Suppose $S=S_I+S_R$ has finite positive variance $v$, and a matching quota share reinsurance contract exists with fraction ${\alpha }^{\ast }=\sqrt{\mathrm{Var}(S_I)/v}\in [0,1]$. The Cauchy-Schwarz inequality bounds $\mathrm{Cov}(S,S_I)\le {\alpha }^{\ast }v$. Expanding $\mathrm{Var}(S-S_I)$ then proves that the matching quota share minimizes the other party's variance, and hence the sum of party variances. Claimwise retentions $0\le h(x)\le x$ in a compound Poisson distribution automatically satisfy the required variance range, because $\mathrm{Var}(S_I)=\lambda \mathbb{E}[h(X{)}^2]\le \lambda \mathbb{E}[X^2]$.

For an $n$-level qudit and primitive root of unity $\omega$, these unitary operators obey $X^n=Z^n=I$ and $ZX=\omega XZ$. The inverse of $X^s{Z}^{-r}$ is $Z^r{X}^{-s}$, with the order reversed. These operators implement the receiver corrections in qudit teleportation.

The least size of a finite set on which the quaternion group acts faithfully is eight. Every nontrivial subgroup of $Q_8$ contains its central involution $-1$. In an action on fewer than eight points, the orbit-stabilizer theorem makes every stabilizer subgroup nontrivial, so $-1$ fixes every point. This rules out a faithful group action, even with several group orbits. The regular action on eight points is faithful by Cayley theorem.

A quaternion of norm $1$ is a unit quaternion. They form the topological group $\mathrm{Sp}(1)$ and identify with the sphere $S^3\subset {\mathbb{R}}^4$. Multiplication and inversion preserve the norm. The Hurewicz theorem identifies ${\pi }_3(S^3)$ with $H_3(S^3;\mathbb{Z})$; hence homotopy-group addition in a topological group shows that $q\mapsto q^m$ has mapping degree $m$, rather than $m^2$.

A quasi-projective variety is an irreducible variety admitting a locally closed embedding in projective space. Thus it can be written as an open subset of a projective variety. This includes both affine varieties and projective varieties.

An open subset of an affine variety is quasi-projective. A blowup of an algebraic variety along a closed center is also quasi-projective when the original algebraic variety is quasi-projective: its morphism to that base is projective, and a projective morphism over a quasi-projective base has a quasi-projective source. This hypothesis on the base is needed; a blowup of an arbitrary algebraic variety need not provide a quasi-projective example.

The complement of a point in the projective line is the affine line, while the projective line itself remains a projective example. The definition organizes open pieces of projective geometry without requiring all their regular functions to come from one global affine chart.

For each fixed $\delta >0$, the quantum typical subspace projector selects eigenvalues of ${\pi }^{\otimes n}$ between $2^{-n(S(\pi )+\delta )}$ and $2^{-n(S(\pi )-\delta )}$. Its probability tends to one by the weak law of large numbers for the spectral information variable. If its probability is at least $1-ϵ$, its dimension is between $(1-ϵ)2^{n(S(\pi )-\delta )}$ and $2^{n(S(\pi )+\delta )}$. These statements give the size and concentration estimates behind Schumacher compression.

Ideal corrected quantum teleportation transfers an input system to the receiver while preserving its entire joint density operator with an arbitrary reference. Expand a joint pure state as ${\sum }_j|j⟩|r_j⟩$. Every teleportation branch acts with the same known correction on each system basis vector and does nothing to the reference vectors; undoing that correction restores the complete sum. Linearity extends the result to mixed states. The sender's measured carrier no longer retains the original reference correlations.

A sender and receiver share $n^{-1/2}{\sum }_j|jj⟩$. The sender measures the input and their resource half in the generalized Bell basis. Outcome $(r,s)$ gives the receiver $X^s{Z}^{-r}|\chi ⟩/n$, with probability $1/n^2$. After classical communication of the outcome, the correction $Z^r{X}^{-s}$ restores the unknown quantum state. The same calculation preserves correlations with a reference, as in teleportation as an identity channel on a reference.

Two initial maximally entangled pairs $A{B}_1$ and $B_{2}C$ can be turned into an entangled pair $AC$ by a generalized Bell basis measurement on $B_1{B}_2$, followed by a local correction at an endpoint using its classical label. This is qudit teleportation of $B_1$, including its correlations with $A$, to $C$. The initial two pairs are consumed. Without the classical label, averaging the conditional endpoint states need not leave entanglement.

The check loss weights positive errors by $\tau$ and negative errors by $1-\tau$. It is convex, continuous and Lipschitz with constant $\max (\tau ,1-\tau )$. At $\tau =1/2$ it is half the absolute error. Its expected value elicits a quantile: asymmetry determines which probability level the fitted location targets.

If the conditional error distribution function is continuous and strictly increasing with $\tau$-quantile zero, the conditional expected check loss is uniquely minimized at zero fitted displacement. If every incorrect parameter differs from the true regression function on an event of positive probability, averaging these nonnegative conditional risk differences gives strict identifiability of the true parameter. Integrable errors and bounded regression functions supply finite risks.

For two masses $M$ at opposite points of a circular orbit, each at radius $R$ from the centre of mass, ${\Omega }^2=GM/(4R^3)$. The trace-free mass quadrupole moment oscillates at twice the orbital frequency, and its third-derivative squared contraction is $128M^2{R}^4{\Omega }^6$. The quadrupole formula gives the displayed phase-independent luminosity. Using separation $2R$ rather than individual radius $R$ avoids a factor-of-32 error.

The dimensionless orbital compactness $GM/(R{c}^2)$ compares the gravitational length scale with the orbital size. For two equal component masses $M$, each moving at radius $R$ about the centre of mass, the quadrupole formula givesHere the total mass is $2M$ and the separation is $2R$. Newtonian gravity gives $v^2=GM/(4R)$ for each component, so increasing orbital compactness also increases the orbital speed relative to the speed of light. Compact stellar objects permit small noncontact orbits and large gravitational wave luminosities. Near merger, the slow-motion quadrupole formula requires a relativistic replacement.

For a quadratic inflaton potential ending at the leading estimate ${ϵ}_V=1$, $\Phi /M_{\mathrm{Pl}}=\sqrt{4N+2}$. Cosmological horizon exit gives $d\log k/dN=-1+{ϵ}_H$. A narrow logarithmic interval therefore probes a field span $2\Delta \log k/(\sqrt{4N+2}(1-{ϵ}_H))$, to the accuracy of the slow-roll approximation. At sixty e-folds, five logarithmic wavenumber units probe about $0.64M_{\mathrm{Pl}}$ around a background field near $15.6M_{\mathrm{Pl}}$.

For a p-adic field with odd residue characteristic, the square map is an isomorphism on $1+{\mathfrak{m}}_K$ by Hensel lemma. The parity of the valuation and the square class of the residue unit therefore determine the square-class group of a field. Its three nontrivial classes give the three quadratic extensions: one unramified extension and two totally ramified extensions.

Trim and fill estimates missing studies from funnel plot asymmetry by trimming extreme studies to locate a centre and filling in counterparts on the less represented side. The adjusted meta-analysis includes these imputed estimates. Its interpretation depends on a symmetry model for missing evidence; it is not a guarantee that genuine publication bias has been removed.

A selection model combines a distribution for all study results with a probability $\pi (y)$ that a result becomes available. The observed distribution is reweighted by this probability and renormalized. Assumed or estimated selection mechanisms permit sensitivity analysis of a meta-analysis, but unavailable results generally prevent a fully assumption-free correction.

This flavour combination is the invariant scalar in the octet and singlet in a fundamental SU3 tensor product. Its isospin and flavour hypercharge vanish. Although it shares the central weight with the Eta octet state, it lies in a different irreducible representation. Physical eta mixing and the singlet axial anomaly distinguish the mass eigenstates from the pure flavour directions.

These neutral isospin-zero pseudoscalar mesons have $J^{PC}=0^{-+}$. Their light-quark components mix the Eta octet state and eta singlet state. The singlet axial anomaly contributes to the unusually large eta-prime mass, while flavour breaking also affects both masses and the mixing. The simple orthogonal one-angle flavour model is useful for explaining the nonet; detailed decay-constant descriptions can require more than one parameter.

A propositional type is a collection of propositional formulas to be jointly realized by a Boolean valuation. A valuation realizes it when every member is true and omits it when some member is false. For propositional omission arguments, the collection need not itself be consistent.