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Allocation concealment prevents those deciding whether to enroll an eligible participant from knowing the upcoming randomized assignment. It protects the randomized controlled trial against selection based on that assignment. It concerns access to the assignment before enrollment; keeping participants or assessors unaware after assignment is a separate design feature.

For a geometric distribution count on the positive integers, independent of independent and identically distributed random variables $X_i$, the moment-generating function of $S={\sum }_{i=1}^N{X}_i$ is the displayed expression. Condition on the count and sum a geometric series. For real $t$, the finite domain requires $M_X(t)<\infty$ and $(1-p)M_X(t)<1$. Positive summands always permit $t\le 0$, even when no positive exponential moment exists; this gives a Laplace transform of a nonnegative random variable. The aggregate has no mass at zero when all summands are strictly positive.

For independent uniform couplings on $[0,g_{\ast }]$ and equatorial initial bath spins,At fixed finite $N$, the mean tends to zero and the mean square tends to $2^{-N}$ as time grows. At any fixed nonzero time, the mean-square bracket is strictly below one; the Markov inequality proves coherence tends to zero in probability as $N$ grows. Individual finite realizations still exhibit finite spin-bath coherence recurrence.

For initially equatorial spins and uniform couplings on $[0,g_{\ast }]$, expand $\log \cos (2g_{k}t)=-2g_k^2{t}^2+O(g_k^4{t}^4)$. The strong law of large numbers then gives $\sum g_k^2\simeq N{g}_{\ast }^2/3$. On the scale $t=O((g_{\ast }\sqrt{N}{)}^{-1})$, the total fourth-order remainder tends to zero, giving the Gaussian coherence envelope. This approximation concerns early times and does not remove finite spin-bath coherence recurrence.

The whole space of infinite subsets of the natural numbers is a meagre set in the Ramsey cone topology. For $D_j=\{X:\min X=j\}$, the closure is $\{X:j\in X\}$, which has empty interior: every infinite ground set can be thinned to omit $j$. Thus all $D_j$ are nowhere dense, while their countable union is the whole space. In contrast, the whole space is not meagre in either the Ellentuck topology or the ordinary topology on infinite subsets.

For $L={\mathbb{Q}}_p({\zeta }_{p^n})$, the uniformiser ${\zeta }_{p^n}-1$ and the uniformizer criterion for lower ramification groups give the displayed valuation for $a\ne 1$. Thus $G_0$ is the full unit group modulo $p^n$, and $G_i$ consists of units congruent to one modulo $p^j$ when $p^{j-1}\le i\le p^j-1$, $1\le j<n$. All groups are trivial for $i\ge p^{n-1}$. The formula includes $p=2$, where the first congruence subgroup is already the full group.

For any aggregate with positive finite variance $v$, quota share reinsurance with retained fraction $\alpha$ has total party variance $v({\alpha }^2+(1-\alpha {)}^2)=v/2+2v(\alpha -1/2{)}^2$. Equal sharing uniquely minimizes this objective. The two party totals remain dependent; adding their individual variances is a different objective from the fixed variance of their sum.

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 $m^2$.

An irreducible variety which admits a locally closed embedding into projective space. Open subsets of affine varieties and blowups of algebraic varieties along closed centres give important examples.

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 ratio of gravitational radius to orbital scale measures how relativistic a bound orbit is. In an equal-mass circular binary with individual orbital radius $R$, the leading quadrupole formula gives $P=(2/5)(c^5/G)[GM/(R{c}^2){]}^5$. Compact objects permit smaller noncontact orbits than extended stars and hence larger orbital compactness; near merger the Newtonian approximation must be replaced.