The piezoelectric contribution to temperature-dependent electric polarization caused by thermal strain.

A device that delivers electrical charge during changes of temperature of a polar material.

The temperature derivative of a component of spontaneous electric polarization under stated mechanical/electrical constraints. An electrode of area $A$ has ideal short-circuit response $I=pAdT/dt$. Mechanical constraints distinguish primary and secondary contributions.

A transient pressure-supported stage of a collapsing cloud after rising optical depth inhibits cooling. Continued accretion and changes of the molecular equation of state can lead to further collapse. It is distinct from the later compact stellar core.

An accelerated contraction phase associated with molecular hydrogen dissociation absorbing compression energy and reducing effective pressure support. A compact stellar core forms when the gas can again oppose contraction. Its timing and geometry depend on the accretion and thermodynamic history.

During continued compression of a first hydrostatic core, molecular hydrogen dissociation absorbs energy and softens the effective equation of state. This can trigger a second rapid contraction before a compact stellar core becomes pressure-supported. It is distinct from the initial nearly isothermal collapse and from later Kelvin-Helmholtz contraction at roughly fixed mass.

A regime in which collisions disrupt aggregates faster than sticking can build larger bodies. The outcome depends on collision speed, porosity and material properties; it is distinct from loss of intact aggregates through radial drift of solids in a gas disk.

An ATP-driven proton pump that acidifies many eukaryotic endomembrane compartments.

Hydrolysis of ATP drives proton movement and helps establish an acidic compartment. The resulting electrochemical gradient supports cargo processing and coupled transport; a low luminal pH is actively maintained rather than an intrinsic property of all membrane vesicles. ^[ref]

A prism with two hexagonal faces and six quadrilateral side faces. Its twelve vertices and eighteen edges provide an alternating aluminium–oxygen cage model when the graph is coloured bipartitely.

Consider a homogeneous optical prism with apex angle $0<A<\pi$, in the same exterior medium on both sides. Let $n>1$ be the relative refractive index, the inside refractive index divided by the exterior refractive index. For a ray crossing both faces in the usual principal section, the internal angles obey $r_1+r_2=A$. Snell's law gives external incidence and emergence angles $i=f(r_1)$ and $e=f(r_2)$, whereThe deviation is $\Delta =i+e-A$. On the interior of this interval,Thus convexity gives $f(r_1)+f(r_2)\ge 2f(A/2)$. The minimum occurs for the symmetric path $r_1=r_2=A/2$, yieldingA path with nongrazing entry and exit requires $n\sin (A/2)<1$. Equality describes only a grazing limit; a larger value prevents transmission across both faces in this section. The condition $n>1$ matters: if $n<1$, the sign of $f^{\prime \prime }$ reverses, so the symmetric path maximizes this signed deviation instead. Measuring the minimum deviation therefore determines $n=\sin ((A+{\Delta }_{\min })/2)/\sin (A/2)$ for an ordinary glass-in-air optical prism.

The prior mean is the expected value of a statistical parameter under its prior distribution, when this expected value exists. The posterior mean instead uses the posterior distribution after conditioning on observations.

A primary battery coupling lithium oxidation to reduction of fluorinated carbon, producing lithium fluoride and carbon.

A battery separator keeps the two electrodes from direct electronic contact while allowing the ionic transport needed for the cell reaction. It is electronically insulating but permeable to relevant ions, commonly through an electrolyte-filled porous structure.

The distinction between ionic and electronic paths makes the circuit work: useful electrons travel through the external load, while ions complete the internal charge balance. Separator thickness and pore structure affect internal resistance and transport. A very low ionic resistance alone is not a complete design goal, because the separator must also maintain separation and compatibility with the cell's chemical environment.

Battery self-discharge is loss of recoverable stored charge through internal processes while a battery supplies no useful external electric current. Parasitic reactions and internal leakage can change the charged state even when the external terminals are open.

A storage experiment compares recoverable capacity after controlled waiting intervals with an otherwise identical freshly charged or prepared cell. Holding temperature and subsequent discharge conditions fixed separates storage loss from a changed capacity test. Terminal voltage alone need not measure the lost capacity, because voltage also depends on chemistry and relaxation. Self-discharge is distinct from the intentional current drawn by a connected sleeping device.

Battery capacity is the amount of electric charge delivered between specified charged and discharged states under stated conditions. It is $Q=\int I(t)dt$, commonly reported in ampere-hours; $1\mathrm{Ah}=3600\mathrm{C}$.

Capacity is distinct from delivered energy, which is $\int V(t)I(t)dt$. Two batteries with equal ampere-hour ratings can deliver different energies because their voltages differ. Discharge rate, temperature, endpoint voltage and aging affect the measured capacity. Comparing ratings therefore requires the same test conditions and a stated choice of active-material or full-device mass when using specific capacity.

A primary battery using metallic lithium and iron disulfide. The overall discharge conversion consumes four lithium atoms and releases four electrons per $\mathrm{Fe}{\mathrm{S}}_2$ formula unit. This is distinct from the rechargeable host-to-host transport in a conventional lithium-ion battery.

The Barabási-Albert model constructs a growing random graph by preferential attachment. Starting from a suitable seed, each new vertex brings $m$ edges and selects existing endpoints with probability proportional to their current degrees. A convention for sampling distinct endpoints must be specified.^[ref]

Older high-degree vertices tend to attract more edges. In the standard large-graph limit, the degree distribution has a tail proportional to $k^{-3}$, unlike the sharply concentrated degrees of a basic Erdős-Rényi model. Both growth and the attachment rule matter. The model supplies a mechanism for hubs; observing a heavy degree tail alone does not establish that this mechanism generated the data.

For power series $f(z)={\sum }_{m\ge 0}{f}_m{z}^m$ and $g(x)={\sum }_{j\ge 1}{g}_j{x}^j$, their function composition is well defined as a formal power series: the coefficient of $x^n$ receives contributions only from $m\le n$, since $g$ has zero constant term. For $m\ge 1$, expand the coefficient asThe $m=0$ term is one for $n=0$ and zero otherwise. The first four coefficients of the function composition are therefore $f_0$, $f_1{g}_1$, $f_1{g}_2+f_2{g}_1^2$, and $f_1{g}_3+2f_2{g}_1{g}_2+f_3{g}_1^3$. For convergent power series, the same calculation holds wherever $g(x)$ lies inside the radius of convergence of $f$.

If $f(x)={\sum }_{n\ge 0}{a}_n{x}^n$ has $a_0\ne 0$, its reciprocal has a unique formal power series $1/f(x)={\sum }_{n\ge 0}{b}_n{x}^n$. Equating coefficients in the Cauchy product $f\cdot (1/f)=1$ gives the displayed recurrence relation. In particular,For a convergent power series, continuity and $f(0)\ne 0$ give a neighborhood without zeros, where the reciprocal is analytic and this series converges. Its radius need not equal the original radius: zeros of $f$ can introduce closer reciprocal singularities.

The local logarithmic slope with the displayed sign convention. Other fields sometimes use the opposite sign, so an index should be accompanied by its defining power-law spectrum.