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The Taylor remainder is the difference between a function and a finite Taylor polynomial. Bounds on higher derivatives or their Hölder continuity control this error.

If $f$ is $n$ times differentiable between $a$ and $x$, then for some $\xi$ strictly between them,

The multivariate Taylor polynomial of total degree $p$ at $x_0$ is

A tail event is an event $A\in \mathcal{T}$, where $\mathcal{T}$ is the tail sigma-algebra of the chosen sequence of random variables. It concerns information available arbitrarily far along the sequence. In the measurable coordinate models described in finite-prefix invariance of tail events, any finite edit preserves whether it occurs.

This is stronger than merely needing a long waiting time. On $\{0,1{\}}^{\mathbb{N}}$, with $1$ denoting heads, let $K=\inf \{j\ge 1:X_j=1\}$, taking $K=\infty$ if there is no head. The eventis not a tail event: changing only $X_1$ from zero to one changes membership. Under fair independent Bernoulli trials its probability is $1/8$. Even the event $\{K<\infty \}$, of probability one in this model, is not a tail event as a literal set: the all-zero sequence and the sequence with just one initial head differ in one coordinate. A probability of zero or one does not characterize tail events.

For scalar perturbations of a generic fluid in cosmic time, momentum conservation in synchronous gauge in cosmology gives

A similarity variable is constant along symmetry-group orbits and parametrizes group-invariant solutions.

The vector fieldgenerates the local transformationsIts second prolongation sends $u_t-u_{xx}-u_x^2$ to $-8t$ times that expression, so it is a Lie point symmetry of the potential Burgers equation.

A group-invariant solution is constant along the prolonged symmetry orbits. For a scaling of two independent variables with fixed dependent variable, it depends only on a ratio such as $x/t$.

For $r\le n/2$, the equivariant map from $(r-1)$-subsets to $r$-subsets that sends each subset to the sum of the $r$-subsets containing it is injective over $\mathbb{C}$. Hence ${\pi }_r-{\pi }_{r-1}$ is the character of a representation.

The sign homomorphism $\mathrm{sgn}:S_n\to \{\pm 1\}$ is the determinant of the corresponding permutation matrix. A transposition has sign $-1$, so this also proves that the parity of any transposition decomposition is well-defined.

The support of a permutation is the set of elements it does not fix. A permutation has finite support when this set is finite.

A permutation cycle $(a_1{a}_2\dots a_k)$ sends $a_i$ to $a_{i+1}$, sends $a_k$ to $a_1$, and fixes every other element. Its order is $k$.

The sign homomorphism sends an even permutation to $1$ and an odd permutation to $-1$; its kernel is the alternating group.

A symmetric group is the group of all bijections from a set to itself, with function composition as its operation.

Frequency differentiation lowers symbol order, position differentiation preserves it, products add orders, and finite sums have order at most the maximum order.

Suppose a finite group $G$ has no index-two subgroup, $P$ is a Sylow $2$-subgroup, $H$ has index two in $P$, and $x\in G$ has order two. The sign of the action on $G/H$ defines a homomorphism $G\to \{\pm 1\}$ and is therefore trivial. Since $[G:H]\equiv 2(\mathrm{mod}4)$, a fixed-point-free involution would be an odd number of transpositions. Thus $x$ fixes a coset $gH$, equivalently $g^{-1}xg\in H$.

Suppose a group acts faithfully and transitively on seven points, every point stabilizer is $S_4$, and every two-point stabilizer is a Klein four-group. Then the group has order $168$ andCount pairs consisting of a point and a Sylow subgroup fixing it for $p=2,3$; the two-point stabilizer ensures uniqueness. For $p=7$, a normal Sylow subgroup would force the group into its order-$42$ normalizer in $S_7$.

If $P$ is a Sylow $p$-subgroup of $G$ and $Q$ is any $p$-subgroup, let $Q$ act on $G/P$. Since $p$ does not divide $|G/P|$, some coset $gP$ is fixed. Hence $g^{-1}Qg\le P$, or equivalently $Q\le gP{g}^{-1}$.

The Sylow $2$-subgroups of $S_3$ are its three transposition subgroups, while its unique Sylow $3$-subgroup is $A_3$. The three Sylow $2$-subgroups of $S_4$ are dihedral groups of order eight and are the normalizers of the three cyclic subgroups generated by inverse pairs of $4$-cycles. The five Sylow $2$-subgroups of $A_5$ are Klein four-groups, one fixing each letter.

A group, or one of its Sylow subgroups, acts on the set of Sylow subgroups by conjugation.

Let $p>q$ be prime numbers. A nonabelian group of order $pq$ has a unique Sylow $p$-subgroup and exactly $p$ Sylow $q$-subgroups. Hence the Sylow count satisfies $p\equiv 1(\mathrm{mod}q)$, so $q\mid p-1$.