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A galaxy red sequence is the relatively narrow red locus in a galaxy colour-magnitude diagram, populated largely by systems with little recent star formation. Its colour, abundance, and galaxy stellar mass evolve. Dust and selection must be distinguished from old stellar ages; growth can involve quenching and later assembly of existing stars.

The observed size distribution of galaxies changes with epoch at a given galaxy stellar mass. Growth of an individual object and changes in which objects enter a selected sample both contribute. Minor galaxy mergers can deposit stars at large radii, increasing size while adding relatively little mass; stellar mass loss and dissipative evolution provide other mechanisms.

Galaxy morphological evolution changes the relative prevalence and internal structure of discs, spheroids, bars, and irregular systems over time. Comparing epochs requires consistent wavelength, spatial resolution, and selection. Galaxy mergers, gas accretion and internal instabilities can all contribute.

A galaxy of characteristic radius $R$ and velocity dispersion $\sigma$ has crossing time $R/\sigma$. Gravitational balance gives $GM/R\sim {\sigma }^2$, hence $t_{\mathrm{dyn}}\sim GM/{\sigma }^3$. A numerical coefficient depends on the radius definition, density profile, and whether one uses a crossing, orbital, or free-fall time; these are not identical clocks.

Stellar asymmetric drift is the lag of a pressure-supported stellar population's mean rotation behind the local circular speed. Neglecting the tilted velocity ellipsoid, radial Jeans equation balance givesA declining density and larger random velocities generally increase the lag; gravitational balance is supplied partly by random-motion stresses.

The intracluster medium is diffuse hot plasma between the galaxies of a galaxy cluster. Its X-ray radiation includes thermal bremsstrahlung and atomic line emission. Measured temperature and density profiles supply a cluster hydrostatic mass estimator when approximately thermal hydrostatic equilibrium holds. The medium supplies much of a cluster's baryonic mass, but this need not equal its total gravitating mass.

The virial theorem balances a relaxed galaxy cluster's orbital kinetic energy, of order $M{\sigma }_v^2$, against its Newtonian gravitational potential energy, of order $-G{M}^2/R$. Their ratio gives the displayed mass scale. The numerical coefficient depends on the density profile, radius definition and conversion from line-of-sight velocity dispersion to three-dimensional orbital motion. Membership contamination, anisotropy and nonequilibrium dynamics can bias the inference.

Gravitational lensing constrains a galaxy cluster's projected mass independently of its light output or dynamical equilibrium. The lensing convergence is projected surface mass density divided by ${\Sigma }_{\mathrm{crit}}=c^2{D}_s/(4\pi G{D}_d{D}_{ds})$, using the appropriate angular diameter distances. Strong gravitational lensing constrains the central mass through multiple-image geometry, and weak gravitational lensing constrains larger-scale projected mass through coherent image distortions. Converting these observations into a three-dimensional mass needs geometrical modelling and the relevant lensing degeneracies.

A plane-symmetric, self-gravitating stellar population with constant vertical velocity dispersion satisfies the vertical Jeans equation and the Poisson equation for Newtonian gravity. With ${\Phi }^{\prime }(0)=0$, their solution isIts surface density is $\Sigma =2{\rho }_0{z}_0$, so $z_0={\sigma }_z^2/(\pi G\Sigma )$. Its outer density tail is exponential with scale height $z_0/2$; its central profile is not exponential.

Galactic disk flaring is an outward increase of vertical thickness. A constant vertical velocity dispersion in an approximately thin-sheet stellar Newtonian gravitational field gives $h\propto 1/{\Sigma }_{\ast }$. An exponential galactic disk then predicts $h\propto e^{R/R_d}$ for a gas tracer. Inside a thick stellar layer the potential is instead approximately harmonic, and the tracer width is ${\sigma }_z/{\nu }_z$ with vertical epicyclic frequency ${\nu }_z$.

Galactic disk heating increases the random velocities of stars through encounters with molecular clouds, nonaxisymmetric gravitational disturbances, and other perturbations. Older stellar populations usually have larger velocity dispersions and greater vertical thickness, although birth conditions and migration also affect the relation.

A thin galactic disk is a stellar component whose vertical extent is small compared with its radial size. Small vertical velocity dispersion and substantial specific angular momentum are characteristic. Its survival constrains cumulative galactic disk heating, but its stars need not all have the same formation epoch.

The glucagon-like peptide-1 receptor is a G protein-coupled receptor activated by GLP-1, an intestinal peptide signal. In pancreatic beta cells, it can enhance glucose-dependent insulin secretion, prominently through cyclic AMP-linked pathways. It is distinct from the glucagon receptor despite their related ligands and receptor family. Tissue expression and physiological context determine which downstream response dominates. ^[ref]

The glucagon receptor is a G protein-coupled receptor activated by glucagon. In liver cells, signalling prominently increases cyclic AMP and helps favour glucose production, including mobilization of stored carbohydrate. The receptor converts a circulating hormonal signal into intracellular pathway changes. Its response depends on nutritional state and available substrates rather than generating glucose independently of cellular metabolism. ^[ref]

For deterministic initial asset values $A_0$ and a finite-dimensional terminal random vector $A_1$, exactly one possibility holds: a portfolio has nonpositive initial cost and nonnegative payoff with strict gain at at least one date, or there is a strictly positive integrable state-price density pricing $A_1$ as displayed. Apply the strictly positive barycentre cone lemma. Outside the closed support cone, strict separation gives a negative-cost portfolio. On its relative boundary, a supporting functional gives a zero-cost portfolio with positive payoff on a set of positive probability. With a traded positive bank account, normalize its discounted density to obtain an equivalent martingale measure.

For a finite-dimensional discounted gain random vector $X$, absence of arbitrage means $h\cdot X\ge 0$ almost surely only when $h\cdot X=0$ almost surely. Let $N=\{h:h\cdot X=0\text{ almost surely}\}$ and minimize $F(h)=\mathbb{E}{e}^{-|X{|}^2-h\cdot X}$ on the orthogonal complement $N^{\perp }$. The Gaussian damping makes $F$ finite and differentiable without physical integrability assumptions on $X$. Every nonzero direction in $N^{\perp }$ has negative gains on an event of positive probability; compactness of directions proves coercivity. At a minimizer, differentiation gives $\mathbb{E}[X{e}^{-|X{|}^2-h_{\ast }\cdot X}]=0$. Normalizing the positive weight produces an equivalent martingale measure with integrable gains. This is a constructive one-period proof of the fundamental theorem of asset pricing even on an infinite probability space.

Let $Y$ be the discounted gains of finitely many financial assets on any probability space. Tilting the physical probability measure by a normalized multiple of $(1+\Vert Y\Vert {)}^{-1}$ makes $Y$ integrable without changing null events. The positive-weight expectation cone alternative then says either a bounded strictly positive weight makes the weighted gain expectation zero, or a deterministic risky holding gives an arbitrage. Normalizing the weight yields an equivalent martingale measure. Conversely, under such a measure a nonnegative terminal gain with zero initial price has zero expectation, and must vanish almost surely. This proves the one-period fundamental theorem of asset pricing without assuming physical integrability of asset gains.

A factorization tree is a rooted tree whose root is an integer $n>1$. Each nonleaf label is a composite number split into two factors greater than one whose product equals that label; the leaves are prime numbers. Multiplying the leaves recovers $n$. Many factorization trees can have the same root, but the fundamental theorem of arithmetic gives them the same leaf multiset.

The fundamental group of a surface. For a closed orientable genus-$g$ surface the fundamental group of a closed orientable surface has $2g$ generators and one relation, the product of their paired commutators.

For a locally compact Hausdorff space $X$, $C_c(X)$ consists of continuous functions whose support is compact. A positive linear functional on this space, bounded on each fixed compact support, is represented by a Radon measure. For a locally compact group with Haar measure, these functions are dense in $L^p$ for $1\le p<\infty$.