If $K$ is a finitely generated group, then the semidirect product $K⋊H$ is a residually finite group exactly when both $K$ and $H$ are residually finite groups. To separate an element whose $H$ coordinate is nontrivial, use a finite quotient of $H$. For $1\ne k\in K$, choose a finite-index normal subgroup $U$ omitting $k$, and intersect all subgroups of index at most $[K:U]$ to obtain a characteristic finite-index subgroup $C\le U$. The induced map into $(K/C)⋊\mathrm{im}(H\to \mathrm{Aut}(K/C))$ has finite target and separates $k$. More generally, finite generation can be replaced by a separating family of finite-index normal subgroups invariant under the given $H$ action.

For quiver representations $X=(X_i,f_{\rho })$ and $Y=(Y_i,g_{\rho })$, a morphism consists of maps $h_i:X_i\to Y_i$ satisfying $h_{t(\rho )}{f}_{\rho }=g_{\rho }{h}_{s(\rho )}$ for every arrow. Composition is vertexwise. These families are the kernel of the extension complex of quiver representations.

The dimension vector of a finite-dimensional quiver representation is $({\dim }_k{X}_i{)}_{i\in Q_0}$. Direct sums add dimension vectors. Fixing the vector gives an affine space of possible arrow matrices, the quiver representation space.

This bilinear form is $⟨\mathbf{n},\mathbf{m}{⟩}_Q={\sum }_i{n}_i{m}_i-{\sum }_{\rho :i\to j}{n}_i{m}_j$. The standard projective resolution of a quiver representation gives its value as $\dim {\mathrm{Hom}}_Q(X,Y)-\dim {\mathrm{Ext}}_Q^1(X,Y)$ for the dimension vectors of $X,Y$. Its diagonal is the Tits form of a quiver.

A finite-dimensional quiver representation is rigid when ${\mathrm{Ext}}_Q^1(X,X)=0$. Its extension complex of quiver representations then has surjective differential. Equivalently, it has an open orbit under base change. Rigidity constrains arrow ranks and path identities.

For a finite-dimensional quiver representation $X$, its automorphism group is the unit group ${\mathrm{End}}_Q(X{)}^{\times }$. It is a connected linear algebraic group: the units of a finite-dimensional algebra form a nonempty Zariski-open set in an affine space, hence a geometrically irreducible variety. This automorphism group is the stabilizer under the base change action on quiver representations.

Over an algebraically closed field $k$, write the Krull-Schmidt decomposition as $X={⨁}_a{M}_a^{\oplus m_a}$ with pairwise nonisomorphic indecomposable modules $M_a$. Reduction modulo the Jacobson radical gives the quotient ${\prod }_a{\mathrm{GL}}_{m_a}(k)$. The field hypothesis matters: over a general field, the corresponding factors are general linear groups over the division rings ${\mathrm{End}}_Q(M_a)/J({\mathrm{End}}_Q(M_a))$, which need not equal $k$.

At fixed dimension vector, ${\mathrm{Rep}}_Q(\mathbf{n})={\prod }_{\rho :i\to j}{\mathrm{Hom}}_k(k^{n_i},k^{n_j})$ is an irreducible affine space of dimension ${\sum }_{\rho }{n}_{s(\rho )}{n}_{t(\rho )}$. Its points are arrow-matrix collections, and its base-change orbits are isomorphism classes.

For $1\to 2\to 3$, $I[a,b]$ has $k$ on the interval of vertices $a,\dots ,b$, zero elsewhere and identity maps along its internal arrows. These six representations are exactly the indecomposables. To decompose $V_1\stackrel{f}{\to }{V}_2\stackrel{g}{\to }{V}_3$, split $\mathrm{im}f\cap \ker g$, its complements inside each of $\mathrm{im}f$ and $\ker g$, and a complement to their sum. Lift the image basis through $f$ and transport the complementary nonkernel basis through $g$. This yields interval blocks directly, without the Gabriel theorem.

A renormalization-group transformation integrates out selected short-distance degrees of freedom and restores chosen cutoff and field-normalization conventions. It defines a map $R_b$ on couplings at a length factor $b$. Its discrete Jacobian matrix multipliers differ from eigenvalues of the generator of a continuous renormalization-group flow; the relation for a scaling direction is $\lambda =b^y$.

A renormalization-group transformation can scale different coordinate directions by different factors to keep distinct gradient terms fixed. If momentum component $p_i$ scales to $b^{w_i}{p}_i$, its coordinate scales by $b^{-w_i}$. The momentum/volume measure has total scaling weight ${\sum }_i{w}_i$, motivating the anisotropic effective dimension.

In a specified scalar-field normalisation, one can define a renormalized squared mass by $R={\tilde{\Gamma }}^{(2)}(0)$. Its value depends on the renormalization prescription and is not automatically the pole mass in a theory with momentum-dependent self-energy. A reference propagator $(p^2+R{)}^{-1}$ and mass counterterm then organise perturbation theory around that prescription.

A renewal equation decomposes a process at its first renewal. For returns of a Markov chain to a fixed state, $f_k$ is the first-return probability and $u_n$ the probability of being at that state at time $n$. Their probability generating functions satisfy $U=1+FU$.

For positive nonarithmetic interarrival times with law $K$ and mean $m\in (0,\infty ]$, let $U={\sum }_{j\ge 0}{K}^{\ast j}$ be their renewal measure. If $g$ is directly Riemann integrable, then ${\int }_{[0,t]}g(t-x)U(dx)\to m^{-1}{\int }_0^{\infty }g(x)dx$, with $1/\infty =0$. Thus a locally bounded solution of $z=g+z\ast K$ has this limit through its renewal representation. Arithmetic laws require the corresponding lattice version.

For a positive interarrival law $K$, the renewal measure $U={\sum }_{j\ge 0}{K}^{\ast j}$ counts the expected number of renewal epochs in a set, including the epoch at zero through $K^{\ast 0}={\delta }_0$. Its cumulative function is one plus the usual renewal function. convolution against $U$ solves a renewal equation.

For independent uniform distribution increments on $(0,1)$ and $0\le x\le 1$, let $N(x)$ be the first index at which their partial sum exceeds $x$. First-step analysis gives $M(x)=\mathbb{E}N(x)=1+{\int }_0^{x}M(s)ds$, hence $M=e^x$. Equivalently, $\mathbb{P}(N(x)>n)=x^n/n!$ is a simplex-volume probability and the tail-sum formula sums to $e^x$. The strict inequality gives $N(0)=1$ almost surely.

For $|{\alpha }_{\theta }⟩=\cos \theta |0⟩+\sin \theta |1⟩$, Alice measures her half of a Bell state in the real basis $\{|{\alpha }_{\theta }⟩,|{\beta }_{\theta }⟩\}$, with $|{\beta }_{\theta }⟩=-\sin \theta |0⟩+\cos \theta |1⟩$. Bob receives one outcome bit and applies $I$ or $iY$. Since $iY|{\beta }_{\theta }⟩=|{\alpha }_{\theta }⟩$ for every $\theta$, the preparation is exact and the correction is independent of the unknown parameter.

A stationary particle of rest mass $m$ absorbing a photon of energy $E$ acquires total invariant massThis follows from four-momentum conservation, since the incoming photon has momentum magnitude $E/c$. If the system splits into two equal particles, the largest kinematically permitted daughter rest mass is $M/2$, attained when both are at rest in the centre-of-momentum frame. Their common laboratory speed at threshold is $cE/(m{c}^2+E)$. Above this daughter-mass threshold there is insufficient centre-of-momentum energy; below it, relative kinetic energy is available.

The invariant future mass-shell measure is $d^{3}p/E$, up to fixed normalization factors absorbed in $f$. Its relation to the Lorentz-invariant phase-space measure follows by integrating $2\theta (p^0)\delta (p_{\mu }{p}^{\mu }+m^2)d^{4}p$ over $p^0$ in a local orthonormal tetrad. A particle carries four-momentum $p^{\mu }$ and crosses a local spatial surface with velocity $p^i/p^0$. Integrating the associated momentum flux gives the displayed stress-energy tensor. For photons, $p^{\hat{a}}=E(1,\mathbf{e})$ and $|\mathbf{e}|=1$.

A scalar function on the future mass shell describing the expected particle count in a phase-space distribution cell. In a local orthonormal tetrad, the number crossing a spacelike surface element is $f{p}^{\mu }d{\Sigma }_{\mu }{d}^{3}p/E$. On a constant-time slice it becomes $f{d}^{3}x{d}^{3}p$. Including degeneracy and phase-space normalization in $f$ gives the stress-energy tensor $T^{\mu \nu }=\int (d^{3}p/E)f{p}^{\mu }{p}^{\nu }$. Other conventions may put the degeneracy or $(2\pi {)}^{-3}$ explicitly in the measure.

With normalized densities $f,g$ and envelope $f\le Mg$, $M>1$, conditioning a proposal on rejection gives density $r=(Mg-f)/(M-1)$. It can be used as an importance sampling proposal only if it covers the relevant target support. If $Mg=f$ on a set of positive target mass, rejected draws never visit that set.