A quantum channel is anti-degradable when its receiver output can be simulated from the complementary environment output. Data processing then forces every coherent information to be nonpositive, and no-cloning prevents perfect quantum transmission.

Suppose $k_{\tau }$ is a positive-semidefinite kernel for every $\tau$ and its diagonal entries are integrable. Positivity of each $2\times 2$ kernel matrix givesThe Cauchy-Schwarz inequality makes every entry integrable, and integrating each nonnegative finite quadratic form shows that $\int k_{\tau }d\tau$ is again a positive-semidefinite kernel.

A free closed-string embedding coordinate is the sum of center-of-mass zero modes and independent periodic left- and right-moving oscillator expansions.

For an ensemble $\{p_x,{\rho }_x\}$ with average $\bar{\rho }$, the Holevo quantity is $\chi =S(\bar{\rho })-{\sum }_x{p}_{x}S({\rho }_x)=I(X:B)$ of the associated classical-quantum state.

The adele ring is the restricted product of the completions $K_v$ with respect to the valuation rings ${\mathcal{O}}_v$ at finite places.

The conductor ${\mathfrak{f}}_{L/K}$ is the smallest modulus through which the Artin reciprocity map for a finite abelian extension $L/K$ factors. Its finite prime divisors are exactly the ramified primes, with exponents measuring the depth of ramification.

Two finite Borel measures on $\mathbb{R}$ with the same characteristic function are equal. One proof convolves both measures with centered Gaussians, applies Fourier inversion to the smooth densities, and then lets the Gaussian variance tend to zero.

A virtual character is an integer linear combination of characters, equivalently an element of the Grothendieck group of finite-dimensional representations.

A simple sufficient independent censoring assumption is that the potential observation-ending time $C$ and survival time $T$ are independent random variables. More generally, appropriate conditional independence given the covariates used in the analysis can suffice for analysis within those conditions. The requirement concerns the potential times, rather than independence of the observed duration $Y=\min (T,C)$ and event indicator $\delta$: those recorded quantities ordinarily depend on each other.

For an ordinary pooled Kaplan–Meier estimator, the observed risk set must represent the future event experience of everyone still event-free at each time. Conditional independence given a covariate does not alone justify ignoring that covariate in a pooled calculation. Also, there must be a positive probability of remaining observed through the horizons being estimated. If follow-up ends at a common closing time, it supplies no observations to identify the later survival function. Independent subjects with a common survival distribution and independent censoring form the basic model in Lu Tian's STAT331 notes, Unit 3, pages 1–4.

For example, subjects who stop attending because their condition is deteriorating can produce informative censoring. A predetermined common closing date can instead give administrative censoring, provided calendar entry and other relevant conditions do not select future survival times. The reason for stopping observation must be justified from study design and context; the recorded event indicators alone cannot verify independent censoring.

A distribution is faithful to a Directed acyclic graph when, for all disjoint vertex sets $A,B,S$, $Z_A\perp Z_B\mid Z_S$ holds if and only if $A$ and $B$ are D-separated by $S$. This includes the graph's global Markov property of a directed acyclic graph and excludes additional independences caused by parameter cancellations. If faithfulness is used to name only the reverse implication, the Markov property must be assumed separately.

A distribution factorizes according to a Directed acyclic graph when its joint density is the product of the conditional density of each variable given its graphical parents. This factorization implies the graph's global Markov property of a directed acyclic graph.

The future or past Cauchy horizon of a partial Cauchy hypersurface is the boundary of the region whose physical evolution is determined by data on that hypersurface. Crossing it can destroy uniqueness of the maximal globally hyperbolic development.

An initial object has a unique morphism to every object.

A monomorphism $m:A\to B$ is left-cancellative: $mu=mv$ implies $u=v$. It represents a subobject of $B$.

A pointed category has a zero object, hence a distinguished zero morphism between every pair of objects.

Regard the proper class of all ordinals as a poset category with $\alpha \to \beta$ when $\alpha \ge \beta$ in the usual order. It is locally small and complete: the product of any set-indexed family is its ordinary supremum, and equalizers are automatic in a poset. It has no initial object because there is no largest ordinal. Consequently every representable functor from it to sets preserves all small limits but has no left adjoint, since a left adjoint would have to send the empty set to an initial object.

A category is connected when it is nonempty and any two objects are joined by a finite zigzag of morphisms after forgetting their directions.

The opposite category reverses every morphism while retaining the same objects. Limits in ${\mathcal{C}}^{\mathrm{op}}$ are colimits in $\mathcal{C}$, and conversely.

A category is locally small when the morphisms between each fixed pair of objects form a set, even if its collection of all objects is a proper class.

A full subcategory $\mathcal{D}\subseteq \mathcal{C}$ is reflective when its inclusion has a left adjoint $L:\mathcal{C}\to \mathcal{D}$, called the reflector. Equivalently, every object $A$ has a universal morphism $A\to LA$ into an object of $\mathcal{D}$.