For a complex character $\chi$ of a finite group, its kernel is the kernel of an affording group representation. Averaging an inner product makes that representation unitary. If its degree is $d$, all eigenvalues of a group element have modulus one, so their sum equals $d$ only when every eigenvalue is one. Thus the displayed character-value criterion agrees with the representation kernel and defines a normal subgroup.

For a group representation $\rho$ with character $\theta$, its determinant character sends $g$ to $\det \rho (g)$. Multiplicativity makes it a linear character, independent of the equivalent matrix realization of $\rho$.

An ordinary character is the trace character of a finite-dimensional group representation in characteristic zero, usually realized over the complex numbers. It contrasts with a Brauer character in positive characteristic, which lifts eigenvalues on p-regular elements to characteristic-zero roots of unity.

For a non-dividend-paying stock in the Black-Scholes model, use density $d{Q}^S/dQ=e^{-\rho T}{S}_T/S_0$. Under $Q^S$, $W_t^S=W_t^Q-\sigma t$ is Brownian motion, and $\log S_t=\log S_0+(\rho +{\sigma }^2/2)t+\sigma W_t^S$. Thus $e^{-\rho T}{\mathbb{E}}_Q[S_{T}H]=S_0{\mathbb{E}}_{Q^S}H$. This change of numeraire is particularly useful for payments delivered in stock units.

For a test mass $M$ and isotropic background particles of mass $m$, the hyperbolic two-body problem uses the relative deflection scale $b_{90}=G(M+m)/w^2$. The test body's longitudinal velocity change in one encounter is $-2mw/[(M+m)(1+b^2/b_{90}^2)]$. Integrating the encounter rate over impact parameter gives a logarithm; replacing its velocity-dependent value by a common Coulomb logarithm in stellar dynamics gives the displayed systematic drag. The approximation $M+m\simeq M$ requires $M\gg m$, not merely $M>m$. Velocity diffusion and exact cutoff dependence are distinct effects omitted by this leading-log drag model.

For an isotropic, integrable background phase-space distribution function, rotational symmetry makes the integral radial in velocity space. Its divergence with respect to $\mathbf{V}$ is $4\pi f(V)$ by the Dirac delta identity for the divergence of the inverse-square vector field. Integrating over a velocity-space sphere and using the divergence theorem proves the displayed identity. Exterior velocity shells have zero contribution, just as in the spherical shell theorem. This explains why the leading-log Chandrasekhar dynamical friction integral contains only particles slower than the test body.

Let a weak tangential deceleration $D>0$ act on a test body moving along a slowly evolving family of circular orbits. Write its circular speed as $V_c(r)>0$ and its specific angular momentum as $j=r{V}_c(r)$. The specific torque is $\dot{j}=-rD$. Defining $q=d\log V_c/d\log r$ givesThis derivation follows angular-momentum balance; simply setting ${\dot{V}}_c=-D$ would omit the angular momentum changed by radial migration.

An adiabatic inward drift along radially stable circular orbits requires $1+q>0$. Indeed, with $\Omega =V_c/r$, the radial epicyclic frequency satisfiesFor this stable branch, the positive local timescale defined as $r/(2|\dot{r}|)$ is $(1+q)V_c/(2D)$. It is $V_c/D$ in a uniform-mass density core with $q=1$, and $V_c/(2D)$ for a flat galaxy rotation curve with $q=0$.

At $q=-1$, the circular-orbit angular momentum is locally independent of radius, so division by $1+q$ fails. For $q<-1$, the algebraic drift formula points outward, but those circular orbits are radially unstable; it is not a stable slow-inspiral solution. The quasistatic approximation also requires the drift time to exceed an orbital time. Finally, this local timescale is not automatically the remaining time to the center, which requires integrating the radius-dependent drift.

On $|{\Phi }^{+}⟩$, measure ${\sigma }_z\cos \theta +{\sigma }_x\sin \theta$, assigning bit $a$ to eigenvalue $(-1{)}^a$. Choose Alice's angles $2(i-1)\delta$ and Bob's angles $(2i-1)\delta$, with $\delta =\pi /(2N)$. The Bell state correlation is $\cos (\theta -\varphi )$. Each adjacent disagreement term and the last agreement term therefore has probability ${\sin }^2(\pi /(4N))$. For $N\ge 2$, their total violates the chained modular Bell inequality bound one, and it tends to zero as $N\to \infty$. At $N=1$ the value is exactly one.

// Target: probability-and-statistics.bigb

For a chain map $f:C\to C^{\prime }$, one mapping-cone convention takes $M(f{)}_i=C_{i-1}\oplus C_i^{\prime }$ with a differential combining $d,d^{\prime }$ and $f$. Its short exact sequence with $C^{\prime }$ and a shift of $C$ produces a long exact sequence in homology.

The chain differential of a homological chain complex is a family of group or module homomorphisms ${\partial }_n:C_n\to C_{n-1}$ satisfying ${\partial }_{n-1}{\partial }_n=0$. The identity implies $\mathrm{im}{\partial }_{n+1}\subset \ker {\partial }_n$, so the quotient $H_n=\ker {\partial }_n/\mathrm{im}{\partial }_{n+1}$ is defined. In a graded complex over $\mathbb{Z}[U]$, the chain differential can have degree $-1$ while $U$ has degree $-2$, with $\partial (Ux)=U\partial x$.

For an endomorphism of a finite-dimensional bounded chain complex over a field, the alternating chain trace equals the alternating trace on homology. Split each chain space into boundaries, representatives for homology, and a complement mapping isomorphically to the preceding boundaries. The chain-map relation makes the traces on the first and third summands cancel in successive degrees, leaving the homology traces. Over a field the induced cohomology maps are dual and have the same traces. This identity turns the absence of diagonal simplex contributions into a vanishing Lefschetz number.

For a field of characteristic other than two and nonzero $a,b$, the quaternion algebra has basis $1,i,j,ij$ with the displayed relations. It is a four-dimensional central simple algebra. After adjoining a square root of $a$, diagonalizing $i$ and taking $j=\left(\begin{array}{cc}0& b\\ 1& 0\end{array}\right)$ identifies it with $M_2$. Quaternion conjugation fixes scalars and negates $i,j,ij$, giving norm $x_0^2-a{x}_1^2-b{x}_2^2+ab{x}_3^2$. An element is invertible exactly when its norm is nonzero. Thus the quaternion algebra is either a division algebra or a split matrix algebra. The familiar quaternion algebra is $(-1,-1{)}_{\mathbb{R}}$.

For independent identically distributed increments of mean zero and variance one, the diffusively rescaled, linearly interpolated random walk converges weakly in $C[0,1]$ to Brownian motion.

For independent identically distributed mean-zero variables with finite positive variance, the central limit theorem gives a positive eventual lower bound for $\mathbb{P}(S_n/\sqrt{n}\ge K)$ for every finite $K$. The decreasing events ${\bigcup }_{n\ge m}\{S_n/\sqrt{n}\ge K\}$ have probabilities bounded below, so continuity from above of a measure gives positive probability that the limsup is at least $K$. The limsup itself is unchanged by removing any finite initial sum, hence is tail measurable. Kolmogorov zero-one law makes each event “limsup at least $K$” certain; intersect over integer $K$. The result is false for zero variance. One should apply the zero-one law to the limsup, rather than assume the exact threshold infinitely-often event is unaffected by a vanishing change.

For each $n$, let $Z_{n,i}$ be independent centered random variables. If the sum of their variances tends to ${\sigma }^2>0$ and ${\sum }_i\mathbb{E}[Z_{n,i}^2{1}_{\{|Z_{n,i}|>\epsilon \}}]\to 0$ for every $\epsilon >0$, their sum converges in distribution to the centered normal distribution with variance ${\sigma }^2$. The second hypothesis is the Lindeberg condition. If the total variance tends to zero, the corresponding degenerate limit follows directly from the Chebyshev inequality.

For particles with masses $m_j$ and positions $r_j$, Jacobi coordinates consist of the total center of mass and successive relative coordinates ${\rho }_j=r_{j+1}-({\sum }_{l=1}^j{m}_l{r}_l)/({\sum }_{l=1}^j{m}_l)$. They separate translation from internal motion and diagonalize the kinetic energy with the corresponding reduced masses.

Somatic cell nuclear transfer introduces a body-cell cell nucleus into an ovum whose own nuclear contribution has been removed or inactivated. The donor nuclear genome and recipient cytoplasm have different origins; successful development does not show that the donor cell nucleus acts independently of the recipient's machinery.

A phosphorimager reads stored radiation exposure from a reusable phosphor screen to form a digital image. In molecular experiments, spatial signal can quantify radiolabel distributed across a gel or blot. Background, saturation, exposure and detector response affect the inferred amount of labelled material. A band intensity therefore requires calibration and normalization before being interpreted as a molecule count or reaction rate. ^[ref]

Histone H1 is the protein used as a substrate for the Cdc2 activity readout in the cell-cycle hysteresis experiment.

A Xenopus laevis egg extract is a preparation of egg contents retaining biochemical machinery of the cell cycle outside intact biological cells.