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When each comparison functor has a left adjoint, iterating the Eilenberg-Moore comparison functor produces the monadic tower. Its monadic length is the least number of comparison steps needed to reach an equivalence; an equivalence has length zero and a non-equivalence that is already monadic has length one.

A left eigenvector satisfies $u^{T}A=\lambda u^T$, equivalently $A^{T}u=\lambda u$ over the real numbers.

A right eigenvector satisfies $Av=\lambda v$.

An eigenbasis is a basis consisting of eigenvectors of one operator.

The binary entropy function isIt is concave on $[0,1]$ and satisfies $H_2(x)\ge 2x$ for $0\le x\le 1/2$ by comparison with the chord from $(0,0)$ to $(1/2,1)$.

Every Dynkin system containing a pi-system $\mathcal{P}$ also contains the sigma-algebra $\sigma (\mathcal{P})$. Equivalently, the Dynkin system generated by a pi-system equals the sigma-algebra it generates.

The dust transparency radius is where a swept shell's ultraviolet optical depth equals one. For a gas fraction $f_g$ in a singular isothermal sphere, $R_{\tau }={\kappa }_{\mathrm{UV}}{f}_g{\sigma }^2/(2\pi G)$.

On the real line, initial data $\delta (x-2\sqrt{D})$ and a source $-A\delta (x+2\sqrt{D})\delta (t-1)$ produceAt $x=0,t=2$, the two terms cancel exactly for $A=\sqrt{e/2}$.

A Schauder basis is shrinking when its dual sequence is a Schauder basis of the whole dual space. Every Schauder basis of a reflexive Banach space is shrinking.

A uniformizer of a discrete valuation ring $R$ is a generator $\pi$ of its unique nonzero maximal ideal. Every nonzero $x$ in the fraction field has a unique expression $x=u{\pi }^n$ with $u$ a unit and $n\in \mathbb{Z}$.

A discrete valuation on a field $K$ is a surjective group homomorphism $v:K^{\times }\to \mathbb{Z}$ satisfying $v(x+y)\ge \min (v(x),v(y))$ whenever $x+y\ne 0$. Its valuation ring is $\{0\}\cup \{x:v(x)\ge 0\}$.

Every principal ideal $I$ of a number field has a generator $\alpha$ satisfyingat every archimedean embedding. Multiply an arbitrary generator by a unit so that its logarithmic unit-lattice coordinates lie in a fixed bounded fundamental parallelepiped.

The relaxation time of a finite reversible lazy Markov chain is the reciprocal of its spectral gap: $t_{\mathrm{rel}}=(1-{\lambda }_2{)}^{-1}$.

For a transient weighted graph, $H_0$ is the completion of the finitely supported functions in the norm $\Vert f{\Vert }_{H_0}=\mathcal{E}(f,f{)}^{1/2}$. Its elements have finite Dirichlet energy and vanish at infinity in the energy sense.

Every closed walkin a directed covering graph of an interval map has a point $x\in J_0$ with $F^j(x)\in J_j$ and $F^n(x)=x$. This follows by successively pulling $J_0$ back through the covering relations and applying the fixed-point property of a closed interval. If the itinerary has least period $n$ and avoids shared endpoints, $x$ lies on an $n$-cycle.

A component outside a longest cycle has at least $k$ attachment vertices in a $k$-connected graph, and no two attachments are consecutive, forcing cycle length at least $2k$.

The minimal subtraction scheme chooses counterterms that remove only the poles in the dimensional regulator $ϵ$, without additional finite terms.

The inverse function theorem answers a practical question: can nearby outputs be used to recover nearby inputs uniquely and smoothly? Its useful surprise is that a nonlinear function can be tested using its linear derivative. If the first-order changes preserve all input directions, then, on a sufficiently small region, the nonlinear map can also be reversed. This is the basis for treating a nonlinear map as a coordinate transformation.

Precisely, let $f:U\to {\mathbb{R}}^n$ be continuously differentiable on an open subset $U$ of Euclidean space, let $a\in U$, and put $b=f(a)$. If $Df(a)$ is an invertible linear map, there are open neighbourhoods $U_0$ of $a$ and $V_0$ of $b$ such that $f:U_0\to V_0$ is a bijection with a continuously differentiable inverse function $g$. Thus this restriction is a diffeomorphism. In coordinates the test is $\det Df(a)\ne 0$, using the Jacobian determinant. The derivative of a local inverse map is obtained by inverting the Jacobian matrix, at the corresponding input. ^[ref]

For a one-dimensional example, $f(x)=x^2$ has $f^{\prime }(1)=2$. Near $x=1$, the inverse function is $g(y)=\sqrt{y}$, and the inverse function rule gives $g^{\prime }(1)=1/2$. But $x^2$ has no inverse function on the whole real line, since $x$ and $-x$ give the same output. The theorem supplies a local restriction, not global injectivity. At zero, the derivative test fails; the nonnegative branch still has a square root inverse, but that inverse has no finite derivative at zero. ^[ref]

Here is a two-dimensional example with an invertible derivative everywhere and infinitely repeated outputs:The exponential function turns $s$ into a positive radius, while $\theta$ is the angle in polar coordinates. Restricting, for example, to $|s|<0.35$ and $|\theta |<0.6$ produces the reversible patch pictured below. On its image, with Cartesian coordinates $(u,v)$ and $u>0$, the inverse function isThe arctangent selects the angle near zero. Without an angle restriction, $f(s,\theta +2\pi k)=f(s,\theta )$ for every integer $k$, so there is no single global inverse function. This map is a local diffeomorphism but is not injective.

The short proof idea is to correct a guessed input using the inverse linear response. With $A=Df(a)$, solving $f(x)=y$ is equivalent to finding a fixed point ofAt $a$, the derivative $D{T}_y(a)=I-A^{-1}Df(a)$ vanishes. Continuity makes $T_y$ a contraction mapping on a sufficiently small closed ball, which it maps into itself when $y$ is sufficiently close to $b$. The contraction mapping theorem then gives existence and uniqueness there. Establishing the smooth dependence on $y$ completes the inverse theorem. This explains why the nonlinearity can be tolerated: its correction has a small local slope.^[ref]

If the line segment from $x$ to $y$ lies in the domain of a differentiable map and $\Vert D{f}_z\Vert \le M$ along it, then

An invertible linear map is a linear bijection; its inverse is also linear.

Invertibility adds a bijection requirement to the definition of a linear map.