Definitions for Topological Entanglement Entropy and Holography

QAlgorithms.Papers.Independent.Intakethird100.P58.D4Parameters

This defines a structure whose data consist of two real numbers, called the coupling $\lambda$ and the circle radius $c$, together with proofs of the strict inequalities $\lambda>0$ and $c>0$. Thus every element of this structure carries both positive real parameters and evidence of their positivity.

QAlgorithms.Papers.Independent.Intakethird100.P58.DirectEquation

For every rotational model $m\in\{\mathrm{conformal},\mathrm{d3},\mathrm{d4}\}$, every function $Z:\mathbb R\to\mathbb R$, and every real $t$, the direct-equation predicate means precisely the equality

$$\left.\frac{d}{du} \left( \frac{u^{n_m}Z'(u)} {Z(u)^3\sqrt{w_m(Z(u))+(Z'(u))^2}} \right)\right|_{u=t} = t^{n_m} \left( \frac{v_m(Z(t))} {2Z(t)^3\sqrt{w_m(Z(t))+(Z'(t))^2}} - \frac{3\sqrt{w_m(Z(t))+(Z'(t))^2}}{Z(t)^4} \right),$$

where $n_{\mathrm{conformal}}=n_{\mathrm{d3}}=1$, $n_{\mathrm{d4}}=2$; $w_{\mathrm{conformal}}(s)=1$, $w_{\mathrm{d3}}(s)=1-s^4$, $w_{\mathrm{d4}}(s)=s(1-s^3)$; and $v_{\mathrm{conformal}}(s)=0$, $v_{\mathrm{d3}}(s)=-4s^3$, $v_{\mathrm{d4}}(s)=1-4s^3$, for all real $s$. No differentiability, positivity, or nonzero-denominator hypotheses are included. Every derivative is assigned value $0$ where the relevant function is not differentiable, division by zero has value $0$, and square roots of nonpositive arguments have value $0$. Thus the equality is meaningful also when $t=0$, when $Z(t)=0$, or when the square-root argument is nonpositive; its right-hand side is $0$ in each of these cases.

QAlgorithms.Papers.Independent.Intakethird100.P58.FirstD4Coefficient

For any function $f:\mathbb R\to\mathbb R$, this defines the proposition that $f(0)=0$, that $f$ is twice continuously differentiable at every real point $z\in[0,1]$, and that at every $z\in(0,1)$,

$$4f'(z)+2z^3f'(z)+5z\bigl(f'(z)\bigr)^3 -7z^4\bigl(f'(z)\bigr)^3 +2z^7\bigl(f'(z)\bigr)^3 -2zf''(z)+2z^4f''(z)=0.$$

The differentiability requirement at both endpoints is with respect to the full real domain, rather than merely within $[0,1]$. The differential equation is required only on the open interval, and no value at $z=1$ is prescribed.

QAlgorithms.Papers.Independent.Intakethird100.P58.InverseEquation

For every rotational model $m\in\{\mathrm{conformal},\mathrm{d3},\mathrm{d4}\}$, every function $r:\mathbb R\to\mathbb R$, and every real $z$, the inverse-equation predicate means precisely the equality

$$\left.\frac{d}{du} \left( \frac{r(u)^{n_m}w_m(u)r'(u)} {u^3\sqrt{1+w_m(u)(r'(u))^2}} \right)\right|_{u=z} = \frac{n_m r(z)^{n_m-1}}{z^3} \sqrt{1+w_m(z)(r'(z))^2},$$

where $n_{\mathrm{conformal}}=n_{\mathrm{d3}}=1$, $n_{\mathrm{d4}}=2$, and $w_{\mathrm{conformal}}(u)=1$, $w_{\mathrm{d3}}(u)=1-u^4$, $w_{\mathrm{d4}}(u)=u(1-u^3)$. Thus the exponent $n_m-1$ is $0$ for the first two models and $1$ for the third, with $r(z)^0=1$ even when $r(z)=0$. No differentiability or positivity assumptions accompany the equality: every derivative displayed is assigned value $0$ where the function being differentiated is not differentiable, division by zero has value $0$, and square roots of nonpositive real numbers have value $0$. The predicate therefore also applies at $z=0$, where its right-hand side is $0$.

QAlgorithms.Papers.Independent.Intakethird100.P58.IsAnnulus

For a model $m$ chosen from $\{\mathrm{conformal},\mathrm{d3},\mathrm{d4}\}$, real numbers $R_i,R_e$, and a function $Z:\mathbb R\to\mathbb R$, this predicate requires $0<R_i<R_e$, continuity of $Z$ on $[R_i,R_e]$, and $Z(R_i)=Z(R_e)=0$. At every $t\in(R_i,R_e)$, it additionally requires that $Z$ be twice continuously differentiable at $t$ as a function on $\mathbb R$, that $Z(t)>0$, that either $m=\mathrm{conformal}$ or $Z(t)<1$, and that

$$\left.\frac{d}{du} \left( \frac{u^n Z'(u)} {Z(u)^3\sqrt{W(Z(u))+Z'(u)^2}} \right)\right|_{u=t} = t^n\left( \frac{V(Z(t))} {2Z(t)^3\sqrt{W(Z(t))+Z'(t)^2}} -\frac{3\sqrt{W(Z(t))+Z'(t)^2}}{Z(t)^4} \right),$$

where the model determines the quantities by

$$(n,W(z),V(z))= \begin{cases} (1,1,0),&m=\mathrm{conformal},\\ (1,1-z^4,-4z^3),&m=\mathrm{d3},\\ (2,z(1-z^3),1-4z^3),&m=\mathrm{d4}. \end{cases}$$

The differential equation and the differentiability requirements are imposed only in the open interval; at its endpoints, the stated requirements are continuity and zero boundary values.

QAlgorithms.Papers.Independent.Intakethird100.P58.IsCylinder

For a model $m\in\{\mathrm{conformal},\mathrm{d3},\mathrm{d4}\}$, a real number $R$, and a function $r:\mathbb R\to\mathbb R$, the cylinder predicate holds exactly when $m=\mathrm{d3}$ or $m=\mathrm{d4}$, $R>0$, the endpoint $1$ is positive, $r(0)=R$, $r$ is continuous on $[0,1]$, $r(y)<r(x)$ whenever $0\le x<y\le1$, and $r$ is twice continuously differentiable at every point of $[0,1)$; for every $z\in(0,1)$, $r(z)>0$ and

$$\left.\frac{d}{du} \left( \frac{r(u)^n w(u)r'(u)} {u^3\sqrt{1+w(u)(r'(u))^2}} \right)\right|_{u=z} = \frac{n r(z)^{n-1}}{z^3} \sqrt{1+w(z)(r'(z))^2};$$

for every real $a$ with $0<a<1$, the function $z\mapsto r(z)^n z^{-3}\sqrt{1+w(z)(r'(z))^2}$ is Lebesgue integrable from $a$ to $1$; and $r(1)>0$ and $r$ is twice continuously differentiable at $1$ as well. Here $n=1$ and $w(u)=1-u^4$ when $m=\mathrm{d3}$, while $n=2$ and $w(u)=u(1-u^3)$ when $m=\mathrm{d4}$. Derivatives in the formulas are assigned value $0$ where differentiability fails, division by zero has value $0$, and square roots of nonpositive arguments have value $0$. The equation is required only for $0<z<1$, and the integrability condition does not include $a=0$.

QAlgorithms.Papers.Independent.Intakethird100.P58.IsDisk

For a model $m\in\{\mathrm{conformal},\mathrm{d3},\mathrm{d4}\}$, real numbers $R,b$, and a function $r:\mathbb R\to\mathbb R$, the disk predicate means that $R>0$, $b>0$, $r(0)=R$, $r$ is continuous on $[0,b]$, $r(y)<r(x)$ for every $0\le x<y\le b$, and $r$ is twice continuously differentiable at every point of $[0,b)$; for every $z\in(0,b)$, $r(z)>0$ and

$$\left.\frac{d}{du} \left( \frac{r(u)^n w(u)r'(u)} {u^3\sqrt{1+w(u)(r'(u))^2}} \right)\right|_{u=z} = \frac{n r(z)^{n-1}}{z^3} \sqrt{1+w(z)(r'(z))^2};$$

and for every real $a$ with $0<a<b$, the function $z\mapsto r(z)^n z^{-3}\sqrt{1+w(z)(r'(z))^2}$ is Lebesgue integrable from $a$ to $b$. Here $(n,w(u))=(1,1)$, $(1,1-u^4)$, or $(2,u(1-u^3))$ according as $m$ is $\mathrm{conformal}$, $\mathrm{d3}$, or $\mathrm{d4}$, respectively. In addition, $r(b)=0$, either $m=\mathrm{conformal}$ or $b<1$, and the function

$$H(t)=\inf\{z\in[0,b]:r(z)=t\}$$

is twice continuously differentiable within $[0,R)$ and has derivative $0$ at $t=0$ when restricted to $[0,\infty)$. The infimum defining $H$ is assigned value $0$ if its defining set is empty; its definition is for every real $t$. Derivatives in the displayed equation and integrand are assigned value $0$ where differentiability fails, division by zero has value $0$, and square roots of nonpositive arguments have value $0$. No equation is imposed at $z=0$ or $z=b$, no integrability down to $a=0$ is required, and the stated regularity of $H$ excludes $t=R$.

QAlgorithms.Papers.Independent.Intakethird100.P58.IsProfile

For a model $m\in\{\mathrm{conformal},\mathrm{d3},\mathrm{d4}\}$, real numbers $R,b$, and a function $r:\mathbb R\to\mathbb R$, the profile predicate holds exactly when the following common requirements hold and at least one of the two alternatives below holds: $R>0$, $b>0$, $r(0)=R$, $r$ is continuous on $[0,b]$, $r(y)<r(x)$ whenever $0\le x<y\le b$, $r$ is twice continuously differentiable at every point of $[0,b)$, and for every $z\in(0,b)$, $r(z)>0$ and

$$\left.\frac{d}{du} \left( \frac{r(u)^n w(u)r'(u)} {u^3\sqrt{1+w(u)(r'(u))^2}} \right)\right|_{u=z} = \frac{n r(z)^{n-1}}{z^3} \sqrt{1+w(z)(r'(z))^2};$$

also, for every real $a$ with $0<a<b$, the function $z\mapsto r(z)^n z^{-3}\sqrt{1+w(z)(r'(z))^2}$ is Lebesgue integrable from $a$ to $b$. Here $(n,w(u))=(1,1)$, $(1,1-u^4)$, or $(2,u(1-u^3))$ according as $m$ is $\mathrm{conformal}$, $\mathrm{d3}$, or $\mathrm{d4}$, respectively. The first alternative is that $r(b)=0$, either $m=\mathrm{conformal}$ or $b<1$, and the function $H(t)=\inf\{z\in[0,b]:r(z)=t\}$ is twice continuously differentiable within $[0,R)$ and has derivative $0$ at $t=0$ within $[0,\infty)$. The second alternative is that $b=1$, $m=\mathrm{d3}$ or $m=\mathrm{d4}$, $r(1)>0$, and $r$ is twice continuously differentiable at $1$. The height function $H$ is defined for every real $t$, with value $0$ when its defining set is empty. Derivatives in the equation and integrand are assigned value $0$ where differentiability fails, division by zero has value $0$, and square roots of nonpositive arguments have value $0$. Neither alternative requires the displayed equation at either endpoint or integrability from $a=0$.

QAlgorithms.Papers.Independent.Intakethird100.P58.RegularGraph

For a model $m\in\{\mathrm{conformal},\mathrm{d3},\mathrm{d4}\}$, real numbers $R,b$, and a function $r:\mathbb R\to\mathbb R$, the regular-graph predicate holds exactly when $R>0$, $b>0$, $r(0)=R$, $r$ is continuous on $[0,b]$, $r(y)<r(x)$ whenever $0\le x<y\le b$, and $r$ is twice continuously differentiable at every $z\in[0,b)$; additionally, for every $z\in(0,b)$, one has $r(z)>0$ and

$$\left.\frac{d}{du} \left( \frac{r(u)^n w(u)r'(u)} {u^3\sqrt{1+w(u)(r'(u))^2}} \right)\right|_{u=z} = \frac{n r(z)^{n-1}}{z^3} \sqrt{1+w(z)(r'(z))^2};$$

finally, for every real $a$ satisfying $0<a<b$, the function

$$z\longmapsto \frac{r(z)^n}{z^3}\sqrt{1+w(z)(r'(z))^2}$$

is Lebesgue integrable on the interval from $a$ to $b$. Here $(n,w(u))=(1,1)$ for $m=\mathrm{conformal}$, $(1,1-u^4)$ for $m=\mathrm{d3}$, and $(2,u(1-u^3))$ for $m=\mathrm{d4}$, so $n-1$ is respectively $0,0,1$. All derivatives in these formulas are totalized by assigning value $0$ where differentiability fails, division by zero has value $0$, and square roots of nonpositive arguments have value $0$. The twice-continuous-differentiability requirement includes $0$ as a point of the full real domain but excludes $b$; the equation and strict positivity of $r$ are required only on $(0,b)$, and no integrability requirement down to $a=0$ is included.

QAlgorithms.Papers.Independent.Intakethird100.P58.RotationalModel

The type of rotational models has exactly three distinct values, named $\mathrm{conformal}$, $\mathrm{d3}$, and $\mathrm{d4}$. Equality between any two values of this type is decidable.

QAlgorithms.Papers.Independent.Intakethird100.P58.SecondD4Coefficient

For any function $g:\mathbb R\to\mathbb R$, this defines the proposition that $g(0)=0$, that $g$ is twice continuously differentiable at every real point $z\in[0,1]$, and that at every $z\in(0,1)$,

$$4+4g'(z)+2z^3g'(z)-2zg''(z)+2z^4g''(z)=0.$$

The differentiability requirement at both endpoints is with respect to the full real domain, rather than merely within $[0,1]$. The differential equation is required only on the open interval, and no value at $z=1$ is prescribed.

QAlgorithms.Papers.Independent.Intakethird100.P58.ShrinkingOscillations

For an arbitrary function $f:\mathbb R\to\mathbb R$, an arbitrary set $S\subseteq\mathbb R$, and arbitrary real numbers $e$ and $L$, this predicate means that $f$ converges to $L$ as its argument approaches $e$ within $S$, and that it takes values strictly below and strictly above $L$ at points of $S$ arbitrarily close to, but distinct from, $e$. More explicitly, the convergence requires that for every $\varepsilon>0$ there is an $\eta>0$ such that every $x\in S$ with $|x-e|<\eta$ satisfies $|f(x)-L|<\varepsilon$; additionally, for every real $\delta>0$, there exist $x_-,x_+\in S$ such that

$$0<|x_--e|<\delta,\qquad f(x_-)<L, \qquad 0<|x_+-e|<\delta,\qquad L<f(x_+).$$

The endpoint $e$ need not belong to $S$; if it does, the convergence requirement forces $f(e)=L$. The witness requirements exclude an empty set or a set with no points distinct from $e$ sufficiently close to $e$.

QAlgorithms.Papers.Independent.Intakethird100.P58.conformalEntropy

For every natural number $N$, including $N=0$, every real number $l$, every function $r:\mathbb R\to\mathbb R$, and every pair of real endpoints $a,b$, this defines

$$\operatorname{conformalEntropy}(N,l,r,a,b) =\frac{4N^2l}{15\pi} \int_a^b\frac{r(z)}{z^3}\sqrt{1+r'(z)^2}\,dz.$$

The natural number $N$ is viewed as a real number in this expression. No sign, differentiability, or integrability assumptions are imposed. The derivative is assigned zero wherever it does not exist, and division by zero is zero, so the integrand is zero at $z=0$. The integral is the oriented Lebesgue interval integral, assigned zero if its integrand is not integrable between the endpoints; equal endpoints give zero. The resulting expression is also zero whenever $N=0$ or $l=0$.

QAlgorithms.Papers.Independent.Intakethird100.P58.conformalF

For any real numbers $R,b,x$ and any function $r:\mathbb R\to\mathbb R$, this defines

$$\operatorname{conformalF}(R,b,r,x) = \frac{\displaystyle\inf\{z\in\mathbb R:0\le z\le b \ \text{and}\ r(z)=R(1-x)\}} {2R\sqrt{x}}.$$

No continuity, monotonicity, positivity, or existence of a point in the displayed set is assumed. If the set is nonempty, it is bounded below and its ordinary real infimum is used; if it is empty, its assigned infimum is zero, including when $b<0$. The real square root is zero for $x\le0$, and division by zero is zero. Thus the defined value is zero whenever $x\le0$, $R=0$, or the displayed set is empty.

QAlgorithms.Papers.Independent.Intakethird100.P58.d3Entropy

For every natural number $N$, including $N=0$, every function $r:\mathbb R\to\mathbb R$, and every pair of real endpoints $a,b$, this defines

$$\operatorname{d3Entropy}(N,r,a,b) =\frac{4N^2}{15} \int_a^b\frac{r(z)}{z^3}\sqrt{1+(1-z^4)r'(z)^2}\,dz.$$

Here $N$ is viewed as a real number. No positivity, differentiability, or integrability hypotheses are required. The derivative is assigned zero wherever it does not exist, division by zero is zero, and the real square root is zero for nonpositive arguments; hence the integrand is zero at $z=0$. The integral is the oriented Lebesgue interval integral, assigned zero when its integrand is not integrable between the endpoints; equal endpoints give zero. For $N=0$, the defined value is zero.

QAlgorithms.Papers.Independent.Intakethird100.P58.d4ConstantDensity

For every real number $z$, this defines

$$\frac{ 4\left[ -\frac{\arctan\!\left(\frac{1+2z}{\sqrt3}\right)}{\sqrt3} -\frac{\log(1+z+z^2)}2 +\frac{\pi}{6\sqrt3} \right] +z(1-z^3)\left(-\frac{1+z}{1+z+z^2}\right)^2 }{2z^3}.$$

Here $\log$ is the natural logarithm. The definition includes $z=0$: division by zero in the real-number operations used here yields $0$, so the defined density has value $0$ at $z=0$.

QAlgorithms.Papers.Independent.Intakethird100.P58.d4Entropy

For every parameter structure $P$, consisting of real numbers $\lambda=P.\mathrm{coupling}>0$ and $c=P.\mathrm{circleRadius}>0$ together with proofs of these inequalities, every natural number $N$, every function $r:\mathbb R\to\mathbb R$, and all real endpoints $a,b$, this defines

$$\operatorname{d4Entropy}(P,N,r,a,b) = \frac{\displaystyle \pi^3(\pi\lambda)^2 \left(\frac{4\pi\lambda}{9c^2}\right)c} {\displaystyle 2\left(\frac{8\pi^6\lambda^2}{N^2}\right)} \int_a^b \frac{r(z)^2}{z^3} \sqrt{1+z(1-z^3)r'(z)^2}\,dz,$$

with $N$ viewed as a real number. No regularity or sign assumptions on $r$, and no restrictions on the endpoints, are imposed. The derivative is assigned zero wherever it does not exist, division by zero is zero, and the real square root is zero for nonpositive arguments; the integrand is therefore zero at $z=0$. The integral is the oriented Lebesgue interval integral and is assigned zero if its integrand is not integrable between the endpoints; equal endpoints give zero. The case $N=0$ is included and makes the prefactor, and hence the defined value, zero.

QAlgorithms.Papers.Independent.Intakethird100.P58.d4G

For every real number $z$, this defines the real-valued function

$$G(z)= -\frac{\arctan\!\left(\frac{1+2z}{\sqrt3}\right)}{\sqrt3} -\frac{\log(1+z+z^2)}2 +\frac{\pi}{6\sqrt3},$$

where $\log$ is the natural logarithm and $\sqrt3$ is the nonnegative square root of $3$. The definition applies to all real $z$; its logarithm has a strictly positive argument for every such $z$.

QAlgorithms.Papers.Independent.Intakethird100.P58.d4GPrime

For every real number $z$, this defines the real-valued function

$$G_{\mathrm{prime}}(z)=-\frac{1+z}{1+z+z^2}.$$

Its denominator is strictly positive for every real $z$, so the formula applies without any restriction on $z$.

QAlgorithms.Papers.Independent.Intakethird100.P58.d4Newton

For every parameter structure $P$, consisting of real numbers $\lambda=P.\mathrm{coupling}>0$ and $c=P.\mathrm{circleRadius}>0$ together with proofs of these inequalities, and every natural number $N$, this defines

$$\operatorname{d4Newton}(P,N)=\frac{8\pi^6\lambda^2}{N^2},$$

where $N$ is viewed as a real number. The circle radius is part of the input structure although it does not occur in the formula. The case $N=0$ is included and gives zero, because division by zero is defined to be zero.

QAlgorithms.Papers.Independent.Intakethird100.P58.d4Prefactor

For every parameter structure $P$, consisting of real numbers $\lambda=P.\mathrm{coupling}>0$ and $c=P.\mathrm{circleRadius}>0$ together with proofs of these inequalities, and every natural number $N$, viewed as a real number in the formula, this defines

$$\operatorname{d4Prefactor}(P,N) = \frac{\displaystyle \pi^3(\pi\lambda)^2 \left(\frac{4\pi\lambda}{9c^2}\right)c} {\displaystyle 2\left(\frac{8\pi^6\lambda^2}{N^2}\right)}.$$

There is no assumption that $N$ is positive. All divisions use the convention that division by zero is zero; consequently, when $N=0$, both the inner quotient in the denominator and the entire displayed quotient evaluate to zero.

QAlgorithms.Papers.Independent.Intakethird100.P58.d4U0

For every parameter structure $P$, consisting of real numbers $\lambda=P.\mathrm{coupling}>0$ and $c=P.\mathrm{circleRadius}>0$ together with proofs of these inequalities, this defines

$$\operatorname{d4U0}(P)=\frac{4\pi\lambda}{9c^2}.$$

QAlgorithms.Papers.Independent.Intakethird100.P58.height

For every function $r:\mathbb R\to\mathbb R$ and every pair of real numbers $b,t$, the height $H_{r,b}(t)$ is defined to be

$$H_{r,b}(t)=\inf\{z\in\mathbb R:0\le z\le b\ \text{and}\ r(z)=t\}.$$

No continuity, monotonicity, positivity, or existence of a solution is assumed. When the set is nonempty, it is bounded and this is its real infimum, which need not itself belong to the set. When the set is empty, the defined value is $0$; this includes every $t$ when $b<0$. For $b=0$, the set is either empty or $\{0\}$, so the height is $0$ for every $t$.

QAlgorithms.Papers.Independent.Intakethird100.P58.large_Nc_scaling

Use either the choice $k=1$, $w(z)=1-z^4$, called the $\mathrm{d3}$ case, or the choice $k=2$, $w(z)=z(1-z^3)$, called the $\mathrm{d4}$ case, and write

$$D_r(z)=\frac{r(z)^k}{z^3}\sqrt{1+w(z)(r'(z))^2}.$$

For each choice, say that real numbers $R,b$ and a function $r:\mathbb R\to\mathbb R$ satisfy the profile conditions when all of the following hold: $R>0$, $b>0$, $r(0)=R$, $r$ is continuous and strictly decreasing on $[0,b]$, $r$ is twice continuously differentiable at every point of $[0,b)$, and, for every $0<z<b$, $r(z)>0$ and

$$\left.\frac{d}{du} \left(\frac{r(u)^k w(u)r'(u)} {u^3\sqrt{1+w(u)(r'(u))^2}}\right)\right|_{u=z} = \frac{k\,r(z)^{k-1}}{z^3}\sqrt{1+w(z)(r'(z))^2}.$$

Also require that $D_r$ be interval integrable with respect to Lebesgue measure from $s$ to $b$ for every real $0<s<b$, and require one of two alternatives: either $r(b)=0$, $b<1$, and the function

$$h(t)=\inf\{z\in[0,b]:r(z)=t\}$$

is twice continuously differentiable on $[0,R)$ and has derivative zero at $t=0$ within $[0,\infty)$; or $b=1$, $r(1)>0$, and $r$ is twice continuously differentiable at $1$. Here the real infimum is the total infimum operation, taking value zero for an empty set. The theorem asserts the conjunction of two statements. First, for every real $R,b$ and every $r:\mathbb R\to\mathbb R$ satisfying these profile conditions in the $\mathrm{d3}$ case, there exists a real $F$ such that, for every natural number $N>0$,

$$\lim_{a\to0^+} \frac{4N^2}{15} \left[ \int_a^b\frac{r(z)}{z^3}\sqrt{1+(1-z^4)(r'(z))^2}\,dz -\frac{R}{2a^2}-\frac{\log(a/R)}{8R} \right] =N^2F.$$

Second, for every parameter pair $P$ with real coupling $\kappa>0$ and circle radius $L>0$, every real $R,b$, and every $r:\mathbb R\to\mathbb R$ satisfying these profile conditions in the $\mathrm{d4}$ case, there exists a real $F$ such that, for every natural number $N>0$, on setting

$$K_{P,N}= \frac{\pi^3(\pi\kappa)^2\left(\frac{4\pi\kappa}{9L^2}\right)L} {2\left(\frac{8\pi^6\kappa^2}{N^2}\right)},$$

one has

$$\lim_{a\to0^+} K_{P,N} \left[ \int_a^b\frac{r(z)^2}{z^3}\sqrt{1+z(1-z^3)(r'(z))^2}\,dz -\frac{R^2}{2a^2}+\frac{3}{2a}-\frac{\log a}{8R^2} \right] =N^2F.$$

In each statement $F$ is chosen independently of $N$, although it may depend on the previously quantified profile and, in the second statement, on $P$; neither statement requires the limit assertion for $N=0$.

QAlgorithms.Papers.Independent.Intakethird100.P58.profileDensity

For every rotational model $m\in\{\mathrm{conformal},\mathrm{d3},\mathrm{d4}\}$, every function $r:\mathbb R\to\mathbb R$, and every real $z$, the profile density is the real number

$$\rho_{m,r}(z)=\frac{r(z)^{n_m}}{z^3} \sqrt{1+w_m(z)\bigl(r'(z)\bigr)^2},$$

where $n_{\mathrm{conformal}}=n_{\mathrm{d3}}=1$, $n_{\mathrm{d4}}=2$, and $w_{\mathrm{conformal}}(z)=1$, $w_{\mathrm{d3}}(z)=1-z^4$, $w_{\mathrm{d4}}(z)=z(1-z^3)$. No differentiability, positivity, or nonzero-denominator assumptions are imposed. Here and in the formula, the derivative is assigned the value $0$ at a point where $r$ is not differentiable, division by zero has value $0$, and the real square root has value $0$ for a nonpositive argument; in particular, the density is defined at $z=0$ and has value $0$ there.

QAlgorithms.Papers.Independent.Intakethird100.P58.radialDegree

For a rotational model $m$, whose possible values are $\mathrm{conformal}$, $\mathrm{d3}$, and $\mathrm{d4}$, the natural number $n_m$ called its radial degree is defined by

$$n_{\mathrm{conformal}}=1,\qquad n_{\mathrm{d3}}=1,\qquad n_{\mathrm{d4}}=2.$$

QAlgorithms.Papers.Independent.Intakethird100.P58.regulatedAction

For any model $m$ among $\mathrm{conformal},\mathrm{d3},\mathrm{d4}$, any function $r:\mathbb R\to\mathbb R$, and any real endpoints $a,b$, this defines

$$\operatorname{regulatedAction}(m,r,a,b)= \begin{cases} \displaystyle\int_a^b \frac{r(z)}{z^3}\sqrt{1+r'(z)^2}\,dz, & m=\mathrm{conformal},\\[4pt] \displaystyle\int_a^b \frac{r(z)}{z^3}\sqrt{1+(1-z^4)r'(z)^2}\,dz, & m=\mathrm{d3},\\[4pt] \displaystyle\int_a^b \frac{r(z)^2}{z^3}\sqrt{1+z(1-z^3)r'(z)^2}\,dz, & m=\mathrm{d4}. \end{cases}$$

These are oriented Lebesgue interval integrals: reversing the endpoints reverses the sign, and equal endpoints give zero. No regularity, integrability, or positivity hypotheses are imposed. Here $r'(z)$ is assigned zero wherever $r$ is not differentiable, division by zero is zero, and the real square root is zero for nonpositive arguments; in particular, each displayed integrand is zero at $z=0$. If the integrand is not integrable on the interval between the endpoints, the interval integral is assigned zero.

QAlgorithms.Papers.Independent.Intakethird100.P58.uv

For any model $m$ among $\mathrm{conformal},\mathrm{d3},\mathrm{d4}$ and any real numbers $R,a$, this defines

$$\operatorname{uv}(m,R,a)= \begin{cases} \displaystyle\frac{R}{2a^2}+\frac{\log(a/R)}{8R}, & m=\mathrm{conformal}\text{ or }m=\mathrm{d3},\\[6pt] \displaystyle\frac{R^2}{2a^2}-\frac{3}{2a}+\frac{\log a}{8R^2}, & m=\mathrm{d4}. \end{cases}$$

There are no positivity or nonzero assumptions on $R$ or $a$. Every division by zero evaluates to zero; the real logarithm satisfies $\log 0=0$ and, for nonzero real $t$, means $\log|t|$. Thus the definition includes zero and negative inputs.

QAlgorithms.Papers.Independent.Intakethird100.P58.warp

For every rotational model $m\in\{\mathrm{conformal},\mathrm{d3},\mathrm{d4}\}$ and every real number $z$, the real number $w_m(z)$ is defined by

$$w_{\mathrm{conformal}}(z)=1,\qquad w_{\mathrm{d3}}(z)=1-z^4,\qquad w_{\mathrm{d4}}(z)=z(1-z^3).$$

There is no restriction on $z$, including no positivity restriction on $z$ or on the resulting value.

QAlgorithms.Papers.Independent.Intakethird100.P58.warpDerivative

For every rotational model $m\in\{\mathrm{conformal},\mathrm{d3},\mathrm{d4}\}$ and every real number $z$, the real number $v_m(z)$ is defined by

$$v_{\mathrm{conformal}}(z)=0,\qquad v_{\mathrm{d3}}(z)=-4z^3,\qquad v_{\mathrm{d4}}(z)=1-4z^3.$$

These formulas define the function for all real $z$.