The structure map $\epsilon :C\to I$ of a comonoid, with $(\epsilon \otimes 1)\Delta$ and $(1\otimes \epsilon )\Delta$ equal to the inverse unitors. This is distinct from the counit of an adjunction.

For a first-order theory $T$, suppose every existential quantifier-free formula over a common substructure has a witness in either of two $T$-models exactly when it has one in the other. Then $T$ has quantifier elimination. One may test a single quantified variable, since elimination of these formulas inductively eliminates all quantifiers. A language with a constant avoids empty-generated-substructure and quantifier-free-sentence conventions.

For $U:\mathcal{C}\to \mathcal{D}$, the left-adjoint solution-set condition requires a weakly initial set in every $(D\downarrow U)$. Equivalently, there is a set of arrows $d_i:D\to U{C}_i$ through which every $d:D\to UC$ factors as $U(h)d_i$. The dual right-adjoint condition asks for weakly terminal solution sets in $(U\downarrow D)$.

For a colorless parent decaying through a color-diagonal interaction into a quark and antiquark, the three orthogonal matching-color states add incoherently. The inclusive flavor decay width is $N_c=3$ times the one-color width. In a parton scattering rate, an initial color average instead cancels the matching final color sum.

This function specifies the fraction of disrupted mass assigned to each fragment size interval. A self-similar law depends on fragment-to-parent size ratios, rather than an absolute size. In logarithmic bins its translation invariance gives the constant processed mass flux of the interior steady collisional cascade. In a removal-dominated interval, it instead sets the fragment source spectrum.

Once escape depletes a diameter interval, collision rates must be recomputed from the depleted projectile population. For a local power law $n(D)=K{D}^{-\alpha }$ and self-similar catastrophic-projectile limits, the rate scales as $K{D}^{3-\alpha }$. Retaining the original rate after changing $K$ or $\alpha$ can give incorrect transition sizes.

With size-independent collision speed and catastrophic disruption threshold $Q_D^{\ast }\propto D^b$, the catastrophic planetesimal collision rate scales as $R\propto D^{3-\alpha +b(1-\alpha )/3}$. Multiplying by mass per logarithmic size bin gives processed mass proportional to $D^{7-2\alpha +b(1-\alpha )/3}$. Constant mass flux in a collisional cascade sets the exponent to zero, yielding the displayed differential-number index when $b\ne -6$. This is the zero-velocity-exponent case of a velocity-dependent collisional cascade.

A bin with mass $m$ loses mass at approximately $m{R}_{\mathrm{cc}}$, so changing an order-one fraction takes $m/(m{R}_{\mathrm{cc}})=1/R_{\mathrm{cc}}$. The catastrophic planetesimal collision rate must be evaluated using the contemporaneous projectile population. It supplies a relaxation scale, not an exact equilibration time for an arbitrary fragment kernel.

If a collision operator is homogeneous of degree two in its bin masses, reducing every mass by $f$ slows the subsequent evolution by $f$. If $m(t)$ is the undepleted trajectory and depletion occurs at $T$, the post-depletion trajectory is $m_d(t)=m(T+(t-T)/f)/f$. Compared with a collision-only model starting with normalization $m(0)/f$, the same observed distribution has inferred age $fT+(t-T)$. The distribution alone cannot distinguish the depletion factor from unknown initial normalization and age.

A simplified collisional cascade tracks large objects $S$ and small fragments $F$. A fragment can disrupt a large object and multiply, while fragment–fragment collisions remove material. With $\dot{S}=-\kappa SF$ and $\dot{F}=\alpha SF-\beta F^2$, elimination of time gives a linear differential equation, $dF/dS-(\beta /\kappa )F/S=-\alpha /\kappa$. It has solution $F=C{S}^k-(\alpha /\kappa )S/(1-k)$ for $k=\beta /\kappa \ne 1$. Its maximum occurs at $F=(\alpha /\beta )S$; early exponential growth does not imply permanent exponential growth after the large-object reservoir is depleted.

The colimit of two arrows $A\to B$ and $A\to C$. A map from the pushout is uniquely a pair of maps from $B$ and $C$ agreeing on $A$. In modules it is $(B\oplus C)/\{(f(a),-g(a)):a\in A\}$.

Modes with $k\gg k_{\mathrm{eq}}$ enter the horizon during radiation domination. Their matter perturbations grow only logarithmically until matter-radiation equality, rather than as $a$. Matching the radiation-era logarithmic growth onto the matter-era growing mode gives the large-$k$ cosmological transfer function scaling shown, up to constants and the matching scale inside the logarithm. For a primordial scalar spectral index $n_s$, the late-time linear cosmological density power spectrum consequently changes from $P(k)\propto k^{n_s}$ at small $k$ to $P(k)\propto k^{n_s-4}{\log }^2(k/k_{\mathrm{eq}})$ at large $k$. These scalings omit baryonic acoustic structure and nonlinear evolution.

Here the cold-dark-matter transfer function is normalized so that ${\delta }_m(k,a)=A(a)k^{2}T(k)\mathcal{R}(k)$, where $\mathcal{R}$ is the primordial comoving curvature perturbation and $A(a)$ is independent of $k$. During radiation domination, horizon entry occurs at $a_{\mathrm{ent}}\propto k^{-1}$. An adiabatic entry amplitude of order $\mathcal{R}$ then acquires a factor $\log (a_{\mathrm{eq}}/a_{\mathrm{ent}})\sim \log (k/k_{\mathrm{eq}})$ by matter-radiation equality. Subsequent matter domination supplies a common growth factor; division by the conventional $k^2$ prefactor therefore gives $T(k)\propto k^{-2}\log (k/k_{\mathrm{eq}})$.

A cohomology operation is stable when it commutes with the suspension isomorphism on reduced cohomology:Products of positive-degree reduced classes on a suspension vanish, but stable operations can still connect its nonzero cohomology classes. The Steenrod squares and, at odd primes, Steenrod reduced powers are examples.

Let $x\in H^2({\mathbb{CP}}^2;{\mathbb{F}}_2)$ be the hyperplane class modulo two. It satisfies ${\mathrm{Sq}}^{2}x=x^2\ne 0$. Stability gives ${\mathrm{Sq}}^2(\sigma x)=\sigma (x^2)\ne 0$ on $\Sigma {\mathbb{CP}}^2$. The suspension of $S^2\vee S^4$ has the same additive groups and vanishing positive-degree products, but its corresponding degree-three class has zero ${\mathrm{Sq}}^2$, by pullback from $S^3$. Thus these suspensions have different cohomology operations even though their additive groups and cup products agree.^[ref]

The equivalence class of a cocycle in a cochain complex, where adding a coboundary leaves its class unchanged.

For a parabolic wave equation with $n=1+\mu W$, a zero-mean stationary real field $W$ of unit variance, and $L_0=i{\partial }_z^2/(2k)$, finite-correlation perturbation theory givesHere $m$ is the coherent field, $C$ is the autocorrelation function of a random field and $G$ is the one-coordinate Fresnel propagator. Expand the random solution once using the Duhamel principle, multiply by $W$ and average to obtain the memory term. The local phase term comes from the ${\mu }^2{W}^2$ term in $n^2$. The expansion is for fixed propagation distances with suitable covariance regularity and moment bounds. A Markov approximation is an additional scale assumption that can turn this integral equation into a local attenuation equation.

For the linear weak-index parabolic wave equation, a Markov approximation with integrated covariance $B(\eta )={\int }_{\mathbb{R}}C(s,\eta )ds$ givesThe drift follows by converting the multiplicative Stratonovich integral $ik\mu E\circ d\mathcal{B}$ to an Itô integral for a Brownian field with transverse covariance $B$. For a Gaussian beam with one transverse coordinate entering the random region at $x_0$, $m(x,z)=e^{-\gamma (x-x_0)}{E}_{\mathrm{free}}(x,z)$. The coherent field retains transverse beam structure. The white-noise closure needs short longitudinal correlations and the corresponding separation of propagation scales; Gaussian one-point statistics alone are insufficient.

A coherent formula is built from atomic relations and equality using $\top$, $\perp$, finite conjunction, finite disjunction and existential quantification. General negation and universal quantification are not constructors. Substitution of well-typed terms preserves coherent formulas.

A coherent sequent is an implication $\varphi {⊢}_{\vec{x}}\psi$ between coherent formulas in a common context. Its semantics is inclusion of the two interpreted subobjects of that context. The context is universally quantified outside the formulas.

A coherent theory is a first-order signature with a set of coherent sequents as axioms. It has a coherent syntactic category containing a conservative syntactic model. Theories of groups, rings and integral domains admit coherent axiomatizations.

When $y$ is not free in $\theta$, the coherent quantifier obeys $\theta \wedge \exists y\varphi ⊣⊢\exists y(\theta \wedge \varphi )$. Categorically this is the interaction of existential images with pullback, ensuring stable image factorizations in a coherent category.