A high-mass X-ray binary has a massive donor star supplying an accreting neutron star or black hole. A strong stellar wind often feeds it, although Roche-lobe overflow and donor equatorial outflows also occur. The short lifetime of the donor associates these systems with young stellar populations.

An accreting X-ray pulsar contains a magnetized neutron star whose magnetic field channels the accreting gas toward its magnetic poles. Rotation carries the bright accretion regions through the observer's view and produces pulses. The input angular momentum can change the spin period; this differs from the rotational energy source of an isolated pulsar.

A Type I X-ray burst is a transient thermonuclear runaway in accreted hydrogen or helium on a neutron star surface. It rises rapidly and cools after the fuel burns. A black hole has no material surface on which the same accumulated surface-fuel event can occur. This is distinct from an accretion-powered X-ray pulse or an accretion instability.

The base group acts independently on each coordinate of a Cartesian power, and the top group permutes coordinates. This is distinct from the imprimitive action on a disjoint union of blocks. It is primitive when the base action is primitive and nonregular and the top group is transitive. The full product-action overgroup is $S_{|\Delta |}\wr S_r$. Socles of almost-simple, diagonal or holomorph bases produce respectively product-action, compound-diagonal or holomorph-compound types.

Use the material-first nominal stress $P_{Ii}$ and $F=RU$. The symmetric Biot stress conjugate to the Biot strain tensor is $T^{(1)}=\mathrm{sym}(PR)$. To prove it, set $M=P^T$ and differentiate $F=RU$. Conservation of angular momentum makes $M{F}^T$ symmetric, so the contraction of $R^{T}M$ with $R^T\dot{R}U$ is zero: it is a contraction of a symmetric tensor with a skew tensor after cyclically moving $U$. Thus $M:\dot{F}=\mathrm{sym}(R^{T}M):\dot{U}=T^{(1)}:{\dot{E}}^{(1)}$. No assumption that $PR$ itself is symmetric is needed.

A finitely generated group $G$ has a soluble word problem for a group if and only if there are a simple group $S$ and a finitely presented group $P$ with embeddings $G\hookrightarrow S\hookrightarrow P$. The theorem does not require $S$ itself to be finitely presented. In particular, a finitely generated subgroup of a finitely presented simple group has soluble word problem. The primary statement is Theorem I of An algebraic characterization of groups with soluble word problem.

If a finitely generated group embeds in a finitely presented group, its null words in any fixed finite generating set form a recursively enumerable set. Choose words for the generator images in the larger group. Enumerate finite products of conjugates of its defining relators and their inverses, freely reduce, and compare with substituted words. Dovetailing over input words enumerates exactly the subgroup's identities. Choosing the finite image words is nonuniform, but their existence suffices for existence of an enumeration algorithm.

For $S\subseteq {\mathbb{N}}_{>0}$, form the Right-angled Artin group on generators $b_i$ indexed by $i\in \mathbb{Z}$, imposing $[b_i,b_j]=1$ precisely when $|i-j|\in S$. Translation $b_i\mapsto b_{i+1}$ is an automorphism. Its semidirect product with an infinite cyclic group generated by $t$ is countable and generated by $b_0,t$. If $n\notin S$, killing all $b_i$ except $b_0,b_n$ retracts the graph group onto the rank-two free group, so their commutator is nonidentity. This proves the displayed equivalence. For non-recursively-enumerable $S$, enumerable identities of finitely generated subgroups rules out an embedding of this two-generated group in any finitely presented group.

An amalgamated free product of finite cyclic groups has an effective normal form theorem for an amalgamated free product: multiplication, subgroup membership and subgroup identifications are finite table operations. Replacing an amalgamated syllable by its image in the neighbouring factor and combining adjacent syllables from one factor strictly decreases syllable count. A remaining alternating product outside the amalgamated subgroup is nonidentity. This supplies a terminating word problem for a group algorithm.

The closed word metric ball of radius $r\ge 0$ centred at a group element $g$ isThis is a closed ball for the word metric. A finite generating set of a group gives a finite word metric ball, because at most ${\sum }_{k=0}^{\lfloor r\rfloor }|S{|}^k$ words over an alphabet can represent its elements. Different words over an alphabet may represent the same element, so this is an upper bound. Word metric balls in the integer lattice give exact counts for two choices of moves.

A nontrivial product of positive-dimensional closed oriented Witt spaces of total dimension four has metabolic rational middle intersection form. For dimensions $1+3$, the two Künneth summands in degree two pair only with one another, so either is a half-dimensional isotropic subspace. For dimensions $2+2$, the outside degree-two summands pair with one another, and the $I{H}^1\otimes I{H}^1$ summand has the tensor pairing of two symplectic forms. A Lagrangian in the first symplectic factor tensored with the entire second factor is a half-dimensional isotropic subspace of that summand. Adding one outside summand gives a Lagrangian in the complete middle form. The Witt bordism classification then makes the product null-bordant.

For positive-definite $R$ and $\nu >p-1$, if $W\sim W_p(\nu ,R^{-1})$ has the Wishart distribution, then $W^{-1}$ has the Inverse-Wishart distribution. For $\nu >p+1$, its expected value is $R/(\nu -p-1)$; below this threshold the matrix mean does not exist. It commonly models a covariance matrix, but couples the prior behavior of variances and correlations.

Observation windows are measurable sets used to count points of a point process. They are pairwise disjoint when $W_i\cap W_j=\varnothing$ for every $i\ne j$. Half-open intervals such as $[0,1)$ and $[1,2)$ meet at a boundary without sharing any point, whereas the closed intervals $[0,1]$ and $[1,2]$ both contain one. A disjoint union makes window counts additive; it does not make their random variables independent.

On a locally path-connected space, on each path component choose a value of a complex logarithm of $f$ at a base point. Continue it along paths using local logarithms. Independence of the path is exactly vanishing of the winding number of a continuous closed path for all loops. This constructs a continuous function $g$ with $f=e^g$. The implication from a logarithm to zero winding requires no local connectivity. For applications to arbitrary planar compact sets, apply the criterion on polygonal neighborhoods rather than assuming the compact set is locally path-connected.

The scalar Wilson-Fisher fixed point is attractive within its critical surface, but has a relevant thermal direction. With $r=m^2{\Lambda }^{-2}$, $\lambda =g{\Lambda }^{-ϵ}$, and $K_D={\Omega }_D/(2\pi {)}^D$, its one-loop equations are $\dot{r}=2r+K_D\lambda /[2(1+r)]$ and $\dot{\lambda }=ϵ\lambda -3K_D{\lambda }^2/[2(1+r{)}^2]$. Their stability matrix at $r_{\ast }=-ϵ/6+O({ϵ}^2)$ and ${\lambda }_{\ast }=16{\pi }^2ϵ/3+O({ϵ}^2)$ has the displayed eigenvalues. Thus an unrestricted claim of infrared attraction is false: the bare mass must be tuned to the critical surface.

For a continuous sample, the statistic also equals ${\sum }_i{1}_{\{X_i>0\}}+{\sum }_{i<j}{1}_{\{X_i+X_j>0\}}$. In each pair the sign of the sum is the sign of the observation with larger absolute value, proving the identity by counting its smaller-rank partners. Under a continuous symmetric null, the pair kernel has projection $F(x)-1/2$ and projection variance $1/12$. The U-statistic central limit theorem yields asymptotic normality of $W_{+}$, with variance asymptotic to $n^3/12$.

The Walsh averages of a sample are $(X_i+X_j)/2$ for $1\le i\le j\le n$, including the diagonal pairs. If the absolute observations are distinct and nonzero, order them as $|X_{(1)}|<\cdots <|X_{(n)}|$. Among the $r$ pairs whose larger absolute observation is $X_{(r)}$, every sum has the sign of $X_{(r)}$. Hence the number of positive Walsh averages is exactly ${\sum }_{r}r{1}_{X_{(r)}>0}$, the positive Wilcoxon signed-rank statistic. This proves equivalence of the corresponding tests; the Walsh averages themselves are generally dependent.

For Planck's law per unit frequency, setting $x=h\nu /(k_{B}T)$ leaves a positive constant times $x^3/(e^x-1)$. Its logarithmic derivative givesThe same uniqueness argument as in the Derivation of Wien's displacement law, with $5$ replaced by $3$, selects the positive root. The maxima differ because corresponding positive spectral intervals obeyThe nonconstant Jacobian determinant $c/{\lambda }^2$ changes density heights. Consequently $c/{\nu }_{\max }=(x_{\lambda }/x_{\nu }){\lambda }_{\max }\simeq 1.75978{\lambda }_{\max }$. A peak is a property of a specified density, rather than an invariant point of the spectrum. For equal logarithmic intervals, the corresponding densities satisfy $\lambda B_{\lambda }=\nu B_{\nu }$ and have matching converted peak positions; this is a third convention, distinct from either linear-interval density.

The Chandrasekhar mass is the limiting mass of a cold nonrotating white dwarf supported by relativistic electron degeneracy pressure, about $1.4M_{\odot }$ for ordinary composition. It bounds the retained remnant's mass, rather than the birth mass of a star that can lose an envelope. The ideal limit assumes fixed mean molecular weight per electron, Newtonian gravity and a noninteracting degenerate electron gas; electron capture, general relativity, stellar rotation and ignition can change the attainable endpoint.

The Chandrasekhar limit belongs to work of the early 1930s. In his 1983 retrospective, Subrahmanyan Chandrasekhar reprints the 1935 mass–radius sequence and then discusses stellar evolution and relativistic stability.^[ref] The 1983 Nobel Prize in Physics recognized his broader theoretical work on stellar structure and stellar evolution; the other half recognized William Alfred Fowler's theoretical and experimental work on nuclear reactions underlying chemical element formation.^[ref]

Chirality distinguishes the two irreducible Weyl representations of the proper Lorentz group. For a massless particle it agrees with helicity up to the particle-antiparticle convention, while a mass term couples opposite chiralities.

The chiral projectors $P_{L,R}=(1\mp {\gamma }^5)/2$ separate the components of a Dirac spinor. The weak charged current selects one chirality. A massive fixed-helicity state generally contains both chiral components; the simple helicity-chirality identification requires the massless limit and a specified particle-antiparticle convention.