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Given finite lists ${\ell }_0,{\ell }_1,\dots$ in a set $X$, their union of entries has an explicit enumeration from a subset of ${\mathbb{N}}^2$: use the existing positions in each list and a Cantor pairing function. If the union is infinite, successively take the first occurrence of an entry not previously taken, in the order of its natural-number code. The well-order on natural numbers makes each step uniquely determined and yields an injection $\mathbb{N}\to X$. No axiom of choice is needed. This does not assert that an arbitrary countable family of unordered finite sets has a countable union in ZF without the axiom of choice.

A closed universe containing only a cosmic string network can remain static. Its Friedmann acceleration equation vanishes identically because $P_s=-{\rho }_s/3$, and its Friedmann equation requires ${\rho }_s{a}^2=3K/(8\pi G)$. Once this conserved normalization is tuned, every positive radius gives a static solution. The usual Hubble-normalized cosmological density parameter is undefined at a static epoch.

Cosmic variance is the sampling uncertainty caused by observing only one realization of a cosmological random field. For a Gaussian isotropic full-sky temperature field and an ideal noiseless estimate, $\mathrm{Var}({\hat{C}}_{\ell })=2C_{\ell }^2/(2\ell +1)$ because there are only $2\ell +1$ independent real angular degrees of freedom.

A helium flash is a thermally unstable ignition of helium in an electron-degenerate stellar core. A nondegenerate intermediate-mass core can expand and regulate ignition, so a usual isolated star of five solar masses begins core helium burning without this flash.

The five-dimensional Horn copositive matrix has diagonal entries $1$, entries $-1$ on the edges of the five-cycle, and entries $1$ on the remaining pairs:For $x\ge 0$, a cyclic relabelling puts a smallest coordinate at $x_5$. The identitythen proves that $H$ is a copositive matrix.

However, $H$ is outside the positive-semidefinite-plus-nonnegative cone. Set $w=(1,2,1,0,0{)}^T$. Its quadratic form is zero. If $H=P+N$ with $P$ a positive semidefinite matrix and $N$ a symmetric nonnegative matrix, both $w^{T}Pw$ and $w^{T}Nw$ must vanish. Positivity of the first three coordinates of $w$ forces every entry of the leading $3\times 3$ block of $N$ to vanish. Applying the argument to all cyclic shifts of $w$ forces every entry of $N$ to vanish, since each pair of indices lies in a cyclic interval of length three. This would make $H$ a positive semidefinite matrix, but zero quadratic form of a positive semidefinite matrix would then give $Hw=0$, whereas $Hw=(0,0,0,2,2{)}^T$.

The sums $P+N$ of a real positive semidefinite matrix $P$ and a symmetric nonnegative matrix $N$ form a convex cone inside the copositive cone. Both terms have nonnegative quadratic forms on the nonnegative orthant. The Horn copositive matrix shows that the inclusion is strict in dimension five; the sum of squares criterion for a biquadratic form explains this cone's relation to semidefinite programming.

If the partial sums of $\sum b_n$ are bounded and $a_n$ decreases to zero, then $\sum a_n{b}_n$ converges. The uniform version holds for functions $b_n(x)$ when their partial sums have one bound $M$ independent of $x$. Indeed partial sums starting at index $N$ have absolute value at most $2M$, and summation by parts bounds every weighted tail from $N$ to $K$ by $2M{a}_N$. The uniformly Cauchy sequence criterion proves uniform convergence. This justifies integrating a conditionally convergent Fourier series on intervals staying away from its endpoint jump.

For a bounded linear regularization operator $R_{\alpha }$ and noisy data with $\Vert f_{\delta }-f\Vert \le \delta$, the triangle inequality givesThe first term is noise amplification; the second is the approximation bias on exact data. Thus $\alpha (\delta )\to 0$ and $\delta \Vert R_{\alpha (\delta )}\Vert \to 0$ suffice for a convergent regularization of an inverse problem, provided $R_{\alpha }$ is consistent on exact data.

If $X_n\stackrel{d}{\to }c$ for fixed real $c$, convergence of the distribution functions at $c-\epsilon$ and $c+\epsilon$ shows $\mathbb{P}(|X_n-c|>\epsilon )\to 0$. Thus convergence in distribution to a deterministic limit implies convergence in probability to that limit, even though the implication fails for nonconstant limits on a specified probability space.

A sequence of random variables $(X_n)$ is bounded in probability when ${\lim }_{K\to \infty }{\sup }_n\mathbb{P}(|X_n|>K)=0$. A sequence converging in probability to a finite random variable has this property, and its product with a sequence tending to zero in probability also tends to zero in probability.

Convergence in L1 means that the expected value of the absolute error tends to zero. It implies convergence in probability and convergence of expected values.

If a smooth posterior density has vanishing boundary terms, its log-posterior gradient satisfies $\mathbb{E}[g]=0$ andThis is a posterior integration-by-parts identity; it concerns differentiation in the random parameter, distinct from the usual mean-zero score identity for sampling distributions. It makes $g$ a vector control variate. Finite moments and known coefficients preserve unbiasedness of the Monte Carlo estimator.

A continuous martingale $M$ is L2-bounded when ${\sup }_{t\ge 0}\mathbb{E}|M_t{|}^2<\infty$. The L2 martingale convergence theorem gives a terminal value $M_{\infty }$ with convergence in $L^2$ and almost sure convergence. The resulting conditional expectation identity is $M_t=\mathbb{E}[M_{\infty }\mid {\mathcal{F}}_t]$. Boundedness on each separate finite horizon is a weaker condition. The subspace with $M_0=0$ is commonly denoted ${\mathcal{M}}_0^2$.

A hypoexponential distribution is the distribution of a sum of independent variables with exponential distributions. For two positive unequal rates $\lambda ,\mu$, convolution of independent random variables gives the probability density functionIt is zero for negative $s$. Its equal-rate limit is the shape-two gamma distribution with rate $\lambda$.

A continuous map injective map $f:X\to Y$ that is a homeomorphism onto its image with the subspace topology. A continuous map injective map from a compact space into a Hausdorff space is a topological embedding.

The operator range is the image of a linear map $K$. It need not be closed in an infinite-dimensional Hilbert space; that failure makes the Moore–Penrose inverse of an operator unbounded on its domain.

A replicating strategy is a self-financing portfolio whose terminal wealth equals the prescribed payoff of a contingent claim. Admissibility specifies the permitted wealth bounds and integrability of its holdings.

A right-linear grammar has productions of the form $A\to wB$ or $A\to w$, where $A,B$ are nonterminals and $w$ is a terminal word, possibly empty. Each right-hand side has at most one nonterminal, at its right end. Such grammars generate exactly regular languages: regard nonterminals as states and expand each terminal word into a finite path to another state or an accepting endpoint. Conversely the transition graph of a deterministic finite automaton gives these productions directly.

For a transitive model $M$ of ZFC and $\alpha \in M\cap \mathrm{Ord}$, the internally computed constructible hierarchy agrees with the ambient level: $L_{\alpha }^M=L_{\alpha }$. Satisfaction for a fixed set structure is computed from the same finite formulas and the same domain in both universes. Transfinite recursion then proves agreement at successors and limits. If $\theta =M\cap \mathrm{Ord}$, it follows that $L_{\theta }\subseteq M$.

For the flux $W_{ij}=-M{\partial }_i{\mu }_j+N_{ij}$ of a vector order parameter, microscopic reversibility requires independent Gaussian white noise components with covarianceThe current changes sign under time reversal. The resulting path-probability ratio contributes $-(k_{B}T{)}^{-1}\int W_{ij}{\partial }_i{\mu }_j$ to its logarithm; integration by parts converts this to the free-energy loss from the conserved dynamics.