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Energy measured at the resource or initial-input stage before downstream conversion to final useful energy. Comparisons with electrical output must state the accounting convention and avoid treating a primary-energy thermal input as the same service as delivered electricity.

A product released by reactions needed for energy acquisition, growth or biomass construction. Methane from energy-yielding microbial reactions and molecular oxygen from oxygenic photosynthesis are examples. Abiotic chemistry can make some of the same molecules, so production mechanism and environmental abundance matter.

A primary producer builds organic chemical compounds from inorganic sources. Photosynthetic algae, plants and cyanobacteria use light energy for this role through photosynthesis and carbon fixation. The role supplies new living material to a food web; it does not mean that the organism creates energy.

A receptor's absorption-dependent response does not independently label the wavelength of an absorbed photon. Different combinations of spectral composition and intensity can yield the same receptor quantum catch. Comparing spectrally distinct receptor classes can supply information unavailable from one alone. In the displayed model, $N$ is a photon distribution and $a$ effective absorptance; an energy spectrum would require the corresponding photon-energy conversion.

On a spacelike constant-target-time slice, integrate the induced spatial line element to obtain the geometric string length. It depends on the embedding and is distinct from the fundamental string length ${\ell }_s=\sqrt{{\alpha }^{\prime }}$. For transverse local motion, energy additionally includes the local Lorentz factor; the geometric length alone does not determine the total energy.

An affine parameter on a timelike geodesic can be scaled to proper time. With metric signature $(+---)$ this fixes the tangent-vector norm to $c^2$. A further constant rescaling preserves the affine geodesic equation but changes that norm; conserved energy and angular-momentum constants must be rescaled consistently. The unit term in a proper-time-normalized radial energy equation is consequently a normalization choice, not a property of every affine parameter.

A membrane transport system that uses an energy source to move protons against their electrochemical gradient.

With bare nuclear masses, the proton separation energy is the energy needed to remove a proton. A negative value means the daughter plus proton has lower rest energy. Proton emission can still have a finite lifetime because of a Coulomb or centrifugal barrier.

A protoplanet is a growing planetary body within a planet-forming system. It has accumulated enough mass for gravity to strongly influence subsequent collisions and accretion, while its final orbit, composition and mass may still change substantially.

Accreting a small mass $\Delta m$ onto a body of mass $M$ and radius $R$ can release energy of order $GM\Delta m/R$. Heating, melting and chemical differentiation can therefore accompany growth. A protoplanet may continue to accumulate, merge with another body, lose material or be scattered; the name describes a stage of evolution rather than guaranteeing a particular final planet.

A protostar is a forming star whose energy comes chiefly from gravitational contraction and accretion, before sustained core hydrogen burning supplies its luminosity. A fixed-mass contraction model approximates the stage after most accretion has ended; an actively accreting protostar need not have fixed mass or the same Hayashi track.

A pseudo-pure state is a density matrix of the formwhere $|\psi ⟩$ is a normalized pure state, $I$ is the identity matrix on a $d$-dimensional space and here $0\le p\le 1$. The term $I/d$ is the maximally mixed state. Under a unitary transformation $U$, this term stays fixed and the selected pure state becomes $U|\psi ⟩$. This is why liquid-state NMR quantum computing can follow pure-state-like transformations in part of a highly mixed ensemble.^[ref]

For a traceless matrix representing an observable $O$, direct evaluation of the trace givesThe identity contribution vanishes because $\mathrm{tr}O=0$. Thus $p$ scales the detectable expectation. It is not itself the purity: expanding ${\rho }^2$ and taking its trace yieldsFor example, for two qubits with $p=1/10$, the purity is $103/400$, close to the maximally mixed value $1/4$, although the selected pure state still controls the traceless signal.

The preparation penalty can also be derived explicitly. Take $n$ independent equal spin-one-half systems with one-spin density matrix $(I+\eta Z)/2$, where $Z$ is a Pauli matrix and $0\le \eta \le 1$ is their individual polarization. Their largest population is $a=((1+\eta )/2{)}^n$. A preparation that preserves this population and averages all other populations to equality has $d=2^n$ andIndeed the selected population of $\rho$ is $[1+(d-1)p]/d=a$. In the weak-polarization regime $n\eta \ll 1$, the binomial theorem gives $p\simeq n\eta /(2^n-1)$. If the energy separation is $\hslash \omega$, set $ϵ=\hslash \omega /(k_{B}T)$; a thermal two-level system has $\eta =\tanh (ϵ/2)\simeq ϵ/2$. This gives $p\simeq nϵ/[2(2^n-1)]$, with every parameter defined. Here $\hslash$ is the reduced Planck constant, $k_B$ the Boltzmann constant and $T$ the temperature. The exponential signal loss concerns this specified averaging preparation; it is not a universal bound on every initialization method.

In a uniform isotropic material, these scalar P-SV displacement potentials separate longitudinal P waves from vertically polarized SV-waves in an $x,z$ plane. They satisfy ${\varphi }_{tt}={\alpha }^2\Delta \varphi$ and ${\psi }_{tt}={\beta }^2\Delta \psi$, with compressional and shear speeds $\alpha ,\beta$. Reversing the sign convention for $\psi$ changes its polarization column but not the physical P-SV directional impedance matrix or the separate modal energy fluxes.

Loss of a rotating neutron star's angular momentum and rotational energy through electromagnetic torque and a magnetized outflow. For constant moment of inertia $I$, differentiating $E_{\mathrm{rot}}=I(2\pi /P{)}^2/2$ gives the displayed positive loss rate when $\dot{P}>0$. It is not the same as the measured radiation luminosity in any one observing band.

A circular closed string contracts and re-expands with $X^0=Rt$ and $X^1+i{X}^2=R\cos t{e}^{i\sigma }$. Its proper length is $2\pi R|\cos t|$, while its energy remains $2\pi TR$. At collapse the induced worldsheet metric degenerates, but the Nambu-Goto phase-space action supplies a regular continuation.

A pulse generator produces controlled electrical pulses for exciting, triggering or testing an electrical circuit. The useful specification includes high and low levels, pulse width, repetition period, edge behavior and the timing relationship to a trigger. A repeating pulse train and a single triggered pulse impose different test conditions even when their individual pulse shapes agree.

For an original ideal train with low level zero, high level $V_p$, pulse width $\tau$ and period $T$, define duty cycle $D=\tau /T$ and repetition frequency $f=1/T$. Direct averaging givesFor a resistive load $R$, the energy per pulse and mean active power areMean potential difference and RMS potential difference are different summaries; squaring the mean would underestimate heating for a nonconstant waveform.

Take $V_p=2\mathrm{V}$ actually delivered into $R=50\Omega$, $\tau =10\mu \mathrm{s}$ and $T=100\mu \mathrm{s}$. Then $D=0.1$, $f=10\mathrm{kHz}$, mean potential difference is $0.2\mathrm{V}$, and RMS potential difference is about $0.6325\mathrm{V}$. The pulse deposits $0.8\mu \mathrm{J}$ and average load power is $8\mathrm{mW}$. These are stipulated rectangular-pulse values, not instrument limits.

Real pulse width needs a threshold definition. Keysight's primary control documentation measures it between the rising and falling 50% crossings and provides duty-cycle entry as an alternative.^[ref] Rise time, edge asymmetry and trigger timing must be measured or specified separately; a nominal width alone does not describe the full waveform.

Termination also changes what reaches the load. The existing Keysight documentation explains a fixed series output impedance and the difference between a selected load setting and the actual connected load. Its example gives twice the displayed amplitude at an open circuit when the display assumes a matched 50-ohm load.^[ref] Use the delivered pulse levels in the energy calculation.

A proposed test connects the intended termination, records the pulse at the load with an oscilloscope, and checks levels, crossing times and repetition intervals. Varying pulse width at fixed amplitude and repetition rate should change resistive average power in proportion to $\tau$ under the ideal model. Loading and finite edge durations explain departures that should not automatically be assigned to the programmed timing.

The pulse repetition frequency counts the number of pulses emitted per unit time in a regular pulse train. If successive pulses are separated by $T_{\mathrm{rep}}$, then $f_{\mathrm{rep}}=1/T_{\mathrm{rep}}$. It describes the spacing of pulses, rather than the rapid oscillation of a carrier inside each pulse or the inverse of the individual pulse duration.

For example, pulses sent every $1\mathrm{ms}$ have pulse repetition frequency $1\mathrm{kHz}$ even if their carrier is in the radio frequency range. With identical pulses of energy $E_p$, the average emitted power is $E_p{f}_{\mathrm{rep}}$. In an elementary pulsed radar model, an echo must return before the next transmission to avoid pulse-order ambiguity, giving an unambiguous range $R<c/(2f_{\mathrm{rep}})$. The factor two comes from outward and return travel. More elaborate pulse coding can change how the instrument resolves that ambiguity.

Pulse width is the time separation of specified crossings on a pulse's leading and trailing edges. The threshold convention is essential when the edges are not instantaneous. Keysight's primary pulse-generator documentation uses the rising and next falling 50-percent crossings, and allows a duty-cycle setting as an alternative control.^[ref]

For low and high levels $V_L$ and $V_H$, define the fractional threshold as $V_L+a(V_H-V_L)$ with $0<a<1$. A positive pulse's width at that level is $\tau (a)=t_{\downarrow }(a)-t_{\uparrow }(a)$. The half-height convention is related to full width at half maximum for an isolated positive pulse above a defined baseline; it is not necessarily the duration of the flat top.

For an original asymmetric trapezoid, let a full linear rise take $t_r=2\mu \mathrm{s}$, the plateau last $H=8\mu \mathrm{s}$, and a full linear fall take $t_f=4\mu \mathrm{s}$. These ramp durations refer to the full low-to-high excursions, not the commonly quoted 10–90 percent rise interval. The crossing times giveThe 50-percent width is $11\mu \mathrm{s}$, while the first departure from the low level to final return lasts $14\mu \mathrm{s}$. At a 90-percent threshold the width is only $8.6\mu \mathrm{s}$. All describe the same waveform under different conventions.

For a repeating pulse train, duty cycle compares the specified width to the repetition period. Finite ramps mean that width times peak amplitude alone need not give the waveform integral or energy. Electrical loading and bandwidth also change the delivered crossings; the pulse generator setting must be checked at the actual load.

A measurement records both edges on an oscilloscope with a common level definition. Pulse-width uncertainty from correlated edge timing depends on $\mathrm{Var}(t_{\downarrow }-t_{\uparrow })$, including their covariance. A common trigger-time translation can cancel from width while remaining in absolute arrival time. Ringing with several crossings, hysteresis, negative pulses, or multiple peaks needs an explicit crossing-selection convention rather than an unspecified single duration.

A pump supplies mechanical energy to a fluid so that it can flow against resistance or gain elevation. For a steady incompressible stream, a rise in pressure $\Delta p$ at similar inlet and outlet heights and speeds represents useful fluid power $P=Q\Delta p$, where $Q$ is the volumetric flow rate. Differences in kinetic energy and gravitational potential energy must be included when those conditions do not hold.

A real installation chooses its flow through the interaction between the pump's pressure-flow response and the connected passages. More resistance generally shifts the pump operating point, so the nameplate or rotational speed alone does not determine mass flow rate. Measuring inlet/outlet pressures, flow and supplied mechanical power separates useful fluid output from dissipative losses. A centrifugal pump uses a rotating impeller; a displacement device instead moves a defined volume during a cycle, with leakage and compressibility affecting the actual delivery.

Water is pumped to an upper reservoir and later discharged through a turbine. For approximately constant head, useful discharge energy is $\eta \rho Vgh$ while power is $\eta \rho Qgh$; reservoir volume and flow rating set different properties.

A four-force preserving the particle's rest mass. For a massive particle, $P=mU$ and $U^2=c^2$ in the mostly-minus convention imply $F\cdot U=c^{2}dm/d\tau$. Thus constant mass is equivalent to orthogonality of the force and four-velocity. Energy transfer in a moving frame is allowed even though rest mass does not change.