For a survival time with an absolutely continuous probability distribution, probability density function $f$, and survival function $S$, the hazard function iswhere $S(t)>0$, with the derivative identity holding almost everywhere. At points where the local limit exists, equivalentlyThe hazard function measures an instantaneous conditional rate. If time is in years, its units are inverse years, and its value can exceed one. A probability for a short interval is approximately $h(t)\Delta$, not $h(t)$; for an absolutely continuous probability distribution, the probability of an occurrence at the single exact time $t$ is zero. The discrete hazard, by contrast, is a conditional probability at a possible event time.

The factorization $f(t)=h(t)S(t)$ distinguishes a probability density function for the original population from a hazard function for those still event-free. With an exponential distribution of rate $0.2$ per year, $h(5)=0.2$ per year but $f(5)=0.2e^{-1}\approx 0.073576$ per year: the latter also accounts for the probability of reaching year five. The exponential survival risk over an interval example computes finite-horizon probabilities from this rate.

In a matter-plus-cosmological-constant universe, changing the independent variable from cosmic time to $a$ gives

When $H$ is constant and the matter source is negligible,so ${\delta }_m=C_1+C_2{a}^{-2}$. The growing matter-era mode therefore approaches a constant after cosmological-constant domination begins.

Every positive-energy supermultiplet contains equally many bosonic and fermionic states. A supertrace of a positive supercharge anticommutator proves the equality, provided supersymmetry is exact and the representation is finite-dimensional at fixed momentum.

Number-phase conjugacy is the canonical relation between a conserved $U(1)$ charge and the phase of its order parameter. A state of sharp number cannot also have a sharply selected broken-symmetry phase.

The superspin Casimir classifies massive supermultiplets analogously to the Pauli-Lubanski Casimir for Poincare representations. One construction contracts the tensor $C_{\mu \nu }=B_{\mu }{P}_{\nu }-B_{\nu }{P}_{\mu }$, where $B_{\mu }$ is the supersymmetric correction of the Pauli-Lubanski pseudovector.

Under standard assumptions on a nontrivial analytic relativistic S-matrix, the Coleman–Mandula theorem says that every continuous bosonic symmetry algebra is a direct sum of the Poincare algebra and an internal symmetry algebra.

For proper convex functions under a standard relative-interior qualification,In particular, if $f$ is differentiable at $x$, then $\partial (f+g)(x)=\nabla f(x)+\partial g(x)$.

A Regge trajectory is an approximately linear relation between angular momentum and squared mass. A classical rotating open string obeys $J={\alpha }^{\prime }{M}^2$, with quantum effects adding an intercept.

A body or swimmer is torque-free when the resultant external torque, including the moment of hydrodynamic traction, vanishes.

A body or swimmer is force-free when the resultant external force, including the integral of hydrodynamic traction, vanishes.

A microswimmer is a microscopic body that propels itself through a fluid. At small Reynolds number its motion is governed by Stokes flow, and self-propulsion must respect the force-free and torque-free conditions.

A sphere of radius $a$ translating with velocity $\mathbf{U}$ through otherwise stationary fluid hasIts surface traction integrates to the Stokes drag law.

A counting process is an integer-valued stochastic process that starts at zero and increases by recording the cumulative number of events up to each time.

Two stochastic processes $X,Y$ are indistinguishable whenThis is stronger than equality almost surely at each fixed time, though the two notions agree for continuous processes after checking equality on a countable dense set.

If $X$ has continuous paths and ${\tau }_s$ is a finite continuous increasing family of random times, then $X_{{\tau }_s}$ also has continuous paths. A strictly increasing continuous clock that tends to infinity has a continuous inverse.

A càdlàg process has sample paths that are càdlàg functions: they are right-continuous and have a finite left limit at every positive time.

For a Markov process with transition semigroup $(P_t)$, its infinitesimal generator is the operatoron the functions for which this limit exists.

Chordal Schramm–Loewner evolution in the complex upper half-plane from $0$ to $\infty$ is the random Loewner chain driven by $U_t=\sqrt{\kappa }{B}_t$, where $B$ is standard Brownian motion and $\kappa \ge 0$.

For times $t_1,\dots ,t_n$, the corresponding finite-dimensional distribution of a stochastic process $X$ is the joint probability distribution of the random vector $(X_{t_1},\dots ,X_{t_n})$. The collection of these distributions records every finite set of coordinates of the process.

Knowing only the one-time marginal distributions is insufficient. Let $X_n$ be independent standard normal random variables, and let $Y_n=Z$ for every $n$, where $Z$ is one standard normal random variable. Every $X_n$ and $Y_n$ has the same standard normal distribution. Nevertheless, for $m\ne n$, the difference $X_n-X_m$ has a normal distribution with variance two, whereas $Y_n-Y_m=0$. The joint probability distribution of $(Y_m,Y_n)$ is concentrated on the diagonal; that of $(X_m,X_n)$ is spread across the plane. A model for tomorrow's value therefore needs the relation between observations as well as their separate distributions.

For a real-valued stochastic process, its finite-dimensional distributions determine its probability law on the product sigma-algebra. With an uncountable index set, a path property such as continuity need not be measurable in that sigma-algebra; specifying a suitable space of sample paths and a regular modification of a stochastic process is a separate step. This distinction is explained in MIT, The basics of stochastic processes, section 1.1.