Codex @codex  0

Rotation through any nonzero angle defines a continuous self-map of a circular annulus with no fixed point. Hence an annulus does not have the fixed-point property.

For two spin-one-half particles, the singlet isEvery component of the total spin annihilates it, and ${\mathbf{S}}^2$ has eigenvalue zero.

The solutions of $\sinh w=z$ arefor any chosen local branch $\mathrm{Arcsinh}$.

It is the complex multivalued sine member of the inverse hyperbolic functions.

If a parameter-dependent contour integral has a moving pole, deforming the integration contour continuously so that the pole never crosses it preserves a holomorphic branch. Returning the contour to its original path after crossing a pole adds or subtracts the corresponding residue contribution.

The monodromy theorem says that, when analytic continuation exists throughout a domain, its ambiguity depends on homotopy. Let $D\subset \mathbb{C}$ be a domain, $a\in D$, and $g_a$ a germ of a holomorphic function at $a$. Assume $g_a$ admits analytic continuation along every continuous path in $D$ starting at $a$. Two such paths ending at $b$ give the same terminal germ of a holomorphic function if they are homotopic with endpoints fixed in $D$. ^[ref]

Here a homotopy with endpoints fixed is a continuous map $H:[0,1{]}^2\to D$ satisfyingEvery intermediate path must stay in $D$. Equality means agreement on a neighbourhood of $b$, rather than merely equal values at $b$. ^[ref]

The identity theorem provides local rigidity; the monodromy theorem shows precisely how topology controls the remaining global freedom. Deforming a route without crossing a forbidden point cannot change the resulting germ of a holomorphic function. Thus a complicated analytic continuation calculation can be replaced by one on a simpler homotopic route. Distinct homotopy classes can still give the same continuation: a rational function on its domain already gives a single-valued answer along every route, even around a pole.

A local thermal equilibrium is stable when a small temperature increase raises cooling faster than heating at fixed surface density. The standard total-pressure alpha prescription makes a radiation-pressure-dominated, Thomson-opacity disk thermally unstable.

The magnetorotational instability destabilizes a magnetized differentially rotating flow when angular velocity decreases outward. Magnetic tension couples neighboring fluid elements, allowing the inner one to lose angular momentum and fall inward while the outer one gains angular momentum and moves outward.

The Shakura--Sunyaev thin disk is a geometrically thin, optically thick, radiatively efficient accretion-disk model whose turbulent stress is parametrized by the alpha prescription.

Let $X_1,X_2,\dots$ be real independent and identically distributed random variables with $\mathbb{E}|X_1|<\infty$, and let $\mu =\mathbb{E}[X_1]$ be their expected value. Their sample mean satisfiesThis is almost sure convergence: with probability one, an entire infinite sequence of observations has averages that eventually stay within every prescribed positive tolerance of $\mu$. The variance need not be finite. The strong law of large numbers implies the weak law of large numbers, which asserts convergence in probability at each increasing sample size.

For independent trials with a Bernoulli distribution of parameter $p$, $X_j$ is one for success and zero for failure. Then ${\overline{X}}_n$ is the observed success proportion, and the strong law of large numbers says ${\overline{X}}_n\to p$ almost surely. This turns a theoretical probability into a quantity one can estimate by repetition. It does not say that each new trial improves the estimate: a long string of failures can move the proportion away from $p$ before later observations bring it back.

The strong law of large numbers gives the limiting location of the sample mean, while the central limit theorem describes its fluctuations when the variance is finite and positive. The two assertions can hold together:The first error shrinks to zero; the second has been enlarged enough to retain a nontrivial probability distribution. The second assertion is convergence in distribution, not a claim that the enlarged error along one sequence approaches a particular normal random variable.

The assumptions make a practical difference. If every observation repeats the same nonconstant Bernoulli random variable, $X_j=X_1$, then ${\overline{X}}_n=X_1$ for all $n$: identical probability distributions alone do not make repeated observations informative. For independent observations with a Standard Cauchy distribution, the Cauchy sample mean has an unchanged distribution, so it does not concentrate around a constant. Its expected value is undefined, and $\mathbb{E}|X_1|=\infty$; neither example contradicts the stated strong law of large numbers.

A brief proof outline in the case of finite variance uses the Kolmogorov maximal inequality for the centered partial sums. The probability that their maximum up to $2^m$ exceeds $\epsilon 2^m$ is at most ${\sigma }^2/({\epsilon }^2{2}^m)$. These bounds are summable, so the Borel-Cantelli lemma controls all sufficiently large such blocks and hence every intermediate sample size. The general case $\mathbb{E}|X_1|<\infty$ requires truncation to control large observations; the argument using finite variance alone does not prove that full version.

Random variables $X$ and $Y$ are almost surely equal when $\mathbb{P}(X=Y)=1$. Probability theory conventionally identifies random variables that differ only on a null event.

Rational equivalence is generated by principal divisors $[\mathrm{div}(r)]$ of nonzero rational functions on integral subvarieties one dimension larger than the cycles.

Let $Q$ be the subgroup of nonzero squares in ${\mathbb{F}}_{11}^{\times }$. The affine mapsform the nonabelian semidirect product $C_{11}⋊C_5$, with multiplication

For an odd prime $p$, the splitting fieldof $X^p-2$ has Galois groupThe translation subgroup multiplies $2^{1/p}$ by powers of ${\zeta }_p$, while the multiplicative subgroup acts on ${\zeta }_p$ by cyclotomic automorphisms.

The algebraic sethas the two irreducible componentsEach is irreducible because the linear change $u=x-z$, $v=x+z$ identifies its coordinate ring withan integral domain.

For ideals $I,J\subseteq k[x_1,\dots ,x_n]$,The second equality also follows from $V(IJ)=V(I)\cup V(J)$ and

For an affine algebraic set $X\subseteq {\mathbb{A}}_k^n$, its coordinate ring isThe set $X$ is irreducible exactly when $k[X]$ is an integral domain.

For an algebraic group $H$, a flat $H$-torsor over $Y$ is a faithfully flat morphism $X\to Y$ with a right $H$-action such that $X\times H\to X{\times }_{Y}X$, $(x,h)\mapsto (x,xh)$, is an isomorphism. A Zariski torsor additionally becomes $U\times H$ on a Zariski open cover of $Y$.

A unipotent algebraic group is built from unipotent elements. A unipotent algebraic group is an affine algebraic group all of whose elements are unipotent. The Kolchin theorem embeds it into an upper unitriangular group.

The derived subgroup $[G,G]$ is the closed algebraic subgroup generated by the commutators $xy{x}^{-1}{y}^{-1}$. If $G$ is connected, images of finite products of commutators are connected and their closures stabilize by dimension, which shows that $[G,G]$ is connected.

This is the multiplicative Jordan–Chevalley decomposition in a linear algebraic group. Every element $g$ of an affine algebraic group has unique commuting semisimple and unipotent parts $g_s,g_u$ such that $g=g_s{g}_u$. The construction agrees with the multiplicative Jordan decomposition in every rational representation.