For additive Gaussian white noise and a time-even order parameter, a midpoint discretization of the Onsager–Machlup functional evaluates drift derivatives at the midpoint configurations. Reversing a trajectory visits the same midpoints in reverse order, so the noise-to-path Jacobian determinant is invariant under reversal. It may depend on the trajectory, and must be distinguished from the constant normalization of the noise measure.

For nonconserved order-parameter dynamics with $\dot{p}=-\Gamma \mu [p]+f$, the factor at one time step, apart from a path-independent power of the step size, isHere the fields have first been restricted to a finite spatial grid, and ${\mathcal{H}}_n$ is the Hessian matrix of the free energy.

For mixed conserved and nonconserved order-parameter dynamics, the required noises for a joint configuration and flux trajectory arewhere $\mathsf{D},\mathsf{G}$ represent the divergence and gradient, and $\Delta =\mathsf{DG}$ represents the Laplacian. The Jacobian matrix isTaking its Schur complement gives the joint factorFor periodic boundary conditions, discretizing the two spatial operators compatibly gives $\mathsf{D}=-{\mathsf{G}}^{\mathsf{T}}$, so $\Gamma I-M\Delta$ is the positive relaxation operator when $\Gamma ,M>0$. Reversal leaves ${\mathcal{H}}_n$ unchanged and reverses the sign of the flux. The product of the joint factors is consequently the same for both histories, justifying its cancellation from the joint path probability of an order parameter and its flux. The functional chain rule used in the action ratio holds in the continuum limit of this midpoint convention.

In the six-dimensional cubic scalar field theory, use full external-propagator amputation and tadpole subtraction. The exchange part of the zero-momentum four-point amplitude isThe triangle corrects either end of each of three exchanges, and the symmetry factor of an internal bubble is $1/2$. Equivalently insert ${\Gamma }^{(3)}=g+g^3{I}_3$ and ${\Gamma }^{(2)}=m^2-g^2{I}_2/2$ into $3[{\Gamma }^{(3)}{]}^2/{\Gamma }^{(2)}$. Without tadpole subtraction, the shift $v=-g{I}_1/(2m^2)$ adds $3g^4{I}_1/(2m^6)$. None of these is an additional local 1PI quartic coupling; zero-momentum bridges are also excluded from a strict high-mode Wilsonian shell. If bare rather than full external propagators are amputated, external-line decorations additionally contribute $6g^4{I}_2/m^4$, and $6g^4{I}_1/m^6$ without tadpole subtraction. Every displayed term has mass dimension minus two. In a renormalized expansion, the exchange diagrams also receive the counterterm insertions $6g\delta g/m^2-3g^2\delta m^2/m^4$. Without enforcing a zero one-point function, a linear counterterm $\delta h\varphi$ contributes $3g^3\delta h/m^6$ through the background shift; choosing $\delta h=-g{I}_1/2$ cancels the attached tadpole. A kinetic counterterm is proportional to the exchanged momentum squared, so its insertion vanishes at this zero-momentum point.

For $X\ne 0$, the image of $t\mapsto e^{tX}$, with its intrinsic immersed Lie subgroup topology, is isomorphic to $(\mathbb{R},+)$ or the circle group. The kernel is respectively zero or $\tau \mathbb{Z}$ for a least positive period $\tau$. The case $X=0$ is the trivial group. An irrational winding in a two-dimensional torus need not be an embedded or closed subgroup, so its intrinsic topology must be distinguished from the subspace topology.

Joule heating is the dissipative conversion of electrical work into internal energy in an electrical conductor. In a metal, the electric field supplies energy to charge carriers, which transfer it to the material. This makes an immersed resistor a useful calibrated heater and makes electrical resistance a source of losses in wiring. Wind-driven electric currents are a proposed source of heating of an inflated giant planet.

For a stationary, purely resistive Ohmic conductor, use the passive sign convention: $V$ is the terminal electric potential difference from the entry to the exit of positive electric current $I$. The absorbed electrical power and Ohm's law giveThe last two expressions assume $V=IR$ at the relevant material temperature. For constant electrical resistance and electric current over a duration $\Delta t$, the dissipated energy is $E_{\mathrm{diss}}=I^{2}R\Delta t$; otherwise use $E_{\mathrm{diss}}=\int V(t)I(t)dt$. In SI, $\mathrm{V}\mathrm{A}=\mathrm{W}=J/s$.^[ref]

Absorbed electrical power $VI$ is more general than Joule heating: a capacitor can store energy, a battery can change its chemical energy, and an electrical circuit can deliver mechanical work or emit light. Even a heated resistor need not show an indefinitely rising temperature: in a thermal steady state, its dissipated power leaves by heat transfer. With the heater inside a calorimeter boundary, electrical input crosses as work; the subsequent heat transfer to the room is a different term in the first law of thermodynamics.

A sigma-model lump is a finite-energy classical field-theory soliton of the two-dimensional O3 nonlinear sigma model. With an orientation chosen so holomorphic maps have positive charge, the Bogomolny bound $E\ge 4\pi |N|$ is saturated by rational maps of degree $N>0$ in a stereographic projection coordinate, and by antiholomorphic maps for negative charge. The field remains smooth at coordinate poles, which are points at the other pole of the target sphere. Scale invariance permits lumps of arbitrary size; topology alone does not prevent concentration to a singular limiting configuration.

The finite-dimensional real vector space $N^1(X{)}_{\mathbb{R}}$ consists of real combinations of Cartier divisors modulo numerical equivalence of divisors. Its elements are numerical divisor classes. The space of curve classes paired with it is $N_1(X{)}_{\mathbb{R}}$.

If a unit tangent makes angle $\theta$ with the first principal direction, the normal curvature is ${\kappa }_n={\kappa }_1{\cos }^2\theta +{\kappa }_2{\sin }^2\theta$. This follows by diagonalizing the second fundamental form relative to the first fundamental form. When the Gaussian curvature is nonnegative, $|{\kappa }_n|\le 2|H|$.

The formal variation of this condition is $\delta F=({\partial }^2+2iA\cdot \partial )\alpha$. Thus its Faddeev-Popov ghost field operator can be chosen as ${\mathcal{O}}_A=-{\partial }^2-2iA\cdot \partial$. With $D_{\mu }={\partial }_{\mu }+i{A}_{\mu }$, one has $D^2={\partial }^2+2iA\cdot \partial +iF[A]$, so ${\mathcal{O}}_A=-D^2$ on the formal gauge slice. The reality obstruction for a complex Euclidean gauge condition prevents interpreting this slice as an ordinary real Euclidean gauge fixing without an additional complex-contour prescription.

Fowler sampling averages several nondestructive reads near the start and end of an integration and differences the two averages. For independent equal read noise, averaging lowers the read-noise variance; correlated noise limits the improvement.

Up-the-ramp sampling measures accumulated charge repeatedly through an exposure and estimates flux from its slope. Cosmic-ray steps, saturation and nonlinearity must be identified or modelled.

For a smooth solution of $\Delta u=f$, the corrected formula ${\Gamma }_{9}u=h^{2}f+(h^2/12){\Gamma }_{5}f$ has unscaled residual $O(h^6)$ and normalized local truncation error $O(h^4)$. The inverse interior matrix has Euclidean operator norm $O(h^{-2})$; on $O(h^{-2})$ grid points this gives unweighted Euclidean nodal error $O(h^3)$. Uniform derivatives up to the boundary are required for a grid-independent remainder bound.

A heat transfer coefficient relates heat flux through a boundary to a temperature difference: $F=\lambda \Delta T$. Its dimensions are power per area per temperature. In a quasistatic ice layer, atmospheric boundary resistance $1/{\lambda }_A$ and conductive resistance $h/k$ add in series, giving effective conductance $(1/{\lambda }_A+h/k{)}^{-1}$.

The muon neutrino is the neutral lepton coupled to a muon by the weak charged current. It carries lepton number $+1$; its antineutrino has the opposite lepton number.

With 24 physical half-integer open-string oscillators, the first three string level operator eigenvalues are $0,1/2,1$. Their states are the vacuum, $a_{1/2}^{i†}|0⟩$, and $a_{1/2}^{i†}{a}_{1/2}^{j†}|0⟩$. The creation operators commute, so the third level is the symmetric square of the 24-dimensional transverse vector, of dimension $24\cdot 25/2=300$. Its trace is a scalar and its trace-free part has dimension 299. Including the vacuum shift gives ${\alpha }^{\prime }{M}^2=1/2,1,3/2$ when the two longitudinal directions are common directions with Neumann boundary conditions. The representation labels refer to the surviving ND rotation group, not full unbroken target Lorentz symmetry.

For one free boson, the mixed ND open-string sector has strip vacuum energy $E_0=1/48$. The strip-to-plane transformation relates this energy to the plane Virasoro algebra zero mode by $E_0=h-c/24$, with central charge $c=1$. Hence its ground conformal weight is $h=1/48+1/24=1/16$. For $d$ independent ND coordinates the weights add to $d/16$. At $d=24$, the plane matter ground weight is $3/2$, while the strip oscillator vacuum energy is $1/2$. The plane physical condition $L_0^{\mathrm{plane}}-1=0$ is therefore equivalent to ${\alpha }^{\prime }{p}_{\parallel }^2+N+1/2=0$. The apparent different constants are coordinate/constraint conventions, not different mass spectra.

A Neumann eigenfunction is an eigenfunction of a spatial differential operator whose normal derivative vanishes on the boundary. A Sturm-Liouville eigenfunction expansion in these functions gives a heat kernel for homogeneous Neumann boundary conditions.

For a nef divisor on an integral projective variety of dimension $n$,Indeed, combine asymptotic Riemann–Roch with cohomology growth for nef twists. Thus its normalized section-growth limit is $D^n/n!$, and it is big exactly when $D^n>0$.

For a natural transformation $\theta :F\Rightarrow G$, naturality is the equation $G(f){\theta }_A={\theta }_{B}F(f)$ for each morphism $f:A\to B$. Thus transforming and then applying a functor gives the same result as applying that functor and then transforming. It makes the component maps a coherent transformation of functors.

The natural parameter space is the set of $\theta$ for which the normalizing integral $\int h(y)e^{{\theta }^{T}T(y)}dy$ is finite. Interior points permit the usual exponential-family derivative identities when differentiation and integration can be interchanged.

For at least two distinct knots $x_1<\cdots <x_n$ and values $v_i$, the unique natural cubic spline matching those values and linear beyond $x_1,x_n$. Its interpolation map is a linear map by uniqueness. Over a larger domain $[a,b]$, the two exterior pieces remain linear. With only one knot, arbitrary slopes of affine functions make uniqueness false.