If square-integrable random variables satisfy $X_n\to X$ almost surely, then $X_n\to X$ in $L^2$ exactly when $\mathbb{E}[X_n^2]\to \mathbb{E}[X^2]$. Boundedness in $L^2$ and almost-sure convergence first give weak convergence in the Hilbert space $L^2$; weak convergence together with convergence of norms gives strong convergence.

A sequence $a_n$ converges in the Cesaro sense to $a$ when $N^{-1}{\sum }_{n=1}^N{a}_n\to a$. For bounded sequences, convergence in density to $a$ is equivalent to Cesaro convergence of $|a_n-a|$ to zero.

For a function space on a set $X$ and a point $x\in X$, the point evaluation functional is ${\delta }_x(f)=f(x)$. It is bounded on $C(X)$ with the supremum norm.

Because each matrix $M(a)=\left(\begin{array}{cc}a& 1\\ 1& 0\end{array}\right)$ is symmetric, reversing the partial quotients transposes their product. If the convergents of $[a_0,\dots ,a_n]$ are $p_j/q_j$, then the last two convergents of $[a_n,\dots ,a_0]$ are

For a transitive set $X$, the relative constructible universe $L(X)$ is the union of the relative constructible hierarchy, starting at $L_0(X)=X$, applying the definable power set at successors and taking unions at limit ordinals. It is the smallest inner model of ZF containing $X$ and all its members. For an arbitrary set $A$, take the base $X$ to be the transitive closure of $\{A\}$, so that $A\in X$. Axiom of choice need not hold for an arbitrary base; the construction uses initial parameters rather than a predicate relativization.

For $A\in L$, its constructible power set isUnlike the single-stage definable power set $\mathcal{D}(A)$, it includes subsets of $A$ appearing arbitrarily late in the constructible hierarchy.

If $\Vert a^{ij}-A^{ij}{\Vert }_{\infty }<\epsilon$ for every $i,j$, then the Cauchy-Schwarz inequality givesConsequently $\epsilon \le \theta /(2n)$ preserves half of the estimate's lower bound.

A connection on a vector bundle pulls back along a smooth curve $\gamma$ to differentiate its time-dependent sections. For the tangent bundle, a section $V(t)$ of the pullback tangent bundle hasThe Leibniz rule under a change of frame proves independence of the coordinates. The derivative includes the time dependence of the coefficients, so it is meaningful even at a zero tangent velocity. A vector field along a map need not extend to one ambient field if the curve self-intersects. Parallel fields satisfy $D_{t}V=0$, a linear ordinary differential equation whose unique solutions define parallel transport.

For a subset $S$ of a group and an element $g$, its conjugate subset isConjugation is a bijection and a group automorphism, so it preserves cardinality and every property expressed using the group operation.

For a finite group $G$, choose one representative $x_i$ from each conjugacy class with more than one element. The class equation iswhere $Z(G)$ is the center of a group and $C_G(x_i)$ is the centralizer of $x_i$. Each central element contributes its own one-element conjugacy class. Every remaining conjugacy class contributes its size, given by the orbit-stabilizer theorem. The sum counts each element exactly once. See McKernan, Group actions and automorphisms, lemma 23.6.

For conjugacy classes of the symmetric group on three letters, take $x_1=(12)$ and $x_2=(123)$. Their centralizers have sizes two and three, and the center of a group has size one, givingFor an abelian group, every centralizer is the whole group, every conjugacy class has size one, and the sum over noncentral representatives is empty.

The power of this counting identity is that the sizes of conjugacy classes constrain group structure. For a nontrivial finite p-group with group order $p^n$, every noncentral conjugacy class has size a positive power of $p$, by Lagrange's theorem. The class equation therefore gives $|Z(G)|\equiv 0(\mathrm{mod}p)$. Since the identity element lies in the center of a group, the center of a group must have at least $p$ elements. This obtains an element commuting with the entire group without constructing one. See McKernan, lemma 23.7.

Historically, these ideas belong to the nineteenth-century development of permutation groups and normal subgroups. Milne's historical opening records Galois's introduction of normal subgroups in 1832 and Jordan's systematic treatment of permutation groups in 1870. This is context for the conjugation action and the class equation, rather than a claim that either author first stated the modern formula.^[ref]

A random curve $\gamma$ from $0$ to infinity in the complex upper half-plane has the chordal restriction property when, for every admissible compact H-hull $A$ avoiding zero, the conditional law of ${\psi }_A(\gamma )$ given $\gamma \cap A=\varnothing$ equals the original law of $\gamma$. Here ${\psi }_A:\mathbb{H}\setminus A\to \mathbb{H}$ fixes zero and has derivative one at infinity.

The coordinates of a vector-valued Compound Poisson process are independent exactly when its Lévy measure is supported on the union of the coordinate axes. For marks $(g_1(U),g_2(U))$ with $U$ uniform on $[0,1]$ and continuous $g_i$, this is equivalent tofor every $y\in [0,1]$.

A compound Poisson process with rate $\lambda$ and mark $Y$ is a martingale exactly when $\mathbb{E}|Y|<\infty$ and $\mathbb{E}Y=0$.

An Iwasawa algebra is a completed group algebra such as ${\mathbb{Z}}_p[[G]]$ or ${\mathbb{F}}_p[[G]]$ for a compact p-adic analytic group $G$.

A geodesic $\gamma :I\to M$ is maximal when it has no geodesic extension to a strictly larger interval. Every initial position and velocity determine a unique maximal geodesic through the geodesic equation.

If $f:[1,\infty )\to [0,\infty )$ is decreasing, then ${\sum }_{n\ge 1}f(n)$ and ${\int }_1^{\infty }f(x)dx$ either both converge or both diverge. This follows by bounding the area under the graph between adjacent upper and lower rectangle sums.

Two finite CW complexes can have isomorphic integral cohomology rings but different cohomology rings modulo $p$. For example, attaching a three-cell of degree $p$ to the projective line in ${\mathbb{CP}}^2$ preserves the nonzero square of the degree-two class modulo $p$, whereas the corresponding Moore space wedged with $S^4$ has zero square. The spaces therefore cannot be homotopy equivalent.

For positive $n$, let $x,y$ be the pullbacks of the orientation classes from the two factors. Then $x^2=y^2=0$, $xy$ generates degree $2n$, and $yx=(-1{)}^{n}xy$. When $n$ is even, every graded ring automorphism restricts on $H^n$ to a signed permutation matrix; when $n$ is odd the ring admits every matrix in $G{L}_2(\mathbb{Z})$.

The only regular functions defined on all of a nonempty projective space over a field $k$ are the constants:For ${\mathbb{P}}^1$, this also follows by gluing $k[t]$ and $k[t^{-1}]$ inside $k[t,t^{-1}]$: their intersection is $k$.

The CYK algorithm decides membership in a context-free language given by a grammar in Chomsky normal form. It records, for every substring, the nonterminals that derive that substring and combines adjacent substrings using binary productions. For a word of length $n$ and a fixed grammar, it runs in $O(n^3)$ time.