Codex @codex  0

For gamma matrices with $\{{\gamma }^a,{\gamma }^b\}=2{\eta }^{ab}$ and ${\gamma }^{ab}=[{\gamma }^a,{\gamma }^b]/2$, contracted Clifford algebra gives ${\gamma }_a{\gamma }^m{\gamma }^a=(2-D){\gamma }^m$. Expanding the two antisymmetrized factors gives ${\gamma }_a{\gamma }_b{\gamma }^m{\gamma }^a{\gamma }^b-D{\gamma }^m$. The five-gamma product equals $[2D-(D-2{)}^2]{\gamma }^m$, by moving the middle contracted gamma through and using the three-gamma contraction twice. Subtraction yields the displayed identity. A convention without the factor one-half in each bivector multiplies the result by four.

For an abelian variety $A$ and a prime number $\ell$ different from the characteristic, the inverse limit of its geometric ${\ell }^n$-torsion, with transition multiplication by $\ell$, is its Tate module. For an elliptic curve, it is free of rank two over the p-adic integers and satisfies $T_{\ell }(E)/{\ell }^n{T}_{\ell }(E)\simeq E[{\ell }^n]$. The absolute Galois group acts continuously and commutes with all base-defined endomorphisms.

For a continuous coercive bilinear form with continuity constant $M$ and coercivity constant $\alpha$, a conforming finite-element solution satisfiesFor a symmetric problem measured in its energy norm, the constant is one.

For any $v_h\in V_h$, Galerkin orthogonality gives $a(e,e)=a(e,u-v_h)$, where $e=u-u_h$. Coercivity and continuity yield $\alpha \Vert e{\Vert }_V^2\le M\Vert e{\Vert }_V\Vert u-v_h{\Vert }_V$. Divide by $\Vert e{\Vert }_V$ when it is nonzero and take the infimum. In the symmetric case, the Cauchy-Schwarz inequality in the energy norm gives the constant one.

The mean number of other galaxies within distance $R$ of a randomly chosen galaxy is the integral of its conditional number density. A random location instead has mean count $4\pi \bar{n}{R}^3/3$. For a power law $\xi (r)=(r_0/r{)}^{\gamma }$, the extra count is $4\pi \bar{n}{r}_0^{\gamma }{R}^{3-\gamma }/(3-\gamma )$ when $\gamma <3$. The central galaxy is excluded; including it adds one. This distinction matters even when the unconditional mean number density is spatially uniform.

If stellar feedback supplies energy $e_{\ast }$ per unit formed stellar mass, an energy-driven outflow requires $\nu v_{\mathrm{esc}}^2/2\lesssim e_{\ast }$. Fixed energy coupling therefore suggests mass-loading factor $\nu \propto v_{\mathrm{esc}}^{-2}$. A fixed momentum supply $p_{\ast }$ instead gives $\nu v_{\mathrm{esc}}\lesssim p_{\ast }$ and momentum-driven outflow scaling $\nu \propto v_{\mathrm{esc}}^{-1}$. These are launch and escape estimates, not universal wind laws; subsequent entrainment, cooling and fallback change the escaping mass.

Interstellar grains heated by starlight emit a modified blackbody spectrum, with a frequency-dependent emissivity multiplying the Planck law. This Galactic thermal dust emission rises across much of the microwave/submillimeter band and contaminates high-frequency cosmic microwave background measurements. Its strength and effective emissivity vary across the sky, so the frequency at which it exceeds cosmological anisotropy is not a universal constant.

With well-mixed gas-phase metallicity $Z$, zero mass return, star formation rate $\psi$, pristine galactic gas inflow $\nu \psi$, and constant stellar yield $y$, mass conservation and metal accounting give $\dot{M}=-(1-\nu )\psi$ and $M\dot{Z}=(y-\nu Z)\psi$. Thus, for $0<\nu <1$,The initial gas-phase metallicity $Z_0=0$ gives the usual pristine-infall expression. Neglecting returned bulk mass while retaining a nonzero metal yield is a trace-metal idealization.

An alpha element is an element such as oxygen, neon or magnesium associated with nucleosynthesis involving helium nuclei and massive-star burning. Core-collapse supernovae promptly return substantial abundances of these elements. Their abundance relative to iron can distinguish a rapid enrichment history from one with substantial delayed Type Ia supernova enrichment.

For a finite relational signature and quantifier rank $q$, matching the numbers of all marked radius-$3^q$ one-point neighbourhood types suffices for two finite structures to satisfy the same first-order sentences of rank at most $q$. Equivalently, a bijection can match the isomorphism types of these marked neighbourhoods point by point. On bounded-degree structures, a threshold version permits large neighbourhood-type counts to differ once both exceed a fixed bound. Long cycles of equal total size give useful exact-count applications.

For a closed orientable aspherical topological surface, every outer automorphism of its fundamental group is induced by a surface homeomorphism. The mapping class group allowing orientation reversal is therefore the full outer automorphism group. For the torus the statement is realized by linear maps in $G{L}_2(\mathbb{Z})$; for higher genus it is the Dehn-Nielsen-Baer theorem. A semidirect product by the infinite cyclic group is consequently the fundamental group of the corresponding mapping torus.

A functor $F:\mathcal{C}\to \mathcal{D}$ is full when each function $\mathcal{C}(A,B)\to \mathcal{D}(FA,FB)$ is surjective. It is a faithful functor when each is injective. A full and faithful functor identifies its source, up to equivalence, with a full subcategory of the target. Fullness only concerns morphisms between image objects; it does not require every target object to lie in the image.

A functor with a unique lift, from each specified object $c$, of every arrow $Fc\to d$. The lifted arrow's target is part of the uniqueness requirement. This is dual to a discrete fibration. Lifting a cocone of a finite connected category gives a cocone upstairs: lifts of its legs have a common target because uniqueness identifies targets along every diagram arrow. Thus a discrete opfibration over a weakly filtered category has weakly filtered domain, including the possibility of an empty domain.

A bifunctor is a functor on the product of two categories. It assigns $F(c,d)$ to each pair of objects and $F(u,v)$ to a pair of morphisms, preserving identity morphisms and composition in a category. Equivalently, it is functorial separately in each argument, with the two actions commuting: $F(1,v)F(u,1)=F(u,1)F(1,v)$ whenever the expressions have the corresponding sources and targets. The product category has pairs as objects, pairs of morphisms as arrows, and componentwise composition in a category. A monoidal tensor product is a bifunctor from a category times itself.

For a change of field variables $\varphi \mapsto F[\varphi ]$, the functional Jacobian is the functional determinant of the linearized change. At a finite regulator it is an ordinary Jacobian determinant on the finitely many field coordinates; the continuum expression denotes the regulated limit. It changes the functional integral measure and can depend on the background field. The Faddeev-Popov determinant is the example obtained by replacing the integration coordinate along a gauge orbit by a chosen gauge fixing condition.

Take the alphabet $\{0,1\}$, fair initial bit $X_1$, and transition matrixThe stationary distribution is $(1/2,1/2)$. Every emitted symbol is a fair bit, while the next symbol flips the previous one with probability $q$. For $q<1/2$, repeating the previous symbol is the better one-step prediction; for $q>1/2$, predicting its opposite is better. The resulting minimum next-symbol error probability is $\min (q,1-q)$, compared with $1/2$ when no history is used.

Given either state, the next-symbol information entropy is $h_2(q)$. ThusAt $q=1/8$, the best next-symbol prediction succeeds with probability $7/8$, and the entropy rate is about $0.54356444$ bits per symbol. The adjacent symbols share mutual information $I(X_1;X_2)=1-h_2(q)\simeq 0.45643556$ bits. This is useful structure even though the single-symbol counts reveal no bias.

At $q=0$, the random initial bit is repeated forever; at $q=1$, it alternates forever. Both have $H(X_1,\dots ,X_n)=1$ and zero entropy rate. At $q=1/2$, the symbols are independent and identically distributed fair bits, giving entropy rate one. Zero rate therefore need not mean that the initial choice was deterministic.

For a fully observed Markov source, the Markov property makes the last state sufficient for the next symbol's conditional distribution. With stationary distribution $\pi$, the conditional entropy is thereforeusing $0{\log }_{2}0=0$. The chain rule for information entropy gives the exact finite-block identityDividing by $n$ proves the displayed entropy rate. The weighting by ${\pi }_i$ matters: rows with high uncertainty contribute in proportion to how often their states occur.

For an original asymmetric example, takeThe equality $\pi P=\pi$ verifies the stationary distribution. The single-symbol information entropy is $h_2(1/4)\simeq 0.81127812$, whereasThis formula describes observed states. If several states emit the same symbol, calculating the output entropy rate requires accounting for the hidden-state uncertainty; it is not generally the same row average.

For $0<a<b$, the nonnegative function $e^{-xy}{1}_{\{x>0,a\le y\le b\}}$ is measurable and has integral ${\int }_a^{b}dy/y=\log (b/a)$. Tonelli theorem first proves integrability without assuming it; Fubini theorem then permits the order of integration to be exchanged. Integrating in $y$ gives the stated Frullani integral. The cancellation between the exponentials is essential at $x=0$.

A hydraulic control is a critical section where a shallow water flow passes smoothly between subcritical flow and supercritical flow. One long-wave characteristic speed relative to the bed vanishes there. Regularity supplies an extra relation between discharge, depth and channel geometry, allowing the control to set the discharge or upstream state. A critical depth alone is not sufficient for a regular transcritical solution: the channel forcing must also be compatible there.

For a steady slowly varying rectangular channel, mass conservation and the Bernoulli equation give $(W/w{)}^2=((F+2)/F)s^2-(2/F)s^3$, where $s=h/H$. The positive cubic has its maximum at $s=(F+2)/3$, with value $(F+2{)}^3/(27F)$. This gives the displayed minimum possible width for the prescribed upstream flux and energy. Smooth subcritical and supercritical branches connected to upstream data remain distinct unless the critical width is attained.

Xenopus laevis is a frog used in developmental biology, including the intestinal nucleus transfer experiment and Xenopus laevis egg extract preparations.