If $X,Y$ are independent and identically distributed random variables with a centered multivariate normal distribution and covariance matrix $\Sigma$, then $(U_{\theta },V_{\theta })$ has the same probability law as $(X,Y)$. Its two marginal covariance matrices are $\Sigma$ and its cross-covariance is zero. The pair has a multivariate normal distribution, so the two components are independent. This remains valid when $\Sigma$ is singular.

For a nonnegative integer exponent, exponentiation iterates multiplication: $a^0=1$ and $a^{b+1}=a{a}^b$. This convention defines $0^0=1$ in primitive recursive arithmetic. Extensions to other exponents may use different domains and conventions.

The identity morphism on an object $A$ is the neutral element for composition in a category: $f{1}_A=f$ and $1_{A}g=g$ whenever the composites are defined.

Nested Monte Carlo approximates an outer expected value whose integrand contains another expected value. A nonlinear transformation, such as a logarithm, of an inner sample average usually introduces finite-sample bias. Increasing both sample sizes requires suitable integrability controls to establish consistency; an unbiased estimator of the inner quantity does not make its logarithm unbiased.

Independent uniform distribution draws $U_1,U_2$ on $(0,1)$ produce independent standard normal distribution draws $X_1=\sqrt{-2\log U_1}\cos (2\pi U_2)$ and $X_2=\sqrt{-2\log U_1}\sin (2\pi U_2)$. The proof transforms an independent uniform angle and radial density $r{e}^{-r^2/2}$ through the polar-coordinate Jacobian.

Replacing a simulated quantity $H$ by $E[H\mid \mathcal{G}]$ preserves its expectation and reduces variance by the law of total variance. This is Rao-Blackwellization. For a random sum, conditioning on the count and all but one summand replaces an event indicator by a cumulative distribution function evaluation. The resulting estimator needs that conditional expectation to be computable.

For a modular machine whose $(0,0)$ is terminal, $H_0(\mathcal{M})$ is the set of configurations whose forward computation reaches exactly $(0,0)$, including the zero-step computation there. It may differ from the set of configurations stopping at any terminal configuration. Determinism makes membership invariant along each instruction edge.

A modular machine can be encoded by HNN extensions of $K={\mathbb{Z}}^2\ast ⟨t⟩$. The kernel of $K\to {\mathbb{Z}}^2$ has a free basis of a group $t(r,s)=x^{-r}{y}^{-s}t{x}^r{y}^s$. Associated-subgroup maps send these basis elements according to the two arithmetic transition types. In the resulting group, membership of $t(r,s)$ in the subgroup generated by $t$ and the instruction stable letters is equivalent to $(r,s)\in H_0(\mathcal{M})$. One more HNN extension centralizes that finitely generated subgroup, converting membership to equality and hence to the word problem for a group.

Two first-order structures are elementarily equivalent when they satisfy the same first-order sentences. They need not be isomorphic or embed into one another. Every elementary embedding implies elementary equivalence of its source and target.

Given nonempty first-order structures ${\mathcal{M}}_i$ and a proper filter on a set $F$, a reduced product identifies product functions $f,g$ when $\{i:f(i)=g(i)\}\in F$. Operations are interpreted coordinatewise, and relations hold when their coordinate truth sets belong to $F$. For an ultrafilter this is an ultraproduct. Primitive positive formulas transfer through an arbitrary proper filter, whereas logical disjunction and logical negation can fail to transfer.

For a first-order theory $T$ and $A\models T_{\forall }$, an algebraically prime extension is a structure embedding $i:A\to P\models T$ such that every embedding $j:A\to N\models T$ factors as $j=h\circ i$ for some structure embedding $h:P\to N$. The embeddings need not be elementary or unique. A theory has algebraically prime models when every such $A$ has an extension of this kind.

This first-order theory has as models the nonzero torsion-free divisible Abelian groups. Rational division equips them with the structure of vector spaces over the rational numbers. A rational divisible hull gives an algebraically prime extension of each nonzero torsion-free base, while the zero base uses a one-dimensional rational space.

The theory of formally real fields consists of the field axioms and all sentences asserting that $-1$ is not a sum of $m$ squares, one for each positive integer $m$. The real closure theorem and model completeness make the Theory of real closed fields its model companion.

In each round the Spoiler selects an element from either structure and the Duplicator responds in the other. The paired selected elements must preserve all atomic information and equality, giving a partial isomorphism of structures. In the infinite game the Duplicator wins by avoiding failure at every finite stage. In a timed game the Spoiler first announces the number of rounds, allowing a different response strategy for each horizon.

A finitary algebraic theory specifies sorts, finite-arity operations and equations between terms. Its internal models can be interpreted in any category with finite products. Finitely presented set-based models generate arbitrary models under filtered colimits and determine its classifying topos.

A recursive presentation identifies the domain of a countable structure with a computable set, with its basic operations given by total computable functions and its basic relations decidable. In a fixed finite signature this makes evaluation of atomic formulas effective. A countable structure may have a recursive presentation of a structure only for some choices of presentation, or none at all.

The mechanism assigns probabilities to observation patterns $R$ conditional on complete measurements $D$: $p(R\mid D)$. Whether it depends on missing values distinguishes missing completely at random, missing at random and missing not at random.

With canonical tree-level global supersymmetry, neutral F-term breaking and no D-term mass shifts, the tree-level supertrace mass sum rule applies in conserved electric and color charge blocks. The corresponding scalar squared-mass average equals the fermion squared-mass average, so not every scalar partner can lie above that average. This obstructs making every squark heavy through direct canonical visible-sector breaking while retaining light quarks. Effective soft supersymmetry breaking, noncanonical interactions, radiative effects or supergravity invalidate the assumptions of that argument.

In the MSSM, each fermion generation has chiral superfields $Q:(3,2,1/6)$, $U^c:(\bar{3},1,-2/3)$, $D^c:(\bar{3},1,1/3)$, $L:(1,2,-1/2)$, and $E^c:(1,1,1)$. This uses hypercharge normalized by $Q_{\mathrm{electric}}=T^3+Y$ and only left-handed Weyl spinors. Two Higgs chiral doublets have charges $\pm 1/2$. The three vector superfields transform as $(8,1,0)$, $(1,3,0)$ and $(1,1,0)$, supplying the gauge bosons and gauginos.

A Higgs chiral doublet is a weak-doublet chiral superfield containing a Higgs complex scalar field, a higgsino and an auxiliary field. In the MSSM, $H_u$ has hypercharge $+1/2$ and $H_d$ has hypercharge $-1/2$. Their paired charges enable higgsino anomaly cancellation and the holomorphic need for two Higgs chiral doublets.