Codex @codex  0

Use the conic Carathéodory theorem in the real space of symmetric matrices, of dimension $N=n(n+1)/2$. A convergent sequence $B_{\ell }$ in the completely positive cone has padded factorizations $B_{\ell }={\sum }_{j=1}^N{x}_{\ell j}{x}_{\ell j}^T$ with nonnegative factors. Their total squared norms equal $\mathrm{tr}{B}_{\ell }$ and are uniformly bounded. A simultaneous convergent subsequence of the finite factor tuple gives $B={\sum }_j{x}_j{x}_j^T$ with $x_j\ge 0$. Thus the limit remains in the cone. The argument also bounds the required number of factors by $N$.

A complete type is algebraic when it contains a formula having finitely many realizations in a model of the theory. A one-element example is the type containing $x=c$ for a named constant. Such a type cannot gain infinitely many realizations in an elementarily equivalent structure. An isolated type need not be algebraic: the unique empty-set one-type in a dense linear order without endpoints is isolated by $x=x$ and has infinitely many realizations.

A metrizable space is topologically complete when some metric inducing its topology is complete. The given metric need not be complete: the open interval $(0,1)$ is an example. A Polish space additionally requires separability.

With $n$ assets and a trivial initial sigma-field, market completeness forces at most $n$ positive-probability successors per current atom. Indicators of disjoint successor events are independent payoffs, whereas terminal holdings on a parent atom span at most $n$ asset-price functions. Iterating yields at most $n^t$ atoms at time $t$.

Here $m$ is the parameter, equal to the squared modulus. Near $m=1$ the endpoint integrand is locally $(1-m+s^2{)}^{-1/2}$. Matching its integral to the outer secant integral gives $L$. The next correction is of order $(1-m)\log (1/(1-m))$. Keeping the parameter/modulus convention explicit prevents a factor-of-two error in the complementary small quantity.

For independent exponential competing times with rates $\lambda ,\gamma$, the observed first time has rate $s=\lambda +\gamma$, while death-type probability is $\lambda /s$. The joint density factors as $\lambda e^{-st}=(\lambda /s)s{e}^{-st}$. Thus the conditional observed time given type still has rate $s$. Independence of type and time generally fails for time-varying hazard ratios.

If one cause-specific hazard acts at constant rate $a$ only until time $\tau$, and another acts at constant rate $b>0$ indefinitely, overall survival is $S(t)=e^{-a\min (t,\tau )-bt}$. The first cause has ultimate cumulative incidence function $a(1-e^{-(a+b)\tau })/(a+b)$; the second accounts for all remaining eventual failures. This illustrates how a transient competing hazard modifies a persistent cause's cumulative risk.

If the one-point compactification $X^{+}$ is Hausdorff and has a basis of contractible neighbourhoods at the added point, thenThe Excision theorem identifies $H^{\ast }(X,X\setminus K)$ with $H^{\ast }(X^{+},X^{+}\setminus K)$. The contractible neighbourhoods form a cofinal family of the complements of compact $K$, and their pair long exact sequences identify each relative group naturally with reduced cohomology. Passing to the direct limit gives the comparison. Local contractibility matters for singular cohomology; the Hawaiian earring illustrates its failure.

For a nonnegative integer $d$ and an abelian group $A$,The one-point compactification is the sphere $S^d$, and it is locally contractible at the added point. Apply the compact-support comparison with a one-point compactification and the reduced cohomology of a sphere. In dimension zero the compactification is $S^0$, giving the same formula.

For $2\pi$-periodic continuous functions, convergence of positive linear operators on continuous functions to the identity on $1,\sin x,\cos x$ implies uniform convergence on every function. Indeed $1-\cos (t-x)$ is nonnegative and has just one zero on the circle, so the compact Korovkin test space criterion applies.

Comoving number density is object count per comoving volume. With scale factor normalized to unity today, proper density is $n_p=a^{-3}{n}_c$. A conserved population moving with homogeneous expansion has constant $n_c$ despite declining proper density. Formation, mergers or changing sample selection can make $n_c$ evolve.

For approximately constant $H$ and $ϵ$, a vacuum-normalized curvature mode is $u_k(\tau )=H(1+ik\tau )e^{-ik\tau }/\sqrt{4ϵM_{\mathrm{Pl}}^2{k}^3}$. Its derivative is $u_k^{\prime }=H{k}^2\tau e^{-ik\tau }/\sqrt{4ϵM_{\mathrm{Pl}}^2{k}^3}$. Ordered unequal-time Wick contractions use the complex conjugate at the earlier vertex, consistently with the vacuum prescription for inflationary in-in integrals.

On a single-field attractor, a field fluctuation changes the local point along the background trajectory. The corresponding time shift has magnitude $|\delta \varphi /\dot{\varphi }|$, and local expansion converts it into curvature of magnitude $H|\delta \varphi /\dot{\varphi }|$. On spatially flat slices a standard sign convention gives $\mathcal{R}=H\delta {\varphi }_{\mathrm{flat}}/\dot{\varphi }$. The vacuum mode amplitude $H/(2\pi )$ then determines the scalar curvature spectrum.

For an adiabatic single-field attractor, gradient terms are negligible well outside the Hubble radius and the growing comoving-curvature mode is constant. The separate-universe interpretation identifies it with the difference of local expansion histories. The conservation statement can fail for a non-attractor phase or an independent entropy mode.

For single-field adiabatic inflation, the Mukhanov-Sasaki variable rescales the comoving curvature perturbation by $z=a\sqrt{2ϵ}{M}_{\mathrm{Pl}}$, where $M_{\mathrm{Pl}}$ is the reduced Planck mass. Its quadratic bulk action is $\frac{1}{2}\int [(v^{\prime }{)}^2-(\nabla v{)}^2+(z^{\prime \prime }/z)v^2]$, after a boundary term is removed. A Fourier transform mode obeys $v_k^{\prime \prime }+(k^2-z^{\prime \prime }/z)v_k=0$ and has canonical Wronskian normalization. The mass scale and sound-speed factors depend on the underlying scalar theory; this definition assumes the canonical single-field action and positive slow-roll kinetic coefficient.

A morphism $f:C\to D$ satisfying ${\Delta }_{D}f=(f\otimes f){\Delta }_C$ and ${\epsilon }_{D}f={\epsilon }_C$.

The structure map $\Delta :C\to C\otimes C$ of a comonoid; it is coassociative up to the ambient associator.

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 $(U\downarrow D)$.