Foundations

Fischer, Lynch and Paterson prove that a deterministic consensus protocol in their fully asynchronous message-passing model can have an admissible nondeciding run with at most one crash.

Raft uses term-numbered leader election, majority voting and log matching to replicate a state machine under crash faults.

A linearizable object admits a sequential explanation in which each completed operation appears to take effect between invocation and response, preserving real-time precedence of nonoverlapping operations.

In Gilbert and Lynch's asynchronous read/write model, an implementation cannot provide both atomic consistency and a terminating response to every request at every nonfailed node during arbitrary partitions.

In the synchronous oral-message Byzantine-generals model, agreement with at most f faulty participants requires at least 3f+1 participants, with identifiable senders and reliable delivery but no unforgeable transferred signature chain.

A normed real or complex vector space is Banach when every norm-Cauchy sequence converges in that space.

A linear map T:X→Y between normed spaces is continuous exactly when ||Tx|| ≤ C||x|| for some finite C.

A Hilbert space is complete in the norm induced by a real or complex inner product.

For a closed linear subspace M of a Hilbert space H, every x has a unique decomposition x=P-M x+z with P-M x in M and z orthogonal to M.

Every continuous linear functional ell on a Hilbert space has a unique representing vector y with ell(x)=<x,y>, and ||ell||=||y||.

Weak convergence x-n→x in a Hilbert space means <x-n,y>→<x,y> for every fixed y.

A distribution on an open U in R-n is a continuous linear functional on smooth compactly supported test functions.

Distribution derivatives transfer differentiation to a test function with the integration-by-parts sign.

A smooth n-manifold has locally Euclidean charts with smooth invertible transition maps, together with Hausdorff and second-countable topology.

A covector is a linear functional on the tangent space, and a tensor is multilinear on tangent/cotangent inputs.

Differential k-forms are smooth alternating covariant tensors.

On a compact oriented smooth n-manifold with boundary, integrating d-omega equals integrating omega on the induced oriented boundary, for a smooth (n−1)-form.

de Rham cohomology quotients closed k-forms by exact k-forms.

On a compact smooth Riemannian manifold without boundary, every de Rham class has a unique harmonic representative.

Over a field of characteristic zero, a polynomial is solvable by radicals exactly when its splitting-field Galois group is solvable.

For x³−2 over Q, the splitting field is Q(cube-root(2),zeta-3), has degree six and Galois group S3.

For a primitive n-th root zeta-n, Gal(Q(zeta-n)/Q) is isomorphic to the units modulo n by a↦(zeta-n↦zeta-n^a), and the degree is Euler phi(n).

A representation of a finite group G is a homomorphism G→GL(V).

For complex finite-group representations, chi(g)=tr rho(g) is constant on conjugacy classes and irreducible characters are orthonormal under the group average.

For integrable X and a sub-sigma-algebra G, E[X|G] is a G-measurable integrable random variable whose integral over every event in G equals that of X.

A discrete martingale M-n is adapted to increasing sigma-algebras F-n, is integrable at every time, and obeys E[M-(n+1)|F-n]=M-n.

A discrete nonnegative supermartingale converges almost surely to a finite integrable limit. Its expectations also converge, but their limit can differ from the expectation of the almost-sure limit.

For a martingale and bounded stopping times S≤T, E[M-T|F-S]=M-S.

W^(k,p)(Omega) consists of L-p functions with distribution derivatives through order k also in L-p; H^k denotes p=2.

On a bounded domain, the homogeneous Dirichlet Poisson problem seeks u in H0^1 with integral grad-u·grad-v=<f,v> for every v in H0^1, where f is in H−1.