
\input config
\input povetron
\input kniha
\pagenumbers

\appendix{A-B-C of equations}\label{sec:A}

This is an alphabetical list of equations on a undergraduate level,
in a easy-to-remember form.
It includes rounded physical constants,
for order-of-magnitude estimates.
At the end, one can find an explanatory supplement.
In this part, we do not use
a dot product,
a cross product, or
complex numbers
(nevertheless, see Appendix~\ref{sec:B}).
A factor in the equations is sometimes denoted simply
as a constant~$K$.

\input{abc1}

\def\s#1{\par\noindent\hbox to 2.5em{$^{\ref{#1}}$\hfil}\hangindent2.5em\hangafter1}

\begingroup
\smallfonts
\baselineskip=\smallbaselineskip

\s{au} astronomical unit

\s{A} expansion of space when the universe was dominated by radiation

\s{A_1} expansion of space when the universe was dominated by matter

\s{A_2} expansion of space when the universe {\em is\/} dominated by vacuum

\s{B} magnetic field around wire with the current~$I$

\s{c} speed of waves with the given length and frequency

\s{c_1} speed of light in vacuum

\s{c_s} speed of sound

%\s{exp} complex exponential

\s{e} Euler number

\s{eV} electronvolt, unit of energy

\s{E} electric field for the given voltage~$U$ and distance~$x-x'$

\s{E_0} rest energy, annihilation

\s{E_k} kinetic energy, zero value in rest

\s{E_g} gravitational potential energy, zero value in infinity

\s{E_e} electric potential energy, zero value in infinity

\s{E_gamma} energy of a photon

\s{E_U} energy of a particle

\s{E_W} work done with the power~$P$ during the time~$t$

\s{E_W_1} work done by the force~$F$ over the path~$x-x'$, in the direction of motion

\s{E_W_2} work done by the pressure~$p$ and the change of volume $V-V'$

\s{E_Q} heat needed for a change of temperature~$T-T'$ of a body with the mass~$m$

\s{E_ij} a difference of energies between levels $i$, $j$ of the hydrogen atom

\s{E_alpha} energy released by a thermonuclear reaction in the Sun

\s{f} frequency

\s{F_ma} force needed for an acceleration~$\vec a$ of a body with mass~$m$

\s{F_mg} gravitational force acting on a mass~$m$ in the gravitational field~$\vec g$

\s{F_E} electric force acting on a charge~$q$ in the electric field~$\vec E$

\s{F_B} magnetic force acting on a {\em moving\/} charge~$q$ in the magnetic field~$\vec B$

\s{F_p} pressure force on an area~$S$

\s{F_ky} string restoring force

\s{F_phi} pendulum restoring force

\s{F_g} gravitational force between two masses $m$, $M$

\s{F_e} electric force between two charges $q$, $Q$

\s{F_c} centripetal force needed for a motion on a circle with the radius~$r$

\s{F_omega} restoring force needed for an oscillatory motion with the angular frequency~$\omega$

\s{F_b} buoyant force (positive) plus gravitational force (negative), Archimedes law

\s{F_f} friction force, proportional to the force~$F_\perp$ perpendicular to the base

\s{F_S} friction force, Stokes drag, $\varrho$~the density of air, $S$~the cross section of a body

%\s{F_B_2} magnetic force acting on a wire with the current~$I$

\s{F_12} law of action and reaction

\s{g} gravitational acceleration on the surface of the Earth

\s{G} gravitational constant

\s{h} Planck constant

\s{H} Hubble constant, the speed of expansion of space

\s{I_Qt} current generated by the flux of charge~$Q$ during time~$t$

\s{I_R} current through a resistor for the given voltage~$U$, Ohm law, direct or alternating circuit

\s{I_L} current through a coil, alternating circuit with the angular frequency~$\omega$, delayed behind the voltage~$U$

\s{I_C} current through a capacitor, alternating circuit with the angular frequency~$\omega$, advanced before the voltage~$U$

\s{k} Boltzmann constant

\s{k_e} Coulomb constant

\s{m} mass from the density~$\rho$

\s{m_} relativistic mass

\s{m_p} proton mass

\s{m_e} electron mass

\s{m_Vega} brightness of Vega

\s{pogson} Pogson eq.

\s{M_g} gravitating mass, Jeans eq.

\s{M_Z} Earth mass

\s{M_S} Sun mass

\s{N} number of particles

%\s{n} concentration

\s{n} index of refraction, Snell law

\s{N_t} radioactive decay

\s{o^-1} lens eq., where $o$~denotes the focal length, $x$~the object distance, $x'$~the image distance

%\s{o^-1_2} focal length

\s{o} focal length of a spherical mirror

\s{p} linear momentum

\s{p_} momentum conservation law

\s{p_gamma} photon momentum

\s{p_gas} ideal gas state eq.

\s{p_solid} solid matter state eq.

\s{p_z} hydrostatic pressure

\s{p_ad} adiabatic pressure

\s{P_Fv} power, force, velocity

\s{P_PhiS} power, flux, area

\s{P_UI} power, voltage, current

\s{pc} parsec, unit of distance

\s{q} elementary charge

\s{r} conic section (ellipse, hyperbola)

\s{r_0} atom radius

\s{r_g} gravitational radius

\s{R_Z} Earth radius

\s{R_S} Sun radius

\s{S} surface area of a sphere

\s{t} hour angle, sidereal time, right ascension

\s{t_star} sidereal time (in seconds), tropical year, UT1 denotes the universal time, J2000 the standard epoch, $\lambda$ the geographic longitude

\s{t_} time dilation

\s{t_H} age of the universe

\s{t_ss} age of the solar system

\s{T_mk} period of a string

\s{T_rg} period of a pendulum

\s{kepler} period of a planet orbiting the Sun, Kepler law

\s{T_LC} period of an LC circuit

\s{T_0} absolute temperature scale

\s{T_S} temperature on the surface of the Sun

\s{v_at} speed from the acceleration

\s{v_omegar} speed from the angular speed

\s{v_Hr} speed from the expansion of space

\s{v_k} Kepler speed

\s{v_T} thermal speed

\s{V} volume of a sphere

\s{x} distance for a uniform motion

\s{x_0.5at^2} distance for an accelerated motion

\s{x_} length contraction

\s{bar_x} arithmetic average

\s{y} harmonic motion

\s{y_2} wave motion

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\s{alpha_} law of reflection

\s{epsilon} relative length difference

\s{epsilon_2} thermal expansion

\s{lambda_j} diffraction on a grating

\s{wien} Wien law

\s{doppler} Doppler effect

\s{muvac} permeability of vacuum

\s{pi} Ludolph number

\s{sigma} stress, modulus, relative length difference

\s{sigma_2} Stefan--Boltzmann constant

\s{sigma^2} standard deviation squared, variance

\s{phi} parallax

\s{phi_2} diffraction limit

\s{phi_3} field of view

\s{Phi} black body flux, to half-space

\s{Phi_} light absorption, opacity~$\kappa$

%\s{Phi_m} magnetic flux

\s{omega} angular frequency

\endgroup

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\appendix{A-B-C of equations (cont.)}\label{sec:B}

This is a continuation on a graduate level.

\input{abc2}

\begingroup
\smallfonts
\baselineskip=\smallbaselineskip

\s{dot} partial derivative with respect to time

\s{nabla} operator of gradient (čes. stoupání),
$\nabla\cdot$ divergence (rozbíhavost),
$\nabla\times$ rotation (stáčení)

\s{,i} partial derivative with respect to coordinate

\s{a.b} dot (scalar) product, $|\vec a\cdot\vec b| = ab\cos\alpha$

\s{axb} cross (vector) product, $|\vec a\times\vec b| = ab\sin\alpha$

\s{gauss} Gauss eq.,
a change of an orbital element due to an acceleration,
transversal component,
unit (${\rm m}\,{\rm s}^{-1}$),
hence the mean motion~$n$

\s{lagrange} Lagrange eq.,
a change of an orbital element due to a perturbation (potential),
unit

\s{fridman} Fridman eq.,
matter,
radiation, and
dark energy
state equations
($A^{-3}$, $A^{-4}$, $A^0$),
remainder from normalisation ($A = 1$)

\s{A_ij_} reduction of a CCD image,
dark, offset, flat

\s{A21} collisional equilibrium,
emission,
absorption,
stimulated emission,
collisions,
level populations in non-lte

\s{A_V} extinction vs. reddening,
$E(B\!-\!V) = B\!-\!V - (B\!-\!V)_0$

\s{divB} Maxwell eq.,
closed magnetic field lines

\s{dotB} Maxwell eq.,
electro--magnetic field,
$\vec B\perp\vec E$,
factor~1,
minus convention

\s{B_f} Planck eq.,
local thermodynamic equilibrium,
\uv{black body},
common exponential $\e^{-\dots}$, or $\e^{+\dots}$ in the denominator,
$-1$ due to an expansion ($\e^x \doteq 1+x+\dots$),
unit (${\rm J}\,\skrt{{\rm s}^{-1}}\,{\rm m}^{-2}\,\skrt{{\rm Hz}^{-1}}$),
hence $hf$ and $f^2/c^2$

\s{cosc} spherical law of cosines,
similar as planar

\s{C_J} Jacobi integral of the 3-body problem,
speed, centrifugal, potential terms

\s{kepler_eq} Kepler eq., transcendental in $E$

\s{E_n} energy levels of the hydrogen atom,
electric energy at the Bohr radius~$r_0$

\s{virial} virial theorem,
bound state,
total energy $\langle E_{\rm k}\rangle + \langle E_{\rm g}\rangle$ negative

\s{divE} Maxwell eq.,
electric field lines emanating from charges,
$\varepsilon$ permittivity (čes. pronikavost)

\s{dotE} Maxwell eq.,
electro--magnetic field,
units must be different for opposite derivatives,
factor $\mu\varepsilon = 1/c^2$,
$\mu$ permeability (čes. prostupnost), from $\vec B\to\vec E$,
charge transport ($\vec j$) also changes field ($\dot\vec E$),
minus convention

\s{taylor} Taylor series,
$n$th derivative,
if not $\infty$, approximation in a limited range

\s{f_;j} covariant derivative of a scalar,
scalar is independent on the basis $\vec e_i$

\s{f^i_;j} covariant derivative of a vector, Christoffel symbol,
correction depends on all vector components (sum over $k$),
remaining indices remain,
minus if covariant ($f_{i;j}$)

\s{f^ij_;k} covariant derivative of a tensor,
simply two indices, two corrections

\s{f_G} Gauss distribution,
thermal broadening

\s{f_L} Lorentz distribution
pressure broadening

\s{f_P} Poisson distribution
photon statistics,
$k$ number of photons,
$\lambda = t/\bar t$,
$\bar t$ mean time between 2 photons

\s{f_Ny} Nyquist frequency,
maximum at a given sampling,
aliasing if $f > f_{\rm Ny}$

\s{gauss2} Gauss theorem

\s{stokes} Stokes theorem

\s{fourier} Fourier transformation

\s{F} 2nd Newton law

\s{F_C} Coriolis force,
in a non-inertial frame!

\s{F_rad} radiation pressure force, Doppler effect, Poynting--Robertson effect

\s{F_ye} Yarkovsky effect force,
thermal emission

\s{tides} tidal force,
gradient times scale

\s{g_ik} covariant metric,
basis vectors components,
functions of coordinates

\s{minkowski} Minkowski metric,
flat spacetime

\s{schwarzschild} Schwarzschild metric,
black hole

\s{flrw} Fridman--Lem\^aitre--Robertson--Walker metric,
homogeneous universe,
expansion of the spatial coordinates,
$K \simeq {1/R^2}$ curvature,
positive or negative

\s{g^ik} contravariant metric,
inverse of

%\s{g_ik;j} covariant derivative of the metric

\s{H_2body} hamiltonian of the 2-body problem,
total energy $E_{\rm k} + E_{\rm p}$,
coordinate~$q$,
conjugate momentum~$p$

\s{H_hydrogen} hamiltonian of the hydrogen atom,
ditto

\s{collision} collisional excitation, deexcitation, bound--bound, quantized

\s{excitation} radiative excitation, deexcitation, bound--bound, quantized

\s{ionisation} ionisation, recombination, bound--free, not quantized

\s{H-} hydrogen anion, bound--free, not quantized

\s{I_alpha} interferometric fringes,
in the focal plane ($\vec\alpha$),
$\mu$~complex visibility,
$\Re\mu$ contrast,
$\Im\mu$ phase,
if $\mu = 1$ unresolved,
otherwise $\mu(\vec B)$

\s{I_f} monochromatic intensity,
unit (${\rm J}\,\skrt{{\rm s}^{-1}}\,{\rm m}^{-2}\,\skrt{{\rm Hz}^{-1}}$)

\s{rte} radiation transfer eq.,
temporal evolution,
advection,
emission,
absorption,
all directions~$\hat k$ interrelated due to scattering ($j_f$)!

\s{rte2} formal solution of the radiation transfer eq.,
stationary,
absorption,
emission (along the same path),
input intensity $I(0)$ is even non-lte!

\s{I} Ohm law,
$I$ on the l.h.s.,
because we choose $U$ (and $R$),
cf. easy-to-remember symmetry of $R$, $L$, $C$!

\s{dotI} inductance (coil) in a non-stationary circuit,
$\dot I$ on the l.h.s, because\dots,
$I$ creates $\vec B$

\s{intI} capacitance (capacitor) in a non-stationary circuit,
$\int I\d t$ on the l.h.s, because\dots,
$Q$ is accumulated

\s{sumI} Kirchhoff circuital law,
charge conservation

\s{j_f} emission coefficient, line transition,
Einstein coefficient,
$A_{21}$ emission,
corresponding concentration,
$\phi_{12}(f)$ line profile

\s{j_f2} emission coefficient, scattering,
all directions~$\hat k$ interrelated

%\s{bessel} Bessel function

\s{vecL} angular momentum (čes. moment hybnosti)

\s{dotL} 2nd Newton law for rotation

\s{dLR_dR} 3rd stellar structure eq.,
thermal equilibrium,
stationary ($\delta Q = 0$, $\d U = 0$, $\delta W = 0$),
specific power~$\varepsilon$ (${\rm W}\,{\rm kg}^{-1}$),
neutrino cooling ($-\varepsilon_\nu$)

\s{dMR_dR} 1st stellar structure eq.,
continuity

\s{vecM} torque (čes. moment síly)

\s{N} photon noise,
for ${\rm e}^{-}$, not adu!
i.e., the smallest number of quanta

\s{N2} sources of noise,
star itself, sky, dark, readout, discretisation

\s{N3} number of particles from the distribution function

\s{bose-einstein} Bose--Einstein distribution,
bosons ($\gamma$, ${\rm W}^{\pm}$, ${\rm Z}^0$, D, $\alpha$)

\s{fermi-dirac} Fermi--Diract distribution,
fermions (${\rm p}^+$, n, ${\rm e}^-$, $\nu$, u, d, T)

\s{Ni_Nj} Boltzmann distribution,
level populations in local thermodynamic equilibrium

\s{maxwell-boltzmann} Maxwell--Boltzmann distribution,
$|\vec v|$ (not $v_x$)

\s{p_rad} radiation pressure, $a = {4\sigma/c}$,
state eq. for *

\s{p_rad2} radiation pressure,
2nd moment

\s{p} pressure from the distribution function,
collisions proportional to concentration~$n$,
velocity~$v$ (frequency),
linear momentum~$|\vec p|$ (pressure force)

\s{dp_dR} 2nd stellar structure eq.,
hydrostatic equilibrium,
pressure gradient,
gravity

\s{tov} Tolman--Oppenheimer--Volkoff eq.,
relativistic hydrostatic equilibrium,
$p/c^2$ as a source of gravity,
$M_R$ corrected similarly, keeping units (kg),
Schwarzschild~$r_{\rm g}$ as in the metric

\s{clausius-clapeyron} Clausius--Clapeyron eq.,
phase transitions on the $P(T)$ diagram,
$l_{\rm h}$ specific heat of evaporation or melting,
$V'-V$ corresponding change of volume (negative for H$_2$O!)

\s{P_20} a Legendre polynomial

\s{bayes} Bayes theorem,
conditional probability of $p(A\,{\rm and}\,B)$

\s{p_Psi} quantum probability, $\Psi$ normalisation

\s{pp1} proton--proton chain, weak interaction

\s{pp2} proton--proton chain, annihilation

\s{pp3} proton--proton chain, strong interaction

\s{pp4} proton--proton chain, strong interaction

\s{dotp} Hamilton eq., conjugate momenta

\s{dotq} Hamilton eq., generalized coordinates

\s{1st} 1st law of thermodynamics,
$Q$~heat added to the system and
$\mu\d N$~chemical reactions (sum of)
can either increase the internal energy~$U$,
or the system can perform some work~$W$

\s{r_H} Hill radius, proportional to $r$, the 3rd square root of $m$'s as in Kepler

\s{calR} perturbation function, cartesian coordinates,
as potential,
direct term,
indirect term (in $\hat r_1$, projected to $\hat r^2$, renormalized)

\s{calR2} perturbation function, orbital elements,
after expansion and averaging over $\lambda$, $\lambda'$

\s{einstein} Einstein eq.,
curvature (measured by $R_{ik}$),
minus curvature due to coordinates,
plus cosmology term,
energy--momentum--stress as a source of gravity,
units ${\rm m}^{-1}$

\s{R^l_ijk} Riemann tensor,
a difference of two vector translations (along $jk$ and $kj$),
sum over $l$

\s{R_ik} Ricci tensor,
just a sum over $l$

\s{R} Ricci scalar,
contraction by the metric

\s{R_z} rotation matrix,
rotation of a vector, not of a base,
counter-clockwise

\s{Re} Reynolds number,
advection/viscosity (as in N.--S.),
$\nabla$ approximated as \uv{$1/x$}

\s{vecr_} orbit orientation,
argument of pericentre,
inclination,
longitude of the ascending node;
\uv{argument} from pericentre,
\uv{longitude} from a fixed direction

\s{2body} the 2-body problem,
in {\em relative\/} coordinates,
hence the sum of masses

\s{S_adu} signal in analog--digital units,
factors are the number of photons,
transmission,
quantum efficiency (${\rm e}^-$),
and amplification (adu).

\s{vecS} deviatoric stress tensor, liquid,
stress only if moving,
kinematic viscosity

\s{dotvecS} deviatoric stress tensor, solid,
stress even if not moving,
Young modulus

\s{S_f} source function

\s{S_f2} Kirchhoff radiative law, source function in the local thermodynamic equilibrium

\s{t'} Lorentz transformation,
both coordinates, factors $\gamma$, $\beta$, units s

\s{dT_dR_rad} 4th stellar structure eq., radiative,
opaque layer of gas,
steep gradient,
flux always multiplied by $\kappa$,
surface area ($R^2$),
and derivative of S.--B. ($T^3$) in the denominator

\s{dT_dR_ad} 4th stellar structure eq., convective,
adiabatic ($\delta Q = 0$),
simple relation ($\d T$, $\d P$),
same units ($C_p$, $V$).

\s{schwarzschild2} Schwarzschild criterion,
opaque is {\em always/} convective,
shallow is more efficient

\s{T^ik} energy--momentum--stress tensor,
fluid, comoving frame ($u^i = 0$)

\s{u} radiative energy density,
0th moment,
over solid angle,
$\int\!\!\!\int \sin\vartheta\d\vartheta\d\varphi$

\s{hde} heat diffusion eq.,
thermal diffusivity $\chi = {K/(\rho C)}$,
conductivity (${\rm W}\,{\rm m}^{-1}\,{\rm K}^{-1}$),
capacity (${\rm J}\,{\rm kg}^{-1}\,{\rm K}^{-1}$),
flux $\oldvec\Phi = -K\nabla T$,
divergence of ($-\nabla\cdot\oldvec\Phi$)

\s{wave} wave eq.,
speed of light (or sound),
2nd derivatives

\s{wave2} planar wave,
complex notation

\s{U} geopotential,
0th term monopole (point mass),
1st dipole (centre of mass),
2nd quadrupole,
associated Legendre polynomials,
spherical functions

\s{poisson} Poisson eq.,
$\vec F = -\nabla U$

\s{sumU} Kirchhoff circuital law

\s{navier-stokes} Navier--Stokes eq.,
temporal evolution,
advection,
pressure,
stress,
i.e., a difference ($\nabla\cdot\vec S$) of viscous forces ($\vec S = \nu\nabla\vec v$)
in neighbouring layers,
hence 2nd derivative

\s{x'} Lorentz transformation
relative speed~$V$, factors $\gamma$, $\beta$, units m

\s{saha} Saha eq., $X = {n_{\rm e}/n}$, $n = {N/V}$,
ratio of ionized to recombined,
common exponential,
ratio of {\em relevant\/} concentrations,
term with ${\rm m}^3$ units,
no~$c$

\s{geodetic} geodetic eq.,
as $\vec F = m\vec a$,
2nd derivative,
proper time,
affine connection (${\rm m}^{-1}$),
same units

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\s{alpha_alpha} 3-$\alpha$ reaction,
unstable beryllium 

\s{Be8_alpha} 3-$\alpha$ reaction,
only a fraction

\s{sinalpha} spherical law of sines,
as planar $\sin\alpha/a$

\s{beta} relativistic factor

\s{gamma} relativistic factor

%\s{Gamma} Gamma function

\s{christoffel} Christoffel symbol,
$g_{ik;j} = 0$, i.e., the metric does not change, if we compensate for the metric change,
the r.h.s. of the affine connection,
if $\Gamma$ unknown, we need to invert it,
hence $g^{im}$,
one upper index is summation,
one lower index is summation,
only 3 combinations of ($j$, $k$, $m$),
minus if asymmetric

\s{heisenberg} uncertainty principle,
the operators of momentum ($\hat p$)
and coordinate ($\hat x$)
do not commutate ($\hbar/2$ is a remainder)

\s{kappa_f} gas opacity, line transition,
Einstein coefficients,
$B_{12}$ absorption,
$B_{21}$ stimulated emission (negative a.),
corresponding concentrations,
$\phi_{12}(f)$ line profile

\s{kappa_f2} dust opacity, absorption, geometric ($\lambda\ll 2\pi r$),
unit ${\rm m^2}\,{\rm kg}^{-1}$,
cross section of spheres $N\pi r^2$, per unit mass

%\s{kappa_f3} dust opacity, scattering, geometric ($\lambda\ll 2\pi r$)

\s{lambda_g} gravitational redshift,
Pound \&\ Rebka (1959)

\s{mu} van Cittert--Zernike theorem, interferometric visibility,
the Fourier of the source ($I(\alpha')$),
spatial frequencies $\vec B/\lambda \equiv (u,v)$,
in \uv{cycles per baseline}

%\s{mu2} chemical potential, dependence on the temperature~$T$ and the molar fraction~$X$

\s{alpha} $\alpha$-parametrisation of viscosity,
unit ${\rm m}^2\,{\rm s}^{-1}$,
maximum speed and maximum size of vortices,
Shakura \&\ Sunyaev (1969)

\s{drho_dt} continuity eq.,
temporal evolution,
advection,
expansion

\s{dtau} optical depth,
thick ($\tau\gg 1$) vs. thin ($\tau\ll 1$)

\s{Phi1} radiative flux,
1st moment,
sometimes $1/(4\pi)$ factor

\s{Phi2} thermal equilibrium on a surface (BC),
irradiation,
conduction,
emission

\s{chi2} comparison of observations with the model,
$\sigma_i$ uncertainties,
$N-M$ degrees of freedom,
cf. probability~$p$ that the value of $\chi^2$ is that high only by chance

\s{schrodinger} Schr\"odinger eq.,
temporal evolution,
diffusion of $\Psi$,
confinement by~$U$,
weighting by $\Psi$,
$\i$~due to a wave solution $\e^{\i(\vec k\cdot\vec r-\omega t)}$,
units of $\hbar$ (${\rm J}\,{\rm s}$)

\s{psi_nlm} wave function of the hydrogen atom, Laguerre polynomials,
quantum numbers $n$, $\ell$, $m$,
exponential cut by the Bohr radius ($r_0$),
atom is bound,
higher energy levels ($n r_0$),
special function,
spherical function,
i.e., orbitals (s, p, d, \dots),
for p, ${\rm e}^-$ oscillates in a figure $\infty$, $8$, \dots,
a state is a superposition

%\s{omega_p} lunisolar precession

%\s{omega_r} relativistic precession

\endgroup

\end


