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Entanglement concentration converts partially entangled pure states into fewer more strongly entangled states using local operations and classical communication. The process can fail on individual attempts, so a complete account includes both the successful output and its probability.

For $|\psi ⟩=\sqrt{p}|00⟩+\sqrt{1-p}|11⟩$ with $1/2\le p<1$, a local successful filter on the first qubit with operator $\mathrm{diag}(\sqrt{(1-p)/p},1)$ produces equal coefficients. After normalization the result is a Bell state, with success probability $2(1-p)$. The other outcome makes the operation complete. Obtaining greater entanglement in a selected output does not contradict the restriction that local operations cannot increase entanglement on average across all outcomes.

The state $a|00⟩+b|11⟩$ is entangled exactly when $ab\ne 0$. Its reduced density matrix has nonzero eigenvalues $|a{|}^2$ and $|b{|}^2$.

For a variation $F(s,t)$ of a curve $\gamma (t)=F(0,t)$, the variation vector field is $V(t)={\partial }_{s}F(0,t)$. A fixed-endpoint variation has $V=0$ at both endpoints.

For $1<p<\infty$ and $u\in W^{1,p}(\Omega )$, the p-energy isIts stationary points under fixed boundary data are weak solutions of the p-Laplacian equation.

If embedded submanifolds $A,B\subseteq M$ meet transversely, then $A\cap B$ is an embedded submanifold of dimension $\dim A+\dim B-\dim M$.

The hypothesis is transversality: tangent spaces to the two submanifolds span the ambient tangent space at each intersection point.

The Z boson is the massive neutral electroweak gauge boson mediating the weak neutral current. In the standard tree-level mixing convention, $Z_{\mu }=(g{W}_{\mu }^3-g^{\prime }{B}_{\mu })/\sqrt{g^2+g^{\prime 2}}$ and $m_Z=v\sqrt{g^2+g^{\prime 2}}/2$.

Its interactions need not change a particle's electric charge. Neutrino scattering through Z exchange can therefore transfer energy to a target without producing a charged lepton of the neutrino flavor. A real Z can decay to charged fermion pairs or neutrino pairs when kinematically allowed. The latter contribute an invisible decay width, so the measured total and visible widths constrain invisible final states. Neutrality alone does not make the Z photon-like: its mass, chiral couplings and short-range low-energy effect differ.

The W bosons $W^{+}$ and $W^{-}$ are massive charged electroweak gauge bosons mediating weak charged currents. Before mass diagonalization, the charged combinations are $W^{\pm }=(W^1\mp i{W}^2)/\sqrt{2}$; electroweak symmetry breaking gives their tree-level mass $gv/2$.

A W-mediated interaction changes the electric charge of its matter partners. For example, neutron beta decay contains a down-to-up quark transition and an electron–antineutrino pair; at low energy the exchanged W is virtual, not a real emitted particle. A real W produced at high energy can decay to a charged lepton and neutrino or an allowed quark pair. The large W mass explains why low-energy weak processes can be approximated by a short-range four-fermion interaction.

Surface charge density is electric charge per unit area. At a conductor in electrostatic equilibrium it equals ${ϵ}_0{\mathbf{E}}_{\mathrm{out}}\cdot \mathbf{n}$ for the normal pointing out of the conductor.

For metric signature $(-,+,+,+)$,In particular, $T^{00}=\frac{1}{2}({ϵ}_0{E}^2+B^2/{\mu }_0)$.

For signature $(-,+,+,+)$ and a consistent orientation,Overall signs depend on tensor and orientation conventions, while their vanishing does not.

The Lorentz force is the force exerted by an electric field $\mathbf{E}$ and a magnetic field $\mathbf{B}$ on a body with electric charge $q$ and velocity $\mathbf{v}$. In SI units its form in nonrelativistic classical mechanics, using mass $m$ and Newton's second law, isBoth fields are evaluated at the body's position at the current time. The electric field contributes $q\mathbf{E}$ even at rest, whereas the magnetic force depends on velocity through a cross product. ^[ref]; ^[ref].

For a concrete charged particle in a uniform magnetic field, use nondimensionalization to stipulate $m=q=B=1$, magnetic field $\mathbf{B}=(0,0,1)$, zero electric field, initial position $(0,0,0)$, and initial velocity $(1,0,0)$. These are rescaled quantities, not an assertion that the mass, electric charge, and magnetic field have the same physical units. At the start, $(1,0,0)\times (0,0,1)=(0,-1,0)$, so the magnetic acceleration points toward $-y$. The resulting circular motion isAt $t=\pi /2$ the position is $(1,-1,0)$ and the velocity is $(0,-1,0)$; at $t=\pi$ the position is $(0,-2,0)$ and the velocity is $(-1,0,0)$. Thus the circle has center $(0,-1,0)$ and radius one. Viewed from $+z$ toward the $xy$ plane, the motion is clockwise. Changing only the electric charge to $q=-1$ gives position $(\sin t,1-\cos t,0)$ and reverses the bending to counterclockwise.

Adding initial velocity $v_z=0.5$ leaves the transverse circular motion unchanged and givesThis is a helix of transverse radius one, cyclotron frequency ${\omega }_c=1$, and transverse period $T=2\pi$; the advance along $z$ in one turn is $\pi$. The parallel component of velocity stays constant because the magnetic force has no parallel component: there is no magnetic acceleration along the magnetic field. If the initial velocity were entirely parallel to the magnetic field, the magnetic force would vanish and the path would be straight. ^[ref].

More generally, with zero electric field, no other forces, and a uniform constant magnetic field of magnitude $B>0$, decompose velocity into perpendicular and parallel components. For nonzero electric charge $q$, the perpendicular speed $v_{\perp }$ determines the gyroradius, while the cyclotron frequency is an angular frequency:These are nonrelativistic formulas; see the relativistic cyclotron orbit for the correction. The beauty is that increasing perpendicular speed increases the radius in exactly the proportion needed to leave the period unchanged. Conversely, known magnetic field, perpendicular speed, and measured radius determine the magnitude of the electric charge divided by mass. This makes the same geometry useful for comparing bodies with different electric charge and mass.

Two short checks explain the result. First, the cross product gives $\mathbf{v}\times \mathbf{B}=(-\sin t,-\cos t,0)$ for both stipulated positive-electric charge paths, exactly their acceleration. Second, the transverse magnetic force supplies the centripetal acceleration, so $m{v}_{\perp }^2/r=|q|v_{\perp }B$; for $v_{\perp }>0$ this gives the gyroradius, and ${\omega }_c=v_{\perp }/r$ gives the cyclotron frequency.

Magnetic forces do no work: the instantaneous power of the magnetic part of the Lorentz force vanishes because a cross product is perpendicular to its first argument. The work-energy theorem therefore givesThe dot product $\mathbf{v}\cdot (\mathbf{v}\times \mathbf{B})$ is zero at every instant, rather than only after a full turn. With zero electric field and no other forces, the kinetic energy and speed remain constant; the electric field can increase or decrease kinetic energy according to the sign of $q\mathbf{E}\cdot \mathbf{v}$. In the two stipulated examples the constant kinetic energy is respectively $K=1/2$ and $K=5/8$ in rescaled units. The extra parallel velocity raises kinetic energy without changing the gyroradius or the cyclotron frequency.

For an electric example in rescaled units, stipulate $m=q=1$, zero magnetic field, constant electric field $\mathbf{E}=(2,0,0)$, no other forces, and rest at $t=0$. The Lorentz force gives acceleration $(2,0,0)$ and velocity $(2t,0,0)$, hence kinetic energy $K=2t^2$ and power $dK/dt=4t=q\mathbf{E}\cdot \mathbf{v}$. At $t=1$, the kinetic energy has risen from zero to two: the electric field has done positive work.

The existing electromagnetic field tensor formulation expresses the same interaction covariantly. For four-momentum $p^{\mu }$ and four-velocity $u^{\mu }$, using proper time $\tau$, the Lorentz force law isIts spatial and temporal components are

Ohm's law states that an Ohmic conductor has electric current proportional to its applied electric potential difference, with constant electrical resistance under fixed material conditions:Here $V=\varphi (A)-\varphi (B)$ is the terminal electric potential difference, and positive $I$ enters at $A$ and exits at $B$. This sign convention makes $VI$ the electrical power absorbed. The straight-line relation through the origin lets one constant predict the response at many applied electric potential differences. Merely defining $R=V/I$ at one nonzero operating point does not establish Ohm's law: a component may have a ratio that varies with applied electric potential difference. Material and temperature must be specified and held fixed; many components have a nonlinear response. ^[ref].

For the ideal resistor model $R=2\Omega$ at electric potential difference $V=5\mathrm{V}$,This power is converted to internal energy by Joule heating; a watt is a joule per second. If the model holds for time $10\mathrm{s}$, the transferred energy is $125\mathrm{J}$. Doubling electric potential difference doubles electric current and quadruples power for the same fixed $R$. The model keeps material temperature fixed even while calculating Joule heating; appreciable changes in temperature may require a changing electrical resistance. ^[ref].

In 1827, Georg Ohm published his mathematical account of electrical transport in Die galvanische Kette, mathematisch bearbeitet. On printed page 36, his relation $S=A/L$ expresses electric current in terms of total driving tension and a reduced length incorporating electrical conductivity and geometry. This is historical notation: his $A$ is not the wire cross-sectional area used here, and his $L$ is not simply the wire's physical length. The striking idea is that material response and geometry can be combined into one quantity governing an electrical circuit. ^[ref].

For a progressive acoustic plane wave in a homogeneous fluid, the pressure-to-particle-speed ratio is $Z=\rho c$, where $\rho$ is mass density and $c$ is the speed of sound. Its units are $\mathrm{Pa}s/m$. This specific impedance differs from pressure divided by a volume flux, whose value also depends on the port area.

At angle $\theta$ from an interface normal, the normal particle-velocity amplitude is $U_n=U\cos \theta$. Therefore the normal acoustic impedance isFor two lossless fluids with propagating incident and transmitted waves, continuity of pressure and normal velocity givesand hence the pressure-amplitude reflection coefficientZero reflection requires equal normal acoustic impedances; equal bulk values $\rho c$ alone suffice only at normal incidence. Using particle displacement rather than velocity multiplies both normal impedances by the same frequency factor and does not replace $Z/\cos \theta$ by $Z\cos \theta$.^[ref]

An obliquely incident in-plane P- or SV-wave generally produces reflected and transmitted P- and SV-waves. Continuity of displacement and traction supplies four scalar conditions for their four amplitudes, while frequency and tangential wavenumber are shared.

For two inviscid media, continuity of normal displacement and normal stress determines the reflected and transmitted longitudinal amplitudes.

At a perfectly bonded interface, all displacement and traction components are continuous.

An S-wave is transverse, carries rotation without dilatation, and has speed $c_S=\sqrt{\mu /\rho }$.

A P-wave is longitudinal, carries nonzero dilatation, and has speed $c_P=\sqrt{(\lambda +2\mu )/\rho }$ in an isotropic solid.

For a homogeneous isotropic solid with constant mass density $\rho >0$ and Lamé parameters $\lambda ,\mu$, the Navier-Cauchy equation isTaking the divergence, $d=\nabla \cdot u$ obeys $d_{tt}=c_P^2\Delta d$ with $c_P^2=(\lambda +2\mu )/\rho$. Taking the curl, $w=\nabla \times u$ obeys $w_{tt}=c_S^2\Delta w$ with $c_S^2=\mu /\rho$, because the curl of a gradient vanishes. These are the longitudinal P wave and transverse S wave speeds.

A suitable Helmholtz decomposition into a scalar-potential gradient and a divergence-free part separates the corresponding motions. In two-dimensional in-plane propagation, the P-SV displacement potentials implement this separation with two scalar wave equations. The constant-coefficient and isotropy assumptions matter: interfaces can couple P and SV modes, and arbitrary heterogeneous coefficients cannot be passed through the divergence and curl as if they were constants.

Bending stiffness is the coefficient relating plate curvature to bending moment. In linear thin-plate theory a transverse displacement $\zeta$ produces a restoring load $B{\nabla }^4\zeta$.