The time to traverse an AU is found to be τ < sub > A </ sub > =, resulting in the astronomical unit in metres as c < sub > 0 </ sub > τ < sub > A </ sub > =.

where N < sub > 2 </ sub >( 0 ) is the number of excited atoms at time t = 0, and τ < sub > 21 </ sub > is the lifetime of the transition between the two states.

If the lifetime of this transition, τ < sub > 21 </ sub > is much longer than the lifetime of the radiationless 3 → 2 transition τ < sub > 32 </ sub > ( if τ < sub > 21 </ sub > ≫ τ < sub > 32 </ sub >, known as a favourable lifetime ratio ), the population of the E < sub > 3 </ sub > will be essentially zero ( N < sub > 3 </ sub > ≈ 0 ) and a population of excited state atoms will accumulate in level 2 ( N < sub > 2 </ sub > > 0 ).

In the following chemical equation with arrows pointing both ways to indicate equilibrium, A and B are reactant chemical species, S and T are product species, and α, β, σ, and τ are the stoichiometric coefficients of the respective reactants and products:

symmetric bilinear form with orthonormal basis v < sub > i </ sub >, the map sending a lattice to its dual lattice gives an automorphism with square the identity, giving the permutation σ that sends each label to its negative modulo n. The image of the above homomorphism is generated by σ and τ and is isomorphic to the dihedral group D < sub > n </ sub > of order 2n ; when n = 3, it gives the whole of S < sub > 3 </ sub >.

Wick rotation also relates a QFT at a finite inverse temperature β to a statistical mechanical model over the " tube " R < sup > 3 </ sup >× S < sup > 1 </ sup > with the imaginary time coordinate τ being periodic with period β.

As an example of this formula, if Δ = 1 / e < sup > 4 </ sup > = 1. 8 %, the settling time condition is t < sub > S </ sub > = 8 τ < sub > 2 </ sub >.

In general, control of overshoot sets the time constant ratio, and settling time t < sub > S </ sub > sets τ < sub > 2 </ sub >.

If α is increased by shortening τ < sub > 2 </ sub >, the settling time t < sub > S </ sub > also is shortened.

If α is increased by lengthening τ < sub > 1 </ sub >, the settling time t < sub > S </ sub > is little altered.

A general chemical reaction in which α moles of a reactant A and β moles of a reactant B react to give σ moles of a product S and τ moles of a product T can be written as

In the specific case in which S is an inverse semigroup σ is the smallest congruence on S such that S / σ is a group, that is, if τ is any

other congruence on S with S / τ a group, then σ is contained in τ.

Given a topological vector space ( X, τ ) over a field F, S is called bounded if for every neighborhood N of the zero vector there exists a scalar α so that

A map τ: X → X is said to be Σ-measurable if and only if, for every σ ∈ Σ, one has.

A map τ is said to preserve the measure if and only if, for every σ ∈ Σ, one has.

Combining the above, a map τ is said to be a measure-preserving transformation of X, if it is a map from X to itself, it is Σ-measurable, and is measure-preserving.

The quadruple ( X, Σ, μ, τ ), for such a τ, is then defined to be a dynamical system.

For continuous dynamical systems, the map τ is understood to be a finite time evolution map and the construction is more complicated.

where τ is the golden ratio, ( 1 +√ 5 )/ 2.

: where E is the energy, τ is magnitude of the torque, and θ is the angle moved ( in radians ).

Since the lifetime of the laser transition L is long compared to that of Ra ( τ < sub > 32 </ sub > ≫ τ < sub > 43 </ sub >), a population accumulates in level 3 ( the upper laser level ), which may relax by spontaneous or stimulated emission into level 2 ( the lower laser level ).

In rotational systems, power is the product of the torque < var > τ </ var > and angular velocity < var > ω </ var >,

The map τ embodies the time evolution of the dynamical system.

A superparamagnetic system will show a characteristic frequency dependence: When the frequency is much higher than 1 / τ < sub > N </ sub >, there will be a different magnetic response than when the frequency is much lower than 1 / τ < sub > N </ sub >, since in the latter case, but not the former, the ferromagnetic clusters will have time to respond to the field by flipping their magnetization.

The Allan variance depends on the time period used between samples: therefore it is a function of the sample period, commonly denoted as τ, likewise the distribution being measured, and is displayed as a graph rather than a single number.

An Allan deviation of 1. 3 × 10 < sup >− 9 </ sup > at observation time 1 s ( i. e. τ = 1 s ) should be interpreted as there being an instability in frequency between two observations a second apart with a relative root mean square ( RMS ) value of 1. 3 × 10 < sup >− 9 </ sup >.

where the average is taken over observation time τ, the y ( t ) is the fractional frequency error at time t and τ is the observation time.

For Allan variance, the time being used has T set to the observation time τ.

A time series taken for one time-difference τ < sub > 0 </ sub > can be used to generate Allan variance for any τ being an integer multiple of τ < sub > 0 </ sub > in which case τ = nτ < sub > 0 </ sub > is being used, and n becomes a variable for the estimator.

* The time between measurements is denoted with T, which is the sum of observation time τ and dead-time.

If taking the time-series and skipping past n − 1 samples a new ( shorter ) time-series would occur with τ < sub > 0 </ sub > as the time between the adjacent samples, for which the Allan variance could be calculated with the simple estimators.

* Conversion of cutoff frequency f < sub > c </ sub > and time constant τ

Under a constant torque of magnitude τ, the gyroscope's speed of precession Ω < sub > P </ sub > is inversely proportional to L, the magnitude of its angular momentum:

where s is the Laplace transform variable, τ is the filter time constant, and K is the filter passband gain.

The product of the resistance and capacitance ( R × C ) is the time constant ( τ ); it is inversely proportional to the cutoff frequency f < sub > c </ sub >, that is,

The intrinsic thermal time constant is τ = C / G.

The capacitor and the load resistance have a typical time constant τ = RC where C and R are the capacitance and load resistance respectively.

Figure 3: The amplitude of a wavepacket whose amplitude changes significantly in time τ < sub > c </ sub > ( red ) and a copy of the same wave delayed by 2τ < sub > c </ sub >( green ) plotted as a function of time t. At any particular time the red and green waves are uncorrelated ; one oscillates while the other is constant and so there will be no interference at this delay.

The mean lifetime ( also called the exponential time constant ) can be looked at as a " scaling time ", because we can write the exponential decay equation in terms of the mean lifetime, τ, instead of the decay constant, λ:

The product of τ ( tau ) = RC is called the time constant of the circuit.

where F < sub > rnd </ sub > is a random force representing the random collisions of the particle and the surrounding molecules, and where the time constant τ reflects the drag force that opposes the particle's motion through the solution.

The drag force is often written F < sub > drag </ sub > = − γv ; therefore, the time constant τ equals m / γ.

However, on long time scales, with t >> τ, the exponential and constant terms are negligible, and the squared distance grows only linearly:

where τ < sub >≤ p </ sub > is a truncation functor in the derived category, and i < sub > k </ sub > is the inclusion of X − X < sub > n − k </ sub > into X − X < sub > n − k − 1 </ sub > and C < sub > X − Xn − 2 </ sub > is the constant sheaf on X − X < sub > n − 2 </ sub >.

where τ is a very small time constant which causes this equation to reduce to the normal form of Darcy's law at " normal " times (> nanoseconds ).

The decay constant, τ, which is the time taken for the intensity of light to fall to 1 / e of the initial intensity, is called the ring-down time and is dependent on the loss mechanism ( s ) within the cavity.

The curves of constant σ and of τ are circles that intersect at right angles.

The corresponding circles of constant σ and τ are shown in red and blue, respectively, and meet at right angles ( magenta box ); they are orthogonal.

The variable q is a constant multiple of the proper time τ for timelike orbits ( which are traveled by massive particles ), and is usually taken to be equal to it.

is excited by some source and than dying away with a time constant τ < sub > D </ sub > of the order of 100 years.

The upper FET gate is electrically grounded, so charge and discharge of stray capacitance C < sub > dg </ sub > between drain and gate is simply through R < sub > D </ sub > and the output load ( say R < sub > out </ sub >), and the frequency response is affected only for frequencies above the associated RC time constant: τ