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The order-$k$ Krylov subspace generated by $A$ and $b$ is

An objective time derivative transforms covariantly under time-dependent rigid changes of observer, so constitutive predictions do not depend on the observer's rotation.

The worldsheet metric equation in conformal gauge sets the stress tensor to zero, $T_{++}=T_{--}=0$. In oscillator language these are the Virasoro constraints.

After conformal gauge fixing, independent reparameterizations ${\sigma }^{+}\mapsto f({\sigma }^{+})$ and ${\sigma }^{-}\mapsto \tilde{f}({\sigma }^{-})$ remain, accompanied by a compensating Weyl transformation.

If $U,V,X$ are independent random variables in ${\mathbb{F}}_2^n$, thenIndeed, conditioning reduces entropy and independence giveIn ${\mathbb{F}}_2^n$, $V=U+(U+V)$, so conditional entropy under a deterministic change of variables shows that the left-hand side is unchanged when $U$ and $V$ are exchanged. Averaging the displayed bound with its exchanged version gives the result.

The simultaneous conditional distance samples one value of the conditioning variable for both arguments:It differs in general from $d_R(X\mid U;Y\mid U)$, which uses two independently sampled conditioning values.

A rearrangement changes the order of a series without changing its multiset of terms. A conditionally convergent real series can be rearranged to approach any prescribed real limit or to diverge.

The Riemann series theorem specifies the possible sums of rearrangements of a conditionally convergent real series.

If $a_n\ge 0$ decreases to zero, then ${\sum }_{n=1}^{\infty }(-1{)}^n{a}_n$ converges.

A centered random variable $X$ is sub-Gamma in the right tail with variance parameter $v$ and scale parameter $c$ whenfor $0\le \lambda <c^{-1}$.

A centered random variable $X$ is sub-Poisson in the right tail with variance parameter ${\sigma }^2$ whenfor every $\lambda \ge 0$.

A transport-entropy inequality controls an optimal expected transportation cost between probability distributions by their Kullback-Leibler divergence.

For a nonnegative function $Z$ of independent random variables, entropy is at most the sum of its conditional coordinate entropies. This tensorization is a basic step of the entropy method.

A nonnegative function $f$ of independent coordinates is self-bounding when there are coordinate-deleted versions $f_i$ such that $0\le f-f_i\le 1$ andThese inequalities make the natural variance proxy no larger than the function itself.

An integer-valued function $f$ is $g$-certifiable when every input $x$ with $f(x)=k$ has a set of at most $g(k)$ coordinates whose values alone guarantee that $f\ge k$.

A function $f(x_1,\dots ,x_n)$ has bounded differences with constants $c_i$ when changing only $x_i$ changes its value by at most $c_i$.

For every $\lambda >0$, Markov inequality applied to $e^{\lambda X}$ givesOptimizing over $\lambda$ gives the Chernoff bound in terms of the moment-generating function of $X$.

The entropy of a binary mixture also satisfieswhere $H$ is binary entropy. This follows by applying the operator monotonicity of logarithm to $p{\rho }_1\le p{\rho }_1+(1-p){\rho }_2$ and its counterpart for ${\rho }_2$.

The complex unit circle is the subgroup $\{z\in \mathbb{C}:|z|=1\}$ under multiplication.

A model is aleph-zero-homogeneous when every finite partial elementary map extends by one more element.

With its standard topology, this type space is a Stone space. The type space $S_n^M(A)$ is the set of complete $n$-types over a parameter set $A$ that are consistent with the complete theory of $M$ together with its diagram over $A$.