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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

The principle of epsilon induction says that every progressive class is universal: ifthen $\phi (x)$ holds for every set $x$. Over the other axioms of ZF, it is equivalent to the Axiom of foundation.

For any sets $X,Y$, either $X$ injects into $Y$ or $Y$ injects into $X$. Applying this to $X$ and its Hartogs ordinal proves the well-ordering theorem, so over ZF cardinal comparability is equivalent to choice.

Every set admits a well-order. In ZF this statement is equivalent to the axiom of choice.

A choice function on a family $\mathcal{A}$ of nonempty sets is a function $f$ with $f(A)\in A$ for every $A\in \mathcal{A}$.

Forthe zero-input stable states are $0$ and $(k+\sqrt{k^2-4})/2$. The switching threshold is the saddle-node bifurcation determined by $f(g_c,s_c)={\partial }_{g}f(g_c,s_c)=0$ on the low-concentration branch.

Even if ${\varphi }_n\sim {\psi }_n$ for every $n$, an expansion in the scale $({\varphi }_n)$ need not induce one in $({\psi }_n)$. A perturbation of ${\psi }_0$ that is smaller than ${\varphi }_0$ but larger than ${\varphi }_1$ supplies a counterexample.