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A horizontal lift is determined by the horizontal distribution of a principal connection or the corresponding connection on a vector bundle. A horizontal lift of a base curve is a curve in the bundle projecting to it and tangent to the horizontal distribution. In a local frame its fiber coordinate solves a linear ordinary differential equation.

A connection on the tangent bundle is torsion-free when ${\nabla }_{X}Y-{\nabla }_{Y}X=[X,Y]$. Its Christoffel symbols are symmetric in their two lower indices.

The dual connection is induced by a connection on a vector bundle through the evaluation pairing. A connection on $E$ induces one on $E^{\ast }$ byIts connection matrices are the negatives of the transposes of those for $E$.

The curvature two-forms are

A splitting sphere meets a knot transversely in two points. It is trivial when one of the resulting one-string tangles is boundary-parallel in its three-ball; otherwise it exhibits a nontrivial connected-sum decomposition.

A composite knot is a connected sum of two nontrivial knots.

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.

For a real symmetric positive-definite matrix $A$, the conjugate gradient method reaches the exact solution in exact arithmetic after at most the number of distinct eigenvalues active in the initial error, hence after at most $n$ steps in dimension $n$.

If those eigenvalues are ${\lambda }_1,\dots ,{\lambda }_m>0$, the polynomial $p(t)={\prod }_{j=1}^m(1-t/{\lambda }_j)$ satisfies $p(0)=1$ and $p(A)e_0=0$. This is an admissible error polynomial at step $m$. Krylov-space minimization in the energy norm does at least as well, so its error must also be zero. Repeated eigenvalues and absent spectral components can make $m$ much smaller than $n$.

The proof assumes exact arithmetic. Roundoff error can spoil exact conjugacy and create residual components absent in the ideal recurrence, so a numerical implementation needs an accuracy-based stopping criterion.

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 compares the cost of moving one probability measure to another with their relative entropy. For a nonnegative transport cost function $c$ and reference probability measure $\mu$, definewhere $\Pi (\nu ,\mu )$ consists of couplings with those marginals. One common convention is ${\mathcal{T}}_c(\nu ,\mu )\le CD(\nu \Vert \mu )$ for every probability measure $\nu$; the orientation of the relative entropy matters.

For a metric $d$, with the measures having finite first moments, the $T_1(C)$ convention is Wasserstein distance $W_1(\nu ,\mu {)}^2\le 2CD(\nu \Vert \mu )$, while $T_2(C)$ replaces $W_1$ by the second Wasserstein distance $W_2$. The first convention bounds fluctuations of all 1-Lipschitz functions: $\log {\mathbb{E}}_{\mu }{e}^{\lambda (f-{\mathbb{E}}_{\mu }f)}\le C{\lambda }^2/2$. The choice of a squared distance or an unsquared transport cost function is part of the statement, rather than a universal normalization.^[ref]