Correlation-based learning among spatially arranged ON and OFF inputs, with synaptic competition and lateral interactions, can select elongated simple cell receptive fields. Neighbor interactions coordinate preferred orientations across cortex, while competition allows different orientations to persist. Selectivity within one cell and a spatial map across many cells are separate predictions, and neither follows from unbounded Hebbian learning alone.

An orientation tuning curve plots a specified response against the stimulus-axis angle $\theta$, with the other stimulus variables controlled. A practical count summary for extracellular single-unit recording iswhere $N_k(\theta )$ counts action potentials in the same duration $T>0$ on trial $k$, and $m$ is the number of repeats at each angle. This estimates spikes per unit time in that window; the window must specify whether it covers onset, sustained stimulation, or both. Record a blank-background response separately, and report whether it has been subtracted. Randomizing angle order and repeating each condition helps separate orientation selectivity from changes over time, including neural adaptation. This is a proposed measurement procedure, not a claim that the original experiments used this estimator.

A simple cell's orientation tuning curve can also change when the stimulus shifts. An angle sweep at fixed phase, a sweep averaging over phase, and a sweep of moving stripes ask different questions. Consequently, a single preferred angle does not specify the whole receptive field. The toy curve under orientation selectivity holds phase fixed and plots dimensionless output, rather than spikes per second.

For a differentiable radial factor and integer $n\ge 0$, the coordinate polynomial obeys $L_3\psi =n\hslash \psi$ and $L_{+}\psi =0$. The identity $L^2=L_{-}{L}_{+}+L_3^2+\hslash L_3$ gives the displayed angular eigenvalue. Lowering gives $L_{-}\psi =-2n\hslash z(x+iy{)}^{n-1}f(r)$, a nonzero eigenfunction with the same angular eigenvalue when $n>0$ and the original state is nonzero. At $n=0$ lowering gives the zero function, which is not an eigenfunction. Negative powers satisfy local differential identities off the axis but are not regular angular states.

The orbifold fundamental group is the group of deck transformations of the universal orbifold cover. It retains the local finite group data that the ordinary fundamental group of the underlying topological space forgets.^[ref]

For an oriented closed topological surface of genus $g$ with cone points of orders $m_1,\dots ,m_r$, the group presentation isEach local cone relation says $c_i^{m_i}=1$; it does not guarantee that $c_i$ has order exactly $m_i$ in the resulting global group. For the teardrop orbifold, a two-sphere with one cone point of order $m>1$, the remaining relation is $c_1=1$, so its orbifold fundamental group is trivial despite nontrivial local isotropy.^[ref]

For a complete hyperbolic surface orbifold, the universal cover is the hyperbolic plane. A torsion-free finite-index subgroup then acts freely and gives a smooth surface cover. The hyperbolic covering model is what identifies local cone stabilizers with actual finite subgroups in this case.

For wealth dynamics $\dot{x}=rx+\ell -c$ and utility $c^{\alpha }(1-\ell {)}^{\beta }$, with $0<\alpha ,\beta <1$ and $\alpha +\beta >1$, the unconstrained interior control equations do not give the maximum. Put $s=1-\ell$. At positive shadow price $p$, maximization over $c$ leaves a function of $s$ proportional to $s^{\beta /(1-\alpha )}-ps$, up to constants. It is convex and hence maximal at $s=0$ or $s=1$. The optimal controls work with zero consumption first, then cease work and consume at $c(t)=C{e}^{rt/(1-\alpha )}$. The switch occurs where this consumption reaches $\alpha /(1-\alpha )$; the full discounted budget fixes $C$ and the switch time. The decreasing-consumption interior stationary candidate violates the Hamiltonian maximization condition.

Quantum optimal control minimizes a state-transfer, observable, or gate error together with resource costs, subject to a controlled Schrödinger equation or Lindblad equation. Forward propagation of the state and backward propagation of a costate compute the gradient efficiently. The optimum depends on the admissible time-dependent fields and laboratory constraints.

For $X_{n+1}=X_n+U_n+{\epsilon }_{n+1}$ and $Y_{n+1}=X_n+{\eta }_{n+1}$ with independent unit-variance normal noises, the equilibrium prediction-error variance is $P=(1+\sqrt{5})/2$. Orthogonality to the next innovation gives the filtering gain $H=P/(1+P)$. Minimizing the quadratic average cost gives a control Riccati coefficient $R$ satisfying $R^2=R+1$, so $R=P$ and $K=R/(1+R)=H$. Completing the control square yields minimal cost $R+(1+R)K^{2}P=1+\sqrt{5}$. The one-step observation delay is essential to the value of $P$.

For bank dynamics $\dot{x}=A+rx-u$ and utility ${\int }_0^T{e}^{-\beta t}\log udt+\mu e^{-\beta T}x(T)$, substituting the terminal balance separates the concave optimization pointwise and gives the displayed control for $\mu >0$. Put $W=e^{rT}x(0)+A(e^{rT}-1)/r$ and $B={\int }_0^T{e}^{\beta s}ds$. Its terminal balance is $W-B/\mu$, so the unique positive multiplier achieving zero terminal wealth is $B/W$. The same control solves the problem with no terminal reward and constraint $x(T)\ge 0$, by the supporting-line inequality for logarithms.

These formulas assume $T>0$, positive available terminal resources $W>0$, and no additional intermediate-state constraints. If $r=0$, interpret $(e^{rT}-1)/r$ by its continuous limit $T$, so $W=x(0)+AT$. If $W\le 0$, strictly positive spending cannot satisfy $x(T)\ge 0$ in this model, and the asserted positive multiplier $B/W$ is not available.

An optical momentum efficiency $Q$ expresses the net axial momentum transferred to a bead relative to incident power $P$ divided by vacuum light speed. It absorbs medium-index factors and the incident/outgoing ray geometry. Refractive-index contrast alone does not determine $Q$ for a transparent bead illuminated by a focused beam. The estimate $Q\sim m-1$ is a weak-contrast scale, not a universal force formula. An effective reflector with power reflectance $\mathcal{R}$ instead gives $Q\simeq 2\mathcal{R}$ in the simple vacuum-momentum convention.

A harmonic optical trap approximates one coordinate near a stable equilibrium by a quadratic potential energy. Measure displacement $x$ from that equilibrium, set its potential energy to zero there, and take a positive effective spring constant $k$, also called the trap stiffness:The restoring force points left when $x>0$ and right when $x<0$. This is a local model: a Taylor series of a smooth potential energy about a nondegenerate local minimum gives $k=U^{\prime \prime }(0)>0$ as its leading curvature. A real optical trap has a finite capture region; the endlessly rising parabola does not describe its escape boundary. A three-dimensional optical trap can have different stiffnesses along different axes, and nonconservative optical forces can limit the potential energy description.

An optical scattering force transfers momentum from light to a particle through scattering of light. In a usual single-beam optical trap, its axial contribution pushes along beam propagation; the optical gradient force must balance it for axial confinement. This balance can place the stable equilibrium away from the beam focus.

An optical gradient force arises from spatial variation in illumination. In a usual optical trap for a suitable transparent particle with higher refractive index than its surroundings, it draws the particle towards high light intensity. It supplies a restoring force near a stable trap, but three-dimensional confinement also depends on the optical scattering force. The direction and size depend on the particle and field; attraction to the brightest point is not universal. ^[ref].

Here butadiene denotes the 1,3-butadiene molecule, with formula ${\mathrm{C}}_4{\mathrm{H}}_6$. It supplies a concrete ultraviolet absorption example: a photon can promote an electron between molecular orbitals with the appropriate energy separation. ^[ref].

Silver chloride is a chemical compound with formula $\mathrm{AgCl}$. Its darkening under illumination supplied the chemical readout in the historical ultraviolet discovery described above. That response depends on the incident optical spectrum and the sample's condition; it does not provide an uncalibrated direct measurement of incident power. ^[ref].

Recoil energy is the kinetic energy gained by an initially stationary atom absorbing a photon of momentum $p_R$. In a standing-wave optical lattice of spacing $d$, the photon wavenumber is $\pi /d$, giving $E_R={\hslash }^2{\pi }^2/(2m{d}^2)$.

Here $n$ is average number density and $\mu$ the ground-state chemical potential. This commonly used condensate response has units of inverse energy; thermodynamic isothermal compressibility is instead $\kappa /n$. In a weakly interacting repulsive condensate, the normalized one-cell ground state $f_s$, with average $|f_s{|}^2=1$, gives $\mu =E_0(s)+UnI(s)+\cdots$, $I(s)=d^{-1}{\int }_0^d|f_s{|}^{4}dx$. Localization raises $I$ and reduces $\kappa$ at fixed $n,U,d$. In deep harmonic wells $I\propto s^{1/4}$, hence $\kappa \propto s^{-1/4}$. This is a ground-band, repulsive-superfluid statement with the held variables specified, not a theorem for arbitrary signs of the interaction or arbitrary excited branches.

An optical power meter measures collected radiant flux using a calibrated detector and readout. With a photodiode, its conversion to watts depends on spectral responsivity, as in calibrated radiant-flux measurement.

A monochromator selects a narrow, adjustable wavelength interval from an optical spectrum. Its finite spectral bandwidth matters when calibrating a photodetector's spectral responsivity in calibrated radiant-flux measurement.

An optical attenuator reduces the transmitted radiant power of a beam. It can use optical absorption, partial reflection, or controlled coupling into a different path. Its transmission $T=P_{\mathrm{out}}/P_{\mathrm{in}}$ must be specified for the relevant wavelength and polarization.

A variable pair of polarizers provides a simple example: for ideal linearly polarized input and a rotated analyzer, Malus's law gives $T={\cos }^2\theta$ relative to aligned transmission. This permits adjustment before a photodetector saturates. A power series measured with several attenuations also tests detector linearity, provided the attenuator itself has been calibrated.

A Ronchi ruling is a grating of alternating transparent and opaque parallel bands. A known band spacing can calibrate the scale of an optical microscope image.