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Doubling ${\mathbb{RP}}^3$ minus an open ball along its spherical boundary gives ${\mathbb{RP}}^3\#{\mathbb{RP}}^3$. Its fundamental group isand its universal cover is $S^2\times \mathbb{R}$.

The integral homology groups are

The complement of an open three-ball in ${\mathbb{RP}}^3$ is the mapping cylinder of the antipodal covering $S^2\to {\mathbb{RP}}^2$. It deformation retracts to ${\mathbb{RP}}^2$ and has boundary $S^2$.

The mod-two cohomology ring of real projective space is

A projective Cremona transformation is a birational self-map of projective space, regular on complementary dense open subsets together with its inverse.

Composition of these birational self-maps forms the Cremona group.

The indeterminacy locus is the set where a rational map has no regular local representative.

Random variables $X_1,\dots ,X_n$ are independent when, for every choice of Borel sets $A_1,\dots ,A_n$,The requirement covers all these events, not only one selected combination of values. Equivalently, the generated sigma-algebras are independent sigma-algebras, or the joint probability distribution is the product measure of the marginal distributions. For an infinite family, the same condition must hold for every finite subfamily. For random elements, replace Borel sets by measurable sets in their specified target spaces.

For discrete random variables, checking every entry of the joint probability mass function is enough:Summing this identity over the desired sets proves the event formulation. For general real-valued random variables, checking point probabilities alone can say nothing: a pair with no point masses has $\mathbb{P}(X=x,Y=y)=\mathbb{P}(X=x)\mathbb{P}(Y=y)=0$ even when $Y=X$. The all-set definition avoids that trap. See Gautam Iyer, Conditional Probability and Independence, definitions 12, 16 and 18 and proposition 19, with the continuous case treated in his ^[ref].

For a concrete comparison, let $X,Y$ be independent fair bits: each has a Bernoulli distribution with parameter $1/2$. Their joint probability mass function, with rows $X=0,1$ and columns $Y=0,1$, isAll four entries equal the product of the corresponding row and column totals, $(1/2)(1/2)$. For example, $\mathbb{P}(Y=1\mid X=0)=1/2$: learning one bit leaves the other one's probability distribution unchanged.

Now keep a fair $X$ but set $Y=X$. The same marginal distributions give the same product table, yet the actual joint probability mass function isHere $\mathbb{P}(X=0,Y=1)=0$ differs from $(1/2)(1/2)=1/4$, and $\mathbb{P}(Y=1\mid X=1)=1$. Thus matching marginal distributions does not establish independence. Grinstead and Snell formulate the discrete product criterion in Introduction to Probability, section 4.1, definition 4.4, printed page 143.

Independence is stronger than zero covariance. For square-integrable independent random variables, $\mathbb{E}[XY]=\mathbb{E}[X]\mathbb{E}[Y]$, so they are uncorrelated random variables. The converse fails: take $X$ with a discrete uniform distribution on $\{-1,0,1\}$ and $Y=X^2$. Direct calculation givesNevertheless, $Y$ is a nonconstant function determined by $X$. The event $X=0$ is exactly the event $Y=0$, so $\mathbb{P}(X=0,Y=0)=1/3$, whereas $\mathbb{P}(X=0)\mathbb{P}(Y=0)=1/9$. The positive and negative contributions to the covariance cancel without erasing the statistical dependence.

The usefulness of independence is that a large joint probability distribution can be assembled from small pieces. For $n$ fair independent bits, each of the $2^n$ patterns has probability $2^{-n}$. Products also simplify expected values, and vanishing cross covariances simplify the variance of a sum. Separate measurable functions of independent random variables remain independent: each output event is already an input event, so the defining product rule still applies. These are consequences of a model's independence assumption; whether the assumption describes actual observations is a question for statistical inference.

A random vector $X=(X_1,\dots ,X_d)$ is a measurable map from one probability space to ${\mathbb{R}}^d$, equivalently a tuple of real random variables defined on the same space. Its joint probability distribution contains their dependence information. For example, if $B$ is a fair Bernoulli random variable, $(B,B)$ takes only $(0,0)$ and $(1,1)$, whereas two independent fair Bernoulli coordinates take all four pairs. Both have identical coordinate marginal distributions. A mean vector and covariance matrix summarize the first two moments when finite but generally do not determine the whole law.

An Exchangeable random partition has a law invariant under finite permutations of the underlying labels.

A Singleton block is a block containing exactly one element.

A restricted-growth string encodes a partition of the ordered label set $\{1,\dots ,n\}$ by numbering blocks in the order in which they first appear. For a nonempty string $r_1,\dots ,r_n$, the rule isEqual entries identify elements in the same block. For example, $12133$ encodes $\{\{1,3\},\{2\},\{4,5\}\}$. At the next position one may reuse an existing block number or introduce exactly the next unused number; skipping a new number would break the convention.

Every set partition has one such canonical encoding, so these strings enumerate set partitions without repeatedly counting permutations of the block names. Their number is the Bell number $B_n$, and the number with exactly $k$ distinct labels is the Stirling number of the second kind $S(n,k)$.

In the $G(n,p)$ model, each of the $\left(\genfrac{}{}{0ex}{}{n}{2}\right)$ possible edges is included independently with probability $p$.

An isolated vertex has degree zero.

A homogeneous linear equation is partition regular when every finite colouring of the positive integers admits a solution whose coordinates all have one colour.

The equation has partition regularity when every finite colouring admits such a monochromatic solution.

For every colouring of the $r$-element subsets of an infinite set, there is an infinite increasing sequence $X$ and a coordinate set $I\subseteq \{1,\dots ,r\}$ such that two increasing $r$-tuples from $X$ have the same colour exactly when they agree in every coordinate indexed by $I$.

For a finite graph $G$, its two-colour Ramsey number $R(G)$ is the least $n$ such that every red-blue colouring of $K_n$ contains a monochromatic copy of $G$. It exists because a monochromatic clique on $|V(G)|$ vertices contains a copy of $G$.

The diagonal Ramsey number $R(t)$ is the least $n$ such that every red-blue edge-colouring of $K_n$ contains a monochromatic $K_t$.

Fix positive integers $r,k,m$ with $m\ge r$. There is an integer $n\ge m$, depending only on $r,k,m$, such that every finite coloringhas a subset $H\subseteq [n]$ of size $m$ for which the restriction of a function $c$ to $\left(\genfrac{}{}{0ex}{}{H}{r}\right)$ is a constant function. Here $[n]=\{1,\dots ,n\}$, and $\left(\genfrac{}{}{0ex}{}{S}{r}\right)$ denotes the collection of all $r$-element subsets of $S$, rather than its binomial coefficient cardinality. The notation $[n]⟶(m{)}_k^r$ abbreviates this property. Thus $H$ is a homogeneous set for a colouring; when $r>1$, the monochromatic set consists of its $r$-element subsets, not its individual elements. This is the finite assertion in Ramsey's original Theorem B, p. 267.

For $r=1$, this is the pigeonhole principle, with sharp threshold $n=k(m-1)+1$. For instance, every red-blue coloring of five objects has three of one color; four objects can split two and two. For $r=2$, $k=2$ and $m=3$, the sharp threshold is six, as shown by the Ramsey number for triangles. Coloring pairs makes this a problem about edges of a complete graph. For larger $r$, the analogous viewpoint uses a hypergraph.

The finite Ramsey theorem lets an existence argument discard most of a large finite set while retaining a prescribed amount of uniform structure. Its threshold applies to every coloring; one does not need an assumption that colors were assigned randomly. A diagonal compactness argument deduces finite existence from Ramsey's theorem, but does not by itself identify the smallest threshold. For pairs, a Ramsey number makes that optimization explicit.

Take plane-parallel optical depth ${\tau }_{\nu }$ increasing inward and an outward ray with $\mu >0$. The radiative transfer equation $\mu d{I}_{\nu }/d{\tau }_{\nu }=I_{\nu }-S_{\nu }$ has the integrating-factor solutionThe first term is radiation transmitted from the lower boundary; the integral adds emission along the ray, weighted by attenuation to the surface. In local thermodynamic equilibrium with thermal absorption and emission, $S_{\nu }=B_{\nu }[T(t)]$, giving the familiar Planck-function expression. Absence of scattering alone does not imply that replacement when the material is out of local thermodynamic equilibrium. The weighting favors depths whose line-of-sight optical depth is of order one, while a sufficiently deep lower boundary is exponentially suppressed.

Local thermodynamic equilibrium assumes matter populations follow thermal distributions at the local temperature: excitation follows the Boltzmann distribution and ionization the Saha equation. It permits a temperature gradient; it does not require the radiation field itself to be a blackbody.

For true absorption and its inverse thermal emission, $j_{\nu }^{\mathrm{abs}}/{\alpha }_{\nu }^{\mathrm{abs}}=B_{\nu }(T)$, the Planck function. The total source function need not equal $B_{\nu }$ when scattering contributes. Distinguishing matter equilibrium from radiation equilibrium is essential near a star's escaping surface.^[ref]