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The logarithms of a fundamental system of $r+s-1$ units form a basis of the hyperplaneand their integer span is a full lattice in $H$.

The kernel of $\mathrm{Log}:{\mathcal{O}}_K^{\times }\to {\mathbb{R}}^{r+s}$ is exactly $\mu (K)$. If every conjugate of a unit has modulus one, all its powers lie in a bounded subset of the discrete Minkowski lattice ${\mathcal{O}}_K$, so two powers coincide and the unit is a root of unity.

If, for some real $u$,then partial summation continues $-{\zeta }^{\prime }(s-iu)/\zeta (s-iu)$ meromorphically to $\Re s>1/2$ with no pole except at $1+iu$. The zeta function has no zero to the right of the critical line, and its functional equation then implies the Riemann hypothesis.

The classical zero-free region and a truncated Perron contour givefor some constant $c>0$.

A scheme is a locally ringed space covered by open subsets isomorphic to spectra of commutative rings. It retains both the points defined by prime ideals and the local algebra of functions around them.

The product topology on $X\times Y$ is generated by the seminorms $(x,y)\mapsto p(x)$ and $(x,y)\mapsto q(y)$. Its continuous dual is canonically $X^{\ast }\oplus Y^{\ast }$.

If $C$ is a nonempty open convex subset of a real locally convex space and $x_0\notin C$, a continuous linear functional strictly separates them: after choosing a sign, $f(c)<f(x_0)$ for every $c\in C$. Apply the Hahn-Banach theorem to the Minkowski functional of a translate of $C$.

For an absorbing convex set $U$ containing zero, its Minkowski functional is $p_U(x)=\inf \{t>0:x\in tU\}$. If $U$ is open and convex, then $p_U$ is sublinear and $U=\{x:p_U(x)<1\}$.

If $X$ is locally compact Hausdorff and $A\cap K$ is closed in every compact subset $K$, then $A$ is closed. Around each point outside $A$, use a compact neighbourhood and the relative openness of its complement of $A\cap K$.

If $R$ is a Dedekind domain and $0\notin S$, then $S^{-1}R$ is an integrally closed Noetherian domain of dimension at most one. It is therefore either a Dedekind domain or, when its dimension is zero, a field.

An improper mixed state is the reduced density matrix of a subsystem entangled with unobserved degrees of freedom. A quantum state readout device returns that reduced state until a relevant collapse lies in its past light cone.

A proper mixed state represents classical uncertainty over states selected by an actual preparation or collapse. Once that selecting event lies in the readout point's past light cone, a quantum state readout device returns the selected component rather than merely the ensemble average.

A quantum state readout device at $x$ hypothetically outputs a classical description of the local quantum state under objective collapse without changing that state. Its output can respond to a remote collapse only after the collapse enters the past light cone of $x$, so the device preserves relativistic causality.

For the conethe first fundamental form isThe local change of polar coordinatesturns this into the Euclidean metric $d{R}^2+R^{2}d{\Theta }^2$.

A Riemannian isometry is a diffeomorphism that preserves the Riemannian metric, and hence lengths, areas, and intrinsic curvature.

A local isometry intertwines the Levi--Civita covariant derivatives. It therefore sends every affinely parametrized geodesic to an affinely parametrized geodesic.

In the two-skeleton of a barycentrically subdivided tetrahedron, the vertex corresponding to the whole tetrahedron has link equal to the connected 1-skeleton of the subdivided boundary, with 14 vertices and 36 edges. Its local $H_2$ therefore has rank $36-14+1=23$. Every other point has smaller local rank, so every self-homeomorphism fixes this vertex.

If a neighborhood of $x$ is a cone on a compact link $L$, excision and the long exact sequence of the cone pair give

An adjoint transport equation transfers the transport operator from a weak solution to a test function. Solving the terminal-value adjoint equation allows arbitrary compactly supported source terms to be used in uniqueness proofs.

A planar equilibrium is a stable spiral when its Jacobian has a complex-conjugate pair of eigenvalues with negative real part. Nearby nonstationary trajectories spiral toward it.