For future-directed unit timelike four-velocities in metric signature $(+---)$, $U_1\cdot U_2/c^2$ is the Lorentz factor of either observer measured in the rest frame of the other. In a common inertial frame it equals ${\gamma }_1{\gamma }_2(1-{\mathbf{\beta }}_1\cdot {\mathbf{\beta }}_2)$. ThereforeExpanding the numerator converts this to the dot-and-cross-product form of relativistic velocity addition. The invariant construction explains why the relative speed is symmetric even when the spatial velocity components belong to different rest frames.

The complexification of a real vector space $V$ allows complex scalar coefficients while preserving its real linear relations. It can be constructed as the tensor product $V{\otimes }_{\mathbb{R}}\mathbb{C}$, or as formal sums $u+iv$ with $u,v\in V$. A real basis of $V$ becomes a complex basis of $V_{\mathbb{C}}$, so its complex dimension equals the original real dimension. Every real linear map extends uniquely by complex linearity. In particular a real representation extends by viewing its real matrices as complex matrices. Complexifying the realification of a complex representation produces the original complex representation together with its complex conjugate; the coordinate change $(u,v)\mapsto (u+iv,u-iv)$ exhibits the two blocks.

A member of a vascular-plant lineage generally reproducing through spores rather than seeds. Most familiar fern bodies are sporophytes; their gametophytes can have different environmental requirements. heterospory and other life-cycle differences prevent treating every fern as one identical model.

A plant lineage reproducing through seeds, including gymnosperms and angiosperms. Reduction and retention of gametophytes and male delivery through pollen distinguish their reproductive ecology from a typical free-living fern gametophyte.

Named after Fritz London, this is an attractive interaction between fluctuating electric dipoles of atoms or molecules, present even without a permanent dipole. At large separation compared with atomic size in the nonretarded regime, the dipole interaction scales as $r^{-3}$ and its second-order ground-state energy correction is negative and scales as $r^{-6}$. Differentiating the energy gives an attractive force of magnitude $6C_6/r^7$. The short-distance overlap regime and the very-long-distance retarded regime require different descriptions.

A pairwise-additive approximation integrating microscopic dispersion interactions over body volumes. For two nonretarded homogeneous half-spaces, it gives interaction energy per area $-A_H/(12\pi d^2)$.

A Van der Waals potential describes the separation-dependent potential energy associated with Van der Waals forces. For two neutral, nonpolar particles at separations where the nonretarded dispersion description applies, a familiar attractive contribution is $U(r)=-C_6/r^6$ with $C_6>0$. Its radial force is $-dU/dr=-6C_6/r^7$, directed toward smaller separation.

At sufficiently short separation, overlapping electronic states produce strong exchange repulsion; continuing the attractive power law to zero separation is not a physical model. A Lennard-Jones potential combines an attractive inverse-sixth-power term with a simple repulsive term to produce a finite-separation minimum. Geometry also changes the effective power law: summing pair interactions over macroscopic bodies gives different distance dependences, as in Hamaker summation. The pair potential and a measured force between extended surfaces must therefore be distinguished.

This version of value iteration updates state values in place, allowing later states in a sweep to use values just computed for earlier states. In a finite discounted Markov decision process, a sweep in the order $1,\dots ,m$ isThis formula uses the old self-component; solving a self-dependent scalar equation would be a different update convention. The ordinary synchronous update uses only old components throughout.

For $0\le \gamma <1$, let $V_{\ast }$ solve the maximizing Bellman equation and let the initial supremum norm error be $E$. Each component update has error at most $\gamma E$, because its transition probabilities average components whose errors are at most $E$. After every state has been updated once, the whole sweep has error at most $\gamma E$. Repeated full sweeps therefore converge to the same fixed point. The order can affect intermediate values and speed, even though it leaves the fixed point unchanged.

An unoccupied oxygen site in a reference crystal structure. Removing an oxide ion gives an effective charge $+2e$ relative to the perfect lattice; the charge of the actual missing ion is $-2e$. A physical sample must satisfy overall charge neutrality through other defects, electronic carriers or changed cation valences.

An electron trapped at an anion vacancy in an ionic crystal.

An unconventional superconductor realizes unconventional superconductivity: its pairing cannot be described by the simple conventional isotropic spin-singlet model. The precise usage should be stated, since unconventional gap symmetry and an unconventional microscopic pairing mechanism are different claims.

A d-wave superconductor supplies a concrete example. Its superconducting gap function changes sign with momentum direction and can vanish at nodes, permitting low-energy Bogoliubov quasiparticles. Nodal quasiparticle power laws probe those excitations, while a phase-sensitive test of d-wave pairing probes the relative gap signs. An unusual superconducting transition temperature alone does not establish a particular pairing symmetry, and a real d-wave gap does not automatically imply broken time reversal.

Pairing with d-wave orbital structure, such as a crystal gap proportional to $k_x^2-k_y^2$. Its sign changes under a quarter-turn rotation, but a compensating overall phase change can leave gauge-invariant observables fourfold symmetric. Gap nodes and phase signs therefore do not by themselves establish nematicity or time-reversal breaking. A phase-sensitive test of d-wave pairing probes relative signs through interference.

A substrate-recognition component of a ubiquitin ligase that recognizes phosphorylated substrates, including Sic1.

As part of an SCF-type assembly, it links a substrate’s phosphorylation state to ubiquitin-dependent turnover. The recognition step explains why adding phosphate can change protein lifetime without directly destroying the modified protein. ^[ref]

A substrate-recognition protein in an iron-sensitive ubiquitin-ligase pathway controlling IRP2. Its stability depends on iron and oxygen conditions.^[ref]

For $\tau \propto N_d^{1/3}$ at fixed liquid-water content and thickness, and $A=\tau /(\tau +7)$, the derivative is $dA/d{N}_d=A(1-A)/(3N_d)$. For a fixed fractional number increase, the initial response is greatest at $A=1/2$. The exact finite-change expression is $A^{\prime }=sA/(1-A+sA)$ with $s=(N_d^{\prime }/N_d{)}^{1/3}$. Uncertain activation of additional particles and changes in water content limit direct application to real clouds.

A model in which each excitation changes volume by a fixed increment $v$. If the all-ground configuration has volume $V_0$, then $V=V_0+v⟨N_{\mathrm{ex}}⟩$. Energies and statistical weights must be specified under the relevant mechanical constraint; volume endpoints alone do not define arbitrary intermediate configurations.

For independently labelled particles with nondegenerate levels, choosing which $N_{+}$ particles occupy the upper level gives a binomial multiplicity. With $x=(N_{+}-N_{-})/N$, its large-number entropy is $N{k}_B[\ln 2-(1+x)\ln (1+x)/2-(1-x)\ln (1-x)/2]$. This counting assumes distinguishable or localized degrees of freedom, not indistinguishable particles sharing one pair of quantum orbitals.

For N independent units with nondegenerate levels 0 and $\Delta >0$, the canonical partition function is $(1+e^{-\beta \Delta }{)}^N$. Its energy is $N\Delta /(1+e^{\beta \Delta })$ and heat capacity is $N{k}_B{x}^2{e}^x/(1+e^x{)}^2$, with $x=\Delta /(k_{B}T)$. Both are activated at low temperature. At high temperature the energy saturates at $N\Delta /2$ and heat capacity decays as $T^{-2}$. The equal-degeneracy Schottky peak is at $x\simeq 2.39936$ and height $0.43923N{k}_B$. A common energy shift leaves heat capacity unchanged; unequal level degeneracies require a different partition function.