text:
\section{Introduction} \label{Introduction}
It has now been formally proven by Lovas and Andai \cite[Thm. 2]{lovas2017invariance} that the separability probability with respect to Hilbert-Schmidt  measure \cite{zyczkowski2003hilbert} \cite[sec. 13.3]{bengtsson2017geometry} of the 9-dimensional convex set of
two-rebit states is $\frac{29}{64}$. Additionally, the multifacted evidence  \cite{slater2017master,milz2014volumes,fei2016numerical,shang2015monte,slater2013concise,slater2012moment,slater2007dyson}--including a recent ``master'' extension \cite{slater2017master} of the Lovas-Andai framework to {\it generalized} two-qubit states--is strongly compelling that the corresponding value for the 15-dimensional convex set of two-qubit states is $\frac{8}{33}$ (with that of the 27-dimensional convex set of two-quater[nionic]bits being $\frac{26}{323}$ [cf. \cite{adler1995quaternionic}], among other still higher-dimensional companion random-matrix related results). (Certainly, one can, however, still aspire to a yet greater
``intuitive'' understanding of these assertions, particularly in some ``geometric/visual'' sense [cf. \cite{szarek2006structure,samuel2018lorentzian,avron2009entanglement,braga2010geometrical,gamel2016entangled,jevtic2014quantum}], as well as further formalized proofs.) It is of interest to compare/contrast these finite-dimensional studies with those other quantum-information-theoretic ones, presented in the recent comprehensive volume of Aubrun and Szarek
\cite{aubrun2017alice}, employing {\it asymptotic geometric analysis}.

By a separability probability, we mean the ratio of the volume of the separable states to the volume  of all (separable and entangled) states with respect to the chosen measure, as proposed, apparently first, by {\.Z}yczkowski, Horodecki, Sanpera and Lewenstein \cite{zyczkowski1998volume} (cf. \cite{petz1996geometries,e20020146,singh2014relative,batle2014geometric}).

In these regards, we present the formulas derived by {\.Z}yczkowski and Sommers
for the {\it total} ($N^2-1$)-dimensional Hilbert-Schmidt (HS) volumes of the $N \times N$ (off-diagonal complex-valued) density matrices \cite[eq. (4.5)]{zyczkowski2003hilbert}
(cf. \cite[eq. (14.38)]{bengtsson2017geometry}),
\begin{equation} \label{ZSComplex}
V_{HS}^{\mathbb{C}}(N)= \frac{\sqrt{N} (2 \pi )^{\frac{1}{2} (N-1) N} \prod_{i=1}^N \Gamma(i)}{\Gamma(N^2) },
\end{equation}
and their $\frac{N^2+N-2}{2}$-dimensional real-valued counterparts \cite[eq. (7.7)]{zyczkowski2003hilbert},
\begin{equation} \label{ZSreal}
V_{HS}^\mathbb{R}(N)=  \frac{\sqrt{N} 2^N (2 \pi )^{\frac{1}{4} (N-1) N} \Gamma[(N+1)/2] \prod_{i=1}^N \Gamma(1+i/2)}{\Gamma(N(N+1)/2) \Gamma(1/2) }.     \end{equation}

Further, Andai alternatively employed Lebesgue measure (yielding results equivalent with the use of the normalization factor, $\sqrt{N} 2^{N(N-1)/2}$ \cite[p. 13648]{andai2006volume} to the Hilbert-Schmidt ones), obtaining in the complex case \cite[Thm. 2]{andai2006volume},
\begin{equation} \label{AndaiComplex}
V_{Lebesgue}^\mathbb{C}(N)=\frac{\pi ^{\frac{1}{2} (N-1) N} \Pi_{i=1}^{N-1} i!}{\left(N^2-1\right)!}.
\end{equation}
For  the real case  (we are only immediately interested here in the even dimensions $N=4, 6, 8, 10$), taking $2 l=N$, Andai gave
\cite[Thm. 1]{andai2006volume},
\begin{equation}  \label{AndaiReal}
V_{Lebesgue}^\mathbb{R}(l)=\frac{2^{-l^2-l} \pi ^{l^2} (2 l)!}{l! \left(2 l^2+l-1\right)!}    \prod_{i=1}^{l-1} (2 i)!.
\end{equation}

To, additionally, obtain the volume formulas with respect to induced measure  \cite{zyczkowski2001induced} \cite[sec. 15.5]{bengtsson2017geometry}, in the two-qubit cases ($N=4$), we must multiply the complex ($\mathbb{C}$) total volume expressions
of {\.Z}yczkowski and Sommers (\ref{ZSComplex}) and of Andai (\ref{AndaiComplex}) for $N=4$ by \cite[eq. (3.7)]{zyczkowski2001induced}
\begin{equation}
 \frac{217945728000 (1)_k (2)_k (3)_k \Gamma (k+4)}{\Gamma (4 (k+4))},
\end{equation}
where the Pochhammer symbol is indicated.
Similarly, for the qubit-qutrit case ($N=6$), we must multiply by
\begin{equation}
\frac{86109566386551207747222094479360000000 (1)_k (2)_k (3)_k (4)_k (5)_k \Gamma
   (k+6)}{\Gamma (6 (k+6))}.
\end{equation}
\subsection{Outline of study}
Our first collection of (qubit-qutrit, qubit-qudit, rebit-retrit, rebit-redit and two-quaterbit) analyses, reported in the immediately following sections (sec.~\ref{qubitqutritsection}-\ref{quaternionicsection}), could be conducted with either  set of the volume formulas referenced above ((\ref{ZSComplex})-(\ref{AndaiReal})). For specificity, we will proceed with the second (Andai/Lebesgue) set (cf. (\ref{MZ1}), (\ref{MZ2})), in investigating higher-dimensional counterparts to the now available extensive collection of very well-supported results for the generalized two-qubit states. In particular, let us note, the conjecture (\ref{qubitqutritconjecture}) that the qubit-qutrit counterpart to the apparent $\frac{8}{33}=\frac{2^3}{3 \cdot 11}$ two-qubit separability probability is $\frac{27}{1000}=(\frac{3}{10})^3 =\frac{3^3}{2^3 \cdot 5^3}$.

In the next secs.~\ref{twoqutritssection} and \ref{Xstates}, we examine separability/PPT-probability issues in the context of the two-qutrits and $X$-states, respectively.

In sec.~\ref{Determinantal}, we study for several scenarios, the division of separability probabilities between  the determinantal inequalities $|\rho^{PT}| > |\rho|$ and $|\rho| > |\rho^{PT} |$, where $\rho^{PT}$ denotes the partial transpose of the density matrix $\rho$. Equidivision occurs with Hilbert-Schmidt measure, but not otherwise.

Further, in secs.~\ref{Bures1}, \ref{Bures2} and \ref{Bures3},  we employ a number of innovative approaches to increase our understanding/estimation of the separability probability with respect to the fundamental Bures (minimal monotone) measure \cite{slater2002priori,slater2005silver}. We also study interpolations of separability probabilities between Hilbert-Schmidt and operator monotone (Bures, $\sqrt{x}$) measures in these sections.

In sec.~\ref{LovasAndaiExtension} and the detailed appendices of Charles Dunkl (sec.~\ref{CFD}) below, we  extend to  induced measure, the ``master Lovas-Andai formula'' for generalized two-qubit states
\begin{equation} \label{MasterFormula}
\tilde{\chi}_{d,0}(\varepsilon) \equiv  \tilde{\chi_d}(\varepsilon)= \frac{\varepsilon ^d \Gamma (d+1)^3 \,
   _3\tilde{F}_2\left(-\frac{d}{2},\frac{d}{2},d;\frac{d}{2}+1,\frac{3
   d}{2}+1;\varepsilon ^2\right)}{\Gamma \left(\frac{d}{2}+1\right)^2},
\end{equation}
reported in \cite[sec. VIII.A]{slater2017master}.
(Here, $\varepsilon$ is a singular-value ratio, and $d$ the random-matrix Dyson index.)
We will find that the extended formula $\tilde{\chi}_{d,k}(\varepsilon)$ naturally breaks into the sum of two parts, both parts simply reducing to one-half of $\tilde{\chi}_{d,0}(\varepsilon)$ when the induced measure index $k$ equals zero, that is, for the case of Hilbert-Schmidt measure. (The original ``unextended'' formula (\ref{MasterFormula}) applies in that specific case, as well as when the measure employed is that based on the operator monotone function $\sqrt{x}$--as Lovas and Andai showed \cite[sec. 4]{lovas2017invariance}.)

One of these two parts, which both together sum to
$\tilde{\chi}_{d,k}(\varepsilon)$, is
\begin{equation} \label{onehalf}
J\left(  \varepsilon\right)     = \frac{(1)_d \varepsilon ^d \Gamma (d+k+1)^2 \,
   _3\tilde{F}_2\left(\frac{d}{2},d,-\frac{d}{2}-k;\frac{d}{2}+1,\frac{3
   d}{2}+k+1;\varepsilon ^2\right)}{d \Gamma \left(\frac{d}{2}\right) \Gamma
   \left(\frac{d}{2}+k+1\right)}.
\end{equation}
The other complementary part of the extended  master formula takes the form
\begin{align} \label{otherhalf}
I\left(  \varepsilon\right)    & =\frac{2\Gamma\left(  1+d+k\right)
^{2}\Gamma\left(  \frac{d}{2}\right)  \Gamma\left(  d\right)  k!\varepsilon
^{d}}{d\Gamma\left(  \frac{d}{2}\right)  ^{3}\Gamma\left(  1+\frac{d}%
{2}+k\right)  \Gamma\left(  1+k\right)  \Gamma\left(  \frac{3d}{2}+1+k\right)
}\\
& \times\sum_{j=0}^{k}\frac{\left(  d\right)  _{k-j}}{\left(  \frac{d}%
{2}+1\right)  _{k-j}}\left(  1-\varepsilon^{2}\right)  ^{k-j}~_{3}F_{2}\left(
\genfrac{}{}{0pt}{}{-d/2-j,d/2,d+k-j}{1+d/2+k-j,1+k+3d/2}%
;\varepsilon^{2}\right)  .
\end{align}
Then, to obtain a generalized ($d$) two-qubit separability with respect to an induced measure corresponding to $k$, one implements the formula
\begin{equation} \label{interpolationGeneral}
\mathcal{P}_{sep/PPT}(d,k)= \frac{\int_{-1}^1  \int_{-1}^x\tilde{\chi}_{d,k} (\sqrt{\frac{1-x}{1+x}}  \sqrt{\frac{1+y}{1-y}})(1-x^2)^{d+k} (1-y^2)^{d+k} (x-y)^d \mbox{d} y \mbox{d} x}{\int_{-1}^1  \int_{-1}^x(1-x^2)^{d+k} (1-y^2)^{d+k} (x-y)^d \mbox{d} y \mbox{d} x}.
\end{equation}
For example, $d=2, k= 1$, with $\tilde{\chi}_{2,1}(\varepsilon)=\frac{1}{4} \varepsilon ^2 \left(3-\varepsilon ^2\right)^2$, yields $\frac{61}{143} =\frac{61}{11 \cdot 13} \approx 0.426573$, as previously reported \cite[eq. (2)]{slater2015formulas} \cite[eq. (4)]{slater2016formulas}. If we replace the $d+k$ terms in (\ref{interpolationGeneral}) by $-\frac{d}{4}+k$, we obtain the separability/PPT-probability function in the operator monotone function $\sqrt{x}$ case.

In our concluding remarks (sec.~\ref{Concluding}), we discuss, among other things, the establshed relevance of {\it Casimir invariants} \cite{slater2016invariance,gerdt20116,byrd2003characterization} to the study of separability probabilities in sec.~\ref{Casimir}.

\section{Qubit-qutrit analyses} \label{qubitqutritsection}
For the two-qubit ($N=4$) case, using (\ref{AndaiComplex}), we have for the 15-dimensional volume of two-qubit states,
\begin{equation} \label{CN4one}
V_{Lebesgue}^\mathbb{C}(4)    = \frac{\pi ^6}{108972864000} =\frac{\pi^6}{2^9 \cdot 3^5 \cdot 5^3 \cdot 7^2 \cdot 11 \cdot 13}.
\end{equation}
Multiplying this by the associated well-supported separability probability $\frac{8}{33}$, we have
\begin{equation} \label{CN4two}
V_{Sep/Lebesgue}^\mathbb{C}(4)    = \frac{\pi ^6}{449513064000} =\frac{\pi^6}{2^6 \cdot 3^6 \cdot 5^3 \cdot 7^2 \cdot 11^2 \cdot 13}.
\end{equation}
So, we see that the same primes (but to different powers) occur in  the denominators of both volume formulas, while the two numerators remain the same.

Let us now see if we can find analogous behavior in the bipartite ($2 \times 3$) qubit-qutrit ($N=6$) case. On the basis of 2,900,000,000 randomly-generated
qubit-qutrit density matrices \cite[sec. 4]{al2010random},\cite{zyczkowski2011generating}, we obtained an estimate (with 78,293,301
separable density matrices found) for an associated separability probability of 0.026997690.
(We incorporate the results for one hundred million density matrices reported in  \cite[sec. II]{slater2016invariance}.
Milz and Strunz give a confidence interval of $0.02700 \pm 0.00016$ for this probability \cite[eq. (33)]{milz2014volumes}. A [narrower] $95\%$ confidence interval
based  on our just indicated calculation is $[0.0269918, 0.0270036]$. In the decade-old 2007 paper
\cite[sec.  10.2]{slater2007dyson}, where the $\frac{8}{33}$ two-qubit conjecture was first formulated, we had advanced a hypothesis of $\frac{32}{1199} =\frac{2^5}{11 \cdot 109} \approx 0.0266889$--subsequently rejected as lying outside the confidence interval reported in \cite[sec.II]{slater2016invariance}. An effort to extend the Lovas-Andai form of analysis
\cite{lovas2017invariance} to the qubit-qutrit and rebit-retrit states has been reported in \cite[App. B]{slater2017master}--but, it now seems, that the separability probabilities reported there were subject to some small, yet not explained,  systematic error.)

We have for the 35-dimensional volume of qubit-{\it qutrit} states,
\begin{equation}
V_{Lebesgue}^\mathbb{C}(6)    =     \frac{\pi ^{15}}{298991549953302804677854494720000000} =
\end{equation}
\begin{displaymath}
\frac{\pi^{15}}{2^{24} \cdot 3^{12} \cdot 5^7 \cdot 7^5 \cdot 11^3 \cdot 13^2 \cdot 17^2 \cdot 19 \cdot 23 \cdot 29 \cdot 31}.
\end{displaymath}
Now, we have found that, for  a separability probability of
\begin{equation} \label{qubitqutritconjecture}
\frac{27}{1000} =\frac{3^3}{2^3 \cdot 5^3} =(\frac{3}{10})^3= 0.027,
\end{equation}
we would have the corresponding volume of separable states,
\begin{equation}
V_{Sep/Lebesgue}^\mathbb{C}(6)    =    \frac{\pi ^{15}}{298991549953302804677854494720000000} =
\end{equation}
\begin{displaymath}
\frac{\pi^{15}}{2^{27} \cdot 3^{9} \cdot 5^{10} \cdot 7^5 \cdot 11^3 \cdot 13^2 \cdot 17^2 \cdot 19 \cdot 23 \cdot 29 \cdot 31}.
\end{displaymath}
So, we see that only the powers of 2, 3 and 5 are modified,  closely following the pattern observed ((\ref{CN4one})-(\ref{CN4two})) in the $N=4$ scenario.

A  point to note here is that in the $4 \times 4$ density matrix setting, the positivity of  the determinant of the partial transpose is sufficient for separability to hold \cite{augusiak2008universal}, but not so in the $6 \times 6$ setting. (The partial transpose for an entangled state might  have {\it two} negative eigenvalues \cite{johnston2013non}--but not, we note, in the corresponding $X$-states scenario \cite[App. A]{mendoncca2017maximally}.) The possibility of a pair of  negative eigenvalues renders it less directly useful to employ determinantal moments of density matrices and of their partial transposes to approximate underlying separability probability distributions, as was importantly done in
\cite{slater2012moment,slater2013concise}, using ``moment-based density approximants'' \cite{provost2005moment}, based on Legendre polynomials.

In App.~\ref{Qubitqutritinduced}, we report parallel qubit-qutrit analyses employing, rather than the Hilbert-Schmidt measure ($k=0$), induced measures ($k \neq 0$). However, at this stage, we do not advance specific conjectures as to their corresponding separability probabilities.

\section{Qubit-qudit  analyses}
\subsection{$2 \times 4$ case}
In \cite[sec. III.B]{slater2016invariance}, we reported a PPT (positive partial transpose) probability, for the $8  \times 8$ density matrices (viewed as $2 \times 4$ systems) of
0.0012923558, based on 348,500,000 random realizations \cite{al2010random}, 450,386 of them having PPT's.  The associated $95\%$ confidence interval is $[0.0128863, 0.0129609]$. (Milz and Strunz did report  an estimate of 0.0013 \cite[Fig. 5]{milz2014volumes}, but gave no
associated confidence interval or sample size.)

Let us interestingly note that the numerator of the ($2 \times 2$) two-qubit separability probability $\frac{8}{33}$ is $2^3$, and of our ($2 \times 3$) qubit-qutrit conjecture, $\frac{27}{1000}$, it is $3^3$. So, we might speculate that in this $2 \times 4$ setting, the numerator of the PPT-probability would be $4^3 = 64$. Proceeding as in sec.~\ref{qubitqutritsection}, using the Andai Lebesgue volume formula (\ref{AndaiComplex}), with $N=8$,  we did find a
candidate PPT-probability (but with a numerator of $4^2$) of  $\frac{16}{12375} =\frac{4^2}{3^2 \cdot 5^3 \cdot 11} \approx 0.001292929$.

It would be of interest to try to examine the issue of what proportion of the  $2 \times 4$ PPT-states are, in fact, separable (cf. \cite[sec. IV]{zyczkowski1999volume}), as opposed to bound entangled, using the methodologies recently presented in \cite{qian2018state,Li2018}.
\subsection{$2 \times 5$ case}
We generated 621,000,000 $10 \times 10$ random such density matrices. Of these, 16,205 had a PPT, giving us as
estimated PPT-probability of 0.0000260950. A possible exact value, in line with the noted numerator phenomenon, might be
$\frac{125}{4790016} = \frac{5^3}{2^8 \cdot 3^5 \cdot 5  \cdot 7 \cdot 11} \approx 0.0000260959$.

In a supplementary analysis, for thirty-six  million $10 \times 10$ density matrices, again randomly generated with respect to Hilbert-Schmidt measure, we  found 950 to have PPT's. Among these,  {\it none} passed the further test for {\it separability from spectrum} \cite[Thm. 1]{johnston2013separability}. That is, for
none, in this 10-dimensional setting, did the condition hold that $\lambda_1< \lambda_9 +2 \sqrt{\lambda_8 \lambda_{10}}$, where the $\lambda$'s are the ten ordered eigenvalues of
the density matrices, with $\lambda_1$ being the greatest (cf. \cite[App. A]{slater2017master}).

\section{Rebit-retrit analysis} \label{Rebitretritanalysis}
For the two-rebit ($l=2, N=4$) case, we have for the 9-dimensional volume of two-rebit states,
\begin{equation} \label{RN4one}
V_{Lebesgue}^\mathbb{R}(2)    = \frac{\pi ^4}{967680} =\frac{\pi^4}{2^{10} \cdot 3^3 \cdot 5^ \cdot 7}.
\end{equation}
Multiplying this by the established (by Lovas and Andai \cite[Cor. 2]{lovas2017invariance})  separability probability $\frac{29}{64}$, we find
\begin{equation} \label{RN4two}
V_{Sep/Lebesgue}^\mathbb{R}(2)    =\frac{29 \pi ^4}{61931520} =\frac{29 \pi^4}{2^{16} \cdot 3^3 \cdot 5 \cdot 7}.
\end{equation}
So, we see that only the power of 2 is modified, and the exponents of 3, 5 and 7 in the denominators are unchanged.

Let us now see if we can find analogous simple behavior in the rebit-retrit ($l=3, N=6)$ case. On the basis of 3,530,000,000 randomly-generated
rebit-retrit density matrices \cite[sec. 4]{al2010random}, with respect to Hilbert-Schmidt measure, we obtained an estimate (with 462,704,503
separable density matrices found) for an associated separability probability of 0.1310777629. The associated
$95\%$ confidence interval is $[0.131067, 0.131089]$.

We have
for the total (20-dimensional) volume of both separable and entangled rebit-retrit states,
\begin{equation}
V_{Lebesgue}^\mathbb{R}(3)    =    \frac{\pi ^9}{1730063650258944000} =\frac{\pi^{9}}{2^{23} \cdot 3^{6} \cdot 5^3 \cdot 7^2 \cdot 11 \cdot 13 \cdot 17 \cdot 19}.
\end{equation}
Then we found that, assuming a very closely fitting separability probability of
\begin{equation}
\frac{860}{6561} =\frac{2^2 \cdot 5 \cdot 43}{3^8} \approx 0.1310775796,
\end{equation}
we would have
\begin{equation}
V_{Sep/Lebesgue}^\mathbb{R}(3)    = \frac{859 \pi ^9}{11338145138337015398400} =
\end{equation}
\begin{displaymath}
\frac{859 \pi^{9}}{2^{38} \cdot 3^{6} \cdot 5^{2} \cdot 7^2 \cdot 11}.
\end{displaymath}
So, we see that only the powers of 2 and now of 5 in the denominator are again modified.

We note, in the case of $\frac{860}{6561}$, a possible parallism with the conjectured numerators in the qubit-qudit $2 \times m$ cases being powers of $m$, while now in the real cases, the denominators would be.

Let us further observe that the two-rebit counterpart to the two-qubit induced measure formula (\ref{qubitinduced})
is \cite[eq. (4)]{slater2015formulas} \cite[eq. (6)]{slater2016formulas},
\begin{equation} \label{rebitinduced}
 P^{2-rebits}_k=1-\frac{4^{k+1} (8 k+15) \Gamma{(k+2)} \Gamma{(2 k+\frac{9}{2}})}{\sqrt{\pi} \Gamma{(3 k+7)}}.
\end{equation}
For the Hilbert-Schmidt $k=0$ case, we obtain the formally demonstrated result, $\frac{29}{64}$.
\section{Rebit-redit analyses}
\subsection{$2 \times 4$ case}
We generated  490,000,000 $8 \times 8$ random density matrices with respect to Hilbert-Schmidt ($k=0$) measure. Of these, 12,022,129 had a PPT, giving us as
estimated PPT-probability of 0.02453496. A good fit is provided by $\frac{201}{8192} = \frac{3 \cdot 67}{2^{13}} \approx 0.0245361$. We note, in light of
our previous analyses, that the denominator $2^{13}$ is obviously also expressible as $4^{6+\frac{1}{2}}$.
\subsection{$2 \times 5$ case}
We generated  620,000,000 $10 \times 10$ random density matrices with respect to Hilbert-Schmidt ($k=0$) measure. Of these, 1,844,813 had a PPT, giving us as
estimated PPT-probability of 0.002975505. A well-fitting candidate PPT-probability is $\frac{29058}{9765625}= \frac{2 \cdot 3 \cdot 29 \cdot 167}{5^{10}} \approx 0.00297554$.
\section{Quaternionic formulas} \label{quaternionicsection}
Let us also note that in \cite[Thm. 3]{andai2006volume}, Andai presented the quaternionic ($\mathbb{H}$) counterpart,
\begin{equation}
 V_{Lebesgue}^\mathbb{H}(N)=   \frac{\pi ^{N^2-N} (2 N-2)!}{\left(2 N^2-N-1\right)!} \prod_{i=1}^{N-2}(2 i)!,
\end{equation}
of the complex ($\mathbb{C}$) and real ($\mathbb{R}$) volume formulas ((\ref{AndaiComplex}), (\ref{AndaiReal})) given above.
We, then have for the 27-dimensional volume of the two-quaterbit states,
\begin{equation}
 V_{Lebesgue}^\mathbb{H}(4)=   \frac{\pi ^{12}}{315071454005160652800000} =
\end{equation}
\begin{displaymath}
\frac{\pi^{12}}{2^{15} \cdot 3^{10} \cdot 5^5 \cdot 7^3 \cdot 11^2 \cdot 13^2 \cdot 17 \cdot 19 \cdot 23}.
\end{displaymath}
Multiplying by the well-supported separability/PPT-probability value (cf. \cite{hildebrand2008semidefinite}) of $\frac{26}{323}$,
we find
\begin{equation}
 V_{Sep/Lebesgue}^\mathbb{H}(4)=  \frac{\pi ^{12}}{3914156909371803494400000} =
\end{equation}
\begin{displaymath}
\frac{\pi^{12}}{2^{14} \cdot 3^{10} \cdot 5^5 \cdot 7^3 \cdot 11^2 \cdot 13^2 \cdot 17 \cdot 19 \cdot 23}.
\end{displaymath}
We would like to extend our earlier analyses above to the (50-dimensional) ``quaterbit-quatertrit'' setting. But it is
clearly a challenging problem
to suitably generate sufficient numbers of random $6 \times 6$ density matrices of such a nature (cf. \cite[App. C]{slater2017master}  of C. Dunkl), in order to obtain the needed probability estimates to attempt to closely fit.

We further note that the two-quaterbit counterpart to the two-qubit  and two-rebit induced measure formulas (\ref{qubitinduced}) and (\ref{rebitinduced}),
is \cite[eq. (3)]{slater2015formulas} \cite[eq. (5)]{slater2016formulas},
\begin{equation} \label{quaterbitinduced}
 P^{2-quaterbits}_k=1-\frac{4^{k+6} (k(k(2 k(k+21)+355)+1452)+2430)\Gamma{(k+\frac{13}{2})} \Gamma{(2 k+15)}}{3 \sqrt{\pi} \Gamma{(3 k+22)}}.
\end{equation}
\section{Two-qutrits} \label{twoqutritssection}
In \cite[sec. III.A]{slater2016invariance}, we reported an estimated Hilbert-Schmidt PPT-probability  of 0.00010218 for the two-{\it qutrit} states \cite{baumgartner2006state}, based on one hundred million randomly generated density matrices. Following the framework employed above, we have made some limited efforts to suggest a possible corresponding exact probability. It is by no means clear, however, if one can hope to extend ($2 \times m$) qubit-based results to a fully qutrit setting. (In any case, we did find that the rational value $\frac{323}{3161088}= \frac{17 \cdot 19}{2^{10} \cdot 3^2 \cdot 7^3} \approx 0.00010218$ provides an exceptional fit.) It would be of interest to try to examine the issue of what proportion of the two-qutrit PPT-states are, in fact, separable (cf. \cite{zyczkowski1998volume}) using the methodologies recently presented in \cite{qian2018state,Li2018}.
\section{$X$-states} \label{Xstates}
We have found that
the Hilbert-Schmidt separability/PPT-probabilities for both the ($6 \times 6$) rebit-retrit and ($9 \times 9$) two-retrit
$X$-states to be, somewhat remarkably, equal to that previously reported \cite[p. 3]{dunkl2015separability} for the lower-dimensional ($4 \times 4$) two-rebit $X$-states, that is, $\frac{16}{3 \pi^2} \approx 0.54038$. (The HS two-qubit $X$-states
separability probability has previously
been shown to equal $\frac{2}{5} = 0.4$ \cite[eq. (22)]{milz2014volumes} \cite[p. 3]{dunkl2015separability}. In \cite[App. B]{slater2017master}, we noted that Dunkl had concluded that the same separability probability did hold for the qubit-qutrit $X$-states.)

We have also found that the equality between two-rebit and rebit-retrit $X$-states separability probabilities continues to hold when the Hilbert-Schmidt measure (the case $k=0$) is
generalized to the class of induced measures \cite{zyczkowski2001induced,bengtsson2017geometry}. In Fig. 1, we present two equivalent formulas that yield these induced measure two-rebit, rebit-retrit separability probabilities.
\begin{figure}
\includegraphics[page=1,scale=0.9]{TwoRebitXstatesFormulas.pdf}
\label{fig:Induced}
\end{figure}
\begin{figure}
\includegraphics[page=2,scale=0.9]{TwoRebitXstatesFormulas.pdf}
\caption{Two-rebit and rebit-retrit $X$-states induced separability probability formulas}
\end{figure}
\section{Determinantal equipartition of Hilbert-Schmidt separability probabilities} \label{Determinantal}
In \cite{slater2016formulas}, a formula (eq. (62) there),
\begin{align*} \label{Hyper1}
Q\left(  k,\alpha\right)   &  =\frac{1}{2}-\frac{\alpha\left(  20\alpha+8k+11\right)
\Gamma\left(  5\alpha+2k+2\right)  \Gamma\left(  3\alpha+k+\frac{3}{2}\right)
\Gamma\left(  2\alpha+k+\frac{3}{2}\right)  }{4\sqrt{\pi}\Gamma\left(
5\alpha+2k+\frac{7}{2}\right)  \Gamma\left(  \alpha+k+2\right)  \Gamma\left(
4\alpha+k+2\right)  }\\
&  \times~_{6}F_{5}\left(
\genfrac{}{}{0pt}{}{1,\frac{5}{2}\alpha+k+1,\frac{5}{2}\alpha+k+\frac{3}{2}%
,2\alpha+k+\frac{3}{2},3\alpha+k+\frac{3}{2},\frac{5}{2}\alpha+k+\frac{19}{8}%
}{\alpha+k+2,4\alpha+k+2,\frac{5}{2}\alpha+k+\frac{7}{4},\frac{5}{2}\alpha+k+\frac{9}{4},\frac
{5}{2}\alpha+k+\frac{11}{8}}%
;1\right)
\end{align*}
was given for that part of the {\it total} induced-measure separability probability, $P(k,\alpha)$, for
generalized (real [$\alpha=\frac{1}{2}$], complex [$\alpha=1$], quaternionic [$\alpha=2$],...) two-qubit states for which the determinantal inequality $|\rho^{PT}| > |\rho|$ holds. For the
Hilbert-Schmidt case ($k=0$) the formula yielded $\frac{P(0,\alpha)}{2}$. (In \cite{szarek2006structure}--making use of Archimedes' formula for the
volume of a D-dimensional pyramid of unit height, and of ``pyramid-decomposability''--it was shown that the
Hilbert-Schmidt separability
probability of the minimially degenerate states is, likewise, {\it one-half} of that of the nondegenerate states.) Our simulations appear to indicate that this
equal division of separability probabilities continues to hold in the rebit-retrit and qubit-qutrit cases. Based on 96,350,607 separable rebit-retrit cases, the estimated proportion for which  $|\rho^{PT}| > |\rho|$ held was 0.499987, and based on 9,450,652 separable qubit-qutrit cases, the companion estimated proportion  was 0.500033.

However, in the non-Hilbert-Schmidt analysis in sec.~\ref{k1K7}, pertaining to the qubit-qutrit states with induced measure parameters $k=1$, $K=7$, we
found that the determinantal inequality $|\rho^{PT}| > |\rho|$ held in only $31.17\%$ (and not $50\%$) of the cases. Further, in sec.~\ref{k2K8}, pertaining to the qubit-qutrit states with induced measure parameters $k=2$, $K=8$, the corresponding percentage was $22.63\%$.
\section{Lovas-Andai-type formula for operator monotone measures}
It is of clear interest to extend the forms of analysis above to measures of interest other than the Hilbert-Schmidt
(flat/Euclidean/Frobenius) one, in particular perhaps, the Bures (minimal monotone) one (cf. \cite{slater2000exact,vsafranek2017discontinuities}). In these regards, in \cite[sec. VII.C]{slater2017master}, we recently reported, building upon analyses of Lovas and Andai \cite[sec. 4]{lovas2017invariance}, a two-qubit separability probability equal to $1 -\frac{256}{27 \pi^2} =1- \frac{2^8}{3^3 \pi^2} \approx 0.0393251$. This was based on another
(of the infinite family of) operator monotone functions, namely
$\sqrt{x}$. (Let us note that the complementary ``entanglement probability'' is simply $\frac{256}{27 \pi^2} \approx 0.960675$. There appears to be no intrinsic reason
to prefer/privilege one of these two forms of probability to the other [cf. \cite{dunkl2015separability}].  We observe that the upper-limit-of-integration variable  denoted $K_s:=\frac{(s+1)^{s+1}}{s^s}$, equalling $\frac{256}{27} =\frac{4^4}{3^3}$, for $s=3$, is frequently employed in the Penson-{\.Z}yczkowski paper, ``Product of Ginibre matrices: Fuss-Catalan and Raney
distributions'' \cite[eqs. (2),  (3)]{penson2011product}.)
\subsection{Operator monotone measure $\sqrt{x}$}
Within the Lovas-Andai framework, employing the previously reported two-qubit ``separability function" $\tilde{\chi}_2(\varepsilon)=\frac{1}{3} \varepsilon^2 (4 -\varepsilon^2)$
\cite[eq. (42)]{slater2017master}, we can interpolate between the computation for the noted ($\sqrt{x}$) operator monotone separability probability of $1- \frac{256}{27 \pi^2}$ ($\eta=-\frac{1}{2}$)
and the computation for  the Hilbert-Schmidt counterpart of $\frac{8}{33}$ ($\eta=2$).
This is accomplishable using the formula (Fig.~\ref{fig:InterpolatedHSmonotone}),
\begin{equation} \label{interpolation}
u(\eta) =   \frac{\int_{-1}^1  \int_{-1}^x\tilde{\chi}_2 (\sqrt{\frac{1-x}{1+x}}  \sqrt{\frac{1+y}{1-y}})(1-x^2)^\eta (1-y^2)^\eta (x-y)^2 \mbox{d} y \mbox{d} x}{\int_{-1}^1  \int_{-1}^x(1-x^2)^\eta (1-y^2)^\eta (x-y)^2 \mbox{d} y \mbox{d} x}=
\end{equation}
\begin{displaymath}
-\frac{-3 \eta  (\eta +4) ((\eta -6) \eta -15)+\frac{16^{2 \eta +3} ((\eta -10) \eta -5)
   \Gamma \left(\eta +\frac{3}{2}\right) \Gamma \left(\eta +\frac{5}{2}\right)^3}{\pi ^2
   (2 \eta +3) \Gamma (4 \eta +5)}+60}{3 (\eta -1)^2 \eta ^2},
\end{displaymath}
where $u(-1)=0$ and $u(1)=\frac{41471}{105}-40 \pi ^2 \approx 0.177729$.
(It is not now clear if any particularly meaningful measure-theoretic/quantum-information-theoretic interpretation can be given to these interpolated values.)
\begin{figure}
    \centering
    \includegraphics{InterpolatedHSmonotone.pdf}
    \caption{Two-qubit separability probability function $u(\eta)$, given by eq.  (\ref{interpolation}), interpolating between the  ($x \rightarrow \sqrt{x}$) operator monotone result ($\eta=-\frac{1}{2}$)
    of $1- \frac{256}{27 \pi^2}$ and the Hilbert-Schmidt result ($\eta=2$) of
    $\frac{8}{33}$. The average of the two is attained at $\eta=0.53544108.$}
    \label{fig:InterpolatedHSmonotone}
\end{figure}
``We argue that from the separability probability point of view, the main difference between the Hilbert-Schmidt measure and the volume form
generated by the operator monotone function $x \rightarrow \sqrt{x}$ is a special distribution on the unit ball in operator norm of
$2 \times 2$ matrices, more precisely in the Hilbert-Schmidt case one faces a uniform distribution on the whole unit ball and for
monotone volume forms one obtains uniform distribution on the surface of the unit ball'' \cite[p. 2]{lovas2017invariance}.
\subsection{Bures measure} \label{Bures1}
Let us, in these regards, also interestingly note that Osipov, Sommers and {\.Z}yzckowski remarked that one can readily interpolate between the generation of random
density matrices with respect to Hilbert-Schmidt and Bures measures \cite[eq. (24)]{al2010random} (cf. \cite[eq. (33)]{borot2012purity}). Their formula took the form
\begin{equation}  \label{JointBuresHSformula}
\rho_x= \frac{(y \mathbb{I} +x U) AA^{\dagger}(y \mathbb{I}+x U^{\dagger})}{\mbox{Tr} (y \mathbb{I}+x U) A A^{\dagger} (y \mathbb{I}   +x U^{\dagger})},
\end{equation}
where $y=1-x$ and $x=0$ yields the Hilbert-Schmidt case, and $x=\frac{1}{2}$, the Bures instance. Let us now perform the change-of-variable
$\eta=2 -5 x$ in eq. (\ref{interpolation}). This allows us to present Fig.~\ref{fig:TwoInterpolations}, in which we simultaneously show
the Hilbert-Schmidt/Bures interpolation--based on 550,000 sets of 101 randomly-generated density matrices--together with the Hilbert-Schmidt-operator monotone function $x \rightarrow \sqrt{x}$ interpolation, shown using
the $\eta$ variable in Fig.~\ref{fig:InterpolatedHSmonotone}.
\begin{figure}
    \centering
    \includegraphics{TwoInterpolations.pdf}
    \caption{Joint plot of the Hilbert-Schmidt/Bures-based on 550,000 sets of 101 randomly-generated density matrices--and Hilbert-Schmidt/operator monotone $x \rightarrow \sqrt{x}$ interpolations. At $x=\frac{1}{2}$, the former curve has an estimated value of 0.073 and the latter, the
    lower exact value $1- \frac{256}{27 \pi^2} \approx 0.0393251$. The curves intersect at $x=0.380241$.}
    \label{fig:TwoInterpolations}
\end{figure}
In Fig.~\ref{fig:OneInterpolation}, we more directly interpolate between the HS and Bures two-qubit separability probabilities employing (\ref{JointBuresHSformula}).
\begin{figure}
    \centering
    \includegraphics{OneInterpolation.pdf}
    \caption{Cosine-like interpolating curve between the Hilbert-Schmidt separability probability of $\frac{8}{33}$ and the (yet not exactly known) Bures separability probability ($x=\frac{1}{2}$) based upon (\ref{JointBuresHSformula}). One million random realizations--with two thousand points in $x \in [0,1]$ per realization--were employed.}
    \label{fig:OneInterpolation}
\end{figure}

Perhaps it is not too unreasonable to anticipate that the Bures two-qubit separability probability
(associated with the operator monotone function $\frac{1+x}{2}$)--though perhaps not so much the two-rebit case in light of \cite[Thm. 4]{lovas2017invariance}--will also be found to assume a strikingly elegant form. (In
\cite{slater2002priori}, we had conjectured a value of $\frac{8}{11 \pi^2} \approx .0736881$. But it was later proposed
in \cite{slater2005silver}, in part motivated by the lower-dimensional {\it exact} results reported in \cite{slater2000exact}, that the value might be $\frac{1680 \sigma_{Ag}}{\pi^8} \approx 0.07334$, where $\sigma_{Ag}= \sqrt{2}-1 \approx 0.414214$ is the ``silver mean''. Both of these studies \cite{slater2002priori,slater2005silver}   were conducted using quasi-Monte Carlo procedures, before the reporting of the  Ginibre-ensemble methodology for generating density matrices, random with respect to the Bures measure \cite{al2010random}. In \cite[Table 1]{slater2005silver} an estimate of 0.0732398 was reported, based on two billion quasi-Monte Carlo [scrambled Tezuka-Faure] points.)
In
\cite[sec. VII]{slater2016invariance}, it was noted that ``on the other hand, clear evidence has been provided
that the apparent $r$-invariance phenomenon revealed by the work of Milz and Strunz,\ldots, does {\it not} continue to hold if one employs,
rather than Hilbert-Schmidt measure, its Bures (minimal monotone) counterpart''. It would be of interest to examine this issue of $r$-invariance in the context of the induced measures (which, of course, include the Hilbert-Schmidt measure as the special $k=0$ case).

Another phenomenon apparently restricted to the Hilbert-Schmidt case, is that the separable states are equally divided (in terms of HS measure) between those for which
$|\rho^{PT}| >|\rho|$, and {\it vice versa}. Based on some 122,000,000 two-qubit density matrices randomly generated with respect
to Bures measure, of the 8,945,951 separable ones, 5,894,648 of them (that is, 65.8918$\%$) had $|\rho^{PT}| >|\rho|$, clearly distinct from simply one-half of them.
\subsubsection{Hilbert-Schmidt-assisted estimation of two-qubit Bures separability probability} \label{Bures2}
Let us attempt to exploit the Bures-Hilbert-Schmidt interpolation formula  (\ref{JointBuresHSformula}) of Osipov, Sommers and {\.Z}yczkowski, in light of our recently-acquired
high degree of certainty that the Hilbert-Schmidt two-qubit separability probability is $\frac{8}{33}$
\cite{slater2017master,lovas2017invariance,milz2014volumes,fei2016numerical,shang2015monte,slater2013concise,slater2012moment,slater2007dyson}. That is, we {\it jointly} estimate the Ginibre matrix $A$ at each
of 4,372,000,000 iterations. (Of these, 1,059,902,370 and 320,546,752 corresponded to separable density matrices, in terms of the Hilbert-Schmidt and Bures measures, respectively.) Then, we apply the average of 1 and the ``correction factor" 1.00002224983, needed to transform the Hilbert-Schmidt estimate to
precisely $\frac{8}{33}$, to the estimated Bures probability. (The average is employed, since a second set of an equal number of 32 [but now independent] standard normal random variates are needed to generate the random matrices with respect to Bures measure.)  Doing so, gives us a very slightly ``corrected'' Bures estimate'' of 0.07331891996, rather than
0.0733181043.
(Since the additional randomly-generated $4 \times 4$ unitary matrix $U$ enters into the Bures computation--but not the Hilbert-Schmidt one, since $x=0$ in that case--it does seem plausible that convergence is slower in the Bures case.)

Performing a parallel (but much smaller) computation in the two-{\it rebit}
case, based on forty million random density matrices (6,286,209 of them being separable), we obtain a corresponding (slightly corrected) Bures separability probability estimate of 0.1571469. (In doing so we take, as required, the Ginibre matrix $A$ in (\ref{JointBuresHSformula}) to be $4 \times 5$ \cite[eqs. (24),(28)]{al2010random}, and not $4 \times 4$ as in the two-qubit calculation.)
\section{Estimation of Bures separability probabilities using  quasirandom sequences} \label{Bures3}
We have begun to explore the question of whether the estimation of the  Bures two-qubit and two-rebit separability probabilities could be accelerated--with faster convergence properties--by, rather than using simply
random generation of normal variates for the Ginibre ensembles, making use of normal variates generated by employing low-discrepancy (quasi-Monte Carlo) sequences \cite{leobacher2014introduction}. In particular, we are employing an ``open-ended'' sequence (based on extensions of the {\it golden ratio}) recently introduced by Martin Roberts in the detailed "blog post", ``The Unreasonable Effectiveness
of Quasirandom Sequences'' \cite{Roberts}.

Roberts notes: ``The solution to the
$s$-dimensional problem, depends on a special constant $\phi_s$, where $\phi_s$ is the value of smallest, positive real-value of x such that''
\begin{equation}
  x^{s+1}=x+1,
\end{equation}
($s=1$, yielding the golden ratio, and $s=2$, the ``plastic constant'' (https://mathematica.stackexchange.com/questions/143457/how-can-one-generate-an-open-ended-sequence-of-low-discrepancy-points-in-3d). (For notational clarity here, we have changed his parameter $d$ to $s$.)
The  $n$-th terms in the quasirandom (Korobov) sequence take the form
\begin{equation} \label{QR}
(\alpha _0+n \vec{\alpha}) \bmod 1, n = 1, 2, 3, \ldots
\end{equation}
where we have the $s$-dimensional vector,
\begin{equation} \label{quasirandompoints}
\vec{\alpha} =(\frac{1}{\phi_s},\frac{1}{\phi_s^2},\frac{1}{\phi_s^3},\ldots,\frac{1}{\phi_s^s}).
\end{equation}
The additive constant $\alpha_0$ is typically taken to be 0. ``However, there are some arguments, relating to symmetry, that suggest that $\alpha_0=\frac{1}{2}$
is a better choice,''  Roberts observes. These points, lying in the $s$-dimensional hypercube $[0,1]^s$, can be converted to (quasirandomly distributed) normal variates using the inverse of the cumulative distribution function \cite[Chap. 2]{devroye1986}.
(Henrik Schumacher developed a specialized algorithm, that we employ, that accelerated the  Mathematica command InverseCDF approximately {\it ten-fold}, as reported at   https://mathematica.stackexchange.com/questions/181099/can-i-use-compile-to-speed-up-inversecdf). We take $s=36$ and 64 to estimate the Bures two-rebit and two-qubit separability probabilities, respectively.

In Fig.~\ref{fig:Roberts2Rebits}, we show the paired estimates of the Bures two-rebit separability probability based on $\alpha_0=0$ and $\alpha_0=\frac{1}{2}$ sequences, and similarly in Fig.~\ref{fig:Roberts2Qubits} for the two-qubit case. We see strong convergence of the paired estimates in both figures. For the (longer) $\alpha_0 =\frac{1}{2}$ sequences, the last recorded estimates are 0.1570962 (based on 3,235,000,000 points) and 0.0733116 (2,280,000,000 points), and for the $\alpha_0= 0$ sequences, the last recorded estimates are 0.1570974 (2,685,000,000 points) and 0.0733143 (2,075,000,000 points).
\begin{figure}
    \centering
    \includegraphics{Roberts2Rebits.pdf}
    \caption{Bures two-rebit separability probability estimates--in units of five million points sampled--based on $\alpha_0=0$ and $\alpha_0=\frac{1}{2}$ quasirandom sequences (\ref{QR}), with the sampling dimension $s=36$. The  $\alpha=\frac{1}{2}$ curve is longer.}
    \label{fig:Roberts2Rebits}
\end{figure}
\begin{figure}
    \centering
    \includegraphics{Roberts2Qubits.pdf}
    \caption{Bures two-qubit separability probability estimates--in units of five million points sampled--based on $\alpha_0=0$ and $\alpha_0=\frac{1}{2}$ quasirandom sequences (\ref{QR}), with the sampling dimension $s=64$. The  $\alpha=\frac{1}{2}$ curve is longer.}
    \label{fig:Roberts2Qubits}
\end{figure}

Our current (significant digits) estimates of the two-rebit and two-qubit Bures separability probabilities, based on these results, are 0.15709 and 0.07331, respectively \cite[Table 1]{khvedelidze2018generation}. We continue with these analyses, aspiring to obtain insights into possible {\it exact} values.

It is of interest to compare/contrast the relative merits of estimation of this pair of Bures separability probabilities in the present 36- and 64-dimensional settings with earlier studies
(largely involving Euler parameterizations of $4 \times 4$ density matrices \cite{tilma2002parametrization}), in which 9- and 15-dimensional integration problems were addressed \cite{slater2005silver,slater2009eigenvalues}. In the higher-dimensional frameworks, the integrands are effectively unity, while not so in the other cases.

``Observe that the Bures volume of the set of mixed states is equal to the volume of an $(N^2-1)$-dimensional hemisphere of radius $R_B=\frac{1}{2}$'' \cite[p. 415]{bengtsson2017geometry}. It is also noted there that $R_B$ times the area-volume ratio asymptotically increases with the dimensionality $D=N^2-1$, which is typical for hemispheres.
\section{Extension of Lovas-Andai formulas to induced measures} \label{LovasAndaiExtension}
Now, let us raise what appears to be a quite interesting research question. That is, can the  Lovas-Andai framework, which has been successfully applied using both Hilbert-Schmidt and operator monotone function $\sqrt{x}$ measures \cite{lovas2017invariance,slater2017master}, be further adopted to the generalization of Hilbert-Schmidt measure to its induced extensions (through the use of the determinantal powers of density matrices in the derivations)? If so, the specific induced separability probabilities reported in
\cite{slater2015formulas} \cite{slater2016formulas}, including formulas (\ref{qubitinduced}), (\ref{rebitinduced}) and (\ref{quaterbitinduced}) above, could be presumably further verified. We now investigate this topic.
\subsection{Extended Lovas-Andai functions $\tilde{\chi}_{d,k}$ for a few specific cases}
To begin, let us replace $\tilde{\chi}_2(\varepsilon)=\frac{1}{3} \varepsilon^2 (4 -\varepsilon^2)$  in
the middle expression in the two-qubit separability probability formula (\ref{interpolation}) for $u(\eta)$ by
\begin{equation} \label{d=2k=1}
\tilde{\chi}_{2,1}(\varepsilon)=\frac{1}{4} \varepsilon ^2 \left(3-\varepsilon ^2\right)^2,
\end{equation}
and set $\eta=3$ (it now being understood, notationally, that $\tilde{\chi}_{2,0}(\varepsilon) \equiv \tilde{\chi}_{2}(\varepsilon)$).

Then, this expression does, in fact, evaluate to the previously-found two-qubit induced $k=1$ value $\frac{61}{143}$ given by formula (\ref{qubitinduced}).
That is,
\begin{equation} \label{interpolation2}
\frac{\int_{-1}^1  \int_{-1}^x\tilde{\chi}_{2,1} (\sqrt{\frac{1-x}{1+x}}  \sqrt{\frac{1+y}{1-y}})(1-x^2)^3 (1-y^2)^3 (x-y)^2 \mbox{d} y \mbox{d} x}{\int_{-1}^1  \int_{-1}^x(1-x^2)^3 (1-y^2)^3 (x-y)^2 \mbox{d} y \mbox{d} x}=  \frac{61}{143}.
\end{equation}
Fig.~\ref{fig:randominducefitk1} shows the residuals from a (clearly close) fit of  $\tilde{\chi}_{2,1}(\varepsilon)$ to an  estimation of it based on
sixty million appropriately generated $4 \times 4$ density matrices.
\begin{figure}
    \centering
    \includegraphics{RandomInducedk1Fit.pdf}
    \caption{Residuals from a fit of the Lovas-Andai-type formula $\tilde{\chi}_{2,1}(\varepsilon)$ to its estimation based on
sixty million randomly generated (with respect to $k=1$ induced measure) $4 \times 4$ density matrices}
    \label{fig:randominducefitk1}
\end{figure}
(We can now further, using $\tilde{\chi}_{2,1}(\varepsilon)$, find the $k=1$ separability probability based on the previously-discussed operator monotone function $x \rightarrow \sqrt{x}$. The result we obtain is $4427-\frac{131072}{3 \pi ^2} =19 \cdot 233 -\frac{2^{17}}{3 \pi^2} \approx 0.209939$.)

Proceeding onward to the $k=2$ case, still in the complex domain ($\mathbb{C}$), we have
\begin{equation} \label{interpolationk2}
\frac{\int_{-1}^1  \int_{-1}^x\tilde{\chi}_{2,2} (\sqrt{\frac{1-x}{1+x}}  \sqrt{\frac{1+y}{1-y}})(1-x^2)^4 (1-y^2)^4 (x-y)^2 \mbox{d} y \mbox{d} x}{\int_{-1}^1  \int_{-1}^x(1-x^2)^4 (1-y^2)^4 (x-y)^2 \mbox{d} y \mbox{d} x}=  \frac{259}{442},
\end{equation}
agreeing with (\ref{qubitinduced}), where, now,
\begin{equation*}
 \tilde{\chi}_{2,2}(\varepsilon)    =\frac{1}{5} \varepsilon ^2 \left(-\varepsilon ^6+8 \varepsilon ^4-18 \varepsilon ^2+16\right).
\end{equation*}
(We can now, using $\tilde{\chi}_{2,2}(\varepsilon)$ find the $k=2$ separability probability based, once again, on the previously-discussed operator monotone measure. The result we obtain is $-\frac{1713917}{3}+\frac{26642219008}{4725 \pi ^2} =
-\frac{61 \cdot 28097}{3}+\frac{2^{26} \cdot 397}{3^2 \cdot 5^2 \cdot 19 \pi^2} \approx 0.399947$.)

Moving from the complex to quaternionic domain ($\mathbb{H}$), again for $k=1$, we have
\begin{equation} \label{interpolationk3}
\frac{\int_{-1}^1  \int_{-1}^x\tilde{\chi}_{4,1} (\sqrt{\frac{1-x}{1+x}}  \sqrt{\frac{1+y}{1-y}})(1-x^2)^5 (1-y^2)^5 (x-y)^4 \mbox{d} y \mbox{d} x}{\int_{-1}^1  \int_{-1}^x(1-x^2)^5 (1-y^2)^5 (x-y)^4 \mbox{d} y \mbox{d} x}=  \frac{3736}{22287},
\end{equation}
agreeing with (\ref{quaterbitinduced}), where, now, we employ
\begin{equation}
 \tilde{\chi}_{4,1}(\varepsilon)    = \frac{1}{21} \varepsilon ^4 \left(-9 \varepsilon ^6+55 \varepsilon ^4-125 \varepsilon ^2+100\right).
\end{equation}
(The separability probability, based on $\tilde{\chi}_{4,1}(\varepsilon)$, in the noted operator monotone case is
$\frac{27637}{168}-\frac{50 \pi ^2}{3} = \frac{29 \cdot 953}{2^3 \cdot 3 \cdot 7}
-\frac{2 \cdot 5^2 \pi^2}{3} \approx 0.0125457$.)

We remark that the two-{\it rebit} ($d=1$)  functions $\tilde{\chi}_{1,k}(\varepsilon)$, and more generally $\tilde{\chi}_{d,k}(\varepsilon)$, for {\it odd} $d$, appear to be of considerably more complicated non-polynomial form, involving inverse hyperbolic, logarithmic and polylogarithmic functions.)
\subsection{General $\tilde{\chi}_{d,k}(\varepsilon)$ formula}
It now seems clear that to obtain--in our extended Lovas-Andai framework--an induced measure-based separability/PPT probability ($\mathcal{P}_{sep/PPT}(d,k)$) in the real ($\mathbb{R}$), complex ($\mathbb{C}$) or
quaternionic  ($\mathbb{H}$) domain, we must set the exponent ($d$) of the
$(x-y)$ terms in the numerators and denominators   to 1, 2 or 4, respectively. While to obtain a specific
$k$-induced measure result, we must take the exponents of the $(1-x^2)$ and $(1-y^2)$ terms to be $d+k$. In other words, we have the general ($(d,k)$-parameterized)
formula $\mathcal{P}_{sep/PPT}(d,k)$ given by (\ref{interpolationGeneral}).
If we replace the $d+k$ terms  in (\ref{interpolationGeneral}) by $-\frac{d}{4}+k$, we obtain the separability/PPT-probability function in the operator monotone function $\sqrt{x}$ case.
\subsection{Analytical background}
Now, let us indicate the general manner in which we obtained the three specific indicated new functions $\tilde{\chi}_{2,1}(\varepsilon)$, $\tilde{\chi}_{2,2}(\varepsilon)$
and  $\tilde{\chi}_{4,1}(\varepsilon)$ above.
In this direction, we have for the complex case, $d=2$, the  general induced measure formula
\begin{equation} \label{(d,k)-parameterized}
 \tilde{\chi}_{2,k}(\varepsilon)   =\frac{\left(-k+\varepsilon ^2-3\right) \left(1-\varepsilon ^2\right)^{k+1}+k+3}{k+3},
\end{equation}
with the denominator of (\ref{interpolationGeneral}) evaluating to $\frac{\pi  \Gamma (k+3)^2}{2 \Gamma \left(k+\frac{7}{2}\right) \Gamma
   \left(k+\frac{9}{2}\right)}$. Alternatively,
\begin{equation}
\tilde{\chi}_{2,k}(\varepsilon)  =    \frac{(k+2) \varepsilon ^2 \left(\sum _{j=0}^{k-1} \left(1-\varepsilon ^2\right)^{k-j}
     {_2F_1}\left(1,-j-1;k+4;\varepsilon ^2\right)+2 \,
   _2F_1\left(1,-k-1;k+4;\varepsilon ^2\right)\right)}{k+3}.
\end{equation}
(For a formal proof by C. Dunkl of this relation, see App. \ref{doublesums}.)
We, then, have for $k=-\frac{5}{2}$,
\begin{equation}
 \tilde{\chi}_{2,-\frac{5}{2}}(\varepsilon)   = 2 \left(\frac{\varepsilon ^2-\frac{1}{2}}{\left(1-\varepsilon
   ^2\right)^{3/2}}+\frac{1}{2}\right),
\end{equation}
which we can interestingly use to replace $\tilde{\chi}_{2}(\varepsilon) $ in (\ref{interpolation}), giving us (again setting $\eta=-\frac{1}{2}$) a result
now of $\frac{21 \pi -64}{21 \pi } \approx 0.0299127$ to compare (in the induced measure framework) with the previously-given ($x \rightarrow \sqrt{x}$) operator monotone result of
$1 -\frac{256}{27 \pi^2}  \approx 0.0393251$. (We can closely approximate this result using $\tilde{\chi}_{2,-2.86051355}(\varepsilon)$.)
Additionally, Charles Dunkl has found--employing rational interpolation--the (quaternionic, $d=4$) counterpart (and also for $d=6$),
\begin{equation}
\tilde{\chi}_{4,k}(\varepsilon)=    \left(\frac{2 \left(2 k^2+14 k+21\right) \varepsilon ^4}{(k+5) (k+6)}-\frac{6 (k+3) \varepsilon
   ^6}{(k+6) (k+7)}-(k+1) \varepsilon ^2-1\right) \left(1-\varepsilon ^2\right)^{k+1}+1.
\end{equation}
For $k=0$, we recover the previously-reported Hilbert-Schmidt formula of $\tilde{\chi}_{4}(\varepsilon)=\frac{1}{35} \varepsilon ^4 \left(15 \varepsilon ^4-64 \varepsilon ^2+84\right)$
\cite[sec. VI]{slater2017master}. The corresponding formula for $k=1$ is (\ref{interpolationk3}).
\subsection{Methodology employed}
To further elaborate upon the general methodology employed to obtain the above results, we refer to the analyses and notation employed in \cite[sec. VII]{slater2017master}. We must, again, perform the constrained integrations  presented there, but now, additionally,  for induced measure of order $k \neq 0$, we must
multiply both the (numerator and denominator) integrands by the $k$-th power of
$\left(\left(r_{14}^2-1\right) \left(r_{23}^2-1\right)-r_{24}^2\right) \varepsilon ^2$. This term is the relevant factor in the determinant
of the $4 \times 4$ density matrices (having three pairs of nullified entries) employed in the cited reference. The additional determinantal factors are all positive and not functions of the $r$'s, and would cancel, so can be ignored in the computations.

To be more specific, in these regards, in \cite[sec. VII]{slater2017master} we employed the set of constraints (imposing--in quantum-information-theoretic terms--the positivity of the density matrix and its partial transpose),
\begin{equation}
r_{23}^2<1\land \left(r_{14}^2-1\right) \left(r_{23}^2-1\right)>r_{24}^2\land r_{23}^2
   \left(\varepsilon ^2 r_{14}^2-1\right)>\varepsilon ^2 \left(\varepsilon ^2
   r_{14}^2+r_{24}^2-1\right).
\end{equation}
Then, subject to these constraints, we had to integrate the jacobian (corresponding to the hyperspherical parameterization
of the three off-diagonal non-nullified entries of the density matrix) $\left(r_{14} r_{23} r_{24}\right){}^{d-1}$
over the unit cube $[0,1]^3$. Dividing the result of the integration by
\begin{equation} \label{denominator}
\frac{\pi  4^{-d} \Gamma \left(\frac{d}{2}+1\right)^2}{d^3 \Gamma
   \left(\frac{d+1}{2}\right)^2},
\end{equation}
yielded the desired $\tilde{\chi_d} (\varepsilon )$. (If we were to take $r_{24}=0$, and a jacobian of
$\left(r_{14} r_{23}\right){}^{d-1}$, we would revert to the X-states setting, and obtain simply
$\varepsilon^d$ as the corresponding  function.)

This last result (\ref{denominator}) was obtained by integrating the same jacobian
$\left(r_{14} r_{23} r_{24}\right){}^{d-1}$ over the unit cube,
subject to the constraints (imposing the positivity of the density matrix),
\begin{equation}
r_{23}^2<1\land \left(r_{14}^2-1\right) \left(r_{23}^2-1\right)>r_{24}^2.
\end{equation}
So to reiterate, to move on to the more general induced measure setting (that is, $k \ne 0$), we must multiply both the indicated (numerator and denominator) integrands by the $k$-th power of
$\left(\left(r_{14}^2-1\right) \left(r_{23}^2-1\right)-r_{24}^2\right) \varepsilon ^2$. The Hilbert-Schmidt ($k=0$) denominator
integration result (\ref{denominator}), then, generalizes to
\begin{equation}
 \frac{\Gamma \left(\frac{d}{2}\right)^3 \varepsilon ^{2 k} \Gamma (k+1) \Gamma
   \left(\frac{d}{2}+k+1\right)}{8 \Gamma (d+k+1)^2}.
\end{equation}
\subsection{Extended master formula investigation}
Our goal now is the development of a still more general  Lovas-Andai ``master formula'' for $\tilde{\chi}_{d,k}(\varepsilon)$ than has been so far reported for  $\tilde{\chi}_{d}(\varepsilon) \equiv \tilde{\chi}_{d,0}(\varepsilon)$ in \cite[sec. VII.A]{slater2017master}, that is, (\ref{MasterFormula}) above.
An expression for the anticipated (induced measure) extended master formula is as the sum of (\ref{onehalf}),
(reducing to one-half of (\ref{MasterFormula}) for $k=0$)
and the two-dimensional integral of the product of
\begin{equation} \label{integrand}
  \frac{1}{\Gamma \left(\frac{d}{2}\right)^2 \Gamma (k+1) \Gamma
   \left(\frac{d}{2}+k+1\right)}
\end{equation}
and
\begin{equation}
Y^{d-1} \left(\frac{1}{r_{14} \varepsilon }\right){}^{d+1} \left(1-r_{14}^2 \varepsilon
   ^2\right){}^{d/2} \left(\left(1-\frac{1}{r_{14}^2}\right) Y^2-r_{14}^2+1\right){}^k
   \left(r_{14}^2 \varepsilon ^2-Y^2\right){}^{d/2}
\end{equation}
and
\begin{equation}
\, _2\tilde{F}_1\left(\frac{d}{2},-k;\frac{d+2}{2};\frac{\left(r_{14}^2 \varepsilon
   ^2-1\right) \left(Y^2-r_{14}^2 \varepsilon ^2\right)}{\left(r_{14}^2-1\right) \varepsilon ^2
   \left(Y^2-r_{14}^2\right)}\right)   .
\end{equation}
The two-dimensional domain of integration is
\begin{equation}
r_{14} \in [0,1], \hspace{.25in}  Y \in [\varepsilon   r_{14}, \varepsilon^2   r_{14}]  .
\end{equation}
The result of this integration must also, as (\ref{onehalf}) does, equal $\frac{1}{2}$  of the master formula (\ref{MasterFormula}) result for $k=0$.
(Questions pertaining to these last discussed issues have been posted at https://mathematica.stackexchange.com/questions/171351/evaluate-over-a-two-dimensional-domain-the-integral-of-hypergeometric-based-f
and https://math.stackexchange.com/questions/2744828/find-five-parameter-values-for-a-3-tildef-2-function-yielding-five-pCFDinolynomi  .)   C. Dunkl has
been able to solve this two-dimensional integration problem (with a change-of-variables, I had suggested, in the integrand), with the
result (App.~\ref{CFDintegration}) given near the end of sec.~\ref{Introduction}.

In \cite[sec. VII.A]{slater2017master}, we reported a formula there (74), which when
multiplied by the ``master  Lovas-Andai'' [Hilbert-Schmidt, $k=0$] formula for $\tilde{\chi}_{d}(\varepsilon)$, (70) there, and appropriately
normalized (by (75) there) would yield the $d$-specific separability/PPT-probability. We have now obtained the
induced measure extension of that Hilbert-Schmidt formula (74)  (thus, reducing to it for $k=0$). It takes the form
\begin{equation} \label{unnormalized}
(-1)^d 2^{5 d+4 k+3} \left(t^2-1\right)^d t^{-4 d-2 k-3} \Gamma (3 d+2 k+2)^2 \,
   _2\tilde{F}_1\left(3 d+2 k+2,3 d+2 k+2;6 d+4 k+4;1-\frac{1}{t^2}\right).
\end{equation}
For its normalization factor, corresponding to \cite[eq. (75)]{slater2017master}, we have found
\begin{equation} \label{normalization}
2^{5 d+4 k+2} \Gamma (d+1) \Gamma (d+k+1) \Gamma (3 d+2 k+2)^2 \, _3\tilde{F}_2(3 d+2
   k+2,3 d+2 k+2,d+1;6 d+4 k+4,2 d+k+2;1).
\end{equation}
As a check on this normalization formula, in the Hilbert-Schmidt ($k=0$) case, we see that it agrees with \cite[eq. (75)]{slater2017master}
for $d=6$, both formulas yielding $\frac{33554432}{4854694845}$. (For a related analysis of Dunkl, see App.~\ref{laststeps}.)
For $d \bmod 4=0$, the normalization factor reduces to
\begin{equation}
\frac{\Gamma \left(\frac{d+1}{2}\right) 2^{\left\lfloor \frac{d}{4}\right\rfloor
   +\frac{11 d}{4}+2 k} \Gamma (d+k+1)^2 \Gamma \left(\frac{3 d}{2}+k+1\right) \Gamma
   \left(\frac{5 d}{4}+k+\left\lfloor \frac{d}{4}\right\rfloor
   +\frac{3}{2}\right)}{\Gamma \left(\frac{3 (d+1)}{2}+k\right)^2 \Gamma \left(\frac{5
   d}{2}+2 k+2\right)} .
\end{equation}
In the operator monotone
($\sqrt{x}$) case at hand, the analogous (hypergeometric-free) normalization factor to (\ref{normalization}) is
\begin{equation}
    \frac{4^{\frac{d}{4}+k} \Gamma \left(\frac{d}{2}+\frac{1}{2}\right) \Gamma
   \left(-\frac{d}{4}+k+1\right)^2 \Gamma \left(\frac{d}{4}+k+1\right)}{\Gamma (2 k+2)
   \Gamma \left(\frac{d}{4}+k+\frac{3}{2}\right)}.
\end{equation}
We have been able to use the two results ((\ref{unnormalized}), (\ref{normalization})) to see that in agreement with a previously-reported formula \cite[eq. (62)]{slater2016formulas} (given above in sec.~\ref{Determinantal})
 that (\ref{onehalf}) accounts for that part of the separability probability for which $|\rho^{PT}| \geq |\rho|$, and the formula of Dunkl (\ref{otherhalf}) for the complementary separability probability. (We have specifically verified this assertion in the $d=2,k=1$ case for which we have $\tilde{\chi}_{2,1}(\varepsilon)=\frac{1}{4} \varepsilon ^2 \left(3-\varepsilon ^2\right)^2$, given by (\ref{d=2k=1}). This decomposes into the sum of the Lovas-Andai separability functions $\frac{1}{20} \varepsilon ^2 \left(\varepsilon ^4-6 \varepsilon ^2+15\right)$ and $\frac{3 \varepsilon ^2}{2}+\frac{1}{5} \left(\varepsilon ^2-6\right) \varepsilon ^4$. These yield, respectively, separability probabilities of $\frac{45}{286}$, for the first complementary part, and  $\frac{7}{26}$, for the second, summing, as known, to $\frac{61}{143}$.)
\section{Concluding remarks} \label{Concluding}
An interesting question is whether it is possible to obtain the Bures (minimal monotone) two-qubit separability probability within the
Lovas-Andai framework, in a somewhat similar manner to how we have extended this framework to the induced measures. Now, in the Bures
case, rather than incorporating simply powers of the determinants of the density matrices into the required integrations, we must include the
reciprocal ($R$) of the product ($P$) of the square root of the determinant and the product term $\Pi_{j<k}^4 (\lambda_i+\lambda_j)$, where the $\lambda$'s
are the eigenvalues of the density matrices \cite[eq. (3.18)]{sommers2003bures}. We were able to employ the symmetric reduction of this last term and re-express it in terms of the traces of matrix powers of the density matrices, and, thus, of the entries of the density matrix. Unfortunately, we have not been able to
perform the requisite integrations. (We should note that both the Hilbert-Schmidt and operator monotone ($\sqrt{x}$) measures lead to separability probabilities invariant over the Bloch radii of the qubit subsystems, while the Bures measure has been demonstrated in \cite[Fig. 31]{slater2015bloch} to not have such behavior. Lovas and Andai have commented \cite[sec. 6]{lovas2017invariance} that in this operator monotone case, the associated  probability is determined by the structure of the surface of the unit ball.)

Of course, it would be most desirable to  rigorously derive the Hilbert-Schmidt/Lebesgue separability/PPT-probabilities for the 35- and 63-dimensional convex sets of qubit-qutrit and qubit-qudit states, among others,  examined above. But, given that the  Hilbert-Schmidt
separability probability of $\frac{8}{33}$ for the 15-dimensional convex set of two-qubit  states has itself proved highly formidable to well establish \cite{slater2017master,lovas2017invariance,milz2014volumes,fei2016numerical,shang2015monte,slater2013concise,slater2012moment,slater2007dyson} (though not yet formally proven),
it seems that major advances would be required to achieve such a goal
in these still higher-dimensional settings (and, thus, confirm or reject the conjectures above).

Implicit in the analytical approach pursued here has been the clearly yet unverified assumption that the separability/PPT-probabilities will continue to be {\it rational-valued} for the higher-dimensional systems, as they have, remarkably, been found to be in the $4 \times 4$ setting.

We can, of course, as future research, continue our various simulations
reported above of random density matrices, hoping to obtain further accuracy in  separability/PPT-probability estimates. One relevant issue of interest would then be the trade-off between the use of  increased precision in the random normal variates employed (we have so far used the Mathematica default precision option), and the presumed consequence, then, of decreased number of  variates to be generated.
\subsection{Casimir invariants} \label{Casimir}
One of our goals here  has been to determine if we could use the $N=4$ results \cite{slater2017master,lovas2017invariance,milz2014volumes,fei2016numerical,shang2015monte,slater2013concise,slater2012moment,slater2007dyson} to gain insight into the $N>4$ counterparts, and, more specifically,
if certain analytical properties continue to hold. We found some encouragement for undertaking such a course from the research reported in
 \cite{slater2016invariance}. There, evidence was provided that a most interesting common characteristic is
shared by two-qubit ($N=4$), qubit-qutrit $(N=6$), qubit-qudit ($N=8$, specifically) and two-qutrit ($N=9$) systems. That is, the associated
(HS) separability/PPT probabilities hold constant over the {\it Casimir invariants} \cite{gerdt20116,byrd2003characterization} of both their subsystems
(such as the lengths of the Bloch radii of the reduced qubit subsystems) (cf. \cite[Corollary 2]{lovas2017invariance}). (A Casimir invariant is a  distinguished element of the center of the universal enveloping algebra of a Lie algebra \cite{gerdt20116}.)

It would be of interest to computationally employ such apparent invariance (formally proved by Lovas and Andai
\cite[Corollary 2]{lovas2017invariance} in the two-rebit $\frac{29}{64}$ case) in strategies to ascertain these various separability/PPT-probabilities. However, we have yet to find
an effective manner of doing so (even after setting the Casimir invariants to zero, leading to
lower-dimensional settings). (In our paper, ``Two-qubit separability probabilities as joint functions of the Bloch radii of the qubit subsystems'' \cite{slater2016two}, we observed a relative repulsion effect between the Casimir invariants of the two reduced systems of several forms of bipartite states.)

Let us, in these regards, also indicate the interesting paper of Altafini, entitled ``Tensor of coherences parametrization of multiqubit density operators for entanglement characterization'' \cite{altafini2004tensor}. In it, he applies the term ``partial quadratic Casimir invariant'' in relation to reduced density matrices. He notes that a quadratic Casimir invariant can be regarded as the specific form ($q = 2$) of Tsallis entropy. Further, he remarks that ``partial transposition is a linear norm preserving operation: $\mbox{tr}(\rho^2) = \mbox{tr}((\rho^{T_1})^2) =  \mbox{tr}((\rho^{T_2})^2)$. Hence entanglement violating PPT does {\it not modify} the quadratic Casimir invariants of the density and the necessary [separability] conditions $[\mbox{tr}(\rho^2_A) \geq \mbox{tr}(\rho^2), \mbox{tr}(\rho^2_B) \geq \mbox{tr}(\rho^2)] $, are
{\it insensible} to it'' (emphasis added).

Let us, relatedly, indicate the pair of formulas (cf. (\ref{ZSComplex}), (\ref{AndaiComplex}))
\begin{equation} \label{MZ1}
  V_{HS}^{2 \times m}(r)=   V_{HS}^{2 \times m}(0) (1-r^2)^{2 (m^2-1)}
\end{equation}
and
\begin{equation} \label{MZ2}
 V_{HS}^{2 \times m}(0)   = \sqrt{m} \cdot 2^{6 m^2 -m -\frac{23}{2}} \cdot \pi^{2 m^2-m -\frac{3}{2}} \cdot
 \frac{\Pi_{k=1}^{2 m} \Gamma{(k)} \cdot \Gamma{(\frac{1}{2}+2 m^2)}}{\Gamma{(4 m^2)} \cdot \Gamma{(-1+2 m^2)}}
\end{equation}
that Milz and Strunz conjectured for the Hilbert-Schmidt volume of the $2\times m$ qubit-qudit states \cite[eqs. (27), (28)]{milz2014volumes}, as a function of the Bloch radius ($r$) of the qubit subsystem. (These appear to have been confirmed for the two-qubit
[$m=2$] case by the analyses of Lovas and Andai
\cite[Cor. 1]{lovas2017invariance}.)

redpajama_set_name: RedPajamaArXiv


After months of speculation on who the final addition to Ultra Street Fighter IV's roster will be, Capcom has finally revealed the identity of the mysterious fighter during the Final Round 17 tournament, as promised in Street Fighter producer Tomoaki Ayano's blog last month.
Alongside Elena, Hugo, Poison and Rolento, joining the cast is none other than Decapre, one of the Cammy dolls that I personally predicted would be the fifth character earlier this year. Despite this underwhelming news, Ultra David and James Chen have revealed during the Final Round 17 that Decapre a charge character and plays much like Juri.
For those who own either Super Street Fighter IV or Super Street Fighter IV: Arcade Edition, you will be able to upgrade to Ultra Street Fighter IV via a DLC expansion pack for $14.99 USD. However, for those who are waiting for a physical copy of Ultra Street Fighter IV, you will have to wait a bit longer to get your hands on it.
Are you looking forward to the five new characters in Ultra Street Fighter IV?


text:
Телесна температура је степен загрејаности организма, људи или животиња. Топлота у организму се ствара мишићним радом, асимилацијом хране и свим животним процеси који доприносе базалном метаболизму, а из организма губи радијацијом и вапоризацијом воде из дисајних путева и коже.

Телесну температуру одређује равнотежа између стварања и одавања топлоте. Пошто се брзина хемијских реакција мења са променом температуре и пошто ензимиски системи у организму имају оптимум деловања у уском распону температуре, нормално функционисање организма је у зависности од релативне константности телесне температуре.

Код хомеотермних животиња нормална температура на којој се организам одржава варира од врсте до врсте и у мањој мери од јединке до јединке. Код људи се сматра да нормална орална (усна) температура износи 37 °C, али је на великом броју нормалних младих одраслих особа установљена просечна јутарња орална телесна температура од 36,7 °C, са стандардном девијацијом од 0,2 °C. Због тога се код већине (95%) младих одраслих особа може очекивати да се јутарња орална температура креће од 36,3 °C – 37,1 °C (+,- 1,95 стандардна девијација).

Физиологија регулације температуре
Уопштено, бескичмењаци не могу да контролишу своју телесну температуру па су због тога зависни од температуре своје околине. Код кичмењака су се развили механизми који одржавају телесну температуру подешавањем производње и одавања топлоте. Код рептила, амфибија и риба, механизми за подешавање су релативно рудиментарни па су ове врсте назване "хладнокрвним" животињама (поикилотерми). Код птица и сисара "топлокрвне" животиње (хомеотерми), низ рефлексних реакција примарно интегрисаних у хипоталамусу дејствују на начин да одрже телесну температуру унутар уских граница промена, упркос широким флуктуацијама температуре у спољашњој средини. Сисари презимари су делимично изузетак. Док су будни, они су хомеотерми, а у току хибернације њихова телесна температура пада.

Као и код других сисара, терморегулација је важан аспект људске хомеостазе. Тело човека топлоту највише генерише у дубоким органима, посебно у јетри, мозгу и срцу, и згрченим скелетним мишићима . Људи кроз еволутивни развој били су приморани да се адаптирају на велику разноврсност климе, укључујући ту и топло-влажну и топло-суву, а и данас то чине. Користећи моћне, савремене, техничке системе и још моћније понашање и знање, човек је успео да преживи и у екстремним условима боравка у космосу, док је био изложен широком температурном опсегу: од -110 °C (за време боравка на површини месеца) до +2.000 °C (ваздуха око Спејс шатла у току његовог повратка на земљу кроз густе слојеве њене атмосфере). Високе температуре представљају озбиљно напрезање за људско тело, стављајући га у велику опасност од озледа па чак и смрти. За људе, прилагођавање различитим климатским условима укључује не само физиолошке механизме које је он стеко кроз еволутивни развој, већ и свесно развијене механизме културне адаптације боравка у животној средини <ref name=" Humanbiology&behaviour4thed">Weiss, M.L., & Mann, A.E. (1985) Human Biology and Behaviour: An anthropological perspective". (4th ed). Boston: Little Brown</ref>.

 Нормална телесна температура

Нормална телесна температура човека се креће у распону од 36,3 °C – 37,1 °C (+,- 1,95 стандардна девијација), с тим што различити делови тела имају различиту температуру, која варира са температуром околине. Удови су углавном хладнији од осталих делова тела. Ректална (чмарна) температура представља унутрашњу температуру тела и најмање варира зависно од спољашње температуре. Нормално је орална (усна) температура за 0,5 °C нижа од оне у завршном цреву, али на њу могу да утичу многи фактори, као што су уношење топлих и хладних напитака, жвакање гуме и дисање на уста.

Нормална унутрашња температура код човека подлеже правилним дневним флуктуацијама од 0,5 °C – 0,7 °C. Код људи који спавају ноћу, а будни су по дану (чак и када мирно леже у постељи), најнижа је око 6 часова ујутру а највиша увече. Температура је најнижа у току спавања, мало виша у будном стању када је човек миран, а повећава се са активношћу.

Код жена поред дневних осцилација, постоји и месечни циклус промена телесне температуре који се карактерише порастом базалне температуре у време овулације.

Прецизност регулације телесне температуре зависи од животне доби и мање је прецизан код;
 Мале деце, код које нормална температура може бити за око 0,5 °C виша од нормалних вредности утврђених за одрасле, због лаког одавања топлоте.
 Најстаријих старосних група, због смањене брзине и ефикасности терморегулационих система.

Телесна температура се незнатно повећава и приликом емоционих узбуђења, вероватно због несвесног напињања мишића. Телесна температура је хронично повећана за око 0,5 °C када је интензитет метаболизма повећан, као у хипертиреози, а снижена је када је интензитет метаболизма снижен, као у микседему. Код неких здравих особа, температура је хронично изнад нормалног распона (констрикционална хипертермија).

 Стварање топлоте
Супстанце и фактори који могу да утичу на степен ћелијског метаболизма утичу и на степен загрејаности организма. То су:
 Унос хране Мишићни-физички рад Тиреоидни хормони, епинефрин и норепинефринТоплота тела потиче од енергије која настаје у току ћелијског дисања – или мноштва основних хемијских реакција - при којима се хемијска енергија из хране претвара у енергетске молекуле АТП и топлоту, и непрестано доприноси загревању организма. Зато сваки унос хране повећава стварање топлоте својим специфичним динамичким дејством.

Међутим главни извор топлоте је грчење (контракција) скелетних мишића, или физички рад.

Стварање топлоте могу мењати ендокрини механизми без обзира на унос хране и мишићни рад. Епинефрин и норепинефрин, изазивају брз али краткотрајан пораст продукције топлоте, док се пораст који изазива тироидни хормон развија спорије и траје дуже

Значајан извор топлоте код мале деце је посебна врста масног ткива, мрко масно ткиво, које се налази између лопатица и око њих и у другим деловима тела. То масно ткиво има интензивнији метаболизам а његова се функција упоређује са функцијом електричног покривача који се у домаћинству користи за загревање тела у хладним данима.

 Изолациони систем тела

Кожа, поткожно ткиво са крвним судовима и наслаге масти у поткожном ткиву су топлотни изолатори тела. Маст је посебно важна јер је њена спроводљивост за топлоту само трећина спроводљивости других ткива.

Када крв из загрејаних унутрашњих органа не долази у кожу изолациона способност тела је здравог мушкарца износи отприлике, три четвртине изолационе способности најчешће употребљаване одеће. Код жена је изолација још боља. Што говори да степен изолације зависи од дебљине поткожног масног ткива (сваки милиметар масти чини да се човек осећа удобније у околини која је за један до два степена хладнија).

Како се топлота ствара у дубљим деловима тела, поткожна изолација представља битну заштитну меру у одржавању нормалне унутрашње температуре, иако допушта да се температура коже приближи температури околине.

 Одавање топлоте
Путеви којима се губи топлота из човековог тела наведени су на слици десно:

Радијација (зрачење) топлоте
У простору у коме влада нормална собна температура, наго тело емитује радијацијом (зрачењем) око 60% укупно одате топлоте.

Одавање топлоте радијацијом (зрачењем) по својој суштини је одавање топлоте у облику инфрацрвеног топлотног зрачења, које чини једна врста електромагнетских таласа.

Већина емитованих инфрацрвених зрака који зраче из тела има таласну дужину од 5 до 20 микрометара, што је 10 до 30 пута више од таласне дужине светлосних зрака. Сва тела која имају температуру већу од апсолутне нуле емитују топлотно зрачење. Зато људско тело зрачи топлотне зраке у свим правцима. Међутим топлотне зраке емитују и зидови и други предмети у непосредном окружењу тела. Ако је температура тела виша од температуре околине, већа количина топлоте зрачи из тела према околини него према телу.

Кондукција (преношење) топлоте
Директним преношењем топлоте са површине тела на друге предмете (нпр столицу или кревет) одаје се само незнатна количина топлоте (око 3%). Ако на столицу седне нага особа, из његовог тела ће се топлота брзо пренети на столицу и за неколико минута ће се изједначити температура столице и његовог тела. Истовремено столица постаје и изолатор који спречава даље значајније одавање топлоте из тела особе која седи на њој.
Конвекција топлоте
Одвођење топлоте из тела струјањем ваздуха обично се назива »одавање топлоте конвекцијом«. Овим механизмом топлота се прво мора пренети на ваздух, а затим се конвекцијским струјањем одводи даље. Незнатно струјање, готово увек постоји у близини тела, јер ваздух који додирује кожу тежи да се са ње издигне чим се загреје. Зато потпуно голо тело, у умерено загрејаној просторији, без јачег струјања ваздуха губи око 12% сопствене топлоте, због одвођења у ваздух те због конвекцијског струјања ваздуха.
 Расхлађивање ветромКада је тело изложено ветру слој ваздуха који се налази у непосредној близини коже, надопуњује се новим ваздухом много брже него иначе, па се губитак топлоте одвођењем зато повећава. Ако је брзина ветра мала, учинак на одавање топлоте приближно је пропорционалан другом корену брзине ветра (нпр. ветар брзине од око 4 км/час расхлађује отприлике два пута интензивније од ветра брзине 1 км/час.
 Кондукција и конвекција топлоте из тела у води.Евапорација (испаравање) топлоте

Кад вода испарава са површине тела са сваким грамом испарене воде ода се око 2,4 кЈ топлоте. Са површине коже и из плућа неприметно испари око 0,6 литара воде на дан. Овим процесом из тела се непрекидно ода око 50 до 60 кЈ топлоте на сат. Ово неосетно испаравање воде кроз кожу и плућа организам не може контролисати у циљу регулације температуре, јер настаје услед непрекидне дифузије молекула воде кроз кожу и кроз респирацијске површине без обзира на површину тела. Међутим регулацијом знојења одавање топлоте испаравањем може се регулисати.Испаравање као нужан механизам за расхлађивање при врло високим температурама. Радијацијом и кондукцијом се топлота одаје из тела радијацијом и кондукцијом, докле год је температура виша од температуре околине. Али када је температура околине више виша од температуре коже тело не одаје топлоту радијацијом и кондукцијом него је прима. У таквим условима је испаравање је једини начин на који се тело може ослободити топлоте. Зато било који чинилац који спречава испаравање када је температура околине виша од температуре коже узрокује да температура тела расте.

КожаКоличина топлоте која се губи преко коже одређена је протоком крви кроз крвне судове коже и активношћу знојних жлезди. Количина топлоте која стиже у кожу из дубљих ткива може се мењати променама протока крви кроз крвне судове коже што утиче на количину топлоте која се губи из тела путем три основна процеса одавања топлоте: зрачењем (радијацијом), провођењем (кондукцијом) и прелазом (конвекцијом).
Проток крви кроз кожу регулише се променом пречника артериолаа дермиса;
 Вазоконстрикција(мањи пречник артериола) смањује проток крви и одавање топлоте у спољашњу средину, јер се топлота задржава у унутрашњости организма,
 Вазодилатација (увећан пречник артериола) повећава проток крви ка површини тела и губитак топлоте у спољну средину.
Када се догоди да нека особа има урођене недостатке знојних жлезда, она је у стању да лако поднесе ниске температуре једнако успешно као и нормална особа, али зато у тропским крајевима може умрети од топлотног удара, јер без система за расхлађивање испаравањем воде, не може спречити пораст температуре тела, када је температура околине виша од температуре тела.

Респираторни систем

Испаравање воде из дисајних путева код људи изазива благи губитак топлоте - којим се губи око 13% створене топлоте у телу. Међутим, код многих животиња, посебно оних који не продукују зној, примарни начин за одавање топлоте из тела је испаравање воде са слузокоже уста и респираторног тракта (нпр. дахтање код пса). Испаравањем 1 грама воде, на овај начин се одстрањује из тела око 0,6 kcal топлоте.

Уринарни и дигестивни тракт

 Механизми животиња за регулацију температуре

 Механизми за регулацију телесне температуре

 Извори

 Литература
 Т. Јовановић. Терморегулација и метаболизам у Медицинска физиологија, 405-416. Графички атеље КУМ, Београд, 2004
 З. Анђелић и сар. Ћелија и ткива, Бонафидес, Ниш, 2002
 Arthur C. Guyton Medicinska fiziologija, Medicinska knjiga-Beograd-Zagreb 1990
 William F.G. Pregled medicinske fiziologije, Savremena administracija,Beograd, 1993.
 Mackowiak PA. Temperature regulation and the pathogenesis of fever. In: Mandell GL, Bennett JE, Dolin R, eds. Principles and Practice of Infectious Diseases. 7th ed. Philadelphia, Pa: Elsevier Churchill Livingstone; 2009:chap 50.
 Power KR. Fever without a focus''. In: Kliegman RM, Behrman RE, Jenson HB, Stanton BF, eds. Nelson Textbook of Pediatrics. 18th Ed. Philadelphia, Pa: Saunders Elsevier; 2007: chap 175.

Спољашње везе

Медицина

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Greg McGrath (born 21 January 1997) is an Irish rugby union player from Wexford, currently playing for Pro14 and European Rugby Champions Cup side Connacht. He previously played for Leinster. His preferred position is tighthead  prop.

Leinster
McGrath was named in the Leinster side for Round 10 of the 2020–21 Pro14 against . He made his debut in the same match, coming on as a replacement.

References

External links
itsrugby.co.uk Profile

1997 births
Living people
Irish rugby union players
Leinster Rugby players
Rugby union props
Connacht Rugby players
Irish expatriate sportspeople in Jersey
Jersey Reds players
Irish expatriate rugby union players
Expatriate rugby union players in Jersey


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Hyundai Santa Fe ›› 2017 ›› 2017 Hyundai Santa Fe
Select another model: Accent ⋅ Azera ⋅ Elantra ⋅ Elantra GT ⋅ Ioniq Electric ⋅ Ioniq Hybrid ⋅ Santa Fe ⋅ Santa Fe Sport ⋅ Sonata ⋅ Sonata Hybrid ⋅ Sonata Plug-in Hybrid ⋅ Tucson ⋅ Veloster
Overview (Final Score: A-)
Often overlooked in the midsize crossover market, the Hyundai Santa Fe is a value-oriented three-row offering that deserves more recognition. There aren't many vehicles in this segment that offer as complete a package as the Santa Fe.
A few changes to the Santa Fe for 2017 bring it into an even more competitive light, remedying one of our primary beefs with the crossover last year. Slight changes to the front fascia, a lot of new technology options for safety and convenience, and a new, higher standard of infotainment for all models. Hyundai really wanted to keep the 2017 Santa Fe highly competitive and these changes have taken an already well-done crossover and pushed it forward towards the front of the pack.
The three-row 2017 Hyundai Santa Fe is a midsize crossover with seating for up to seven and one of the industry's best warranties. The 2017 Santa Fe is offered in four trim levels, not including the smaller Santa Fe Sport.
What We Love About the 2017 Hyundai Santa Fe:
Excellent value proposition for the packages offered and goodies included
Great highway drive presence and around-town mobility
One of the industry's best warranties at 10 years, 100,000 miles as standard
What We Don't Love About the 2017 Hyundai Santa Fe:
Good cargo space but access is not as well done as in some rivals
Fuel economy is only about average for the segment
Exterior View (9/10)
The 2017 Hyundai Santa Fe has a revised front fascia that sharpens the crossover's looks a bit, adding more sport to the Santa Fe's already good-looking exterior. The Santa Fe doesn't necessarily stand out from the crowd for eye-catching looks, but it's a great-looking vehicle as is with a contemporary, sharp appeal. The Santa Fe has an air of class that is similar to some premium models.
Clean lines and a forward-leaning cant are the basis for the Santa Fe's design theme and are augmented by a sharper nose and thinner-looking, slotted grille. Raked headlamps and upward-leaning body lines continue this forward movement. These relatively simple design elements come together to make for a lot of appeal.
We like the Hyundai Santa Fe's look and the new changes for 2017 are good updates. The Santa Fe isn't the most standout or interesting to look at, but it's definitely a smooth operator.
Interior Comfort, Quality, Ergonomics (10/10)
The 2017 Hyundai Santa Fe has one of the best interiors in the midsize crossover segment. High-quality materials and an overall upscale affect to the interior's feel give the Santa Fe a lot of goodness inside.
This is especially true of the dashboard and front seating, which are all first rate no matter the trim level of the Santa Fe one finds oneself in. Adjustments and controls layout are all very good and make everyday life in the 2017 Santa Fe a pleasant experience. The infotainment screen, audio and climate controls on the center stack are nicely placed with an intuitive angle that pushes the most-used items forward. The slight cant to the audio controls below the infotainment screen make them more easily accessed by both driver and front passenger while on the go.
Seating in the second row is also spacious and well done. The Santa Fe's roofline would suggest a pinched experience in the headroom department, but that is definitely not the case for the second row. Depending on options package and trim, the Santa Fe's second row is either a bench seat for three or two pilot seats with a pass-through in between for third row access.
Speaking of the third row, it has seating for two and is sized just about right for child safety seats, boosters and young teenagers. Most adults will find the third row cramped with high leg positions and little headroom, but as an extra place to put some kiddos, it's a great design.
Cargo is good in the 2017 Hyundai Santa Fe at 13.5 cubic feet behind the third row. Forty-one cubic feet are opened up with the third row folded, and 80 cubic feet with both the second and third rows down. Those numbers are roughly on par with the rest of the segment, but don't indicate how easily the space can be used.
One of our complaints about the 2017 Santa Fe is the limited access to the cargo that is afforded through the narrow rear hatch and its pinched top. Seats don't always fold down flat either, making the loading floor a little less ergonomic. But for the most part and for most people's needs, the Santa Fe will accommodate.
Technology (10/10)
Another very strong area of consideration for the 2017 Santa Fe is in its technology offerings. The base models now include a 7-inch touchscreen as standard and plenty of basic connectivity and use with that. Any trim level can be upgraded to the more tech-laden infotainment option with its larger and more crisp 8-inch display. Both systems come standard with Hyundai's Blue Link telematics system.
Navigation comes standard with that upgrade as does app integration for smartphones. What's more, both of the infotainment options for the Santa Fe are easy to use and have a low learning curve. There isn't a lot of bloat with over-done options and seldom-used services, making this one of the cleaner and more functional infotainment systems on the market.
Fuel Economy (8/10)
The 2017 Hyundai Santa Fe has one engine option and one transmission choice. These return fuel economy that is about average for the segment. The EPA estimates the front-wheel drive 2017 Santa Fe to give 21 mpg combined, with 18 mpg in the city and 25 mpg on the highway. The all-wheel drive option lowers those estimates to 20 mpg combined (18 city, 24 highway). The Ultimate trim package adds some weight to the Santa Fe, dropping fuel economy estimates by 1 point.
Our real-world test of the 2017 Santa Fe Ultimate bore out those EPA numbers with an average of 19 mpg overall. Last year a lower-end model returned similar numbers.
Predicted Reliability, Initial Quality Ratings (9/10)
The 2017 Santa Fe is rated as "Better Than Most" by the J.D. Power and Associates Vehicle Dependability Study. A few recalls on nominal items for the Santa Fe have kept us from giving it a top score here, however, despite it having the industry's best bumper-to-bumper and powertrain warranties.
The Santa Fe is backed by a 10-year/100,000-mile powertrain warranty and a 5-year/60,000-mile bumper to bumper warranty. Only Kia offers a similar warranty.
Safety (10/10)
A major upgrade for the 2017 Hyundai Santa Fe over the 2016 model is with safety. The Santa Fe now houses the same advanced driver aids and safety systems we've come to expect as a gold standard for any automotive segment. This has helped boost the Santa Fe's safety scores in crash tests as well.
The National Highway Traffic Safety Administration (NHTSA) rates the 2017 Hyundai Santa Fe with five stars overall (out of five) and gave the Santa Fe a full suite of perfect fives through its crash testing.
The Insurance Institute for Highway Safety has also given the 2017 Santa Fe its top rating of "Top Safety Pick+" thanks to revisions to the Santa Fe's front crumple zone, which boosted its formerly marginal small overlap crash result to a best-possible "Good" rating. So a perfect set of "Good" results on all of the IIHS crash tests plus the available advanced crash safety items resulted in a best-possible TSP+ rating for the Santa Fe.
Performance (8/10)
The 2017 Hyundai Santa Fe comes with a 3.3-liter 6-cylinder engine that outputs 290 horsepower and 252 foot pounds of torque. A 6-speed automatic transmission is standard, as is front-wheel drive. All-wheel drive is available at all trim levels. Towing for the Santa Fe is rated at up to 7,000 pounds when properly equipped.
We like the acceleration and highway speed readiness of the 2017 Santa Fe's V6 and drivetrain. This crossover isn't the sportiest or fastest on the market, but it's not a slouch and is feels confident and capable at all times. It accelerates up and on ramp, around for the pass, etc. very well.
Another high point to the Santa Fe's build is in its tight chassis design. The 2017 Santa Fe is very maneuverable for its size and the standard rear parking camera is a boon when getting into tight spaces. The Santa Fe is unusually easy to parallel park for this class thanks to the high amounts of visibility for the driver and the tight steering.
Pricing and Value (9/10)
The 2017 Hyundai Santa Fe continues with the Korean automaker's mission of offering affordable, well-made, honest products. The competitive Santa Fe has competitive price points for all of its trim levels and the availability as ala carte package options for technology and safety are a big deal in a segment that usually opts for trim-plus-limited-package only offerings.
Most buyers are likely to find that the mid-level SE trim is a good fit with a smart array of technology and comfort for the $30,000-ish price point. The lower price of entry for most Santa Fe models make up for its generally lackluster resale value down the line.
Total Score and Competitive Comparison (73/80, 91%)
The three-row crossover segment is full of good options for today's buyer. The Toyota Highlander rules the roost, for the most part, with its excellent ergonomics and very family-oriented appeal. Yet against that, the Hyundai Santa Fe is more than competitive thanks to a better price point and more upscale appeal. Not to mention that warranty.
Another good choice for the cargo-oriented is the Chevrolet Traverse, though it lacks in other things the Santa Fe can boast. The Ford Flex is the best in terms of third row and good fuel economy, but lacks the maneuverability and smart looks of the Santa Fe. The Kia Sorento is a compelling offer that has much the same appeal as the Santa Fe, making it well worth a cross-shop.
The 2017 Hyundai Santa Fe is a full package of goodness in the three-row segment, with a well-rounded design that is both upscale and intelligent. It's functional, ergonomic and economical on several fronts. The Santa Fe should not be overlooked when shopping in the midsize three-row crossover segment.
SE SE 3.3L Auto AWD V6 AWD 32550
Limited Limited 3.3L Auto AWD V6 AWD 36700
SE Ultimate SE Ultimate 3.3L Auto AWD V6 AWD 40450
Limited Ultimate Limited Ultimate 3.3L Auto AWD V6 AWD 41150
SE SE 3.3L Auto V6 FWD 30800
Limited Limited 3.3L Auto V6 FWD 34950
SE Ultimate SE Ultimate 3.3L Auto V6 FWD 38700
Limited Ultimate Limited Ultimate 3.3L Auto V6 FWD 39400
Comparison of 2017 Hyundai Santa Fe with Similar Cars
2017 Hyundai Santa Fe $30,800 $29,335
2017 Ford Explorer $31,660 $30,553 2017 Hyundai Santa Fe VS 2017 Ford Explorer
2017 Jeep Grand Cherokee $30,395 $30,270 2017 Hyundai Santa Fe VS 2017 Jeep Grand Cherokee
2017 Toyota Highlander $30,630 $27,965 2017 Hyundai Santa Fe VS 2017 Toyota Highlander
2017 Honda Pilot $30,745 $28,118 2017 Hyundai Santa Fe VS 2017 Honda Pilot
2017 Nissan Pathfinder $30,290 $28,158 2017 Hyundai Santa Fe VS 2017 Nissan Pathfinder
Interested to see how the 2017 Hyundai Santa Fe ranks against similar cars in terms of key attributes? Here are the 2017 Hyundai Santa Fe rankings for MPG, horsepower, torque, leg room, head room, shoulder room, hip room and so forth.
4 2017 Ford Explorer 22
5 2017 Jeep Grand Cherokee 21
6 2017 Hyundai Santa Fe 19
2 2017 Hyundai Santa Fe 290
5 2017 Ford Explorer 280
6 2017 Jeep Grand Cherokee 240
2 2017 Ford Explorer 41.4
3 2017 Jeep Grand Cherokee 39.9
4 2017 Hyundai Santa Fe 39.9
Hyundai Santa Fe SUVs for Sale
2021 Hyundai Santa Fe - 3,903 Listings
2020 Hyundai Santa Fe - 564 Deals Out of 8,652 Listings
2018 Hyundai Santa Fe - 114 Deals Out of 397 Listings
Used Ford Explorer - 10,954 Deals Out of 42,291 Listings
Used Jeep Grand Cherokee - 8,599 Deals Out of 35,394 Listings

redpajama_set_name: RedPajamaCommonCrawl


Tender Arts Studio: Happy Halloween!
Be sure to keep the lanterns lit!
Didn't October fly by? I would like a repeat- it was a great month. I got down to Pleasanton to visit with dear friends Sandra and Monica. Had a great time at their antique fair, attended a gem fair, ate monkey brains and stopped in at my favorite store in the whole world, American Harvest.
Mr. Max turned 14! I told him yesterday that he really needs to stop growing because soon he will be bigger than the house and that will just be awkward!
I had my birthday too! Good heavens- 47! I wished for at least 47 more. Eric said I ruined my chances by telling, but I don't think so. I'm too excited to see what's around the next corner!
I've been stitching up a storm, but no photos yet. The sun comes out Monday through Friday and hides on the weekend when I have time to take some shots! Isn't that always the way?
Anyway, hope you all have a frightfully good (and safe) Halloween tonight!


text:
Q: How to escape html placeholder attribute in rendered vue I've setup a jsFiddle to showcase the issue.

  {{title}}

  <input type="text" :placeholder="icon" />

new Vue({
  el: '#app',
  data: {
    title: 'Vue Rendered Info',
    icon: ''
  }
});

This is just a sample (check the jsFiddle for a working sample) of what I'm trying to do. In my real project I have a .vue component; regardless of my icon being in the data or directly in the  I have the same issue. but for the sake of stackoverflow I've made a "working" example of the issue in the jsFiddle.
I'm attempting to place a Font-Awesome icon inside of the placeholder text of an input.
Any insight on how to solve this issue would be great.

A: To use Unicode characters in Javascript you must properly escape them. To quote Microsoft's page on Special Characters (JavaScript)

You can specify a Unicode character by using the \uhhhh escape
  sequence, where hhhh is a four-digit hexadecimal number.

The unicode identifier for the font-awesome magnify-glass icon is F002.
As such you'll want the following instead:
new Vue({
    el: '#app',
    data: {
        title: 'Vue Rendered Info',
        icon: '\uf002'
    }
});

redpajama_set_name: RedPajamaStackExchange


Planning on getting an abortion Joel?Use the morning after pill right after, instead of killing a baby.
So you're saying God didn't precede the universe? Or that he isn't alive? BLASPHEMY!
Hard to argue with people who believe everything they find on the Internet. https://www.youtube.com/watch?v=FZouZJSG4icMinor hint here, there's no use for a petri dish in any abortion procedure. Petri dishes are for growing single celled organisms in culture media.
Bad laws have been voided and these bad laws must be voided and replaced with ethical and moral laws. Or God will let the USA fall on history's trash pile.
….unless it's immigration law; to the left that one is flexible.