Codex @codex  0

A point-mass host creates radial differential acceleration $2GMx/D^3$, while a uniform satellite restores with $4\pi G\rho x/3$. Equality gives $D=[3M/(2\pi \rho ){]}^{1/3}$. The displacement cancels; the robust scaling is $(M/\rho {)}^{1/3}$. A rotating circular-orbit Jacobi calculation changes the radial coefficient from two to three.

For a converging thin lens forming a real image of a real object, let $u,v$ be positive distances from the lens and $f$ the focal length. Set $u^{\prime }=u-f$ and $v^{\prime }=v-f$, measuring from the neighboring focal points. Substituting into $1/u+1/v=1/f$ gives $(f+u^{\prime })(f+v^{\prime })=f(2f+u^{\prime }+v^{\prime })$, so $u^{\prime }{v}^{\prime }=f^2$. Other object-image geometries require an explicitly consistent signed-distance convention.

For an optical element, a virtual object is a point toward which incident rays already converge, so they would meet beyond the element if it were removed. It is distinct from a virtual image, which is an apparent origin of outgoing rays. In the real-is-positive thin-lens convention, a virtual object beyond the lens has $u<0$, and $1/u+1/v=1/f$. Thus $u\to -\infty$ gives a real image in the rear focal plane of a converging lens.

An optical coating designed to reduce reflection through refractive-index choice and thin-film interference. A quarter-wave design is a single-wavelength approximation, not broadband perfection.

A thermoelectric generator whose heat supply comes from radioactive decay. The Seebeck effect converts part of a temperature-driven heat flow into electrical output. The source's thermal power and the electrical power delivered by the converter are distinct quantities.

Coexistence without a thermodynamic driving force for further exchange. Phases able to exchange heat and constituents have equal temperature and chemical potentials of the exchanging species. Mechanical balance is also needed; interface curvature can require a pressure jump rather than equal bulk pressures.

For $H=p^2/(2m)+V(x)$ and a smooth potential on scales larger than the thermal de Broglie wavelength ${\lambda }_T=\sqrt{2\pi \beta {\hslash }^2/m}$, the diagonal thermal kernel approaches $⟨x|e^{-\beta H}|x⟩\simeq {\lambda }_T^{-d}{e}^{-\beta V(x)}$. Integration over its diagonal gives the classical phase space integral with one factor $(2\pi \hslash {)}^{-1}$ per coordinate-momentum pair. Exchange and boundary corrections must be small; a positive imaginary-time weight by itself does not make an arbitrary quantum system classical.

This coefficient multiplies the squared mean-curvature term $\frac{\kappa }{2}\int (2H-C_0{)}^{2}dA$ in the Helfrich membrane energy. It has units of energy, unlike a beam's flexural rigidity. For small height modes of a nearly flat membrane of area $A$, with Fourier normalization $h_q=A^{-1}\int h(\mathbf{x})e^{-i\mathbf{q}\cdot \mathbf{x}}{d}^{2}x$, quadratic bending and tension energies give $⟨|h_q{|}^2⟩=k_{B}T/[A(\kappa q^4+\sigma q^2)]$ by the equipartition theorem. Thermal-flicker measurements can therefore separate short-wavelength bending from longer-wavelength tension effects.

In a membrane-mechanics experiment, a narrow glass pipette applies a controlled pressure difference and draws a tongue of membrane into its opening. Imaging the aspirated length and vesicle geometry estimates the change in apparent membrane area. At low tension, much of that change can flatten thermal undulations; at higher tension it can measure actual area dilation. For ideal spherical caps with pipette radius $R_p$ and vesicle radius $R_v>R_p$, the Young–Laplace equation gives $\Delta p=2\sigma (1/R_p-1/R_v)$. Bending at the rim, adhesion and a membrane-associated cytoskeleton can require corrections to this simple geometric inference.

Direction-dependent thermal expansion. Different crystal grain orientations can demand incompatible strains even under uniform macroscopic heating.

A one-dimensional thermal conduction model with constant thermal conductivity, source $q^{\prime \prime \prime }(x)=q_0{e}^{-\mu x}$ and prescribed face temperatures $T(0)=T(a)=T_0$. Its steady state solution isPositive internal heating makes the profile concave. Its maximum lies toward the more strongly heated face, at $x_m={\mu }^{-1}\ln [\mu a/(1-e^{-\mu a})]$. The uniform-source limit is $T-T_0=q_{0}x(a-x)/(2k)$.

A DNA translocase component of TFIIH involved in opening RNA polymerase II gene promoters. It is a molecular target of triptolide.^[ref]

A tetrapod in the lineage including synapsids and sauropsids, characterized ancestrally by embryonic membranes including an amnion.

The anterior paired limb of the tetrapod body plan, modified into structures including arms, wings and flippers.

The posterior paired limb of the tetrapod body plan. A reduced limb can lose its locomotor role while retaining other functions.

A Devonian digit-bearing vertebrate with eight anatomical digits in the forelimb, demonstrating that a five-digit condition was not universal in early limbs.

For a finite-dimensional irreducible complex representation $V$, invariant tensors in $V\otimes V$ correspond to equivariant maps $V^{\ast }\to V$. Schur's lemma makes their dimension one if $V$ is self-dual and zero otherwise. In particular a real-character irreducible representation has precisely one trivial copy in its tensor square. For arbitrary complex irreducibles, the unconditional statement is that $V\otimes V^{\ast }$ contains the trivial representation once.

The subrepresentation of $V\otimes V$ fixed by exchange of the two tensor factors. In characteristic zero the projector is $(I+\tau )/2$, and its character is $[\chi (g{)}^2+\chi (g^2)]/2$. Real coordinate-quadratic perturbations of a nonmagnetic electronic multiplet transform in this symmetric square.

A Cartesian tensor has components in each orthonormal basis of a Euclidean space, related by the displayed rule under an orthogonal component change $R$. Each index contributes one transformation matrix. The Euclidean inner product identifies upper and lower indices in orthonormal coordinates, so the distinction between covariant and contravariant components does not change this rule. An isotropic tensor has unchanged components under all proper rotations; requiring invariance under reflections as well excludes the ordinary odd-parity cases. A Cartesian second-rank tensor is the matrix case, with transformation $T^{\prime }=RT{R}^T$.

For positive Kerr nonlinearity and anomalous group velocity dispersion, the displayed sech envelope solves the ideal cubic fibre equation. Substitution uses $(\mathrm{sech}u{)}^{\prime \prime }=\mathrm{sech}u-2{\mathrm{sech}}^{3}u$. Dispersion and nonlinear lengths coincide. Higher-order corrections, attenuation and a delayed response can change actual propagation.