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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.

Attaching cells of dimension $i+1$ to a connected CW complex, for $i\ge 2$, preserves all homotopy groups below degree $i$. In degree $i$ it takes the quotient by the subgroup generated by the attaching maps and their translates under the fundamental group action. This follows from the relative homotopy sequence and the generation of the first nonzero relative group by the new cells. In particular, a single attaching map representing a generator of ${\pi }_i\cong \mathbb{Z}$ kills that group even when the fundamental group is nontrivial. Higher attachments can create higher homotopy groups, which may then be killed at later stages.

For continuous boundary data on the counterclockwise unit circle and its Cauchy transform $\Phi$, $\Phi (r\omega )-\Phi (r^{-1}\omega )=P\varphi (r\omega )$ for $0<r<1$. This follows by subtracting the two Cauchy kernels and obtaining the Poisson kernel on the circle. Consequently the paired interior-minus-exterior difference tends to $\varphi (\omega )$ as $r\uparrow 1$. For $r\downarrow 1$ from above, the same written difference has the opposite sign. This paired limit does not require each individual boundary value of the transform to exist.

// Destination: analysis.bigb

The principal stresses are the eigenvalues of the symmetric Cauchy stress tensor. In their orthonormal eigenbasis, each traction is normal to its corresponding principal plane.

The component of traction normal to a surface with unit normal $n$. For an incompressible Newtonian fluid, $\sigma =-pI+2\mu e$, so ${\sigma }_{nn}=-p+2\mu n\cdot en$. A free-interface normal-stress jump includes the curvature force from surface tension.

If a sequence in a metric space has ${\sum }_{j=1}^{\infty }d(x_{j+1},x_j)<\infty$, the triangle inequality gives $d(x_m,x_n)\le {\sum }_{j=n}^{m-1}d(x_{j+1},x_j)$ for $m>n$. The tail of this convergent series tends to zero and bounds every later pair, proving the Cauchy sequence condition. In particular, an increment bound $d(x_{j+1},x_j)\le C{r}^j$ with $C>0$ and $0<r<1$ gives $d(x_m,x_n)\le C{r}^n/(1-r)$ by the geometric series. Unlike checking a small latest step, this bounds all remaining movement.

This sufficient condition is not necessary. The real sequence $x_n=(-1{)}^n/n$ converges to zero and is therefore a Cauchy sequence, although $|x_{n+1}-x_n|=1/(n+1)+1/n$ has a divergent sum. Cancellation allows convergence despite infinite accumulated absolute values of the successive changes.

The tail diameter of a sequence in a metric space is $D_N=\sup \{d(x_m,x_n):m,n\ge N\}$, allowing $+\infty$. The bounds are nonincreasing as $N$ grows because fewer pairs remain. A sequence is a Cauchy sequence exactly when $D_N\to 0$. The supremum need not be attained: for $q_n=1-2^{-n}$, $D_N=2^{-N}$ even though no two finite terms achieve that difference. For $(-1{)}^n$, $D_N=2$ for every $N$. For the harmonic numbers, $D_N=+\infty$ for every $N$, because each remaining tail is unbounded.

The absolute value of a centered Cauchy distribution variable with scale $\gamma >0$ has this probability density function. Its positive support and doubled density distinguish it from the two-sided distribution. Its unit-scale case arises as the ratio of two independent, equally scaled half-normal distribution variables.

independent standard Cauchy distribution variables sum to a Cauchy law of scale $n$, so their sample mean has its original distribution for every sample size. This follows by induction using convolution of probability densities. It does not contradict the usual weak law of large numbers, since the Cauchy law has no finite first absolute moment and its expectation is undefined. Symmetry does not turn a divergent expectation into zero.

An axisymmetric smooth connected soap film spanning equal radius-$R$ rings at heights $\pm H$ has stationary profile $r=k^{-1}\cosh (kz)$, $k>0$. The boundary equation is $R/H=\cosh u/u$, $u=kH$. Its unique minimum occurs at $u\tanh u=1$, because the derivative numerator is $u\sinh u-\cosh u$. If $A$ is this root, the minimum is $\cosh A/A=\sinh A$. There are no catenoids below that ratio, one at equality, and two above it. Existence of stationary connected films does not assert global minimality against pinching or disconnected competitors.

Take two distinct objects and one nonidentity arrow for each element of $S$, without arrows between different copies. A functor from this category to $\mathcal{C}$ is exactly a choice of one arbitrary arrow of $\mathcal{C}$ for each element of $S$. Consequently this construction is left adjoint to the functor taking the set of all arrows of a small category. Its free object has three arrows per generator, including the two identity arrows.

The discrete category on a set has that set as its objects and only identity morphisms. Functors from it are exactly assignments of objects in the target category.