For real $u$, this scalar S-matrix block satisfies $S_u(\theta )S_u(-\theta )=1$ and Hermitian analyticity of a two-particle S-matrix. It therefore has modulus one for real rapidity differences. For $0<u<\pi$ its pole at $\theta =iu$ has residue $2i\sin u$ and admits a direct-channel bound-state pole interpretation. Its crossed block is $\cosh [(\theta -iu)/2]/\cosh [(\theta +iu)/2]$. The endpoints $u=0$ and $u=\pi$ are removable degeneracies giving constant amplitudes, so a particle cannot be inferred there from the uncancelled denominator alone. This block specifies an amplitude; a complete theory must also obey the remaining crossing symmetry and bootstrap fusion constraints.

Writing $\varphi ={\varphi }_0+2\pi m\tau /\beta +\eta$ with endpoint-fixed $\eta$ separates winding action and Gaussian fluctuations. The latter give the free return kernel on the lifted line, $Z_0=\sqrt{I/(2\pi {\hslash }^2\beta )}$. The starting-angle integral supplies $2\pi$. Zero winding is not the same as the zero-momentum ground-state sector.

Poisson resummation relates the momentum sum ${\sum }_n{e}^{-\beta {\hslash }^2{n}^2/(2I)}$ to the winding sum $\sqrt{2\pi I/(\beta {\hslash }^2)}{\sum }_m{e}^{-2{\pi }^{2}I{m}^2/(\beta {\hslash }^2)}$. Gaussian Fourier transformation proves equality, including the prefactor. The momentum form converges rapidly at low temperature and the winding form at high temperature.

For fixed $\theta >0$, a sufficiently large finite set of integers with additive energy at least $\theta |A{|}^3$ has a nonconstant three-term arithmetic progression. The Balog-Szemerédi-Gowers theorem and Ruzsa modelling lemma reduce the problem to the Roth theorem on three-term arithmetic progressions in a dense cyclic model.

An energy and orbital angular momentum eigenstate for a real two-dimensional central potential can be chosen with integer $k$ and real radial function $f$. Its radial equation isQuantum degeneracy permits other stationary states, including $f(r)\cos (k\varphi )/\sqrt{\pi }$, which are not single angular exponentials. Separation of variables supplies a basis rather than describing every superposition in a degenerate eigenspace.

A planar rotation about the origin is the linear map represented by the rotation matrix $R_{\theta }=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)$. It preserves inner products, lengths, oriented angles and area, with positive angles measured anticlockwise. Its inverse is $R_{-\theta }$ and its determinant is one.

The rotation gate about the Pauli X gate axis commutes with $X$. The J gate convention gives $J(\alpha )J(0)=H\mathrm{diag}(1,e^{i\alpha })H=e^{i\alpha /2}{R}_x(\alpha )$, differing from the standard axis rotation only by global phase.

The exponential generating function for labelled rooted trees satisfies $T=z{e}^T$: remove the root and obtain an unordered set of smaller labelled rooted trees. For $0\le z<1/e$, the convergent series is the solution in $[0,1)$ of $T{e}^{-T}=z$, rather than the larger real solution. Alternatively, tree-component expectation in the Erdős-Rényi model and subcritical exploration give $T(\theta e^{-\theta })=\theta$ for $0<\theta <1$: their limiting probabilities for the order of the tree component of a uniform vertex sum to one. The component tail bound makes this passage through the infinite sum valid.

When matter leaves a moving body with zero instantaneous relative velocity, it carries the body's current momentum per unit mass but creates no additional recoil term. A short-interval momentum balance including the newly released material gives $q\dot{v}=F_{\mathrm{ext}}$ for the remaining body's mass $q(t)$. The expression $\frac{d}{dt}(qv)=F_{\mathrm{ext}}$ omits the outgoing momentum flux. Nonzero relative ejection velocity instead produces the thrust term of the rocket equation.

A subset closed under addition, additive inverses and multiplication, with the induced operations. Under the unital convention used here it also contains the identity of $R$. A subring of an integral domain is again an integral domain. The ring of invariants is one example: fixed elements remain fixed under all ring operations.

Extend the cylinder premeasure from a positive functional to a Borel probability measure and use cylinder-function density in Cantor space. The integral and functional agree on cylinder simple functions, and both are continuous in the supremum norm, so they agree on all of $C(\Omega )$. Uniqueness on the generating cylinder algebra gives uniqueness of the Borel measure.

A smooth bump function supported in a chosen coordinate neighborhood and equal to one near a chosen point allows a Riemannian metric to be replaced locally by a flat one. The displayed convex combination stays positive definite and glues smoothly to the original metric outside the support. This is a local modification of the metric, not a claim that the original metric is flat.

A smooth Riemannian metric with no nonidentity local isometry between open subsets. Equivalently distinct nonempty open subsets never have isometric induced metrics. This strong local rigidity is also called bumpy in some spectral-geometry treatments; it is distinct from the closed-geodesic nondegeneracy convention for bumpy metrics.

A smooth local basis of vector fields whose inner products under a Riemannian metric are the identity matrix. Tracing the Riemannian Hessian in such a frame gives the Laplace-Beltrami operator, with the chosen sign.

The symmetric covariant two-tensor defined using the Levi-Civita connection by ${\mathrm{Hess}}_{g}f(X,Y)=X(Yf)-({\nabla }_{X}Y)f$. Its negative metric trace is the positive Laplace-Beltrami operator applied to $f$. Along a geodesic it gives the second derivative of $f$.

The hypothesis asks for growth smaller than every fixed positive power on the critical line. The implied constant may depend on the chosen exponent. The Riemann hypothesis implies the Lindelöf hypothesis through a smoothed explicit formula and strip convexity.

For $\sigma >1$, the logarithm of the displayed product is a prime-power sum with coefficients $3+4\cos \theta +\cos 2\theta =2(1+\cos \theta {)}^2\ge 0$. A zero of order $a\ge 1$ at $1+it$, with $t\ne 0$, would make the product vanish like $O((\sigma -1{)}^{4a-3})$: the real factor has a pole of order three, and the third factor is bounded. This contradicts its lower bound one. Hence the Riemann zeta function has no zeros on its line of real part one; its pole at one is not a zero.

For a finite group generated by $A,B$, glue sheets labeled by the group over the three-punctured Riemann sphere using left monodromy multiplication. Fill the punctures by the local power charts dictated by the orders $m,n,r$ of $A,B,AB$. Right multiplication by inverses defines the commuting deck action. Its nontrivial stabilizers are conjugates of the corresponding cyclic subgroups. The Riemann-Hurwitz formula gives the displayed Euler characteristic. A compatible metric can be averaged over the group; in genus at least two the invariant hyperbolic metric is canonical by the uniformization theorem.

For dimension $n\ge 3$, the Schouten tensor packages the Ricci tensor and scalar curvature in the trace part of the Riemann curvature tensor. In dimension three it is $S_{ij}=R_{ij}-R{g}_{ij}/4$, and the full Riemann curvature tensor is reconstructed from it.

A rigid rod with moving centre and changing orientation has material velocity $\dot{\mathbf{X}}+s\dot{\theta }\mathbf{n}$. Its centred rotational velocity is odd in arclength and has no resultant force, while its orientation-dependent resistance tensor couples translation to transverse force. Resistive-force theory uses per-length coefficients $c_{\parallel },c_{\perp }$.