An element $x$ of a C-star algebra is normal when $x^{\ast }x=x{x}^{\ast }$.

An element $u$ of a unital C-star algebra is unitary when $u^{\ast }u=u{u}^{\ast }=1$.

An element $x$ of a C-star algebra is Hermitian when $x=x^{\ast }$. Its spectrum is real, its norm equals its spectral radius, and it is the difference of two positive elements.

Brownian loop measure is a conformally invariant sigma-finite measure on unrooted planar Brownian loops. Taking outer boundaries connects it to conformal-restriction measures on self-avoiding loops.

For Brownian motion, $B_t/t\to 0$ almost surely as $t\to \infty$. Apply the strong law of large numbers to unit-time increments and control the oscillations on the intervening unit intervals.

Planar Brownian motion is the random vector $(B_t^{(1)},B_t^{(2)})$ formed from two independent standard one-dimensional Brownian motions.

The occupation time of a measurable set $A$ through time $t$ is ${\int }_0^t{1}_{\{B_s\in A\}}ds$. The occupation-times formula expresses such integrals through local time of a semimartingale. Recurrence and the Strong Markov property imply that one-dimensional Brownian motion spends an unbounded total time in every nonempty open interval.

One-dimensional reflected Brownian motion on $[0,\infty )$ has the same law as $|B_t|$. The Tanaka formula represents it as a Brownian motion plus a nondecreasing local time of a semimartingale that grows only at zero.

A Bessel process of dimension $d$ is the radial part of $d$-dimensional Brownian motion. Away from zero it solves the stochastic differential equationThis equation also defines noninteger dimensions through its weak solutions.

The zero set $\{t\ge 0:B_t=0\}$ of one-dimensional Brownian motion is almost surely closed, uncountable, and has zero Lebesgue measure. The last fact follows from Tonelli theorem because $\mathbb{P}(B_t=0)=0$ for every $t>0$.

On every compact time interval, a Brownian path is almost surely a Hölder continuous function of every exponent $\alpha <1/2$, and of no exponent $\alpha \ge 1/2$. In particular, $B_t=o(t^{\gamma })$ as $t\downarrow 0$ for every $\gamma <1/2$ after choosing a Hölder exponent strictly between $\gamma$ and $1/2$.

Wiener measure is the probability distribution of Brownian motion on a continuous-path space, usually $C_0([0,T])$ or $C_0([0,\infty ))$.

A parametric bootstrap simulates repeated datasets from a fitted parametric model and refits the procedure to each dataset. The resulting empirical distribution approximates the sampling distribution under that fitted model.

A bootstrap sample of size $n$ is drawn by sampling with replacement from the $n$ observed data slots. Some observations occur repeatedly. For a given slot, every draw misses it with probability $1-1/n$, so its omission probability after $n$ independent draws isBy linearity of expectation, the expected number of omitted slots is $n(1-1/n{)}^n$, whether or not their omission events are independent. For the three-observation exact bootstrap distribution for three observations, an individual slot is omitted with probability $8/27$ and the expected number of omitted slots is $8/9$. This finite-sample calculation explains both repeated values and incomplete coverage; $e^{-1}$ is a limiting expected proportion, not an exact fraction in every resample.

For an adiabatic gas with $\gamma =5/3$, asymptotic density ${\rho }_0$, and sound speed $v_{s0}$ accreting onto a Newtonian point mass $M$, the largest globally subsonic steady rate is

For an oscillator whose positive frequency jumps from ${\omega }_{-}$ to ${\omega }_{+}$, continuity of the normalized mode and its first derivative gives the Bogoliubov coefficientsThe initial vacuum therefore contains $|\beta {|}^2$ final particles in that mode.

A formula $\Phi (x,\kappa )$ defines a $\beta$-stable property of $\kappa$ when it is absolute between the universe and every transitive $M\supseteq V_{\kappa +\beta }$.

The posterior mean is the expected value of a parameter under its posterior distribution; under squared-error loss it is the Bayes estimator.

A posterior density is the density of a posterior distribution with respect to a chosen reference measure.

In vacuum geodesic slicing, discarding nonnegative shear leaves ${\partial }_{t}K=K^2/3$. Positive initial $K_0$ therefore gives $K=K_0/[1-K_0(t-t_0)/3]$ and blows up at $t=t_0+3/K_0$.