The lower half-plane is the set of complex numbers with negative imaginary part. It is the reflection of the complex upper half-plane across the real axis.

For a compact set $K\subset {\mathbb{C}}^n$, its polynomial hull is the set displayed, where $p$ ranges over all complex multivariate polynomials. It contains $K$. In one complex dimension, bounded complementary components are filled in: the maximum modulus principle bounds every polynomial inside those components by its boundary values. Polynomial approximation and separation distinguish points outside the hull.

A boundary limit taken while approaching within a cone that stays away from tangency. At zero in the complex upper half-plane, every fixed cone $|\mathrm{Re}z|\le A\mathrm{Im}z$ is allowed. A nontangential limit need not be a limit along all paths approaching the boundary.

// Destination: analysis.bigb

Homogeneous mixing assumes that contacts sample the whole modelled population without differences between groups or locations. In the SIR model, the chance that a sampled contact belongs to the infectious compartment is $i=I/N$. This assumption makes a frequency-dependent transmission rule depend on the overall infectious compartment rather than a graph of individual contacts. Structured contacts require a different mathematical model, such as a discrete SIR epidemic on a graph.

Disease prevalence is the proportion of a population that is infected at a specified time. In a susceptible-infective model with total population $N=S+I$, it is $\theta =I/N$.

An SIS model divides a host population into a susceptible compartment and an infectious compartment. Recovery returns hosts directly to susceptibility, without lasting immunity or a separate recovered stage. Let $N$ be a constant population size and $I(t)$ the infectious fraction, so the infectious count is $NI$ and the susceptible count is $N(1-I)$.

For homogeneous frequency-dependent transmission at rate $\beta$ and recovery at rate $\gamma >0$, the count-level infection rate is $\beta (NI)N(1-I)/N$. Dividing the net count rate by $N$ givesThe disease-free state is $I=0$. For ${\mathcal{R}}_0=\beta /\gamma >1$, the positive endemic equilibrium is $I_{\ast }=1-1/{\mathcal{R}}_0$. The derivative of the right-hand side at $I_{\ast }$ is $-(\beta -\gamma )<0$, proving local stability. If ${\mathcal{R}}_0<1$, every positive fraction decreases toward zero; at ${\mathcal{R}}_0=1$, $\dot{I}=-\beta I^2$ gives algebraic decay. The scalar model illustrates a transmission/recovery threshold; heterogeneous contacts or different infectious-duration assumptions require other equations.^[ref]

The epidemiological abbreviation is distinct from the singular-isothermal-sphere abbreviation in galaxy dynamics.

Any deterministic comparison sort correct for all orders of $n$ distinct sort keys needs at least $\lceil {\log }_2(n!)\rceil$ comparisons on some input. Here $n!$ is a factorial, ${\log }_2$ a logarithm, and the brackets denote the ceiling function. This is a worst-case requirement, not a requirement on every input.^[ref]

For three distinct sort keys, the diagram has one leaf for each possible ordering of the original entries $a,b,c$. Some paths need two comparisons, others three; two comparisons cannot distinguish all six orders.

For the proof outline, represent adaptive comparisons by a binary tree: distinct sort keys give only two outcomes per comparison. A leaf fixes an output permutation; one leaf cannot correctly handle two different strict orders. Hence there must be at least $n!$ leaves. Maximum depth $h$ permits at most $2^h$ leaves, so $2^h\ge n!$ gives the bound. Even a three-way comparator has only two reachable outcomes on this distinct-key family.

The growth follows without a precise approximation to the factorial. At least $\lfloor n/2\rfloor$ of its factors are at least $n/2$, so for $n\ge 2$,The brackets denote the floor function. This gives a constant times $n\log n$ lower bound for large $n$, matching the worst-case comparison growth of merge sort.

Distinct sort keys are essential to the $n!$ counting argument. Identical sort keys do not create $n!$ different strict orders. If all sort keys coincide, stable sorting returns the arrival order. This does not remove the worst-case bound for a comparison sort required to handle arbitrary distinct-key inputs.

If $B$ is a commutative $A$-algebra which is a finite free module over $A$, its norm sends $b$ to the determinant of multiplication $m_b:B\to B$. The determinant is independent of the chosen basis and is multiplicative because $m_{bc}=m_b{m}_c$. It commutes with scalar extension. On a separable field extension it becomes the product over embeddings, recovering the field norm. An invertible $b$ has invertible norm.

A combinatorial structure whose elements carry distinct labels from a prescribed finite set. For example, a rooted tree can have labelled vertices; assigning labels is part of the structure counted by an exponential generating function.

Electron impact can excite an ion into a state above an electron-emission threshold, which then ejects a bound electron by autoionization. This two-stage excitation-autoionization contributes to collisional ionization in addition to direct impact removal.

For a discrete memoryless channel, operational capacity equalsRates below $C$ admit codes whose error tends to zero, while rates above $C$ cannot have vanishing error.

A block code assigns source messages to words of a fixed length over an alphabet. Its codewords form a subset of that fixed-length word space; a binary block code uses the two-symbol finite field ${\mathbb{F}}_2$.

A codeword is a member of the specified set of words used to encode messages. For a linear code specified by a parity-check matrix $H$, the codewords are precisely the vectors satisfying $Hc=0$. The minimum distance between distinct codewords controls error correction.

For codes over an alphabet of size $q$, the greedy packing argument gives $A_q(n,d)\ge q^n/V_q(n,d-1)$, where $V_q(n,r)={\sum }_{j=0}^r\left(\genfrac{}{}{0ex}{}{n}{j}\right)(q-1{)}^j$. A maximal distance-$d$ code has radius-$(d-1)$ Hamming balls covering the entire space; counting their union proves the lower bound. The linear refinement is the Varshamov bound for linear codes.

// Target: algebra.bigb

Code encoding maps messages injectively to codewords. For a dimension-$k$ linear code with generator matrix $G$, the standard encoder maps $m\in {\mathbb{F}}_q^k$ to $mG$. The redundancy makes error detection and recovery possible; for a Reed-Solomon code, encoding can instead evaluate a message polynomial at the chosen field elements.

Channel coding represents messages with codewords chosen to withstand transmission errors. An error-correcting code deliberately supplies redundancy, so a receiver can distinguish nearby corrupted outputs and recover the message. Lossless source coding instead removes predictable redundancy; practical communication can use both operations. The Shannon second coding theorem relates achievable rates on a discrete memoryless channel to its channel capacity.

Complex cobordism is the generalized cohomology theory represented by the stable Thom spaces of universal complex vector bundles. Its geometric cycles use maps with chosen stable complex normal data. Its complex Thom classes give Chern classes in complex cobordism, and proper oriented maps have a Gysin map in complex cobordism.

Framed cobordism is the generalized cohomology theory of stable cohomotopy. Its geometric maps have a stable framing of a vector bundle on their normal data. A finite covering has canonically framed zero-dimensional relative tangent data; its Gysin class of $1$ is the stable transfer of the cover. A trivial $d$-sheeted cover gives $d\cdot 1$ in degree zero.

A compact smooth cobordism is an h-cobordism when both boundary inclusions $M_i\hookrightarrow W$ are homotopy equivalences. The name does not itself impose simple connectivity. A trivial cobordism is an h-cobordism because a product cylinder retracts onto either end.

A cobordism is trivial when it is diffeomorphic to a cylinder with the prescribed incoming boundary identification. Both boundary inclusions are homotopy equivalences and its relative homology with respect to either end vanishes. It can have critical points for a chosen cobordism Morse function; those must form an acyclic relative Morse chain complex.