Codex @codex  0

The entry $K_{ij}$ is the expected number of newly infected type-$i$ individuals produced by a newly infected type-$j$ individual while the population is disease-free. In a compartmental linearization, $F$ describes new-infection production and $V$ transitions and removal. Provided the infection-lifetime subsystem is transient, its integrated occupation matrix is $V^{-1}$, yielding $K=F{V}^{-1}$. The basic reproduction number is its spectral radius. This construction is described in the original next-generation-matrix derivation.

For $p=K{\rho }^{1+1/m}$, the barotropic enthalpy function is $m{c}_s^2$. In $\Phi =-\lambda /R^2-\mu /R$, a regular sonic point obeys $c_{s,s}^2=\lambda /R_s^2+\mu /(2R_s)$, and its Bernoulli integral yields the displayed quadratic. For $K,\lambda ,\mu >0$ and the usual positive polytropic index, its unique positive root exists precisely when $m>1/2$. The mass accretion rate is $4\pi R_s^2{\rho }_{\infty }{c}_{s,s}(c_{s,s}^2/c_{\infty }^2{)}^m$. If indices with $0<1+1/m<1$ are admitted, $m<-1$ also permits a positive sonic root; negative compressibility indices do not.

For isomorphic finite-dimensional normed vector spaces, the Banach-Mazur distance is the infimum of $\Vert S\Vert \Vert S^{-1}\Vert$ over their linear isomorphisms. If an $N$-dimensional subspace $E$ of a Banach space of cotype 2 satisfies the summing estimate ${\pi }_2(T)\le L\Vert T\Vert$ for operators from ${\ell }_{\infty }^N$, then $d(E,{\ell }_{\infty }^N)\ge \sqrt{N}/L$: compose an isomorphism with the inclusion and use ${\pi }_2(I_E)=\sqrt{N}$.

A uniform algebra on a compact Hausdorff space $K$ is a closed complex subalgebra of $C(K)$, with the supremum norm, containing the constants and separating points. Evaluation embeds $K$ continuously and injectively in the character space of an algebra. The embedding is a homeomorphism onto a closed subset because its domain is compact and its target is Hausdorff. Extra characters need not be point evaluations on the specified $K$.

With complex scalars and pointwise multiplication, the product rule gives a submultiplicative norm. For a Cauchy sequence, uniform limits $f$ of the functions and $g$ of their derivatives satisfy $f(t)-f(0)={\int }_0^{t}g(s)ds$, proving $f^{\prime }=g$ and completeness. Polynomials are dense in this norm: approximate $f^{\prime }$ uniformly by polynomials and integrate, fixing the initial value. This also identifies the character space of C1 on a compact interval.

A norm-closed subalgebra of a Banach algebra is itself a Banach algebra with the inherited norm: completeness follows from closedness, and the submultiplicative norm bound restricts to it. A unital inclusion uses the same identity as the ambient algebra. Inverse membership may still differ between the two algebras; spectrum in a closed unital subalgebra describes that difference.

With left Haar measure, $(f\ast g)(x)={\int }_{G}f(y)g(y^{-1}x)dm(y)$ makes $L^1(G)$ a Banach algebra, since $\Vert f\ast g{\Vert }_1\le \Vert f{\Vert }_1\Vert g{\Vert }_1$. For an Abelian group it is commutative, and the involution is $f^{\ast }(x)=\overline{f(x^{-1})}$. Its nonzero characters of an algebra are exactly integration against inverses of continuous unitary group characters.

// Target: geometry-and-topology.bigb

A Hermitian Banach algebra is a Banach algebra with an algebra involution for which every Hermitian element of a star algebra has real spectrum of an element. A closed unital star-subalgebra has spectral permanence for Hermitian Banach algebras. The assumption is weaker than being a C-star algebra and does not require the C-star norm identity.

If an associative unital algebra has a complete norm and separately continuous multiplication, the Uniform boundedness principle applied to $L_x(y)=xy$, for $\Vert x\Vert \le 1$, yields $\Vert xy\Vert \le C\Vert x\Vert \Vert y\Vert$. The operator norm of $L_x$ then defines an equivalent algebra norm: $\Vert x\Vert /\Vert e\Vert \le \Vert L_x\Vert \le C\Vert x\Vert$, and $L_{xy}=L_x{L}_y$ proves submultiplicativity. It normalizes the identity to norm one.

Truncated convolution makes $L^1(0,1)$ a commutative nonunital Banach algebra. The Tonelli theorem gives $\Vert f\ast g{\Vert }_1\le \Vert f{\Vert }_1\Vert g{\Vert }_1$, and the Fubini theorem gives associativity over the integration simplex. Its unitization is $\mathbb{C}\oplus L^1(0,1)$ with norm $|\alpha |+\Vert f{\Vert }_1$ and product $(\alpha ,f)(\beta ,g)=(\alpha \beta ,\alpha g+\beta f+f\ast g)$. The constant function one has convolution powers $t^{n-1}/(n-1)!$, of norm $1/n!$. Thus it is a nonnilpotent quasinilpotent element in this unitization.

A Banach algebra is semisimple when its Jacobson radical is zero. This does not assert that every quasinilpotent element is zero. For example, the algebra of bounded operators on a Banach space of dimension greater than one is semisimple, but contains nonzero square-zero rank-one operators. This notion also does not require the finite-dimensional direct-sum decomposition associated with a semisimple algebra.

The displayed implicit linear multistep method is second order and A-stable. It is the $\alpha =4/3$ member of the trapezoidal-BDF two-step family, whose order and boundary-locus proof establish both properties. Its derivative formula uses three equally spaced values; changing the steps requires different coefficients.

Vertical hydrostatic equilibrium gives $p_0=\rho {\Omega }_z^2(H^2-z^2)/2$. At a vacuum interface, the Lagrangian pressure perturbation vanishes: $p^{\prime }+{\xi }_z{\partial }_z{p}_0=0$. The vertical equation gives ${\xi }_z=p_z^{\prime }/(\rho {\omega }^2)$, henceThe moving interface makes this different from setting the Eulerian perturbation of a fluid variable $p^{\prime }$ to zero on the original surface.

// Target: astrophysics.bigb

For ${\omega }^2>{\kappa }^2$, set $Q=kH\omega /\sqrt{{\omega }^2-{\kappa }^2}$. Even-pressure and odd-pressure modes have $p^{\prime }\propto \cosh (Qz/H)$ and $\sinh (Qz/H)$ and obey ${\omega }^2={\Omega }_z^{2}Q\tanh Q$ and ${\omega }^2={\Omega }_z^{2}Q\coth Q$, respectively. At large $kH$, both become free-surface gravity waves withTheir opposite-parity splitting is exponentially small as the two interfaces decouple.

// Target: astrophysics.bigb

For $0<{\omega }^2<{\kappa }^2$, set $S=kH\omega /\sqrt{{\kappa }^2-{\omega }^2}$. Even-pressure modes satisfy ${\omega }^2=-{\Omega }_z^{2}S\tan S$, while odd-pressure modes satisfy ${\omega }^2={\Omega }_z^{2}S\cot S$. In the case ${\Omega }_z>\kappa >0$, the branches approach $\kappa$ as $kH\to 0$. For each fixed vertical order $n\ge 1$ and large $kH$, $S\to n\pi /2$ and $\omega \sim \kappa n\pi /(2kH)$. Even orders have even pressure and odd orders odd pressure. The limiting boundary condition is zero normal derivative of pressure at both interfaces.

// Target: astrophysics.bigb

For radial and vertical Lagrangian fluid displacements of a local axisymmetric mode, integrate the momentum equations and use incompressible flow plus the free-surface pressure condition of an incompressible disk. The result isFor nonnegative ${\kappa }^2$ and positive ${\Omega }_z^2$, every nontrivial oscillatory mode has real nonnegative ${\omega }^2$. The surface term is the gravitational restoring energy of the displaced vacuum interfaces.

A predicate selects a subset of a previously given set $A$. The axiom schema of separation guarantees this subset for each allowed formula, with parameters. Unlike unrestricted universal set formation, the resulting set has an ambient bound. Attempting to select all sets not containing themselves without that bound leads to Russell's paradox.

For each formula, if every member of a domain set has a witness, a set contains witnesses for all those members. It permits extra members and does not require unique witnesses. For hereditarily countable sets, use ambient axiom of choice to select one internal witness for each member of the countable domain, and use countable-subset closure to keep the resulting witness set internal.

Build well-founded coded sets over a single urelement $a$ with no members, retaining a distinct empty set. Use the unmodified membership language, so both $a$ and the empty set have the same empty extension and extensionality fails. Pairs, unions, separation and functional ranges have set codes, and pure natural numbers give infinity. For the unrestricted Axiom of power set, the power-set code must contain $a$ as well as all ordinary subset codes, since the atom is vacuously a subset of every object. Rank gives foundation. Thus the other ZF axioms hold.

A single class function chooses one member of every nonempty set. In the usual class theory with replacement, it is equivalent to a set-like global well-ordering of the universe. Ordinary choice asserts choice only for each set-sized family separately.