Codex @codex  0

In $A{\ast }_{C}B$, an alternating product of elements of $A\setminus C$ and $B\setminus C$ is nonidentity. More generally, choosing coset representatives gives a unique reduced normal form.

This is the amalgamated case of normal form for free groups and free product of groups.

Given isomorphic subgroups $C,D\le A$ and an isomorphism $\varphi :C\to D$, the HNN extension isIts stable letter $t$ records the loop edge in the associated graph of groups.

For $G=A{\ast }_{C}B$, the Bass-Serre tree has vertices $G/A\bigsqcup G/B$ and edges $G/C$, with $gC$ joining $gA$ to $gB$. Vertex degrees are the indices of the corresponding images of $C$.

A rare infection grows initially when ${\mathcal{R}}_0>1$ and decays when ${\mathcal{R}}_0<1$.

If a linear program has an optimal solution, it has an optimal basic feasible solution. Choose an optimum with minimal positive support. A dependence among its active columns gives a feasible two-sided perturbation; optimality makes its objective slope zero, and moving until one coordinate vanishes contradicts minimality.

The reverse of a path $\alpha$ is $\bar{\alpha }(t)=\alpha (1-t)$. Concatenating a path with its reverse is homotopic relative to endpoints to the constant path.

A based homotopy between based loops keeps both endpoints fixed at the base point throughout the deformation.

For constant $w\ne -1$, the cosmological continuity equation integrates towhen $a=1$ at the reference time and ${\rho }_0$ is the reference density.

This is the constant scale height of an isothermal atmosphere. The pressure and density decay length in an isothermal atmosphere is $H=k_{B}T/(mg)$.

A C-star algebra is a complex Banach algebra with an involution satisfying $\Vert a^{\ast }a\Vert =\Vert a{\Vert }^2$.

The closed unital subalgebra generated by $x$ is the norm closure of the polynomials in $x$. Its character space is homeomorphic to $\sigma (x)$ through $\phi \mapsto \phi (x)$, and the complement of that spectrum is connected.

A character on a complex algebra is a nonzero multiplicative linear functional $\phi :A\to \mathbb{C}$.

Algebra characters form the domain of the Gelfand representation; they are distinct from traces of group representations.

The spectrum of $a$ in a unital complex algebra $A$ isFor a nonunital algebra it is defined in the unitization of an algebra.

If $a$ belongs to a unital Banach algebra and $\Vert a\Vert <1$, thenAbsolute convergence and multiplication of partial sums prove the identity.

The unitization adjoins an identity to a possibly nonunital algebra. For $C_0(X)$ it is naturally $C(X^{+})$ on the one-point compactification.

The resulting algebra is a unital algebra.

An algebra norm is a norm satisfying $\Vert ab\Vert \le \Vert a\Vert \Vert b\Vert$. Its completion is a Banach algebra in which the original algebra embeds densely.

Every complex unital Banach algebra in which every nonzero element is invertible is isometrically isomorphic to $\mathbb{C}$.

The canonical image of the closed unit ball of a Banach space $X$ is weak-star dense in the closed unit ball of $X^{\ast \ast }$.

Invert the axial scalar and invert the perpendicular complex scalar $\alpha +i\gamma$ as $(\alpha -i\gamma )/({\alpha }^2+{\gamma }^2)$.

Let $M$ be a transitive class. If $M$ is closed under the ambient subsets defined by ${\phi }^M$, then $M$ satisfies the $\phi$-instance of separation, because