For a linear group action of $G$ on a state space, an equivariant dynamical system has a vector field satisfyingThus applying a symmetry before computing a velocity gives the same result as applying it to the computed velocity. An equivariant vector field need not be invariant as a vector: its value transforms along with the point where it is evaluated.

If the ordinary differential equation has unique solutions, its flow map obeys ${\Phi }_t(gx)=g{\Phi }_t(x)$ whenever both sides exist. Indeed, $g{\Phi }_t(x)$ has derivative $gF({\Phi }_t(x),\mu )=F(g{\Phi }_t(x),\mu )$ and starts at $gx$, so Picard-Lindelöf theorem identifies it with ${\Phi }_t(gx)$.

Consequently, every subgroup's fixed-point subspace of a group action is invariant under the flow map. If $hx=x$ for all $h$ in the subgroup, then $h{\Phi }_t(x)={\Phi }_t(hx)={\Phi }_t(x)$. This reduces some symmetry-preserving motions to a smaller state space. It also constrains normal forms: the wavevector selection rule for equivariant monomials and the cubic equivariants of the full cube symmetry group are concrete coefficient tests.

A Smale horseshoe is a compact hyperbolic invariant set of a smooth map with expanding and contracting directions and at least two strips that stretch across a common rectangle. Iterating the strips and their inverse images produces a Cantor-set product on which the dynamics are conjugate to a full shift. It supplies infinitely many saddle periodic orbits and chaotic itineraries; hyperbolicity alone does not make it an attracting invariant set. The geometric construction explains why phase-dependent reinjection near a Shilnikov bifurcation can create chaos even when the associated symmetric amplitude equations have no such mechanism.

Let $f$ be a smooth diffeomorphism and $K$ a compact invariant set. It is uniformly hyperbolic if its tangent spaces have a continuous splitting $T_{x}M=E_x^s\oplus E_x^u$ preserved by the derivative, with constants $C>0$ and $0<\rho <1$ such thatThe same constants must work for every $x\in K$. Thus the stable directions contract forward and the unstable directions contract backward, uniformly over the set. A sign observed in a finite-time numerical Lyapunov exponent calculation does not establish these uniform estimates.

For the cat map on a two-dimensional torus, the constant matrix $\left(\begin{array}{cc}2& 1\\ 1& 1\end{array}\right)$ has eigenvalues $(3\pm \sqrt{5})/2$. Its two eigenvectors give fixed stable and unstable directions; the smaller eigenvalue is the reciprocal of the larger. With the flat Euclidean metric, $C=1$ and $\rho =(3-\sqrt{5})/2$ give the required estimates everywhere. The whole torus is therefore hyperbolic.

A Smale horseshoe supplies a different example: its hyperbolic invariant set contains infinitely many saddle periodic orbits, yet need not attract a neighborhood. Hyperbolicity describes stretching and contraction, while being an attractor concerns nearby trajectories. For a nonsingular flow map, the analogous splitting also has the neutral direction along the vector field; it should not be omitted by copying the map definition unchanged.

The two-domain system recognizes bacteria and archaea as the two primary lineages, placing the nuclear ancestry of eukaryotes within archaea. In this model, archaea as a complete clade includes that eukaryotic lineage; restricting the name to prokaryotic members describes a different grouping. Williams and colleagues' study, published online in 2019 and in a 2020 journal issue, found support for this model and a close relationship to Asgard archaea. See Williams et al., 2020: Phylogenomics provides robust support for a two-domains tree of life.

This updates the interpretation of the 1977 and 1990 classifications rather than undoing their discovery of deep differences between bacteria and archaea. Deep phylogenetic trees depend on the sampled genes, organisms and inference model; the precise closest archaeal relatives and early sequence of events require further evidence. Bacterial ancestry of mitochondria through endosymbiosis is a separate part of eukaryotic history.

The three-domain system recognizes bacteria, archaea and eukaryotes as three domains. In 1990, Woese, Kandler and Wheelis proposed the domain rank and the names Bacteria, Archaea and Eucarya, using molecular comparisons to argue that the older prokaryote/eukaryote division hid important differences. See Woese, Kandler and Wheelis, 1990: Towards a natural system of organisms.

The associated three-domain ancestry model places the nuclear lineage of eukaryotes alongside archaea, rather than within them. This historically influential interpretation should be distinguished from the two-domain system; listing three familiar groups does not establish three independent primary branches.

Online text: diveintosystems.org/book/C11-MemHierarchy/caching.html

Cited passages: Dive into Systems, section 11.4.2: Cache Misses and Associative Designs, compulsory, capacity and conflict misses; Dive into Systems, section 11.4: CPU Caches, opening discussion of hits, misses and filling the cache.

A prion disease is associated with self-propagating abnormal folding of a protein. Its epidemiological interpretation requires distinguishing exposure, infection, clinical onset and diagnostic detection.

A group of metabolic disorders characterized by persistently elevated blood glucose due to inadequate insulin secretion, action or both. Its forms differ in mechanism despite the shared glycemic result.^[ref]

Online text: ocw.mit.edu/courses/6-262-discrete-stochastic-processes-spring-2011/3a19ce0e02d0008877351bfa24f3716a_MIT6_262S11_chap02.pdf

Cited passages: Chapter 2, section 2.2.4, printed pp. 78-79: counting-process definition and its equivalence to exponential interarrival times.

Coordinating CPU and device access to a DMA buffer, including any required memory cache maintenance and ordering. A completed device write must be made visible before the CPU consumes it. The Linux kernel distinguishes coherent mappings from streaming mappings needing synchronization at ownership changes; even coherent mappings can require ordering of writes. ^[ref][ref]