{\sum }_{i<j}\log |{\lambda }_i-{\lambda }_j|$ to the eigenvalue action, encoding repulsion between distinct eigenvalues. Diagonal zero modes and constant normalization factors must be treated separately.

Consider the Helmholtz equation $(\Delta +k^2)u=0$ in $z>0$, with $k>0$, boundary value $u(x,0)=g(x)$, and upward outgoing or decaying propagation. Taking the tangential Fourier transform, with frequency $\xi$, givesThus the upward normal derivative is the Fourier multiplier $i\beta (\xi )$. The outward unit normal of the upper half-space points downward, so its Dirichlet-to-Neumann map has multiplier $-i\beta (\xi )$ instead. The normal orientation and the square-root branch determine both signs.

For $|\xi |<k$ the modes propagate; for $|\xi |>k$ they are evanescent waves decaying with height. At high frequency $\beta (\xi )\sim i|\xi |$, so the map grows in magnitude like one tangential derivative. Small-amplitude boundary roughness can therefore have a large derivative effect when it varies on a short spatial scale.