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The index $[G:H]$ is the number of left cosets of $H$ in $G$.

Let an automorphism $\sigma$ of an algebraically closed field $L$ have finite order and fixed field $K$. Artin's fixed-field theorem makes $L/K$ finite cyclic Galois. Every finite extension $M/K$ embeds in $L$: adjoin finitely many algebraic generators successively and send each one to a root of the transported minimal polynomial in $L$. This works without assuming separability or using a primitive element. Its image is an intermediate field of a cyclic extension and is therefore Galois and cyclic over $K$.

For $r>1$, choose a prime $\ell \equiv 1(\mathrm{mod}r)$ by Dirichlet's theorem. The cyclic grouphas a subgroup of index $r$, whose fixed field is cyclic Galois of degree $r$ over $\mathbb{Q}$.

A connected graph is unicyclic when it contains exactly one cycle, equivalently when its numbers of vertices and edges agree.

A pseudoforest permits both these components and acyclic tree components.

The Araki–Lieb inequality bounds the difference of the two subsystem Von Neumann entropies by their joint Von Neumann entropy. In finite dimensions, all terms are finite:

For a bipartite density operator ${\rho }_{AB}$,

To prove one direction, take a purification of a density operator ${\rho }_{AB}$ using a system $C$. Equality of complementary entropies gives $S(A)=S(BC)$ and $S(C)=S(AB)$. Then Subadditivity of Von Neumann entropy gives $S(A)\le S(B)+S(AB)$; exchanging $A$ and $B$ proves the absolute-value bound. A Bell state has $S(AB)=0$ despite nonzero equal marginal entropies. ^[ref]

A finite abelian group is indecomposable exactly when it is cyclic of prime-power order. The structure theorem proves necessity. Conversely, every nonzero subgroup of $C_{p^n}$ contains its unique subgroup of order $p$, so two nonzero subgroups cannot be complementary direct summands.

An elementary divisor is a prime power $p^e$ occurring in the cyclic decomposition of a finitely generated torsion module.

Every finitely generated torsion-free module over a principal ideal domain is free. In the structure theorem, torsion-freeness removes every cyclic torsion summand and leaves only the free summand.

The Bessel potential of order $s$ is the Fourier multiplier $(1+|\xi {|}^2{)}^{-s/2}$. Sufficiently high order turns a compactly supported finite-order distribution into a bounded continuous function.

Let $i\ne j$ belong to an irreducible recurrent Discrete-time Markov chain. Write $T_k=\inf \{n\ge 0:X_n=k\}$ and $T_k^{+}=\inf \{n\ge 1:X_n=k\}$. DefineThe positive return convention is essential: $T_i$ itself is zero when starting at $i$. Let $N$ count visits to $j$ before the first positive return to $i$, including repeated holding visits at $j$. ThenWith Probability $a$ the excursion reaches $j$. From each visit there, the Strong Markov property gives Probability $b$ of hitting $i$ before the next return to $j$. recurrence ensures there is no third possibility of escaping both states forever. Thus, conditional on reaching $j$, the visit count has a positive-support geometric distribution with mean $1/b$.

For a two-state chain which chooses either state independently with Probability $1/2$ at every step, $a=b=1/2$. Half of the return cycles from zero never visit one; the rest visit it a geometrically distributed number of times of mean two. The unconditional expected visit count is therefore one.

If $H>0$ at one time, the inequality $d(H^{-1})/dt\ge 1$ forces $H\to +\infty$ and $a\to 0$ within finite time to the past. If $H<0$, it forces $H\to -\infty$ and $a\to 0$ within finite time to the future.

A machine computes the characteristic function of a language $X$ when it halts on every input with the one-letter output $1$ for inputs in $X$ and $0$ for inputs outside $X$.

The induced worldsheet metric is the pullback of the target-space metric:

A strictly convex function has at most one minimizer: if distinct points attained the same minimum, every strict convex combination of them would have a smaller value.

On the positive orthant,is strictly concave. Its unique maximizer $x^{\ast }$ on a compact convex feasible set satisfies the first-order inequalityequivalently

A stratum is a subgroup assigned its own nuisance baseline.

In discrete time, let $V$ be a supermartingale and let $\sigma \le \tau$ be bounded stopping times. Then the stopped values are integrable andFor a martingale, equality holds. The same bounded-time statement holds for a càdlàg supermartingale under the usual conditions for a filtration. In discrete time, expand stopped increments and use that $1_{\{\tau \ge k\}}$ is known at time $k-1$; continuous time follows by approximation and the path regularity.

Almost-sure finiteness of $\tau$ alone is insufficient. A simple symmetric random walk started at zero hits one almost surely, but its value there is one rather than its initial expectation zero. Conditions such as uniform integrability of a stopped uniformly integrable martingale, or bounded-increment optional stopping with integrable time, justify appropriate unbounded-time versions.

For a discrete martingale and stopping time $T$,The indicator is measurable at time $n$, so the stopped process is a martingale. If it is uniformly bounded and $T<\infty$ almost surely, bounded convergence gives $\mathbb{E}{M}_T=\mathbb{E}{M}_0$ directly.

A stochastic differential equation specifies a stochastic process through a combination of ordinary drift and random increments. In the scalar Itô integral convention, the notationmeans the stochastic integral equation $X_t=X_0+{\int }_0^{t}b(s,X_s)ds+{\int }_0^t\sigma (s,X_s)d{W}_s$, with the required measurability and local integrability. The driver $W$ is a Brownian motion; $b$ is the drift coefficient and $\sigma$ the diffusion amplitude.

For example, the Ornstein-Uhlenbeck process satisfies $d{X}_t=-\lambda X_{t}dt+\sigma d{W}_t$: linear restoring drift competes with persistent noise. A strong solution of a stochastic differential equation uses a prescribed driver, whereas a weak solution of a stochastic differential equation may construct the probability space and driver. Pathwise uniqueness, uniqueness in law and the lifetime of a maximal local solution of a stochastic differential equation are separate questions. If the equation uses a Stratonovich integral, its convention changes the drift conversion and must be stated.

The stochastic integral $\int HdX$ integrates a predictable process against a semimartingale. It extends pathwise integration against finite-variation processes and the Itô integral against local martingales.