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A one-particle-irreducible Feynman diagram is a connected Feynman diagram that remains connected after any one internal line is cut. The vertices of the quantum effective action generate precisely these diagrams.

These Feynman diagrams generate the vertices of the quantum effective action.

For a subgroup $H$ of a group $G$, the normalizer isIt is the stabilizer subgroup of $H$ under the conjugation action on subgroups. Its elements can permute members of $H$; they need not commute with them. For $r=(123)$ in the symmetric group $S_3$, the cyclic subgroup $H=⟨r⟩$ has normalizer $S_3$, although the centralizer of $r$ has only three elements, as illustrated in centralizer and normalizer.

Every element of $H$ preserves $H$ under conjugation, so $H\subseteq N_G(H)$. Moreover, $H$ is a normal subgroup of $N_G(H)$, and this is the largest subgroup of $G$ with that property. Indeed, any subgroup in which $H$ is a normal subgroup consists entirely of elements preserving $H$ under conjugation. In particular, $N_G(H)=G$ exactly when $H$ is a normal subgroup of $G$.^[ref]

The orbit-stabilizer theorem counts distinct conjugate subsets of a subgroup byFor $H=⟨(123)⟩$ in $S_3$, there is just one, since $[S_3:N_{S_3}(H)]=6/6=1$. For the subgroup $K=⟨(12)⟩=\{e,(12)\}$, preserving $K$ forces its unique nonidentity element to stay fixed. Hence $N_{S_3}(K)=C_{S_3}((12))=K$, and its three distinct conjugate subsets are $\{e,(12)\}$, $\{e,(13)\}$ and $\{e,(23)\}$. This is the subgroup counterpart of counting a conjugacy class with a centralizer: the object being stabilized determines which denominator belongs in the count.^[ref]

The equality in the definition is essential. For a subgroup $H$ that is a finite group, an inclusion $gH{g}^{-1}\subseteq H$ already gives equality by equal cardinalities. For an infinite subgroup, that inference can fail: conjugation might carry it onto a proper subgroup. The normal subgroup criterion is therefore, with the explicit hypothesis $H\le G$,No finiteness or commutativity assumption is needed. A normal subgroup is preserved as a set; its elements need not lie in the center of a group.

There is a second useful count, distinct from counting conjugate subgroups. The normalizer acts on $H$ itself by group automorphisms, giving the group homomorphismIts group kernel is $C_G(H)$, so the centralizer of a subset is a normal subgroup of $N_G(H)$ and the first isomorphism theorem givesThis quotient group records the different rearrangements of $H$ induced by the surrounding group. It gives a faithful group action on $H$, but need not be the entire automorphism group of $H$. For example, take the ambient group $G=H=⟨(123)⟩$. Since $G$ is an abelian group, $N_G(H)=C_G(H)=G$, so the quotient group is trivial; nevertheless $r\mapsto r^{-1}$ is a nonidentity group automorphism of $H$. Returning to the ambient symmetric group $S_3$ makes that group automorphism available through a reflection, and $N_{S_3}(H)/C_{S_3}(H)\cong C_2$. Its two cosets are the three rotations and the three reflections.

The subgroup $H$ itself need not be contained in $C_G(H)$: their intersection is the center of a group $Z(H)$. It is contained precisely when $H$ is an abelian group. Thus $N_G(H)/H$, which exists because $H$ is a normal subgroup of its normalizer, and $N_G(H)/C_G(H)$ answer different questions and should not be substituted for one another.

In exact arithmetic, conjugate gradients reaches the exact solution in at most the number of distinct eigenvalues of $A$, and therefore in at most $n$ steps for an $n$-dimensional system.

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 ${\rho }_2$.