text stringlengths 14 5.77M | meta dict | __index_level_0__ int64 0 9.97k ⌀ |
|---|---|---|
Elected to the Senate for Queensland 2004. Re-elected 2010. Resigned 8.8.2013. Elected to the House of Representatives for New England, New South Wales, 2013. Re-elected 2016. Election ruled void 27.10.2017 under section 44 of the Constitution; ruled ineligible from 2.7.2016. Elected to the House of Representatives for New England, New South Wales at by-election 2017, vice Hon. B Joyce (resigned).
Minister for Agriculture from 18.9.2013 to 21.9.2015.
Cabinet Minister from 18.9.2013 to 27.10.2017.
Minister for Agriculture and Water Resources from 21.9.2015 to 27.10.2017.
Deputy Prime Minister from 18.2.2016 to 27.10.2017.
Minister for Resources and Northern Australia from 27.7.2017 to 27.10.2017.
Minister for Agriculture and Water Resources from 6.12.2017 to 20.12.2017.
Cabinet Minister from 6.12.2017 to 26.2.2018.
Deputy Prime Minister from 6.12.2017 to 26.2.2018.
Minister for Infrastructure and Transport from 20.12.2017 to 26.2.2018.
Senate Standing: Library from 16.8.2005 to 7.12.2005.
Senate Select: Fuel and Energy from 26.6.2008 to 18.3.2010.
Senate Legislative and General Purpose Standing: Legal and Constitutional: References from 16.8.2005 to 11.9.2006; Foreign Affairs, Defence and Trade: Legislation from 16.8.2005 to 13.9.2005; Foreign Affairs, Defence and Trade: References from 6.9.2005 to 11.9.2006; Economics from 11.9.2006 to 14.5.2009; Economics: Legislation from 14.5.2009 to 2.2.2010; Economics: References from 14.5.2009 to 2.2.2010.
Joint Standing: National Capital and External Territories from 16.8.2005 to 11.3.2010.
Joint Select: Parliamentary Budget Office from 23.11.2010 to 23.3.2011; Broadcasting Legislation from 14.3.2013 to 24.6.2013.
House of Representatives Standing: Industry, Innovation, Science and Resources from 1.3.2018 (Chair from 1.3.2018).
Leader of The Nationals in the Senate from 17.9.2008 to 8.8.2013.
Member of the Opposition Shadow Ministry from 8.12.2009 to 18.9.2013.
Shadow Minister for Finance and Debt Reduction from 8.12.2009 to 25.3.2010.
Shadow Minister for Regional Development, Infrastructure and Water from 25.3.2010 to 14.9.2010.
Shadow Minister for Regional Development, Local Government and Water from 14.9.2010 to 18.9.2013.
Deputy Leader of the Federal Parliamentary Nationals from 13.9.2013 to 12.2.2016.
Leader of the Federal Parliamentary Nationals from 12.2.2016 to 27.10.2017.
Leader of the Federal Parliamentary Nationals from 6.12.2017 to 26.2.2018.
Location: northern New South Wales highlands; it includes the centres of Aberdeen, Armidale, Ashford, Barraba, Bingara, Glenn Innes, Guyra, Inverell, Manilla, Merriwa, Murrurundi, Nundle, Quirindi, Satur, Scone, Tamworth, Tenterfield, Tingha, Urbenville and Walcha.
Industries: cattle, sheep, grain, forestry, dairying, oats, maize, fruit, vegetables, tin-mining, bricks, trout hatchery, tourism, concrete pipe works, leather goods, tannery, gemstones, tobacco, lucerne, honey, grain processing, rubber industry, plaster works and concrete. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,963 |
{"url":"http:\/\/mathoverflow.net\/revisions\/84864\/list","text":"MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06\/24\/2013) at approximately 9:00 PM Eastern time (UTC-4).\n\n3 edited title\n\n# Cyclic cubic numbers as rational linear comboscombinations of roots of unity\n\n2 added 28 characters in body\n\nIn the written version of a talk Barry Mazur gave to Friends of the Harvard Mathematics Department on May 5, 2009, there is an interesting question in Footnote 5 (page 8).\n\nHe recalls how Gauss wrote $\\sqrt p$ (where $p$ is an odd prime) as an explicit rational linear combination of roots of unity (using Gauss sums) and says that he doesn't know any such explicit expression for the roots $\\alpha$ of an irreducible cubic polynomial $T^3+bT+c\\in\\mathbf{Q}[T]$ whose discriminant is a square (so that $\\mathbf{Q}(\\alpha)$ is a cyclic extension of $\\mathbf{Q}$, and hence contained in $\\mathbf{Q}(\\zeta)$ for some root of unity $\\zeta$).\n\nQuestion. Does anyone know such an explicit expression for the roots of irreducible cubic polynomials whose discriminant is a square ?\n\n1\n\n# Cyclic cubic numbers as linear combos of roots of unity\n\nIn the written version of a talk Barry Mazur gave to Friends of the Harvard Mathematics Department on May 5, 2009, there is an interesting question in Footnote 5 (page 8).\n\nHe recalls how Gauss wrote $\\sqrt p$ as an explicit rational linear combination of roots of unity (using Gauss sums) and says that he doesn't know any such explicit expression for the roots $\\alpha$ of an irreducible cubic polynomial $T^3+bT+c\\in\\mathbf{Q}[T]$ whose discriminant is a square (so that $\\mathbf{Q}(\\alpha)$ is a cyclic extension of $\\mathbf{Q}$, and hence contained in $\\mathbf{Q}(\\zeta)$ for some root of unity $\\zeta$).\n\nQuestion. Does anyone know such an explicit expression for the roots of irreducible cubic polynomials whose discriminant is a square ?","date":"2013-06-19 09:27:13","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8922688364982605, \"perplexity\": 176.3630553633809}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2013-20\/segments\/1368708546926\/warc\/CC-MAIN-20130516124906-00031-ip-10-60-113-184.ec2.internal.warc.gz\"}"} | null | null |
Sladeniaceae is een botanische naam, voor een familie van tweezaadlobbige planten. Een familie onder deze naam wordt zelden erkend door systemen voor plantentaxonomie, maar wel door het APG-systeem (1998) dat de familie ongeplaatst laat (incertae sedis).
In het APG II-systeem (2003) wordt de mogelijkheid geboden deze familie te erkennen, maar de betreffende planten mogen ook worden ingedeeld in familie Pentaphylacaceae. De Angiosperm Phylogeny Website [16 november 2007] erkent deze familie wel. Het gaat om heel kleine familie van houtige planten.
Het Cronquist systeem (1981) plaatste de betreffende planten in de familie Theaceae.
Externe links
Ericales | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,104 |
package checkers
import (
"go/ast"
"go/types"
"github.com/go-lintpack/lintpack"
"github.com/go-lintpack/lintpack/astwalk"
)
func init() {
var info lintpack.CheckerInfo
info.Name = "rangeExprCopy"
info.Tags = []string{"performance"}
info.Params = lintpack.CheckerParams{
"sizeThreshold": {
Value: 512,
Usage: "size in bytes that makes the warning trigger",
},
"skipTestFuncs": {
Value: true,
Usage: "whether to check test functions",
},
}
info.Summary = "Detects expensive copies of `for` loop range expressions"
info.Details = "Suggests to use pointer to array to avoid the copy using `&` on range expression."
info.Before = `
var xs [2048]byte
for _, x := range xs { // Copies 2048 bytes
// Loop body.
}`
info.After = `
var xs [2048]byte
for _, x := range &xs { // No copy
// Loop body.
}`
info.Note = "See Go issue for details: https://github.com/golang/go/issues/15812."
collection.AddChecker(&info, func(ctx *lintpack.CheckerContext) lintpack.FileWalker {
c := &rangeExprCopyChecker{ctx: ctx}
c.sizeThreshold = int64(info.Params.Int("sizeThreshold"))
c.skipTestFuncs = info.Params.Bool("skipTestFuncs")
return astwalk.WalkerForStmt(c)
})
}
type rangeExprCopyChecker struct {
astwalk.WalkHandler
ctx *lintpack.CheckerContext
sizeThreshold int64
skipTestFuncs bool
}
func (c *rangeExprCopyChecker) EnterFunc(fn *ast.FuncDecl) bool {
return fn.Body != nil &&
!(c.skipTestFuncs && isUnitTestFunc(c.ctx, fn))
}
func (c *rangeExprCopyChecker) VisitStmt(stmt ast.Stmt) {
rng, ok := stmt.(*ast.RangeStmt)
if !ok || rng.Key == nil || rng.Value == nil {
return
}
tv := c.ctx.TypesInfo.Types[rng.X]
if !tv.Addressable() {
return
}
if _, ok := tv.Type.(*types.Array); !ok {
return
}
if size := c.ctx.SizesInfo.Sizeof(tv.Type); size >= c.sizeThreshold {
c.warn(rng, size)
}
}
func (c *rangeExprCopyChecker) warn(rng *ast.RangeStmt, size int64) {
c.ctx.Warn(rng, "copy of %s (%d bytes) can be avoided with &%s",
rng.X, size, rng.X)
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,888 |
Nestled deep in the mountain forests of North Carolina, you can roll into South Mountains State Park ready for a true outdoors trip.
Camping on Carolina Beach at Freeman Park! August 9, 2013 So so fun!
Experience the beauty of the Great Smoky Mountains from Greenbrier Campground! Enjoy hiking in the Smokies, relaxing by the river and so much more.
Distillery where full hookup RV site is located.
Best Campgrounds in North Carolina | The Perfect Camping Place for Hiking, Fishing and Relaxation. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,716 |
\section{}
\section{introduction}
The black hole information paradox \cite{Hawking:1976ra} is one of the most profound problems
in physics, which might lead to a deeper understanding of the relation between
general relativity and quantum theory.
If radiation from a black hole is thermal, the final state of black hole should be a mixed state even
for a black hole originating from a gravitationally collapsing pure state.
This process is forbidden in unitarity of quantum mechanics.
Therefore, it is expected that the radiation from a black hole would be non-thermal.
Susskind, Thorlacius and Ugrum proposed the \textit{black hole complementarity principle} \cite{Susskind:1993if,tHooft:1990fkf,Susskind:1993mu},
which gives the phenomenological picture for the evaporation of black hole that
explains how the non-thermal radiation could be emitted from a black hole.
This proposal is consistent with three postulates, which are briefly given as follows:
(postulate 1) Hawking radiation is in a pure state,
(postulate 2) outside the region near the horizon of a massive black hole,
physics can be described by an effective field theory of general relativity plus quantum field theory,
and (postulate 3) a black hole is regarded as a quantum system with discrete energy levels whose number
is the exponential of the Bekenstein entropy \cite{Bekenstein:1974ax} of black hole.
In 2012, however, Almheiri, Marolf, Polchinski and Sully (AMPS) pointed out in
Ref. \cite{Almheiri:2012rt} that
postulate 1, postulate 2, and the equivalence principle are mutually inconsistent
for an \textit{old black hole} \cite{Page:1993df, Page:1993wv, Page:2013dx} and idea that we briefly review here.
Let us consider an old black hole with early Hawking radiation A, late Hawking radiation B and infalling quanta
behind the horizon C.
A and B have to be fully entangled so that the final state of the black hole is a pure state (postulate 1).
On the other hand, according to quantum field theory in curved spacetime, B and C, pair-created particles, are also fully entangled (postulate 2).
That is, according to postulate 1 and 2, B should be fully entangled simultaneously with both A and C.
This contradicts with the {\it monogamy of entanglement}
that forbids any quantum system being entangled with two independent systems fully and simultaneously.
AMPS then proposed ``{\it firewalls}", high-energy quanta at horizons
energetic enough to break the entanglement of Hawking pairs, which would get rid of the inconsistency between postulate 1 and 2.
However, the existence of firewalls implies that the free falling observer going across the horizon
has a dramatic experience: the observer burns up at the horizon.
That is, firewalls amounts to abandoning the equivalence principle.
\begin{figure}[t]
\begin{center}
\includegraphics[keepaspectratio=true,height=50mm]{fig1.eps}
\end{center}
\caption{
The infalling mode near the horizon, C on the hyper surface $\Sigma$, can hold coherence,
whereas the infalling mode in the vicinity of the singularity, C on the hyper surface $\Sigma'$,
exits the particle horizon (dashed line) and loses causal contact as a whole, which leads to the decoherence of the infalling mode.
As a result, the entanglement of the Hawking pairs disappears and its state becomes separable.
}%
\label{061001fig}
\end{figure}
In this paper a sufficient reason for rejecting the AMPS firewall
concept as a solution to the black hole information paradox is
presented. It was previously pointed out that the black hole
information paradox only manifests limitations of the semiclassical
theory, rather than presents a conflict between any fundamental
principles \cite{Nomura:2016qum}. It was proved that firewalls are excluded by
Einstein's field equations for black holes of mass exceeding the
Planck mass \cite{Abramowicz:2013dla}, and demonstrated that the AMPS argument is based on
an over-counting of internal black hole states including those that
are singular in the past \cite{Page:2013mqa}.
Here we show that an infalling mode inside a black hole C is infinitely squeezed due to the gravitational effect of a black hole,
which makes the infalling mode highly sensitive to decoherence
\cite{footnote1}
and leads to the loss of its entanglement with the
outgoing mode B (Fig. \ref{061001fig}).
This means that there would be no violation of monogamy of entanglement around a black hole and
the black hole complementarity principle can be consistent with the equivalence principle.
The plan of this paper is as follows. In Sec. II we will introduce a quantum state around a
black hole formed from gravitational collapse and will describe how we calculate the
time evolution of the quantum state.
The resolution to the firewall paradox is described in Sec. III.
We will show that the quantum state of a Hawking pair, which is initially entangled state,
would become a separable state due to environment-induced decoherence (see e.g.,
\cite{Kieferdeco2} for the review of decoherence).
In Sec. IV we will confirm the consistency between our proposal, explaining how the
Hawking pair evolves to a separable state from an initially entangled state, and the previous works that investigated how the
purity of the Hawking radiation will be realized.
Sec. V is dedicated to conclusions.
\section{formalism}
The Unruh vacuum state \cite{Unruh:1976db} is the quantum state on an eternal black hole spacetime which models the late time properties of the \textit{in vacuum}
of a collapsing star, which is denoted by $\ket{\text{in}}$, that contains no Hawking particle at the past infinity.
The Unruh vacuum is associated with the infalling modes and the outgoing modes that are positive frequency with respect to the Killing vector $\partial_t$ and $\partial_T$ respectively,
where $t$ is the Schwarzschild time and $T$ is the Kruskal time.
Introducing the vacuum state $\ket{0}_c$ for the infalling modes and $\ket{0}_b$ for the outgoing modes,
the Unruh vacuum state can be expressed as $\ket{U} = \ket{0}_c \ket{0}_b$, and the relation between the in vacuum state $\ket{\text{in}}$ and the Unruh vacuum state $\ket{U}$ has the form \cite{Brout:1995rd}
\begin{eqnarray}
\ket{\text{in}} \propto \frac{1}{\sqrt{Z_\omega}} \displaystyle \left( \sum_{n=0}^{\infty} e^{-\pi \omega n(\omega)/\kappa} (b^{\dag}_{\omega})^n (c^{\dag}_{\omega})^n / n! \right)\ket{0}_c \ket{0}_b,
\label{060801}
\end{eqnarray}
where $b^{\dag}_{\omega}$ and $c^{\dag}_{\omega}$ are creation operators for the state $\ket{0}_b$ and $\ket{0}_c$ respectively,
$n(\omega)$ is the number of particles with mode $\omega$, $\kappa \equiv (4 G M)^{-1}$ is the surface gravity, and $Z_{\omega} \equiv (1-e^{-\pi \omega/ \kappa})^{-1}$.
In the following, we will use the Unruh vacuum state as a quantum state around a black hole
although modeling the quantum state around the collapsing star with the (outgoing) Kruskal mode
has not been fully successful and may demand us to take into account the technical issues,
e.g., the backscattering effect in the definition of $\ket{0}_c$ and $\ket{0}_b$
\cite{footnote2}.
The relation (\ref{060801}) implies that the infalling modes are fully entangled with the outgoing modes, which is the problematic entanglement and should be broken for the purity of the Hawking
radiation as is pointed out by AMPS \cite{Almheiri:2012rt}.
In the following, we will neglect multi-pair creations
because the states of {\it n}-particles are suppressed by the exponential factor $e^{-\pi \omega n/\kappa}$
and their cumulative contribution to the entanglement entropy (EE) between the infalling and outgoing mode is negligibly small
\cite{footnote3}.
For simplicity and to grasp the essence, we here consider a generically entangled state
\begin{align}
\ket{\text{in}} &\to \sqrt{1-p^2} \ket{0}_c \ket{0}_b + p \ket{1}_{c} \ket{1}_{b},
\label{061203}\\
\ket{1}_c &= \int_{0}^{\infty} d \omega \varphi_c (\omega) \ket{1,\omega}_c,
\label{170712}\\
\ket{1}_b &= \int_{0}^{\infty} d \omega \varphi_b (\omega) \ket{1,\omega}_b,
\end{align}
where $\ket{1, \omega}_{c} \equiv c^{\dag}_{\omega} \ket{0}_c, \ \ket{1, \omega}_{b} \equiv
b^{\dag}_{\omega} \ket{0}_b$, $p$ is a real number satisfying $0 < |p| < 1/{\sqrt{2}}$,
and $\varphi_c (\omega)$ $(\varphi_b (\omega))$ is a function satisfying $\int d \omega
|\varphi_c(\omega)|^2 = 1$ $(\int d \omega
|\varphi_b(\omega)|^2 = 1)$, which ensures that
$\ket{1}_c$ $(\ket{1}_b)$ is a one-particle state of an infalling (outgoing)
localized wave packet \cite{footnote4}.
In the latter part of this paper, we will show that this entanglement is broken by the existence of the singularity,
which is caused by the decoherence of an infalling mode.
An infalling mode inside a black hole is redshifted \cite{footnote5}
as $\lambda = \lambda_0 \sqrt{2GM/r-1}$, where $\lambda_0$ is the initial wavelength,
and it diverges in the limit of $r \to 0$. Therefore, the infalling mode exits the particle horizon
near the singularity and loses causal contact as a whole (Fig. \ref{061001fig}), which is responsible for the squeezing
(EPR-like correlation) of the infalling mode, that has the role to retain its coherent structure
\cite{Kiefer:2006je}, and decoherence as is discussed in Sec. III.
We consider a massless scalar field $\phi$ on the Schwarzschild spacetime with a mass $M$ whose metric is given as
$ds^2 = f(r) dt^2 - f^{-1}(r) dr^2 -r^2 d\Omega_2^2$ with $f(r) \equiv 1-2GM/r$,
\ where $d\Omega_2^2$ denotes the line element of a two-sphere $d\Omega_2^2 \equiv d\theta^2 +\sin^2{\theta} d\varphi^2$.
Using the tortoise coordinate $r^{\ast} = r + 2GM \ln \left| 1-r/(2GM) \right|$, we can rewrite it as
$
ds^2 = g_{\mu \nu} dx^{\mu} dx^{\nu} \equiv f(r)\left[ dt^2 -dr^{\ast}{}^2 \right] -r^2 d\Omega_2^2.
$
In order to describe the infinite squeezing of an infalling mode, let us investigate the dynamics of the vacuum $\ket{0}_c$ inside the black hole $r < 2GM$.
The action $S$ is given as
\begin{widetext}
\begin{eqnarray}
S = \int d^4x {\mathcal L}
= \frac{1}{2} \int d^4x \sqrt{-g} g^{\mu \nu} \partial_{\mu} \phi \partial_{\nu} \phi
= \frac{1}{2} \int d^2 x \displaystyle \sum_{l,m} \left[ \chi' {}_{lm}^2 -2 \chi_{lm} \chi' {}_{lm} {\mathcal G} + {\mathcal G}^2 \chi^2_{lm}
- \dot{\chi}_{lm}^2 +f(r) \frac{l (l+1)}{r^2} \chi^2_{lm} \right],
\label{060805}
\end{eqnarray}
\end{widetext}
where we decompose the field $\phi$ into partial waves with an angular momentum $l$ as $\phi \equiv \displaystyle \sum_{l,m} \chi_{lm} Y_{lm} /r$,
a prime and a dot denote differentiation with respect to $r^{\ast}$ and $t$ respectively,
and ${\mathcal G} \equiv r'/r$.
From the action (\ref{060805}), the Euler-Lagrange equation can be derived as
\begin{eqnarray}
\left[ \frac{\partial^2}{\partial r^{\ast} {}^2} - \frac{\partial^2}{\partial t^2} -f(r) \left( \frac{2GM}{r^3} + \frac{l(l+1)}{r^2} \right) \right] \chi_{lm} =0.
\label{060806}
\end{eqnarray}
We find that the mode functions satisfying (\ref{060806}) are almost independent of the angular momentum $l$ in the vicinity of the singularity
because $l(l+1)/r^2$ in (\ref{060806}) can be ignored for $r \ll 2GM$.
We are interested in the behavior of an infalling mode near the singularity, and
therefore, we set $l=0$ and omit the suffixes $(l,m)$ in the following.
The time like coordinate inside the black hole is $r^{\ast}$, therefore, the conjugate momentum
$\pi$ of the field $\chi$ is given as \cite{Yajnik:1997su}
\begin{eqnarray}
\pi \equiv \partial {\mathcal L}/\partial \chi' = \chi' - {\mathcal G} \chi
\label{063001}
\end{eqnarray}
and then the Hamiltonian is
\begin{eqnarray}
H = \int dt \displaystyle \frac{1}{2} \left[ \pi^2 + \dot{\chi}^2 + 2{\mathcal G} \chi \pi \right].
\label{060803}
\end{eqnarray}
We can decompose the field $\chi$ and its conjugate momentum $\pi$ as
\begin{widetext}
\begin{eqnarray}
&&\chi \equiv \int^{+ \infty}_{-\infty} \frac{d \omega}{\sqrt{2 \pi}} \bar{\chi}_{\omega} (r^{\ast}) e^{-i \omega t} + \text{(O.M.)}
\equiv \int^{+ \infty}_{-\infty} \frac{d \omega}{\sqrt{2 \pi}} \left[ c_{\omega} \tilde{\chi}_{\omega} (r^{\ast}) e^{-i\omega t} + c_{\omega}^{\dag} \tilde{\chi}^{\ast}_{\omega} (r^{\ast}) e^{+i \omega t} \right] \theta(\omega)+\text{(O.M.)},\label{063003} \\
&&\pi \equiv \int^{+ \infty}_{-\infty} \frac{d \omega}{\sqrt{2 \pi}} \bar{\pi}_{\omega} (r^{\ast}) e^{-i \omega t} + \text{(O.M.)}
\equiv -i \int^{+ \infty}_{-\infty} \frac{d \omega}{\sqrt{2 \pi}} \left[ c_{\omega} \tilde{\pi}_{\omega} (r^{\ast}) e^{-i \omega t} - c_{\omega}^{\dag} \tilde{\pi}^{\ast}_{\omega} (r^{\ast}) e^{+i \omega t} \right] \theta(\omega)+ \text{(O.M.)}, \label{063004}
\end{eqnarray}
\end{widetext}
where (O.M.) denotes the outgoing modes and $\theta (\omega)$ is a step function: $\theta (\omega)=1$ for $\omega > 0$
and $\theta (\omega) = 0$ for $\omega < 0$.
The canonical commutation relation is
$[\bar{\chi}_{\omega}, \bar{\pi}^{\dag}_{\omega'}] = i \delta (\omega-\omega')$.
In the following, we will omit the suffix $\omega$
for simplicity.
From (\ref{063001}) and the canonical commutation relation, we obtain the Wronskian condition as
$\left(\tilde{\chi}'^{\ast} \tilde{\chi} - \tilde{\chi}' \tilde{\chi}^{\ast} \right) =i$.
The third term in (\ref{060803}) is responsible for the squeezing of infalling modes \cite{Polarski:1995jg,Kiefer:2006je,Lesgourgues:1996jc,Kiefer:2008ku,Kiefer:1998qe}, which becomes stronger as $r^{\ast} \to 0$ as is shown later.
To investigate the dynamics of the states $\ket{0}_c$ and $\ket{1,\omega}_c$,
we first derive the wave functions for them,
$\Psi_0[\bar{\chi}]$ and $\Psi_1[\bar{\chi}]$, that satisfy $c \ket{0}_c=0$
and $\ket{1,\omega}_c =c^{\dag} \ket{0}_c$ respectively.
From (\ref{063003}) and (\ref{063004}), we can rewrite the former in the Schr$\ddot{\text{o}}$dinger representation as
$\left[ \bar{\chi} + i {\gamma}^{-1}(r^{\ast}, \omega) \bar{\pi} \right] \ket{0}_c =0$,
where $\gamma (r^{\ast}, \omega) \equiv \tilde{\pi}^{\ast} / \tilde{\chi}^{\ast}$.
Replacing the conjugate momentum $\bar{\pi}$ by $-i \partial/ \partial \bar{\chi}^{\dag}$, we obtain the wave function
$\Psi_0 [\bar{\chi}]$ of the state $\ket{0}_c$ as
\begin{eqnarray}
\Psi_0 [\bar{\chi}] = \sqrt{\frac{2 \gamma_R}{\pi}} \exp{\left[ -\gamma (r^{\ast}, \omega) \bar{\chi} \bar{\chi}^{\dag} \right]},
\label{061201}
\end{eqnarray}
where $\gamma_R \equiv \text{Re}[\gamma (r^{\ast}, \omega)]$.
On the other hand, $\ket{1,\omega}_c$ satisfies $\ket{1,\omega}_c = c^{\dag} \ket{0}_c$, and hence we obtain
$\Psi_1[\bar{\chi}] \propto \left( \bar{\chi} -\gamma^{\ast} {}^{-1} (r^{\ast}) \partial / \partial \bar{\chi}^{\dag} \right) \Psi_0 [\bar{\chi}]$,
which leads to
\begin{eqnarray}
\Psi_1[\bar{\chi}] = \frac{2\gamma_R}{\sqrt{\pi}} \bar{\chi} \exp{\left[
- \gamma (r^{\ast}, \omega) \bar{\chi} \bar{\chi}^{\dag} \right]}.
\label{061202}
\end{eqnarray}
The function $\gamma$ can be calculated numerically from (\ref{060806}).
\section{decoherence near a black hole singularity}
In the following we show that
the density matrix $\rho_{co}$ of the quantum state (\ref{061203})
is reduced to a separable \cite{footnote6}
density matrix $\rho_{de}$ due to the decoherence
once the infalling mode reaches the vicinity of the singularity,
namely, $\rho_{co} \to \rho_{de}$ for $r^{\ast} \to 0$.
To this end, we first show that the infalling mode becomes highly squeezed as the mode approaches the singularity, and secondly,
that the squeezed state is highly sensitive to decoherence.
The density matrix $\rho_{co}$ can be written as
\begin{eqnarray}
\rho_{co} \equiv
(1-p^2) \ket{0}_c \bra{0}_c \otimes \ket{0}_b \bra{0}_b +p^2 \ket{1}_c \bra{1}_c \otimes \ket{1}_b \bra{1}_b&& \nonumber\\
+p\sqrt{1-p^2} \left( \ket{1}_c \bra{0}_c \otimes \ket{1}_b \bra{0}_b+\ket{0}_c \bra{1}_c \otimes \ket{0}_b \bra{1}_b \right),~~~&&
\label{061407}
\end{eqnarray}
and as is shown later, the separable density matrix $\rho_{de}$ is
\begin{eqnarray}
\rho_{de} = (1-p^2) \ket{0}_c \bra{0}_c \otimes \ket{0}_b \bra{0}_b
+p^2 \ket{1}_c \bra{1}_c \otimes \ket{1}_b \bra{1}_b.\nonumber \\
\label{061204}
\end{eqnarray}
Hence, we will show that the third and fourth
terms in (\ref{061407}) disappear, that is, $\rho_{co} \to \rho_{de}$, as the infalling mode approaches the vicinity of the singularity.
Let us consider the time evolution of the non-diagonal terms of $\rho_{co}$.
Using (\ref{170712}), $\ket{0}_c \bra{1}_c$ and $\ket{1}_c \bra{0}_c$ in the non-diagonal terms
can be decomposed as
\begin{align}
\begin{aligned}
\ket{0}_c \bra{1}_c &= \int d\omega \varphi_c^{\ast} (\omega) \ket{0}_c \bra{1,\omega}_c,\\
\ket{1}_c \bra{0}_c &= \int d\omega \varphi_c (\omega) \ket{1,\omega}_c \bra{0}_c
\label{062817}
\end{aligned}
\end{align}
respectively, and we will show the decay of $\ket{0}_c \bra{1}_c$ and $\ket{1}_c \bra{0}_c$
by calculating the time evolution of $\ket{0}_c \bra{1, \omega}_c$ and $\ket{1, \omega}_c \bra{0}_c$.
$\ket{0}_c \bra{1,\omega}_c$ and $\ket{1,\omega}_c \bra{0}_c$ component of the Wigner function of $\rho_{co}$, $W_{01}^{(c)}$ and $W_{10}^{(c)}$, are given as
\begin{widetext}
\begin{eqnarray}
\begin{split}
&&W_{01}^{(c)} = W_{10}^{(c)} {}^{\ast}
= \int \int \frac{d x_{R} dx_I}{(2 \pi)^2} e^{-i (\bar{\pi}_{R} x_R + \bar{\pi}_{I} x_I)}
\bra{\bar{\chi} - \frac{x}{2}} \ket{0}_c \bra{1,\omega}_c \ket{\bar{\chi} + \frac{x}{2}}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \\
&&= \frac{1}{\pi^2} \left( \sqrt{2 \gamma_{R}} \bar{\chi} - i \sqrt{\frac{2 \gamma_{I}^2}{\gamma_{R}}} (\bar{\chi} + \frac{\bar{\pi}}{2 \gamma_{I}}) \right) \exp{\left[ -2 \gamma_{R} |\bar{\chi}|^2 \right]}
\exp{\left[ -\frac{2 \gamma_{I}^2}{\gamma_{R}} \left| \bar{\chi} + \frac{\bar{\pi}}{2 \gamma_{I}} \right|^2 \right]}, \label{070501}
\end{split}
\end{eqnarray}
\begin{figure}[t]
\begin{center}
\includegraphics[keepaspectratio=true,height=82mm]{fig2.eps
\end{center}
\caption{(a), (b), and (c) are the imaginary parts of the non-diagonal components $W_{01}^{(c)}$,
and (a'), (b'), and (c') are the imaginary parts of the coarse-grained non-diagonal components
${\mathcal W}_{01}^{(c)}$, where we set
$|r^{\ast}|/2GM = 10$ (for (a), (a')), $|r^{\ast}|/2GM = 0.1$ (for (b), (b')),
$|r^{\ast}|/2GM = 0.001$ (for (c), (c')), and $2GM \omega = 0.5$.
The non-diagonal term
$W_{01}^{(c)} = W_{10}^{(c)}{}^{\ast}$ has the form of $X \delta (X)$ in the limit of $r^{\ast} \to 0$,
and therefore the coarse-grained distribution ${\mathcal W}_{01}^{(c)} = {\mathcal W}_{10}^{(c)} {}^{\ast}$ disappears.
This leads to the transition from the entangled Hawking pair to the separable Hawking pair in the vicinity of the singularity.
}%
\label{0630nonfig}
\end{figure}
\end{widetext}
where we used (\ref{061201}) and (\ref{061202}) and the suffixes $R$ and $I$ represent the real and imaginary
part respectively.
We numerically confirmed that they are infinitely squeezed in the limit of $r^{\ast} \to 0$ with $2GM \omega = 0.5$
(Fig.\ref{0630nonfig} (a), (b), and (c))
and the ratio $\gamma_I/\gamma_R \propto \sinh{2s}$ diverges in the vicinity of the singularity, $\gamma_I/\gamma_R \to - \infty$, where $s$ is the squeezing parameter. This means that $s$ also diverges, $|s| \to \infty$, as
$r^{\ast} \to 0$ (see e.g., \cite{Polarski:1995jg}).
Secondly, we will show that an infinitely squeezed state with an environment is highly fragile against decoherence,
in which the environment plays an important role.
For instance, let us consider a double-slit experiment with electrons in which
they create an interference pattern (non-diagonal density matrix).
If they are exposed to thermal noise (environment), the pattern will be coarse-grained
and will disappear (decoherence).
This is the intuitive interpretation for the role of environment in decoherence.
We here take into account the environment as follows.
The field $\chi$ can be
separated into two parts, the long-wavelength part as the system (an infalling Hawking particle) and the short-wavelength part as
the environment (vacuum fluctuations).
We here regard only the modes with wavelengths much shorter than the gravitational
curvature radius of black hole as the short-wavelength part, as in the \textit{stochastic inflation} scheme
\cite{Starobinsky:1994bd,Burgess:2006jn,Calzetta:nqft}.
Therefore, the environment can be regarded as a coherent state with a good approximation
and we can consider the decoherence by tracing out the coherent environment.
It is shown that the tracing out the coherent environment is corresponding to convolving (coarse-graining)
the system's Wigner function (\ref{070501}) with that of a coherent state $W_{\text{E}}$ \cite{Kanada-En'yo:2015pra}
(see also \cite{K.Husimi1940,K.Takahashi1986}),
\begin{eqnarray}
W_{\text{E}} \equiv \pi^{-2} \exp{\left(- |\bar{\chi}|^2- |\bar{\pi}|^2 \right)}.
\label{100501}
\end{eqnarray}
Taking the convolution of (\ref{070501}) and (\ref{100501}), the non-diagonal term of the coarse-grained Wigner function ${\mathcal W}_{01}^{(c)} = {\mathcal W}_{10}^{(c)} {}^{\ast}$ is obtained as
\begin{widetext}
\begin{eqnarray}
{\mathcal W}_{01}^{(c)}
\equiv (W^{(c)}_{01} \ast W_E) =\frac{Q |Q|^2}{\pi^2} (\bar{\chi}-i\bar{\pi}) \exp{\left[ -|Q|^2 \left\{ (|\bar{\chi}|^2 + |\bar{\pi}|^2) +2 \gamma_{R}
(|\bar{\chi}|^2 + \left| \bar{\pi}/(2 \gamma_R) + (\gamma_I/\gamma_R) \bar{\chi} \right|^2) \right\} \right]},
\label{070502}
\end{eqnarray}
\end{widetext}
where $Q \equiv \sqrt{2 \gamma_{R}}/(1 + 2 \gamma)$. In the limit of $r^{\ast} \to 0$,
the real and imaginary parts of the function $\gamma (r^{\ast}, \omega)$ diverge and hence $Q$ asymptotically approaches zero.
Therefore, the non-diagonal term ${\mathcal W}_{01}^{(c)}$ is decaying as approaching the
singularity (Fig.\ref{0630nonfig} (a'), (b'), and (c')), which means that the Hawking pair
will experience decoherence as the infalling mode approaches the singularity
since the effect of decoherence on a density matrix is essentially the decay of its non-diagonal
terms, see e.g., \cite{Kieferdeco2}.
Although general relativity and quantum field theory are, of course, no longer valid
near the singularity at $r \lesssim r_{\text{Pl}} = 2GM (M_{\text{Pl}} / M)^{2/3}$
\cite{footnote7}, where $M_{\text{Pl}}$ is the Planck mass,
the decoherence is almost completed
at $r \gg r_{\text{Pl}}$ in the case of interest, namely a massive black hole $M \gg M_{\text{Pl}}$ (remember postulate 2).
That is, the above estimates suggest that the squeezing becomes so strong
that the decoherence can take place well before the modes reach $r \sim r_{\text{Pl}}$,
and therefore using a (semi)classical spacetime picture of the mode evolution should
still be reliable.
As is shown above, the intense squeezing leads to the decay of the non-diagonal terms.
Therefore, the third and fourth terms in (\ref{061407}), containing the non-diagonal components
$\ket{1,\omega}_c \bra{0}_c$ and $\ket{0}_c \bra{1,\omega}_c$ (see (\ref{062817})),
decay due to the decoherence and this leads to the transition of the state
$\rho_{co} \to \rho_{de} = (1-p^2) \ket{0}_c \bra{0}_c \otimes \ket{0}_b \bra{0}_b +
p^2 \ket{1}_c \bra{1}_c \otimes \ket{1}_b \bra{1}_b$.
This implies that the entanglement of Hawking pairs
decays as the infalling mode approaches the singularity.
\section{microscopic picture of information recovery}
We can apply the loss of the entanglement between a Hawking pair to
the black hole information paradox.
According to our proposal, the entanglement between B and C is broken
when C approaches the singularity. Therefore, the timescale on which the entanglement is broken
is of the order of
the free fall timescale, $t_{F} \sim 2GM$, measured by a freely falling observer
\cite{footnote8}.
In other words, we cannot avoid the entanglement between B and C only during the moment of the free fall $\sim t_{F}$.
Therefore, we have to discuss how the scenario proposed here is consistent with
the monogamy of entanglement and the previous works \cite{Saini:2015dea,Kawai:2015uya}, in which
the timescale of information recovery is carefully discussed in the microscopic level.
In Ref. \cite{Saini:2015dea}, the radiation around a gravitationally collapsing shell was analytically investigated and it was shown that the correlations between the Hawking
particles (between A and B) are initially zero but grow on the timescale of $t_{F}$ for an observer far from the black hole.
Ref. \cite{Kawai:2015uya} also pointed out that the microscopic timescale of information recovery may be of the order of $t_{F}$
by considering the interaction between a collapsing shell and the Hawking radiation.
For these reasons, we can conclude that the entanglement between A and B would be initially zero and
gradually appears on the timescale of $t_{F}$,
and B can be allowed to be entangled with C only for the short time $ \sim t_F$, which is quite consistent with our scenario.
This implies that B would not be fully entangled with A and C simultaneously (Fig. \ref{091102}), and therefore there is no any violation of the monogamy of entanglement.
\begin{figure}[b]
\begin{center}
\includegraphics[keepaspectratio=true,height=50mm]{fig3.eps}
\end{center}
\caption{
The schematic picture showing
how the microscopic picture of information recovery \cite{Saini:2015dea,Kawai:2015uya}
is consistent with our proposal.
B is initially entangled with C and its entanglement will decay on the timescale of $t_{F}$.
On the other hand, the entanglement between A and B is initially zero and may grow on the timescale of $t_{F}$.
}%
\label{091102}
\end{figure}
\section{conclusions}
We showed that a Hawking pair becomes a separable state from an entangled state
by pointing out that the high squeezing and decoherence occur inside a black hole.
The analysis was done with a simplified state (\ref{061203})
and the environment interacting with the infalling Hawking modes whose Wigner function
is given by (\ref{100501}).
The interaction with the environment can be effectively taken into account by smearing out
the Wigner function of the infalling mode with that of the environment (\ref{070502}).
As a result, we showed that the non-diagonal terms of the density matrix for
the Hawking pair would decay quickly compared to the black hole evaporation timescale,
which implies that the decoherence would be caused by the interior gravitational effect and
that the entanglement between Hawking pairs will be broken.
It should be emphasized that although general relativity and quantum field theory would
break down near the singularity, our proposal is valid as long as the mass of black hole
is much larger than the Planck mass, $M \gg M_{\text{Pl}}$ \cite{Susskind:1993if,Almheiri:2012rt}.
According to our proposal, we would no longer need firewalls.
We believe that our work can be important for the understanding of how the states of
Hawking pairs of particles become separable, and how the black hole information paradox
can be solved.
\section*{acknowledgements}
The author thanks T.~Nakama, Y.~Tada, D.~Yamauchi, and J.~Yokoyama for helpful comments.
This work was partially supported by Grant-in-Aid for JSPS Fellow No.16J01780.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,130 |
Витков () је значајно брдо у Прагу. Налази се источно од самог центра града, између Карлина и Жижкова. На врх брда био је у време прве чехословачке републике постављен велики функционалистички споменик народне слободе са великом статуом Јана Жишке. После 1948. комунистичке власти су га променили ово у маузулеум првог председника "радничке класе" - Клемента Готтвалда. Унутрар брда има неколико тунела - један железнички и један за пешаке. Око споменика на свим странима брда налази се данас парк и шума.
Спољашње везе
Праг | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,152 |
Q: How to rearrange a pandas dataframe having N columns and append N columns together in python? I have a dataframe df as shown below,A index,B Index and C Index appear as headers
and each of them have sub header as the Last price
Input
A index B Index C Index
Date Last Price Date Last Price Date Last Price
1/10/2021 12 1/11/2021 46 2/9/2021 67
2/10/2021 13 2/11/2021 51 3/9/2021 70
3/10/2021 14 3/11/2021 62 4/9/2021 73
4/10/2021 15 4/11/2021 47 5/9/2021 76
5/10/2021 16 5/11/2021 51 6/9/2021 79
6/10/2021 17 6/11/2021 22 7/9/2021 82
7/10/2021 18 7/11/2021 29 8/9/2021 85
I want to transform the to the below dataframe.
Expected Output
Date Index Name Last Price
1/10/2021 A index 12
2/10/2021 A index 13
3/10/2021 A index 14
4/10/2021 A index 15
5/10/2021 A index 16
6/10/2021 A index 17
7/10/2021 A index 18
1/11/2021 B Index 46
2/11/2021 B Index 51
3/11/2021 B Index 62
4/11/2021 B Index 47
5/11/2021 B Index 51
6/11/2021 B Index 22
7/11/2021 B Index 29
2/9/2021 C Index 67
3/9/2021 C Index 70
4/9/2021 C Index 73
5/9/2021 C Index 76
6/9/2021 C Index 79
7/9/2021 C Index 82
8/9/2021 C Index 85
How can this be done in pandas dataframe?
A: The structure of your df is not clear from your output. It would be useful if you provided Python code that creates an example, or at the very lest the output of df.columns. Now let us assume it is a 2-level multindex created as such:
columns = pd.MultiIndex.from_tuples([('A index','Date'), ('A index','Last Price'),('B index','Date'), ('B index','Last Price'),('C index','Date'), ('C index','Last Price')])
data = [
['1/10/2021', 12, '1/11/2021', 46, '2/9/2021', 67],
['2/10/2021', 13, '2/11/2021', 51, '3/9/2021', 70],
['3/10/2021', 14, '3/11/2021', 62, '4/9/2021', 73],
['4/10/2021', 15, '4/11/2021', 47, '5/9/2021', 76],
['5/10/2021', 16, '5/11/2021', 51, '6/9/2021', 79],
['6/10/2021', 17, '6/11/2021', 22, '7/9/2021', 82],
['7/10/2021', 18, '7/11/2021', 29, '8/9/2021', 85],
]
df = pd.DataFrame(columns = columns, data = data)
Then what you are trying to do is basically an application of .stack with some re-arrangement after:
(df.stack(level = 0)
.reset_index(level=1)
.rename(columns = {'level_1':'Index Name'})
.sort_values(['Index Name','Date'])
)
this produces
Index Name Date Last Price
0 A index 1/10/2021 12
1 A index 2/10/2021 13
2 A index 3/10/2021 14
3 A index 4/10/2021 15
4 A index 5/10/2021 16
5 A index 6/10/2021 17
6 A index 7/10/2021 18
0 B index 1/11/2021 46
1 B index 2/11/2021 51
2 B index 3/11/2021 62
3 B index 4/11/2021 47
4 B index 5/11/2021 51
5 B index 6/11/2021 22
6 B index 7/11/2021 29
0 C index 2/9/2021 67
1 C index 3/9/2021 70
2 C index 4/9/2021 73
3 C index 5/9/2021 76
4 C index 6/9/2021 79
5 C index 7/9/2021 82
6 C index 8/9/2021 85
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,623 |
As Sydney Pollack's film "Sketches of Frank Gehry" premieres, Charlie looks back at two interviews he did with the architect, one in 1998 and another in 2001.
November 2015 November 2015 May 2009 May 2009 June 2008 May 2008 May 2008 July 2006 April 2006 April 2006
Sydney Pollack; Frank Gehry; Will Shortz
Entertainment, Art and design, Media
Sydney Pollack on his documentary about Frank Gehry; Charlie re-airs two interviews with Gehry; Will Shortz, NY Times crossword editor. 53:06
Remembering Hamilton Jordan
Charlie remembers political strategist Hamilton Jordan with a look back at Jordan's appearance on the show in 2000. 06:17
Remembering Robert Rauschenberg
Charlie remembers artist Robert Rauschenberg with a look back at his appearances on the show. 15:03
Charles Durning; 20th Century Abstract...
Entertainment, Books, Art and design
On working with Tennessee Williams and his current play, "Inherit the Wind." 20th century abstract art. Three poets remember the fmr. Poet Laureate. 54:13
McChrystal and Petraeus
Charlie looks back at interviews with General McChrystal and with the man who replaced him, General David Petraeus. 18:34 | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,341 |
Huntington Convention Center of Cleveland Debuts Lactation Suites for Nursing Moms
Nursing mothers now have a couple of private sanctuaries to retreat to when attending conventions and meetings at the Huntington Convention Center of Cleveland.
The SMG-managed facility has installed two portable lactation suites for female attendees and exhibitors wishing to breast feed their infants or use breast pumps in privacy.
"We are offering the new lactation suites during conventions, meetings, and trade shows to offer comfort and convenience for our guests," said Mark J. Leahy, general manager of the HCCC.
He continued, "Young mothers have had to utilize public restroom spaces or find a private space in the Convention Center, which is often difficult during a large convention or trade show with thousands of attendees and exhibitors. The new lactation suites create a private place within a public environment, enhancing our amenities for women and families."
Created by Mamava, a Vermont-based company that designs solutions for nursing mothers on the go, the portable lactation suites are self-contained mobile pods that include benches, an electrical outlet and a door that can be locked for privacy.
Measuring 4 x 8 feet, the suites are designed for one person at a time but also provide enough space to simultaneously hold mom, infant, breast pumps and luggage.
New mom Brittany Wire Gloer, director of supplier relationship solutions at PartsSource was thrilled to have access to the lactation stations while attending the UBM Advanced Design in Manufacturing Expo at the HCCC.
"I was concerned about what I was going to do when I got to the conference, so I called the convention center ahead of time to find out if they had a private area where I could nurse and was told they had recently installed a couple of dedicated areas," Gloer said.
She added, "I was pleasantly surprised by the lactation pods and how private and clean they were. I wish every convention center had this type of convenience for nursing moms."
The two suites are positioned near the entrance to Exhibit Hall C and on the concourse level near Meeting Room 1, as well as at the Global Center Atrium escalator landing and the entrance to Exhibit Hall A.
As the first wave of convention centers to incorporate such spaces, the HCCC is the second SMG facility to install lactation suites, preceded by the Philadelphia Convention Center, which introduced one in late 2015.
"Until now, women have often had to pump milk in the bathroom while they are away from their babies working a trade show or attending conferences if the show planners haven't designated a space," explained Sascha Mayer, co-founder of Mamava.
She added, "We believe that all mamas deserve a private, clean and comfortable place to use a breast pump or breastfeed – anywhere, anytime. Mamava pods provide flexibility for facilities and easy access for moms."
Cleveland's Huntington Convention Center Opens Art Gallery
The Huntington Convention Center of Cleveland and the Global Center for Health Innovation, managed by SMG, opened the Cleveland Convention Gallery, an exhibition space for…
Huntington Convention Center of Cleveland Diverts Massive Amounts of Waste in 2017
Last year was an especially green one for the Huntington Convention Center of Cleveland and the Global Center for Health Innovation, which knocked it out of the ballpark in…
New 600-room Hilton Hotel Opens Next to Huntington Convention Center of Cleveland
Meeting and convention-goers holding events in Cleveland just received a nice boost, with a new 600-room Hilton opening adjacent to the Huntington Convention Center of…
Detroit's Convention Center Renamed Huntington Place; Commits to City's Continued Growth
ASM Global announced a new name for the 723,000-square-foot convention center that it has managed in downtown Detroit since 2010. Formerly named Cobo Center, then TCF Center…
Cleveland's I-X Center Opens Doors to Outside Contractors for Trade Shows
For the past 30 years, the 9th largest convention center in the United States hasn't permitted outside contractors, until now. Cleveland, Ohio's 2.2 million square foot I-X… | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,856 |
\subsection*{Lemma \arabic{lemmacounter}}}
\def\TarskiFiveSegmentFigure{%
\psset{unit=2.25cm}
\pspicture(2.5,0)(2.2, 0.95)
\pspolygon[fillstyle=solid,fillcolor=lightblue,linestyle=none]
(1,0.85)(2,0.15)(0.7,0.15)
\pspolygon[fillstyle=solid,fillcolor=lightblue,linestyle=none]
(3.5,0.85)(4.5,0.15)(3.2,0.15)
\qline(0.0,0.15)(2.0,0.15)
\qline(0.0,0.15)(1.0,0.85)
\psline[linestyle=dashed](1.0,0.85)(2.0,0.15)
\qline(1.0,0.85)(0.7,0.15)
\put(1,0.9){$d$}
\put(0.0,0){$a$}
\put(0.7,0){$b$}
\put(2.0,0){$c$}
\qline(2.5,0.15)(4.5,0.15)
\qline(2.5,0.15)(3.5,0.85)
\psline[linestyle=dashed](3.5,0.85)(4.5,0.15)
\qline(3.5,0.85)(3.2,0.15)
\put(3.5,0.9){$D$}
\put(2.5,0){$A$}
\put(3.2,0){$B$}
\put(4.5,0){$C$}
\psset{unit=3cm}
\endpspicture}
\def\SameSideFigure{%
\pspicture(0.55,0.45)(2,1.5)
\qline(0.7,1)(2,1)
\put(0.6,0.97){$p$}
\put(2.05,0.97){$q$}
\psdot(0.9,0.5)
\put(0.8,0.46){$a$}
\psdot(1.4,0.5)
\put(1.31,0.46){$b$}
\psline(1.8,1.5)(1.4,0.5)
\psline(1.8,1.5)(0.9,0.5)
\pscircle[fillstyle=solid,fillcolor=white](1.6,1){0.03}
\put(1.6,0.87){$y$}
\pscircle[fillstyle=solid,fillcolor=white](1.346,1){0.03}
\put(1.3,0.87){$x$}
\pscircle[fillstyle=solid,fillcolor=white](1.8,1.5){0.03}
\put(1.85,1.5){$c$}
\endpspicture
}
\def\OppositeSideFigure{%
\pspicture(0.55,0.45)(2,1.5)
\qline(0.7,1)(2,1)
\put(0.6,0.97){$p$}
\put(2.05,0.97){$q$}
\psdot(0.9,0.5)
\put(0.8,0.46){$a$}
\psdot(1.8,1.5)
\put(1.85,1.5){$b$}
\psline(1.8,1.5)(0.9,0.5)
\pscircle[fillstyle=solid,fillcolor=white](1.346,1){0.03}
\put(1.27,1.05){$x$}
\endpspicture
}
\def\CrossbarFigure{%
\pspicture(-0.5,0)(0.8,0.6)
\SpecialCoor
\qline(0,0)(0.45;60)
\qline(0,0)(0.7;120)
\qline(0,0)(0.9;30)
\psdot(0.5;30)
\psdot(0.75;30)
\psdot(0.5;120)
\psdot(0.3;120)
\psdot(0.292;60)
\psdot(0,0)
\qline(0.5;30)(0.3;120)
\put(-0.38,0.4){$J$}
\put(0.67,0.32){$K$}
\put(0.26,0.44){$X$}
\put(-0.3,0.2){$U$}
\put(-0.13,-0.05){$B$}
\put(0.47,0.17){$V$}
\put(0.09,0.3){$P$}
\put(-0.47,0.54){$A$}
\put(0.8,0.47){$C$}
\psline(0.5;120)(0.75;30)
\pscircle[fillstyle=solid,fillcolor=white](0.46;60){0.03}
\endpspicture
}
\def\InnerOuterPaschFigure{%
\pspicture(0,0.2)(3.6,1.6)
\pspolygon[fillstyle=solid,fillcolor=lightblue,linestyle=none]
(0,0.6)(1.1,1.4)(1.3,0.6)
\pspolygon[fillstyle=solid,fillcolor=lightblue,linestyle=none]
(2,0.3)(3.0,1.05)(3.5,0.3)
\psdot(0,0.6)
\put(0,0.47){$A$}
\pscircle[fillstyle=solid,fillcolor=white](1,0.6){0.03}
\put(1,0.47){$x$}
\qline(0,0.6)(0.97,0.6)
\psdot(1.3,0.6)
\put(1.35,0.55){$C$}
\qline(1.03,0.6)(1.3,0.6)
\psdot(1.1,1.4)
\put(1.1,1.45){$B$}
\qline(0,0.6)(1.1,1.4)
\qline(1.1,1.4)(1.375,0.3)
\psdot(1.375,0.3)
\put(1.43,0.25){$E$}
\psdot(0.52,0.98)
\put(0.43,1.03){$F$}
\qline(1.375,0.3)(1.015,0.584)
\qline(0.52,0.98)(0.979,0.62)
\psdot(2,0.3)
\put(1.95,0.16){$A$}
\psdot(3.5,0.3)
\put(3.45,0.16){$C$}
\qline(2,0.3)(2.47,0.3)
\qline(2.53,0.3)(3.5,0.3)
\pscircle[fillstyle=solid,fillcolor=white](2.5,0.3){0.03}
\put(2.45,0.19){$x$}
\psdot(2.7,1.5)
\put(2.65,1.57){$E$}
\qline(2.7,1.5)(3.5,0.3)
\psdot(3.0,1.05)
\put(3.07,1.02) {$B$}
\qline(2.7,1.5)(2.5,0.33)
\qline(2,0.3)(3.0,1.05)
\psdot(2.566,0.722)
\put(2.4,0.7){$F$}
\endpspicture}
\def\GuptaMidpointFigureOne{%
\psset{unit=2cm}
\pspicture(0,-0.2)(2,1.4)
\pspolygon[fillstyle=solid,fillcolor=yellow,linestyle=none](1.414,1.414)(0,0)(2,0)(1.414,0.5857)(1.414,1.414)
\psdot(0,0)
\put(-0.15,-0.17){$C$}
\psdot(1.414,0)
\put(1.4,-0.17){$B$}
\psdot(1.414,1.414)
\put(1.48,1.4){$E$}
\psdot(2,0)
\put(2,-0.17){$D$}
\psdot(1,1)
\put(0.9,1.1){$A$}
\qline(0,0)(2,0)
\qline(0,0)(1.414,1.414)
\qline(1.414,0)(1.414,0.545)
\qline(1.414,0.62)(1.414,1.414)
\qline(1,1)(1.397,0.60)
\qline(2,0)(1.43,0.57)
\qline(1,1)(1.414,0)
\qline(0,0)(1.39,0.575)
\qline(1.45,0.594)(2,0.8284)
\pscircle(1.414,0.5857){0.04}
\put(1.55, 0.5){$F$}
\psset{unit=3cm}
\endpspicture}
\def\GuptaMidpointFigureTwo{%
\psset{unit=2cm}
\pspicture(0,-0.2)(2,1.4)
\pspolygon[fillstyle=solid,fillcolor=yellow,linestyle=none](0,0)(1.414,1.414)(1.414,0)(1.207,0.5)(0,0)
\psdot(0,0)
\put(-0.15,-0.17){$C$}
\psdot(1.414,0)
\put(1.4,-0.17){$B$}
\psdot(1.414,1.414)
\put(1.48,1.4){$E$}
\psdot(2,0)
\put(2,-0.17){$D$}
\psdot(1,1)
\put(0.9,1.1){$A$}
\qline(0,0)(2,0)
\qline(0,0)(1.414,1.414)
\qline(1.414,0)(1.414,1.414)
\qline(1,1)(1.397,0.60)
\qline(2,0)(1.43,0.57)
\qline(1,1)(1.20,0.53)
\qline(1.225,0.47)(1.414,0)
\qline(0,0)(1.187,0.491)
\qline(1.222,0.512)(1.39,0.575)
\qline(1.44,0.594)(2,0.8284)
\pscircle(1.207,0.5){0.04}
\put(1.05,0.3){$M$}
\psdot(1.414,0.5857)
\put(1.55, 0.5){$F$}
\psset{unit=3cm}
\endpspicture}
\def\ProclusBisectionFigure{%
\psset{unit=2cm}
\pspicture (2,0.9)(-1,-0.7)
\pspolygon[fillstyle=solid,fillcolor=yellow,linestyle=none]
(-0.25,-0.433)(0.5,0.866)(1.25,-0.433)(0.5,-0.176)
\psdot(0,0)
\psdot(1,0)
\psdot(0.5,0.866)
\psdot(-0.25,-0.433)
\psdot(1.25,-0.433)
\qline(-0.25,-0.433)(0.5,0.866)
\qline(0.5,0.866)(1.25,-0.433)
\qline(1.25,-0.433)(0,0)
\qline(-0.25,-0.433)(1,0)
\qline(0,0)(1,0)
\qline(0.5,0.866)(0.5,-0.176)
\pscircle[fillstyle=solid,fillcolor=white](0.5,-0.176){0.04}
\put(-0.2,-0.03){$A$}
\put(1.06,-0.03){$B$}
\put(-0.47,-0.48){$D$}
\put(1.3,-0.48){$E$}
\put(0.57,0.83){$C$}
\put(0.44,-0.38){$F$}
\psset{unit=3cm}
\endpspicture}
\def\ProclusBisectionFigureTwo{%
\psset{unit=2cm}
\pspicture (2,0.9)(-1,-0.7)
\pspolygon[fillstyle=solid,fillcolor=yellow,linestyle=none]
(0,0)(0.5,0)(0.5,0.866)(1.25,-0.433)
\psdot(0,0)
\psdot(1,0)
\psdot(0.5,0.866)
\psdot(-0.25,-0.433)
\psdot(1.25,-0.433)
\qline(-0.25,-0.433)(0.5,0.866)
\qline(0.5,0.866)(1.25,-0.433)
\qline(1.25,-0.433)(0,0)
\qline(-0.25,-0.433)(1,0)
\qline(0,0)(1,0)
\psdot(0.5,-0.176)
\qline(0.5,0.866)(0.5,-0.176)
\pscircle[fillstyle=solid,fillcolor=white](0.5,0){0.04}
\put(-0.2,-0.03){$A$}
\put(1.06,-0.03){$B$}
\put(-0.47,-0.48){$D$}
\put(1.3,-0.48){$E$}
\put(0.57,0.83){$C$}
\put(0.44,-0.38){$F$}
\put(0.42,0.08){$M$}
\psset{unit=3cm}
\endpspicture}
\def\AlternateInteriorAnglesFigure{%
\pspolygon[fillstyle=solid,fillcolor=yellow](0.7,0.7)(1.3,0.7)(0.8,0.425)
\pspolygon[fillstyle=solid,fillcolor=yellow](0.9,0.15)(0.3,0.15)(0.8,0.425)
\pspicture(3.2, 0.75)
\qline(0.0,0.15)(2,0.15)
\qline(0.0,0.7)(2,0.7)
\psdot(0.7,0.7)
\put(0.67,0.78){$p$}
\psdot(0.9,0.15)
\put(0.88,0.04) {$q$}
\psdot(0.3,0.15)
\put(0.3,0.04){$s$}
\psdot(0.8,0.425)
\put(0.85,0.36){$t$}
\qline(0.3,0.15)(1.3,0.7)
\qline(0.7,0.7)(0.9,0.15)
\psdot(1.3,0.7)
\put(1.3,0.78) {$r$}
\put(-0.15,0.12) {$L$}
\put(-0.15,0.68) {$K$}
\endpspicture}
\def\FigureOneThirtyFiveColored{%
\psset{unit=1cm}
\pspicture(0,-0.5)(4,2.3)
\pspolygon[fillstyle=solid,fillcolor=yellow,linestyle=none]
(0,2)(0.5,0)(2.5,2)
\pspolygon[fillstyle=solid,fillcolor=yellow,linestyle=none]
(2,2)(4.5,2)(2.5,0)
\pspolygon[fillstyle=solid,fillcolor=red,linestyle=none]
(2,2)(2.5,2)(2.1,1.6)
\psdot(0.5,0)
\put(0.4,-0.35){$B$}
\psdot(0,2)
\put(-0.1,2.15){$A$}
\psdot(2,2)
\put(1.85,2.15){$D$}
\psdot(2.5,2)
\put(2.36,2.15){$E$}
\psdot(4.5,2)
\put(4.3,2.15){$F$}
\psdot(2.5,0)
\put(2.3,-0.35){$C$}
\qline(0,2)(4.5,2)
\qline(0,2)(0.5,0)
\qline(0.5,0)(2.5,2)
\qline(2,2)(2.5,0)
\qline(4.5,2)(2.5,0)
\qline(0.5,0)(2.5,0)
\psdot(2.1,1.6)
\put(2.24,1.47){$G$}
\endpspicture
}
\def\ProclusPostulateFourFigure{%
\pspicture(2, 0.8)(0,-0.06)
\qline(0,0.1)(1,0.1)
\qline(0.5,0.1)(0.5,0.75)
\qline(0,0)(1,0.2)
\qline(1.5,0.1)(2,0.1)
\qline(1.5,0.1)(1.5,0.75)
\put(-0.15,0.1){$H$}
\put(-0.15,-0.05){$K$}
\put(0.45,-0.02){$B$}
\put(0.95,-0.02){$C$}
\put(0.38,0.7){$A$}
\put(1.37,0.7){$D$}
\put(1.42,-0.02){$E$}
\put(1.92,-0.02){$F$}
\put(0.95, 0.23){$G$}
\endpspicture}
\def\OneSixteenInnerFigure{%
\pspicture(0,-0.05)(2,0.75)
\pspolygon[fillstyle=solid,fillcolor=yellow](1.8,0)(0,0)(2,0.8)(1.4,0.2)(1.8,0)
\pspolygon[fillstyle=solid,fillcolor=lightblue](0,0)(2,0.8)(1.2,0)(0,0)
\psline[linecolor=red](1,0.4)(1.8,0)
\psline(0,0)(1.8,0)
\qline(0,0)(0.8,0.8)
\qline(0.8,0.8)(1.2,0)
\psdot(0.8,0.8)
\put(0.65,0.77){$A$}
\psdot(1.2,0)
\put(1.17,-0.13){$C$}
\psdot(1.8,0)
\put(1.75,-0.13){$D$}
\psline[linecolor=blue](1.2,0)(2 ,0.8)
\psline[linecolor=blue](0,0)(1.19,0.477)
\psline[linecolor=blue](1.21,0.483)(2,0.8)
\psdot(1,0.4)
\put(0.83,0.4){$E$}
\psdot(2,0.8)
\put(2,0.67){$F$}
\pscircle[fillstyle=solid,fillcolor=white](1.4,0.2){0.03}
\put(1.5,0.19){$H$}
\psdot(0,0)
\put(-0.04,-0.13){$B$}
\endpspicture}
\def\OneSixteenOuterFigure{%
\pspicture(0,-0.15)(2,0.85)
\pspolygon[fillstyle=solid,fillcolor=yellow](0.8,0.8)(1.8,0)(0,0)(1,0.4)(0.8,0.8)
\pspolygon[fillstyle=solid,fillcolor=lightblue](0.8,0.8)(1.8,0)(1.2,0)(0.8,0.8)
\psline[linecolor=red](0.8,0.8)(1.17,0.50)
\psline[linecolor=red](1.23,0.46)(1.8,0)
\psline(0,0)(1.8,0)
\qline(0,0)(0.8,0.8)
\qline(0.8,0.8)(1.2,0)
\psdot(0.8,0.8)
\put(0.65,0.77){$A$}
\psdot(1.2,0)
\put(1.17,-0.13){$C$}
\psdot(1.8,0)
\put(1.75,-0.13){$D$}
\psline[linecolor=blue](1.2,0)(2 ,0.8)
\psline[linecolor=blue](0,0)(1.19,0.477)
\psline[linecolor=blue](1.21,0.483)(2,0.8)
\psdot(1,0.4)
\put(0.83,0.4){$E$}
\pscircle[fillstyle=solid,fillcolor=white](1.2,0.48){0.03}
\put(1.17,0.55){$J$}
\psdot(2,0.8)
\put(2,0.67){$F$}
\psdot(0,0)
\put(-0.04,-0.13){$B$}
\endpspicture}
\def\OneSixteenFigureTwo{%
\pspicture(0,-0.15)(2,0.75)
\pspolygon[fillstyle=solid,fillcolor=yellow](0.8,0.8)(1.2,0)(2,0.8)(1.2,0.48)(0.8,0.8)
\pspolygon[fillstyle=solid,fillcolor=lightblue](1.2,0)(2,0.8)(1,0.4)
\psline[linecolor=red](0.8,0.8)(1.8,0)
\psline(0,0)(1.8,0)
\qline(0,0)(0.8,0.8)
\qline(0.8,0.8)(1.2,0)
\psdot(0.8,0.8)
\put(0.65,0.77){$A$}
\psdot(1.2,0)
\put(1.17,-0.13){$C$}
\psdot(1.8,0)
\put(1.75,-0.13){$D$}
\psline[linecolor=blue](1.2,0)(2 ,0.8)
\psline[linecolor=blue](0,0)(1.19,0.477)
\psline[linecolor=blue](1.21,0.483)(2,0.8)
\psdot(1,0.4)
\put(0.83,0.4){$E$}
\psdot(1.2,0.48)
\put(1.17,0.55){$J$}
\psdot(2,0.8)
\put(2,0.67){$F$}
\pscircle[fillstyle=solid,fillcolor=white](1.47,0.27){0.03}
\put(1.54,0.24){$H$}
\psdot(0,0)
\put(-0.04,-0.13){$B$}
\endpspicture}
\def\FigureParallelDefFour{%
\psset{unit=2cm}
\pspicture(2,1.4)
\psline[linecolor=red](0.42,1.185)(0.935,0.715)
\psline[linecolor=red](0.965,0.685)(1.485,0.215)
\psline[linecolor=red](0.52,0.22)(0.935,0.685)
\psline[linecolor=red](0.965,0.71)(1.4,1.2)
\qline(0,0.2)(0.47,0.2)
\qline(0.53,0.2)(1.47,0.2)
\qline(1.53,0.2)(2,0.2)
\qline(0,1.2)(0.37,1.2)
\qline(0.43,1.2)(1.37,1.2)
\qline(1.43,1.2)(2,1.2)
\psdot(0,0.2)
\put(-0.1,0.02){$C$}
\pscircle[fillstyle=solid,fillcolor=white](0.5,0.2){0.03}
\put(0.45,0.02){$P$}
\pscircle[fillstyle=solid,fillcolor=white](1.5,0.2){0.03}
\put(1.45,0.02){$Q$}
\pscircle[fillstyle=solid,fillcolor=white](0.4,1.2){0.03}
\put(0.35,1.27){$R$}
\pscircle[fillstyle=solid,fillcolor=white](1.4,1.2){0.03}
\put(1.35,1.27){$S$}
\psdot(0,1.2)
\put(-0.1,1.27){$A$}
\psdot(2,0.2)
\put(1.95,0.02){$D$}
\psdot(2,1.2)
\put(1.95,1.27){$B$}
\pscircle[fillstyle=solid,fillcolor=white](0.95,0.7){0.03}
\put(0.88,0.8){$H$}
\endpspicture
\psset{unit=3cm}}
\def\ConnectivityFigure{%
\pspicture(2,0.6)
\psdot(0,0.25)
\psdot(2,0.25)
\psdot(1,0)
\psdot(1,0.5)
\qline(0,0.25)(0.2,0.25)
\psbezier(0.2,0.25)(0.4,0.25)(0.7,0.5)(1,0.5)
\psbezier(1,0.5)(1.3,0.5)(1.6,0.25)(1.8,0.25)
\qline(1.8,0.25)(2,0.25)
\psbezier(0.2,0.25)(0.4,0.25)(0.7,0)(1,0)
\psbezier(1,0)(1.3,0)(1.6,0.25)(1.8,0.25)
\put(-0.05,0.12){$A$}
\put(0.95,0.55){$B$}
\put(0.95,0.05){$C$}
\put(2.02,0.17){$D$}
\endpspicture}
\def\PizzaFigure{%
\pspicture(0.6,0.15)(0,-0.2)
\SpecialCoor
\pswedge[fillstyle=solid,fillcolor=lightblue,linestyle=none](0,0){0.3}{-30}{0}
\pswedge[fillstyle=solid,fillcolor=lightblue,linestyle=none](0,0){0.3}{15}{45}
\pswedge[fillstyle=solid,fillcolor=red,linestyle=none](0,0){0.3}{0}{15}
\pswedge[fillstyle=solid,fillcolor=red,linestyle=none](0,0){0.3}{-45}{-30}
\psarc(0,0){0.3}{-45}{45}
\qline(0,0)(0.3;0)
\qline(0,0)(0.3;15)
\qline(0,0)(0.3;45)
\qline(0,0)(0.3;-30)
\qline(0,0)(0.3;-45)
\endpspicture}
\def\PizzaFigureTwo{%
\pspicture(0.6,0.15)(0,-0.2)
\SpecialCoor
\pswedge[fillstyle=solid,fillcolor=lightblue,linestyle=none](0,0){0.3}{-15}{45}
\pswedge[fillstyle=solid,fillcolor=red,linestyle=none](0,0){0.3}{-30}{-15}
\pswedge[fillstyle=solid,fillcolor=red,linestyle=none](0,0){0.3}{-45}{-30}
\psarc(0,0){0.3}{-45}{45}
\qline(0,0)(0.3;45)
\qline(0,0)(0.3;15)
\qline(0,0)(0.3;-15)
\qline(0,0)(0.3;-30)
\qline(0,0)(0.3;-45)
\endpspicture}
\def\AngleEqualityFigureOne{%
\pspicture(0.5,0.4)(-0.5,0)
\SpecialCoor
\pspolygon[fillstyle=solid,fillcolor=lightblue,linestyle=none](0,0)(0.3;60)(0.3;120)
\qline(0,0)(0.5;60)
\qline(0,0)(0.5;120)
\psdot(0.5;60)
\psdot(0.5;120)
\psdot(0.3;60)
\psdot(0.3;120)
\psdot(0,0)
\qline(0.3;60)(0.3;120)
\put(-0.38,0.4){$A$}
\put(0.29,0.4){$C$}
\put(-0.13,-0.05){$B$}
\put(-0.27,0.2){$U$}
\put(0.18,0.2){$V$}
\endpspicture}
\def\AngleEqualityFigureTwo{%
\pspicture(0.5,0.4)(-0.5,0)
\SpecialCoor
\pspolygon[fillstyle=solid,fillcolor=lightblue,linestyle=none](0,0)(0.3;60)(0.3;120)
\qline(0,0)(0.4;60)
\qline(0,0)(0.4;120)
\psdot(0.4;60)
\psdot(0.4;120)
\psdot(0.3;60)
\psdot(0.3;120)
\psdot(0,0)
\qline(0.3;60)(0.3;120)
\qline(0.3;60)(0.3;120)
\put(-0.3,0.32){$a$}
\put(0.24,0.32){$c$}
\put(-0.09,-0.05){$b$}
\put(-0.27,0.2){$u$}
\put(0.18,0.2){$v$}
\endpspicture}
\def\AngleOrderFigure{%
\pspicture(0.5,0.4)(-0.5,0)
\SpecialCoor
\pspolygon[fillstyle=solid,fillcolor=lightblue,linestyle=none](0,0)(0.3;60)(0.3;120)
\qline(0,0)(0.45;60)
\qline(0,0)(0.5;120)
\qline(0,0)(0.75;30)
\psdot(0.5;30)
\psdot(0.46;60)
\psdot(0.75;30)
\psdot(0.5;120)
\psdot(0.4;120)
\psdot(0,0)
\qline(0.3;60)(0.3;120)
\put(-0.38,0.4){$J$}
\put(0.67,0.32){$K$}
\put(0.29,0.4){$X$}
\put(-0.33,0.3){$P$}
\put(-0.13,-0.05){$Q$}
\put(0.47,0.17){$R$}
\psline[linecolor=red](0.5;120)(0.75;30)
\endpspicture}
\def\OneTwentyFigure{%
\pspicture(-0.2,-0.2)(0.5,0.5)
\SpecialCoor
\pspolygon[fillstyle=solid,fillcolor=lightblue,linestyle=none](0,0)(0.5;-20)(0.25;240)
\qline(0,0)(0.5;60)
\qline(0,0)(0.5;-20)
\qline(0.5;60)(0.5;-20)
\qline(0,0)(0.25;240)
\qline(0.5;-20)(0.25;240)
\psdot(0,0)
\psdot(0.5;-20)
\psdot(0.25;240)
\psdot(0.5;60)
\put(-0.15,0){$A$}
\put(-0.28,-0.23){$B$}
\put(0.3,0.4){$D$}
\put(0.5,-0.25){$C$}
\endpspicture}
\title{Euclid after Computer Proof-Checking}
\markright{Euclid after Computer Proof-Checking}
\author{Michael Beeson}
\date{\today
}
\begin{document}
\maketitle
\begin{abstract}Euclid pioneered the concept of a mathematical
theory developed from axioms by a series of justified proof steps.
From the outset there were critics and improvers. In this century
the use of computers to check proofs for correctness sets a new
standard of rigor. How does Euclid stand up under such an examination?
And what does the exercise have to teach us about geometry,
mathematical foundations, and the relation of logic to truth?
\end{abstract}
\section{Introduction.} In approximately 300 BCE,
a research institute known as the {\em Museum}
was founded at Alexandria. A community of scholars lived
and worked at the Museum, holding property in common and having
a common dining room.%
\footnote{According to Strabo's description
(\cite{strabo}, Book~XVII,Chapter~ I).
See also see \cite[pp.~30,38]{sarton2}. }
Euclid joined this community,
and became the author of several
influential books.%
The most famous of these is his
{\em Elements}.%
\footnote{The story of how the {\em Elements} survived
until modern times is fascinating but not strictly
relevant to our present topic; for the Library of Alexandria see \cite{charlesriver,el-abbadi2020,
sarton2, strabo}; for its travels to India,
see \cite[Chapters 7 and 8]{oleary} and \cite[Book~XVII, Chapter 8]{strabo}.
For the Arabic flowering and return to Europe, see \cite{oleary} and \cite[p.~595]{deRisi2016b}.
For its printing in 1482 see \cite[p.~364]{heath-vol1}.
}
No other scientific book has had
an equal influence on the world.
The authors of the United States Declaration of Independence had
studied Euclid: hence the ``self--evident truths'' from which
the principles of the Declaration were deduced in Euclidean style.
Abraham Lincoln took several months studying Euclid%
\footnote{See \cite[pp.~64-65]{ketcham} and \cite[Chapter 14]{hirsch}.}
and thus learned what it means to demonstrate a proposition in court.
He put it to good use in the Lincoln--Douglas debates.
Bertrand Russell studied Euclid at age eleven and ``did not know there
was anything so delicious in the world.''%
\footnote{See \cite{russell-autobiography} (Prologue). But by 1902, when Russell was thirty, he
had changed his mind, and published an article pointing out errors
in Euclid~ I.1, I.4, I.7, and I.16 \cite{russell1902}.}
From at least 450 CE (and probably before that!) there was no shortage
of critics, and suggestions for repair and improvement. Dozens of
these have been surveyed by De Risi \cite{deRisi2016b}. Proclus already
criticized the parallel postulate (Euclid 5), saying that it did not
deserve to be a postulate but should be proved, since we know that some
kinds of (curved) lines can approach each other without meeting, so
why can't straight lines do that too? In the nineteenth
century there was an increasing focus on the problem of whether the
parallel postulate (Euclid 5) could be eliminated, by proving it from
the other axioms. Several famous mathematicians thought they had
done so; Legendre published three different mistaken proofs. This
effort focused attention on rigor and careful deduction, and
led to careful axiomatic developments. At the same time, logic
was also being developed; Boole's {\em Laws of Thought} was
published in 1853. A milestone was reached with Pasch's work
in 1882 \cite{pasch1882}, which introduced the notion of ``betweenness'' missing
in Euclid, and the idea that one should justify the existence of
points where lines cross lines or circles, or circles cross other circles.
The geometers eventually realized that there is such a thing as
non-Euclidean geometry, in which Euclid 5 fails but the other postulates
hold. This was the world's first independence proof. The efforts
to codify systems of deduction, stimulated by the pressing need to
ensure correctness in geometry, led directly to the logical systems
of Peano
and Frege at the end of the nineteenth century,
which are the wellsprings of modern logic. Geometry was the
midwife of logic.
In 1899, Hilbert published his very influential book \cite{hilbert1899}, in which he attempted to
bring new standards of rigor to geometry.
Hilbert's system mixed first-order and second-order
axioms with a little set theory thrown in for good measure, but
he clearly understood that axioms could be dependent or independent
and a given axiom system might have different models. Hence his
famous dictum ``tables, chairs, and beer mugs,'' the point of which was
that reasoning should
be fundamentally syntactic, so that the steps could be checked independently
of the meaning of the terms; hence if you substitute ``tables, chairs, and
beer mugs'' everywhere for ``points, lines, and planes,'' then everything
should still be correct.
Hilbert's book was about plane geometry, but he did not follow
Euclid. His aim (as stated in the Introduction to \cite{hilbert1899})
was to ``establish for geometry a complete, and as simple as possible,
set of axioms.'' His method to achieve that goal was to show that
arithmetic (addition and multiplication) can be defined geometrically.
This was first done more than two centuries earlier, in Descartes's famous
{\em La G\'eom\'etrie}, but Descartes's work was not based on
specific axioms. The Greeks never tried to multiply two
lines to get another line; the product of two lines was a rectangle.
Descartes broke through this conceptual barrier, and Hilbert
made Descartes's arguments rigorous, using the theorems of the
nineteenth-century jewel ``projective geometry.''
Hilbert's book put in the mathematical bank the profits of the
discoveries about non-Euclidean geometry: geometry was no longer about
discovering the truth about points, lines, and planes, but instead it
was about what theorems follow from what axioms. Rather than deduce
Euclid's theorems directly from his axioms, Hilbert relied on the
indirect argument that every geometrical theorem could be derived by
analytic geometry, and since coordinates could be introduced by
the geometric definition of arithmetic, every theorem had a proof from
his axioms.
In 1926--27, Tarski lectured on his first-order theory of geometry.%
\footnote{According to \cite[p.~175]{tarski-givant}. This reached
journal publication only in \cite{tarski1959}. }
But Tarski and his collaborators too ignored Euclid, focusing as Hilbert did on the development of geometric arithmetic and the characterization of models.
Less than a decade after the first electronic computers were
available, there were efforts to apply them to the problem of finding
proofs of geometry theorems (\cite{gelernter1959, gelernter1960}). Later, the verification
of geometry theorems by algebraic calculation became an advanced art form.
(See the introduction to \cite{beeson2019} for citations and history). On the other hand, since the 1990s there has been a body of
work in ``proof-checking,'' which means that a computer program
checks that the reasoning of a given proof is correct (as opposed to
``automated deduction,'' in which the computer is supposed to find the
proof by itself). In recent years there have been several high-profile
cases of proof-checking important theorems whose large proofs involved
many cases. Tarski's work on geometry, as presented in \cite{schwabhauser},
has also been the subject of proof-checking and proof-finding experiments
\cite{beeson2017a,boutryphd,narboux2017b,narboux2017c,narboux2017}.
Until 2017, nobody had tried to proof-check Euclid directly.
That omission was remedied in \cite{beeson2019}.
We produced formal proofs of 245
propositions, including the 48 propositions of Euclid Book I, and
numerous other propositions that were needed along the way, and
translated those proofs into the languages of
two famous proof-checkers, Coq and HOL Light.
The purpose of this article is to consider what is to be learned from
that proof-checking experiment. That work invites the
question, {\em How wrong was Euclid?} That
question will be answered below. Here we consider as well:
\begin{itemize}
\item {\em What was Euclid thinking?}
\item {\em What is the relation between logic and truth?}
\item {\em How should we choose the primitive notions for a formalization?}
\item {\em What level of mathematical infrastructure should be visible?}
\end{itemize}
Relevant to those questions are the following principles of
mathematical practice (which are sometimes followed and sometimes not):
\begin{itemize}
\item {\em If it can be defined, instead of taken primitive, it should be defined.}
\item {\em If it can be proved, instead of assumed, it should be proved.}
\end{itemize}
Tarski clearly believed in both these principles: he famously sought to
minimize the number of primitives and the number of axioms, and
tried to find axioms that could be stated in the primitive language without
preliminary definitions. Hilbert evidently wasn't such a strong
believer; his axioms were stronger by far than they had to be,
because
he wanted to reach his goal of defining arithmetic as quickly as possible.
Of course, it was the second of those principles that drove people to
work so hard trying to prove Euclid~5 from the other axioms, and
over the centuries there were many claims that Euclid~4 (all right
angles are equal) can be proved (this will be discussed below).
I have come to think of the proof-checking of Euclid as the
enterprise of {\em providing infrastructure}. Euclid's proofs
are like mountain railroads: there is a clear origin and destination,
but there are a lot of possibilities to go off the rails. There are
places in Euclid where the train would fall into the abyss; there are
other places where the trestles are very weak. After \cite{beeson2019},
we have 245 propositions to support Euclid's 48. It is a much
sturdier system. Here I will discuss it and consider its
relation to Euclid's {\em Elements}, Book~I, which it is meant to formalize.
\section{Structure of Euclid's {\em Elements}.}
Euclid had {\em definitions}, {\em common notions}, {\em axioms}, and
{\em postulates}.
Nowadays, the common notions,
axioms, and postulates would be lumped together and considered axioms.
In Euclid, the common notions were intended to be principles of
reasoning that applied more generally than just to geometry.
For example, what
we would now call equality axioms, or such principles as
``the part is not equal to the whole'' or ``the whole is equal to
the sum of the parts.'' Any kind of thing might have parts, which were
always the same kind of thing: the parts of a line were smaller lines, etc.
The axioms and postulates were about geometry.
The distinction between an ``axiom'' and a ``postulate,'' according
to Proclus \cite[\S 201, p.~157]{proclus}, who attributes the distinction
to the earlier mathematician Geminus, is that a postulate asserts that some point can be constructed,
while an axiom does not. In modern terms, an ``axiom'' is purely universal,
while a postulate has an existential quantifier. Euclid's Postulate~4
(all right angles are equal) had no associated construction, and for that
reason, critics in antiquity (such as Proclus, in \cite[\S 192, p.~150]{proclus} said it did not deserve
the name ``postulate.''
Heath's translation lists five common notions, five postulates, and zero
axioms. Simson's translation \cite{simson} lists three postulates, twelve axioms,
and zero common notions. The extra axioms are discussed by Heath \cite[p.~223]{euclid1956},
where they are rejected. During the centuries since Euclid, a great many
axioms have been put forward as ``should have been included.'' The
axiomatization of \cite{beeson2019},
discussed in this article, is thus the last elephant in a long parade.%
\footnote{See \cite{deRisi2016b} for a thorough discussion. In that
paper, it takes De Risi twenty pages just to list the different axioms
that various authors have added to Euclid's.
}
But the
seal of correctness given by computer proof-checking does support the
claim to be the ``last elephant.'' That axiomatization followed the consensus of the
centuries on these points: Postulate~4 should be proved, the SAS criterion of
Proposition~I.4 should be an axiom, and Postulate~5 has to remain.
It also followed the consensus of the twentieth century that the concept
of betweenness is needed, and it adopted axioms similar to those of
Hilbert and Tarski about betweenness.
\section{Lines.}
Euclid's lines were all finite lines; a line could be extended, but
after the extension it was still a finite line, just longer.%
\footnote{Nowadays lines are infinite; Euclid's (finite) lines are
today's ``segments.'' A ``line'' for Euclid could also be curved; hence
``straight line'' adds something. However, the default in Euclid, though
not necessarily in all Greek geometry, is that lines are straight.
We follow Euclid.}
How far
could it be extended? Later Hilbert and Tarski both answered that question
by taking it as an axiom that $AB$ could be extended by an amount equal to a given
line $CD$. This is sometimes known as the {\em rigid compass}: we can
measure off $CD$ with a compass, then move the compass to the end of $AB$
and lay off the measured amount along a straightedge. By contrast,
Euclid used a {\em collapsible compass}: you could only extend a line
by putting one tip of the compass on $B$ and the other tip on some
existing point, then drawing a circle; then $AB$ would be
``drawn through'' to meet that circle. Euclid's second proposition,
I.2, gives a beautiful proof that the rigid compass can be simulated
by a collapsible compass; that conclusion is axiomatized away by
Hilbert and Tarski, violating the principle ``if it can be proved, it
should be proved.''%
\footnote{A referee pointed out \cite{rigby1970}, which gives an axiomatic
framework for a part of geometry in which the collapsible compass is preserved.}
In Euclid's time it had not yet been realized that
not everything can be defined: certain notions must be taken as primitive,
because otherwise there will be nothing to use in the first definition.
Euclid defined a line as that which has length but no breadth. Other
sources make clear that the Greeks regarded lines (curved or straight) as
the traces of a motion.%
\footnote{This view of lines goes back at least to
Aristotle; see \cite[p.~79]{proclus}, where Proclus says Aristotle
regarded a line as ``the flowing of a point.''}
In particular, most modern mathematicians,
steeped in set theory from their youth, think that a line is equal to
the set of its points. This was definitely not the Greek conception.
This viewpoint was regarded as thoroughly refuted by the paradoxes
of Zeno, which were already centuries old by the time of Euclid.
These paradoxes depended on the conception that, if a line were made up
of its points, the points would be like tiny beads on a string.
In mathematics there is another principle: {\em Perhaps you do not need to
know what it is, if you know how to use it.} Euclid never once appeals
to his definition of a line, so in essence he did treat it as a primitive
notion. He always refers to a line by two distinct points, as in ``line $AB$.''
You never see ``line $\ell$\,'' in Euclid. Lines are used to construct other
points, as the intersection points of lines with other lines or with circles.
Euclid spoke of two lines being ``equal.'' He meant by this what
Hilbert called ``congruent.'' Probably he had in mind that a rigid motion
could move one line to coincide with the other. It seems certain that
he did not mean that they had the same length, as measured by a number.
The other obvious primitive notion about lines (besides equality)
is the notion ``point $P$ lies on line $AB$.''
This is the notion of {\em betweenness}, which was explicitly axiomatized
by Pasch only in 1882. (One can either use {\em strict} betweenness, as
Hilbert did, or nonstrict betweenness, as Tarski did.) Both Hilbert and
Tarski had some axioms about betweenness. Euclid, on the other hand,
had none.
Euclid did use the notion that one line is ``less than'' or ``greater than''
another. That notion can be defined in terms of betweenness and equality:
$AB < CD$ if there is some $X$ between $C$ and $D$ such that $AB$ is equal
to $CX$. Conversely, we could define (for collinear points)
$X$ is between $C$ and $D$ if $CX < CD$ and $XD < CD$.
For Euclid, ``less than'' was a natural notion, because $AB < CD$ meant
that $AB$ was equal to a part of $CD$. The notion of one thing being
part of another thing was regarded as a ``common notion,'' i.e., a notion
that applied to things in general, as opposed to notions specific to geometry.
Euclid took ``part'' as a primitive notion, making no attempt to define it,
but it was clear that the parts of a thing were the same kind of thing: the parts of a line were lines, the parts of an angle were angles. The definition of
{\em point} is {\em that which has no parts}. The common notions probably
seemed completely precise to Euclid, because he felt that the notion of ``part''
was clear. From the modern point of view, there are questions: if $AB$ is
divided into two parts by its midpoint $M$, what about point $M$ itself?
Is it in both parts or neither part? Is the whole $AB$ really the ``sum'' of
$AM$ and $MB$? This problem comes back ``in spades'' when we divide a
triangle into two parts: what about the separating line? When we put
the parts back together, there would be no cut remaining.
We chose to follow Pasch, Hilbert, and Tarski in using betweenness as a
primitive notion; we followed Hilbert in using strict betweenness, because
Euclid has no ``null lines'': when he says ``line $AB$,'' $A$ and $B$ are
always distinct points. Of course, nonstrict betweenness is easily
defined in terms of strict betweenness, so it is really inconsequential which
is taken as primitive. We write $\B abc$ for ``$b$ is between $a$ and $c$.''
The three betweenness axioms and their names are given here:
\begin{center}
\begin{tabular}{l l}
symmetry & $\B abc \leftrightarrow \B cba$ \\
identity & $\neg\, \B aba$ \\
inner transitivity & $\B abd \ \land \ \B bcd \rightarrow \B abc$
\end{tabular}
\end{center}
It turns out that the theory of betweenness is not quite trivial. There are
delicate and interesting questions about the axiomatization of this notion.
Tarski's original version of his theory had 16 axioms. Working with
his students, in 1956--57 several of these axioms were proved
dependent on others. That the three axioms given above are enough is
remarkable, but it is not very important for formalizing Euclid,
since
Euclid never even mentioned betweenness. We must add the required
infrastructure, but it won't matter exactly how we do it, so we do not
go further into the matter here.%
\footnote{See \cite{tarski-givant} for a complete history of Tarski's
axiom systems, including the discoveries of dependencies among the axioms.}
If five betweenness axioms had been required instead of three, we would have
just added five betweenness axioms.
The axiomatization of \cite{beeson2019} supplemented Euclid's
axioms, as given in the Heath translation, with the axiom the authors called {\em connectivity}: if $B$ and $C$ are
both between $A$ and $D$, but neither $\B BCD$ nor $\B CBD$,
then $B=C$. In other words, the picture in Figure~\ref{figure:connectivity}
is impossible. This axiom is found in several of the Greek manuscripts
used by Heiberg in preparing his influential translation.%
\footnote{See \cite[p.~638]{deRisi2016b}. One of these manuscripts
is the oldest copy of Euclid to survive into the modern age.}
The ancients
did not use betweenness; they expressed this axiom as ``two lines intersect in at most one point,'' or as ``two lines cannot enclose space.'' The latter
lacks precision as ``space'' is not defined; the former is equivalent to
our axiom. Zeno of Sidon attacked Proposition~I.1 on the grounds that
it is not conclusive unless it first be assumed that neither two straight lines
nor two circumferences can have a common segment.%
\footnote{See \cite[p.~359]{heath-vol1}, who in turn cites Proclus.}
\begin{figure}[ht]
\caption{The connectivity axiom: two lines cannot enclose a space.}
\label{figure:connectivity}
\center{\ConnectivityFigure}
\end{figure}
The connectivity axiom is used to prove the uniqueness of the midpoint,
and the basic property of collinearity, if $a$, $b$, and $c$ are collinear
and $a$,$b$, and $d$ are collinear,
and $b \neq c$ and $a \neq b$, then $b$, $c$, and $d$ are collinear.
That fact is used in almost
every one of our formal proofs.
How did Euclid get by without that axiom?
By not proving the uniqueness of the midpoint, and using the properties
of collinearity without explicit mention.
If we had adhered to the principle, ``If it can be proved, it should
be proved and not assumed,'' we would have formalized Gupta's
1965 proof of the connectivity axiom, which was proved in
his thesis \cite{gupta1965} and can more easily be found as Satz~5.3 in
\cite{schwabhauser}. But that proof, with its elaborate counterfactual
diagram, is not ``in the spirit of Euclid,'' so we chose to include
the axiom of connectivity, which so many over the centuries have thought
should be included.%
\footnote{Hilbert smuggled that axiom into his system by requiring
uniqueness in his angle-copying axiom. Pasch had it explicitly
as an axiom, Kernsatz~V, \cite[p.~5]{pasch1882}.
}
Collinearity, which is defined from betweenness by enumerating the cases,
plays a larger-than-life role in our computer formalization; mostly
in the form of noncollinearity. There are many cases in which to
state a theorem formally, one must add to the hypothesis statements
like ``$ABC$ is a triangle.'' Since we define a triangle as three
noncollinear points, this amounts to ``$A$, $B$, and $C$ are not collinear.''
More than
half the individual inferences in our formalization turned out to
be statements of collinearity or noncollinearity. These statements
are ``pure infrastructure,'' absolutely necessary to prevent the
theorems from collapsing into the abyss, but absent in Euclid,
and serving only to ensure that the diagram does not degenerate.
Euclid simply assumed that points that appear to be on a line are
indeed on that line, and points that appear noncollinear, are noncollinear.
\section{Angles.} \label{section:angles}
The Greek concept of angle was more general than the modern concept,
as we see from Euclid's definitions of {\em planar angle} (the inclination
to one another of two lines in a plane that meet one another and do not
lie in a straight line), from which we see that one could also consider
angles that do not lie in a plane; and of {\em rectilinear angle},
when the lines are straight lines. In particular, two touching
circles form an angle that is not rectilinear; modern mathematics does
not use the word ``angle'' in that situation. Neither, as it turns out,
does most of Euclid; and here we use ``angle'' for ``rectilineal angle.''
Euclid always refers to angles by three points, as in ``angle $ABC$,''
never using the more modern notation ``angle $\alpha$.'' As with lines,
he uses the word ``equal'' instead of Hilbert's ``congruent.''
As with lines, Euclid never once appeals to the definition of ``angle,''
so we must ask how angles are used, rather than what they are.
It turns out that (in addition to the equality relation) there are
relations of ``less than'' and ``greater than'' between angles,
and two angles can be ``taken together,'' or sometimes ``added,''
in such a way that the common notions ``if equals are added to equals
the results are equal,'' and ``the whole is equal to the sum of its parts''
are applicable to angles. In effect, then, Euclid treats angles
as a primitive notion, with an ordering relation. One consequence of
Euclid's definition is that all his angles are ``less than two right angles,''
or in modern terminology, less than 180 degrees. For otherwise
angle $ABC$ would not be determined, unless we were to insist that
angle $ABC$ is not the same as angle $CBA$; but Euclid clearly considers
$ABC$ equal to $CBA$.
Although Euclid refers to angles by the names of three points, sometimes
we have to consider different ways of choosing those three points.
To give a specific example, consider angles $BAE$ and $BAC$, where
$E$ lies between $A$ and $C$.
(See Figure~\ref{figure:OneSixteenOne}). In the proof of I.16, Euclid
implicitly equates those angles. Did he consider them to be
the {\em same}
angle (identical), or merely {\em equal} angles? This question
cannot be answered by reading Euclid, since he never explained what he
meant by ``equality.'' In practice, it isn't going
to matter whether we consider them to be equal angles,
or are two names for the same angle. The same missing steps will have
to be supplied.
Euclid never mentions ``rays,'' because all his lines are finite.
Therefore Euclid's angles all have finite (but extensible)
sides. Nevertheless the language of ``rays'' is useful; we say
$x$ lies on the ray $AB$ if $\B ABx$ or $x = B$ or $\B AxB$.
At some point it was realized that angles could just be {\em defined}
as triples of points.%
\footnote{
As far as I can determine, Mollerup \cite{mollerup1904} deserves the credit
for these definitions.
}
Angles then are a case in point for the principle, ``if it can be defined,
it should be defined.'' In our computer proof-checking of Euclid, we
used Tarski's points-only language, so lines became pairs of points,
and angles became triples of points.
Then the notions of
equal angles and ``less than'' for angles can also be defined.
See Figure~\ref{figure:angles}, in which the blue triangles are congruent.
\begin{figure}[ht]
\caption{Definitions of angle equality and angle less than.}
\label{figure:angles}
\center{\AngleEqualityFigureOne = \AngleEqualityFigureTwo
$<$ \AngleOrderFigure}
\end{figure}
Angles $ABC$ and $abc$ are equal if
there exist points $U$, $V$, $u$, and $v$ on rays $BA$, $BC$, $ba$, and $bc$
respectively, such that $BU = bu$ and $BV = bv$ and $UV = uv$.
And angle $abc$ is less than $PQR$ if there are $X,J,K$ with
$\B JXK$ and $J$ on ray $QP$ and $K$ on ray $QR$
such that $abc = PQX$.
If we view formalization as providing infrastructure, there is a lot
of infrastructure connected with equality and inequality of angles.
To start with, we must verify that angle equality is an equivalence
relation; that requires Euclid I.4, the SAS congruence criterion,
which is discussed in the next section. This particular piece of
infrastructure results from having to prove what Euclid took as
``common notions''. But not all the required angle infrastructure
is of that nature.
For example, in the proof of Euclid~III.20 there is
an unjustified step; I mean by ``unjustified'' that
Euclid did not write any justification for it, in the sense of a
reference to an earlier proposition or definition. To justify it,
he would have had to prove a proposition something like ``the sum
of the doubles is the double of the sum,'' or more explicitly,
if two angles are doubled, then their doubles taken together equal
the double of the angles taken together. In more modern terms,
if we call the angles $1$ and $2$, twice the sum of angles $1$ and $2$
is the sum of twice angle $1$ and twice angle 2.
\begin{figure}[ht]
\caption{The sum of the doubles is the double of the sum.}
\label{figure:pizza}
\center{\PizzaFigureTwo \PizzaFigure}
\end{figure}
If someone had pressed Euclid on this point, he would have justified this
step by ``the whole is equal to the sum of the parts.'' If we have
four slices of pizza labeled 1,2,1,2 in order, and we take them
out of the box and then put them back in the order 1,1,2,2, lo and behold,
they fit into the original angle exactly. See Figure~\ref{figure:pizza}. Now try to prove it
by the methods of high school geometry. That is ``infrastructure.''
You see from this example that it is logic and the choice of axioms that
give rise to the need for infrastructure. Computer proof-checking
only shines a light on the situation. You cannot convince the computer
by a story about pizza.
To formalize Euclid~III.20, one has to define addition and subtraction
of angles; Euclid thought that ``taken together'' was clear enough without
a definition. It was a common notion, applying to any kind of ``thing,''
geometric or not. This was not an ``operation'' in the modern sense, so
the questions of commutativity and associativity were not considered.
But they now require proof. Since Euclid had no notation for addition
(of any kind of object), his notation was completely relational; that is,
he had to say $ABC$ and $abc$ taken together are equal to $PQR$, since
he could not say $ABC + abc$. Algebraic (functional) notation for
addition and multiplication was introduced some 1800 years
after Euclid.
To discuss the sum of four angles with different order and associativity
is extremely awkward in Euclid's notation, which was, of course, reflected
in our formal notation since we purposely tried to match our notation to
Euclid's.
Whether we take angles as primitive (as Hilbert did) or defined (as we do,
and Tarski did) doesn't matter very much for the formalization of Euclid,
as if one takes angles as defined, then one proves their basic properties and
from then on things look pretty much the same as if angles were primitive.
The exact choice of axiom system is not philosophically or mathematically important
(though it might make certain proofs easier or harder); what is important is the complete
precision of detailed proofs.
\section{The SAS congruence theorem and the 5-segment axiom.}
Euclid attempted, in Proposition I.4, to prove the side-angle-side
criterion for angle congruence (SAS). But his ``proof'' appeals to
the invariance of triangles under rigid motions, about which there is
nothing in his axioms, so for centuries it has been recognized that
in effect SAS is an axiom, not a theorem.
Before discussing SAS, we discuss the notion of triangle congruence.
Euclid does not define either ``triangle'' or ``congruent triangles,''
but taking I.4 to define the SAS criterion, the conclusion includes
the pairwise equality of corresponding sides, and the pairwise equality
of corresponding angles. As we discussed in Section~\ref{section:angles},
equality of angles is taken as a defined notion, rather than primitive.
With the definitions given there, if two triangles have all pairs of
corresponding sides equal, they automatically have corresponding angles equal too.
Hence SAS needs only to mention the equality of corresponding sides in
its conclusion. Moreover, two of those pairs of sides are equal by hypothesis,
so the conclusion of SAS just needs to be one equality of lines. Next
we discuss how to formulate SAS without explicitly mentioning angles.
To do that, we use an axiom known as the ``five-line
axiom.'' This axiom is illustrated in Figure~\ref{figure:5-segment}.
The point of this axiom is to express SAS without mentioning angles at all.
To understand the relationship of SAS to the five-line axiom, let us
express Euclid I.4 (which is SAS) using Figure~\ref{figure:5-segment}.
The hypothesis is that $db = DB$, $dc=DC$, and angles $dbc$ and $DBC$
in Figure~\ref{figure:5-segment} are equal. The conclusion
is that $dc = DC$. The point of the 5-line axiom is to replace the
hypothesis ``angle $dbc$ and $DBC$ are equal'' by the hypothesis that
triangles $abd$ and
$ABD$ are congruent, i.e., $ab=AB$, $bd=BD$, and $ad=AD$. To rephrase the matter:
The hypothesis of the five-line axiom expresses the congruence (equality, in Euclid's
phrase) of angles $dbc$
and $DBC$ by means of the congruence of the exterior triangles $abd$ and
$ABD$.
It is not difficult to derive Euclid's I.4 from the five-line axiom. It is
also not difficult to derive the five-line axiom from I.4. So, it just
a choice whether to take the five-line axiom, or I.4, as an axiom, and
after deriving one from the other, it makes little difference to the
subsequent development. We chose to follow Tarski in using the five-line
axiom, since it can be stated succinctly using the primitive notions of the
language (without abbreviations).
\begin{figure}[ht]
\center{\TarskiFiveSegmentFigure}
\caption{If the four solid lines on the left are equal to the
corresponding solid lines on the right, then the dashed lines
are also equal.}
\label{figure:5-segment}
\end{figure}
\FloatBarrier
This version of the five-line axiom was introduced by Tarski,
although we have changed nonstrict betweenness to strict betweenness.%
\footnote{The history of this axiom is as follows.
The key idea (replacing reasoning about angles by reasoning
about congruence of segments) was
introduced (in 1904) by J. Mollerup \cite{mollerup1904}.
His system has an axiom closely related to the 5-line axiom,
and easily proved equivalent. Tarski's version \cite{tarski-givant}, however, is
slightly simpler in formulation. Mollerup (without comment)
gives a reference to
Veronese \cite{veronese1891}. Veronese does have a theorem
(on page 241) with the same diagram as the 5-line axiom, and
closely related, but he does not suggest an axiom related to this
diagram.}
\section{Line-circle and circle-circle continuity.}
Euclid's first proposition, Proposition~I.1,
constructs an equilateral triangle by drawing two circles
of radius $AB$ with centers at $A$ and $B$, respectively. A meeting point
of these circles is a point $C$ equidistant from $A$ and $B$. But
why do the circles meet? Euclid smuggles the point $C$ into the proof
by using a ``definite description'': ``the point $C$ at which the circles
cut one another.'' The modern consensus is that this is a case of
a ``missing axiom.'' We have to supply the {\em circle-circle continuity axiom},
according to which, if one circle has points inside and outside the other
circle, then the two circles meet.%
\footnote{De Risi \cite[p.~614]{deRisi2016b} gives credit to Richard
for first recognizing (in 1645) that an axiom would be required, though others
previously noted the gap in the proof of I.1, without filling it.}
The words {\em inside} and {\em outside}
are defined as follows: $X$ is inside a circle centered at $O$
if $OX$ is less than some
radius, and outside if $OX$ is greater than some radius. A radius is a line
connecting the center with a point on the circle.
There is also the { line-circle continuity axiom}, asserting that if
$A$ is inside a circle, and $P$ is any point different from $A$,
then there are two points collinear with $AP$ lying on the circle,
and one of them has $A$ between it and $P$. This axiom is needed
twice in Euclid Book~I, in I.2 and I.12, where the ``dropped perpendicular''
to a line from a point not on the line is constructed by drawing
a sufficiently large circle, which must meet the line in two points,
forming a line whose perpendicular bisector is the desired perpendicular.
These axioms are used seldom, but crucially, in Euclid. Specifically,
circle-circle is used in Euclid~I.1, which is the ``bootstrap'' proposition
for the first ten; it is used again in I.22, to construct a triangle out
of three given lines. And dropped perpendiculars, constructed by
line-circle, are of course fundamental.
In modern times it has been shown that, using only the other axioms,
line-circle implies circle-circle and vice versa. The only purely
geometric proof known of these facts makes use of the ``radical axis''
and is a little complicated; see \cite{hartshorne}. According to
the principle ``if it can be proved, it should be proved,'' we should
have just taken one of these axioms; but instead we took both
line-circle and circle-circle. Had we taken only one, we would have
not been proof-checking Euclid, but proof-checking the modern theorems
about the radical axis. That could clearly be done, but would not
have added anything significant.
There is another axiomatic question about line-circle and circle-circle.
They were mentioned by Tarski in his original paper, but not included
in his full axiom list; he probably thought they followed from his
Dedekind-style continuity schema (A11), which asserts that first-order
Dedekind cuts are filled. This seems plausible until you actually try
to prove it. But to do so you need to drop perpendiculars to a line,
and to do that you need, if you follow Euclid, circle-circle
intersection. So the argument is circular. This is fixed by
appealing to the 1965 thesis of Gupta \cite{gupta1965}, whose proofs
were finally published in \cite{schwabhauser}. Gupta showed, amazingly, how
to construct both dropped and erected perpendiculars
{\em without using circles at all}. So it does turn out to be correct
to omit circle-circle in favor of (A11), but Tarski certainly didn't
have a proof of that in 1927 or even 1959.
\section{Book Zero.}
It may surprise the reader when I say that even after the point $C$ in
Proposition~I.1 has been admitted to exist, there is yet another defect
in Euclid's proof. Namely, although we now know $AC=AB=BC$, in order
to prove $ABC$ is a triangle, we must also prove that $A$, $B$, and $C$
are not collinear. To prove this we used a lemma we called
{\em partnotequalwhole}: if $\B ABC$ then $AB \neq AC$.%
\footnote{The name of the lemma is taken from Euclid's Common Notion 5;
but Euclid does not cite CN5, in I.1 or anywhere else, and De Risi
\cite{deRisi2020} after careful study reaches the conclusion that the original Euclid
had only the first three of the five common notions given in the Heath
translation. It seems, however, that he {\em should}
have had CN5, and should have cited it in I.1.}
The proof
needs nine inferences, starting with Euclid's extension axiom to
extend $ABC$ to another point $D$ left of $A$, i.e., with $\B DAB$.
Then we need to
show that $DABC$ occur in that order, in particular $\B DAC$,
using one of the lemmas about
betweenness alluded to in the section on betweenness. These
lemmas are part of what we might call ``Book Zero''; Book Zero
contains infrastructure that is more fundamental than the propositions
of Book I. It consists of the betweenness lemmas, several lemmas with
a similar flavor to {\em partnotequalwhole}, and several trivial lemmas
that reflect the fact that we represented lines as pairs of points.
Thus if $AB=CD$, we also have $BA=CD$ and $AB=DC$ and $BA=DC$,
expressing the fact that these are unordered lines, not vectors.
The fundamental properties of collinearity and noncollinearity, which
are never mentioned in Euclid, should also be considered part of Book Zero.
\section{Pasch's axiom.}
Pasch \cite{pasch1882} not only introduced betweenness, but also
the axiom that later was given his name.%
\footnote{Specifically, Kernsatz~IV, \cite[p.~20]{pasch1882}.
Pasch called his axioms ``Kerns\"atze.'' The ``kernel'' of a theory
consisted of kernel concepts and kernel theorems, but Pasch had a
modern understanding of completeness and consistency, as p.~18 indicates.
}
Pasch's axiom (Figure~\ref{figure:InnerOuterPaschFigure})
says that if $ABC$ is a triangle, and line $DE$ lies in a
plane with $ABC$ and meets $AB$
in a point $F$ between $D$ and $E$, then $DE$ or an extension of $DE$
meets $AC$ or $BC$.%
\footnote{In fact the axiom was formulated two centuries earlier
by Roberval; see \cite[p.~632, axiom C18, and discussion
p.~615]{deRisi2016b}. One may wonder why it took two millenia
for this axiom to be formulated. See \cite{deRisi2019} for a
penetrating historical and philosophical discussion of that question.
In Pasch's statement, the first ``between'' here used ``innerhalb''
and so was strict; the second did not use ``innerhalb'' so was not
strict betweenness. Pasch used both.)
}
Pasch's requirement that $DE$ lie in a plane with $ABC$ of
course cannot be dropped, since the line might not lie in the plane
of the triangle. In order to drop that hypothesis, obtaining
a statement that mentions only betweenness, one must strengthen the hypothesis
so that the line certainly lies in the plane of the triangle. There
are two ways to do this, resulting in axioms known as ``outer Pasch''
and ``inner Pasch.'' See Figure~\ref{figure:InnerOuterPaschFigure}.
In Pasch's own axiom (and figure) there is no requirement for point $E$
to be collinear with $BC$; that was added by Peano to make the coplanarity
hypothesis explicit.
\begin{figure}[ht]
\caption{ Inner Pasch (left) and outer Pasch (right). Line $EF$ meets triangle $ABC$ in one side $AB$, and meets an extension of side $BC$. Then
it also meets the third side $AC$.
The open circles show the points asserted to exist. }
\label{figure:InnerOuterPaschFigure}
\InnerOuterPaschFigure
\end{figure}
These planar forms of Pasch's axiom were invented by Peano and published
in 1889, seven years after Pasch.%
\footnote{
Axiom XIII in \cite{peano1889} is outer Pasch, with $\B abc$
written as $b \in ac$.
Axiom XIV is inner Pasch. Peano wrote everything in formal symbols
only, and eventually bought his own printing press to print his books himself. See \cite{kennedy2002}.}
Inner Pasch has a certain symmetry: In Figure~\ref{figure:InnerOuterPaschFigure},
we could just as well have shaded triangle $BEF$ instead of triangle
$BAC$. One soon becomes accustomed to noticing (and shading) the whole
quadrilateral.
Gupta's thesis (which contained enough material for three theses)
contains proofs that outer Pasch implies inner Pasch, and vice versa,
using the other axioms (but not continuity). Of course Euclid,
who never mentions betweenness, did not explicitly use either version of
Pasch, but they both came up naturally when we formalized Euclid.
As with the continuity axioms, we could have picked just one, but then
we would have been proof-checking Gupta's thesis, in addition to Euclid;
so we just took both inner and outer Pasch as axioms.
The need for the Pasch axioms is pervasive: we used
inner Pasch 36 times and outer Pasch 31 times in formalizing
Euclid Book~I.
It is instructive to see how inner and outer Pasch are needed to provide
infrastructure for Euclid. We illustrate with Proposition~I.16, the exterior
angle inequality. That proposition says,
\begin{quote} In any triangle, if one of the sides be produced, the exterior
angle is greater than any of the interior and opposite angles.
\end{quote}
Refer to Figure~\ref{figure:OneSixteenOne}. $ABC$ is the triangle and
$ACD$ is the exterior angle, which is asserted to be greater than angle $BAC$
and greater than angle $ABC$. To prove that,
Euclid constructs $F$ with $EF=EB$, and
proves triangle $AEB$ is equal to triangle $CEF$,
so in order to prove that angle $ACD > BAC$ it suffices to prove
that $ECF < ACD$. Euclid justified that with Common Notion~5, the
whole is greater than the part. But long before Pasch, one might
have objected, how do we know that angle $ECF$ actually {\em is}
a part of $ACD$? That question needs the same answer that the
modern definition of angle ordering requires: the construction
of the point $H$.
In the proof we gave, inner Pasch is used to construct $H$,
as shown in the figure.
\begin{figure}[ht]
\center{\OneSixteenInnerFigure}
\caption{Proof of I.16 using inner Pasch.}
\label{figure:OneSixteenOne}
\end{figure}
One can also prove I.16 from outer Pasch, instead of inner Pasch.
It requires two applications of outer Pasch, as shown in
Figure~\ref{figure:OneSixteenTwo}.
\begin{figure}[ht]
\center{\OneSixteenOuterFigure \OneSixteenFigureTwo}
\caption{Proof of I.16 using outer Pasch.}
\label{figure:OneSixteenTwo}
\end{figure}
\section{Angle bisection.}
Euclid~I.9 gives a construction to bisect a given angle $PQR$. Namely,
lay off equal segments on the two sides of the angle, connect their
endpoints $AB$, and use I.1 to construct an equilateral triangle
$ABC$. Then $QC$ is the angle bisector. Ah, but to determine a line,
we must have $Q \neq C$. And if the original triangle is equilateral,
$C$ {\em will} be $Q$. So what then? Well, says the devil, then
just take the {\em other} intersection point of the two circles in
Proposition~I.1. That is, we should modify the circle-circle continuity
axiom to say there are {\em two} points of intersection of the circle.
All right, let us suppose that is done. Then we confront the real
difficulty of the proof: why is $QC$ the angle bisector? In fact,
why does it even lie in the plane of angle $PQR$?
For this proposition to be correct, the definition of the bisector
of an angle must say that there are points $U$ and $V$ on the
sides of the angle, and the bisector connects the vertex with some
point between $U$ and $V$. Now, even if we assume there are two
equilateral triangles on $AB$, and one of them has a third vertex
$C$ different from $Q$, there is no apparent reason why $QC$ must meet
$AB$, as the revised hypothesis requires.
Proclus already pointed out and attempted to repair
some of these difficulties in 450 CE,
see \cite[p.~214]{proclus} and Heath's commentary on I.9 \cite{euclid1956}.
Of course Proclus did not have Pasch's axiom at his disposal; but
his proof is easily completed using inner Pasch.
See Figure~\ref{figure:proclus}. (In that figure, we shade
the whole quadrilateral formed by the four points to which inner Pasch
is applied, since the choice of three of them is arbitrary.)
\begin{figure}[ht]
\caption{To bisect angle $ACB$, given $AC$ equal to $BC$, make
$AE = BD$, and then
construct $F$ by inner Pasch. One more application of Pasch bisects $AB$.}
\label{figure:proclus}
\center{\ProclusBisectionFigure \ProclusBisectionFigureTwo}
\end{figure}
\FloatBarrier
Proclus observes that triangles $ABE$ and $BAD$ are
equal (congruent), since angles $EAB$ and $DBA$ are equal by
Proposition~I.5. Then $AF$ and $BF$ are equal, by I.6.
Then triangles $ACF$ and $BCF$ are equal. Then (by definition of
equal angles) $CF$ is the desired bisector.
After that,
we can proceed
directly from Proclus's proof of I.09 to I.10, since the midpoint
of $AB$ can be constructed with one more application of inner Pasch,
as shown in the second part of Figure~\ref{figure:proclus}.
Proclus also noticed and repaired the problem that it needs to be
proved that $F$ lies in the interior of the angle. Since Proclus
did not have inner Pasch available, he made another argument; but
inner Pasch solves that problem too, as well as the problem of
showing that the bisector lies in the plane of the angle,
which neither Proclus nor Heath noticed.%
\footnote{In \cite{beeson2019}, not having studied Proclus enough, we
used instead a proof from Gupta's 1965 thesis \cite{gupta1965},
which can also be seen as Lemma~7.25 in \cite{schwabhauser}. With the
same figure, Gupta proves that $M$ is the midpoint of line $AB$.
We used Gupta's proof to prove Euclid~I.10 (line bisection) and
then used I.10 to prove I.09. But Gupta's proof is complicated,
because it avoids using circles. Proclus's proof is simpler, and
allows us to preserve
Euclid's order of the propositions.}
\section{Two dimensions or three?}
Euclid Books I--IV are commonly thought to be about plane geometry,
but consider:
\begin{itemize}
\item There is a definition of {\em plane}.
\item The definition
of {\em parallel} mentions that the two lines must be in the same plane.
\item There is no ``dimension axiom,'' such as Tarski's axiom that three
points each equidistant from points $P,Q$ must be collinear, which
guarantees that all points lie in a plane.
\item In the last Book, Euclid takes up the Platonic solids, and certainly
uses the results of Book~I.
\end{itemize}
Nevertheless all the diagrams in Books~I--IV
appear to be planar figures. We conclude that Euclid's intention
was to present theorems (and proofs) that are valid in every plane.
Remember that Euclid did not have our modern conception of ``model'' of
a geometrical theory. There was just one true space, and it was three-dimensional, containing many planes.
But there are several places in Book~I where this seems to have been
forgotten. For example,
Proposition~7 says that if $ABC$ and $ABD$ are
two triangles with $AC=AD$ and $BC=BD$, and $C$ and $D$ are on the
same side of $AB$, then $C=D$.
The figure Euclid gives is supposed to be impossible,
but as soon as you remember it might be in three dimensions, it looks
very possible. It is saved from being mistaken by the hypothesis
mentioning ``same side,'' which forces the diagram to be planar; but
Euclid did not define ``same side,'' nor did he use that hypothesis
in the proof. The definition of ``same side'' is discussed
below; for now we are focussing on the dimension issue.
Euclid's last definition in Book~I is
\begin{quote}
Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.
\end{quote}
The inclusion of the requirement that the lines be in the same plane
shows that Euclid's omission of a dimension
axiom was not a simple oversight: he meant to allow
for the possibility that lines might {\em not} lie in
the same plane. But he did
not define ``lie in the same plane.''
Whatever Euclid meant by his definition, he found it
obvious that if $AB$ is parallel to $CD$, then
$CD$ is parallel to $AB$. I say that because this fact is used
(in the proof of Proposition~I.30)
without even being mentioned (let alone proved).
This property is called the ``symmetry of parallelism.''
We repair this omission by defining two (distinct) lines $AB$
and $CD$ to lie in the same plane if they are linked by a
``crisscross'' configuration, as shown in Figure~\ref{figure:crisscross}.
\begin{figure}[ht]
\center{\FigureParallelDefFour}
\caption{$AB$ and $CD$ are coplanar, as witnessed by $RSPQH$.}
\label{figure:crisscross}
\end{figure}
With this definition of ``coplanar,'' we can take Euclid's
definition of ``parallel'' literally: $AB$ is parallel to
$CD$ if they are coplanar and do not meet. This definition makes
the symmetry of parallelism an immediate consequence, and it
also makes it evident that if $AB$ is
parallel to $CD$, and any or all of the four points are moved
along their respective lines to new (distinct) positions, then
$AB$ is still parallel to $CD$. %
\footnote{
We define ``Tarski-parallel''
by ``$AB$ and $CD$ do not meet, and $C$ and $D$ lie on the same side of
$AB$.'' This is clearly not what Euclid intended, as to Euclid it
seems obvious that if $AB$ is parallel to $CD$ then $CD$ is parallel to
$AB$, but it requires the plane separation theorem to prove that
about Tarski-parallel.
On the other hand,
the two definitions can be proved equivalent. It follows that if $AB$
and $CD$ are parallel then $A$ and $B$ are on the same side of $CD$, which
is quite often actually necessary, but never remarked by Euclid.
This is another example of ``infrastructure.''
}
Euclid's Postulate~5, the ``parallel postulate,'' mentions the
concepts of alternate interior angles, and the concept that the
two angles on the same side of a transversal ``make together'' more
than or less than two right angles. The intention clearly is that
the lines involved all lie in the same plane, which will have to
somehow involve intersection points of some lines.
It is sometimes possible to reduce theorems about angles directly
to statements about points and the equality relation between
segments. In particular, it is not
necessary to develop the theory of angle ordering to state Euclid's parallel postulate.
In Figure~\ref{figure:AlternateInteriorAnglesFigure}, we show how to
translate the concept ``equal alternating interior angles'' into the
formal language we used.
\smallskip
\begin{figure}[ht]
\caption{Transversal $pq$ makes alternate interior angles equal with $L$ and $K$, if $pt=tq$ and
$rt=st$.}
\label{figure:AlternateInteriorAnglesFigure}
\hskip 2.5cm
\AlternateInteriorAnglesFigure
\end{figure}
Another place where Euclid forgot about three dimensions is
in Proposition~I.9, the bisection of an angle. This proof was
discussed above, but here we take up the 3-dimensional aspect
of it. The problem is that the equilateral triangle constructed
by Proposition~I.1 should introduce a new point by its vertex, not
just give back the original angle's vertex.
One may wish to take ``the other'' equilateral triangle, but
Proposition~I.1 only says there is one, not two. But in the
absence of a dimension axiom, there is a whole
circle $C$ of intersection points, on the plane that bisects $AB$.
In the absence of a dimension axiom,
{\em we have to think of circles as spheres}. The circle-circle
axiom is still valid, but even if we assume the two intersection points are on a diameter
of that circle $C$, it might be tilted out of the plane of $PQR$.
I do not say that Euclid had that mental picture; only that he
did not have a dimension axiom, and apparently quite purposely,
so that we who formalize his work must remember that
there is no dimension axiom.
Euclid does not define
``rectangle.'' One would like to define it as a quadrilateral with four right
angles. It is a theorem that such a figure must lie
in a plane. However, the proofs we found involve reasoning ``in three dimensions.''
Even though Euclid Book I has no dimension axiom, and we must therefore
be careful not to assume one, nevertheless all the {\em proofs} in Book~I
deal with planar configurations. We therefore define ``rectangle'' to be a quadrilateral with four
right angles, whose diagonals cross, that is, meet in a point. This condition
is one way of specifying that a rectangle lies in a plane. We can then prove
that a rectangle is a parallelogram.
Euclid defines a square to be a quadrilateral with at least one
right angle, in which
all the sides are equal.
But in I.46 and I.47 the proofs work as if the definition
required all four angles to be right, so we take that as the definition.
He does not specify that all four vertices lie in
the same plane. This is not trivial to prove, but we did prove it, so Euclid's
definition does not require modification.
\section{Sides of a line and the crossbar theorems.}
We have already discussed Proposition~I.7, which mentions the undefined
``same side'' but never uses it in the proof.
Since Euclid never defined {\em same side},
there is no obvious way to fix it. Hilbert worked in plane geometry
in the strong sense, so he did not need to define {\em same side}
in a way that works in space.
That notion was, apparently, first defined by M. Pasch in 1882
\cite[p.~27]{pasch1882}, but only under the assumption
that the points lie in the same plane. To remove co-planarity as a
primitive notion from the definition was first done by Tarski (as far
as I can determine). He defined two points $a$ and $b$ to be on opposite sides
of $pq$ if there is a point between $a$ and $b$ collinear with $pq$,
and defined $a$ and $b$ to be on the same side of $pq$ if they are
both on the opposite side of $pq$ from the same point $c$.
(See Figure~\ref{figure:sides}.)
\begin{figure}[ht]
\center{\OppositeSideFigure\quad \SameSideFigure}
\caption{(Left) $a$ and $b$ are on the opposite side of $pq$.
(Right) $a$ and $b$ are on the same side of $pq$ if there exist
points $x$ and $y$ collinear with $pq$, and a point $c$,
such that $\B(a,x,c)$ and $\B(b,y,c)$.}
\label{figure:sides}
\end{figure}
Once these concepts are defined, one can use (both inner and outer)
Pasch to prove the {\em plane separation theorem}: if
$C$ and $D$ are on the same side of $AB$, and
$D$ and $E$ are on opposite sides of $AB$, then $C$ and $E$
are on opposite sides. Since neither Pasch nor ``same side''
occurs in Euclid, this is not a Euclidean theorem; it is
infrastructure provided by Tarski and Szmielew, 2300 years later.
Yet we needed it to correct the proofs of not only Euclid~I.7, but
also Propositions~11, 27, 28, 29, 30, 35, 42, 44, 45, 46, and 47.
These corrections would still be required even if we did add a
dimension axiom, as, even in the plane, a line has two sides and
the plane separation theorem requires a proof.
A similar piece of infrastructure, also closely related to
Pasch's axiom, is the crossbar theorem. See Figure~\ref{figure:crossbar}.
\begin{figure}[ht]
\center{\CrossbarFigure}
\caption{Crossbar theorem: drawing $BP$ through to $JK$.}
\label{figure:crossbar}
\end{figure}
This theorem says that
if $P$ is a point in the interior of angle $ABC$ (which means that
$P$ lies between two points $U$ and $V$ on the rays $BA$ and $BC$,
respectively), and if $J$ and $K$ are any two points on those rays,
then the ray $BP$ will meet the ``crossbar'' $JK$ in some point $X$.
If Euclid needs such a point, he simply says that $BP$ is ``drawn through''
to $X$. One
place where the crossbar theorem is needed is to prove the
uniqueness of the angle bisector.
To the modern mathematical eye, having proved existence of angle
bisectors, the next question should be the uniqueness of the angle
bisector. This never occurs as a proposition in Euclid, and its
omission causes no harm in Book~I. But it is definitely required
to fix the proof of III.20, and it takes more than 110 inferences,
most of which are ``infrastructure'' steps,
concerning the collinearity or noncollinearity of points, the equality
of various angles, the transitivity of the less-than relation on angles,
etc. These are all things that Euclid usually did not mention.
There are several versions of the crossbar theorem. More than
one version is needed, because sometimes we need to know the order
of the points $BPX$. If we assume $\B BUJ$ and $\B BVK$, then
the conclusion can be
strengthened to $\B BPX$; another version has those betweenness relations
reversed. One is proved with two applications of inner Pasch, the other
with two applications of outer Pasch. But sometimes we do not know
the order of the points $BUJ$, so we need the version stated with rays, too.
All this infrastructure is required because of the modern insistence on
requiring the existence of points to be proved, rather than producing
the required points by ``drawing through.''
\section{Euclid~4: all right angles are equal} \label{section:Euclid4}
Over the centuries there were many claims that Euclid~4 is a theorem,
and hence should not be taken as a postulate. For example, we find
a proof already in Proclus (450 CE) \cite[p.~148]{proclus}, illustrated
in Figure~\ref{figure:proclusI.4}.
\begin{figure}[ht]
\caption{Proclus's proof of I.4.}
\label{figure:proclusI.4}
\center{\ProclusPostulateFourFigure}
\end{figure}
Assume that $ABC$ and $DEF$ are right angles. Proclus says,
``If $DE$ be made to coincide with $AB$, the line $EF$ will fall within
the angle, say, at $BG$.'' Then $H$ and $K$ are constructed, and
we have $ABC = ABH < ABK = ABG < ABC$, so $ABC < ABC$, which is
impossible. It is the first step that is problematic: the proof
appears to depend on the invariance of angles under a rigid motion,
the same flaw that bedevils Euclid's ``proof'' of SAS in I.4. The
remedy for that is the angle-copying Proposition I.23. But
Euclid appealed to Postulate 4 in proving I.23. Thus Proclus's
proof is not correct as it stands. Moreover, the proof culminates
by saying it is impossible that $ABC < ABC$. Euclid never proved it;
apparently he considered it to be part of the common notion
``the part is less than the whole.''
In our formalization,
this theorem is called {\em angletrichotomy2}. And its (rather lengthy
at 463 lines)
proof uses I.23.%
\footnote{Hilbert smuggled this principle into his axiom system
without explicit mention, by postulating the uniqueness of the copied
angle in his angle-copying axiom.}
So there is a {\em second} circularity in Proclus's
proof, from the modern point of view. Our formalization
followed Szmielew's proof, probably discovered at Berkeley in the 1960s,
but published only in 1983 \cite{schwabhauser}; see especially Satz~10.12.
The idea of Szmielew's proof is not so different from Proclus's,
but she replaced the illegal rigid-motion argument with a careful study
of isometries, including reflections in a point or in a line. Rotations
can be built from reflections in lines. Of course, this has to be done
without Postulate~4. Szmielew's Satz~10.12 is this: {\em If two right angles have corresponding legs
equal, then the hypotenuses are also equal.} That theorem
is easily seen to be equivalent to Postulate~4. The idea of the proof
is to construct an isometry that takes the corresponding legs onto
each other. First a translation brings the two vertices together,
say at point $b$.
Then a rotation makes one leg coincide with the corresponding leg.
This amounts to a correct formalization of Proclus's first step,
without a circularity.
In \cite{beeson2019}, we followed Szmielew, but in doing so, used Euclid's
construction of perpendiculars based on line-circle and circle-circle continuity.
Szmielew did it without circles, using Gupta's circle-free construction
of perpendiculars from his thesis \cite{gupta1965}.
\section{Equal figures.}
The word {\em area} almost never occurs in Euclid's {\em Elements}, despite the
fact that area is clearly a fundamental notion in geometry. Instead, Euclid speaks of ``equal figures.''
Apparently a ``figure'' is
a simply connected polygon, or perhaps its interior. The notion is
neither defined nor illustrated by a series of examples; for example, it
is never made clear whether a figure has to be convex, or even whether
a circle is a figure, or whether a figure has an interior, or is just
made of lines.
The notion of ``equal figures'' plays a central role in Euclid. For
example, the culmination of Book~I is the Pythagorean theorem.
Nowadays we would, if required to express the theorem without algebraic
formulas, say that given a right triangle,
the area of the square on the
hypotenuse is the sum of the areas of the squares on the sides. But
Euclid said instead, that the square on the hypotenuse is equal to the
squares on the sides, taken together. His proof shows how the two squares
can be cut up into pieces that can be rearranged to make this equality of figures evident,
given earlier propositions about equal figures.
Nor was Euclid alone in avoiding the word ``area.''
A century later when Archimedes calculated the area of a circle,
he did not express his result by saying that the area of the circle is
$\pi$ times the square of the radius. Instead, he said that circle is
equal to the rectangle whose sides are the radius and half the circumference.
(So a circle did count as a figure for Archimedes!)
Why did Euclid avoid the word {area}? Not because he did not know
that area can be measured; it must have been for more abstract, mathematical
reasons. Let us consider his problem:
if he were to use the word, he would either have to {\em define} it,
or put down some {\em postulates} about it. Both choices offer some
difficulties. Area involves assigning a {\em number} to each figure, to
measure its area. It is therefore not a purely geometric concept.
Moreover, even if one is willing to introduce numbers, that just pushes
the problem back one step: one must then define or axiomatize numbers.
Euclid's proofs, starting from I.35, use the notion of
``equal figures'' without either definition or explicit axiomatization.
He allows himself to paste equal triangles onto equal figures,
concluding that the results are equal, and justifies it by the common
notion {\em if equals be added to equals, the
wholes are equal}. He allows himself to cut
equal triangles off of equal figures, and justifies it by {\em if equals be subtracted from equals, the
remainders are equal.} If, with a modern eye, we interpret ``equal figures''
to mean ``figures with equal area,'' these properties look like the
additivity of area.
Common Notion 5, ``the whole is greater than
the part,'' could be taken to imply that a figure cannot be equal to
a part of itself, and Common Notion 4, ``things which coincide with
one another are equal to one another,'' could be interpreted to imply
that congruent figures are equal.
Actually, Euclid needed one more property: halves of equal
figures are equal, used in Proposition~I.39. The step that (implicitly) uses that property
occurs in Euclid's text without justification.
We will give an example of how Euclid reasoned about equal figures,
namely Euclid I.35. See Figure~\ref{figure:I.35colored}.
\begin{figure}[ht]
\caption{Euclid's proof of I.35.}
\FigureOneThirtyFiveColored
\label{figure:I.35colored}
\end{figure}
Euclid wants to prove the parallelograms $ABCD$ and $BCFE$
are equal. He proves the triangles $ABE$ and $DCF$ are
congruent. Implicitly, he assumes $DEG$ and $DGE$ are
equal figures (that is, the order of listing the vertices
does not matter). Then ``subtracting equals from equals,'' the
yellow quadrilaterals are equal. Then, ``adding equals to equals,''
he adds triangle $BCG$ (implicitly assuming $BCG$ is equal to $BGC$)
to arrive at the desired conclusion.
Later generations of mathematicians were not willing to accept Euclid's
over-liberal interpretation of the common notions in support of
``equal figures.'' See the summary
discussion with many references \cite[pp.~327--328]{euclid1956}. In
particular, once mathematicians had some experience with axiomatization,
it became obvious that ``equal figures'' is not a special case of
equality, since equal figures cannot be substituted for each other
in every property. Instead, it is a new relation, and the original
choice that Euclid finessed faced us directly when we wanted to
proof-check Euclid: we had either to define
or to axiomatize the notion. We chose to axiomatize it.%
\footnote{We do not take space here to describe attempts to define it
by Hilbert and others. See \cite{beeson2020,hartshorne}
for a full discussion; \cite{beeson2020} also gives a definition
that ``Euclid could have given.''}
Following
the lead of Hartshorne \cite{hartshorne}, we wrote down fifteen axioms for the
three primitive notions of ``equal triangles'' and ``equal quadrilaterals''
and ``triangle equal to quadrilateral.'' No figures with more than four
sides occur in Book~I, so that was sufficient. These were ``cut-and-paste''
axioms as described above, plus two axioms (first invented
by de Zolt, see~\cite{hartshorne}) saying that if you cut off
something, the result is not equal to what you had before the cut.
\section{Making Formal Proofs Readable.}
A proof has two purposes: to establish beyond doubt that a theorem
is correct, and to communicate to a human reader {\em why} it is correct.
Our formal proofs improve on Euclid in the first respect, but in
the second respect they need improvement. We address that issue now.
\begin{comment}
"Neither the referee nor any reader can check the proofs
and has to believe the statements which are not proven in the paper,
but only affirmed by the authors to be true."
\end{comment}
We begin by saying something about what it
is like to formalize Euclid's proofs. What one finds is that one
needs a large number (more than sixty percent) of ``invisible infrastructure''
steps, proving statements that Euclid would take for granted because
they appear so in the diagram. Let me give one example. Often
to verify the hypotheses of some proposition or lemma we wish to apply,
we need to know that
an angle $ABC$ is equal to itself. Before we can write that down,
we have to verify that the three points $A$, $B$, and $C$ are noncollinear,
so that they really do form an angle. Euclid always takes that for
granted if it appears so in the diagram, but it often requires many
formal steps, using for example the lemma that if $A$, $B$, $C$ are noncollinear
and $U$ and $V$ are distinct and both are collinear with $AB$, then
$B$, $U$, $V$ are noncollinear. A chain of three or four applications of that
lemma may be required to verify that a given triangle really is
a triangle. After a while, one can do this ``diagram chasing''
as fast as one can type,
but it means that the formal proofs are at least double the length of
Euclid's and contain many uninteresting steps. Euclid generally takes
for granted statements of collinearity and betweenness that appear obvious
in the diagram.
To address the problem of too many and too-detailed ``infrastructure'' steps,
we created a list of ``trivial'' inferences, or more accurately trivial
justifications, which are to be suppressed
on output. For example, the application of lemmas about collinearity
and noncollinearity. By putting more or fewer justifications on the
``trivial list,'' we can vary the ``step-size'' or ``grain'' of the
output proofs. We call the result a ``proof skeleton''. We found
that with a suitable list of trivial steps, the proof skeletons contained
Euclid's steps, and only really essential additions (such as uses of
Pasch's axiom). For example, in Prop.~I.16, the proof skeleton has 15 inferences,
while the full formal proof has about 120 inferences, most of which are about
collinearity, non-collinearity, and distinctness of points that appear
distinct in the diagram.
A second obstacle to human readability is the fact that
formal proofs contain only symbols (no words).
Euclid used no symbols except names for points; so
if we want to compare our proofs to Euclid's, we ought to write them
in Euclid's way. We
therefore experimented with machine-generating English-language proofs%
\footnote{Greek-language proofs might have been more authentic. Output
in any human language can be generated once the ``stock phrases'' are translated.}
from our formal proofs, ``in the style of Euclid.''
Euclid's natural language consists of a small number
of stock phrases that connect the assertions. We wrote a script
that translates proof skeletons (or proofs) into \LaTeX\ code for an English
version of the proof. The script follows Euclid by
prefacing each construction of a new point with a sentence about how it
is constructed. For example, it inserts a
phrase ``Let $AC$ be bisected at $E$'', when Prop.~I.10 (the midpoint theorem)
is about to be applied.
That script first
extracts a proof skeleton by deleting trivial steps, and then
produces a \LaTeX\ document, translating the proof into
``Euclidean English.''
In a few seconds, it processes all 245 formal proofs,
producing a computer-generated version of Euclid Book~I.
Here we present just two samples from this document. The reader
is invited to compare them with Euclid's proofs. In particular, the
proof of I.16 supplies the application of Pasch's axiom that is missing
from Euclid's proof; and the proof of I.20 illustrates the formal treatment
of angle ordering, appealing to a definition rather than Common Notion~5.
These proofs demonstrate, I
assert, that the translation from formal proofs to human-readable proofs,
while it may be formidable for humans, is trivial for computers.%
\footnote{The other direction, from human-readable proofs to formal proofs,
is far from trivial.}
The
programs that make these transformations are short and simple, and were
easy to write.
\section*{ Proposition 20 }
{\em In any triangle two sides taken together
are greater than the third side.}
\begin{figure}[ht]
\center{\OneTwentyFigure}
\end{figure}
\medskip
Let $ABC$ be a triangle.
It is required to show that $BA$, $AC$ are together greater than $BC$.
\medskip
\noindent
\hspace{0pt}Let $BA$ be produced in a straight line to $A$, making $AD$ equal to $CA$. Then
\hspace{0pt}$A$ is between $B$ and $D$ and $AD$ is equal to $CA$.\justification{extension}
\hspace{0pt}Triangle $ADC$ is isoceles with base $DC$.\justification{defn:isosceles}
\hspace{0pt}Angle $ADC$ is equal to angle $ACD$.\justification{I.5}
\hspace{0pt}Angle $ADC$ is equal to angle $DCA$.\justification{equalanglestransitive}
\hspace{0pt}Angle $DCB$ is greater than angle $ADC$.\justification{defn:anglelessthan}
\hspace{0pt}Angle $ADC$ is equal to angle $BDC$.\justification{defn:equalangles}
\hspace{0pt}Angle $DCB$ is greater than angle $BDC$.\justification{angleorderrespectscongruence2}
\hspace{0pt}Angle $DCB$ is greater than angle $CDB$.\justification{angleorderrespectscongruence2}
\hspace{0pt}Angle $BCD$ is greater than angle $CDB$.\justification{angleorderrespectscongruence}
\hspace{0pt}$BD$ is greater than $BC$.\justification{I.19}
{\flushright{ \vspace{-15pt} Q.E.D.}\flushleft}
\section*{Proposition 16}
{\em In any triangle, if one of the sides be produced,
the exterior angle is greater than either of the interior
and opposite angles.}
\begin{figure}[ht]
\center{\OneSixteenInnerFigure}
\label{figure:OneSixteenRepeat}
\end{figure}
Let $ABC$ be a triangle, and let one side of it $BC$ be produced
to $D$; then $C$ is between $B$ and $D$. \\
It is required to show that the exterior angle $ACD$ is greater than
the interior and opposite angle $BAC$.%
\footnote{\,I.16 asserts that the exterior angle is greater than both
interior angles. Like Euclid, we here present only the proof for one exterior angle.}
\medskip
\noindent
\hspace{0pt}Let $AC$ be bisected at $E$. Then
\hspace{0pt}$E$ is between $A$ and $C$ and $EA$ is equal to $EC$.\justification{I.10}
\hspace{0pt}Let $BE$ be produced in a straight line to $E$, making $EF$ equal to $EB$. Then
\hspace{0pt}$E$ is between $B$ and $F$ and $EF$ is equal to $EB$.\justification{extension}
\hspace{0pt}Let $AC$ be produced in a straight line to $C$, making $CG$ equal to $EC$. Then
\hspace{0pt}$C$ is between $A$ and $G$ and $CG$ is equal to $EC$.\justification{extension}
\hspace{0pt}Angle $BEA$ is equal to angle $CEF$.\justification{I.15}
\hspace{0pt}Angle $AEB$ is equal to angle $CEF$.\justification{equalanglestransitive}
\hspace{0pt}$AB$ is equal to $CF$, and Angle $EAB$ is equal to angle $ECF$, and Angle $EBA$ is equal to angle $EFC$.\justification{I.4}
\hspace{0pt}Angle $BAC$ is equal to angle $BAE$.\justification{equalangleshelper}
\hspace{0pt}Angle $BAC$ is equal to angle $EAB$.\justification{equalanglestransitive}
\hspace{0pt}Angle $BAC$ is equal to angle $ECF$.\justification{equalanglestransitive}
\hspace{0pt}Angle $ECF$ is equal to angle $ACF$.\justification{equalangleshelper}
\hspace{0pt}Angle $BAC$ is equal to angle $ACF$.\justification{equalanglestransitive}
\hspace{0pt}Let $CF$ and $ED$ meet at $H$. Then
\hspace{0pt}$H$ is between $D$ and $E$ and $H$ is between $F$ and $C$.\justification{Pasch-inner}
\hspace{0pt}Angle $BAC$ is equal to angle $ACH$.\justification{equalangleshelper}
\hspace{0pt}Angle $BAC$ is equal to angle $ACF$.\justification{equalangleshelper}
\hspace{0pt}Angle $BAC$ is equal to angle $ACH$.\justification{equalanglestransitive}
\hspace{0pt}Angle $ACD$ is greater than angle $BAC$.\justification{defn:anglelessthan}
{\flushright{ \vspace{-15pt} Q.E.D.}\flushleft}
In these examples, only the italicized informal statement at the top
and the diagram are human-generated. The rest, including all the English, all
the references, and the typesetting,
is machine-generated. At last, we have achieved,
and certified by computer,
the goal that Gerolamo Saccheri stated in the title of his
1733 book, {\em Euclid Vindicated from Every Blemish}.%
\footnote{Other translators have chosen {\em Euclid Freed of
Every Flaw}. The title above is the one chosen by de Risi. \cite{saccheri-deRisi}
}
\section{Euclid Vindicated.}
Table~\ref{table:1} compares Euclid's {\em Elements}
(Books I to IV) with the changes we made in formalizing Euclid.
\begin{table}[ht]
\caption{Changes to Propositions and Axioms.}
\label{table:1}
\begin{center}
\begin{tabular}{l|l|l}
Issue & Euclid & Changes made \\
\hline
Postulate IV &long thought provable & proved by Szmielew \\
Proposition I.4 & rejected since antiquity & replaced by 5-line axiom \\
definition of ``parallel'' & ``in the same plane'' & supplied definition \\
Postulate V & ``alternate interior angles'' & supplied definition \\
connectivity axiom & missing & added \\
betweenness notion & missing & added \\
betweenness axioms & missing & identity, symmetry,\\
& & and transitivity \\
betweenness, basic theorems & missing & proved \\
definition of ``same side'' & missing & added Tarski's definition \\
2 or 3 dimensions? & valid in any plane & use Tarski's ``same side'' \\
Pasch & missing axiom & inner and outer Pasch \\
line-circle and circle-circle & missing axioms & added both \\
common notions for lines & & 3,5 proved; 4 dropped\\
equality of angles & primitive notion & defined notion \\
less than for angles & primitive notion (?) & defined notion \\
common notions for angles & & proved, instead \\
& &of assumed \\
equal figures & no definition or axioms & added 15 axioms \\
rectangle definition & omitted & four right angles \\
& & and diagonals meet
\end{tabular}
\end{center}
\end{table}%
Table~\ref{table:2} shows some of the propositions in Book~I
that needed corrections.
We do not include as ``corrections'' the provision of ``infrastructure''
steps about collinearity and noncollinearity, nor elementary reasoning
about betweenness, nor the many cases where Pasch was needed,
or some lemmas had to be proved, nor proofs where Euclid
treated only one of several cases (for example I.35).
\begin{table}[ht]
\caption{Corrections to Proofs (refer to Euclid's diagrams).}
\label{table:2}
\begin{center}
\begin{tabular}{p{13pt} l|l|l}
Prop.& Description & Difficulty & Correction \\
\hline
I.1& equilateral triangle & existence of $C$ & circle-circle \\
I.1& equilateral triangle & $ABC$ might be collinear & connectivity axiom \\
I.4& SAS & superposition & 5-line axiom \\
I.7& triangle uniqueness & same side not defined & use Tarski's definition \\
I.7& triangle uniqueness & angle trichotomy & proved as theorem \\
I.9& angle bisection & $A$ and $F$ might coincide & use Proclus's proof \\
I.12& dropped perpendicular & Why do $G$ and $E$ exist? & line-circle \\
I.16& exterior angle & Why is $ECD > ECF$? & Pasch \\
I.22& triangle construction & why does $K$ exists? & circle-circle \\
I.22& triangle construction & why does $DE$ meet circles?& line-circle \\
I.27& parallel construction & ``alternate angles'' undefined & $AD$
and $EF$ \\
& & & must meet \\
I.32& exterior angle & see I.16 & \\
I.33& parallelogram constr. & ``same direction'' & diagonals must meet\\
I.35& parallelograms & Why is $DEG = EGD$? & an equal-figures axiom \\
I.35& parallelograms & several other steps & equal-figures axioms\\
I.39& equal triangles & same-side undefined, unused &
use Tarki's definition \\
I.46& square definition & definition doesn't match use
& changed definition
\end{tabular}
\end{center}
\end{table}
\FloatBarrier
\section{Conclusions.}
The two most characteristic features of Euclid are geometrical
diagrams, and chains of
logical reasoning about those diagrams. The exact relation between these two features
has been a concern of every thoughtful reader of Euclid, right from the beginning.
The reasoning is guided by the diagram; but sometimes it is led astray by the diagram,
too! It took two millennia to {\em separate} the two features,
inventing {\em symbolic logic}--meaningless chains of symbols representing correct
inferences made according to precise rules. As the logician J. Barkley Rosser expressed
it \cite[p.~7]{rosser1978}:
\begin{quote} This does not mean that it is now any easier to discover a proof for a difficult
theorem. This still requires the same high order of mathematical talent as before.
However, once the proof is discovered, and stated in symbolic logic, it can be checked
by a moron.
\end{quote}
It can even be checked by a computer. Rosser's teacher, Kleene, was once asked why he
wrote his proofs so formally. He replied, ``How else can I be sure they are right?''
And that is the most obvious, and most important,
result of proof-checking Euclid: Now we are {\em sure} that
the proofs are correct. Though some of the gaps and errors we uncovered were known
for a long time, others were not, so mere human checking did not really do the job.
One striking feature of these formal proofs is that they are longer than the proofs
mathematicians write, usually by a factor of about four.
I have called those extra steps ``infrastructure.'' In Euclid they are mostly about
collinearity, noncollinearity, and betweenness. They represent facts that a human reader infers from
the diagram and takes for granted without explicit proof. Even
when we {\em think} we are checking a proof carefully, we are skipping many necessary small
steps, jumping over those steps to reach a conclusion that we believe {\em on some other grounds}, for example, on the appearance of a diagram.
When we see that the evidence of our eyes or intuition (the diagram) is confirmed by the successful
completion of a long chain of logical reasoning, that produces a feeling of
satisfaction that is the heart of mathematics. That's why eleven-year old Bertrand
Russell called Euclid ``delicious.'' Neither a diagram without reasoning,
nor a meaningless chain of inferences, deserves the name, ``mathematics.''
\begin{acknowledgment}
{Acknowledgments}
I am indebted to
John Baldwin,
Pierre Boutry,
Erwin Engeler,
Vincenzo de Risi,
Julien Narboux,
Victor Pambuccian,
Dana Scott,
Albert Visser,
and
Freek Wiedijk,
for many conversations and emails about Euclid and formalization.
I dedicate this paper {\em in memoriam} to Marvin Jay Greenberg,
who first introduced me to axiomatic geometry.
\end{acknowledgment}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 88 |
Das Hintelestal ist ein nördliches Seitental der Donau bei Kolbingen im Landkreis Tuttlingen.
Lage
Das Tal liegt im Naturraum Baaralb, Oberes Donautal und Hohe Schwabenalb und findet sich etwa 1,5 Kilometer südöstlich von Kolbingen im Landkreis Tuttlingen.
Schutzgebiet
Kenndaten
Das Tal wurde per Verordnung am 1. Oktober 1937 als Naturschutzgebiet ausgewiesen und hat eine Fläche von 19,1 Hektar. Es wird unter der Schutzgebietsnummer 3.081 beim Regierungspräsidium Freiburg geführt und ist in die IUCN-Kategorie IV eingeordnet. Der CDDA-Code lautet 81881 und entspricht zugleich der WDPA-ID.
Das Schutzgebiet wurde von ursprünglich im Jahr 1937 ausgewiesenen 50 Hektar Fläche per Verordnung vom 20. Dezember 1989 auf die heutige Fläche verkleinert und setzt sich seitdem weiter südlich des Hintelestals als Naturschutzgebiet Buchhalde-Oberes Donautal (NSG 3.171) fort.
Flora und Fauna
Das Schutzgebiet mit steilen, stellenweise felsigen Talhängen, bestockt mit Hangbuchen- und Schluchtwäldern, ist bekannt für umfangreiche Bestände der auch Märzenbecher genannte Frühlingsknotenblume. Im weiteren Verlauf des Jahres blühen hier unter anderem der Hohle Lerchensporn und das Wilde Silberblatt.
Siehe auch
Liste der Naturschutzgebiete in Baden-Württemberg
Liste der Naturschutzgebiete im Landkreis Tuttlingen
Literatur
Weblinks
Einzelnachweise
Naturschutzgebiet im Landkreis Tuttlingen
Kolbingen
Schutzgebiet (Umwelt- und Naturschutz) in Europa
Schutzgebiet der Schwäbischen Alb | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,360 |
The Ideology of Reason in Service to Humanity
July 7, 2013 | Author Steve Harvey
In the continuing debate against Libertarians (and all other ideologues of all stripes, for that matter), here's the bottom line: There's only one rational ideology to adhere to, and that is to strive to be rational; there's only one humane ideology to adhere to, and that is to strive to be humane.
Striving to be rational is not a vague, relative term: We have centuries of experience in the development of disciplined, methodical reasoning. We've developed scientific methodology and a wide spectrum of variations of it adapted to situations in which variables can't be isolated, statistical data analysis, research techniques designed to rigorously minimize the influence of bias and to maximize accuracy. We've developed legal procedure based on a debate between competing views framed by a set of rules designed to ensure maximum reliability of the evidence being considered and to identify the goals being pursued (adherence to formally defined laws). We've developed formal logic and mathematics, rules of deduction and induction, which maximize the soundness of conclusions drawn from premises, the premises themselves able to be submitted to the same rules for verifying raw data and drawing conclusions from that data.
Not everyone is trained in these techniques, but everyone can acknowledge their value and seek to participate in privileging them over other, more arbitrary and less rational approaches to arriving at conclusions. A commitment to democracy and pluralism does not require a commitment to stupidity and ignorance. The mechanisms by which we balance the need for all to have their say and all interests to be represented with the need for the best analyses to prevail in the formation of our public policies is an ongoing challenge, but we can all agree that we should meet that challenge head-on, rather than pretend that the drowning out of the cogent arguments of informed reason by the relentless and highly motivated noise of irrational ignorance is the height of self-governance.
Striving to be humane is not a vague, relative term either: We have centuries of development of thought concerning what that means, including John Rawls's "A Theory of Justice", which provides a pretty good heuristic guideline of what humane policies should look lie (they should be the kinds of policies that highly informed and rational people would choose if they didn't know what situation they were going to be born into or what chances of life they were going to encounter). This is basically a derivation and elaboration of the Golden Rule, which exists in some form or another in virtually every major religion on Earth. We all understand that justice requires that everyone be assured the same opportunity to thrive, and while we can agree that that is a formidable challenge that is more of an ideal toward which we can continue to strive than a finished achievement we can expect to accomplish in the near future, and that important counterbalancing imperatives must be considered and pursued simultaneously (in other words, that we need to balance the challenges of creating an ever-more more robust, fair, and sustainable social institutional framework), we can also agree that it is one of the guiding principles by which we should navigate as we forge our way into the future.
So, guided by our humanity, we have a clear objective that all of our public policies should strive to serve: Maximizing the robustness, fairness, and sustainability of our social institutional landscape to the greatest extent possible, such that no individual, if fully informed and rational, would want to change any aspect of it if they did not know where or when or into what situation they would be born or what chance occurrences they would encounter in life. And we have a clear means of most effectively pursuing that objective: Robust public discourse in which we allow the most cogent, information-intensive, methodologically and analytically sound arguments regarding how best to maximize the robustness, fairness and sustainability of our social institutional landscape, on a case-by-case, issue-by-issue basis, to prevail.
And THAT, what I just described above in the preceding five paragraphs, is really the only ideology we need, the only ideology we should adhere to as we move forward as a polity, wise enough to know that none of us knows all that much, humane enough not to blithely dismiss –whether implicitly or explicitly– the suffering and gross injustices endured by numerous others, intelligent enough to know that the appropriate role of a democratically and constitutionally circumscribed government in the modern world cannot be intelligently reduced to a handful of platitudes, informed enough to recognize that the rule of law is predominantly a procedural rather than substantive ideal, and smart enough to recognize that it is our commitment to these procedural and methodological disciplines of informing and devising public policies that will define how intelligently, humanely, and effectively we govern ourselves.
What continues to stand against this simple and clear ideology of a commitment to reason and humanity realized through disciplined procedures and methodologies are the plethora of blind dogmas, substantive false certainties, and precipitous conclusions that litter our shared cognitive landscape. Whether it is Marxism, politically active evangelical Christianity, politically active fundamentalist Islam, Libertarianism, or any other substantive dogma which presumes to know what we are in reality continuing to study, debate, and discover, this perennial need by so many to organize in an effort to impose a set of presumptive substantive conclusions on us all, one ideological sledgehammer or another with which to "repair" the machinery of government, is an obstacle rather than productive contribution to truly intelligent and humane self-governance.
It doesn't matter if any given adherents to such an ideology are right about some things and those arguing from a non-ideological perspective are wrong about some things; it would be extraordinary if that were not the case, because disciplined analysis seeks to track a subtle and elusive object (reality), while blind dogma, like a broken clock, stands in one place, and thus is right on those rare occasions when reality happens to pass through that spot. What matters is that we all say, "I am less committed to my tentative conclusions than to the process for arriving at them, and would gladly suspend any of my own tentative conclusions in exchange for a broad commitment by all engaged in political discourse and political activism to emphasize a shared commitment to reason in service to humanity."
The claim made by some that libertarians aren't against using government in limited ways to address our shared challenges and seize our shared opportunities, while insisting that the problem now is that we have "too much government," ignores the incredible breadth and depth of challenges and opportunities we face, challenges and opportunities that careful economic analysis clearly demonstrate often require extensive use of our governmental apparatus to meet and to seize. That is why every modern, prosperous, free nation on Earth has a large administrative infrastructure, and why every single modern, prosperous, free nation on Earth has had such a large administrative infrastructure in place since prior to participating in the historically unprecedented post-WWII expansion in prosperity and liberty: Because, as an empirical fact, that is what has thus far worked most effectively. But that does not preclude the possibility that the approach I've identified would lead to an overall reduction in the size and role of government; it only requires that in each instance the case be made, with methodological rigor, that any particular reduction in government actually does increase the robustness, fairness, and sustainability of our social institutional framework.
The challenge isn't to doggedly shrink government in service to a blind ideological conviction, but rather to wisely, with open eyes and informed analyses, refine our government by shrinking that which should be shrunk and expanding that which should be expanded, an ongoing endeavor which requires less ideological presumption and more analytical intelligence. We neither need nor benefit from neatly packaged blind dogmas; we need and benefit from an ever-greater commitment to disciplined reason in service to unflagging humanity.
Now, the legitimate contention arises that that is fine in theory, but in the real world of real people, ideological convictions and irrational decision-making prevail, and to refuse to fight the irrational and inhumane policies doggedly favored by some by any and all means possible, including strategies that do not hamstring themselves by seeking an ideal that does not prevail in this world today, is to surrender the world to the least enlightened and most ruthless. To that I respond that I do not oppose the strategic attempts by those who are informed by reason and humanity to implement the products of their discipline and conviction through strategic and realistic political means, but only implore of them two things: 1) That they take pains to ensure that their conclusions actually are the product of reason in service to humanity, and not simply their own blind ideological dogma, and 2) that they invest or encourage the investment of some small portion of our dedicated resources, some fraction of our time and money and energy directed toward productive social change, toward cultivating subtler cultural changes that increase the salience of reason and humanity in future political decision-making processes. I have outlined just such a social movement in A Proposal: The Politics of Reason and Goodwill.
Another legitimate contention is the recognition of our fallibility, and the need to rely on bedrock principles rather than arrogate to ourselves a case-by-case, issue-by-issue analysis, much as we limit our democratic processes with bedrock Constitutional principles that we can't elect to violate. There is much truth in this, but it either becomes one more rational consideration that we incorporate into our ongoing effort to do the best we can in a complex and subtle world, or it displaces our reason and humanity entirely and reduces us to automatons enslaved by a historically successful reduction of reality. We see these alternatives in regards to how the Bible and Constitution are utilized, by some as guides which inform their own reason and humanity and require conscious interpretation and application, and by others as rigid confirmation of their own dogmatic ideology, the latter often through selective or distorted interpretations of their own.
We've seen the value of improved methodology and increased commitment to methodological discipline in the realm of science, which has bestowed on us a greatly invigorated ability to make sense of a complex and subtle universe. We've seen the value of improved procedures and procedural discipline in law, which has increased the justness of our criminal justice system (certainly an improvement over "trial by ordeal," or the Inquisitor's securing of a confession by means of torture, for instance). We've seen the value of improved methodologies in selecting and holding accountable political leaders, through carefully monitored "free and fair" elections and the supremacy of the rule of law over individual power. To be sure, all of these are mere steps forward, not completed journeys; the human foibles they partially mitigated are not entirely erased from the new paradigms they preside over. But they are steps forward.
And, though it's more debatable, with more and greater atrocities seeming without end challenging the assertion, I think our humanity has grown in recent centuries as well. Historians almost universally agree that a larger proportion of the human population suffered violent death the further back in time you go. Even while exploitation and inhumanities persist, they are increasingly viewed as morally reprehensible by increasing numbers of people in increasing regions of the Earth. We have, indeed, as a national and international society, improved our formal commitment to human rights, even if our realization of that commitment has woefully lagged behind. It remains incumbent on us to close that gap between the ideal and the reality.
What, then, are the logical next steps for civilization? How do we advance the cause of reason in service to humanity? The answer, I believe, is to extend and expand the domains of these methodologies and attitudes, to increase the degree to which they are truly understood to be the defining vehicle of human progress. If it's good to have a small cadre of professionals engaging in science, it's even better to have many more incorporating more of that logic into their own opinion formation process. If it's good for the election of office holders to be conducted through rational procedures, it's even better for the knowledge and reasoning of those who vote in those elections to be fostered through more rational procedures as well. And if it's good for some of us to include larger swathes of humanity in the pronoun "we," then it's even better for more of us to do so to an ever greater degree.
Even if the effort to cultivate a movement in this direction only succeeds, over the course of generations, in making the tiniest marginal increase in the use of disciplined reason, and the tiniest increases in the degree of commitment to our shared humanity, by the tiniest marginal fraction of the population, that would be a positive achievement. And if, alongside such marginal increases in the reliance on disciplined reason and commitment to humanity, there is also a marginal increase in the acknowledgement that the products of disciplined reason are more useful to us as a society and a people than the products of arbitrary bigotries and predispositions, and that the recognition of the humanity of others unlike us is more morally laudable than our ancient tribalistic and sectarian reflexes, that, too, would be a positive achievement.
The influence of reason in our lives has been growing steadily for centuries and has had a dramatic impact on our social institutional and technological landscape, though it has only really ever been employed in a disciplined way by a small minority of the human population. The increase in our humanity as well, in such forms as the now nearly universal condemnation of slavery, the increasing recognition of the value of equal rights for all, the generational changes in our own society with some bigotries withering with time, can also be discerned. Even marginal increases in the employment of reason and its perceived legitimacy, and of our shared humanity being the ends to which it is employed, can have very dramatic effects on the robustness, fairness, and sustainability of the social institutional and technological landscape of the future, and on the welfare of human beings everywhere for all time. This is the path that all of our most laudable achievements of the past have followed and contributed to, and it is the path we should pursue going forward ever more consciously and intentionally, because that is what the ever fuller realization of our humanity both requires of us and offers us the opportunity to do.
Posted in Economics, Issues, Law and Morality, Political Forum, Science and Technology, Society and Culture, The Human Mind | Tags: cult, cults, discipline, evangelical christianity, fanaticism, fundamentalist Islam, humanity, ideology, John Rawls, legal procedure, libertarian, marxism, methodology, procedure, reason, rule of law, scientific methodology, theory of justice | No Comments »
How Much Racism Is There On The Far Right?
March 3, 2013 | Author Steve Harvey
Buy my e-book A Conspiracy of Wizards
One of the subtexts running through the current meta-debate between the Left and the Right is a constant volleying back and forth of accusations and refutations of racism. The Left accuses the Right of racism for a variety of reasons that I partially capture below. The Right indignantly denies it, retaliating with accusations back, insisting that "playing the race card" is the real expression of racism.
Personally, I think this discussion is generally overdone and often distracting, but the thread of validity in the criticism by the Left of the Right, and the reinforcement of irrationality and counterfactuality in the Right's response, motivates me to give it a comprehensive treatment.
First, it is important to explore the concept of "racism" itself. If, by "racism," we mean only explicit, overt, self-conscious antipathy toward members of another race, then I'd say that only a small minority of politically active people of either major partisan camp are "racist." The vast majority denounce such crude racism, and the extant but dwindling population of such unreconstituted racists in the population at large are not a significant political force anymore.
Before I turn to the more implicit forms of racism that I think do continue to play a significant, if not central, role in political affairs, I'd like to emphasize that I think that the ideological thread most prominent in right-wing thought isn't racism proper at all, but rather what I'll call "quasi-racism," an intense in-group/out-group bias, informing a set of beliefs and positions that are very tribalistic, and very dismissive of "the other." The antagonistic attitude toward numerous non-racial outgroups (though sometimes with strong racial associations), such as gays, Muslims, undocumented immigrants, foreigners in general, the poor, atheists, and, basically, anyone who isn't perceived to be an in-group member, is one of the most prominent defining characteristics of modern right-wing thought.
Explicit racism, however, is not absent from the right-wing echo-chamber. On a Facebook thread following one posting of the statistic that a gun in the home is 43 times more likely to be the instrument of the death of a member of the household than to be used in self-defense, for instance, one commenter responded to another by referring to "a group of n*****s raping your boyfriend" (the point being that you'd want to have a gun handy in that apparently representative scenario). On another thread at another time, a southern Tea Partier included among the problems besetting us "ungrateful blacks." These are not isolated examples: While such explicit expressions of racism are not the norm, they recur at a constant rate on such threads, always, of course, by right-wing commenters slipping over a line many others approach without crossing.
In the wake of the Trayvon Martin shooting, there was a Facebook wall post of a news story about a trio of "scary" black violent offenders, apparently being used to make the argument that it is understandable that armed vigilantes should go out in their neighborhoods and pursue unarmed black teens walking home from the store –even if the price of such "liberty" is the occasional shooting death of one such unarmed black teen– because, in their unself-aware but deep-rooted world view, it's rational to be afraid, it's rational to presume that a hoodie-wearing black teen walking through your neighborhood is up to no good, and so it is, implicitly, rational to provoke a deadly encounter with said black teen under those circumstances.
In other words, the right-wing insistence that it's a non-issue that their ideology can lead to instances of overzealous vigilantes pursuing and killing unarmed black teens walking home from the store is an astounding illustration of an underlying –and effectively racist– defect in their ideology. (The contention that it's a non-issue because it was allegedly self-defense on the shooter's part neglects the fact that the alleged need for self-defense was indisputably created by the decision to go out with a gun and pursue the arbitrarily "suspicious looking" unarmed black teen in the first place.)
These same people champion Jim-Crow-like voter suppression laws (on a discredited pretext and repeatedly struck down by the courts as unconstitutional), use code words like "Chicago politics" and "ACORN" and other allusions to blacks-as-inherently-corrupt, advocate discrimination against Muslims (and denial of their first amendment freedom of religion rights), frequently vilify and denegrate Hispanics, want to deny civil rights to gays, and, in general, are committed to a tribalistic orientation to the world, in which the small in-group of overwhelmingly white, mostly male, almost exclusively Judeo-Christian bigots opposes the rights and aspirations of the myriad out-groups surrounding them, denying the reality of a legacy of historical injustices and of current inequities, fighting for a regressive, aggressive, compassionless, irrational, barbaric society, in which those who feel well-served by the status quo (or, more precisely, by the status quo of a previous era) fight to recover an archaic -if all too recent– social order more preferential to their in-group statuses.
And they do so by disregarding fact and reason; by dismissing as bastions of liberalism precisely those professions that methodically gather, verify, analyze, and contemplate information (which, as a liberal, I take as a complement and as an affirmation of how much more rational our ideology is than theirs); by selecting, revising, and ignoring historical data to serve their fabricated ideological narrative; by ignoring the weight of professional economic theory and analysis (prompting the free-market-advocacy Economist magazine to label them "economically illiterate and disgracefully cynical"); by cherry-picking, reinterpreting, and selectively disregarding constitutional provisions and phrases in service to that same ideological narrative; and, in general, by defying fact and reason in service to ignorance and bigotry.
Whether we emphasize the racist overtones, the more explicit in-group/out-group tribalism in general, or just the prevailing ignorance and brutality of their ideology, the final evaluation is the same: It's a perfect storm of organized irrationality in service to implicit and explicit inhumanity. And it's not who and what we should choose to be as a people and a nation.
So, how much racism is there on the far right? It's a moot point; the racism is enveloped by so much more that is the very cloth from which racism is cut that the accusation of racism is too narrow a focus and too much of a distraction. Emphasizing the broader irrational inhumanity that defines this ideological camp both captures and goes beyond the identification of the racist overtones within it.
(For more on these themes, see The New Face Of American Racism, The Tea Party's Neo-"Jim Crow", The History of American Libertarianism, The Presence of the Past, Godwin's Law Notwithstanding, Basal Ganglia v. Cerebral Cortex, Basal Ganglia Keeping Score, and "Sharianity")
Posted in Civil Rights and Responsibilities, Immigration and Multiculturalism, Law and Morality | Tags: ACORN, anti-muslim, attribution theory, bigotry, chicago style politics, civil rights, conservatives, evangelical christianity, evangelicals, gay rights, hispanics, immigration, implicit racism, in-groups, jim crow, Muslims, out-groups, racism, right-wing, voter suppression | No Comments »
Small Government Idolatry
November 9, 2010 | Author Steve Harvey
(This is the fourth in a series of four posts which discuss Tea Party "Political Fundamentalism", comprised of the unholy trinity of "Constitutional Idolatry", Liberty Idolatry, and Small Government Idolatry.)
To recap briefly, "Political Fundamentalism" is the mutation of christian fundamentalism that allows it to appeal more broadly to the highly secularized by equally dogma-reliant anti-intellectual populism that permeates our culture. Whereas there has long been cause for some concern about the fanaticism and cooptation by the Republican Party of right-wing evangelicals, I had always maintained that dogmatic ideology rather than merely religious fanaticism was the real problem, and that religious fanaticism in our highly secularized society could only go so far. This mutation into a secular fanaticism, equally rigid and dysfunctional, equally tyrannical, and equally anti-intellectual, is far greater cause for concern.
Political Fundamentalism is the continuation of the Inquisition, adapting to a changing world in an attempt to prevent the world itself from adapting to changing circumstances and insights, creating an obstruction to the continuation of the growth and application of the Scientific Revolution and the Enlightenment. Political Fundamentalism can be found all over the political ideological spectrum, just as religious fundamentalism can be found all over the religious spectrum, and, in both cases, the differences in ideological particulars are less compelling than the similarities in attitude. But the currently most dangerous form of Political Fundamentalism in America is the right-wing version, comprised of the three elements already named.
"Constitutional Idolatry," the first element I wrote about, is the conversion of an historical document meant to provide a somewhat flexible legal doctrine and framework into a sacred text the caricature of which must be rigidly adhered to according to some non-existent and impossible literal interpretation. And "Liberty Idolatry," the second element I wrote about, is the reduction of the concept of "liberty" to one divorced from consideration of interdependence and mutual responsibility, defending freedoms independently of consideration of the harm they may inflict on others or on all.
The third element in the unholy trinity of Political Fundamentalism is Small Government Idolatry. It is a fixed belief that smaller government is always better, that lower taxes and less spending are always better, that "government is the problem" (as Ronald Reagan famously proclaimed, ushering in a movement that will long be the bane of our attempts at designing and implementing reasonable proactive policies and public investments). Like its strongly intertwined fellow travelers, Constitutional Idolatry and Liberty Idolatry, it is a fixed belief, impervious to reason and evidence, insulated from compelling counterarguments or sensible attempts to achieve balance and moderation. It is a force for the contraction of the human mind, opposition to reason and knowledge, and obstruction of progress, at a very real and tragic cost in increased human suffering and decreased human welfare.
An argument against Small Government Idolatry is not an argument for big government (just as an argument against Constitutional Idolatry is not an argument against the Constitution, and an argument against Liberty Idolatry is not an argument against liberty). It is an argument in favor of doing the analysis, in favor of applying our principles knowledgeably and rationally in the context of a complex and subtle world, on a case-by-case basis. It is an argument for facing the responsibilities we have to one another and to future generations, utilizing authentic economic analyses rather than ideological pseudo-economic platitudes to balances the demands imposing themselves on government against the real economic and fiscal constraints that must discipline how these demands are met.
A blind commitment to "small government" is both humanly and fiscally irresponsible, for most economists, other social scientists, and lawyers recognize the inevitably large role that modern governments must play in modern economies, even independently of the demands that a commitment to social justice and improved equity impose on them. I've frequently referenced the role of information asymmetries in creating an absolute imperative that we continue to develop our regulatory infrastructure to keep pace with the opportunities to play the market system to individual advantage at sometimes catastrophic public expense. We've seen examples in the Enron-engineered California energy crisis of 2000-2001, and the financial sector collapse that nearly catalized a second Great Depression in 2008. Designing, implementing, and enforcing functional rules of the game for our complex market economy is an essential function of government, and one which already destroys the notion that a government too small too meet that need is preferable to one large enough to do so.
It is also fiscally, as well as humanly, irresponsible to let the problems of extreme poverty, child abuse and neglect, frequently unsuccessful public schools, high rates of violent crime, poor public health and inadequate healthcare for many, and other similar and related social problems, all of which form a mutually reinforcing matrix of dysfunctionality and growing problems that both undermine the safety and welfare of us all, and end up costing us far more to react to (with astronomical rates of very expensive incarceration, and other costs of dependency and predation) than it would have cost us to proactively address.
The fiscal concerns that the Political Fundamentalists identify are not to be disregarded, or treated as irrelevant, but rather are one set of considerations among many, to be included in a complete analysis rather than treated as always and forever dispositive independently of any application of reason or knowledge to the question of whether it is actually dispositive or not. The challenge of self-governance requires utilizing our fully developed and focused cognitive capacities, applied to all available information, in pursuit of intelligent and well-conceived policies. It is undermined by the imposition of an a priori set of fixed certainties that are impervious to both knowledge and reason.
We need, in our political discourse, less fundamentalism and more analysis, less idolatry and more (and better) methodology, less false certainty and more foundational humility. We need less deference to fixed and static beliefs, and more to our process by which we test our beliefs and improve upon them. We need less commitment to ideologies, and more commitment to working together as reasonable people of goodwill, doing the best we can to confront the challenges and opportunities of a complex and subtle world.
Posted in Civil Rights and Responsibilities, Law and Morality, Political Forum | Tags: administrative state, anti-intellectualism, constitutional idolatry, economic analysis, evangelical christianity, fiscal responsibility, information asymmetries, Political Fundamentalism, populism, proactive policies, reactive policies, small government, social problems, Tea Party | No Comments » | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,020 |
package org.jbpm.kie.services.impl.admin.commands;
import org.jbpm.services.api.model.UserTaskInstanceDesc;
import org.jbpm.services.task.commands.TaskContext;
import org.jbpm.services.task.commands.UserGroupCallbackTaskCommand;
import org.jbpm.services.task.events.TaskEventSupport;
import org.jbpm.services.task.exception.PermissionDeniedException;
import org.jbpm.services.task.utils.ContentMarshallerHelper;
import org.kie.api.runtime.Context;
import org.kie.api.task.model.Content;
import org.kie.api.task.model.OrganizationalEntity;
import org.kie.api.task.model.Task;
import org.kie.internal.task.api.ContentMarshallerContext;
import org.kie.internal.task.api.TaskModelProvider;
import org.kie.internal.task.api.TaskPersistenceContext;
import org.kie.internal.task.api.model.ContentData;
import org.kie.internal.task.api.model.InternalContent;
import org.kie.internal.task.api.model.InternalTask;
import org.kie.internal.task.api.model.InternalTaskData;
import java.util.ArrayList;
import java.util.List;
import java.util.Map;
public class UpdateTaskCommand extends UserGroupCallbackTaskCommand<Void> {
private static final long serialVersionUID = -1856489382099976731L;
private UserTaskInstanceDesc userTask;
private Map<String, Object> inputs;
private Map<String, Object> outputs;
public UpdateTaskCommand(Long taskId, String userId, UserTaskInstanceDesc userTask, Map<String, Object> inputs, Map<String, Object> outputs) {
super();
setUserId(userId);
setTaskId(taskId);
this.userTask = userTask;
this.inputs = inputs;
this.outputs = outputs;
}
@SuppressWarnings("unchecked")
@Override
public Void execute(Context cntxt) {
TaskContext context = (TaskContext) cntxt;
TaskEventSupport taskEventSupport = context.getTaskEventSupport();
TaskPersistenceContext persistenceContext = context.getPersistenceContext();
Task task = persistenceContext.findTask(taskId);
// security check
if (!isBusinessAdmin(userId, task.getPeopleAssignments().getBusinessAdministrators(), context)
&& !isOwner(userId, task.getPeopleAssignments().getPotentialOwners(), task.getTaskData().getActualOwner(), context)) {
throw new PermissionDeniedException("User " + userId + " is not business admin or potential owner of task " + taskId);
}
taskEventSupport.fireBeforeTaskUpdated(task, context);
// process task meta data
if (userTask.getFormName() != null) {
((InternalTask) task).setFormName(userTask.getFormName());
}
if (userTask.getName() != null) {
((InternalTask) task).setName(userTask.getName());
}
if (userTask.getDescription() != null) {
((InternalTask) task).setDescription(userTask.getDescription());
}
if (userTask.getPriority() != null) {
((InternalTask) task).setPriority(userTask.getPriority());
}
if (userTask.getDueDate() != null) {
((InternalTaskData) task.getTaskData()).setExpirationTime(userTask.getDueDate());
}
// process task inputs
long inputContentId = task.getTaskData().getDocumentContentId();
Content inputContent = persistenceContext.findContent(inputContentId);
Map<String, Object> mergedContent = inputs;
if (inputs != null) {
if (inputContent == null) {
ContentMarshallerContext mcontext = context.getTaskContentService().getMarshallerContext(task);
ContentData outputContentData = ContentMarshallerHelper.marshal(task, inputs, mcontext.getEnvironment());
Content content = TaskModelProvider.getFactory().newContent();
((InternalContent) content).setContent(outputContentData.getContent());
persistenceContext.persistContent(content);
((InternalTaskData) task.getTaskData()).setOutput(content.getId(), outputContentData);
} else {
ContentMarshallerContext mcontext = context.getTaskContentService().getMarshallerContext(task);
Object unmarshalledObject = ContentMarshallerHelper.unmarshall(inputContent.getContent(), mcontext.getEnvironment(), mcontext.getClassloader());
if(unmarshalledObject != null && unmarshalledObject instanceof Map){
((Map<String, Object>)unmarshalledObject).putAll(inputs);
mergedContent = ((Map<String, Object>)unmarshalledObject);
}
ContentData outputContentData = ContentMarshallerHelper.marshal(task, unmarshalledObject, mcontext.getEnvironment());
((InternalContent)inputContent).setContent(outputContentData.getContent());
persistenceContext.persistContent(inputContent);
}
((InternalTaskData)task.getTaskData()).setTaskInputVariables(mergedContent);
taskEventSupport.fireAfterTaskInputVariablesChanged(task, context, inputs);
}
if (outputs != null) {
// process task outputs
context.getTaskContentService().addOutputContent(taskId, outputs);
}
persistenceContext.updateTask(task);
// finally trigger event support after the updates
taskEventSupport.fireAfterTaskUpdated(task, context);
return null;
}
protected boolean isOwner(String userId, List<OrganizationalEntity> potentialOwners, OrganizationalEntity actualOwner, TaskContext context) {
List<String> usersGroup = new ArrayList<>(context.getUserGroupCallback().getGroupsForUser(userId));
usersGroup.add(userId);
if (actualOwner != null) {
boolean isOwner = userId.equals(actualOwner.getId());
if (isOwner) {
return true;
}
}
return potentialOwners.stream().anyMatch(oe -> usersGroup.contains(oe.getId()));
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,094 |
Q: How to access contacts API with Service Account I am in the process of upgrading my Marketplace applications to support the new marketplace api and OAUTH 2.
I have managed to migrate most APIs but am stuck on the contacts api. With the the previous marketplace version we used 2LO and client key/client secret to authenticate across the Google Apps domain. My understanding is that the only way to do this in current version is with Service Accounts and OAuth 2.
Based on the V3 calendar API I'm assuming something like this (although the contacts API does not support it from what I can see) -
ServiceAccountCredential credential = new ServiceAccountCredential(
new ServiceAccountCredential.Initializer(serviceAccountEmail)
{
Scopes = new[] { "https://www.google.com/m8/feeds" },
User = "admin@domain.co"
}.FromCertificate(certificate));
var service = new ContactsService(new BaseClientService.Initializer()
{
HttpClientInitializer = credential,
ApplicationName = "Contact API Sample",
});
If anybody has done this, your advice will be appreciated!
A: I know this question has been answered, but adding an alternative that doesn't require the AssertionFlowClient or a specific version of DotNetOpenAuth. This allows you to use the same ServiceAccountCredential code from your original question (ie, using the same method to query a user's gmail)
Thank you for following up on your own post, it helped me find this solution!
private static ServiceAccountCredential GenerateCred(IEnumerable<string> scopes, string delegationUser)
{
var certificate = new X509Certificate2(certLocation, certPassword, X509KeyStorageFlags.Exportable);
var credential = new ServiceAccountCredential(
new ServiceAccountCredential.Initializer(serviceAccountEmail)
{
Scopes = scopes,
User = delegationUser
}.FromCertificate(certificate));
return credential;
}
var credential = GenerateCred(new[] { "https://www.googleapis.com/auth/contacts.readonly" }, userToDelegate);
//Get the token for this scope and user
await credential.RequestAccessTokenAsync(new CancellationToken());
//use the token to initalize the request
var rs = new RequestSettings(projectName)
{
OAuth2Parameters = new OAuth2Parameters()
{
AccessToken = credential.Token.AccessToken
}
};
var request = new ContactsRequest(rs);
Feed<Contact> f = request.GetContacts();
foreach (var c in f.Entries)
{
//process each contact
}
A: Well I got no response anywhere on this so I can only assume that the standard client libraries don't have support for this.
I have come up with a workaround based on the following post for email access.
http://www.limilabs.com/blog/oauth2-gmail-imap-service-account
You will need to follow his post and note the following:
*
*this must use the DotNetOpenAuth version that he specifies. The latest version will not work.
*You will only need his AssertionFlowClient classes which can be added to the below code I have provided here:
X509Certificate2 certificate = new X509Certificate2(
serviceAccountCertPath,
serviceAccountCertPassword,
X509KeyStorageFlags.Exportable);
AuthorizationServerDescription server = new AuthorizationServerDescription
{
AuthorizationEndpoint = new Uri("https://accounts.google.com/o/oauth2/auth"),
TokenEndpoint = new Uri("https://accounts.google.com/o/oauth2/token"),
ProtocolVersion = ProtocolVersion.V20,
};
AssertionFlowClient provider = new AssertionFlowClient(server, certificate)
{
ServiceAccountId = serviceAccountEmail,
Scope = string.Join(" ", new[] { "https://mail.google.com/", "https://www.google.com/m8/feeds/" }),
ServiceAccountUser = userEmail,
};
IAuthorizationState grantedAccess = AssertionFlowClient.GetState(provider);
RequestSettings rs = new RequestSettings("iLink");
rs.OAuth2Parameters = new OAuth2Parameters();
rs.OAuth2Parameters.AccessToken = grantedAccess.AccessToken;
rs.OAuth2Parameters.RefreshToken = null;
rs.OAuth2Parameters.ClientId = null;
rs.OAuth2Parameters.ClientSecret = null;
rs.OAuth2Parameters.RedirectUri = null;
ContactsRequest cr = new ContactsRequest(rs);
Feed<Contact> f = cr.GetContacts();
foreach (var c in f.Entries)
{
Response.Write(c.Name.FullName);
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,309 |
Der Italiensperling (Passer italiae) ist eine Populationsgruppe von Sperlingen, die eine große Ähnlichkeit gegenüber dem Weidensperling (Passer hispaniolensis) und dem Haussperling (Passer domesticus) aufweist. Anzutreffen ist der Italiensperling auf der Apenninhalbinsel, auf Korsika sowie auf Kreta. Er kommt dort weitestgehend ohne die beiden nahe verwandten Arten Weidensperling und Haussperling vor und vertritt somit diese beiden Arten in dem Gebiet. Der taxonomische Status des Italiensperling ist äußerst umstritten, er wurde und wird neben der Klassifizierung als eigenständige Art sowohl als Unterart des Haus- als auch des Weidensperlings angesehen. Auch die evolutionäre Entstehung des Italiensperlings ist umstritten, seit längerer Zeit wird unterstellt, er sei durch eine stabilisierte Hybridisierung aus Haus- und Weidensperling entstanden, was allerdings heute stark bezweifelt wird.
Aussehen
Im Aussehen liegt der Italiensperling zwischen seinen beiden vermuteten Elternarten und ist von anderswo spontan entstehenden Hybriden kaum zu unterscheiden: Wie beim Weidensperling sind Kopf, Stirn und Nacken lebhaft kastanienbraun, zuweilen auch rötlichbraun. Die Wangen sind fast reinweiß, nicht schmutziggrauweiß wie beim Haussperling. Ein feiner weißer Überaugenstreif ist meist deutlich erkennbar. Der Kehlfleck ist rein schwarz, der Brustlatz deutlicher schwarz geflockt als beim Haussperling. Im sonstigen Körpergefieder ähnelt der Italiensperling wieder sehr dem Haussperling, doch sind bei ihm Hinterrücken und Bürzel ebenfalls bräunlich und nicht grau. Die Weibchen des Italiensperlings lassen sich im Feld von denen des Haussperlings kaum, von denen des Weidensperlings nur sehr schwer unterscheiden.
Verbreitung
Auf der Apenninhalbinsel ist der Italiensperling die häufigste Vogelart. Seine Verbreitung nach Norden wird durch den Alpenbogen begrenzt. Südlich der Alpen besteht eine recht abrupte, ungefähr 35 bis 40 Kilometer breite Übergangszone zwischen den Populationen des Haus- und Italiensperlings, in der es auch häufig zu Bastardisierungen kommt. Phänotypisch reine Italiensperlinge erreichen auch Tiroler Täler nördlich der Alpen und erscheinen vereinzelt auch in Südkärnten.
Im Gegensatz zum recht abrupten Artübergang zum Haussperling im Norden ist in der Mitte und im Süden Italiens der Italiensperling durch eine breite fließende Übergangszone mit dem Weidensperling verbunden.
Die Italiensperlinge Korsikas ähneln in ihrem Aussehen denen Norditaliens, und der Übergang zu den Weidensperlingen auf Sardinien ist hier recht deutlich. Auch Kreta und Peloponnes sind vom Italiensperling besiedelt.
Lebensraum
Wie der Haussperling ist der Italiensperling ein Bewohner der Städte, Dörfer und landwirtschaftlichen Anwesen. Er ist ein ausgesprochener Standvogel ohne Tendenz zur nomadisierenden Lebensweise des Weidensperlings.
Ernährung
Seine Nahrung besteht aus Sämereien aller Art; tierische Nahrung wird je nach Verfügbarkeit in unterschiedlichem Ausmaße aufgenommen, ihr Anteil an der Gesamtmenge übersteigt jedoch 10 % nicht wesentlich.
Taxonomische Diskussion
Die taxonomische Einordnung des Italiensperlings ist seit seiner Beschreibung vor nahezu 200 Jahren stets umstritten gewesen und auch heute noch keinesfalls geklärt.
Das Aussehen des Federkleids der Männchen des Italiensperlings liegt eindeutig zwischen Haus- und Weidensperling; die Weibchen aller drei Arten hingegen sind sich sehr ähnlich. Haus- und Weidensperling gelten als unterschiedliche Arten, da sie in vielen Gebieten sympatrisch vorkommen, ohne dass es zu Hybridisierungen kommen würde. Alles Weitere ist umstritten, die wesentlichen Fragen werden im Folgenden erörtert.
Wie ist der Italiensperling entstanden?
Die auch heute noch gängigste Hypothese wurde vor allem durch Wilhelm Meise geprägt, der 1936 die These aufstellte, der Italiensperling sei eine stabilisierte Hybridform aus Haus- und Weidensperling. Er begründete diese These unter anderem damit, dass es in Oasen in Ostalgerien und Tunesien auch heute noch zur Hybridisierung zwischen Weidensperling und der Unterart tingitanus des Haussperlings kommt, und dass dortige Hybride dem Italiensperling im Aussehen sehr ähneln. Auch gibt es im Süden Italiens eine breite Übergangszone zwischen Weiden- und Italiensperling sowie im Norden eine Zone, in der Italien- und Haussperling hybridisieren. Weiterhin ähneln die Italiensperlinge im Norden mehr den Haussperlingen, die im Süden mehr den Weidensperlingen, und es gibt dabei einen graduellen Übergang. Somit scheint der Italiensperling beiden vermeintlichen Vorläuferarten sehr nahezustehen.
Alle Szenarien zur Erklärung der hybridogenen Entstehung des Italiensperling unterstellen dabei drei Phasen: In der ersten Phase gibt es eine ausgiebige Hybridisierung zwischen beiden Elternarten, in der zweiten Phase kommt es zu einer geographischen Isolation der Hybridform (beispielsweise durch zunehmende Vergletscherung während des Pleistozän), die sich dadurch stabilisieren kann. In der dritten Phase kommt die Hybridform wieder in Kontakt mit beiden Elternarten, und es kommt wiederum zu Hybridisierungen, die die Abgrenzungen der Arten beeinflussen.
In der Pflanzenwelt ist eine Artbildung durch Hybridisierung keine Seltenheit, jedoch in der Vogelwelt haben sich bislang alle derartigen Vermutungen nicht bestätigen lassen.
Dennoch wurde diese These der Entstehung des Italiensperlings akzeptiert und sogar als Paradebeispiel der Artentstehung durch stabilisierte Hybridisierung angesehen.
Die Arbeit Meises beeinflusste die Ornithologen in der Folgezeit sehr. Erst lange Zeit später wurde zuerst von Burkard Stephan darauf hingewiesen, dass die Argumentation Meises einen Zirkelschluss enthält, da dieser bereits in der Ausgangsposition seiner Arbeit einen Merkmalsindex so definiert, dass der Italiensperling genau zwischen Weiden- und Haussperling angesiedelt ist und damit die Artbildung durch Hybridisierung implizit schon unterstellt.
Alternativ wird deshalb auch eine "herkömmliche" Artentstehung des Italiensperlings angenommen, wobei dabei weiterhin ungeklärt ist, ob dieser eine eigenständige Art oder eine Unterart von Haus- oder Weidensperling ist. Eine weitere Hypothese in diesem Zusammenhang ist, dass der Italiensperling ein Zwischenstadium bei der Artentrennung von Haus- und Weidensperling darstellt.
Wem steht der Italiensperling näher?
Unabhängig von Entstehung und taxonomischem Status ist auch die Frage, ob der Italiensperling dem Haus- oder dem Weidensperling näher steht, bis heute umstritten.
Aus ökologischer Sicht scheint der Italiensperling dem Haussperling näher zu stehen, denn er ist, wie dieser, auch ein ausgesprochener Kulturfolger. Die breite Zone fließenden Übergangs zwischen Italien- und Weidensperling im Süden Italiens sowie der recht abrupte Übergang zwischen Italien- und Haussperling in den Alpen lassen allerdings eher auf ein näheres Verwandtschaftsverhältnis zwischen Weiden- und Italiensperling schließen, wobei anzumerken ist, dass Wilhelm Meise 1936 den abrupten Übergang zwischen Haus- und Italiensperling in den Alpen etwas willkürlich als Grund ansah, den Italiensperling dem Haussperling als Unterart zuzuordnen, was wiederum viele Forscher in der Folgezeit beeinflusste.
In der Vergangenheit durchgeführte wissenschaftliche Untersuchungen kommen zu unterschiedlichen Ergebnissen. Viele am Ende des 20. Jahrhunderts veröffentlichte Forschungsarbeiten, die neben den klassischen Arbeitsweisen auch Bioakustik, Reproduktionsbiologie, Molekulargenetik und Chromosomenstudien einbezogen, zeigen, dass der Italiensperling viel mehr Gemeinsamkeiten mit dem Weidensperling als mit dem Haussperling hat.
Im Gegensatz dazu steht eine 1988 durchgeführte Untersuchung der Genabschnitte von 15 polymorphen Isozymen, die für die Abspaltung des Italien- vom Haussperling eine Zeitspanne von nur 15.300 Jahren und im Gegensatz dazu die Abspaltung des Italien- vom Weidensperling auf vor 113.300 Jahren datiert.
Im Jahr 2001 wurden die mitochondriale Gen-Sequenz des Cytochrome-b und weitere mitochondriale Pseudogene von mindestens zwei verschiedenen Individuen der beteiligten Arten untersucht. Auch diese Analysen zeigen weniger Unterschiede zwischen Italien- und Haussperling (Abweichung 0,54 %) als zwischen Italien- und Weidensperling (Abweichung 2,66 %) und legen somit eine engere Verwandtschaftsbeziehung zwischen Haus- und Italiensperling nahe. Diesen beiden molekulargenetischen Arbeiten widerspricht wiederum eine 2002 durchgeführte DNA-Analyse von Mikrosatelliten, die den Ursprung des Italiensperlings aus einer Weidensperlingspopulation nahelegt.
Ist der Italiensperling eine eigenständige Art?
Ursprünglich beschrieben wurde der Italiensperling als eigenständige Art (Fringilla italiae Vieillot 1817). In der Zwischenzeit wurde er – und wird noch immer – als Unterart des Haussperlings (P. domesticus italiae) oder als Unterart des Weidensperlings (P. hispaniolensis italiae) betrachtet. Der These der hybridogenen Entstehung Rechnung tragend, wird er auch als Passer x italiae klassifiziert. Um dem Problem der unklaren Artentstehung aus dem Wege zu gehen, sieht man ihn vor allem aus praktischen Gründen auch als eigene Art (Passer italiae).
Für die Klassifizierung als Unterart des Weidensperlings spricht heute vieles, auch der fließende Übergang zwischen den Sperlingspopulationen im Süden Italiens. Anzumerken ist hierbei, dass entsprechend den taxonomischen Prioritätsregeln die korrekten Bezeichnungen für Italien- und Weidensperling eigentlich Passer italiae italiae und Passer italiae hispaniolensis lauten müssten, da nomenklatorisch bei Arten mit Unterarten der ältere Name als Artname verwendet wird.
Interessant ist weiterhin, dass in den 1990er Jahren im mittleren und nördlichen Italien bei Foggia und im Delta des Po je eine reine Weidensperlingspopulation eingewandert ist, deren Bestand stetig anwächst. Diese Weidensperlinge scheinen sich nicht mit den dort lebenden Italiensperlingen zu vermischen. Entsprechend den aufgestellten Regeln ist aber für Neozoen ein Zeitraum von wenigstens 25 Jahren oder 3 Generationen abzuwarten, um diese als etabliert zu betrachten. Dann wäre der Artstatus des Italiensperlings neu zu überdenken.
Einzelnachweise
Literatur
Till Töpfer: Die Geschichte vom Italiensperling; In: Der Falke 54; 250–256; Ausgabe 07/2007;
U. N. Glutz von Blotzheim, K. M. Bauer: Handbuch der Vögel Mitteleuropas (HBV). Band 14-I: Passeriformes. 5. Teil. AULA-Verlag 1997, ISBN 3-923527-00-4.
Weblinks
Eintrag bei der Schweizerischen Vogelwarte
Federn des Italiensperlings
Sperlinge | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,156 |
Album: "Into the Dying of Time"
Valyria: «Valyria is a ruined city in the world of Essos, the world in which George R.R. Martin's "A Song of Fire and Ice" takes place. Valyria is a city of wonderment, magic and untold power. The aesthetic Martin creates within his novels is a similar atmosphere to that we as a band try to conjure with our music.
Δελτίο τύπου: «With the world becoming a darker place by the day, we need more metal bands that deal in Realistic Escapism; that is escapism that draws from events in the real world (see Joseph Campbell's seminal "Hero With A Thousand Faces" for more) and helps the listeners cope heroically with their own daily battles. The aural equivalent of the work of George R.R. Martin! | {
"redpajama_set_name": "RedPajamaC4"
} | 318 |
Q: Get current element from a service (cycle Hook problems) Hello i have some troubles with angular cycle hook.
this is the Stackblitz of my test:
https://stackblitz.com/edit/angular-vpoyqm
This is the logic:
I have two components and 1 service for store data
1) billings.component
2) billing.component
In billings.component i have a method setCurrentBilling for sent the current billing to the service and show the billing (ngIF).
In billing.component i need to get this current component from the service. if i try to console it in the ngOnInit, the property current is undefined the first time.
what is the step for get the current billing at the first click please ?
Thank you for your precious help
A: Your template:
<div *ngFor="let billingItem of billings" class="LG_invoice-item" #billing (click)="setCurrentBilling(billingItem)" >
<div class="LG_form-title" (click)="billing.value = !billing.value">
{{billingItem.month}} {{billingItem.years}}
</div>
<app-billing *ngIf="billing.value" ></app-billing>
</div>
So you have a parent div and a nested one, both with a click listener.
The first one to be executed is the inner one:
(click)="billing.value = !billing.value"
Which makes the template to be re-rendered, showing now a new element:
<app-billing *ngIf="billing.value" ></app-billing>
So the ngOnInit of this element is executed:
ngOnInit() {
this.billingsService.log();
setTimeout(() => {
this.billingsService.log();
}, 0);
//console.log('currentelem: ', );
}
Here you still haven't set current. And then, after this first cycle is completed, the other click handler comes into action:
(click)="setCurrentBilling(billingItem)"
And after that, the setTimeout from the first click will be resolved.
Solution: a click handler could do both tasks:
<div *ngFor="let billingItem of billings" class="LG_invoice-item" #billing
(click)="billing.value = !billing.value;setCurrentBilling(billingItem)" >
<div class="LG_form-title">
{{billingItem.month}} {{billingItem.years}}
</div>
<app-billing *ngIf="billing.value" ></app-billing>
</div>
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,020 |
Officers were dispatched to the [Toys R Us] store shortly before 10 p.m., Wednesday on reports that the man had already assaulted three customers with [two 'Star Wars' light sabers]. None of them was injured.
Before officers arrived, dispatchers were told the man had walked out into the parking lot, still swinging the swords. Police found the man in the lot talking incoherently.
An officer tried to use a taser, but the device failed. A second Taser also failed after the man used the light sabers to break one of the wires, Simpson said.
You doubted The Juice? The light saber neutralized the taser! So, it had to be done the old-fashioned way.
Officers then rushed the man, taking him to the ground as he struggled violently and continued to shout nonsense.
David A. Canterbury, 33, was placed on a police hold at a hospital for a mental evaluation.
He was cited for three counts of fourth-degree assault, second-degree disorderly conduct, third-degree theft, resisting arrest and interfering with a police officer. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,286 |
Commentary: Moving on from Windows 98
Windows XP is a major upgrade from Windows 98 in the home market, providing improved stability, increased performance and a series of extras designed to help home and small-business users more easily manage their Web use, Web pages and, eventually, .Net services.
By META Group | August 27, 2001 -- 10:25 GMT (03:25 PDT) | Topic: Windows
It will appeal to consumers with its enhanced features for accessing rich media, plus its attractive "toys" such as tools for video editing and CD burning.
Under the covers, Windows XP is the final step in merging the two separate Windows lines that Microsoft was forced to create a decade ago. Essentially, it is a consumer-friendly version of the Windows 2000 code base, which marked a move to a unified 32-bit environment on the corporate desktop and thus the start of that merger. Now, the stability and maturity of Windows NT is being extended to home and small-business desktops.
Microsoft is preparing a media blitz for the retail release of Windows XP--planned for October--that will rival the launch of Windows 95. Microsoft and the PC makers hope that this will contribute to a major revival of the consumer PC market in the Christmas buying season.
We expect some upturn in PC sales driven by Windows XP and rock-bottom prices, but not the huge volumes that Microsoft and PC makers are hoping for. Much will depend on the mood of consumers. If they are still concerned about the economy, they will be slow to make major purchases, regardless of pricing. If there is any uptick in consumers' confidence, they will be more likely to buy new PCs.
Windows XP includes several new facilities designed to improve users' experiences on the Web and office networks. For instance, the Web Digital Authoring and Versioning (WebDAV) technology is designed to facilitate finding, retrieving and manipulating files on the Web or intranet. Windows XP also contains a NetCrawler agent that automatically locates and installs printers on the Web or office network when a user connects a laptop to the network.
While these new features are not by themselves compelling reasons to upgrade, they do provide incremental improvements, especially for consumer PC users. These new features are also intended, in part, to facilitate use of .Net services when they begin to appear.
What businesses need to know
Windows XP has some attractive features for office users, although it does not represent a compelling upgrade for enterprises that have already moved to Windows 2000. For example, the operating system contains a standard 802.11 wireless networking stack, which makes accessing multiple networks easier and eliminates the need for third-party administrative tools to manage the connections, and a new remote desktop tool that enables support staff to remotely take over a PC to diagnose a problem. Companies, however, need to carefully control use of this feature to avoid security exposures.
One tactic Microsoft will be using to promote Windows XP, particularly in the corporate market, is to tie it to the company's.Net initiative for Web services. However, while the new software has features that complement .Net, the linkage is mostly marketing. Future versions beyond the current Windows 2000 and XP desktops will offer more concrete facilities to link the desktop to these Web services. Windows XP is not a necessary step toward .Net, and users can skip this particular version without compromising their ability to exploit .Net.
We believe that home and small-business users still running their PCs on Windows 98 should consider upgrading to XP, either by installing the US$99 upgrade on their existing hardware or by buying a new XP-based machine. For an optimal home experience, a relatively new machine with 256MB of RAM should be preferred to make use of all the new features. Users currently in the market for a new PC should, if possible, delay their purchase until mid-October, when the new XP machines should be plentiful. We expect pricing on those new machines to be very favorable by that time.
Companies that have not yet started upgrading to Windows 2000 should move directly to XP instead and consider it the latest Windows release once they have completed appropriate testing and certification. Companies that have already upgraded or are in the process of upgrading their desktops to Windows 2000 will probably have no compelling reason to do a second, unplanned upgrade to XP. However, they should consider switching new purchases of desktops and laptops to Windows XP Professional by the third quarter of 2002 to ensure access to the latest drivers and updates for those systems.
We do not believe a mixed Windows 2000/Windows XP environment, if managed carefully, will place any significant burden on IT groups. The newer version can be set up to look exactly like Windows 2000, and the two operating systems can coexist well on corporate desktops. In effect, enterprises should treat Windows XP as the next service pack for Windows 2000.
Meta Group analysts William Zachmann, Steve Kleynhans, Dale Kutnick, David Cearley and Val Sribar contributed to this article.
Visit Metagroup.com for more analysis of key IT and e-business issues.
Entire contents, Copyright © 2001 Meta Group, Inc. All rights reserved.
Microsoft Enterprise Software Windows 10 PCs Reviews
More from META Group
Pros and cons of the Linux PDA
Microsoft is enabling enterprise features by default in the Dev Channel builds of Chromium-based Edge. Here's what's available to IT pro testers now and what's coming later. ...
Microsoft gives IT pros the signal to start testing Chromium-based Edge
Microsoft has yet to release the promised 'beta' channel for its Chromium-based Edge browser, but it's now telling IT pros the Dev channel is ready for them to start testing it. ...
Recent Windows zero-day used by Buhtrap gang for cyber-espionage
Old school cybercrime-focused hacker group returns with cyber-espionage campaign.
Microsoft stirs suspicions by adding telemetry files to security-only update
This month's Security-only update package for Windows 7 includes an unexpected compatibility/telemetry component that has some skeptical users up in arms. ...
Microsoft is reorging its field sales team, laying off some 'Modern Desktop' salespeople
Microsoft is cutting a number of field sales jobs as part of one of its regular reorgs, specifically ones involving pushing Windows as a key part of their charter. ...
Microsoft to Windows 10 users: Patch Secure Boot now against 'critical' bug
Microsoft's latest SSU helps fix a bug in Secure Boot that interferes with Windows' BitLocker encryption system.
Windows 10 security: Bad bug in our CPU diagnostics app, so patch now, says Intel
Intel fixes high-severity security flaw in its CPU performance-testing software for Windows 10 systems.
Microsoft July 2019 Patch Tuesday fixes zero-day exploited by Russian hackers
Microsoft patches 77 security flaws, including 15 rated "critical." | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,288 |
JAKARTA (Yosefardi) – Plantation firm PT Eagle High Plantations Tbk (BWPT) allocates Rp140-160 billion for building crude palm oil (CPO) mill in East Kalimantan.
The CPO mill will have production capacity of 30 tons per hour. The construction will start in mid next year, to commence operation in late 2017 or early 2018.
BWPT is now building CPO mills in West Kalimantan and Papua which cost Rp200 billion and Rp250 billion respectively. The mills are designed to have production capacity of 90 tons per hour.
BWPT posted net loss of Rp94 billion in the third quarter of 2015 on falling prices of CPO and squeezed operating margins. Sales revenue only grew by 10.7% while gross profit and operating profit dropped 15% and 79% respectively. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,173 |
{"url":"http:\/\/mathhelpforum.com\/geometry\/81249-chords.html","text":"1. ## chords\n\nHello everyone,\n\nCould someone please show me how to do this one?\n\nFInd the chord(120 degrees) when the radius of the triangle is 1 and then when the radius is 3438. I know that the answer is the square root of 3 for the first one, but I'm not sure how to go about solving it.\n\nMy attempt:\n\nI know that the angles would have to be 120, 30, 30, but I'm not sure what to do next.\n\nThank you very much\n\n2. Originally Posted by Chocolatelover2\nHello everyone,\n\nCould someone please show me how to do this one?\n\nFInd the chord(120 degrees) when the radius of the triangle is 1 and then when the radius is 3438. I know that the answer is the square root of 3 for the first one, but I'm not sure how to go about solving it.\n\nMy attempt:\n\nI know that the angles would have to be 120, 30, 30, but I'm not sure what to do next.\n\nThank you very much\nThe \"radius of the triangle\"? Do you mean that the triangle is inscribed in a circle of radius 1? If so, then since 120= 360\/3, you have an equilateral triangle. Let x be the length of the chord (one side of the triangle) and let r be the radius of the circle. Dropping a perpendicular to one side from the opposite angle divides the triangle into two right triangle having hypotenuse of length x and one leg of length x\/2. By the Pythagorean theorem, the other leg, an altitude of the triangle, has length $\\frac{\\sqrt{3}}{2}x$.\n\nNow draw a line from the center of the circle to one of the other two vertices of the triangle. That, together with the altitude just drawn, gives a right triangle with hypotenuse r (the radius of the circle) and one leg of length x\/2. The other leg, along the altitude, again by the Pythagorean theorem, has length $\\sqrt{r^2- \\frac{x^2}{4}}$. But that altitude is that leg plus a radius: $\\frac{\\sqrt{3}}{2}x= r+ \\sqrt{r^2- \\frac{x^2}{4}}$ so that $\\frac{\\sqrt{3}}{2}x- r= \\sqrt{r^2- \\frac{x^2}{4}}$.\n\nSquaring both sides, $\\frac{3}{4}x^2- \\sqrt{3}xr+ r^2= r^2- \\frac{x^2}{4}$. Now the \" $r^2$\" terms cancel giving $\\frac{3}{4}x^2+ \\frac{1}{4}x^2= x^2= \\sqrt{3}xr$. Dividing both sides by x, $x= \\sqrt{3}r$.","date":"2016-12-03 08:07:44","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 8, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8490802645683289, \"perplexity\": 108.16683714324202}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-50\/segments\/1480698540909.75\/warc\/CC-MAIN-20161202170900-00395-ip-10-31-129-80.ec2.internal.warc.gz\"}"} | null | null |
package com.mbh.materialdialog;
import android.app.AlertDialog;
import android.content.Context;
import android.content.DialogInterface;
import android.graphics.Color;
import android.graphics.drawable.Drawable;
import android.os.Build;
import android.support.annotation.ColorInt;
import android.support.annotation.DrawableRes;
import android.support.annotation.LayoutRes;
import android.support.annotation.StringRes;
import android.view.LayoutInflater;
import android.view.View;
import android.view.ViewGroup;
import android.view.WindowManager;
import android.widget.Button;
import android.widget.ImageView;
import android.widget.RelativeLayout;
import android.widget.TextView;
import static android.view.View.GONE;
import static com.mbh.materialdialog.Utils.getColorDrawable;
import static com.mbh.materialdialog.Utils.getLLDrawable;
import static com.mbh.materialdialog.Utils.isGoodString;
/**
* Created By MBH on 2016-10-17.
*/
public class MD {
private final Context mContext;
private final OnMDButtonClicked positiveListener, negativeListener, neutralListener;
private final OnMDDismissed dismissedListener;
private final OnMDCanceled canceledListener;
private final
@StringRes
int positiveTextRes, negativeTextRes, neutralTextRes;
private final Drawable iconDrawable;
private final
@DrawableRes
int iconDrawableRes, backgroundDrawableRes;
private final
@LayoutRes
int customViewRes;
private final
@ColorInt
int positiveColor, negativeColor, neutralColor,
titleColor, messageColor, backgroundColor;
private String positiveText;
private String negativeText;
private String neutralText;
private String title, message;
private
@StringRes
int titleRes;
private
@StringRes
int messageRes;
private Drawable backgroundDrawable;
private View customView;
private AlertDialog mAlertDialog;
private AlertDialog.Builder mAlertDialogBuilder;
private View contentView;
private boolean cancalable = true, autoDismiss = true;
private MD(Builder builder) {
mContext = builder.context;
positiveColor = builder.positiveColor;
negativeColor = builder.negativeColor;
neutralColor = builder.neutralColor;
titleColor = builder.titleColor;
messageColor = builder.messageColor;
positiveListener = builder.positiveListener;
negativeListener = builder.negativeListener;
neutralListener = builder.neutralListener;
dismissedListener = builder.dismissedListener;
canceledListener = builder.canceledListener;
cancalable = builder.cancalable;
autoDismiss = builder.autoDismiss;
positiveText = builder.positiveText;
positiveTextRes = builder.positiveTextRes;
negativeText = builder.negativeText;
negativeTextRes = builder.negativeTextRes;
neutralText = builder.neutralText;
neutralTextRes = builder.neutralTextRes;
title = builder.title;
titleRes = builder.titleRes;
messageRes = builder.messageRes;
message = builder.message;
iconDrawable = builder.iconDrawable;
iconDrawableRes = builder.iconDrawableRes;
customView = builder.customView;
customViewRes = builder.customViewRes;
backgroundColor = builder.backgroundColor;
backgroundDrawableRes = builder.backgroundDrawableRes;
backgroundDrawable = builder.backgroundDrawable;
initAlertDialog();
initContentView();
}
/**
* Create a dialog with two button (quick dialog solution)
*
* @param context: Context
* @param title: dialog title
* @param message: dialog message
* @param btnTxtPositive: Positive button text
* @param positiveListener: positive button listener
* @param btnTxtNegative: negative button text
* @param negativeListener: negative button listener
* @return MD
*/
public static MD simpleDoubleBtnMD(Context context,
String title, String message,
String btnTxtPositive,
OnMDButtonClicked positiveListener,
String btnTxtNegative,
OnMDButtonClicked negativeListener) {
return new MD.Builder(context)
.title(title)
.message(message)
.positiveText(btnTxtPositive)
.positiveListener(positiveListener)
.negativeText(btnTxtNegative)
.negativeListener(negativeListener)
.build();
}
/**
* Create a dialog with two button (quick dialog solution)
* @param context: Context
* @param title: dialog title resource
* @param message: dialog message resource
* @param btnTxtPositive: Positive button text resource
* @param positiveListener: positive button listener
* @param btnTxtNegative: negative button text resource
* @param negativeListener: negative button listener
* @return MD
*/
public static MD simpleDoubleBtnMD(Context context,
@StringRes int title, @StringRes int message,
@StringRes int btnTxtPositive,
OnMDButtonClicked positiveListener,
@StringRes int btnTxtNegative,
OnMDButtonClicked negativeListener) {
return simpleDoubleBtnMD(context,
context.getString(title),
context.getString(message),
context.getString(btnTxtPositive), positiveListener,
context.getString(btnTxtNegative), negativeListener);
}
/**
* Create a dialog with one button (quick dialog solution)
*
* @param context: Its better with context
* @param title: dialog title
* @param message: dialog message
* @param btnTxt: button text
* @param onBtnClicked: Button click listener (if not important, it can be null)
* @return MD
*/
public static MD simpleBtnMD(Context context,
String title, String message,
String btnTxt, OnMDButtonClicked onBtnClicked) {
return new MD.Builder(context)
.title(title)
.message(message)
.positiveText(btnTxt)
.positiveListener(onBtnClicked)
.build();
}
/**
* Create a dialog with one button (quick dialog solution)
*
* @param context: Its better with context
* @param title: dialog title resource
* @param message: dialog message resource
* @param btnTxt: button text resource
* @param onBtnClicked:
* @return MD
*/
public static MD simpleBtnMD(Context context,
@StringRes int title, @StringRes int message,
@StringRes int btnTxt, OnMDButtonClicked onBtnClicked) {
return simpleBtnMD(context, context.getString(title),
context.getString(message), context.getString(btnTxt), onBtnClicked);
}
private void initAlertDialog() {
mAlertDialogBuilder = new AlertDialog.Builder(mContext);
mAlertDialogBuilder.setCancelable(cancalable);
}
private void initContentView() {
contentView = LayoutInflater.from(mContext)
.inflate(R.layout.layout_material_dialog, null);
mAlertDialog = mAlertDialogBuilder.setView(contentView).create();
mAlertDialog.getWindow().setBackgroundDrawableResource(R.drawable.material_dialog_window);
mAlertDialog.getWindow().clearFlags(WindowManager.LayoutParams.FLAG_NOT_FOCUSABLE |
WindowManager.LayoutParams.FLAG_ALT_FOCUSABLE_IM);
mAlertDialog.getWindow().setSoftInputMode(WindowManager.LayoutParams.SOFT_INPUT_MASK_STATE);
mAlertDialog.setCanceledOnTouchOutside(cancalable);
// mAlertDialogWindow.setContentView(contentView);
contentView.setFocusable(true);
contentView.setFocusableInTouchMode(true);
final View backgroudContainer = contentView.findViewById(R.id.material_background);
if (backgroundColor != Color.WHITE) {
backgroundDrawable = getColorDrawable(mContext, backgroundColor);
// backgroundDrawable = getColorDrawable(mContext, backgroundColor);
} else if (backgroundDrawableRes > 0) {
if (Build.VERSION.SDK_INT >= Build.VERSION_CODES.LOLLIPOP) {
backgroundDrawable = mContext.getResources().getDrawable(backgroundDrawableRes, mContext.getTheme());
} else {
backgroundDrawable = mContext.getResources().getDrawable(backgroundDrawableRes);
}
}
if (backgroundDrawable != null) {
backgroundDrawable = getLLDrawable(mContext, backgroundDrawable);
if (Build.VERSION.SDK_INT >= Build.VERSION_CODES.JELLY_BEAN) {
backgroudContainer.setBackground(backgroundDrawable);
} else {
backgroudContainer.setBackgroundDrawable(backgroundDrawable);
}
} else {
backgroudContainer.setBackgroundResource(R.drawable.material_card);
}
if (titleRes > 0 || isGoodString(title) || iconDrawable != null || iconDrawableRes > 0) {
final TextView mTitleView = (TextView) contentView.findViewById(R.id.title);
final String fTitle = titleRes > 0 ? mContext.getString(titleRes) : title;
if (isGoodString(fTitle)) {
mTitleView.setText(title);
mTitleView.setTextColor(titleColor);
} else {
mTitleView.setVisibility(GONE);
}
final ImageView iv_icon = (ImageView) contentView.findViewById(R.id.iv_icon);
if (iconDrawableRes > 0) {
iv_icon.setImageResource(iconDrawableRes);
} else if (iconDrawable != null) {
iv_icon.setImageDrawable(iconDrawable);
} else {
iv_icon.setVisibility(GONE);
}
} else {
View ll_titleContainer = contentView.findViewById(R.id.ll_titleContainer);
ll_titleContainer.setVisibility(GONE);
}
final TextView mMessageView = (TextView) contentView.findViewById(R.id.message);
String text = messageRes > 0 ? mContext.getString(messageRes) : message;
if (isGoodString(text)) {
mMessageView.setText(message);
mMessageView.setTextColor(messageColor);
} else {
mMessageView.setVisibility(GONE);
}
final RelativeLayout mButtonLayout = (RelativeLayout) contentView.findViewById(R.id.buttonLayout);
// final LinearLayout buttonPNLayout = (LinearLayout) mAlertDialogWindow.findViewById(R.id.buttonPNLayout);
Button mPositiveButton = (Button) mButtonLayout.findViewById(R.id.btn_positive);
Button mNegativeButton = (Button) mButtonLayout.findViewById(R.id.btn_negative);
Button mNeutralButton = (Button) mButtonLayout.findViewById(R.id.btn_neutral);
initButton(mNeutralButton, neutralColor, neutralText, neutralTextRes, neutralListener);
initButton(mNegativeButton, negativeColor, negativeText, negativeTextRes, negativeListener);
initButton(mPositiveButton, positiveColor, positiveText, positiveTextRes, positiveListener);
if (mNeutralButton.getVisibility() == GONE &&
mNegativeButton.getVisibility() == GONE &&
mPositiveButton.getVisibility() == GONE) {
mAlertDialog.setCancelable(true);
}
if (customViewRes > 0) {
customView = LayoutInflater.from(mContext)
.inflate(customViewRes, null);
}
if (customView != null) {
ViewGroup mMessageContentRoot = (ViewGroup) contentView.findViewById(
R.id.message_content_view);
if (customView != null) {
mMessageContentRoot.removeAllViews();
mMessageContentRoot.addView(customView);
}
}
if (dismissedListener != null) {
mAlertDialog.setOnDismissListener(new DialogInterface.OnDismissListener() {
@Override
public void onDismiss(DialogInterface dialog) {
dismissedListener.onDismissed(MD.this);
}
});
}
if (canceledListener != null) {
mAlertDialog.setOnCancelListener(new DialogInterface.OnCancelListener() {
@Override
public void onCancel(DialogInterface dialog) {
canceledListener.onCancel(MD.this);
}
});
}
}
private void initButton(Button button, @ColorInt int color, String text, @StringRes int textRes, final OnMDButtonClicked listener) {
if (textRes > 0)
text = mContext.getString(textRes);
if (!isGoodString(text)) {
button.setVisibility(GONE);
return;
}
button.setText(text);
setTextColorToButton(button, color);
setListenerToButton(button, listener);
}
private void setTextColorToButton(Button button, @ColorInt int color) {
button.setTextColor(color);
if (isLollipop()) {
button.setElevation(0);
}
}
private void setListenerToButton(Button button, final OnMDButtonClicked listener) {
button.setOnClickListener(new View.OnClickListener() {
@Override
public void onClick(View v) {
if (listener != null) {
listener.onClick(MD.this);
}
checkAutoDismiss();
}
});
}
private boolean isLollipop() {
return Build.VERSION.SDK_INT >= Build.VERSION_CODES.LOLLIPOP;
}
private void checkAutoDismiss() {
if (autoDismiss) {
dismiss();
}
}
/**
* To Dimiss the dialog if showing
*/
public void dismiss() {
if (mAlertDialog != null && mAlertDialog.isShowing()) {
mAlertDialog.dismiss();
}
}
/**
* Check if dialog is showing
*
* @return true if showing false if null or not showing
*/
public boolean isShowing() {
return mAlertDialog != null && mAlertDialog.isShowing();
}
/**
* Change the title of the Dialog
*
* @param title: title string resource id
* @return MD
*/
public MD title(@StringRes int title) {
this.titleRes = title;
return title(mContext.getString(title));
}
/**
* Change the title of the Dialog
*
* @param title: title string
* @return MD
*/
public MD title(String title) {
this.title = title;
if (contentView != null) {
final TextView tv = (TextView) contentView.findViewById(R.id.title);
tv.setText(title);
tv.setVisibility(View.VISIBLE);
}
return this;
}
/**
* Change the message of the dialog
*
* @param message: message string resource id
* @return MD
*/
public MD message(@StringRes int message) {
messageRes = message;
return message(mContext.getString(messageRes));
}
/**
* Change the message of the dialog
*
* @param message: message string
* @return MD
*/
public MD message(String message) {
this.message = message;
if (contentView != null) {
final TextView msg = (TextView) contentView.findViewById(R.id.message);
msg.setText(message);
msg.setVisibility(View.VISIBLE);
}
return this;
}
/**
* Show Dialog
*
* @return MaterialDesign
*/
public MD show() {
mAlertDialog.show();
return this;
}
/**
* Building The Material Dialog with Builder pattern
*/
public static final class Builder {
private final Context context;
private
@ColorInt
int positiveColor = Color.BLUE;
private
@ColorInt
int negativeColor = Color.RED;
private
@ColorInt
int neutralColor = Color.BLACK;
private
@ColorInt
int titleColor = Color.BLACK;
private
@ColorInt
int messageColor = Color.DKGRAY;
private
@ColorInt
int backgroundColor = Color.WHITE;
private OnMDButtonClicked positiveListener;
private OnMDButtonClicked negativeListener;
private OnMDButtonClicked neutralListener;
private OnMDDismissed dismissedListener;
private OnMDCanceled canceledListener;
private boolean cancalable = true;
private boolean autoDismiss = true;
private String positiveText;
private
@StringRes
int positiveTextRes = -1;
private String negativeText;
private
@StringRes
int negativeTextRes = -1;
private String neutralText;
private
@StringRes
int neutralTextRes = -1;
private String title;
private
@StringRes
int titleRes = -1;
private String message;
private
@StringRes
int messageRes = -1;
private Drawable iconDrawable, backgroundDrawable;
private
@DrawableRes
int iconDrawableRes = -1;
private
@DrawableRes
int backgroundDrawableRes = -1;
private View customView;
private
@LayoutRes
int customViewRes = -1;
private MD md = null;
public Builder(Context context) {
this.context = context;
}
public Builder positiveColor(@ColorInt int val) {
positiveColor = val;
return this;
}
public Builder negativeColor(@ColorInt int val) {
negativeColor = val;
return this;
}
public Builder neutralColor(@ColorInt int val) {
neutralColor = val;
return this;
}
public Builder titleColor(@ColorInt int val) {
titleColor = val;
return this;
}
public Builder messageColor(@ColorInt int val) {
messageColor = val;
return this;
}
public Builder positiveListener(OnMDButtonClicked val) {
positiveListener = val;
return this;
}
public Builder negativeListener(OnMDButtonClicked val) {
negativeListener = val;
return this;
}
public Builder neutralListener(OnMDButtonClicked val) {
neutralListener = val;
return this;
}
public Builder dismissedListener(OnMDDismissed val) {
dismissedListener = val;
return this;
}
public Builder canceledListener(OnMDCanceled val) {
canceledListener = val;
return this;
}
public Builder cancalable(boolean val) {
cancalable = val;
return this;
}
public Builder autoDismiss(boolean val) {
autoDismiss = val;
return this;
}
public Builder positiveText(String val) {
positiveText = val;
return this;
}
public Builder positiveText(@StringRes int val) {
positiveTextRes = val;
return this;
}
public Builder negativeText(String val) {
negativeText = val;
return this;
}
public Builder negativeText(@StringRes int val) {
negativeTextRes = val;
return this;
}
public Builder neutralText(String val) {
neutralText = val;
return this;
}
public Builder neutralText(@StringRes int val) {
neutralTextRes = val;
return this;
}
public Builder title(String val) {
title = val;
return this;
}
public Builder title(@StringRes int titleRes) {
this.titleRes = titleRes;
return this;
}
public Builder message(@StringRes int messageRes) {
this.messageRes = messageRes;
return this;
}
public Builder message(String val) {
message = val;
return this;
}
public Builder iconDrawable(Drawable val) {
iconDrawable = val;
return this;
}
public Builder icon(@DrawableRes int iconDrawableRes) {
this.iconDrawableRes = iconDrawableRes;
return this;
}
public Builder customView(View val) {
customView = val;
return this;
}
public Builder customView(@LayoutRes int val) {
customViewRes = val;
return this;
}
public Builder backgroundDrawable(@DrawableRes int backgroundDrawable) {
this.backgroundDrawableRes = backgroundDrawable;
return this;
}
public Builder backgroundDrawable(Drawable backgroundDrawable) {
this.backgroundDrawable = backgroundDrawable;
return this;
}
public Builder backgroundColor(@ColorInt int backgroundColor) {
this.backgroundColor = backgroundColor;
return this;
}
public MD build() {
if(md == null)
md = new MD(this);
return md;
}
public MD show() {
if(md == null) build();
return md.show();
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,517 |
Elias Childe (1778 - Lambeth, 13 de abril de 1849) foi um pintor de paisagens britânico. Ele era um artista prolífico, trabalhando tanto em óleos quanto em aquarelas.
Vida
Ele era irmão mais velho do artista James Warren Childe e Henry Langdon Childe, que desenvolveu a lanterna mágica. Ele exibiu pela primeira vez em 1798 na Royal Academy, quando morava na 29 Compton Street, Soho, com seu irmão James. Ele se concentrou na paisagem, um campo em que ele foi um sucesso. Em 1825, ele foi eleito membro da Sociedade de Artistas Britânicos.
Childe exibiu pela última vez em 1848 e morreu em 1849.
Trabalho
Childe exibiu mais de 500 fotos nas exposições da Society of British Artists, da Royal Academy e da British Institution . Suas fotos eram populares e vendiam bem. Ele se destacou particularmente nos efeitos de luar, e um exemplo desse estilo foi para a Galeria Nacional de Arte Britânica em South Kensington.
Mortos em 1849
Nascidos em 1778
Pintores do Reino Unido | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,751 |
La cuenca hidrográfica del Plata, también conocida como la región platina, es la parte de América del Sur drenada por el propio Río de la Plata y sus afluentes. Cubre un área de unos , lo que la convierte en la quinta más grande del mundo, por detrás de las cuencas de los ríos Amazonas, Congo, Nilo y Misisipi, y abarca territorios de cinco paísesː Argentina, Bolivia, Brasil, Paraguay y Uruguay.
Las precipitaciones que caen en su ámbito se reúnen en dos grandes cursos, los ríos Paraná y Uruguay, que luego vierten sus aguas en el Río de la Plata, el que finalmente desemboca en el océano Atlántico Sur.
El conjunto fluvial de la cuenca del Plata forma el principal sistema de recarga del acuífero Guaraní, una de las mayores reservas continentales de agua dulce del mundo. Los gobiernos de los países implicados estudian el modo de aprovecharlo de forma sustentable, asegurando así la provisión de agua potable a sus habitantes. La cuenca sirve de asiento a una población de más de 100 millones de habitantes, por lo que la interacción humana con la misma a lo largo del tiempo en forma incontrolada produce cambios significativos, tanto para la cuenca como para la calidad de vida de sus habitantes.
Los dos grandes ríos de la cuenca, el Paraná y el Uruguay, tienen una densa red de afluentes, subafluentes y tributarios menores, como los ríos Paraguay, Pilcomayo, Bermejo, Pilagá, Salado del Norte, Carcarañá, Tercero, Cuarto, Iguazú, Salado del Sur, Gualeguay, Corrientes, Santa Lucía, Guayquiraró, San Javier, Samborombón (sobre el Paraná), Ibicuí, Mocoretá, Gualeguaychú, Miriñay, Aguapey, Negro, Queguay, Arapey, arroyo Nogoyá (sobre el Uruguay), entre otros.
Ubicación geográfica
La Cuenca del Plata se encuentra en los paralelo 15° latitud sur y 35° latitud sur y los meridianos 68° longitud oeste y 44° longitud oeste, con aproximadamente 3 170 000 km², abarcando parte de Brasil y Argentina, la totalidad del Paraguay, el sur y este de Bolivia,y gran parte del Uruguay . Es geopolíticamente importante en América del Sur, pues abarca zonas con distintas características hidrográficas, económicas y socioculturales: la cuenca del río Paraguay, la del Alto Paraná, la del río Uruguay, y la del Paraná Medio e inferior.
Principales ríos del sistema
La cuenca del Plata se compone de cuatro importantes subcuencas, las de los ríos: Uruguay, Paraná, Paraguay, y la propia cuenca del Río de la Plata. El conjunto comprende las cuencas de los tributarios de estos, como el río Bermejo, el río Pilcomayo, y el Salado del norte, entre otros. En tan enorme extensión se pueden encontrar distintos ambientes acuáticos naturales, que van desde los trópicos de agua dulce hasta aquellos en que esta se mezcla con agua de mar, formando un ecosistema estuarial, en la desembocadura del Río de la Plata.
Características generales de la cuenca del Plata
La Cuenca del Plata tiene 3 200 000 km², en la que se destacan el río Paraná, uno de los más grandes y caudalosos del mundo, y los ríos Paraguay y Uruguay. Este conjunto hidrográfico desemboca en el Río de la Plata y, por intermedio de este, en el Océano Atlántico. El caudal medio de la cuenca es de 23 000 m³. La mayoría de estos cursos son navegables por buques de mediano porte y casi todos ellos por barcazas.
Existen numerosas represas hidroeléctricas en operación, principalmente en la cuenca del río Paraná. En territorio uruguayo, sobre el río Negro oriental se ubican Rincón del Bonete, Rincón de Baygorria, y Paso del Palmar. En el río Uruguay, en la frontera entre Uruguay y Argentina, se encuentra la represa de Salto Grande. También en Argentina se encuentra el embalse de Cabra Corral, en el Saladillo, provincia de Salta; y en la provincia de Corrientes, en la frontera con Paraguay, se encuentra la Represa de Yacyretá, sobre el río Paraná. Aguas arriba de este curso, en la frontera del Paraguay con el Brasil se emplaza la mayor del mundo: Itaipú. Otras represas en territorio brasileño son Paranoa, Dourada, Furnas, Isla Soltera, y Jupiá.
El río Iguazú también cuenta con varias represas, todas ellas en territorio brasileño: Gobernador Bento Munhoz da Rocha Neto, Gov. Ney Aminthas de Barros Brag, Represa de Salto Caxias, Represa de Salto Santiago, Represa de Salto Osório, y Represa de Foz de Areia.
En las nacientes del río Paraná la precipitación media anual son del orden de 1200 a 1700 mm. En el río Paraguay la precipitación media anual varía entre 1000 y 1400 mm. Ambos ríos se unen en confluencia a 50 km aguas arriba de la ciudad de Corrientes, en la zona conocida como Paso de la Patria, y drenan cuencas de aproximadamente el mismo tamaño. El aporte del Alto Paraná en la zona de Paso de la Patria es de 12 000 m³/s, y el Río Paraguay contribuye con un caudal medio anual de 4000 m³/s, totalizando un caudal medio anual de 16 000 m³/s.
Hasta dicho punto la cuenca del río Paraná tiene pendientes bien marcadas y una red de drenaje bien desarrollada; en cambio, en el río Paraguay las pendientes son mucho menores y la red de desagües es menos desarrollada. En la cuenca superior del río Paraguay existe una región de grandes pantanos, de una extensión de unos 400 000 km², llamada El Gran Pantanal, donde las aguas remansan demorando su escurrimiento.
El parque nacional Iberá y la reserva natural del Iberá se encuentra en la zona centro-norte de la provincia argentina de Corrientes; la componen los esteros y las lagunas del Iberá con una extensión de 13 000 km², drenando al río Paraná por intermedio del río Corriente.
Toda la cuenca es frecuentada por pescadores, tanto los comerciales, como los que lo hacen como deporte, pues encuentran en ella un importante y variado conjunto de especies de peces deportivos.
Las subcuencas y sus afluentes
Cuenca del río Paraná
La cuenca del río Paraná es la de mayor superficie (1 510 000 km²) y el río, nacido de la unión de los ríos Paranaíba y Grande, es, a la vez, el curso más largo: 2570 km. Si se le suman los 1200 km del río Paranaíba, la longitud total asciende a 3770 km. Desde su nacimiento hasta la desembocadura pueden diferenciarse tres tramos: el superior o Alto Paraná (hasta la confluencia del río Paraguay, 1550 km), el Paraná Medio (722 km) hasta la ciudad de Diamante, y el Paraná inferior o Delta (hasta la confluencia con el río Uruguay, 298 km).
El Alto Paraná
El punto clave en el curso del Alto Paraná —en el que recibe, por la margen izquierda, al último de sus grandes tributarios, el Iguazú, que se vierte en él después de recorrer 1320 km bajando desde la Sierra del Mar con rumbo este-oeste, junto el Tieté— integra el grupo de las grandes vías fluviales que permitieron la expansión de la colonización portuguesa a expensas de los dominios hispánicos.
El Alto Paraná es un río de meseta que corre encajonado entre abruptas barrancas labradas en las coladas de meláfiros (rocas efusivas permo-carboníferas) que dan origen a la formación de rápidos, correderas y cataratas, entre las que se destacan las cataratas del Guayrá, ubicadas 193 km aguas arriba de la afluencia del Iguazú. Sobre su margen derecha se localiza el territorio del Paraguay y sobre su margen izquierda se localizan los territorios de Brasil primero y Argentina después.
Aguas arriba de Corpus, donde el lecho rocoso se halla a solo 5,50 m de profundidad, el Alto Paraná tiene profundidad suficiente para la navegación de regular calado hasta Puerto Méndez, en Brasil. En Puerto Iguazú, el último de los puertos argentinos, donde el hidrómetro marca con frecuencia alturas superiores a los 20 m, el ahondamiento del cauce por el volumen de aguas que arrastra ha sido más intenso que el de sus afluentes, de modo que estos forman saltos de diferente magnitud antes de desaguar en él.
El caso más notorio es el del río Iguazú, por el ensanchamiento del cauce aguas arriba de las cataratas que llevan su nombre, pues estas son incapaces de absorber a través de la Garganta del Diablo todo su caudal; de esta manera, se origina un hemiciclo de derrames de 2,7 km de longitud (de los cuales 2,1 corresponden a Argentina) que vierten hacia el estrecho valle de no más de 100 m de ancho por el que, a través de un trayecto de 28 km, el Iguazú se vuelca al Paraná. Esta es la consecuencia de la erosión diferencial de cuatro tipos distintos de saltos, tres de ellos salvados mediante dos escalones en basalto muy resistente a la erosión, en tanto el cuarto tipo, correspondiente al Salto Unión, presenta una escarpa basáltica de un material menos compacto, esponjoso y por ende más desgastable, asentada sobre arenisca triásica de Botuc, que facilita la erosión en la base y el efecto retrocedente que alarga la Garganta del Diablo, cuya extensión se acrecienta cada día.
La caída de agua forma dos bucles, uno que erosiona el pie de la catarata y otro que se pulveriza y eleva en forma de nube, originando el frecuente espectáculo del arco iris con la luz del sol. Esta garganta, enmarcada entre paredes de más de 60 m de altura y menos de 100 m de ancho, solo es navegable en los 20 km finales. Aguas arriba de las cataratas el caudaloso Iguazú, que a lo largo de 115 km es frontera entre la Argentina y Brasil, ve interrumpida su navegabilidad por la presencia de correderas y el ensanchamiento del cauce, que disminuye su profundidad. Estos obstáculos fueron los que impulsaron a Álvar Núñez Cabeza de Vaca, el primero que exploró este río, a continuar su trayecto por tierra, descubriendo las cataratas en 1542.
El río Iguazú recibe un pequeño tributario, el San Antonio, río fronterizo cuyo curso sinuoso puede seguirse por el vivo contraste que presenta la ocupación humana a una y otra margen: desmontada y densamente poblada la del Brasil, despoblada y con el bosque casi virgen la de la Argentina, donde en las estribaciones de la sierra de la Victoria se desarrolla el parque nacional Iguazú.
La alta pluviosidad de esta zona da origen a incontables ríos y arroyos de curso meandroso, identificables solo en los casos en que el desmonte ha facilitado la erosión hídrica, dejando al descubierto los faldeos desnudos, pues el resto de estos cursos de agua quedan ocultos bajo el espeso manto de la selva desarrollado en este ambiente subtropical.
El Alto Paraná y el Iguazú se caracterizaban como ríos de meseta, por el tono claro de sus aguas, pero la intensificación del desmonte, la frecuente roturación de los suelos por la expansión agrícola y el aceleramiento de la erosión han contribuido a que tengan en el presente un color acentuadamente rojizo debido a los materiales que llevan en suspensión. Contrastan, por ejemplo, con los tonos más oscuros de las decantadas aguas del río Acaray, que en territorio paraguayo han sido represadas a pocos kilómetros de su desembocadura en el Alto Paraná para la producción de energía hidroeléctrica cuyo principal mercado, en 1973, era la provincia de Misiones.
Desde 1973 el curso del Alto Paraná ha sido desviado con motivo de la construcción de la presa brasileño-paraguaya de Itaipú, incrementándose también en forma explosiva la población de Foz de Iguazú. Esta obra provoca considerables cambios en el régimen del río como consecuencia de las exigencias del funcionamiento de las turbinas de la central hidroeléctrica, lo que incide sobre los proyectos técnicos elaborados por la Argentina y Paraguay para los aprovechamientos de Corpus y Yacyretá.
También la vida del río podría ser alterada si muchos de sus peces no encuentran ya las condiciones propicias para su existencia, en tanto otros ejemplares de la fauna fluvial podrían ver que se amplía su hábitat, entre ellos el vector de la esquistosomiasis (que habita en las aguas someras de lento escurrimiento), flagelo que ya aflige al Brasil.
El Alto Paraná posee un clima tropical con precipitaciones concentradas en los meses de verano, que establecen el régimen del río hasta su desagüe en el río de La Plata, con predominio de caudales de verano-otoño. Las lluvias de la alta cuenca, que se producen de diciembre a abril, con un máximo en febrero, derraman los mayores montos en las nacientes del río Tieté, en la Serra do Mar, donde supera los 4000 mm anuales. El Alto Paraná discurre por un lecho tortuoso y de ancho variable, que presenta los caracteres de un río de meseta, con variación de amplitudes desde la angostura de la garganta de Jupiá, hasta el remanso que precede a los saltos de Guayrá, donde el lecho se expande a 4 km de ancho.
Responsables en parte de estas variaciones son los mantos de basalto que propician en el lecho la formación de valles estrechos, rápidos y cascadas, entre las que descuellan las cataratas del Guayrá o Sete Quedas, descubiertas por Irala y que han desaparecido por la construcción de la represa. Sus caídas estrepitosas provocaban la pulverización del agua, que formaba densas nieblas al tiempo que labraba por erosión retrocedente sucesivos peldaños de 40 m de altura, en la gran columna basáltica de la sierra de Amanbay que atraviesa el curso superior del río, y originaban la profundización y el estrechamiento del cauce. En territorio brasileño el Paraná recibe afluentes de importancia que proceden de las sierras costeras: Tieté, Paraná Panema, Ivaí e Iguazú, que establece en su tramo final el límite internacional argentino-brasileño, formando las cataratas homónimas unos 28 km antes de su desembocadura.
Desembocadura del Iguazú en el Alto Paraná
El Iguazú (vocablo que significa "agua grande") posee una longitud de 1320 km, y una cuenca de alimentación de 62 000 km²; es uno de los afluentes más largos que tiene el río Paraná en Brasil, al que pertenecen 1205 km. Nace en el planalto paranaense, a 900 m s. n. m., y cruza una región tropical que recibe un promedio de 1900 mm anuales de precipitaciones. Al desembocar en el río Paraná corta por erosión retrocedente los derrames basálticos, formando un conjunto de cascadas de gran magnitud, conocidas como Cataratas del Iguazú o Santa María, descubiertas en 1542 por el Adelantado Álvar Núñez Cabeza de Vaca, en su itinerario desde Santa Catarina a Asunción. La longitud de la línea de cresta de los saltos alcanza a 2700 m, de los cuales 600 m pertenecen al Brasil. Su origen se vincula al largo cañón labrado por el río Paraná, profundamente encajado en los mantos de basalto desde Posadas a Guayrá. Su intensa erosión retrocedente dejó a su afluentes, y entre ellos al Iguazú, corriendo a un nivel superior, obligándolos a volcar sus aguas al colector por medio de saltos. Distintos tipos de caídas de agua tallan el perfil rocoso, con ritmos más o menos veloces, entre las que descuellan el Salto Unión Americana por precipitar la máxima corriente del río a la Garganta del Diablo, dividida en dos partes por el límite internacional.
Otros afluentes del Alto Paraná
Aguas abajo de su confluencia con el río Iguazú el encajamiento lineal del río Paraná también origina saltos en sus afluentes misioneros. Entre ellos se destaca por su extensión el río Uruguay o Marambas y otros de menor longitud (Aguaray Guazú, Piray Guazú, Paranay Guazú, Cuñapirú, Yabebiry, etc.) y el Itaembé, que sirve de límite entre las provincias de Misiones y Corrientes. Esta característica de los ríos, cuyos lechos forman saltos rápidos y correderas, los hace aptos para la producción de energía, aunque limita su navegabilidad. Las obras realizadas por Brasil en el Alto Paraná comprometen la posibilidad de otros aprovechamientos energéticos del río, ya que cualquier alteración artificial que se provoque en una de sus partes influye inexorablemente sobre el resto del sistema, situación que se torna estratégica por tratarse de un río de curso sucesivo y soberanía compartida.
Las grandes represas construidas y proyectadas por Brasil en su territorio (Jupiá, Ilha Solteria, Itaipú y otras) pueden ejercer un papel beneficioso como reguladores del flujo de agua durante todo el año, pero su contaminación amenaza provocar graves daños a las áreas cercanas a "la desembocadura de la Cuenca del Plata", como consecuencia de que la gran cantidad de energía hidroeléctrica producida se destina a numerosos proyectos industriales para la región centro sur de ese país que han de generar fuertes concentraciones de población y afluentes urbanos e industriales contaminantes; por otra parte, ya se han detectado residuos, principalmente pesticidas provenientes de la zona de expansión de la frontera agropecuaria del Brasil.
Otro factor de preocupación es la propagación, hacia toda el área de la cuenca, de la esquistosomiasis, enfermedad transmitida por ciertos caracoles (caramujo) y peces (moncholos - Pimelodus albicans) que proliferan, en especial, en la aguas lénticas, por lo que las zonas de lento escurrimiento de las represas artificiales se convierten en su hábitat. Otro elemento de perturbación es la intensificación del proceso de acumulación de sedimentos en los embalses de capacidad limitada ocasionada por la erosión hídrica y acentuado por la pérdida de la masa boscosa y las praderas de la alta cuenca, que mantenían las aguas de los ríos límpidas y sin sedimentos.
El manto de basaltos que obstruyó el curso del Alto Paraná dio origen a los llamados rápidos de Apipé, a la vez que se formaban varios brazos que rodean las islas de Ibicuy, Talavera, Apipé y otras, entre las que se destaca la de Yaciretá con 415 km², que emerge de las aguas cubierta de árboles y pastos graminosos con una altura que impide su inundación. El proyecto de aprovechamiento múltiple mediante las obras del complejo Apipé-Yaciretá acordado con Paraguay provoca un fuerte impacto sobre el albardón ribereño, con gran expansión de la zona inundada especialmente sobre la margen paraguaya y aun sobre el área de derrames del río en los Esteros del Iberá, a través de la zanja de trasvasamiento de caudales de San Miguel, con obras de regulación que permitirán el aprovechamiento de los caudales excedentes y la recuperación de casi tres millones de hectáreas aptas para la agricultura.
Las características del Río Paraguay, tanto por el brusco cambio de rumbo como por la magnitud de los caudales que aporta, siendo estos colectados en una cuenca apenas inferior a la del Paraná, a la que llegan aportes desde los relieves andinos, le otorgan características que son tratadas aparte.
El Paraná Medio
Hasta Diamante se extiende el Paraná Medio a lo largo de aproximadamente 60 km, con diferencias estructurales en ambas márgenes; el valle es más estrecho que aguas abajo y, por ende, está sujeto con mayor intensidad a los efectos de las crecientes que invaden islas y terrazas fluviales. Recibe escasos afluentes que derramen sus caudales especialmente del lado correntino con rumbo noreste-sudoeste; los más importantes son los ríos Santa Lucía, Corrientes y Guayquiraró, este último, límite natural entre las provincias de Corrientes y Entre Ríos.
Por la escasa profundidad del lecho, la navegación de este tramo del Paraná se halla restringida a naves de cabotaje, pero su desnivel de 34 m ha llevado a Agua y Energía Eléctrica a formular el proyecto de aprovechamiento energético del Paraná Medio, cuya construcción modificará la dinámica hídrica al inundar el valle en su totalidad produciendo un impacto no evaluado aún. Entre los beneficios secundarios que se podrán obtener cuentan los derivados de la formación de los espejos de agua por la construcción de las represas, que superarán 1 300 000 ha, las que se constituirán en hábitat propicio para el desarrollo de camalotales —plantas acuáticas que producen biomasa renovable apta para la generación de energía química en forma de gas metano— y residuos semisólidos ricos en componentes nitrogenados utilizables en el acondicionamiento y fertilización de los suelos.
Desde la confluencia con el Paraguay, el Paraná controla su curso a través de una falla cuyo labio levantado corresponde a la margen izquierda. Su permanente proceso de erosión socava la base de la barranca a causa del ensanchamiento del cauce requerido por el proceso permanente del deltificación interna. La profusión de islas de carácter deltaico, implantadas en el lecho del río, impulsa la formación de riachos laterales —denominados "saladillos"— que acompañan al curso principal del río. Los procesos de sedimentación y erosión lateral del cauce ocasionan inconvenientes para la navegación y para las construcciones ubicadas sobre las barrancas. Al norte de la ciudad de Santa Fe se localiza una importante cuenca lacustre, de contorno irregular, que presenta tres sectores, las lagunas San Pedro, Leyes y Setúbal, a la cual concurren los ríos Saladillo, Dulce y Amargo.
En el paraje Las Cuatro Bocas recibe las aguas del Salado Norte (o: Pasaje-Juramento-Salado), de curso interprovincial (1500 km), cuya cuenca cubre 247 000 km². Sus aguas constituyen un recurso de valor estratégico para las provincias que atraviesa. Sus numerosos afluentes captan corrientes desde los nevados del borde de la Puna que integra el río Las Conchas Guachipas (Calchaquí-Santa María), cuyos recorridos reciben diferentes denominaciones y presentan en sus cursos sucesivos fenómenos de captura por erosión retrocedente.
Esto da a la cuenca superior del Salado un raro diseño, con pronunciados cambios de rumbo, como en el caso de la quebrada de Las Conchas-Guachipas (en los Valles Calchaquíes), cuya confluencia con el río Lerma da nacimiento al curso del Pasaje o Juramento que, al entrar en la provincia de Santiago del Estero, recibe finalmente el nombre de Salado del Norte. Sus caudales, incrementados con el aporte de las precipitaciones, se ven fuertemente disminuidos por los usos económicos del agua para irrigación y por las pérdidas por evaporación e infiltración, que determinan pronunciadas variaciones entre diferentes tramos de su curso. Así, en El Arenal se registra un caudal medio de 20,8 m³/s mientras que, en Suncho Corral, aguas abajo de los aprovechamientos del embalse Los Figueroas, disminuye a 15,73 m³/s.
El Bajo Paraná
Numerosos esteros y bañados jalonan su curso (Pellegrini, Figueroa, Añatuya), cegado por los materiales fangosos que el río arrastra durante las crecientes y cuya acumulación en el lecho ha provocado los desplazamientos horizontales del curso en busca de una mayor pendiente.
El segundo afluente de importancia a este tramo es el río Carcarañá, formado por los ríos Tercero y Cuarto, provenientes de la zona montañosa (Sierras de Córdoba) y cuya cuenca imbrífera abarca aproximadamente 48.000 km². El río Tercero nace en la sierra de Comechingones y en su cuenca superior recibe numerosos afluentes que se nutren de las precipitaciones de área montañosa (600 a 1000 mm anuales), otorgándole al curso principal grandes volúmenes de agua disponibles —con un caudal medio de 27,17 m³/s— para propósitos múltiples (energía, riego y control de crecientes). Los ríos San Miguel, Santa Rosa, Grande, de la Cruz y otros drenan aguas claras hacia el río Tercero, que corta con curso antecedente el cordón de la Sierra Chica.
El río Cuarto vuelca al Tercero las aguas de un conjunto de drenajes que descienden de la sierra de Comechingones, transformándose en un río de llanura al norte de la ciudad de Río Cuarto, originando una zona de bañados en la proximidad de La Carlota en la que sus aguas se salinizan tomando el nombre de Saladillo.
En su tramo Inferior el río Paraná discurre dividido en varios brazos anastomosados entre sí hasta su confluencia con el río Uruguay. El Delta del Paraná, con 14000 km², se extiende a partir de la ciudad de Diamante. Con una longitud de 320 km y un ancho variable —18 km frente a Baradero y más de 60 km entre los ríos Luján y Gutiérrez— representa la más colosal manifestación del acarreo de sedimentos de la cuenca. Se halla profundamente influido por las crecidas desfasadas del río Uruguay, el régimen mareológico y, particularmente, por los procesos atmosféricos de sudestada, que provocan grandes inundaciones sobre el Bajo Delta.
A la altura del puerto de Baradero el río Paraná se divide en dos cursos, el Paraná de las Palmas al oeste y el Paraná Guazú al este, que abrazan la red anastomósica de canales. El sistema de fallas de rumbo norte-sur que atraviesan la provincia de Entre Ríos controla la dirección de los principales ríos que vuelcan en el curso inferior del Paraná — Nogoyá, Gualeguay y Paranacito —, los que organizan una compleja red hídrica cuyas aguas, sujetas a crecientes extraordinarias de graves consecuencias para la población, son aprovechadas con tajamares que aseguran su acopio en la época estival. Por la margen derecha el Paraná recibe en el ámbito de la provincia de Buenos Aires una serie de ríos y arroyos que desaguan la pampa ondulada: son el Ramallo, el Tala, el Arrecifes, el Areco y el Luján.
Cuenca del río Paraguay
Las cabeceras
El río Paraguay tiene sus cabeceras en la meseta de Mato Grosso, al sur de la Chapada de Parecis, a 3000 m s. n. m., y recorre 2600 km antes de desembocar en el Paraná. Es la principal vía de acceso para los países mediterráneos del sistema del Plata (Bolivia y Paraguay) y ocupa una vasta cuenca de 1 095 000 km². Con exclusión de sus cabeceras, su curso atraviesa una vasta llanura de escasa pendiente, con grandes planos de inundación, entre los que se destaca por su magnitud e influencia en el régimen de la cuenca la extensa depresión del Pantanal de Xarayes (60 000 km²), que retiene durante dos o tres meses las aguas de las crecientes, provocadas en su curso alto por las abundantes lluvias estivales. De este modo, la onda de creciente llega al tramo inferior del Paraguay entre mayo y junio. Precisamente la presencia de los bañados asegura su régimen permanente, caracterizado por la regularidad, con máximos caudales en invierno y estiaje estival y un caudal medio anual de 5000 m³/s. En sus tramos medio e inferior el río Paraguay divide dos regiones morfológicamente diferenciadas: al este la zona montañosa —que constituye el reborde oriental de la meseta de Brasilia— y al oeste la llanura chaqueña —surcada por ríos de caudal marcadamente estacional de origen local, excepto el Pilcomayo y el Bermejo, que le aportan gran cantidad de sedimentos, en especial el segundo, que transporta anualmente 100 millones de toneladas de sólidos en suspensión—, provenientes de la Cordillera de los Andes.
La desembocadura en el río Paraná
Al desembocar en el Paraná, el Río Paraguay produce un "remanso", ocasionado por el movimiento de hélice o voluta de las aguas del Paraná, y vierte sus aguas por tres brazos, Humaitá, Atajo y Paso de la Patria, donde se advierte el contraste de color entre las aguas claras del alto Paraná y las rojizas del Paraguay producto de la descarga de sedimentos del río Bermejo, que ponen en evidencia la significativa importancia de la erosión hídrica en la cuenca que amenaza la productividad de los suelos y desencadena procesos de sedimentación que hacen peligrar los aprovechamientos hidroeléctricos y las vías de comunicación fluvial. La ampliación de la frontera agrícola a expensas de bosque y la falta de prácticas conservacionistas facilitan la erosión hídrica.
Los afluentes del río Paraguay
El río Pilcomayo
El nombre del río Pilcomayo es de origen quechua (pisku - mayu) y significa: pisku= pájaros, mayu= río " es decir "río de los pájaros". Su curso compartido por tres países (Argentina, Bolivia y Paraguay) es inconstante, sujeto a las grandes variaciones de caudal que han dilatado la dilucidación de las cuestiones fronterizas entre Argentina y Paraguay. Nace en las estribaciones de la cordillera de los Frailes (Bolivia) y capta los derrames de un amplio frente andino alimentado por el deshielo. La longitud de su curso alcanza los 1.070 km, en él pueden distinguirse cuatro secciones:
La Cuenca de Alta Montaña tiene sus fuente más austral en el río San Juan en la Argentina, pero la mayor parte de él se desarrolla en territorio de Bolivia, donde colecta la mayoría de los caudales y drena una vasta zona que recibe nevadas y lluvias de alrededor de 700 mm anuales.
El Tramo Superior recibe precipitaciones menores, es de carácter alóctono y se interna hacia el sureste en la llanura chaqueña con un cauce bien definido, pero con grandes variaciones de ancho y altura de las barrancas que lo ciñen.
El Pilcomayo Medio es un río divagante y conflictivo que alimenta numerosos esteros y bañados como el Bañado la Estrella en la Provincia de Formosa. Cuenta con diversos afluentes temporarios; uno de ellos es el río Confuso, por el que en 1927 corría la mayor parte del caudal y que en la actualidad presenta las cabeceras desecadas.
El Abanico Deltaico, de cauces cambiantes que se extiende hasta la desembocadura en el Paraguay, desangrando parte del caudal hacia el río Negro a través de El Reventón, con la subsiguiente imposibilidad de aprovechamiento para la navegación aguas abajo y con perjuicio para poblaciones como Clorinda.
Las crecientes del Pilcomayo se producen en verano y el estiaje a fines de invierno y principios de primavera, como corresponde a su régimen de alimentación pluvial. Los caudales varían en forma apreciable a lo largo de su curso, en especial en su tramo inferior, agostado por la pérdidas en los bañados por evaporación e infiltración, resultando el módulo medio de 200 m³/s (1941-1956: Hemiciclo Seco).
El río Bermejo
El río Bermejo —llamado así por el color rojizo de su aguas debido a la gran cantidad de sedimento que estas llevan— (1.450 km) es uno de los ríos interiores de mayor potencialidad de la Argentina, con una cuenca de 133.000 km². Sus nacientes reúnen las corrientes que descienden de los contrafuertes de la cordillera Oriental.
La alta cuenca del Bermejo presenta ejemplos de procesos erosivos de dimensiones extraordinarias, como el que existe en el valle de Tarija que, a la inestabilidad geológica de los depósitos cuaternarios y al régimen pluviométrico, suma una casi absoluta ausencia de tapiz vegetal y se caracteriza por el inadecuado uso agro pastoril del suelo. Aproximadamente el 34% de su superficie total se halla afectada. En Zanja del Tigre, el río Bermejo transporta un elevado monto de material en suspensión —un promedio de 64 millones de t/año entre 1945/6 y 1962/3— lo que afecta la estabilidad de los lechos fluviales, embanca los canales y colma precozmente los embalses artificiales, obligando a costosas obras de dragado en el resto del sistema fluvial platense.
Desde Bolivia, donde drena la sierra de Santa Victoria, donde nace el río Santa Rosa y la confluencia de este, hasta las Juntas de San Antonio , donde recibe al Grande de Tarija, el Bermejo lleva la frontera internacional. Ya en territorio argentino recibe varios tributarios. Por su margen derecha recibe el Iruya, con su afluente el Pescado, el Blanco o Zenta, gran colector de las aguas del borde de la Puna de Atacama. El Iruya le aporta más del 70% del material sólido que el río transporta en suspensión aguas abajo, producto de la potencia erosiva de su cauce, que socava las altas barrancas de areniscas blandas, cuya coloración justifica su nombre. La estacionalidad e intensidad de las precipitaciones en la alta cuenca (900 mm anuales), que concreta en verano la disponibilidad de agua para alimentar su trayecto, también influyen sobre el grado de erosión, en especial donde los suelos desprovistos de vegetación quedando expuestos a la corriente. A estos ríos en esa zona se suman en importancia el Pilaya, el Itiyuro y el Baritú.
Cuenca del río Uruguay
El río Uruguay, eje de circulación y frontera natural de la Argentina con el Uruguay y el Brasil, es el segundo en importancia dentro del sistema del Plata, con su amplia cuenca imbrífera que cubre aproximadamente 365 000 km². Su curso recorre 1779 km, desde su naciente en las sierras Del Mar y General hasta su desembocadura en la confluencia con el Paraná Bravo.
Su amplia cuenca de alimentación se localiza en zonas que reciben 2000 mm anuales de lluvias en los meses de invierno y primavera, y que provocan crecientes retardadas en uno a dos meses. Aunque el régimen del río es muy irregular, pueden identificarse dos crecientes separadas por los estiajes de enero y agosto. El caudal medio anual en Concordia es de aproximadamente 4000 m³/s, aunque se han registrado caudales máximos de 17 720 m³/s (1965). En su curso son frecuentes los derrames basálticos que crean rápidos, saltos y restingas talladas por la erosión hídrica. Precisamente, 15 km al norte de Concordia la ruptura de pendiente de Salto Grande (afloramiento de meláfiros), que marca un límite para la navegación aguas arriba, se ha aprovechado para llevar a cabo el proyecto hidroeléctrico homónimo.
La obra del complejo Salto Grande afecta el escurrimiento y los procesos erosivos aguas abajo, regulando los caudales. Ha motivado la formación de un gran lago de embalse y la inundación y traslado de poblaciones.
Las costas del río Uruguay son diferentes. Sobre la margen izquierda posee altas barrancas —lo que también se observa en la zona de Misiones, donde forma los saltos del Moconá— pero en la zona de Corrientes y Entre Ríos son generalmente bajas. Sus principales afluentes son el Aguapey, Miriñay, Mocoretá y Gualeguaychú en la Argentina, y el río Negro en Uruguay.
Cuenca del Río de la Plata
El Paraná finalmente desemboca en el Río de la Plata. No muy lejos de su desembocadura se encuentra la isla Martín García, un promontorio rocoso de solo 2 km² de superficie, formada por un afloramiento del basamento cristalino.
El río Uruguay es arrinconado contra la banda oriental por el voluminoso aporte sedimentario transportado por el Paraná, que no solo forma un espacioso delta que avanza a razón de 70 a 90 m por año sino también la Playa Honda o Placer de las Palmas.
El Río de la Plata ocupa una amplia cubeta enmarcada por las líneas de falla del Uruguay, el Paraná Guazú y el Paraná de las Palmas, que constituyó por mucho tiempo la principal vía navegable en la que se fundaron puertos como los de Campana y Zárate, este último de importancia crucial por constituir la cabeza de puente del ferrobarco que vinculaba a la costa pampeana con Puerto Ibicuy, en Entre Ríos. En el año 1973 el cauce estaba prácticamente obliterado por los sedimentos y el tránsito de ultramar era derivado al Paraná Bravo. En la actualidad el dragado del canal Mitre ha reactivado el tránsito por el Paraná de las Palmas y el puente Zárate-Brazo Largo ha relevado al viejo ferrobarco. El Río de la Plata se caracteriza por la existencia de un delta fluvial (Delta del Paraná), probablemente fruto de un delta decapitado durante la ingresión marina del Querandinense.
El Río de La Plata posee una superficie de 35 000 km²; nace de la confluencia del brazo principal del río Paraná con el río Uruguay, y desemboca en el mar formando un amplio estuario, que desagota el extraordinario caudal de su vasta cuenca (entre 16 000 y 23 000 m³/s). Es compartido por la República Argentina y la República Oriental del Uruguay. Su longitud hasta la línea imaginaria que une la punta Norte del Cabo San Antonio (Argentina) con Punta del Este (Uruguay) es de 275 km, presentando un ancho variable que alcanza 40 km, entre Buenos Aires y Colonia del Sacramento, y más de 270 km en su desembocadura.
El régimen del río está influenciado por los caudales de sus dos principales tributarios y por la acción de las mareas y la participación de las típicas situaciones del tiempo: sudestadas y pamperos que empujan sus aguas respectivamente hacia la costa argentina o uruguaya.
Por la margen derecha recibe una serie de afluentes — Arroyo del Medio, Luján, Reconquista, Matanza o Riachuelo, Maldonado, Santiago, Samborombón entre muchos otros arroyos y ríos— entre los que se destaca por su mayor extensión el Salado. Por la margen izquierda el principal afluente es el río Santa Lucía.
El Salado del Sur, típico río de llanura, traza gran cantidad de meandros que ocupan su valle plano y muy amplio con numerosas lagunas: Chañar, La Picasa, Mar Chiquita de Junín, Gómez, del Carpincho, Encadenadas del este, y muchas más. Tras un recorrido de 650 km, con rumbo noroeste-sudeste por la provincia de Buenos Aires, desemboca en la bahía de Samborombón, oficiando dificultosamente de desagüe para una cuenca cercana a los 40 000 km². En el pasado geológico, su curso se hallaba unido al río Quinto —como atestiguan la serie de cañadas y depresiones que los conectan— pero actualmente constituye un emisario lacunar cuya escasa pendiente y profundidad provocan frecuentes inundaciones y desbordes en ocasión de lluvias copiosas, siendo un río no apto para riego, navegación o, generación de energía hidroeléctrica, ni siquiera como colector eficiente de los aportes que recibe principalmente por su margen derecha: arroyo Vallimanca, arroyo Las Flores y arroyo El Gualicho. Una serie de obras de defensa para evitar las episódicas inundaciones, principalmente canales, articulan el drenaje de otros cursos que desaguan en la bahía de Samborombón (arroyo de los Huesos, arroyo Langueyú, arroyo Chapaleofú y arroyo Tapalquén). En mayo de 1980 la inundación desplazó casi 30 000 millones de toneladas de agua a todo lo ancho de la cuenca del Salado (7 000 000 ha), anegando centros poblados y campos, cuyo lavado y posterior salinización, y la pérdida del pastizal destinado a la cría de animales provocó ingentes daños a la población.
Problemática ambiental
Los problemas con las especies exóticas
Una especie exótica puede interactuar de distintas maneras con el ambiente en el que ingresa. En el mejor de los casos, se adapta al nuevo medio y termina en relativo equilibrio con la comunidad preexistente, sin alterarla de modo apreciable. Sin embargo, cuando se caracteriza por tener alta tasa de crecimiento, gran energía reproductivo-adaptativa, fuerte capacidad de dispersión y, además, por carecer de enemigos naturales en el nuevo ecosistema, ya fuesen depredadores o competidores por los recursos, se expande rápidamente y ocupa de modo efectivo el territorio. Este es el comportamiento típico de las especies invasoras, que pueden producir alteraciones importantes en el ambiente que invaden, ya sea este natural o humano.
Al río ha llegado una especie de exótica de bivalvos de agua dulce, Limnoperna fortunei, que por esta vía accedió a todo el continente americano, así como dos especies más de la familia Corbiculidae: Corbicula largillierti y C. fluminea.
La distribución y modo de vida de los bivalvos exóticos llegados al Río de la Plata —C. fluminea y L. fortunei— se consideran especies invasoras porque, además de ser exóticas, se caracterizan por una temprana maduración sexual, una gran capacidad reproductora y un considerable poder de adaptación a los ambientes que colonizan, ya sean naturales o creados por el hombre.
Los procedimientos usados con moluscos en otros países son muy variados; incluyen descargas eléctricas, tratamiento de las aguas con cloro (o cloración), venenos muy tóxicos, electromagnetismo, altas temperaturas y ultrasonido. Muchos son caros y, en el caso de los venenos, hay que considerar los riesgos de su toxicidad residual.
Hoy la presencia de estas nuevas especies no solo se registra en el área propia del río sino, también, en muchos otros sitios de su cuenca mayor, como los ríos Carcarañá, Paraná y Uruguay, igual que cuerpos de agua que les están conectados o les son adyacentes, lo cual incluye a las provincias argentinas de Córdoba, Santa Fe, Entre Ríos, Corrientes, Misiones y Chaco.
La sedimentación en el Río de la Plata
Su lecho está ocupado por bancos de arena y arcilla que obligan a su permanente dragado para posibilitar la navegación. Dicho método no suprime las causas de la potente sedimentación sino que solamente atenúa sus efectos, reflejo de procesos de erosión hídrica que ocurren a miles de kilómetros de distancia. Algunos signos reproducen espectacularmente su impacto sobre las economías vinculadas a los puertos de los ríos Paraguay, Paraná y de la Plata, que exportan aproximadamente 20 millones de m³
Véase también
Anexo:Lista de tratados internacionales de recursos hídricos
Impacto ambiental potencial de proyectos hidroeléctricos
Anexo:Cuencas del mundo por superficie
Cuenca del Orinoco
Acuífero Guaraní
Acuífero Puelche
Autoridad de Cuenca Matanza Riachuelo
Referencias
Enlaces externos
Tratado de la Cuenca del Plata
Comité Intergubernamental Coordinador de los Países de la Cuenca del Plata (CIC)
FONPLATA - Fondo Financiero para el Desarrollo de la Cuenca del Plata
OEA: Cuenca del Plata - Estudio para su Planificación y Desarrollo - República Argentina - Cuenca del Río Bermejo II - Cuenca Inferior
OEA: Coordinación de Acciones Locales para el Futuro Sostenible de la Cuenca del Plata
Mapa de la cuenca del Plata y la Hidrovía Paraguay-Paraná
La cuenca del Plata, en riesgo
Mapa de la Cuenca del Plata
Crean grupo sobre humedales de la Cuenca del Plata en el marco de la Convención Ramsar
Cuenca del Plata e Hidrovía Paraná Paraguay | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,831 |
Why did you leave your job at McKesson?
Low pay, not interested in the employees opinion or safety from management. Not enough information when an employee request details of insurance, bonus or personal benefits.
What is the most stressful part about working at McKesson?
Not knowing specific expectations. Not being informed about new policies and procedures to work more effectively. The communication is not the greatest.
What is the work environment and culture like at McKesson?
How would you describe the pace of work at McKesson? | {
"redpajama_set_name": "RedPajamaC4"
} | 31 |
Ursa lunula is een spinnensoort in de taxonomische indeling van de wielwebspinnen (Araneidae).
Het dier behoort tot het geslacht Ursa. De wetenschappelijke naam van de soort werd voor het eerst geldig gepubliceerd in 1849 door Hercule Nicolet.
Wielwebspinnen | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,077 |
and it is at work now is 27.5 tflop and has cost 8 million euro.
that's roughly 27.5 gflop / 8k euro = far over 3 gflop per 1000 euro.
So your estimate is not far off.
applications are pathetic slow at their cheap processors.
other manufacturer that produces a great cpu.
I'm a bit worried about RAMBUS being involved into cell i have to admit.
DDR ram came, P4 recovered quite some.
major bandwidth that we cannot afford.
Yet the main point is that a CELL processor serves not only a certain market.
an 8 processor Xeon MP 2.8Ghz.
8 processor Xeon MP never will have.
>how expensive your Unix desktop or departmental compute server was).
>100 hour accounts that basically wasted 90% of the resource.
>cache-defeating form of memory access). Rob Peter, pay Paul.
>performance of a Cray on my code, not on a measley 100 hour "research"
>more easily managed by just running the damn jobs independently via e.g.
>interruption of any particular node -- is Not Easy.
>optimization energy go into the library (or not).
>> that the compiler can optimize for a given arch.
>Absolutely, dead on the money. Especially since there is no "COTS"
>units that are neither standard nor code-portable.
>Ahh, it's good to be alive.
>> Cell's (PS3) speed will be hindered greatly by code not vectorizing.
>> This could be a great opportunity to make a large sum of money.
>> scandal's web site makes it look like it is a research language only.
>> MPI is standard, UPC has compilers from many vendors (including Cray).
>> for strengths and weaknesses.
>> > > their loops that inhibit code from vectorizing.
>> > use a language or library which directly supports vector commands?
>> > like CVL or NESL, right, so, are they not useful? | {
"redpajama_set_name": "RedPajamaC4"
} | 2,036 |
Tourcelles-Chaumont-Quilly-et-Chardeny est une ancienne commune française, située dans le département des Ardennes en région Grand Est.
Elle a existé de 1828 à 1871.
Géographie
Histoire
La commune est issue de la fusion des trois communes de Tourcelles-Chaumont, Quilly et Chardeny, en 1828. Ces communes reprirent leur indépendance en 1871. Tourcelles-Chaumont devient le chef-lieu de la commune.
Administration
Démographie
Voir aussi
Articles connexes
Liste des anciennes communes du département des Ardennes
Tourcelles-Chaumont
Quilly
Chardeny
Notes et références
Ancienne commune dans les Ardennes | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,249 |
Q: Why I can' see new version google apps script editor? I'm in Japan. By now I still just can see the legacy version. Anyone know why?
I googled but got no answer. And there is no button for switch between legacy version and the new version.
A: Updates from workspace blog say
Update
This post has been updated to reflect a change in rollout pace. It is now an extended rollout (longer than 15 days for feature visibility), which we expect to complete in January 2021. We previously stated it would be a gradual rollout (up to 15 days for feature visibility) starting on December 7, 2020.
It is normal for you to not see the button until 31 January 2021.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 782 |
{"url":"https:\/\/www.prepanywhere.com\/prep\/textbooks\/calculus-and-vectors-mcgraw-hill\/chapters\/chapter-4-derivatives-of-sinusodial-functions\/materials\/4-4-applications-of-sinusoidal-functions-and-their-derivatives","text":"4.4 Applications of Sinusoidal Functions and Their Derivatives\nChapter\nChapter 4\nSection\n4.4\nSolutions 15 Videos\n\nAn AC-DC coupled circuit produces a current described by the function I(t) = 60cost + 25, where t is time, in seconds, and I is the current, in amperes, at time t.\n\na) Find the maximum and minimum currents and the times at which they occur.\n\nb) For the given current, determine\n\n\u2022 i) the period, T, in seconds\n\u2022 ii) the frequency, f, in hertz\n\u2022 iii) the amplitude, A, in amperes\nQ1\n\nThe voltage signal from a standard North American wall socket can be described by the equation V(t) = 170\\sin 120\\pi t, where t is time, in seconds, and V is the voltage, in volts, at time t.\n\na) Find the maximum and minimum voltage levels and the times at which they occur.\n\nb) For the given signal, determine\n\n\u2022 i) the period, $T$, in seconds\n\u2022 ii) the frequency, f, in hertz\n\u2022 iii) the amplitude, A, in volts\nQ2\n\nConsider a simple pendulum that has a length of 50 cm and a maximum horizontal displacement of 8 cm.\n\na) Find the period of the pendulum.\n\nb) Determine a function that gives the horizontal position of the bob as a function of time.\n\nc) Determine a function that gives the velocity of the bob as a function of time.\n\nd) Determine a function that gives the acceleration of the bob as a function of time.\n\nQ3\n\nConsider a simple pendulum that has a length of 50 cm and a maximum horizontal displacement of 8 cm.\n\na) Find the maximum velocity of the bob and the time at which it first occurs.\n\nb) Find the maximum acceleration of the bob and the time at which it first occurs.\n\nc) Determine the times at which\n\n\u2022 i) the displacement equals zero\n\u2022 ii) the velocity equals zero\n\u2022 iii) the acceleration equals zero\n\nd) Describe how the answers in part c) are related in terms of when they occur. Explain Why these results make sense.\n\nQ4\n\nA marble is placed on the end of a horizontal oscillating spring.\n\nIf you ignore the effect of friction and treat this situation as an instance of simple harmonic motion, the horizontal position of the marble as a function of time is given by the function b(t) = A cos 2\\pi t, where A is the maximum displacement from rest position, in centimetres, f is the frequency, in hertz, and t is time, in seconds. In the given situation, the spring oscillates every 1 s and has a maximum displacement of 10 cm.\n\na) What is the frequency of the oscillating spring?\n\nb) Write the simplified equation that expresses the position of the marble as a function of time.\n\nc) Determine a function that expresses the velocity of the marble as a function of time.\n\nd) Determine a function that expresses the acceleration of the marble as a function of time.\n\nQ5\n\na) Sketch a graph of each of the following relations over the interval from O to 4 s. Align your graphs vertically.\n\n\u2022 i) displacement versus time\n\u2022 ii) velocity versus time\n\u2022 iii) acceleration versus time\n\nb) Describe any similarities and differences between the graphs. Find the maximum and minimum values for displacement. When do these values occur? Refer to the other graphs and explain why these results make sense.\n\nQ6\n\nA piston in an engine oscillates up and down from a rest position as shown.\n\nThe motion of this piston can be approximated by the function b(t) = 0.05\\cos(13t), where t is time, in seconds, and la is the displacement of the piston head from rest position, in metres, at time t.\n\na) Determine an equation for the velocity of the piston head as a function of time.\n\nb) Find the maximum and minimum velocities and the times at which they occur.\n\nQ7\n\nA high\u2014power distribution line delivers an AC\u2014DC coupled voltage signal whose\n\n\u2022 AC component has an amplitude, A, of 380 kV\n\u2022 DC component has a constant voltage, V, of 120 kV\n\u2022 frequency, f, is 60 Hz\n\na) Add the Ac component, V_{AC}, and DC component, V_{DC}, to determine an equation that relates voltage, V, in kilovolts, to time, t, in seconds.\n\nUse the equation V_{AC}(t) = A \\sin 2\\pi ft to determine the AC component.\n\nb) Determine the maximum and minimum voltages and the times at which they occur.\n\nQ8\n\nA differential equation is an equation involving a function and one or more of its derivatives. Determine whether the function y = \\pi \\sin \\theta + 2\\pi \\cos \\theta is a solution to the equation \\displaystyle \\frac{d^2y}{d\\theta^2} + y = 0 .\n\nQ9\n\na) Determine a function that satisfies the differential equation \\displaystyle \\frac{d^2y}{dx^2} = - 4y .\n\nb) Explain how you found your solution.\n\nQ10\n\nAn oceanographer during a storm and modelled the vertical displacement of a wave, in metres, using the equation h(t) = 0.6\\cos 2t + 0.8\\sin t, where t is the time in seconds.\n\na) Determine the vertical displacement of the wave when the velocity is 0.8 m\/s.\n\nb) Determine the maximum velocity of the wave and when it first occurs.\n\nc) When does the wave first change from a \u201chill\u201d to a \u201ctrough\u201d? Explain.\n\nQ12\n\nPotential energy is energy that is stored, for example, the energy stored in a compressed or extended spring. The amount of potential energy stored in a spring is given by the equation \\displaystyle U = \\frac{1}{2}k x^2 , where\n\n\u2022 U is the potential energy, in joules\n\u2022 k is the spring constant, in newtons per metre\n\u2022 x is the displacement of the spring from rest position, in metres\n\nUse the displacement equation from question 5 to find the potential energy of an oscillating spring as a function of time.\n\nQ13\n\nKinetic energy is the energy of motion. The kinetic energy of a spring is given by the equation \\displaystyle K = \\frac{kv^2 T^2}{8\\pi^2} , where K is the kinetic energy, in joules; k is the spring constant, in newtons per metre; v is the velocity as a function of time, in metres per second; and T is the period, in seconds.\n\nUse the velocity equation from question 5 to express the kinetic energy of an oscillating spring as a function of time.\n\nQ14\n\nAn oscillating spring has a spring constant of 100 N\/s, an amplitude of 0.02 m, and a period of 0.5s.\n\na) Graph the function relating potential energy to time in this situation. Find the maxima, minima, and zeros of the potential energy function, and the times at which they occur.\n\nb) Repeat part a) for the function relating kinetic energy to time.\n\nc) Explain how the answers to parts a) and b) are related,\n\nQ15\n\nFor any constants A and B, the local maximum value of A \\sin x + B \\cos x is\n\nA. \\displaystyle \\frac{1}{2}|A + B| \n\nB. \\displaystyle A + B| \n\nC. \\displaystyle \\frac{1}{2}(|A| + |B)| \n\nD. \\displaystyle |A| + |B) \n\nE. \\displaystyle \\sqrt{A^2 + B^2} `","date":"2021-06-16 22:27:36","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.842215359210968, \"perplexity\": 612.7940408403022}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-25\/segments\/1623487626122.27\/warc\/CC-MAIN-20210616220531-20210617010531-00354.warc.gz\"}"} | null | null |
Chinese sword with Qianlong reign mark
108.5 cm / 42.7 inch
88.2 cm / 34.7 inch
forte 5.5 mm
middle 5 mm
near tip 3 mm
forte 38 mm
middle 37 mm
near tip 31 mm
Weight without scabbard
Point of balance
21 cm from base of guard
(handle-side)
Steel, iron, silver, brass, bone. Scabbard: wood, ray-skin, brass
Circe 1890's - 1930's
This sword is part of a group of very similar swords. Some are shorter double swords, other single swords, some large, like this one. What they all have in common are blades with extensive overlays in silver, sometimes also with copper and brass. The decor always incorporates figures, Sanskrit symbols, constellations, and a reign mark, all contained within a framed border following the contours of the blade.
The reign mark is in zhuanshu (seal script) that says 大清乾隆年制 (Da Qing Qianlong nian zhi) or "Great Qing Qianlong Period Manufacture". The Qianlong emperor's long reign lasted from 1736-1796. When looking at the style and workmanship of the swords in this group however, they seem to date to the late 19th to early 20th century. More specific, probably around 1890 - 1920.
These swords have fooled collectors, auctioneers, curators, dealers, and authors alike, who took these marks at face value. A case in point is a sword kept in the Royal Armories in Leeds, under accession number XXVIS.190. Following the description in the museum, it was described as a "Ceremonial sword from the Qianlong reign" in a number of publications, including Osprey's "Late Imperial Chinese Armies 1520 - 1840".1
The Royal Armories sword in Osprey's Late Imperial Chinese Armies 1520 - 1840.
Our sword is laid out on top of the book.
It is rare to find a reign mark on an antique Chinese weapon, let alone one that doesn't correspond to its date of manufacture. It is not without precedent, though. Some 19th century rebel groups marked their weapons with Ming dynasty reign marks, harking back to the last period the Han were ruled by Han, and not by Manchus as had been the case in the Qing from 1644 onwards. It was an act of rejecting the present Qing, and idealizing a distant past.
Imperial reign marks of previous periods are very common on Chinese porcelain though. Kate Hunt, Head of Sale for Chinese Works of Art at Christie's South Kensington:
"...for hundreds of years Chinese artisans copied reign marks from earlier dynasties out of a respect and reverence for these earlier periods. These marks are often referred to in auction catalogue descriptions as 'apocryphal' marks. These marks were not necessarily intended to fool buyers into thinking they were buying a genuine earlier work of art."2
During the Qianlong period only few weapons would bear the Qianlong mark and these were all made in the imperial workshops, adhering to the highest standards of craftsmanship of the period. Such items are auctioned for tens of thousands up to millions. Like with porcelain, the key to looking through such markings is to assess the style and quality of the piece itself and assess whether the presented item is indeed of the quality that this mark would suggest. An example:
Our Qing sword with apocryphal Qianlong mark (left),
compared to a real Qianlong period imperial piece (right).
The lesson here for the student of antique arms and armor is to always keep questioning. Museums descriptions, books, auctioneers, long-time collectors, they can all be wrong from time to time. Museums, in particular, are often held in high regard by the layman, but they too make mistakes. Their collections span thousands of items and there is only so much expertise in-house. A curator with a master's on Japanese lacquerware may be appointed in a position where he's head of Chinese arms as well. Some museums do a great job cooperating with experts from many fields to increase their understanding, while other institutions, large and small, find it hard to change something even when it is pointed out to them again and again. It is down to us, the arms and armor community as a whole, private and institutionalized, to keep studying and sharing to raise the level of our understanding.
Notes to introduction
1. C.J. Peers; Late Imperial Chinese Armies 1520 - 1840, Osprey Publishing, Oxford, 2005 reprint, page 33. The accession number in the publication is XXVI-90s, which is perhaps an older accession number because today the exact same sword turns up under XXVIS.190.
2. See: Kate Hunt, Demystifying Chinese reign marks — everything you need to know to get started.
Now we have established what it is not, let's now focus on what it is: An exceptionally large and heavy Chinese straightsword of the late Qing dynasty, possibly even early Republic. For this period, it's actually quite a nice piece. Most were of much simpler manufacture with little to no attempts to decorate them at all.
Blade decoration
The most striking feature of the piece is, of course, the blade. It is crosshatched to form a background to which silver wire would adhere when hammered on. Using this method, the blade is decorated in full with a framing border around its contours. Within the border, we find on each side a Qianlong reign mark, two Sanskrit characters, four figures, a dragon, and a constellation.
Four figures on one side of the blade are women in Chinese dress that we can identify as the "Four Great Beauties" (四大美女) of Chinese folklore. They were allegedly so beautiful that one, Xi Shi, made fish forget how to swim, Wang Zhaojun made the birds forget how to fly, Diachan eclipsed the moon, and the last, Yang Guifei brought flowers to shame.
The four figures on the other side are the probably Heruka or "Blood Drinkers" of Tibetan Buddhism. These are considered enlightened and fierce protectors, known as "Enlightened Kings" (明王) in Chinese. The one on the top is hard to identify. The second from above is easily identifiable as Vajrayana or "The King of knowledge having conquered three worlds" (降三世明王). The third, sitting on a lotus, is probably Yamantaka (大威德金剛) "The Defeater of Death" while the fourth, standing figure is Kuṇḍali (軍荼利明王), dispenser of the nectar of immortality.
Near the tip, the blade has, on each side, a dragon and seven stars of the big dipper, a constellation that was very important in traditional Chinese thought, among others in Daoism. It represented the seat of the celestial king around which all revolves, and it was the constellation in which the time of death was determined. This constellation is often seen on antique Chinese swords, primarily straightswords but sometimes sabers, spearheads, and other weapons.
All in all, the decor is eclectic. The mark of an illustrious emperor, harking back to the most prosperous times the Chinese empire had ever had until then. Four beauties from Chinese folklore. Four fierce deities, protectors of Tibetan Buddhism, a religion practiced by the Qianlong emperor himself and many of his Tibetan, Mongol, and Manchu subjects. Add to that constellation most prominent in another religion, Daoism. Finally, a five-clawed dragon on each side depicted just like it is on the Chinese flag of the last decades of the Qing. It is almost as if the sword is trying to commemorate "the good old days", a testament of the greatness of Chinese culture in a time that everything was falling apart. Despite all the decoration, the blade shows hints of a forge-folded construction, possibly with an inserted hardened edge, yet was never sharpened.
The hilt consists of an "ace of spades" style guard, typical for the late Qing, with matching lobed pommel. The fittings are engraved with designs of archaic dragons. The bone handle is engraved on both sides, one side showing what looks to be a monk in fighting pose, possibly Bodhidharma, who brought Chan Buddhism to China, which in turn spread to Japan and is now known as Zen. The other side shows an underwater scene with fish, snail, and shrimp. Original peening at the pommel intact. There is also some play in the handle.
It comes in its original wooden scabbard, covered in ray-skin, dyed green and polished. Ray-skin in great condition, no losses. Its brass fittings pierced and with dragon engravings. A section of the band of the upper shield is missing.
Why were these made?
The most common description given for them is ceremonial, yet to my knowledge, the only Chinese ceremony that uses a straightsword is a Daoist ceremony. Swords used in these ceremonies typically have a strictly Daoist theme in their decoration. This leads to the conclusion that they were probably made as purely decorative pieces, possibly aimed at a predominantly foreign market.
Comparable examples
As mentioned, several of these swords, either in sets or as large, single swords, circulate. Most of them are in private hands and those I've tracked have changed hands for as little as $100 and as much as $20.000 in the last decade or so.
One nearly identical example, mentioned in the introduction, is in the Royal Armories in Leeds under accession number XXVIS.190. It is remarkably similar, the only difference is that ours has a ray-skin covered scabbard and the one in the Royal Armories has a black lacquered scabbard. It is still represented as Qianlong era sword on the website and was published in a variety of publications. The Royal Armories note: "There are numerous examples from the same series; for example, those in the collection of Prof. Dr. W Uhlmann, Gerbrunn, Germany, no. 229; Mr M Watts, Crawley Down, West Sussex."
Other notable published examples include a set and a single sword, published in Alex Huangfu's Iron and Steel Swords of China (中國刀劍), pages 208, 209, and the foldout in the back. In this book he describes both the set and the single long sword as "Qing Qianlong period ceremonial court dress swords" (清乾隆款宫廷礼仪佩剑). As evidence for this attribution he makes he reproduces a photo of the Royal Armories example , and a photo of the sign with which it was displayed.
A good, representative example of a type of Chinese sword that is widely thought to date from the Qianlong reign. Its many misrepresentations aside, these are quite interesting items in their own right. It remains somewhat unclear why they were made, but I suspect they were made as presentation pieces for foreigners, or items intended for the curio trade.
SOLD FOR € 2500,- in August, 2017
I keep the price here deliberately, so the collector knows what to reasonably pay for them.
China Jian Not what it seems
Ancient Ba-Shu dagger
From approximately the 5th to 3rd century B.C.
Chinese Shan presentation dha
Presented by the local Dai nobility to a British customs officer in 1936.
Tibetan style pierced saddle plate
With designs of four dragons in scrollwork around a "wish-granting-jewel"
Antique Chinese jian
A fine Chinese straightsword blade, of typical Qing form with a rather wide profile.
A Chinese hook sword
A rather well-made example of its type.
Qing dynasty imperial edict box
Used to move imperial orders from the emperor's quarters to the recipient.
A typology of Chinese sabers
Introduction Historical references on Chinese saber types are scar...
A saber glossary in Manchu
An overview of Manchu saber terminology.
Making a Chinese rattan shield
Ever since I acquired an antique Chinese tengpai
Antique Vietnamese swords
A reference list of antique Vietnamese swords and sabers I've ... | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,990 |
Q: How to disable multiple select on lists in sharepoint 2010? thank you for reading my question!
I have a little problem with lists in Sharepoint 2010. The goal is to disable the checkboxes on the left of each item, but to keep them selectable.
I know that it's possible to disable the checkboxes at all in the view settings (Modify View -> Tabular View -> Allow individual item checkboxes). But when I do this the items are not selectable any more.
Thank you very much!
LMW
A: If you really want to accomplish this, you can use client side scripting to convert the behavior of the check boxes to radio buttons (i.e. a user clicks a check box, all others are deselected).
I just ran a test where I hide all the check boxes using JavaScript, but doing so prevented me from making any selections...so the check boxes need to stay, but there's no reason you can't limit the number of selected check boxes to 1.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,459 |
Eduardo Nava ist der Name folgender Personen:
* Eduardo Nava (Leichtathlet) (* 1968), mexikanischer Sprinter
Eduardo Nava (Tennisspieler) (* 1997), US-amerikanischer Tennisspieler | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,529 |
@interface ListPersonViewController : RefreshTableViewController
@property (nonatomic, retain) NSMutableArray *arrUsers;
@property (assign) ListType listType;
@property (nonatomic, assign) NSString *user_id;
@property (nonatomic, assign) NSString *post_id;
@property (nonatomic, retain) NSString *nickname;
@end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,606 |
Teens Stream More Than Listen to AM/FM
Edison's Larry Rosin says findings "could be a lens into the future of audio usage"
U.S. teens now spend more time with streaming audio like Pandora and Spotify, than they do listening to AM/FM. That's according to the latest Share of Ear study from Edison Research.
A quick graph from the study shows teens age 13 to 17 spend an average of 64 minutes a day with streaming audio, compared to 53 minutes a day listening to AM/FM, both over-the-air and the online streams of AM/FM stations.
"While AM/FM radio listening leads by a significant margin among all other age groups, much of teens' listening time has shifted to pureplay Internet audio services like Pandora and Spotify and others," said Edison Research President Larry Rosin. "This could be a lens into the future of audio usage."
The findings mark the first public release from the fall study, a twice-yearly tracking of forms of audio, including AM/FM radio, streaming audio, owned music, podcasts, SiriusXM satellite radio, TV "cable radio" channels and others.
The study included 2,000+ Americans age 13 and older; they completed a 24-hour diary of audio listening on an assigned day.
The diaries were completed online last fall in English and in Spanish.
Reports Offer Insights on the Podcast Listener
Podcasting is one area that continues to see significant growth, so say Westwood One and Edison Research | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,101 |
\section{Introduction }
When discussing quantum field theory in the eternal Schwarzschild black hole background, one usually considers the Boulware, Unruh, or Hartle-Hawking states \cite{Boulware:1974dm}, \cite{Unruh:1976db}, \cite{Hartle:1976tp}, \cite{DeWitt:2003pm}, \cite{Candelas:1980zt}. While the Boulware state is defined by taking the exact modes to be positive--definite w.r.t. $\partial / \partial t$ Killing vector, which would give the usual Poincare--invariant ground state in flat--spacetime QFT, the Unruh and Hartle-Hawking states are defined by introducing Kruskal coordinates (canonical affine parameters on the past and future horizons) and taking modes to be positive--definite w.r.t. these Kruskal coordinates. The outcome of the two latter definitions of positive-definite modes is that either both or one of the out--going or in--going modes are in a thermal state with the temperature that is equal to the Hawking one. Which state is more suitable to a certain physical situation is analyzed by the calculation of the energy-momentum tensor on the horizon and at spatial infinity \cite{Candelas:1980zt}, \cite{Akhmedov:2015xwa}. However, there was no consideration of the situation in which both out--going and in--going modes are in thermal states with temperatures which generically do not coincide and which are not equal to the Hawking temperature. It would be natural to assume that this state is the best approximation for a physical situation when one considers a black hole in a box of a gas with a temperature which is not equal to the Hawking one. The same question can be posed analogously in the Rindler and de Sitter space--times with respective canonical temperatures of the horizons.
The de Sitter space, Rindler space, and two-dimensional black hole examples were analysed in \cite{Akhmedov:2020ryq}. It was shown that when one considers fields in a thermal state with temperature which is not equal to the canonical temperature of the horizon of the background geometry, then propagators, both points of which are located at the horizon, acquire an anomalous singularity. This singularity is anomalous in a sense that the standard light--like separation divergence comes from high frequencies and does not depend on the temperature of the state, while at the horizon the singularity comes from the infrared region in the integral over frequencies and there is an explicit dependence on the temperature of the state in which the propagator is evaluated.
We consider the following quantum field theory
\begin{align*}
S=\frac{1}{2}\int d^4 x \sqrt{-g}\Big[ \partial_\mu \phi \partial^\mu \phi -\mu^2 \phi^2\Big]
\end{align*}
on the Schwarzschild and Reissner-Nordström black hole backgrounds.
In \cite{Akhmedov:2020ryq} only the two--dimensional analogue of the Schwarzschild black hole was considered. In this paper these results are extended to the four--dimensional black holes. We start in the section \ref{modes_wight_sch} by constructing modes and the Wightman function in the Schwarzschild black hole background. In the section \ref{canonical_sing} it is shown that the singularity of the Wightman function is canonical when its points are light-like separated outside the horizon, i.e. the divergent term in the Wightman function has the standard coefficient. Then, in the section \ref{anom_sing} we demonstrate that the Wightman function acquires anomalous singularity when both of its points are located on the horizon, i.e. the coefficient before the divergent term differs from the canonical one and explicitly depends on the temperature of the out-going modes. Finally, we conclude in section \ref{RNsection} by showing that the anomalous singularity of the Wightman function also occurs in the Reissner-Nordström black hole background.
\section{Schwarzschild black hole}
In this section we consider both massive and massless fields in Schwarzschild spacetime. First, we set the notations and define modes and thermal Wightman functions. For more detailed review the reader may refer to \cite{Boulware:1974dm}, \cite{Unruh:1976db}, \cite{Hartle:1976tp}, \cite{DeWitt:2003pm}, \cite{Candelas:1980zt}. Then, the Wightman function is computed when both of its points are located at the event horizon.
\subsection{Modes and Wightman function} \label{modes_wight_sch}
\begin{figure}[!h]
\centering
\includestandalone[width=0.7\textwidth]{picSCHW}
\caption{Penrose diagram of the Schwarzschild black hole. Here and further "PNI" and "FNI" stand for "past null infinity" and "future null infinity" correspondingly.} \label{picschw}
\end{figure}
The Schwarzschild black hole background is given by the metric:
\begin{align}
&ds^2 = \bigg( 1-\frac{2M}{r} \bigg) dt^2 - \frac{dr^2}{1-\frac{2M}{r}}-r^2(d \theta^2 + \sin^2 \theta d \varphi^2).
\end{align}
Here Eddington-Finkelstein coordinates are defined as:
\begin{align}
&u = t- r_*, \qquad v=t+r_*, \nonumber \\
&r_* = r+2M \log \big( r/2M-1 \big).
\end{align}
In Sec. \ref{canonical_sing} it will be shown that the Wightman function has a standard divergence when its points are lightlike separated near the horizon. Then, in Sec. \ref{anom_sing} I will consider the situation in which both points of the Wightman function are located on the future horizon, which is denoted as $H_{future}$ in Fig. \ref{picschw}.
\subsubsection{Massless case}
Solutions of the corresponding d'Alembertian equation are as follows:
\begin{equation} \label{split}
u_{\omega lm} (x) = (4 \pi |\omega|)^{-1/2} r^{-1} f_{\omega l} (r,t) Y_{lm} (\Omega),
\end{equation}
where $\Omega$ denotes the angular coordinates.
To define the set of Boulware modes, we impose the condition that the modes are positive-definite w.r.t. ${\partial}/{\partial t}$ Killing vector, so one can represent the function $f_{\omega l} (r,t)$ in the form
\begin{equation} \label{f_split}
{f}_{\omega l} (t,r) = e^{-i|\omega| t} {R}_l (\omega | r) ,
\end{equation}
in which the radial function $R_l (\omega | r)$ solves the equation
\begin{align} \label{radial}
&\frac{d^2R}{dr_*^2}+\big[ \omega^2 - V_l (r) \big] R = 0,
\end{align}
with the potential
\begin{equation} \label{potential}
V_l (r) = \bigg( 1 - \frac{2M}{r} \bigg) \bigg( \frac{l(l+1)}{r^2} + \frac{2M}{r^3} \bigg).
\end{equation}
The complete set of modes in the outer region of the Schwarzschild black hole is given by the out-going and in-going modes, denoted accordingly as:
\begin{eqnarray} \label{set1}
&\overrightarrow{u}_{\omega lm} (x) = (4 \pi |\omega|)^{-1/2} e^{-i |\omega| t} r^{-1} \overrightarrow{R}_l(\omega | r) Y_{lm} (\theta, \varphi), \nonumber \\
&\overleftarrow{u}_{\omega lm} (x) = (4 \pi |\omega|)^{-1/2} e^{-i |\omega| t} r^{-1} \overleftarrow{R}_l(\omega | r) Y_{lm} (\theta, \varphi),
\end{eqnarray}
where the radial functions satisfy the boundary conditions as follows
\begin{equation}
\overrightarrow{R}_l(\omega| r) = \begin{cases}
e^{i \omega r_*} + \overrightarrow{A}_l(\omega) e^{-i \omega r_*},
& r \rightarrow 2M \\
{B}_l (\omega) e^{i \omega r_*}, & r \rightarrow \infty
\end{cases} \nonumber
\end{equation}
and
\begin{equation} \label{boundary_cond}
\overleftarrow{R}_l(\omega |r) = \begin{cases}
{B}_l ({\omega}) e^{ - i \omega r_*}, & r \rightarrow 2M \\
e^{-i \omega r_*} + \overleftarrow{A}_l ({\omega}) e^{i \omega r_*}, & r \rightarrow \infty
\end{cases}
\end{equation}
As we have only two types of modes, the mode decomposition of the field operator is as follows
\begin{equation}
\phi (x) = \sum_{lm} \int_0^{\infty} d \omega \bigg( a_{\omega lm} \overrightarrow{u}_{\omega lm} (x) + b_{\omega lm} \overleftarrow{u}_{\omega lm} (x) + h.c. \bigg).
\end{equation}
Then for the Wightman function in a generic state with zero anomalous quantum averages (i.e. with $\langle a_{p} a_{q}\rangle = \langle a^{\dagger}_{p} a^{\dagger}_{q}\rangle =\langle b_{p} b_{q}\rangle = \langle b^{\dagger}_{p} b^{\dagger}_{q}\rangle = 0 $), but which is not necessary Fock ground state, one has:
\begin{multline} \label{wightman_general}
W (x,x') \equiv \langle \phi (x) \phi(x') \rangle
= \int^{+\infty}_{0} {d \omega} \int^{+\infty}_{0} {d \omega'} \bigg[ \langle a_{\omega} a_{\omega'}^{\dagger} \rangle \overrightarrow{u}_{\omega}(x) \overrightarrow{u}^*_{\omega'}(x')
+ \langle a_{\omega}^{\dagger} a_{\omega'} \rangle \overrightarrow{u}^*_{\omega}(x) \overrightarrow{u}_{\omega'}(x')
+ \\
+\langle b_{\omega} b_{\omega'}^{\dagger} \rangle\overleftarrow{u}_{\omega}(x) \overleftarrow{u}^*_{\omega'}(x') + \langle b_{\omega}^{\dagger} b_{\omega'} \rangle
\overleftarrow{u}^*_{\omega}(x) \overleftarrow{u}_{\omega'}(x') \bigg].
\end{multline}
Here we have omitted spherical harmonic indices to simplify the equation.
Allowing the temperatures of out--going and in--going modes to differ from each other, in this paper we will study states in which
\begin{equation}
\langle a_{\omega}^{\dagger} a_{\omega'} \rangle = \frac{ \delta(\omega -\omega')}{e^{\beta_R |\omega|}-1}, \qquad \langle b_{\omega}^{\dagger} b_{\omega'} \rangle = \frac{\delta(\omega -\omega')}{e^{\beta_L |\omega|}-1},
\end{equation}
so the Wightman function can be written as
\begin{multline}
W(x,x') = \sum_{lm} \int^{+\infty}_{0} {d \omega} \bigg[ \frac{\overrightarrow{u}_{\omega lm}(x) \overrightarrow{u}^*_{\omega lm}(x')}{1-e^{-\beta_R |\omega|}}
+ \frac{\overrightarrow{u}^*_{\omega lm}(x) \overrightarrow{u}_{\omega lm}(x')}{e^{\beta_R |\omega|}-1} +
\frac{\overleftarrow{u}_{\omega lm}(x) \overleftarrow{u}^*_{\omega lm}(x')}{1-e^{-\beta_L |\omega|}}+ \\ + \frac{\overleftarrow{u}^*_{\omega lm}(x) \overleftarrow{u}_{\omega lm}(x')}{e^{\beta_L |\omega|}-1} \bigg] = \\
=\sum_{lm} \int^{+\infty}_{0} \frac{d \omega}{4\pi |\omega|} \frac{1}{rr'}\bigg[ \frac{e^{i |\omega| (t-t')} \overrightarrow{R}^*_l (\omega|r) \overrightarrow{R}_l (\omega|r') }{e^{\beta_R | \omega|}-1}
+\frac{e^{i |\omega| (t-t')} \overleftarrow{R}^*_l (\omega|r) \overleftarrow{R}_l (\omega|r') }{e^{\beta_L |\omega|}-1} \bigg] Y_{lm} (\Omega) Y^*_{lm} (\Omega') + \\+
\sum_{lm} \int^{0}_{-\infty} \frac{d \omega}{4\pi |\omega|} \frac{1}{rr'}\bigg[ \frac{e^{-i |\omega| (t-t')} \overrightarrow{R}^*_l (\omega|r) \overrightarrow{R}_l (\omega|r') }{1-e^{-\beta_R | \omega|}}
+\frac{e^{-i |\omega| (t-t')} \overleftarrow{R}^*_l (\omega|r) \overleftarrow{R}_l (\omega|r') }{1-e^{-\beta_L |\omega|}} \bigg] Y_{lm} (\Omega) Y^*_{lm} (\Omega'),
\end{multline}
where in the last relation we have made the substitution $\omega \to -\omega$ for the first and the third terms in the first line. Also we have used the relation $\overrightarrow{R}_l(-\omega|r) = \overrightarrow{R}^*_l(\omega|r)$, which can be deduced from \eqref{near_horizon_mode} or \eqref{pt_reflection} with \eqref{boundary_cond}. Then we put $|\omega| = +\omega$ in the first integral of the last equality and $|\omega| = -\omega$ in the second integral to obtain the following form of the Wightman function:
\begin{multline} \label{wightman}
W(x,x') = \sum_{lm} \int^{+\infty}_{-\infty} \frac{d \omega}{4\pi \omega} \frac{1}{rr'}\bigg[ \frac{e^{i \omega (t-t')} \overrightarrow{R}^*_l (\omega|r) \overrightarrow{R}_l (\omega|r') }{e^{\beta_R \omega}-1}
+\\+\frac{e^{i \omega (t-t')} \overleftarrow{R}^*_l (\omega|r) \overleftarrow{R}_l (\omega|r') }{e^{\beta_L \omega}-1} \bigg] Y_{lm} (\Omega) Y^*_{lm} (\Omega').
\end{multline}
For the case $\beta_R = \beta_L = \infty$ we obtain the Boulware state, while $\beta_R=\frac{2\pi}{\kappa}, \beta_L = \infty$ corresponds to the Unruh state, and $\beta_R=\beta_L=\frac{2\pi}{\kappa}$ results in the Hartle-Hawking state. Note that in the limit $\beta=\infty$ Bose-Einstein distribution reduces to Heaviside theta-function $\theta(-\omega)$ and the Wightman function reduces to what one would have obtained from \eqref{wightman_general} by taking the state to be the Fock vacuum. Here $\kappa = (4M)^{-1}$ is the surface gravity of the black hole.
\subsubsection{Massive case}
In the case of massive field ($\mu \neq 0$), the equation for the radial function is
\begin{align} \label{radial}
&\frac{d^2R}{dr_*^2}+\big[ \omega^2 - V_l (r) \big] R = 0,
\end{align}
with the potential
\begin{equation} \label{potential}
V_l (r) = \bigg( 1 - \frac{2M}{r} \bigg) \bigg( \frac{l(l+1)}{r^2} + \frac{2M}{r^3} + \mu^2 \bigg),
\end{equation}
which tends to $\mu^2$ in the $r^* \to \infty$ limit. So as in \cite{Akhmedov:2020ryq}, \cite{Akhmedov:2015xwa}, \cite{Akhmedov:2016uha} there are modes with $\omega^2 < \mu^2$ that are localised near the horizon and which are exponentially decreasing in the classically inaccessible domain.
Rewriting the equation \eqref{radial} in terms of coordinate $r$, taking the near-horizon limit and introducing the new coordinate $\xi^2 = \frac{r}{2M}-1$ leads to the equation
\begin{equation}
\frac{d^2 R}{d\xi^2} + \frac{1}{\xi} \frac{dR}{d\xi} + \bigg[ \frac{(4M\omega)^2}{\xi^2} -4l(l+1)-(4M\mu)^2 \bigg] R = 0,
\end{equation}
solutions of which are given by the modified Bessel functions of the second kind. Keeping in mind the condition of exponential decrease in the classically inaccessible domain, one obtains
\begin{equation} \label{radial_massive_sch}
R \approx \sqrt{\frac{\omega \sinh({4\pi M \omega})}{\pi M}} K_{4iM\omega} \big( 2 \sqrt{(2M\mu)^2+l^2+l} \, \xi \big).
\end{equation}
Then, the modes are denoted as
\begin{align}
&\varphi_{\omega lm} (x) = \frac{1}{\sqrt{ \pi |\omega|}} e^{-i |\omega| t} R_{ l} (\omega | r) Y_{lm} (\Omega),
\end{align}
where the radial part is given by \eqref{radial_massive_sch}, so that near the horizon one has
\begin{gather} \label{delta_sch}
R_{l} (\omega | r) \approx \frac{1}{r} \cos (\omega r_* + \delta_{\omega l}), \\
\delta_{\omega l} \approx \frac{\pi}{2} +2M\omega \log (4M^2 \mu^2 +l^2+l) - 2M\omega - \text{arg} \, \Gamma(1+4iM\omega).
\end{gather}
The modes with $\omega^2>\mu^2$ are similar to the massless case:
\begin{align} \label{f_modes}
&\overrightarrow{F}_{\omega lm} (x) = (4 \pi |\omega|)^{-1/2} e^{-i |\omega| t} r^{-1} \overrightarrow{F}_l(\omega | r) Y_{lm} (\Omega), \nonumber \\
&\overleftarrow{F}_{\omega lm} (x) = (4 \pi |\omega|)^{-1/2} e^{-i |\omega| t} r^{-1} \overleftarrow{F}_l(\omega | r) Y_{lm} (\Omega),
\end{align}
where their radial parts obey the following boundary conditions:
\begin{align}
&\overrightarrow{F}_l ({\omega} | r) = \begin{cases}
e^{i \omega r_*} + \overrightarrow{C}_{\omega} e^{-i \omega r_*},
& \qquad r \rightarrow 2M \\
\sqrt{\frac{\omega}{p}} {D}_{\omega} e^{i p r_*}, & \qquad r \rightarrow \infty
\end{cases} \\
&\overleftarrow{F}_l ({\omega} |r) = \begin{cases}
{D}_{\omega} e^{ - i \omega r_*}, & r \rightarrow 2M \\
\sqrt{\frac{\omega}{p}} [ e^{-ipr_*} + \overleftarrow{C}_{\omega} e^{ipr_*}], & r \rightarrow \infty
\end{cases}
\end{align}
with $p=\text{sgn} (\omega) \cdot \sqrt{\omega^2 - \mu^2}$, \, $\omega^2 > \mu^2$.
The mode decomposition of the field operator for the massive case is as follows
\begin{multline} \label{mode_massive_sch}
\phi(x) = \sum_{lm} \int_0^{\mu} d\omega \big( \alpha_{\omega lm} \varphi_{\omega lm} + \alpha_{\omega lm}^{\dagger} \varphi_{\omega lm}^* \big) + \\
+\sum_{lm} \int_{\mu}^{\infty} d \omega \big( \beta_{\omega lm} \overrightarrow{F}_{\omega lm} + \beta_{\omega lm}^{\dagger} \overrightarrow{F}^*_{\omega lm} + \gamma_{\omega lm} \overleftarrow{F}_{\omega lm} + \gamma_{\omega lm}^{\dagger} \overleftarrow{F}_{\omega lm}^* \big),
\end{multline}
and the Wightman function has the following form:
\begin{multline}
W(x,x') = \sum_{lm} \int^{+\mu}_{-\mu} \frac{d \omega}{\pi \omega} \frac{e^{i \omega (t-t')} R^*_l(\omega|r) R_l (\omega|r') }{e^{\beta_0 \omega}-1} Y_{lm}(\Omega) Y^*_{lm}(\Omega') + \\
+\sum_{lm} \int_{|\omega|>\mu} \frac{d \omega}{4\pi \omega} \frac{1}{rr'} \bigg[ \frac{ e^{i \omega (t-t')} \overrightarrow{F}^*_{ l}(\omega | r) \overrightarrow{F}_{l}(\omega|r')}{e^{\beta_R \omega}-1}
+
\frac{ e^{i \omega (t-t')} \overleftarrow{F}^*_{l}(\omega|r) \overleftarrow{F}_{l}(\omega|r')}{e^{\beta_L \omega}-1} \bigg] Y_{lm}(\Omega) Y^*_{lm}(\Omega') .
\end{multline}
Note that we have added here the inverse temperature $\beta_0$ for the modes with $\omega^2 \leq \mu^2$, which does not have to be equal neither to $\beta_R$ nor to $\beta_L$.
\subsection{Canonical singularity} \label{canonical_sing}
First we want to explicitly show in what sense the divergences derived further in this section are anomalous. To do so, we consider the Wightman function \eqref{wightman} when both of it's points are located in the vicinity of the horizon, but not exactly on it. We will take light-like separation of these two points, keeping their angular coordinates equal. We restrict ourselves to the massless fields in this subsection.
The metric near the horizon can be written as
\begin{equation}
ds^2 \approx \rho^2 d \eta^2 - d \rho^2 -r^2(\rho) \, d\Omega^2,
\end{equation}
where
\begin{align}
\rho = 4M \sqrt{\frac{r}{2M}-1} ,\qquad \eta = \frac{t}{4M}.
\end{align}
Then, the square of the geodesic distance between the two points under consideration is written as
\begin{equation} \label{geodesic}
L^2 \approx 2 \rho_1 \rho_2 \cosh (\eta_1-\eta_2) - \rho_1^2-\rho_2^2.
\end{equation}
One can expand the potential near the horizon and find a solution (in terms of modified Bessel function of the second kind) that satisfies boundary condition \eqref{boundary_cond}:
\begin{equation} \label{near_horizon_mode}
\overrightarrow{R}_l (\omega|r) \approx \frac{2}{\Gamma (-i 4 M \omega)} e^{-2iM \omega \log (l^2+l+1)+2iM\omega} K_{4iM\omega} (2 \sqrt{l^2+l+1} e^{\frac{r_*-2M}{4M}}),
\end{equation}
from which one can also find that
\begin{equation}
\overrightarrow{A}_l (\omega) \approx - e^{4iM\omega - 4iM\omega \log(l^2+l+1) + 2i \text{arg} \Gamma(1+4iM \omega) }.
\end{equation}
Now, plugging \eqref{near_horizon_mode} into \eqref{wightman}, one obtains that
\begin{multline} \label{prop_non_horizon}
W(x,x') \approx \frac{1}{\pi^2} \frac{1}{(2M)^2} \sum_{lm} Y_{lm} (\Omega) Y_{lm}^* (\Omega) \int^{\infty}_{-\infty} d\omega \frac{e^{\frac{i \omega (t-t')}{4M}}}{e^{\frac{\beta_R \omega}{4M}}-1} \sinh( \pi \omega) \cdot \\ \cdot
K_{i\omega} (2 \sqrt{l^2+l+1} e^{\frac{r_*-2M}{4M}}) K_{i\omega} (2 \sqrt{l^2+l+1} e^{\frac{r_*'-2M}{4M}}).
\end{multline}
The simple formula is available if we take $\beta_R = \frac{8\pi M}{n}$ with $n$ being integer \cite{Akhmedov:2019esv}, \cite{Akhmedov:2020qxd}. Namely, we employ the fact that in this case
\begin{equation}
\frac{\sinh(\pi \omega)}{e^{\frac{\beta_R \omega}{4M}}-1} = \frac{e^{-\pi \omega}}{2} \sum_{k=1}^{n} e^{\frac{2\pi \omega (k-1)}{n}}.
\end{equation}
Plugging this expressions into \eqref{prop_non_horizon} and evaluating the integral over $\omega$, we obtain that
\begin{multline} \label{wightman_discrete_sum}
W(x,x') \approx \frac{1}{2 \pi} \frac{1}{(2M)^2} \sum_l \frac{(2l+1)}{4\pi} \times \\ \times
\sum_{k=1}^{n} K_0 \bigg( 2 \Tilde{l} \sqrt{e^{\frac{r_*-2M}{2M}} + e^{\frac{r_*'-2M}{2M}} +2 e^{\frac{r_*-2M}{4M}} e^{\frac{r_*'-2M}{4M}} \cosh \bigg[ -\frac{t-t'}{4M} +i \pi - i \frac{2\pi (k-1)}{n} \bigg] } \bigg),
\end{multline}
where $\Tilde{l} = \sqrt{l^2+l+1}$.
Here only the $k=1$ term depends on the geodesic distance between the two points \eqref{geodesic}, so detaching this terms from the sum and writing the Wightman function as:
\begin{equation} \label{wightman_k}
W(x,x') = W_{k = 1} (x,x') + W_{k \neq 1} (x,x'),
\end{equation}
we obtain that for a light-like separated $x$ and $x'$, i.e. with $L=0$ (but not yet sitting at the horizon):
\begin{multline}
W_{k=1}(x,x') \approx \frac{1}{2\pi} \frac{1}{(2M)^2} \sum_l \frac{(2l+1)}{4\pi}
K_0 \bigg( \frac{\sqrt{l^2+l+1}}{2M} \sqrt{-{L^2}} \bigg) \approx \\
\approx \frac{1}{4\pi^2} \frac{1}{(2M)^2} \int_0^{\infty} dl \, l K_0 \bigg( \frac{l}{2M} \sqrt{-L^2} \bigg) = -\frac{1}{4 \pi^2 L^2},
\end{multline}
and as the terms in $W_{k \neq 1} (x,x')$ do not depend on the geodesic distance between $x$ and $x'$, they are negligible when we are considering light-like separation of the points out of the horizon. Hence, we get that
\begin{equation}
W(x,x') \approx -\frac{1}{4 \pi^2 L^2},
\end{equation}
i.e. we have the standard coefficient of $(4\pi^2)^{-1}$ in front of the divergence.
However, when the points are located on the horizon, in \eqref{wightman_k} $k \neq 1$ terms also have divergent contributions. We can place the two points on the future horizon by taking the following limit:
\begin{align} \label{limit}
&t=-\lambda, \qquad r_*=\lambda, \nonumber \\
&t'=-\lambda, \qquad r_*'=\lambda+c, \qquad \lambda \to -\infty.
\end{align}
Now we demonstrate that in this limit the Wightman function acquires additional divergent terms. In the case when $n=2$, the Wightman function has the following form:
\begin{multline}
W(x,x') \approx \frac{1}{2\pi} \frac{1}{(2M)^2} \sum_l \frac{(2l+1)}{4\pi} \bigg\{
K_0 \bigg( \frac{\sqrt{l^2+l+1}}{2M} \sqrt{-{L^2}} \bigg) + \\ + K_0 \bigg( \frac{\sqrt{l^2+l+1}}{2M} \sqrt{-{L^2 \Lambda}} \bigg) \bigg\} \approx -\frac{1}{4 \pi^2 L^2} \frac{1+\Lambda}{\Lambda},
\end{multline}
where we have expressed the argument of the second term in the curly bracket through the geodesic distance in the limit \eqref{limit} via the coefficient $\Lambda$, which has the form
\begin{equation}
\Lambda = \frac{1+e^{c/2M}+2 e^{c/4M}}{1+e^{c/2M}-2e^{c/4M}}.
\end{equation}
However, in this approach it is not clear how to obtain the answer for the Wightman function even in the case of the discrete inverse temperature $\beta_R = \frac{8\pi M}{n}$ with an arbitrary $n$, let alone the case of the arbitrary $\beta_R$. This difficulty comes from the fact that as opposed to the static patch of de Sitter space, in this case one cannot express the Wightman function with the non-canonical temperature as the sum of the canonical temperature Wightman functions. In the following section we calculate the Wightman function using the asymptotic form of the modes near the horizon, and the result is that the coefficient of the singularity of the Wightman function depends explicitly on the temperature (as it is shown in \eqref{sing_schw_massive} and \eqref{schw_res} below). Furthermore, by taking high-frequency limit in \eqref{prop_non_horizon}, it can be shown that the singularity outside the horizon is an UV effect, while at the horizon the singularity comes from the IR region of the frequencies. This justifies the fact that everywhere outside the horizon the divergence does not depend on the temperature, while at the horzion there is an explicit dependence on the temperature (i.e. on the low laying state).
\subsection{Anomalous singularity} \label{anom_sing}
In this section the anomalous singularity in the Wightman function on the horizon for generic thermal state is derived.
\subsubsection{Massless case}
Using the asymptotic behavior of the modes \eqref{boundary_cond} in the limit \eqref{limit},
the Wightman function \eqref{wightman} can be written as:
\begin{multline} \label{prop}
W(x,x') \approx \frac{1}{(2M)^2} \sum_{lm} Y^*_{lm} (\Omega) Y_{lm} (\Omega') \int_{-\infty}^{\infty} \frac{d \omega }{4 \pi \omega} \bigg[ \frac{1}{e^{\beta_R \omega }-1} \bigg( e^{-i\omega c} + \overrightarrow{A}_l (\omega) e^{-2 i \lambda \omega } + \\ + \overrightarrow{A}_l^* (\omega) e^{2 i \lambda \omega } + | \overrightarrow{A}_l (\omega) |^2 e^{i\omega c} \bigg) + \frac{1}{e^{\beta_L \omega}-1} | B_l (\omega) |^2 e^{i \omega c} \bigg].
\end{multline}
To integrate over $\omega$, analytic expression for reflection coefficient $\overrightarrow{A}_l (\omega)$ is needed. To compute it, one needs to know transition coefficients between solutions of confluent Heun equation near the horizon and infinity. However, there is no knowledge about these coefficients \cite{Fiziev:2005ki}, but we can approximate potential for radial function by P{\"o}schl-Teller potential \cite{Chirenti:2017mwe},
\cite{Berti:2009kk}, which has the form
\begin{equation}
\Tilde{V}_l(r) = \frac{V_0}{\cosh^2 \alpha (r_* - \Tilde{r}_*) },
\end{equation}
where $\Tilde{r}_*$ is the position of the peak and
\begin{align}
V_0 = V_l (\Tilde{r}_*), \qquad \alpha^2 = - \frac{1}{2V_0} \frac{d^2 V_l}{dr_*^2}_{|r_*=\Tilde{r}_*}.
\end{align}
In this potential we can easily compute the reflection coefficient $\overrightarrow{A}_l (\omega)$, which is found to be
\begin{equation} \label{pt_reflection}
\overrightarrow{A}_l (\omega) = \frac{\Gamma({i \omega}/{\alpha}) \Gamma(-i {\omega}/{\alpha}-s) \Gamma(1+s-i {\omega}/{\alpha})}{\Gamma(-i {\omega}/{\alpha}) \Gamma(1+s) \Gamma(-s) }, \qquad s = -\frac{1}{2} + \frac{i}{2} \sqrt{\frac{4V_0}{\alpha^2}-1}.
\end{equation}
So we have to evaluate the integral
\begin{multline}
\int_{-\infty}^{\infty} \frac{d \omega }{4 \pi \omega} \frac{ \overrightarrow{A}_l (\omega) e^{-2 i \lambda \omega }}{e^{\beta_R \omega}-1} = \int_{-\infty}^{\infty} \frac{dz}{4 \pi z} \frac{1}{e^z-1} \frac{\Gamma(iz/\alpha \beta_R) \Gamma(-iz/ \alpha \beta_R-s) \Gamma(1+s-iz/ \alpha \beta_R)}{\Gamma(1+s) \Gamma(-s) \Gamma(-iz/ \alpha \beta_R)} e^{2 i \frac{| \lambda |}{\beta_R} z} = \\
= \pi i \text{Res}(f(z),0) + 2\pi i \text{Res}(f(z), \overline{0}),
\end{multline}
where the second term $\text{Res}(f(z), \overline{0})$ denotes the sum over all residues on the upper half-plane poles.
The residue at $z=0$ in the limit $\lambda \to -\infty$ is computed to be
\begin{equation}
\pi i \text{Res}(f(z),0) = \frac{|\lambda|}{2 \beta_R}.
\end{equation}
Here an additional term of the form $\Gamma'(0)/\Gamma(0)$ could arise from derivatives of gamma functions, but it is discarded in the limiting procedure as it is of the form
\begin{equation} \label{gammaterms}
\frac{\Gamma(-iz / \alpha \beta_R) \Gamma'(iz/ \alpha \beta_R)}{\Gamma^2 (iz/ \alpha \beta_R)} + \frac{\Gamma(iz / \alpha \beta_R) \Gamma'(-iz/ \alpha \beta_R)}{\Gamma^2 (iz/ \alpha \beta_R)}.
\end{equation}
In fact, using that
\begin{equation}
\Gamma(\epsilon) = - \Gamma(-\epsilon) \frac{\Gamma(1+\epsilon)}{\Gamma(1-\epsilon)}
\end{equation}
one concludes that \eqref{gammaterms} is zero.
Furthermore
\begin{equation}
2\pi i \text{Res}(f(z), \overline{0})= -\frac{i}{2} \sum_{n>0} \frac{(-1)^n}{(n!)^2} \frac{1}{e^{i \alpha \beta_R n}-1} \frac{\Gamma(n-s) \Gamma(1+s+n)}{\Gamma(1+s) \Gamma(-s)} e^{-2 |\lambda| \alpha n},
\end{equation}
which tends to zero as $\lambda \to -\infty$.
So in the limit when both points of the propagator are located on the horizon, we obtain
\begin{equation}
\int_{-\infty}^{\infty} \frac{d \omega }{4 \pi \omega} \frac{\overrightarrow{A}_l (\omega) e^{-2 i \lambda \omega }}{e^{\beta_R \omega}-1} \approx \frac{|\lambda|}{2 \beta_R}, \qquad \lambda \to -\infty.
\end{equation}
The other complex conjugated term in \eqref{prop} gives the same contribution. Then, using that
\begin{align}
P_l (\vec{x} \cdot \vec{y} ) = \frac{4 \pi}{2l+1} \sum_{m=-l}^l Y_{lm} (\vec{x}) Y_{lm}^*(\vec{y}), \qquad \sum_{l=0}^{\infty} \frac{2l+1}{2} P_l (x) P_l(y) = \delta(x-y),
\end{align}
the Wightman function can be rewritten as
\begin{equation} \label{schw_res}
W(x,x') \approx \frac{|\lambda|}{ \beta_R} \frac{\delta(\Omega,\Omega')}{(2M)^2} , \qquad \lambda \to - \infty.
\end{equation}
Actually, one can obtain the same answer without approximating the potential and employing exact solution in fitted potential. Namely, one can use the approximate near-horizon solution \eqref{near_horizon_mode},
but ultimately the result \eqref{schw_res} remains the same. However, the near-horizon approximation suffers from the fact that the potential for the radial function in this case is infinitely growing, so that one can only introduce the out-going modes that are fully reflected back from the potential, i.e. in this situation one has $|\overrightarrow{A}_l (\omega)|=1$. The approximation with the P{\"o}schl-Teller potential shows that deformation of the potential from the infinitely growing with no in-going modes to the bump-like with the presence of the transmitted in-going modes does not change the answer for the anomalous singularity.
This shows that this anomalous singularity is not a peculiarity for a given solution of Einstein's equations, but actually its presence does not depend on the form of the potential.
Also, one can see that for the discrete temperature $\beta = \frac{8 \pi M}{n}$, the result \eqref{schw_res} can be obtained from \eqref{wightman_discrete_sum} by taking the asymptotic form of the $K_0(x)$ as $x \to 0$. Dependence on $l$ splits off from the rest of the argument of this function because of the logarithm function in this asymptotic, so the leading terms in the horizon limit are given by the terms that depend on the geodesic distance, and the sum over $l$ gives the angular delta-function in \eqref{schw_res}.
\subsubsection{Massive case}
As the anomalous singularity on the horizon comes from the low-frequency modes (see previous subsection or \cite{Akhmedov:2020ryq}), we neglect the contribution from high-frequency modes in \eqref{mode_massive_sch}. Then,
the Wightman function in the horizon limit is as follows:
\begin{equation}
W(x,x') \approx \sum_{lm} \int^{+\mu}_{-\mu} \frac{d \omega}{\pi \omega} \frac{e^{i \omega (t-t')} R^*_l(\omega|r) R_l (\omega|r') }{e^{\beta_0 \omega}-1} Y_{lm}(\Omega) Y^*_{lm}(\Omega')
\end{equation}
Assuming that $\mu \beta_0$ is sufficiently large for the integrand to be zero at the endpoints of the integration domain, we extend the domain of integration to the whole real axis, and the integral evaluated in complete analogy with the two-dimensional case \cite{Akhmedov:2020ryq}. Namely, in the limit \eqref{limit}, the Wightman function is written as
\begin{multline}
W(x,x') \approx \sum_{lm} \frac{Y_{lm}(\Omega) Y_{lm}^*(\Omega')}{(2M)^2} \int^{\infty}_{-\infty} \frac{d\omega}{4\pi \omega} \frac{1}{e^{\beta_0 \omega}-1} \bigg(e^{-i\omega c} + e^{2 i \omega \lambda+2i \delta_{\omega l}}
+ e^{i\omega c} + e^{-2i \omega \lambda-2i \delta_{\omega l}} \bigg),
\end{multline}
where $\delta_{\omega l}$ is given by \eqref{delta_sch}.
Evaluating the contribution of the double pole at the $\omega=0$ as in the massless case,
we obtain that
\begin{equation} \label{sing_schw_massive}
W(x,x') \approx \frac{| \lambda |}{ \beta_0} \frac{\delta(\Omega,\Omega')}{(2M)^2}, \qquad \lambda \to -\infty,
\end{equation}
where $\delta(\Omega,\Omega')$ is the delta function on the two-sphere.
\section{Reissner-Nordström black hole} \label{RNsection}
In this section first we construct modes and Wightman functions outside of the outer horizon. We investigate behavior of the Wightman function on the outer horizon and evaluate the anomalous singularity when both points are located there. The resulting singularity is analogous to the one in the Schwarzschild black hole.
\subsection{Modes and Wightman function}
\begin{figure}[!h]
\centering
\includegraphics[width=0.6\textwidth]{picRN}
\caption{Penrose diagram of the Reissner-Nordström black hole.} \label{picRN}
\end{figure}
Geometry of the Reissner-Nordström black hole is given by the metric
\begin{equation}
ds^2 = \bigg(1-\frac{2M}{r}+\frac{Q^2}{r^2}\bigg) dt^2 - \frac{dr^2}{\big(1-\frac{2M}{r}+\frac{Q^2}{r^2}\big)}-r^2(d \theta^2 + \sin^2 \theta d \varphi^2).
\end{equation}
This geometry has two horizons at
\begin{equation}
r_{\pm} = M \pm \sqrt{M^2-Q^2},
\end{equation}
where the surface $r=r_+$ is the event horizon and the surface $r=r_-$ is a Cauchy horizon (we consider non-extremal case $Q<M$). Surface gravity corresponding to these horizons is equal to:
\begin{align}
\kappa_{\pm} = \frac{r_+-r_-}{2r^2_{\pm}},
\end{align}
and the tortoise coordinate may be defined as
\begin{align}
r_* = r+\frac{1}{2\kappa_+} \log \bigg( \frac{|r-r_+|}{r_+-r_-} \bigg) -\frac{1}{2\kappa_-} \log \bigg( \frac{|r-r_-|}{r_+-r_-} \bigg).
\end{align}
Also, it will be useful to define the Eddington-Finkelstein coordinates, which are written in the exterior region as
\begin{align}
u = t-r_*, \qquad v = t+r_*,
\end{align}
In the following I consider the situation in which both points of the Wightman function are located on the outer horizon $r=r_+$, which is denoted as $H_R$ in Fig. \ref{picRN}.
\subsubsection{Massless case}
In the exterior region we can proceed analogously as in the Schwarzschild spacetime. Namely, we split variables as in \eqref{split}
and represent these functions in the form \eqref{f_split}, where the radial functions $R(r)$ satisfy the equation as follows
\begin{align} \label{exterior_radial}
&\qquad \qquad \qquad \frac{d^2R}{dr_*^2}+\big[ \omega^2 - V_l (r) \big] R = 0, \nonumber \\
&V_l(r) = \bigg(1-\frac{2M}{r}+\frac{Q^2}{r^2}\bigg) \bigg( \frac{l(l+1)}{r^2} + \frac{2M}{r^3} - \frac{2Q^2}{r^4} \bigg).
\end{align}
Again we distinguish two types of solutions \eqref{set1} of this equation with boundary conditions \eqref{boundary_cond}, corresponding to out-going and in-going modes. Then, as in the Schwarzschild spacetime, the Wightman function in the generic thermal state is written as
\begin{multline} \label{outer_hh}
W(x,x') = \sum_{lm} \int^{+\infty}_{-\infty} \frac{d \omega}{4\pi \omega} \frac{1}{rr'}\bigg[ \frac{e^{i \omega (t-t')} \overrightarrow{R}^*_l (\omega|r) \overrightarrow{R}_l (\omega|r') }{e^{\beta_R \omega}-1}
+\\+\frac{e^{i \omega (t-t')} \overleftarrow{R}^*_l (\omega|r) \overleftarrow{R}_l (\omega|r') }{e^{\beta_L \omega}-1} \bigg] Y_{lm} (\Omega) Y^*_{lm} (\Omega').
\end{multline}
\subsubsection{Massive case}
In the massive case the equation for the radial function is
\begin{align} \label{radial_rn}
&\frac{d^2R}{dr_*^2}+\big[ \omega^2 - V_l (r) \big] R = 0,
\end{align}
with the potential
\begin{equation} \label{potential}
V_l (r) = \bigg( 1 - \frac{2M}{r} + \frac{Q^2}{r^2} \bigg) \bigg( \frac{l(l+1)}{r^2} + \frac{2M}{r^3} - \frac{2Q^2}{r^4} + \mu^2 \bigg).
\end{equation}
Again, the modes with $\omega^2 \leq \mu^2$ are
\begin{align}
&\varphi_{\omega lm} (x) = \frac{1}{\sqrt{ \pi |\omega|}} e^{-i |\omega| t} R_{ l} (\omega | r) Y_{lm} (\Omega), \nonumber \\
&R_{ l} (\omega | r) \approx \frac{1}{r} \cos (\omega r_* + \delta_{\omega l}),
\end{align}
where the expression for the phase acquires the form
\begin{align}
&\delta_{\omega l} \approx \frac{\pi}{2} +\frac{\omega}{2 \kappa_+} \log \big( \Tilde{l}^2+\mu^2 \big) -\omega r_+ - \text{arg} \, \Gamma(1+\frac{i\omega}{\kappa_+}), \nonumber \\
&\qquad \qquad \qquad \qquad \Tilde{l}^2 = l^2+l+1-\frac{r_-}{r_+}.
\end{align}
The modes with $\omega^2 > \mu^2$ are defined as in \eqref{f_modes}. Then, the Wightman function is as follows:
\begin{multline}
W(x,x') = \sum_{lm} \int^{+\mu}_{-\mu} \frac{d \omega}{\pi \omega} \frac{e^{i \omega (t-t')} R^*_l(\omega|r) R_l (\omega|r') }{e^{\beta_0 \omega}-1} Y_{lm}(\Omega) Y^*_{lm}(\Omega') + \\
+\sum_{lm} \int_{|\omega|>\mu} \frac{d \omega}{4\pi \omega} \frac{1}{rr'} \bigg[ \frac{ e^{i \omega (t-t')} \overrightarrow{F}^*_{ l}(\omega | r) \overrightarrow{F}_{l}(\omega|r')}{e^{\beta_R \omega}-1}
+
\frac{ e^{i \omega (t-t')} \overleftarrow{F}^*_{l}(\omega|r) \overleftarrow{F}_{l}(\omega|r')}{e^{\beta_L \omega}-1} \bigg] Y_{lm}(\Omega) Y^*_{lm}(\Omega') .
\end{multline}
\subsection{Anomalous singularity}
\subsubsection{Massless case}
As in the previous section, we take the near outer horizon limit $r \to r_+$ \eqref{limit} of the expression \eqref{outer_hh}, evaluate the reflection coefficients $\overrightarrow{A}_l (\omega)$ by solving the radial equation near the outer horizon to obtain
\begin{align}
&\overrightarrow{R}_l (\omega |r) \approx 2 \Tilde{l}^{-\frac{i\omega}{\kappa_+}}\frac{e^{i \omega r_+}}{\Gamma(-i \omega / \kappa_+)} K_{\frac{i\omega}{\kappa_+}} \big( 2 \Tilde{l} e^{\kappa_+ (r_*-r_+)} \big), \nonumber \\
&\overrightarrow{A}_l (\omega) \approx - \frac{\kappa_+}{\pi \omega} e^{- \frac{2i\omega}{\kappa_+} \log \Tilde{l} +2i \omega r_+} \sinh \bigg( \frac{\pi \omega}{\kappa_+} \bigg) \Gamma \bigg( 1+\frac{i\omega}{\kappa_+} \bigg).
\end{align}
Substituting this reflection coefficient into the expression for the Wightman function in the outer horizon limit
\begin{multline}
W(x,x') \approx \frac{1}{(2M)^2} \sum_{lm} Y^*_{lm} (\theta, \varphi) Y_{lm} (\theta',\varphi') \int_{-\infty}^{\infty} \frac{d \omega }{4 \pi \omega} \bigg[ \frac{1}{e^{\beta_R \omega }-1} \big( e^{-i\omega c} + \overrightarrow{A}_l (\omega) e^{-2 i \lambda \omega } + \\ + \overrightarrow{A}_l^* (\omega) e^{2 i \lambda \omega } + | \overrightarrow{A}_l (\omega) |^2 e^{i\omega c} \big) + \frac{1}{e^{\beta_L \omega}-1} | B_l (\omega) |^2 e^{i \omega c} \bigg],
\end{multline}
and evaluating the integrals as in the Schwarzschild spacetime, one obtains
\begin{equation}
W(x,x') \approx \frac{|\lambda| }{\beta_R} \, \frac{ \delta(\Omega,\Omega')}{r_+^2} \, , \qquad \lambda \to -\infty.
\end{equation}
\subsubsection{Massive case}
As is the Schwarzschild spacetime, the leading contribution in the horizon limit \eqref{limit} comes from the low frequencies. Then, for the Wightman function in the horizon limit one has:
\begin{multline}
W(x,x') \approx \sum_{lm} \int_{-\mu}^{\mu} d \omega \, \frac{\varphi_{\omega lm}^* (x) \, \varphi_{\omega lm} (x')}{e^{\beta_0 \omega}-1} \approx \\
\approx \sum_{lm} \frac{Y_{lm}(\Omega) Y_{lm}^*(\Omega')}{(2M)^2} \int^{\infty}_{-\infty} \frac{d\omega}{4\pi \omega} \frac{e^{i \omega (t-t')}}{e^{\beta_0 \omega}-1} \bigg( e^{i \omega (r_*+r_*')+2i \delta_{\omega l}} + e^{-i \omega (r_*+r_*')-2i \delta_{\omega l}} \bigg).
\end{multline}
Evaluating the integrals as in the Schwarzschild spacetime, one obtains the result
\begin{equation}
W(x,x') \approx \frac{| \lambda |}{ \beta_0} \frac{\delta(\Omega,\Omega')}{r_+^2}, \qquad \lambda \to -\infty.
\end{equation}
Thus, again we encounter the anomalous singularity at the outer horizon.
\section{Conclusions and discussions}
It is shown that the anomalous singularity also occurs in the four--dimensional Schwarzschild and Reissner-Nordström black hole backgrounds.
Furthermore, as in Rindler and in de Sitter $d$--dimensional examples \cite{Akhmedov:2020ryq}, it is shown that anomalous singularity does not occur when the coordinates additional to $(t,r)$ do not coincide, and the divergence is amplified if they do. As we can represent delta function of zero argument as a linear divergence, we obtain that the power of the divergence is quadratic, as is expected from the analyisis in the Rindler and de Sitter spacetimes where in $d$ dimensions the divergence was of power $(d-2)$. These anomalous singularities lead to the explosive behavior of the expectation value of the stress-energy tensor, which means that with non-canonical temperature the backreaction of the QFT on the background geometry is strong. Does that mean that one cannot place a black hole in a bath with non-canonical temperature? This is the question for the further study.
One more possible outcome of this anomalous singularity is that it can affect the beta function as the coefficient before the light-like divergence is different than the canonical ones.
It is interesting to see whether this anomalous singularity affects loop contributions to the two-point function in an interacting theory. Also, it is tempting to see how the results of this paper affect computation of the entropy of the fields outside the black hole. However, these problems are out of the scope of this paper and will be considered elsewhere.
\section{Acknowledgements}
I would like to thank E.T. Akhmedov, K.V. Bazarov, and D.V. Diakonov for valuable discussions.
This work was supported by the grant from the Foundation for the Advancement of Theoretical Physics and Mathematics "BASIS" and by Russian Ministry of education and science.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,500 |
Q: Which of types Mat or vector is better to use with function estimateRigidTransform()? As known, we can pass to the function estimateRigidTransform() two parameters with one of two types: Mat estimateRigidTransform(InputArray src, InputArray dst, bool fullAffine)
*
*cv::Mat frame1, frame2;
*std::vector<cv::Point2f> frame1_features, frame2_features;
I.e., for example, to implement video-stabilization (shake remove) we can use one of two approach:
*
*with cv::Mat: video stabilization using opencv
cv::Mat frame1 = imread("frame1.png");
cv::Mat frame2 = imread("frame2.png");
Mat M = estimateRigidTransform(frame1, frame2, 0);
warpAffine(frame2, output, M, Size(640,480), INTER_NEAREST|WARP_INVERSE_MAP);
*with std::vector<cv::Point2f> features;
vector <uchar> status;
vector <float> err;
std::vector <cv::Point2f> frame1_features, frame2_features;
cv::Mat frame1 = imread("frame1.png");
cv::Mat frame2 = imread("frame2.png");
goodFeaturesToTrack(frame1 , frame1_features, 200, 0.01, 30);
goodFeaturesToTrack(frame2 , frame2_features, 200, 0.01, 30);
calcOpticalFlowPyrLK(frame1 , frame2, frame1_features, frame2_features, status, err);
std::vector <cv::Point2f> frame1_features_ok, frame2_features_ok;
for(size_t i=0; i < status.size(); i++) {
if(status[i]) {
frame1_features_ok.push_back(frame1_features[i]);
frame2_features_ok.push_back(frame2_features[i]);
}
}
Mat M = estimateRigidTransform(frame1_features_ok, frame2_features_ok, 0);
warpAffine(frame2, output, M, Size(640,480), INTER_NEAREST|WARP_INVERSE_MAP);
Which of these approach is better to use, and why?
I.e. which of types Mat or vector<Point2f> is better to use with function estimateRigidTransform()?
A: In the first case OpenCV will perform implicitly a calcOpticalFlowPyrLK() inside the function estimateRigidTransform(). See the implementation in lkpyramid.cpp @ line 1383.
This is the only difference between the two methods. If finding correspondences between frame1 and frame2 matters then use version #2 otherwise #1.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,923 |
\section{Introduction}
\label{sec:intro}
The reionization of the Universe is the cosmic phase transition in which the hydrogen in the inter galactic medium (IGM), which was neutral ever since the recombination epoch ended at $z \sim 1100$, transformed into an ionized state. Recent {\it Planck}-based measurements of the IGM's Thomson optical depth to cosmic microwave background (CMB) photons suggest $z_{\mathrm{0.5}}=7.68\pm0.79$ as the redshift at which half of the hydrogen was ionized, and points to a rather ``late'' reionization, occurring mostly at $z\lesssim10$ \citep{Planck18VI}.
Complementary and independent measurements based on Gunn-Peterson troughs in quasars \cite[e.g.,][]{Becker01,Djorgovski01,Fan03,Songaila04},
dark gap statistics in quasar spectra \cite[e.g.,][]{McGreer15}, and surveys of Ly$\alpha$-emitting galaxies \cite[e.g.,][]{Schenker14}, all point to reionization being (nearly) complete by $z\sim6$, with some recent studies suggesting that reionization has ended as late as $z\approx5.3$ \cite[e.g.,][]{Eilers18,Kulkarni19,Keating20,Bosman18,Bosman21,Zhu21}.
One key question concerning the reionization of the Universe is the nature of the sources of ionizing radiation that drive it. One type of such sources are young and hot stars in early star-forming galaxies. Multi-wavelength surveys of the high-redshift galaxy population
have allowed to determine the evolution of the galaxy ultraviolet (UV) luminosity function at $z=6-10$ \cite[see, e.g.,][and references therein]{Stark16}, which, combined with assumptions about the ionizing emissivity and escape fraction of ionizing photons, can be used to calculate the production rate of ionizing photons by the high redshift galaxy population \cite[e.g.,][]{Robertson13,Bouwens15,Ishigaki18}. Such studies find that galaxies' ionizing radiation production rate is consistent with early galaxies being the main driver of reionization, with ionizing photon production rate densities at $z=6$ higher than $10^{50.5}\,\ensuremath{\mathrm{s}^{-1}\mathrm{Mpc}^{-3}}\xspace$ , which is higher than the production rate required to keep the IGM ionized at that redshift (assuming a clumping factor in the range of $2-4$; \citealt{Madau99}).
Another source of ionizing radiation are high redshift active galactic nuclei (AGNs) and quasars, powered by accreting super massive black holes (SMBHs).
While quasars are far less common than star-forming galaxies, they are significantly more powerful sources of ionizing radiation.
The radiatively efficient accretion disks that power AGN produce high luminosities (reaching $\Lbol\sim10^{47}\,\ergs$) over long timescales ($\gg10^6\,\rm{yr}$), with ionizing radiation emitted from their hot inner parts (which can reach temperatures of $\sim 10^5\,\rm{K}$).
Over 480 quasars have been observed at $z>5$, 170 of which are at $z>6$ \cite[][and references therein]{RC20}, harboring SMBHs with masses at the range of $\sim10^{8-10}\,\ifmmode M_{\odot} \else $M_{\odot}$\fi$ \cite[e.g.,][]{Willott10, Trakhtenbrot11, DeRosa14, Mazzucchelli17,Onoue19, Shen19}. The existence of these quasars, observed at the end of the epoch of reionization, makes them viable contributors to, or perhaps even the main drivers of, cosmic reionization.
One key outstanding issue related to the contribution of AGNs to reionization revolves around the uncertainty in the space density of low-luminosity ($\lesssim10^{45}\,\ergs$) AGNs at high redshifts, which are expected to be the most common among accreting SMBHs.
The claim of a high number density of $z>4$ low-luminosity AGNs, by \citep[][followed by the more recent work \citealt{Giallongo19}]{Giallongo15}, combined with the {\it Planck}-based finding of a low Thompson scattering optical depth \citep{Planck18VI}, have resurfaced the idea that AGNs could have significantly contributed---or even dominated---cosmic hydrogen reionization \cite[e.g.,][]{MH15,Grazian20}.
However, the \cite{Giallongo15} result was challenged by other studies \cite[e.g.,][]{Parsa18}, and it has become clear that it stands in some contrast with other observational efforts to identify $z\gtrsim5$, low-luminosity AGNs in appropriately deep surveys \cite[e.g.,][]{Weigel15,Cappelluti16,Akiyama18,Matsuoka18,McGreer18,Niida20}.
Consequently, the space density of such systems and their contribution to reionization at $z\gtrsim6$ was claimed to remain limited \cite[e.g., ][]{Wang19,Shen20}.
The contribution of AGNs to reionization naturally depends on the output rate of ionizing photons from the accretion process, and on the corresponding spectral energy distribution (SED).
Many studies assume either a fixed-shape SED, or at least a narrow range of possible SEDs, motivated by observations in the relevant UV regime, \cite[e.g.,][]{Berk01,Telfer02,Lusso15}, and indeed across the EM spectrum \cite[e.g.,][]{Elvis1994,Marconi04}.
While such SEDs can be scaled to fit different bolometric luminosities, they naturally result in the production rate of ionizing photons (denoted in this work as \ensuremath{Q_{\mathrm{ion}}}\xspace) being a simple, often fixed fraction of the total photon production rate (i.e., the bolometric luminosity, \Lbol).
In order to determine the total ionizing flux density of AGNs, these SEDs are combined with the quasar luminosity function (QLF), which at the relevant redshift regime often heavily relies on unobscured AGNs, i.e., quasars (i.e., the $z\sim6$ QLF; \citealt{Wang19,Shen20,Grazian20}).
This pragmatic approach has two drawbacks.
First, as the QLF is currently unknown beyond $z\sim6$, it is extremely challenging to calculate the contribution of AGNs to reionization over the relevant, earlier period.
In practice, this necessitates either extrapolating the (highly uncertain) redshift-dependent trends seen in the $z\lesssim6$ QLF \cite[e.g.,][]{HaimanLoeb98,Fontanot12, Grissom14, MH15, Garaldi19}, or relying on models of the (active) SMBH population \cite[e.g.,][]{BL01, TG11, Feng16, Qin17}.
Second, such assumptions do not take into account the possible variation in the SED, which is expected given the dependence of the accretion disk SED on the black hole (BH) parameters.
Specifically, in our work we focus on radiatively efficient, geometrically-thin, optically-thick accretion disks (\cite{SS73}), which show a good agreement to observed (low-$z$) AGN SEDs and spectra (see, e.g., \citealt{Davis07, Lusso15, Shang15, Capellupo15}, but see also \citealt{KB99, Jin12}).
The high level of similarity between the broad-band SEDs \& UV spectra of $z\sim6$ quasars and those of lower redshift quasar samples \cite[e.g.,][]{Shen19,Vito2019_z6_Xrays,Pons2020_Xray_z65} suggests that the thin-disk model is also appropriate for the population of quasars at the end of the epoch of reionization.
As illustrated in Figure~\ref{fig:diff_SEDs}, for such thin-disk SEDs, \ensuremath{Q_{\mathrm{ion}}}\xspace varies significantly with BH mass, accretion rate and spin\footnote{These dependencies are further discussed in Section~\ref{sec:methods_single_BH}.} -- all of which may evolve over the timescales relevant for the emergence of the first generation of SMBHs, and thus for their contribution to reionization.
This variation and/or evolution cannot be captured by simple, essentially fixed-shape (power-law) SEDs.
Finally, a challenge that every reionization scenario focusing on AGNs has to address is the very formation and early growth of SMBHs. It is currently unknown how the observed $z\sim 5-7$ SMBHs have reached their high BH masses, over such a short period (i.e., $<10^9$ yr after the Big Bang; see, e.g., reviews by \citealt{Volonteri10, Volonteri12, VB12, Haiman13,LF16,JH16,Valiante17,Gallerani17,Inayoshi20}, and references therein).
One possible explanation involves stellar-mass BH seeds ($\lesssim10^2\,\ifmmode M_{\odot} \else $M_{\odot}$\fi$) accreting at Eddington-limited accretion rates with a high duty-cycle.
While possible in principle (as illustrated by, e.g., \citealt{Trakhtenbrot20}), accretion at such high duty cycles was shown to be unlikely in hydrodynamical simulations (see, e.g., \citealt{Inayoshi20} and references therein). Alternatively, several works have put forward the possibility of the formation of massive seeds ($10^4\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$<$}}} M_{\rm seed}/\ifmmode M_{\odot} \else $M_{\odot}$\fi \mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$<$}}} 10^6$; see, e.g., \citealt{KBD04, Begelman06, SS06} and reviews by \citealt{Natarajan2011,Volonteri12}).
Another intriguing alternative explanation is that of stellar mass BH seeds accreting at super-Eddington rates, which can account for the observed $z=6$ SMBH masses even during short periods of accretion time \cite[e.g.,][]{Madau14,Volonteri15}.
In the context of reionization it is important to stress that super-Eddington accretion could be highly luminous (i.e., consistent with the Eddington luminosity) even if radiatively inefficient \cite[e.g.,][]{McKinney14,SN16}.\footnote{This is in contrast to extremely low-rate accretion flows, which are both radiatively inefficient {\it and} extremely faint \cite[e.g.,][]{YuanNarayan14}.}
Note, however, that the observed accretion rates of $z\sim6$ quasars are consistent with high but sub-Eddington accretion \cite[e.g.,][]{Mazzucchelli17,Trakhtenbrot17,Onoue19,Shen19,Yang21}.
In our work we aim to investigate the contribution of AGNs to reionization by interpreting the QLF at $z=6$ in a physical way, i.e., as a population of SMBHs with various SMBH-related parameters and corresponding, physically-motivated SEDs.
We then trace their growth back in time by considering various growth scenarios, while keeping track of their evolving SEDs and the corresponding ionizing radiation output. In Section~\ref{sec:methods} we present the methods used in this work to model the evolution of the ionizing output of a single SMBH, as well as the population of accreting SMBHs at high redshifts.
Moreover, we incorporate a specific radiatively efficient slim-disk model to account for super-Eddington accretion. Section~\ref{sec:results} presents the results obtained using these methods, followed by a detailed discussion, as well as a comparison of the ionizing output of SMBHs and galaxies, in Section~\ref{sec:dicussion}. We conclude with Section~\ref{sec:summary}, where we summarize the key results of our work and discuss its limitations and possible future extensions.
Throughout this work we assume a $\Lambda\mathrm{CDM}$ cosmology with $\Omega_m=0.3$, $\Omega_\Lambda=0.7$, $\Omega_b=0.047$, and $H_0=70\,\mathrm{km}\,\mathrm{s}^{-1}\mathrm{Mpc}^{-1}$.
\section{Methods}
\label{sec:methods}
\subsection{The ionizing output of a single SMBH}
\label{sec:methods_single_BH}
The main goal of this work is to evaluate the contribution of actively accreting $z\geq6$ SMBHs, powered by radiatively efficient accretion disks to cosmic reionization, by tracing an evolving population of AGNs, each contributing with an SED that is appropriate for the properties of the accreting SMBH.
To this end, we will mainly rely on two markedly different SED models:
\begin{itemize}
\item A fixed-shape SED --- an observationally motivated SED based on the template constructed in \cite{Marconi04}.
This SED consists of a broken power-law ($L_\nu \propto \nu^{\alpha_\nu}$), with $\alpha_\nu = 2$ for $\lambda < 1\,\mic$ (the Rayleigh-Jeans tail of a blackbody), $\alpha_\nu =-0.44$ in the range $1200\,\ensuremath{\mathrm{\AA}}\xspace < \lambda < 1\,\mic$, and $\alpha_\nu = -1.76$ in the range $1\,\kev < \lambda < 1200\,\ensuremath{\mathrm{\AA}}\xspace$.
Since our aim is to model the accretion disk's SED we will ignore the part of the SED template blueward of 1 \kev, which is attributed to the hot corona.
This model has only one parameter --- $\Lbol$ --- which is used to normalize the SED, i.e. by requiring that $\Lbol\equiv\int L_\nu d\nu$.
\item Standard thin disk --- the SED of a standard, \cite{SS73} like geometrically-thin, optically-thick accretion disk.
In particular, we construct these SEDs using a series of annular regions that follow the temperature profile as given in \cite{SS73}, and applying additional relativistic corrections \citep{NT73,PT74,RH95}.
In principle, this model has five parameters: The mass of the BH ($M$), the disk mass accretion rate (\ensuremath{\dot{M}}\xspace), the dimensionless BH spin ($a$), and the inner and outer radii of the disk (\ensuremath{r_{\mathrm{in}}}\xspace and \ensuremath{r_{\mathrm{out}}}\xspace, respectively).
We take \ensuremath{r_{\mathrm{in}}}\xspace to be the innermost stable orbit (ISCO), which is set by $a$. In addition, \ensuremath{r_{\mathrm{out}}}\xspace is taken to be $500\,r_{\rm g}$ throughout this work.
Thus, in practice, the variety of SEDs we consider are determined by {\it three} parameters in total ($M$, \ensuremath{\dot{M}}\xspace, and $a$).
\end{itemize}
We assume a radiatively efficient accretion flow for both models. Thus, the bolometric luminosity, \Lbol, relates to the accretion rate through
\begin{equation}
\label{eq:Lbol}
\Lbol = \eta \ensuremath{\dot{M}}\xspace c^2 \, ,
\end{equation}
where the radiative efficiency $\eta$ is a non-linear function of $a$.
In Section~\ref{sec:method-superEdd} below we describe an additional {\it slim}-disk model SED that is applicable in the super-Eddington regime.
For any given SED, the number of ionizing photons emitted per second, \ensuremath{Q_{\mathrm{ion}}}\xspace, is given by
\begin{equation}
\label{eq:Q=integral}
Q_{\rm ion} = \int\displaylimits_{13.6\,\mathrm{eV}}^\infty \frac{L_{\nu}}{h\nu}\mathrm{d}\nu\,,
\end{equation}
where the lower integration limit is set by the hydrogen ionization threshold frequency.
Figure~\ref{fig:diff_SEDs} shows the SEDs of thin-disks for different BH parameters, with the appropriate \ensuremath{Q_{\mathrm{ion}}}\xspace shown along each SED.
The upper panel shows SEDs of non-spinning BHs ($a=0$) with $\ensuremath{\dot{M}}\xspace=0.1\,\ifmmode \Msun\,{\rm yr}^{-1} \else $\Msun\,{\rm yr}^{-1}$\fi$ and various BH masses. Note that changing the BH mass from $10^{10}$ to $10^{8}\,\ifmmode M_{\odot} \else $M_{\odot}$\fi$ increases \ensuremath{Q_{\mathrm{ion}}}\xspace by more than $10$ orders of magnitude, as a result of the accretion disk getting hotter for lower-mass black holes.
Decreasing the mass even further, to $10^6\,\ifmmode M_{\odot} \else $M_{\odot}$\fi$, increases the disk temperature even still, but increases the number of emitted ionizing photons by only $\sim25\%$.
This is due to the increase of the average energy of the emitted photons combined with the equal energy budget of each SED (Eq. \ref{eq:Lbol}).
The middle panel in Figure~\ref{fig:diff_SEDs} presents SEDs for different accretion rates, where $M=10^9\,\ifmmode M_{\odot} \else $M_{\odot}$\fi$ and $a=0$. In this case raising the accretion rate from $1$ to $15\,\ifmmode \Msun\,{\rm yr}^{-1} \else $\Msun\,{\rm yr}^{-1}$\fi$ increases \ensuremath{Q_{\mathrm{ion}}}\xspace by a factor of ${\sim}60$, again due to the accretion disk having a higher temperature for higher values of accretion rates. In the lower panel the spin varies from a stationary BH to the maximal allowed spin for a thin accretion disk ($a=0.998$), for a $M=10^9\,\ifmmode M_{\odot} \else $M_{\odot}$\fi$ BH with $\dot{M}=1\,\ifmmode \Msun\,{\rm yr}^{-1} \else $\Msun\,{\rm yr}^{-1}$\fi$, which results in an increase by a factor of ${\sim}30$ in \ensuremath{Q_{\mathrm{ion}}}\xspace.
For comparison, the upper panel shows the fixed-shape SED, normalized to have the same \Lbol\, as the other SEDs in this panel (i.e., $\Lbol\simeq3\times10^{44}\,\ergs$ for $\ensuremath{\dot{M}}\xspace=0.1\,\ifmmode \Msun\,{\rm yr}^{-1} \else $\Msun\,{\rm yr}^{-1}$\fi$ and $a=0$). This clearly demonstrates that the constant shape SED, which can only be scaled, cannot capture the more complicated SED behavior as embodied by the standard thin-disk model, and specifically cannot fully account for the dependence of \ensuremath{Q_{\mathrm{ion}}}\xspace on the BH parameters.
\begin{figure}
\centering
\includegraphics[width=1.0\columnwidth]{{SEDs_panels_different_parameters.png}}
\caption{SEDs of standard geometrically-thin, optically-thick accretion disks with various parameters. Upper panel: all models have $a=0$ and $\ensuremath{\dot{M}}\xspace=0.1\,\ifmmode \Msun\,{\rm yr}^{-1} \else $\Msun\,{\rm yr}^{-1}$\fi$, but various BH masses (the $M=10^{6}\,\ifmmode M_{\odot} \else $M_{\odot}$\fi$ model slightly exceeds the Eddington limit).
Middle panel: various accretion rates, for $M=10^{9}\,\ifmmode M_{\odot} \else $M_{\odot}$\fi$ and $a=0$. Lower panel: various BH spins, for $M=10^{9}\,\ifmmode M_{\odot} \else $M_{\odot}$\fi$ and $\ensuremath{\dot{M}}\xspace=1\,\ifmmode \Msun\,{\rm yr}^{-1} \else $\Msun\,{\rm yr}^{-1}$\fi$.
In all panels, the dashed vertical line marks the ionization frequency of hydrogen. For each SED, we note the corresponding value of \ensuremath{Q_{\mathrm{ion}}}\xspace.
The black solid line in the upper panel illustrates the fixed-shape SED used in this work (as well as other relevant studies), here normalized to the same \Lbol$\,$ as the rest of the SEDs in that panel.
}
\label{fig:diff_SEDs}
\end{figure}
Note that the assumption of an isotropic radiation field (excluding the geometrical factor due to the disk's flat geometry) is generally not valid when relativistic effects are taken into account. These effects become more pronounced the closer the emission region is to the BH. Since for highly spinning BHs the ISCO, and hence the inner radius, are exceedingly small, the angle dependence of the emitted radiation will be stronger for high-spin BHs. To account for these relativistic effects we use the \texttt{KERRTRANS} code (presented in \citealt{Agol97}) to derive inclination angle dependent SEDs, and then integrate over all inclinations to derive the angle-integrated, frequency-resolved luminosity of the source (i.e., $L_\nu$).
However, as demonstrated in Appendix~\ref{Appendix0} and Figure \ref{fig:kerrtrans vs regular}, for non-spinning BHs the differences between the SEDs calculated using \texttt{KERRTRANS} and those calculated while ignoring angle dependent relativistic effects are negligible.
As the spins of high-redshift SMBHs are essentially unknown, or at least highly uncertain (e.g., \citealt{Trakhtenbrot17, Jones20}; see also the review by \citealt{Reynolds20} and references therein), we assume $a=0$ throughout most of this work.
We therefore use SEDs generated using \texttt{KERRTRANS} only when exploring scenarios with highly spinning BHs.
In this work we consider several BH growth scenarios, which describe how the key BH properties evolve with (cosmic) time. These, in turn, determine the evolution of the SED, given an SED model (i.e., thin-disk or fixed-shape). In the growth scenarios considered in this work we take the BH spin to stay constant. Thus, the evolving SED would depend on the evolution of $M$ and \ensuremath{\dot{M}}\xspace. A related quantity is the Eddington ratio, \ensuremath{f_{\mathrm{Edd}}}\xspace, defined as the ratio between the bolometric luminosity and the Eddington luminosity and which for radiatively-efficient (thin) accretion flows can be expressed as:
\begin{equation}
\label{eq:fedd}
\ensuremath{f_{\mathrm{Edd}}}\xspace=\frac{\Lbol}{\Ledd}\propto\frac{\ensuremath{\dot{M}}\xspace}{\dot{M}_{\rm Edd}}\propto\frac{\ensuremath{\dot{M}}\xspace}{M}
\end{equation}
Currently, quasars are observed up to the early epochs when reionization is mostly complete ($z\simeq6-7.5$; \citealt{Banados18,Wang21}).
Thus, when considering the evolution of a single SMBH, we will track its growth back in time, starting at the end-point of a $10^9\,\ifmmode M_{\odot} \else $M_{\odot}$\fi$ SMBH at $z=6$ in order to be consistent with the typical observed mass at that redshift \cite[e.g,][]{TN12,DeRosa14,Trakhtenbrot17,Shen19}.
Note, however, that we will explore various choices for \ensuremath{\dot{M}}\xspace and $a$ at $z=6$, and the associated trends in BH growth and ioinizing output.
In our work we will rely on standard, thin-disk SEDs to consider two simple scenarios of BH growth:
\begin{itemize}
\item
Constant $\ensuremath{f_{\mathrm{Edd}}}\xspace$ --- starting with the mass boundary condition at $z=6$, we evolve the BH parameters back in time at a fixed Eddington ratio.
This results in an exponential growth with a Salpeter timescale \citep{Salpeter64}, given by
\begin{equation}
\label{eq:salpeter}
\tau_{\rm{Salpeter}}{\simeq}0.4\,\rm{Gyr}\,\frac{\eta}{(1{-}\eta)}\,\frac{1}{\ensuremath{f_{\mathrm{Edd}}}\xspace}.
\end{equation}
\item
Eddington-limited, constant $\ensuremath{\dot{M}}\xspace\,$ (the \emph{``Mixed''} scenario hereafter) --- starting with the mass boundary condition at $z=6$, we evolve the BH parameters back in time with a fixed physical accretion rate (i.e., in \ifmmode \Msun\,{\rm yr}^{-1} \else $\Msun\,{\rm yr}^{-1}$\fi).
Since growth at a constant \ensuremath{\dot{M}}\xspace results in an increase in \ensuremath{f_{\mathrm{Edd}}}\xspace when going back in time (i.e. as $M$ decreases; Eq. \ref{eq:fedd}), the Eddington limit ($\ensuremath{f_{\mathrm{Edd}}}\xspace = 1$) will be reached at some time $t_\mathrm{Edd}$. Since the thin-disk models we consider are not valid in the super-Eddington regime, we will switch the growth for $t<t_\mathrm{Edd}$ to a growth at a constant $\ensuremath{f_{\mathrm{Edd}}}\xspace = 1$ (see above).
The physical motivation for such a scenario can be a galaxy that feeds mass at some average constant rate to the SMBH at its center, with the accretion rate itself being regulated by the Eddington limit.
\end{itemize}
We also explore a model that allows for accretion at super-Eddington rates (see Section~\ref{sec:method-superEdd} immediately below), thus enabling the BH to accrete at a constant \ensuremath{\dot{M}}\xspace throughout its growth history (i.e., also at $t<t_\mathrm{Edd}$).
Since the SED depends on the disk's temperature range, which in turn depends on the mass and accretion rate, the time step in each of the above growth schemes is set so that the difference in mass between two consecutive steps is at most 10\%.
This way the temperature difference between two consecutive steps does not exceed $\sim$5\%. For each time-step we calculate the SED and \ensuremath{Q_{\mathrm{ion}}}\xspace of the appropriate accretion disk, keeping track of their evolution as the BH grows.
\subsection{SEDs of super-Eddington slim disks}
\label{sec:method-superEdd}
Following the discussion in Section~\ref{sec:intro}, we are interested in modeling SMBH accreting at super-Eddington rates, for which the standard thin-disk model is inappropriate. While there are many different ideas for modeling super-Eddington accretion flows, in our work we focus on ``slim'',\footnote{The notion of ``slim'' disks commonly refers to a height-to-radius ratio of $H/R\sim1$, compared with $H/R \ll 1$ for ``thin'' disks (see \cite{Netzer13}).} luminous accretion disks, which have long been considered to be able to sustain super-Eddington flows onto BHs (at least at mild super-Eddington rates; e.g., \citealt{Abramowicz88}).
Specifically, we use the \texttt{AGNslim} model presented in detail in \cite{KD19}. We chose this specific model due to the availability of associated SED-generating tools.
Detailing the physics behind the model, and/or its validity, are beyond the scope of this work, and the interested reader is referred to \citet[and references therein]{KD19} for a detailed discussion.
Here we briefly mention that \texttt{AGNslim} allows the disk to become slim due to increased radiation pressure, taking into account the increase in optical depth for high accretion rates, which causes `photon-trapping' near the mid-plane of the disk. These photons are then advected to the SMBH before they're able to escape (vertically) from the surface of the disk. In simple terms, this allows the disk to supply mass to the SMBH at super-Eddington rates while not violating the local Eddington limit near the disk surface. In addition, this model includes additional Comptonization which causes an excess of UV radiation. This model has $14$ parameters in total.
Except for $M$, \ensuremath{f_{\mathrm{Edd}}}\xspace (which determines \ensuremath{\dot{M}}\xspace) and $a$, we take all parameters to have the default values.\footnote{The full list of parameters can be viewed at \url{https://heasarc.gsfc.nasa.gov/xanadu/xspec/manual/node132.html}.}
The \texttt{AGNslim} model does {\it not} consider outflows launched from the accretion flow \cite[e.g.,][]{DotanShaviv2011}.
A comparison of the \texttt{AGNslim} SED and the standard thin disk SED, for various choices of \ensuremath{f_{\mathrm{Edd}}}\xspace, $M=10^9\,\ifmmode M_{\odot} \else $M_{\odot}$\fi$ and $a=0$, is presented in Figure~\ref{fig:SED_AGNslim_vs_thin}.
Note that although the slim disk SED is much harder than the standard disk SED, there is no significant difference in the number of ionizing photons produced by each SED.
This is because, for these and similar parameters, many of the photons being Comptonized to higher energies in the \texttt{AGNslim} model are already above the Lyman limit, in addition to the slim disk producing a slightly lower total number of photons. The ionizing flux of the \texttt{AGNslim} model \textit{could} be significantly higher than that of the thin-disk case, for higher masses and/or lower accretion rates.
\begin{figure}
\centering
\includegraphics[width=1.0\columnwidth]{SED_AGNslim_vs_standard.png}
\caption{A comparison of the standard thin disk SED (dashed) and the slim disk SED produced by \texttt{AGNslim} (solid), for various choices of \ensuremath{f_{\mathrm{Edd}}}\xspace (colors).
The Lyman limit is marked by a vertical dotted line, and we note the values of \ensuremath{Q_{\mathrm{ion}}}\xspace derived for the two SED types, assuming $\ensuremath{f_{\mathrm{Edd}}}\xspace=0.1$.
All the SEDs in this plot assume $M=10^9\,\ifmmode M_{\odot} \else $M_{\odot}$\fi$ and $a=0$.
The difference in \ensuremath{Q_{\mathrm{ion}}}\xspace is negligible despite the apparent excess UV radiation in the \texttt{AGNslim} model, since many of the photons that are being further Comptonized in the slim disk ``envelope'' are already above the Lyman limit, and due to a slightly lower total number of photons produces by the slim disk.}
\label{fig:SED_AGNslim_vs_thin}
\end{figure}
In order to calculate the ionizing photon production rate of a super-Eddington accreting flow we follow the same methods used for standard, Eddington-limited disks (described in the preceding Section~\ref{sec:methods_single_BH}).
However, note that when the SED is described by the \texttt{AGNslim} model, the growth of the BH can proceed at a constant \ensuremath{\dot{M}}\xspace and go over the Eddington limit at early times. Since the maximal allowed value for the Eddington ratio in this model is $1000\times\dot{M}_\mathrm{Edd}$, we will switch to a growth at a constant Eddington ratio equal to that value whenever this value is reached.
As we show in the next Section, the exact choice of this new maximal accretion rate does not affect our results, since the growth is already extremely fast.
\subsection{The ionizing output of a population of SMBHs}
\label{sec:methods_population}
In this work we use a bolometric QLF to describe the population of active SMBHs at $z=6$ (and beyond), which is parameterized as a double power-law of the form
\begin{equation}
\label{eq:double power-law}
\phi_{L}\left(L\right) = \frac{\phi^{\ast}}{\left(L/L^{\ast}\right)^{\gamma_{1}}+\left(L/L^{\ast}\right)^{\gamma_{2}}}\,,
\end{equation}
where $\phi^{\ast}$ is the comoving number density normalization, in units of $\rm{Mpc}^{-3}\,\rm{dex}^{-1}$;
$L^{\ast}$ is the ``break'' luminosity;
and $\gamma_{1}$ and $\gamma_{2}$ are the faint-end and bright-end power-law slopes, respectively.
Specifically, for this work, we use QLFs from the recent study by \cite{Shen20}, as detailed in Section~\ref{sec:res_population}.
Our analysis aims to trace the active SMBH population, and thus the QLF, to $z>6$.
Instead of extrapolating the highly uncertain, empirical redshift-dependent trends seen at $z\sim4-6$ \cite[see detailed discussion in][]{Shen20}, we instead trace the evolution of the QLF based on a physically-motivated interpretation, coupled with our physical BH growth scenarios.
Specifically, we associate every luminosity bin along the QLF with a certain set of BH parameters.
We assume the same spin, and hence the same radiative efficiency, for the entire BH population.
Thus, the bolometric luminosity at each point on the QLF will be associated with a certain accretion rate (Eq. \ref{eq:Lbol}).
In our work we take 30 BH mass bins, log-uniformly spread across the range $10^{6-10}\,M_{\odot}$ ($\Delta\log M\simeq0.13$ dex), to represent the reasonable mass range of the SMBH population at $z = 6$.
In order to associate the QLF with this BH mass range we further assume a universal Eddington ratio for the entire active $z=6$ SMBH population.
This assumption allows us to effectively deduce an active SMBH mass function (BHMF) from the QLF, with the most massive BH being the most luminous and rarest systems.
We can then use the mass growth schemes described in Sections~\ref{sec:methods_single_BH} and ~\ref{sec:method-superEdd} to trace back the evolving QLF and BHMF for any $z>6$.\footnote{Note that the SMBH masses at $z>6$ will be necessarily smaller than those imposed at $z=6$, i.e. there will be no SMBHs with $M=10^{10}\,\ifmmode M_{\odot} \else $M_{\odot}$\fi$ at $z>6$.}
The space density of quasars as a function of \ensuremath{Q_{\mathrm{ion}}}\xspace can be derived from the QLF by
\begin{equation}
\label{eq:L-to-Q density}
\phi_{Q}\left(Q_{\rm ion}\right) = \phi_{L}\left(L\right)\frac{\Delta \log L}{\Delta \log Q_{\rm ion}}\,,
\end{equation}
where $\Delta\log L$ and $\Delta\log Q_{\rm ion}$ are small logarithmic intervals of bolometric luminosity and production rate of ionizing photons, respectively, associated with consecutive BH mass bins within the (evolving) mass range.
By integrating the $Q_{\rm ion}$ distribution over the entire range of masses, at any specific redshift, we get the ionizing photon flux density of the redshift-resolved progenitor population of the $z=6$ SMBHs:
\begin{equation}
\label{eq:Nion}
\dot{N}_{\rm ion} = \int^{Q_{\rm ion}\left(M_{\rm max}\right)}_{Q_{\rm ion}\left(M_{\rm min}\right)} \phi_{Q}\left(Q_{\rm ion}\right) \mathrm{d}Q_{\rm ion}\,.
\end{equation}
This quantity can be directly compared to the ionizing photon flux needed for reionizing the Universe and/or to the output of other ionizing sources (i.e., early galaxies).
\section{Results}
\label{sec:results}
In this Section we rely on the methods outlined above to trace the ionizing output of a single SMBH through $z=6-20$ under several Eddington-limited growth scenarios, and assess the output of the evolving SMBH population.
We then re-examine the radiative outputs when super-Eddington growth is allowed.
\subsection{How many ionizing photons does a single SMBH produce?}
\label{sec:res_single_bh}
In this section we will present the evolution of \ensuremath{Q_{\mathrm{ion}}}\xspace for growing BHs with various parameters and growth schemes, as detailed in Section~\ref{sec:methods_single_BH}. In all the scenarios outlined in this section the BHs reach a mass of $\mbh=10^9\,\ifmmode M_{\odot} \else $M_{\odot}$\fi$ at $z=6$.
Figure~\ref{fig:Qvst_const_fEdd} presents the evolution of \ensuremath{Q_{\mathrm{ion}}}\xspace over $6 \leq z \leq 20$ for spinless BHs with three different choices of a {\it constant} Eddington-ratio, $\ensuremath{f_{\mathrm{Edd}}}\xspace=0.1, 0.5$ and $1$.
We show calculations carried out both with the standard, thin-disk model (solid lines) and with the fixed-shape SED model (dashed lines).
Note that the various models do {\it not} result in identical ionizing outputs at $z=6$, as the different choices of \ensuremath{f_{\mathrm{Edd}}}\xspace\ and SED model would necessarily change \ensuremath{Q_{\mathrm{ion}}}\xspace.
In the latter fixed-shape SED scenario, the constant \ensuremath{f_{\mathrm{Edd}}}\xspace\ leads to an exponential growth in \Lbol\ (Eq. \ref{eq:salpeter}), and thus we expect an exponential growth in \ensuremath{Q_{\mathrm{ion}}}\xspace.
We expect the standard thin-disk to result in a more complex behavior, since now not only \Lbol\ is changing but also the shape of the SED. As Figure~\ref{fig:Qvst_const_fEdd} shows, however, this model still results in an approximately exponential growth.
By integrating the number of photons over time, we get the total number of ionizing photons produced by the BH during its entire growth. For our choice of parameters, this total time-integrated number of ionizing photons for the thin-disk model is higher by a factor of ${\sim}1.6$ and ${\sim}1.75$ compared to the fixed-shape SED model for $\ensuremath{f_{\mathrm{Edd}}}\xspace=0.5$ and $1$, respectively. For $\ensuremath{f_{\mathrm{Edd}}}\xspace=0.1$ there is no significant difference in the total ionizing output between the two models.
\begin{figure}[t]
\centering
\includegraphics[width=1.0\columnwidth]{const_fEdd_log.png}
\caption{Evolution of \ensuremath{Q_{\mathrm{ion}}}\xspace for a spinless BH growing through a constant Eddington ratio accretion, for different choices of \ensuremath{f_{\mathrm{Edd}}}\xspace.
Calculations with a standard thin-disk model are presented in solid lines and those with a fixed-shape SED in dashed lines.
By construction, all growth scenarios result in a $10^9\,\ifmmode M_{\odot} \else $M_{\odot}$\fi$ SMBH at $z=6$. The differences in \ensuremath{Q_{\mathrm{ion}}}\xspace at this ``final'' epoch are due to the different SEDs associated with each of the models.
}
\label{fig:Qvst_const_fEdd}
\end{figure}
For Eddington limited accretion with a constant \ensuremath{\dot{M}}\xspace the difference between the two models is more striking.
The evolution of \ensuremath{Q_{\mathrm{ion}}}\xspace for this ``Mixed'' growth scheme, for spinless BHs with different choices of $\ensuremath{\dot{M}}\xspace(z=6)$, is presented in Figure~\ref{fig:Qvst_Mixed}, where we again show both thin-disk and fixed-shape calculations, with \ensuremath{Q_{\mathrm{ion}}}\xspace plotted in linear scale on the left and log-scale on the right.
In this growth scenario both SED models exhibit two distinct regimes of \ensuremath{Q_{\mathrm{ion}}}\xspace evolution:
\begin{enumerate}
\item
At late times, starting at $z=6$ and going backwards in time, accretion proceeds with a constant, sub-Eddington \ensuremath{\dot{M}}\xspace, resulting in a constant \ensuremath{Q_{\mathrm{ion}}}\xspace for the fixed-shape SED and a \ensuremath{Q_{\mathrm{ion}}}\xspace that decreases with increasing time for the thin-disk SED.
This latter trend is a unique consequence of the thin-disk SED model, and is driven by the increase in disk temperature, and thus in ionizing photons, with decreasing BH mass. Note that this trend is {\it not} caused by an increase in the luminosity, but rather by the changes to the SED (see Fig.~\ref{fig:SED_evolution_demo}).
\item
At earlier times, when the system reaches its Eddington limit, the BH can no longer maintain the assumed $\ensuremath{\dot{M}}\xspace(z=6)$ and thus the accretion switches to a constant $\ensuremath{f_{\mathrm{Edd}}}\xspace=1$. In this case the change in \ensuremath{Q_{\mathrm{ion}}}\xspace is, again, exponential for both SED models.
\item
The transition between the late, constant \ensuremath{\dot{M}}\xspace regime and the early, constant \ensuremath{f_{\mathrm{Edd}}}\xspace\ regime for the thin-disk model manifests itself in a pronounced peak in \ensuremath{Q_{\mathrm{ion}}}\xspace. This peak occurs at the time and BH mass in which the assumed $\ensuremath{\dot{M}}\xspace\left(z=6\right)$ corresponds to the Eddington limit (with the latter quantity being mass- and thus time-dependent).
\end{enumerate}
The Eddington-limited, constant \ensuremath{\dot{M}}\xspace scenario, coupled with the thin-disk spectral evolution, produces a factor of ${\sim}2$ more ionizing photons at the peak of emission, as compared to the fixed-shape SED model.
The total, time-integrated, number of ionizing photons produced by the former model is higher by a factor of ${\sim}1.6$, ${\sim}1.5$ and ${\sim}1.3$ compared to the latter, for \ensuremath{\dot{M}}\xspace of $5$, $10$ and $15$ $\ifmmode M_{\odot} \else $M_{\odot}$\fi/\rm{yr}$ at $z=6$, respectively.
Figure~\ref{fig:SED_evolution_demo} (in Appendix \ref{Appendix1}) demonstrates the time-evolving SEDs calculated for one of the growth scenarios depicted in Figure~\ref{fig:Qvst_Mixed} (the blue curve, with $\ensuremath{\dot{M}}\xspace=10\,\ifmmode \Msun\,{\rm yr}^{-1} \else $\Msun\,{\rm yr}^{-1}$\fi$ at $z=6$).
At late times, the SEDs evolve ``horizontally'' due to the increasing mass, causing a decrease in \ensuremath{Q_{\mathrm{ion}}}\xspace, while in earlier time, during the constant Eddington ratio accretion growth phase, the SEDs evolve both ``horizontally'' and ``vertically'', causing an exponential increase in \ensuremath{Q_{\mathrm{ion}}}\xspace.
\begin{figure*}
\centering
\includegraphics[width=0.48\textwidth]{Mixed.png}
\hfill
\includegraphics[width=0.48\textwidth]{Mixed_log.png}
\caption{\textit{Left}: Evolution of \ensuremath{Q_{\mathrm{ion}}}\xspace in linear-scale for a spinless BH growing through a constant, Eddington-limited \ensuremath{\dot{M}}\xspace accretion, for various choices of \ensuremath{\dot{M}}\xspace.
Calculations with a standard thin-disk model are presented in solid lines and those with a fixed-shape SED in dashed lines.
All growth scenarios result in a $10^9\,\ifmmode M_{\odot} \else $M_{\odot}$\fi$ SMBH at $z=6$. \textit{Right}: Same but \ensuremath{Q_{\mathrm{ion}}}\xspace is in log-scale.
}
\label{fig:Qvst_Mixed}
\end{figure*}
\begin{figure}
\centering
\includegraphics[width=1.0\columnwidth]{different_spins_no_spinning_up.png}
\caption{Evolution of \ensuremath{Q_{\mathrm{ion}}}\xspace for a BH accreting at a constant $\ensuremath{f_{\mathrm{Edd}}}\xspace=1$ for various choices of (constant) BH spin ($a$).
All growth scenarios result in a $10^9\,\ifmmode M_{\odot} \else $M_{\odot}$\fi$ SMBH at $z=6$.}
\label{fig:Qvst_diff_spin}
\end{figure}
Another parameter that affects \ensuremath{Q_{\mathrm{ion}}}\xspace is the BH spin.
Figure~\ref{fig:Qvst_diff_spin} presents the evolution of \ensuremath{Q_{\mathrm{ion}}}\xspace for different choices of constant BH spin, where the mass growth proceeds at a constant $\ensuremath{f_{\mathrm{Edd}}}\xspace=1$.
Note that a more realistic model should take into account the spin evolution during steady and/or intermittent accretion episodes \cite[e.g.,][see discussion in Section~\ref{sec:discussion_extending}]{King08,Dotti13}.
Nonetheless, our simplistic assumption of a constant spin can still demonstrate the basic effects that different spin values have on the accretion disk's production rate of ionizing photons. We recall that higher BH spins provide higher radiative efficiencies.
This results in a total, time-integrated, higher number of ionizing photons for high spinning BHs,
with the $a=0.998$ BH producing a factor of ${\sim}6$ higher \ensuremath{Q_{\mathrm{ion}}}\xspace than a spinless BH of an identical mass and accretion rate.
One important caveat to this result is that higher spins imply higher radiative efficiencies, and thus slower mass growth.\footnote{The assumption of a fixed \ensuremath{f_{\mathrm{Edd}}}\xspace also implies that a higher spin (i.e., higher $\eta$ in Eq.~\ref{eq:Lbol}) BH will have a lower \ensuremath{\dot{M}}\xspace, which is another contributor to its slower mass growth.}
Assuming a certain final mass at $z=6$, this causes the \textit{implied} masses we get at any given earlier epoch to be higher for highly spinning BHs, also implying higher BH seed masses (see discussion in Section~\ref{sec:discussion_extending}).
\subsection{The evolution of the ionizing flux density of SMBHs}
\label{sec:res_population}
In this section we will follow the methods described in Section~\ref{sec:methods_population} to trace the evolution of the QLF, and the corresponding \ensuremath{Q_{\mathrm{ion}}}\xspace density distribution function, from $z=6$ to earlier epochs. We will then derive the ionizing flux density, \ensuremath{\dot{N}_{\mathrm{ion}}}\xspace, of the entire population of actively accreting SMBHs over the range $z=6-15$.
Since there is currently a lack of observations of faint AGNs at $z=6$, we adopt two different QLFs which differ mainly in their faint-end slopes in order to represent the uncertainty in this regime. In the present work we use QLFs that were constrained by multiwavelength observations and presented in the recent paper by \cite{Shen20}.
Specifically, we use the ``global fit A'' and ``B'' QLFs in \citet[see their Fig.~5]{Shen20}---hereafter referred to simply as ``QLF A'' and ``QLF B'', respectively.
``QLF A'' corresponds to a considerably higher number of low-luminosity AGNs at $z\sim6$, compared to ``QLF B''.
At the faint end, ``QLF A'' is comparable to the QLF derived by \cite{Giallongo19}.
For example, at $\Lbol=10^{44}\,\ergs$, both the QLF denoted as ``Model 3'' in \cite{Giallongo19} and ``QLF A'' give $\phi_L\simeq4.8\times10^{-5}\,{\rm dex}^{-1}\,{\rm cMpc}^{-3}$, which is a factor $\sim20$ higher than the corresponding value for ``QLF B''.
We thus use ``QLF A'' to represent the most ``optimistic'' case in terms of the number of faint AGNs at $z>6$.
The parameters of the QLFs we consider are listed in Table \ref{table:QLF parameters}.
An example of the evolution of a QLF from $z=6$ to $z=15$ is shown in the left panel of Figure~\ref{fig:QLF_ evo}.
Here we used ``QLF A'' at $z=6$, and the Eddington ratio and spin of the entire SMBH population were fixed at $\ensuremath{f_{\mathrm{Edd}}}\xspace=0.6$ and $a=0$, respectively.
In this case, the standard thin disk and the constant shape SED produce almost identical curves, so only the standard thin disk curves are shown. Figure~\ref{fig:QLF_ evo} shows that under the above assumptions, the QLF exhibits a pure luminosity evolution, meaning that the evolving QLF can be described solely by the evolution of $L^*$.
In contrast, as can be seen in the right panel of Figure \ref{fig:QLF_ evo}, the resulting evolution of the \ensuremath{Q_{\mathrm{ion}}}\xspace density function differs for the two SED models, with the thin-disk model \ensuremath{Q_{\mathrm{ion}}}\xspace density function becoming ``steeper'' the lower the redshift, as well as displaying a faster-evolving ``horizontal shift'', than with the fixed-SED model.
\begin{figure*}
\centering
\includegraphics[width=0.48\textwidth]{QLF_evolution_fedd=06.png}
\hfill
\includegraphics[width=0.48\textwidth]{QQF_evolution_fedd=06.png}
\caption{The evolution of the QLF (left) and of the \ensuremath{Q_{\mathrm{ion}}}\xspace density function (right) between $z=6$ to $z=15$.
Here we assume constant $\ensuremath{f_{\mathrm{Edd}}}\xspace=0.6$ and $a=0$ for the entire population of SMBHs, and ``QLF A'' at $z=6$. Solid lines mark calculations with the standard thin disk SED and dashed lines represent the fixed-shape SED model.}
\label{fig:QLF_ evo}
\end{figure*}
The ionizing flux density of the entire SMBH population at any given redshift can be obtained by integrating the \ensuremath{Q_{\mathrm{ion}}}\xspace density function at that redshift (Eq. \ref{eq:Nion}) over the range of \ensuremath{Q_{\mathrm{ion}}}\xspace. Figure \ref{fig:Qvsz} shows the result of this calculation, assuming $\ensuremath{f_{\mathrm{Edd}}}\xspace=0.6$ and $a=0$ as before. Here we plot \ensuremath{\dot{N}_{\mathrm{ion}}}\xspace for the standard thin-disk (solid blue) and the fixed-shape (dashed blue) SED models, assuming ``QLF A'' at $z=6$.
The resulting \ensuremath{\dot{N}_{\mathrm{ion}}}\xspace at $z=6$ for the thin-disk model is higher by a factor of $\sim1.8$ than that of the fixed-shape SED model. Moreover, the total number of ionizing photons emitted (per $\mathrm{Mpc}^3$) over the entire redshift range for the thin-disk model is higher than the fixed-shape one by a factor of $\sim1.75$. These results are consistent with the results obtained for a single SMBH (Section~\ref{sec:res_single_bh}), which showed that accounting for the spectral evolution of the SED increases the ionizing output by similar factors.
Figure~\ref{fig:Qvsz} also shows, in solid red line, a similar calculation done with the thin-disk SED model and ``QLF B'' (instead of ``A'').
As this QLF has far fewer low-luminosity AGNs, the region between the two solid lines in Figure~\ref{fig:Qvsz} reflects the underlying uncertainty in the ionizing flux density produced by the AGN population due to the unknown number of such low-luminosity AGNs at $z\gtrsim6$. In this regard, we recall that both QLF models from \cite{Shen20} used in this work are essentially unconstrained by actual measurements of low-luminosity ($L\lesssim10^{45}\,\ergs$) AGNs at $z\sim6$ (see their Fig.~5).
At $z=6$, this uncertainty gives rise to a factor of $\sim 1.8$ higher ionizing flux density for the higher number of faint AGNs scenario. Integrated in time, the population with the more numerous number of faint AGNs produces a factor of $\sim 1.7$ higher total number of ionizing photons (per $\mathrm{Mpc}^3$) than the population with the lower number of faint AGNs. This minor difference in the ionizing flux density between the two populations, as compared to the seemingly significant difference in faint AGNs (a factor of $\sim20$ for AGNs powered by a SMBH with $M
\simeq 10^6\,\ifmmode M_{\odot} \else $M_{\odot}$\fi$), is discussed in more detail in Section~\ref{sec:faint AGNs}.
\begin{deluxetable}{c|cccc}
\tablenum{1}
\label{table:QLF parameters}
\tablecaption{The QLF parameters used in this work}
\tablewidth{0pt}
\tablehead{
\colhead{Name} & \colhead{$\log\,\phi^{\ast}$} & \colhead{$\log\,L^{\ast}$} &
\colhead{$\gamma_1$} & \colhead{$\gamma_1$} \\
\colhead{} & \colhead{$[\mathrm{dex}^{-1}\,\mathrm{cMpc}^{-3}]$} & \colhead{$[\mathrm{erg}\,\mathrm{s}^{-1}]$} &
\colhead{} & \colhead{}
}
\startdata
QLF A & 2.702 & 46.06 & 0.9671 & 1.694 \\
QLF B & 2.962 & 45.91 & 0.2196 & 1.699 \\
\enddata
\tablecomments{See \cite{Shen20} for more details.}
\end{deluxetable}
To put our calculations in the context of cosmic (hydrogen) reionization, Figure~\ref{fig:Qvsz} finally shows the ionizing flux required to keep the Universe ionized as a function of redshift (as a gray-shaded region).
The lower (upper) boundary of the region shown corresponds to a clumping factor of $C=2$ \cite[3, respectively; see][]{Madau99}.
\begin{figure}
\centering
\includegraphics[width=1.0\columnwidth]{Nion_vs_z_ONE_panel.png}
\caption{The evolution of \ensuremath{\dot{N}_{\mathrm{ion}}}\xspace derived for the population of accreting SMBHs, for different QLFs.
All calculations assume that all SMBHs accrete with constant $\ensuremath{f_{\mathrm{Edd}}}\xspace=0.6$ and $a=0$.
Blue and red lines represent $z=6$ ``QLF A'' and ``B'' from \cite[][respectively]{Shen20}. In each set of calculations, solid lines represent the thin disk model while dashed lines represent the fixed-shape SED model.
Gray-shaded regions are the ionizing flux density required to keep the Universe ionized, with the lower and upper boundaries of the light-gray region corresponding to a clumping factor of $C=2$ and $C=3$, respectively.}
\label{fig:Qvsz}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=1.0\columnwidth]{Nion_spins_fEdds.png}
\caption{The evolution of \ensuremath{\dot{N}_{\mathrm{ion}}}\xspace derived for the population of accreting SMBHs, for different values of spin and \ensuremath{f_{\mathrm{Edd}}}\xspace, assuming accretion at a constant Eddington ratio. All calculations assume $z=6$ ``QLF A''.
Gray-shaded regions are the ionizing flux density required to keep the Universe ionized (as in Fig.~\ref{fig:Qvsz}).}
\label{fig:Nion_diff_spin_fEdd}
\end{figure}
Figure \ref{fig:Qvsz} shows that for our particular choices of fixed $\ensuremath{f_{\mathrm{Edd}}}\xspace=0.6$ and $a=0$ the contribution of SMBHs to reionization can be significant only at late stages, near $z\simeq6$, with \ensuremath{\dot{N}_{\mathrm{ion}}}\xspace dropping sharply by $2$ orders of magnitude already by $z=7$. The impact of different choices of parameters on the contribution of SMBHs to reionization is explored further in Figure \ref{fig:Nion_diff_spin_fEdd}, which shows the evolution of \ensuremath{\dot{N}_{\mathrm{ion}}}\xspace for different BH spins and Eddington ratios, and assuming the fiducial ``QLF A''. These parameters determine the steepness of the curve, with higher spins and lower (universal) Eddington ratios showing ``flatter'' curves and thus a possible extension of the contribution of accreting SMBHs to reionization to earlier times (higher $z$).
These trends are mainly driven by the fact that fast-spinning BHs have higher radiative efficiency, causing the BH to grow more slowly, maintaining a high ionizing output during their growth up to $z=6$.
Similar considerations explain the effect of changing \ensuremath{f_{\mathrm{Edd}}}\xspace: a lower \ensuremath{f_{\mathrm{Edd}}}\xspace means a lower \ensuremath{\dot{M}}\xspace to $M$ ratio, which results in a slower mass growth.
\subsection{How does super-Eddington accretion affect the ionizing output?}
\label{sec:res_supEdd}
In this section we apply the methods outlined in Section~\ref{sec:methods_single_BH} and Section~\ref{sec:methods_population} to the \texttt{AGNslim} SED model, described in Section~\ref{sec:method-superEdd}, to account for the possibility of super-Eddington accretion and fast growth of the first generation of SMBHs.
Throughout this work we will refer to this model simply as the ``slim disk'' or the ``super-Eddington'' model.
In what follows, we will use \ensuremath{\Mdot/\Mdot_{\rm{Edd}}}\xspace instead of \ensuremath{f_{\mathrm{Edd}}}\xspace (i.e., \ensuremath{\Mdot/\Mdot_{\rm{Edd}}}\xspace\ instead of \lledd), since in super-Eddington accretion flows \Lbol\ saturates and thus no longer scales linearly with \ensuremath{\dot{M}}\xspace (as in Eq. \ref{eq:Lbol}; see discussion in Section~\ref{sec:discussion_superEdd}).
\begin{figure}
\centering
\includegraphics[width=1.0\columnwidth]{Q_t_agnslim_mixed.png}
\caption{The evolution of \ensuremath{Q_{\mathrm{ion}}}\xspace for the slim disk model, using \texttt{AGNslim}.
Solid lines represent super-Eddington growth scenarios, where accretion proceeds at a constant \ensuremath{\dot{M}}\xspace that can exceed the Eddington limit.
Dashed lines represent the corresponding Eddington-limited growth scenarios, where the accretion follows the same (constant) \ensuremath{\dot{M}}\xspace but only as long as it does {\it not} exceed the Eddington limit.
The different colors represent different choices of $\ensuremath{\dot{M}}\xspace\left(z=6\right)$.
For the solid curves, \ensuremath{\Mdot/\Mdot_{\rm{Edd}}}\xspace at the peak of ionizing output emission is marked by a black vertical line, and the points where $\dot{M}/\dot{M}_{\rm Edd}$ reaches the values of $10$, $100$ and $1000$ are marked by different symbols. All the calculations shown here assume $M\left(z=6\right)=10^9\,\ifmmode M_{\odot} \else $M_{\odot}$\fi$ and a constant $a=0$.}
\label{fig:Qvst_AGNslim}
\end{figure}
Because the slim disk model is valid in the super-Eddington regime, it allows to describe BHs that accrete at a constant \ensuremath{\dot{M}}\xspace throughout their entire mass growth. Figure \ref{fig:Qvst_AGNslim} shows the ionizing output histories of BHs with $a=0$ and $M\left(z=6\right)=10^9\,\ifmmode M_{\odot} \else $M_{\odot}$\fi$ for different values of $\ensuremath{\dot{M}}\xspace\left(z=6\right)$.
We show growth scenarios with accretion fixed at a constant \ensuremath{\dot{M}}\xspace that may exceed the Eddington limit (in solid lines), as well as the corresponding Eddington-limited scenarios (dashed lines), which we have previously described as the ``Mixed'' growth scenario.
Starting at $z=6$ and going backwards in time, both accretion modes produce similar, increasing ionizing outputs as long as the accretion is below the Eddington limit. When the Eddington limit is reached, the two sets of curves diverge: the Eddington-limited accretion stays at $\ensuremath{\Mdot/\Mdot_{\rm{Edd}}}\xspace=1$, resulting in an exponential evolution of mass and \ensuremath{Q_{\mathrm{ion}}}\xspace, as discussed in Section~\ref{sec:res_single_bh}, while the slim-disk constant \ensuremath{\dot{M}}\xspace accretion continues with further increase in \ensuremath{\Mdot/\Mdot_{\rm{Edd}}}\xspace and in \ensuremath{Q_{\mathrm{ion}}}\xspace. This increase is not maintained for long, with \ensuremath{Q_{\mathrm{ion}}}\xspace peaking at $\ensuremath{\Mdot/\Mdot_{\rm{Edd}}}\xspace \approx1.5-2.3$.
Beyond this peak (i.e., earlier times), the ionizing output drops sharply---by more than two orders of magnitude within $\Delta t\sim10\,\mathrm{Myr}$---while \ensuremath{\Mdot/\Mdot_{\rm{Edd}}}\xspace continues to increase, until the BH reaches the seed mass of $10\,\ifmmode M_{\odot} \else $M_{\odot}$\fi$.
This sharp change in \ensuremath{Q_{\mathrm{ion}}}\xspace is caused by the combined effect of (1) advection of ionizing photons in the super-Eddington regime, and (2) the more generic decrease in ionizing output with decreasing BH mass, which is indeed very dramatic in the super-Eddington regime (see further discussion in Section~\ref{sec:discussion_superEdd}).
Note also, that despite the apparently significant differences between the Eddington-limited, ``Mixed'' scenario and the super-Eddington, constant \ensuremath{\dot{M}}\xspace scenario seen in Figure~\ref{fig:Qvst_AGNslim}, the total (time integrated) number of ionizing photons produced in the former scenario is higher by only $\sim$10\% than in the latter.
The slim disk model can be also applied to the population analysis as outlined in Section~\ref{sec:methods_population}. Figure~\ref{fig:Nion AGNslim} shows the evolution of \ensuremath{\dot{N}_{\mathrm{ion}}}\xspace for a population of (spinless) SMBHs, which is described by ``QLF A'' at $z=6$ from \cite[see Table \ref{table:QLF parameters}]{Shen20} and by different values of $\ensuremath{f_{\mathrm{Edd}}}\xspace\left(z=6\right)$, setting the accretion rate for each BH mass interval.
Going backwards in time, the accretion of all BHs proceeds at a constant \ensuremath{\dot{M}}\xspace. This results in the \ensuremath{\dot{N}_{\mathrm{ion}}}\xspace curves looking almost like a step function: the SMBH population produces a steady, relatively high supply of ionizing photons ($>10^{50}\,\ensuremath{\mathrm{s}^{-1}\mathrm{Mpc}^{-3}}\xspace$) at late times and is sharply terminated at a certain redshift. This ``termination redshift'' is higher for lower values of $\ensuremath{f_{\mathrm{Edd}}}\xspace(z=6)$, for similar reasons to the ones discussed in Section~\ref{sec:res_population}: a lower \ensuremath{f_{\mathrm{Edd}}}\xspace (i.e., a lower $\ensuremath{\dot{M}}\xspace/M$ ratio) translates to slower mass growth (while maintaining the same total integrated space luminosity density at $z=6$).
This result emphasizes that the contribution of accreting SMBHs to the reionization of the Universe can be significant at late times ($z\sim6$), and can be extended to higher redshifts by means of relatively low (typical) accretion rates.
\begin{figure}
\centering
\includegraphics[width=1.0\columnwidth]{Nion_AGNslim_updated.png}
\caption{The evolution of the total ionizing flux density of all accreting SMBHs, whose SEDs are described by the slim disk model, where the accretion proceeds with a constant \ensuremath{\dot{M}}\xspace. \ensuremath{\dot{M}}\xspace for each mass interval is set by the value of \ensuremath{f_{\mathrm{Edd}}}\xspace at $z=6$ (represented in various colors).
We also assume $a=0$ and ``QLF A'' at $z=6$.
Gray-shaded regions are the ionizing flux required to keep the Universe ionized (as in Fig.~\ref{fig:Qvsz}).}
\label{fig:Nion AGNslim}
\end{figure}
\section{Discussion}
\label{sec:dicussion}
In what follows we discuss in more detail several aspects of our results. First, in Section~\ref{sec:faint AGNs}, we discuss the impact of the unknown number of faint AGNs at $z=6$ on \ensuremath{\dot{N}_{\mathrm{ion}}}\xspace. Next, in Section~\ref{sec:discussion_superEdd}, we discuss some insights related to the ionizing output of super-Eddington, slim accretion disks. We then discuss how various growth scenarios can extend the contribution of SMBHs to reionization to higher redshifts in Section~\ref{sec:discussion_extending}. We follow with a comparison between the ionizing flux density of SMBHs and galaxies in Section~\ref{sec:discussion_galaxies}. Finally, in Section~\ref{sec:discussion_ionized_regions}, we explore the implications of evolving SEDs on the sizes of proximity zones around quasars, by performing simplistic (1D) IGM radiative transfer calculations.
\subsection{The contribution of faint AGNs to reionization}
\label{sec:faint AGNs}
In Figure~\ref{fig:Qvsz} we show calculations of the ionizing flux density of AGNs using two $z=6$ QLFs: ``QLF A'' (blue) and ``B'' (red). These two QLFs differ mainly in the number of faint AGNs, with the former having an AGN density at $\Lbol =10^{44}\,\ergs$ that is higher by a factor of $\sim20$ than that of the latter.
In light of this, our result that the difference in \ensuremath{\dot{N}_{\mathrm{ion}}}\xspace at $z=6$ between the two QLFs is only ${
\sim}80\%$ (see Section~\ref{sec:res_population}) may seem surprising.
This contrast can be explained by considering that (1) the {\it total} space density of AGNs, across $10^{44} < \Lbol/\ergs < 10^{48}$, for the QLF with the high number of faint AGNs is higher only by a factor of ${\sim}7$ compared to the other QLF;
and (2) that the additional AGNs in the higher-density QLF indeed have relatively low luminosities ($\Lbol \mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$<$}}} 10^{45}\,\ergs$) and thus relatively low \ensuremath{Q_{\mathrm{ion}}}\xspace, as compared to the higher luminosity AGNs.
In other words, the bulk of the ionizing radiation for both QLFs is produced by the AGNs in the high luminosity regime, where the two QLFs are rather similar, being well constrained by observations.
Lower luminosity AGNs, irrespective of their abundance, do not contribute significantly to the \ensuremath{\dot{N}_{\mathrm{ion}}}\xspace of the whole AGN population.
We postulate that the difference in \ensuremath{\dot{N}_{\mathrm{ion}}}\xspace will be more significant for QLFs with a higher number of {\it intermediate} luminosity AGNs (i.e., if $\phi_\star$ and/or $L_\star$ are higher; see Table \ref{table:QLF parameters}).
\citealt{Giallongo15, Giallongo19} has indeed derived such a QLF based on observations that are focused on $4.5 \mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$<$}}} z \mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$<$}}} 5$, but there is currently no robust evidence that the $z=6$ QLF has comparable features \cite[e.g.,][]{Shen20, Kim20, Niida20}.
\subsection{Super-Eddington accretion}
\label{sec:discussion_superEdd}
We next turn to the growth scenarios that involve super-Eddington accretion. Figure~\ref{fig:Qvst_AGNslim} shows the evolution of ionizing photon flux for growth scenarios that allow for super-Eddington accretion through slim disks.
It demonstrates that when looking back in time, super-Eddington accretion produces a very sharp drop in \ensuremath{Q_{\mathrm{ion}}}\xspace, of over 2 dex, within less than 10 Myr, when the SMBH reaches (mass accretion) Eddington ratios of ${\sim}1.5{-}2.3$.
This drop can be explained by two effects. First, there is a ``saturation'' in \ensuremath{Q_{\mathrm{ion}}}\xspace at high Eddington ratios due to a saturation in the luminosity of the slim disk, which is driven by the advection of photons from the optically thick regions of the accretion flow onto the BH (horizon), before they are able to escape the flow. We note that this photon advection (or ``trapping''; including in outflows) is a rather generic feature of super-Eddington accretion models, and is not specific to the model we employ here (see discussion in \citealt{KD19} and also, e.g., \citealt{Ohsuga2005,DotanShaviv2011,McKinney14,SN16}, and references therein). This saturation in \ensuremath{Q_{\mathrm{ion}}}\xspace is demonstrated in Figure \ref{fig:Q_saturation}, which presents \ensuremath{Q_{\mathrm{ion}}}\xspace as a function of \ensuremath{\Mdot/\Mdot_{\rm{Edd}}}\xspace for non-spinning black holes with masses of $10^7\,\ifmmode M_{\odot} \else $M_{\odot}$\fi$ (red) and $10^8 \,\ifmmode M_{\odot} \else $M_{\odot}$\fi$ (blue), for the slim-disk model. As can be seen in Figure~\ref{fig:Q_saturation}, beyond $\ensuremath{\dot{M}}\xspace\simeq{\rm few}\times\dot{M}_{\rm Edd}$ the accretion rate can increase while the ionizing output remains (roughly) constant.
The second effect contributing to the sharp change in \ensuremath{Q_{\mathrm{ion}}}\xspace is driven by the decrease in BH mass. As was shown in the top panel of Figure \ref{fig:diff_SEDs}, decreasing the BH mass beyond a certain value does not increase the ionizing output of the disk, and may even decrease it.
As mentioned in Section~\ref{sec:methods_single_BH}, this occurs since for lower BH masses and higher disk temperatures the average energy of the (ionizing) photons increases, and thus the {\it total} number of ionizing photons {\it decreases} (for a given $\Lbol\propto\ensuremath{\dot{M}}\xspace$).
We performed a simplified calculation, presented in Appendix \ref{Appendix2}, that suggests the two effects---of photon advection and of decreasing BH mass---can be of comparable significance to the drop in ionizing output in the low-$M$, high-$\ensuremath{\Mdot/\Mdot_{\rm{Edd}}}\xspace$ regime.
We finally note that there are other models of super-Eddington accretion flows onto SMBHs that, contrary to \texttt{AGNslim}, suggest a {\it suppression}, not enhancement, of UV radiation, driven by the dominance of advection in the inner parts of the flow \cite[e.g.,][]{Ohsuga2005,Pognan20}.
Such a scenario would naturally lead to yet lower ionizing outputs, and thus yet sharper drops in \ensuremath{Q_{\mathrm{ion}}}\xspace as SMBHs exceed their Eddington limit, further limiting the period during which AGNs could have contributed to (late-stage) reionization.
To conclude, our calculations show that slim accretion disks produce (roughly) similar ionizing outputs to those of standard, Eddington-limited thin disks, even when taking into account the excess UV emission produced by the additional Comptonization in the slim disk model we used.
However, slim disks {\it can} show drastic changes of orders of magnitude in their ionizing output during periods of significantly super-Eddington accretion periods, on relatively short time scales.
\begin{figure}
\centering
\includegraphics[width=1.0\columnwidth]{Q_saturation.png}
\caption{The saturation of \ensuremath{Q_{\mathrm{ion}}}\xspace for high values of $\ensuremath{\dot{M}}\xspace/\dot{M}_{\rm Edd}$ for spinless black holes with masses of $10^7\,\ifmmode M_{\odot} \else $M_{\odot}$\fi$ (red) and $10^8\,\ifmmode M_{\odot} \else $M_{\odot}$\fi$ (blue). All the calculations presented here use the slim disk model.}
\label{fig:Q_saturation}
\end{figure}
\subsection{Extending the contribution of AGNs to reionization to earlier times}
\label{sec:discussion_extending}
In Figure \ref{fig:Nion_diff_spin_fEdd} we demonstrated that a slowly growing population of SMBHs, due to low \ensuremath{f_{\mathrm{Edd}}}\xspace and/or high spin, will produce a high number of ionizing photons at high redshifts, as compared to a faster-evolving population (which may be even our fiducial case with $\ensuremath{f_{\mathrm{Edd}}}\xspace=0.6$ and $a=0$). However, such slow mass growth scenarios, would inventively lead to require that BHs have high initial masses, that is, massive BH seeds.
The seed masses at $z=20$ required to produce a $10^9\,\ifmmode M_{\odot} \else $M_{\odot}$\fi$ SMBH at $z=6$ for each of the scenarios shown in Figure~\ref{fig:Nion_diff_spin_fEdd} are listed in Table \ref{tab:BH_seeds}.
Specifically, a scenario where all SMBHs have a constant $\ensuremath{f_{\mathrm{Edd}}}\xspace=0.1$ and/or a constant maximal spin would imply BH seeds with unreasonably high masses, exceeding $10^7\,\ifmmode M_{\odot} \else $M_{\odot}$\fi$ (or even $10^8\,\ifmmode M_{\odot} \else $M_{\odot}$\fi$ if \ensuremath{f_{\mathrm{Edd}}}\xspace\ is low {\it and} the spin is high). Such extreme masses exceed even the most massive seeds considered in most works \cite[see reviews by][but see also \citealt{Mayer15} and references therein.]{Natarajan2011,Volonteri12,Inayoshi20}.
On the other hand, according to our calculations all the growth scenarios shown in Figure \ref{fig:Nion_diff_spin_fEdd} that imply more reasonable BH seed masses ($M\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$<$}}}{\rm few}\times10^4\,\ifmmode M_{\odot} \else $M_{\odot}$\fi$) would produce an ionizing output that---even by $z=7$---is at least an order of magnitude lower than what is required to keep the IGM ionized (c.f. the gray-shaded area in Fig.~
\ref{fig:Nion_diff_spin_fEdd}.)
\begin{deluxetable}{ccc}
\tablenum{2}
\label{tab:BH_seeds}
\tablewidth{1.0\columnwidth}
\tablecaption{BH seed masses for the scenarios depicted in Fig.~\ref{fig:Nion_diff_spin_fEdd}.}
\tablehead{
\colhead{$L/L_{\mathrm{Edd}}$} & \colhead{$a$} & \colhead{$M_{\mathrm{seed}}/\ifmmode M_{\odot} \else $M_{\odot}$\fi$} \\
\colhead{} & \colhead{} & \colhead{at $z{=}20$}
}
\startdata
$0.1$ & $0$ & $4.2\times10^7$ \\
$0.1$ & $0.7$ & $1.9\times10^8$ \\
$0.1$ & $0.998$ & $6.6\times10^8$ \\
$0.6$ & $0$ & $10$ \\
$0.6$ & $0.7$ & $4.8\times10^4$ \\
$0.6$ & $0.998$ & $8.7\times10^7$\\
$1.0$ & $0$ & $10$ \\
$1.0$ & $0.7$ & $60$ \\
$1.0$ & $0.998$ & $1.7\times10^7$
\enddata
\tablenotetext{}{\small{We list the seed masses at $z=20$ required to produce a $10^9\,\ifmmode M_{\odot} \else $M_{\odot}$\fi$ SMBH at $z=6$.}}
\end{deluxetable}
One way for accreting BHs to maintain a high ionizing output at high redshifts while still having reasonable (implied) seed masses at $z=20$ is to have an extremely early rapid relative growth phase --- immediately after seed formation, followed by a prolonged slow relative growth throughout the epoch of reionization (i.e., most of $z\lesssim6$).
One such scenario, for a single SMBH, was already explored in Figure \ref{fig:Qvst_Mixed}. In that case, the growth starts with a constant $\ensuremath{f_{\mathrm{Edd}}}\xspace=1$ at first and then transitions to a growth at a constant \ensuremath{\dot{M}}\xspace (i.e., ``Mixed'' growth).
This scenario can be applied to the entire SMBH population, described by a QLF, by assuming accretion at a constant spin and some Eddington ratio at $z=6$, as before.
The results for such a calculation with $a=0$ and various values of $\ensuremath{f_{\mathrm{Edd}}}\xspace\left(z=6\right)$ are presented in Figure~\ref{fig:Nion_mixed_diff_fEdd}.
In these scenarios the ionizing flux of the entire population remains approximately constant during the period of accretion at a constant \ensuremath{\dot{M}}\xspace, extending further back in time as the Eddington ratio decreases.
Specifically, for $\ensuremath{f_{\mathrm{Edd}}}\xspace\left(z=6\right)=0.06$, the SMBH population maintains an ionizing flux density higher than $10^{50}\,\ensuremath{\mathrm{s}^{-1}\mathrm{Mpc}^{-3}}\xspace$ up to $z\sim9$.
Note that for all parameter choices in Figure \ref{fig:Nion_mixed_diff_fEdd}, the implied BH seed masses at $z=20$ are always $\leq10\,\ifmmode M_{\odot} \else $M_{\odot}$\fi$, while still allowing for the population to include SMBHs with $M=10^9\,\ifmmode M_{\odot} \else $M_{\odot}$\fi$ by $z=6$ (consistent with the observed $z\simeq6$ quasar population).
Another scenario that achieves a similar effect of high ionizing flux and small BH seeds is that of ``spinning-up'' BHs. In this scenario, accretion begins when the BH has a low spin, followed by the BH being spun-up by the (residual) angular momentum of the disk, until the BH reaches the maximally allowed value ($a=0.998$; see \citealt{King08} and references therein for details of the process).
We explore a simplified version of such a scenario in Figure~\ref{fig:Nion_spin_up} (dashed magenta), where the spin of the entire BH population is set to $a=0$ for $20<z<10$ ($\Delta t\simeq0.29\,\mathrm{Gyr}$) and then switches to $a=0.998$ for $10<z<6$ ($\Delta t\simeq0.45\,\mathrm{Gyr}$). During the entire growth, the Eddington ratio is fixed to $\ensuremath{f_{\mathrm{Edd}}}\xspace=0.6$ for the entire BH population.
This setup allows for a $10^9\,\ifmmode M_{\odot} \else $M_{\odot}$\fi$ SMBH at $z=6$ to grow from a $z=20$ BH seeds of $\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$<$}}}\,10^6\,\ifmmode M_{\odot} \else $M_{\odot}$\fi$, which is consistent with the maximal mass proposed for so-called ``direct-collapse'' BH seeding models \citep{Inayoshi20}.
Compared with the constant high-spin scenario, where the entire BH population has a maximal spin throughout its entire growth history and therefore requires unreasonably high seed masses ($>10^6\,\ifmmode M_{\odot} \else $M_{\odot}$\fi$ at $z=20$), the ``spin-up'' scenario results in only $10\%$ less ionizing photons in total (i.e., time integrated).
For comparison, Figure~\ref{fig:Nion_spin_up} also shows the ionizing flux density of a population with $a=0$, and the same \ensuremath{f_{\mathrm{Edd}}}\xspace. It can be clearly seen that in such a scenario the drop in ionizing flux with increasing redshift is much more significant. Both scenarios in Figure~\ref{fig:Nion_spin_up} also show the time and redshift at which a $z=6$ $10^9\,\ifmmode M_{\odot} \else $M_{\odot}$\fi$ SMBH reaches a mass of $10^6\,\ifmmode M_{\odot} \else $M_{\odot}$\fi$.
Obviously, more complex, physically motivated spin evolution scenarios may be considered \cite[e.g.,][]{King08,Dotti13}, however these are beyond the scope of the present paper.
We note that in all growth scenarios explored in this work we only considered continuous BH accretion, i.e. a duty cycle of 100\%.
Any lower duty cycle that results in comparable ``final'' BH masses (at $z=6$) would necessarily extend the accretion to higher redshifts, and thus increase the AGN contribution to reionization over these earlier epochs.
This is similar to the effect of assuming a lower \ensuremath{f_{\mathrm{Edd}}}\xspace and/or higher BH spin (i.e., slower mass growth).
However, we stress that in any Eddington-limited scenario, the duty cycle of the highest mass SMBHs cannot be too low, or else they will not reach their observed high masses at $z\simeq6$.
\begin{figure}
\centering
\includegraphics[width=1.0\columnwidth]{Nion_mixed.png}
\caption{The evolution of \ensuremath{\dot{N}_{\mathrm{ion}}}\xspace for the population of accreting SMBHs, where the accretion proceeds at an Eddington-limited \ensuremath{\dot{M}}\xspace, set by the value of \ensuremath{f_{\mathrm{Edd}}}\xspace at $z=6$ (different colors).
These calculations assume $a=0$ and ``QLF A'' at $z=6$.}
\label{fig:Nion_mixed_diff_fEdd}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=1.0\columnwidth]{Nion_spinning_up2.png}
\caption{The evolution of the total ionizing flux density of all accreting SMBHs for the simplified ``spin-up'' scenario: $a=0$ for $20<z<10$ and $a=0.998$ for $10<z<6$ (dashed magenta).
For comparison, we show again the constant $a=0$ scenario (in dotted gray).
For both scenarios we use ``QLF A'' at $z=6$ and assume that the accretion proceeds with a constant $\ensuremath{f_{\mathrm{Edd}}}\xspace=0.6$. For each of the scenarios, we also mark the epoch at which a SMBH with $M\left(z=6\right)=10^{9}\,\ifmmode M_{\odot} \else $M_{\odot}$\fi$ would reach a mass of $10^{6}\,\ifmmode M_{\odot} \else $M_{\odot}$\fi$.}
\label{fig:Nion_spin_up}
\end{figure}
\subsection{How do accreting SMBHs compare to galaxies?}
\label{sec:discussion_galaxies}
Many studies have tried to identify the sources that have contributed the most to the reionization the Universe, with most of these studies (but not all) concluding that accreting SMBHs are a subdominant source of ionizing radiation as compared to young, star-forming galaxies \cite[e.g.,][and references therein]{Robertson15, Bouwens15, Parsa18, Dayal20, Ananna20}.
In this section we compare the results of our SMBH-focused calculations to the ionizing contribution of such galaxies, as reported by the comprehensive study by \cite{Bouwens15}.
That study derived the \ensuremath{\dot{N}_{\mathrm{ion}}}\xspace attributed to early star forming galaxies, according to
\begin{equation}
\label{eq:Bouwens Nion}
\ensuremath{\dot{N}_{\mathrm{ion}}}\xspace = \ensuremath{f_{\mathrm{esc}}}\xspace\ensuremath{\xi_{\mathrm{ion}}}\xspace \rho_\mathrm{UV}\,,
\end{equation}
where $\rho_\mathrm{UV}$ is the observed rest-frame UV luminosity density of Lyman-break galaxies ($\mathrm{erg}\,\mathrm{s}^{-1}\,\mathrm{Hz}^{-1}\,\mathrm{Mpc}^{-3}$);
\ensuremath{\xi_{\mathrm{ion}}}\xspace is the Lyman-continuum photon production efficiency;
and \ensuremath{f_{\mathrm{esc}}}\xspace is the fraction of hydrogen-ionizing photons that escape the galaxies and affect the IGM (the escape fraction).
It then calculated the empirical evolution of \ensuremath{\dot{N}_{\mathrm{ion}}}\xspace, as constrained by measurements of the Thompson optical depth of CMB photons and various other astrophysical probes of the ionization state of the Universe. Comparing the two ionizing emissivities, \cite{Bouwens15} concluded that there is a good agreement between the observed \ensuremath{\dot{N}_{\mathrm{ion}}}\xspace and the one derived from the observed galaxy population, in both redshift evolution, as well as in normalization, provided that $\log\ensuremath{f_{\mathrm{esc}}}\xspace \ensuremath{\xi_{\mathrm{ion}}}\xspace =24.50$ is assumed for all galaxies.
There are significant uncertainties on both \ensuremath{f_{\mathrm{esc}}}\xspace and \ensuremath{\xi_{\mathrm{ion}}}\xspace \cite[e.g.,][]{Siana10, Vanzella12, Mostardi13, DC15}, with the \cite{Bouwens15} study considering the ranges $\log\ensuremath{f_{\mathrm{esc}}}\xspace\simeq(-1.3)-(-0.4)$ and $\log\left(\ensuremath{\xi_{\mathrm{ion}}}\xspace/\rm{s}^{-1}/\left(\rm{erg}\,\rm{s}^{-1}\,\rm{Hz}^{-1}\right)\right)\simeq25.2-25.5$.
Figure~\ref{fig:Bouwens15} shows the evolving \ensuremath{\dot{N}_{\mathrm{ion}}}\xspace derived for young, star-forming $6<z<10$ galaxies by \cite{Bouwens15}, under the assumption of a universal $\log\ensuremath{f_{\mathrm{esc}}}\xspace\ensuremath{\xi_{\mathrm{ion}}}\xspace=24.50$ (solid black line), along with the extrapolation to earlier epochs (dashed black line), as well as the appropriate uncertainty ranges (shaded regions; see caption).
Figure~\ref{fig:Bouwens15} also shows the most ``optimistic'' scenarios for the population of growing SMBHs explored in the present work, judged by their ionizing emissivity over relatively long periods.
Specifically, we show the ``Mixed'' and super-Eddington scenarios with $\ensuremath{f_{\mathrm{Edd}}}\xspace\left(z=6\right)=0.06$ (red and blue lines, respectively) and the simplified ``spin-up'' scenario (magenta).
For all the scenarios plotted here we assumed the ``QLF A'' at $z=6$.
To ease the comparison between the ionizing output of AGNs (following our calculations) and that of galaxies (following \citealt{Bouwens15}), the top panel in Fig.~\ref{fig:Bouwens15} shows the ratio between the former and the latter (i.e., $\dot{N}_{\mathrm{AGN}}/\dot{N}_{\mathrm{galaxies}}$).
\begin{figure}
\centering
\includegraphics[width=1.0\columnwidth]{Bouwens_comparison_with_ratio_panel.png}
\caption{Comparison of \ensuremath{\dot{N}_{\mathrm{ion}}}\xspace for the most ``optimistic'' AGN-focused scenarios presented in this work: the ``Mixed'' (red) and super-Eddington (blue) scenarios with $\ensuremath{f_{\mathrm{Edd}}}\xspace\left(z=6\right)=0.06$ and the ``spin-up'' scenario (magenta).
All these scenarios use ``QLF A'' at $z=6$.
The \ensuremath{\dot{N}_{\mathrm{ion}}}\xspace of galaxies, as derived by \cite{Bouwens15} assuming $\log\ensuremath{f_{\mathrm{esc}}}\xspace\ensuremath{\xi_{\mathrm{ion}}}\xspace=24.50$, is shown in solid black for $6<z<10$, while the dashed line traces its extrapolation over $10<z<20$. Dark- and light-shaded gray regions correspond to the reported range of \ensuremath{\xi_{\mathrm{ion}}}\xspace and the additional range of \ensuremath{f_{\mathrm{esc}}}\xspace, respectively. The top panel presents the ratio between the \ensuremath{\dot{N}_{\mathrm{ion}}}\xspace of AGNs and that of galaxies, for the three AGN scenarios.}
\label{fig:Bouwens15}
\end{figure}
Several conclusions can be drawn from Figure \ref{fig:Bouwens15}. First, provided that $\log\ensuremath{f_{\mathrm{esc}}}\xspace\ensuremath{\xi_{\mathrm{ion}}}\xspace=24.50$, the ionizing flux density of galaxies is very high at $z=6$, at roughly (${\sim}7\times10^{50}\,\ensuremath{\mathrm{s}^{-1}\mathrm{Mpc}^{-3}}\xspace$) and remains high up to $z{\simeq}10$ ($\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$>$}}} 2.7\times10^{50}\,\ensuremath{\mathrm{s}^{-1}\mathrm{Mpc}^{-3}}\xspace$). For comparison, for AGNs in the ``Mixed'' and super-Eddington scenarios, the \ensuremath{\dot{N}_{\mathrm{ion}}}\xspace is in the range of $(1-2)\times10^{50}\,\ensuremath{\mathrm{s}^{-1}\mathrm{Mpc}^{-3}}\xspace$ between $6<z<9$, which is subdominant to the one found by \cite{Bouwens15} by a factor of few.
For the ``spin-up'' scenario the AGNs-to-galaxies ratio is slightly lower, as the AGN \ensuremath{\dot{N}_{\mathrm{ion}}}\xspace drops below $10^{50}\,\ensuremath{\mathrm{s}^{-1}\mathrm{Mpc}^{-3}}\xspace$ before reaching $z=7$. Note, however, that for the lower end of the uncertainty in $\log\ensuremath{f_{\mathrm{esc}}}\xspace\ensuremath{\xi_{\mathrm{ion}}}\xspace$ the ionizing flux density of AGNs can be comparable to that of galaxies, and can even surpass it at higher redshifts.
An important caveat here is that throughout our analysis we assumed that all the ionizing radiation of accreting SMBHs escapes to the IGM. In this regard the results presented here serve as an upper limit on the AGNs' \ensuremath{\dot{N}_{\mathrm{ion}}}\xspace. In addition, the AGN scenarios in Figure \ref{fig:Bouwens15} are somewhat contrived: the ``spin-up'' scenario requires very massive seeds (see Section~\ref{sec:res_population}), while the ``Mixed'' and super-Eddington scenarios require a very low \ensuremath{f_{\mathrm{Edd}}}\xspace of $0.06$ at $z=6$, in tension with observations of the most luminous quasars. To demonstrate the latter point, we note that \cite{Shen19} found an average $\ensuremath{f_{\mathrm{Edd}}}\xspace\simeq0.3$ for a large sample of $z\gtrsim5.7$ quasars, and that \cite{Onoue19} found $0.16\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$<$}}}\ensuremath{f_{\mathrm{Edd}}}\xspace\lesssim1.1$ for a sample of six fainter quasars at $z>5.8$ \cite[see also][]{Mazzucchelli17}. It is not impossible, however, that the more typical \ensuremath{f_{\mathrm{Edd}}}\xspace of the entire $z=6$ AGN population, including the yet-to-be-detected low-luminosity AGNs \cite[e.g.,][]{Weigel15,Cappelluti16,Vito2016}, is indeed closer to the $\ensuremath{f_{\mathrm{Edd}}}\xspace = 0.06$ assumed in our ``optimistic'' scenarios.\footnote{See, e.g., \cite{Schulze2015} for an example of how the ``typical'' \ensuremath{f_{\mathrm{Edd}}}\xspace is determined for complete AGN samples at lower redshifts.}
Thus, we conclude that it is unlikely that accreting SMBHs were the main drivers of reionization at high redshifts. However, significant contribution at $z\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$<$}}} 8$ is plausible, and can even be dominant at higher redshifts, provided that (1) galaxies have a relatively low $\log\ensuremath{f_{\mathrm{esc}}}\xspace\ensuremath{\xi_{\mathrm{ion}}}\xspace$, and (2) SMBHs grow through certain scenarios while the escape fraction for the emergent AGN radiation remains high.
\subsection{Ionized regions around quasars}
\label{sec:discussion_ionized_regions}
To demonstrate the applicability of the framework developed in this work, we use the ionizing fluxes derived in several growth scenarios to calculate the sizes of `quasar near zones', which are H\,{\sc ii} regions around quasars with sizes of order ${\sim}0.1-10\,\mathrm{Mpc}$. These ionized regions can be probed by rather direct (spectroscopic) observations, with their sizes potentially probing the quasar activity timescales \cite[e.g.,][]{Eilers17, Eilers20, Davies20, Chen21}.
In addition, high IGM neutral fractions leave an imprint in the quasar spectra in the form of a Ly$\alpha$ ``damping-wing'', the detection of which may thus probe the high neutral fraction regime, at $z\gtrsim7$ \cite[e.g.,][]{Banados18, Wang20}.
The calculations presented here are simplistic and serve both to demonstrate the implications of our framework for the size evolution of large-scale H\,{\sc ii} regions, as well as provide an independent test for the relevance of our calculations.
The near zones calculations presented here are based on the method described in detail by \cite{Davies16}.
In brief, we solve one-dimensional radiative transfer ordinary differential equations, assuming a central ionizing source embedded in pure-hydrogen, fully neutral, uniform- and constant-density IGM, in order to calculate the ionized region size as a function of time.
We have experimented with several slightly more elaborate assumptions (e.g., various choices for the IGM density and ionization fraction), but these did not significantly affect the main results and trends we discuss below.
Figure \ref{fig:Rion} presents the evolution of the proper size of the quasar near-zone, $\ensuremath{R_{\mathrm{ion}}}\xspace$, for a spinless SMBH with $M=10^9\,\ifmmode M_{\odot} \else $M_{\odot}$\fi$ and $\ensuremath{f_{\mathrm{Edd}}}\xspace=0.6$ at $z=6$, along with luminosity-corrected measurements of proximity zones around quasars at $6\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$<$}}} z\lesssim7$ taken from \cite{Carilli10}, \cite{Eilers17}, \cite{Eilers20} and \cite{Ishimoto20}.
Here we define \ensuremath{R_{\mathrm{ion}}}\xspace as the distance from the ionizing source to the point where the hydrogen ionization fraction reaches a value of $0.9$.\footnote{In practice, the ionization state transition is sufficiently sharp so that any analysis is indifferent to this specific choice.}
We show the results of calculations that assume either the thin-disk (dashed line) and fixed-shape (dotted line) SED models, where the accretion proceeds with a constant \ensuremath{f_{\mathrm{Edd}}}\xspace, as well as the slim disk model which proceeds at a constant \ensuremath{\dot{M}}\xspace, including in the super-Eddington regime (solid line).
Inspecting Figure~\ref{fig:Rion}, we first note that our calculated $z\simeq6$ proximity zone sizes are reassuringly in general agreement with observations.
The most striking difference between the super-Eddington, slim-disk model and the two Eddington-limited models is that the size evolution of $\ensuremath{R_{\mathrm{ion}}}\xspace$ for the former is much steeper, going from ${\sim}10\,\mathrm{kpc}$ to ${\sim}10^3\,\mathrm{kpc}$ within less than 40 Myr---a factor of ${\sim}15$ faster than for the standard thin-disk, Eddington-limited model (which, in turn, is ${\sim}15\%$ faster as compared to the fixed-shape SED model).
The evolution of the proximity zone is thus much more recent for the super-Eddington model, occurring mostly at $z\lesssim6.5$. In addition, the final proximity zone size is the largest for the slim-disk model, followed by the thin-disk (smaller by ${\sim}10\%$) and then by the fixed-shape SED (smaller by ${\sim}20\%$).
\begin{figure}
\centering
\includegraphics[width=1\columnwidth]{ionized_bubbles_one_panel_masses.png}
\caption{The evolution of $R_\mathrm{ion}$ around a spinless SMBH with $M=10^9\,\ifmmode M_{\odot} \else $M_{\odot}$\fi$ and $\ensuremath{f_{\mathrm{Edd}}}\xspace=0.6$ at $z=6$, for different SED models: slim-disk (solid), thin-disk (dashed) and fixed-shape (dotted). The slim-disk model evolves with a constant \ensuremath{\dot{M}}\xspace while the thin-disk and fixed-shape models evolve with a constant \ensuremath{f_{\mathrm{Edd}}}\xspace.
The different colored symbols mark measurements of luminosity-corrected proximity-zone sizes from \cite{Carilli10}, \cite{Eilers17}, \cite{Eilers20} and \cite{Ishimoto20} (see legend).
The dashed gray line shows the relationship $R_{\rm{ion}}\propto\Lbol^{1/3}\propto\exp\left(t/3\tau_{\rm Edd}\right)$, as expected for a fixed-shape SED. The points at which $\log M_{\rm{BH}}/ \ifmmode M_{\odot} \else $M_{\odot}$\fi$ reaches various values are marked for the slim and thin-disk scenarios.}
\label{fig:Rion}
\end{figure}
The slightly faster growth of the H\,{\sc ii} region for the standard thin disk than for the constant shape SED is consistent with the sharper increase in \ensuremath{Q_{\mathrm{ion}}}\xspace for the former (e.g., Fig. \ref{fig:Qvst_const_fEdd}), with the larger ``final'' \ensuremath{R_{\mathrm{ion}}}\xspace being due to the overall higher number of ionizing photons emitted (see Section~\ref{sec:res_single_bh}).
The much faster growth of the H\,{\sc ii} region for the slim disk model stems from the fast \ensuremath{Q_{\mathrm{ion}}}\xspace evolution for accretion at super-Eddington rates (Fig. \ref{fig:Qvst_AGNslim}), with the final region size being the largest of the three models due to the hard spectrum of the slim disk SED (Fig. \ref{fig:SED_AGNslim_vs_thin}).
One of the most striking differences between the various scenarios explored in Figure \ref{fig:Rion} is the relation between \ensuremath{R_{\mathrm{ion}}}\xspace and the BH mass $M$. The points at which $\log M_{\rm BH}/ \ifmmode M_{\odot} \else $M_{\odot}$\fi$ reaches 6,7 and 8 are marked on the curves of the thin and slim-disk scenarios.
Evidently, although the final $z=6$ \ensuremath{R_{\mathrm{ion}}}\xspace size (i.e., when the BH reaches a mass of $10^9\,\ifmmode M_{\odot} \else $M_{\odot}$\fi$) is similar for the two accretion scenarios, for any lower BH mass (i.e. for any epoch at $z>6$) the proximity zones for the slim-disk, super-Eddington scenario are much smaller than for the thin-disk, Eddington-limited one.
For example, in the thin-disk case the proximity zone reaches a size of $1\,{\rm Mpc}$ when the BH has reached only $M\simeq4\times10^6\,\ifmmode M_{\odot} \else $M_{\odot}$\fi$, while for the slim-disk case the same \ensuremath{R_{\mathrm{ion}}}\xspace is reached only when the BH mass is as massive as $\sim10^8\,\ifmmode M_{\odot} \else $M_{\odot}$\fi$.
Our results echo, and provide further support for, the idea that $z\simeq6$ quasars with relatively compact proximity zones are ``young'', as explored in several recent works \cite[e.g.,][]{Andika20, Eilers20,Eilers2021,Morey2021}.
Within our framework and the calculations presented in Figure~\ref{fig:Rion}, the only way to obtain small proximity zones ($\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$<$}}}$1 Mpc) around $z\simeq6$ quasars that are powered by ``mature'' SMBHs with $M\gtrsim10^8\,\ifmmode M_{\odot} \else $M_{\odot}$\fi$, is if their growth has started at an extremely late stage ($z<7$) and proceeded through extremely fast, super-Eddington accretion, to be able to reach the ultimate high BH masses.
Unless the escape fraction is extremely low (i.e., heavily obscured systems), the key for producing compact proximity zones is a low (mass-averaged) radiative efficiency of the accretion process, as is the case for super-Eddington accretion.
Alternatively, a compact proximity zone could result from low duty cycle BH activity (i.e., sub-Eddington accretion turning on and off intermittently), however the resulting ``final'' BH mass may be significantly lower than what is observed for the $z=6$ quasar population.
At any rate, the calculations highlighted in this Section demonstrate that our framework can be used as a predictor of (trends in) $R_\mathrm{ion}$, and as an efficient way to link various BH growth scenarios to proximity zone observations.
\section{Summary and Conclusions}
\label{sec:summary}
Any attempt to assess the contribution of accreting SMBHs to the reionization of the Universe has to consider several (sometimes contradictory) aspects: a SMBH produces ionizing radiation only during vigorous accretion episodes; the faster it grows --- the more ionizing radiation it produces; grow \textit{too} fast, and the contribution to reionization is limited to a short period; grow too \textit{slow}, and the high observed BH masses cannot be explained.
To complicate things further, additional (subtle) effects are expected when considering a population of growing SMBHs, and their spectral evolution.
In this work we presented a framework for the calculation of the ionizing output of accreting SMBHs in a physically motivated way that accounts for the growth of the SMBH and the spectral dependence on BH mass, accretion rate, and spin. We then extended the framework to a population of SMBHs, assuming a QLF representation and a universal \ensuremath{f_{\mathrm{Edd}}}\xspace (defined as $L_\mathrm{bol}/L_\mathrm{Edd}$) at $z=6$. After modeling the mass and spectral evolution of the population, we derived the history of the ionizing flux density of the entire SMBH population.
The key results from the application of this framework are:
\begin{itemize}
\item Accounting for the spectral evolution of accreting SMBHs can increase their total ionizing output by $\sim30-80\%$, compared to a (commonly-used) fixed-shape SED model (Figs. \ref{fig:Qvst_const_fEdd}, \ref{fig:Qvst_Mixed}, \ref{fig:Qvsz}).
\item Accreting SMBHs can probably contribute significantly to cosmic (hydrogen) reionization only at late times ($z\lesssim7$; Figs. \ref{fig:Qvsz}, \ref{fig:Nion_diff_spin_fEdd}).
\item Slower mass growth of the SMBH population, by means of a low \ensuremath{f_{\mathrm{Edd}}}\xspace and/or high spin, increases significantly the population's ionizing output at high redshifts ($z>7$), and may allow for a non-negligible contribution of accreting SMBHs to reionization at up to $z\approx9$ (Figs. \ref{fig:Nion_diff_spin_fEdd}, \ref{fig:Nion_mixed_diff_fEdd}, \ref{fig:Nion_spin_up}).
\item Growth scenarios with periods of super-Eddington accretion only highlight the previous points: a late contribution to reionization, which can be extended to higher redshifts by means of a lower accretion rate, but with a yet sharper drop in the ionizing output at earlier epochs (Fig. \ref{fig:Nion AGNslim}).
\item Accounting for the spectral evolution of accreting SMBHs can slightly increase the size of H\,{\sc ii} regions around quasars (by ${\sim}10\%$). The super-Eddington slim disk model can increase the size by a further ${\sim}10\%$, and --- more importantly --- lead to a very fast growth (a factor of $\times15$ faster than for the standard thin disk; Fig. \ref{fig:Rion}).
\item SMBHs are probably a sub-dominant source of ionizing radiation, as compared to galaxies. However, the relative contribution of SMBHs to reionization can be increased by a slow mass evolution at low redshifts ($6\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$<$}}} z \lesssim9$); and/or an exceptionally high space density of moderate-luminosity AGNs; and/or if the ionizing radiation of star-forming galaxies is suppressed (intrinsically and/or by a low escape fraction; see Fig. \ref{fig:Bouwens15}).
\end{itemize}
The last point notwithstanding, it is important to note that the recent measurement of a low Thompson optical depth of CMB photons made by \cite{Planck18VI} is consistent with a late ($z<9$) reionization, with \cite{Mason19}, for example, finding $z_{0.5}=6.93\pm0.14$ as the redshift of the mid-point of reionization.
Moreover, other recent studies, which are based on observations (Ly$\alpha$ forest measurements) or models (radiative transfer or hydrodynamical calculations), have concluded that reionization may have extended up to $z\sim5.3$ \cite[e.g.,][]{Eilers18,Kulkarni19,Keating20,Bosman18,Bosman21,Zhu21}. When taken together with the results of this work, namely the late contribution of AGNs to reionization and possible extension of the AGN contribution towards $z\sim9$, this makes the scenario in which AGN contribution to reionization is non-negligible (and even comparable to that of galaxies) more plausible.
We note that all of our \ensuremath{Q_{\mathrm{ion}}}\xspace and \ensuremath{\dot{N}_{\mathrm{ion}}}\xspace calculations can be trivially shifted to earlier times (e.g., setting the end-point of the BH evolution at some $z>6$) with the only caveat being higher implied seed masses.
There is a great degree of uncertainty in the population analysis presented in Section~\ref{sec:res_population} due to uncertainties involving the QLF, specifically at $z\gtrsim6$. Currently there is very limited knowledge of the lower-luminosity shape of the QLF due to a lack of observations of relevant AGNs beyond $z\simeq4.5$ \cite[e.g., ][]{Shen20}.
Other uncertainties are related to the simplifying assumptions we made throughout the present work, each of which could be the focus of future investigations. One such assumption is that of a constant spin, in contrast to a scenario of a BH with a self-consistent spin evolution \cite[e.g.,][]{King08,Dotti13,Volonteri13}. In addition, when interpreting the QLF, we assumed a fixed \ensuremath{f_{\mathrm{Edd}}}\xspace and spin for the entire population, while the actual population may have a wide range of Eddington ratios \cite[e.g.,][]{Mazzucchelli17,Shen19} and spins, and also assumed that new BHs are not being formed, nor that AGNs turn their accretion ``on'' or ``off''.
Furthermore, a more detailed analysis of the radiative outputs of SMBHs should consider the total number of ionizations in the IGM, including secondary ones, in contrast to the simpler approach taken here (focusing only on \ensuremath{Q_{\mathrm{ion}}}\xspace and \ensuremath{\dot{N}_{\mathrm{ion}}}\xspace).
This may be particularly relevant for quasars, due to their hard SEDs, and even more relevant for slim-disk, super Eddington models (such as \texttt{AGNslim}), where the SEDs are even harder.
In the present work, we have not considered the physics of circumnuclear, interstellar, and/or intergalactic media, which would include obscuration, attenuation and gas geometry. Perhaps the best way to address these complex processes is by incorporating the SEDs and considerations that were highlighted in this work as ``sub-grid'' components in large cosmological hydrodynamic simulations.
Such simulations, and/or semi-analytical models, can also be used to explore various other BH formation and early growth scenarios, which are not captured by the QLF-based population analysis presented here.
Going beyond hydrogen ionization, our framework could be also extended to investigate the contribution of accreting SMBHs to (later) helium reionization and to early cosmic heating.
The detection of ever larger and more complete populations of early accreting SMBHs, beyond $z\simeq7$, coupled with advances in the understanding of their accretion flows and growth histories, should lead to further re-assessment of the contribution of SMBHs to the reionization of the Universe.
\begin{acknowledgments}
We thank the anonymous reviewer for their constructive and insightful comments, which helped us improve this paper.
We thank Smadar Naoz for early discussions that motivated key parts of our work, and Rennan Barkana for useful comments that helped improve this paper.
We thank Aya Kubota and Chris Done for their assistance in incorporating the slim-disk model into our framework and for helpful comments.
We also thank Shane Davis for his assistance with some of the \texttt{KERRTRNAS} calculations.
We finally thank Steven Furlanetto for kindly providing the energy deposition fractions for the radiative transfer calculations.
We acknowledge support from the Israel Science Foundation (grant number 1849/19) and from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement number 950533).
\end{acknowledgments}
\software{{\tt AstroPy} \citep{Astropy2013,Astropy2018},
{\tt Matplotlib} \citep{Matplotlib2007},
{\tt NumPy} \citep{NumPy20}, {\tt SciPy} \citep{SciPy20}, {\tt Xspec and \tt PyXspec} \citep{Xspec96}}
\clearpage
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,584 |
#include <stdarg.h>
#define NONAMELESSUNION
#define NONAMELESSSTRUCT
#include "windef.h"
#include "winbase.h"
#include "winreg.h"
#include "winerror.h"
#include "winternl.h"
#include "objbase.h"
#include "shlwapi.h"
#include "wine/list.h"
#include "wine/debug.h"
#include "wine/unicode.h"
#include "mapival.h"
WINE_DEFAULT_DEBUG_CHANNEL(mapi);
BOOL WINAPI FBadRglpszA(LPSTR*,ULONG);
/* Internal: Check if a property value array is invalid */
static inline ULONG PROP_BadArray(LPSPropValue lpProp, size_t elemSize)
{
return IsBadReadPtr(lpProp->Value.MVi.lpi, lpProp->Value.MVi.cValues * elemSize);
}
/*************************************************************************
* PropCopyMore@16 (MAPI32.76)
*
* Copy a property value.
*
* PARAMS
* lpDest [O] Destination for the copied value
* lpSrc [I] Property value to copy to lpDest
* lpMore [I] Linked memory allocation function (pass MAPIAllocateMore())
* lpOrig [I] Original allocation to which memory will be linked
*
* RETURNS
* Success: S_OK. lpDest contains a deep copy of lpSrc.
* Failure: MAPI_E_INVALID_PARAMETER, if any parameter is invalid,
* MAPI_E_NOT_ENOUGH_MEMORY, if memory allocation fails.
*
* NOTES
* Any elements within the property returned should not be individually
* freed, as they will be freed when lpOrig is.
*/
SCODE WINAPI PropCopyMore(LPSPropValue lpDest, LPSPropValue lpSrc,
ALLOCATEMORE *lpMore, LPVOID lpOrig)
{
ULONG ulLen, i;
SCODE scode = S_OK;
TRACE("(%p,%p,%p,%p)\n", lpDest, lpSrc, lpMore, lpOrig);
if (!lpDest || IsBadWritePtr(lpDest, sizeof(SPropValue)) ||
FBadProp(lpSrc) || !lpMore)
return MAPI_E_INVALID_PARAMETER;
/* Shallow copy first, this is sufficient for properties without pointers */
*lpDest = *lpSrc;
switch (PROP_TYPE(lpSrc->ulPropTag))
{
case PT_CLSID:
scode = lpMore(sizeof(GUID), lpOrig, (LPVOID*)&lpDest->Value.lpguid);
if (SUCCEEDED(scode))
*lpDest->Value.lpguid = *lpSrc->Value.lpguid;
break;
case PT_STRING8:
ulLen = lstrlenA(lpSrc->Value.lpszA) + 1u;
scode = lpMore(ulLen, lpOrig, (LPVOID*)&lpDest->Value.lpszA);
if (SUCCEEDED(scode))
memcpy(lpDest->Value.lpszA, lpSrc->Value.lpszA, ulLen);
break;
case PT_UNICODE:
ulLen = (strlenW(lpSrc->Value.lpszW) + 1u) * sizeof(WCHAR);
scode = lpMore(ulLen, lpOrig, (LPVOID*)&lpDest->Value.lpszW);
if (SUCCEEDED(scode))
memcpy(lpDest->Value.lpszW, lpSrc->Value.lpszW, ulLen);
break;
case PT_BINARY:
scode = lpMore(lpSrc->Value.bin.cb, lpOrig, (LPVOID*)&lpDest->Value.bin.lpb);
if (SUCCEEDED(scode))
memcpy(lpDest->Value.bin.lpb, lpSrc->Value.bin.lpb, lpSrc->Value.bin.cb);
break;
default:
if (lpSrc->ulPropTag & MV_FLAG)
{
ulLen = UlPropSize(lpSrc);
if (PROP_TYPE(lpSrc->ulPropTag) == PT_MV_STRING8 ||
PROP_TYPE(lpSrc->ulPropTag) == PT_MV_UNICODE)
{
/* UlPropSize doesn't account for the string pointers */
ulLen += lpSrc->Value.MVszA.cValues * sizeof(char*);
}
else if (PROP_TYPE(lpSrc->ulPropTag) == PT_MV_BINARY)
{
/* UlPropSize doesn't account for the SBinary structs */
ulLen += lpSrc->Value.MVbin.cValues * sizeof(SBinary);
}
lpDest->Value.MVi.cValues = lpSrc->Value.MVi.cValues;
scode = lpMore(ulLen, lpOrig, (LPVOID*)&lpDest->Value.MVi.lpi);
if (FAILED(scode))
break;
/* Note that we could allocate the memory for each value in a
* multi-value property separately, however if an allocation failed
* we would be left with a bunch of allocated memory, which (while
* not really leaked) is unusable until lpOrig is freed. So for
* strings and binary arrays we make a single allocation for all
* of the data. This is consistent since individual elements can't
* be freed anyway.
*/
switch (PROP_TYPE(lpSrc->ulPropTag))
{
case PT_MV_STRING8:
{
char *lpNextStr = (char*)(lpDest->Value.MVszA.lppszA +
lpDest->Value.MVszA.cValues);
for (i = 0; i < lpSrc->Value.MVszA.cValues; i++)
{
ULONG ulStrLen = lstrlenA(lpSrc->Value.MVszA.lppszA[i]) + 1u;
lpDest->Value.MVszA.lppszA[i] = lpNextStr;
memcpy(lpNextStr, lpSrc->Value.MVszA.lppszA[i], ulStrLen);
lpNextStr += ulStrLen;
}
break;
}
case PT_MV_UNICODE:
{
WCHAR *lpNextStr = (WCHAR*)(lpDest->Value.MVszW.lppszW +
lpDest->Value.MVszW.cValues);
for (i = 0; i < lpSrc->Value.MVszW.cValues; i++)
{
ULONG ulStrLen = strlenW(lpSrc->Value.MVszW.lppszW[i]) + 1u;
lpDest->Value.MVszW.lppszW[i] = lpNextStr;
memcpy(lpNextStr, lpSrc->Value.MVszW.lppszW[i], ulStrLen * sizeof(WCHAR));
lpNextStr += ulStrLen;
}
break;
}
case PT_MV_BINARY:
{
LPBYTE lpNext = (LPBYTE)(lpDest->Value.MVbin.lpbin +
lpDest->Value.MVbin.cValues);
for (i = 0; i < lpSrc->Value.MVszW.cValues; i++)
{
lpDest->Value.MVbin.lpbin[i].cb = lpSrc->Value.MVbin.lpbin[i].cb;
lpDest->Value.MVbin.lpbin[i].lpb = lpNext;
memcpy(lpNext, lpSrc->Value.MVbin.lpbin[i].lpb, lpDest->Value.MVbin.lpbin[i].cb);
lpNext += lpDest->Value.MVbin.lpbin[i].cb;
}
break;
}
default:
/* No embedded pointers, just copy the data over */
memcpy(lpDest->Value.MVi.lpi, lpSrc->Value.MVi.lpi, ulLen);
break;
}
break;
}
}
return scode;
}
/*************************************************************************
* UlPropSize@4 (MAPI32.77)
*
* Determine the size of a property in bytes.
*
* PARAMS
* lpProp [I] Property to determine the size of
*
* RETURNS
* Success: The size of the value in lpProp.
* Failure: 0, if a multi-value (array) property is invalid or the type of lpProp
* is unknown.
*
* NOTES
* - The size returned does not include the size of the SPropValue struct
* or the size of the array of pointers for multi-valued properties that
* contain pointers (such as PT_MV_STRING8 or PT-MV_UNICODE).
* - MSDN incorrectly states that this function returns MAPI_E_CALL_FAILED if
* lpProp is invalid. In reality no checking is performed and this function
* will crash if passed an invalid property, or return 0 if the property
* type is PT_OBJECT or is unknown.
*/
ULONG WINAPI UlPropSize(LPSPropValue lpProp)
{
ULONG ulRet = 1u, i;
TRACE("(%p)\n", lpProp);
switch (PROP_TYPE(lpProp->ulPropTag))
{
case PT_MV_I2: ulRet = lpProp->Value.MVi.cValues;
/* fall through */
case PT_BOOLEAN:
case PT_I2: ulRet *= sizeof(USHORT);
break;
case PT_MV_I4: ulRet = lpProp->Value.MVl.cValues;
/* fall through */
case PT_ERROR:
case PT_I4: ulRet *= sizeof(LONG);
break;
case PT_MV_I8: ulRet = lpProp->Value.MVli.cValues;
/* fall through */
case PT_I8: ulRet *= sizeof(LONG64);
break;
case PT_MV_R4: ulRet = lpProp->Value.MVflt.cValues;
/* fall through */
case PT_R4: ulRet *= sizeof(float);
break;
case PT_MV_APPTIME:
case PT_MV_R8: ulRet = lpProp->Value.MVdbl.cValues;
/* fall through */
case PT_APPTIME:
case PT_R8: ulRet *= sizeof(double);
break;
case PT_MV_CURRENCY: ulRet = lpProp->Value.MVcur.cValues;
/* fall through */
case PT_CURRENCY: ulRet *= sizeof(CY);
break;
case PT_MV_SYSTIME: ulRet = lpProp->Value.MVft.cValues;
/* fall through */
case PT_SYSTIME: ulRet *= sizeof(FILETIME);
break;
case PT_MV_CLSID: ulRet = lpProp->Value.MVguid.cValues;
/* fall through */
case PT_CLSID: ulRet *= sizeof(GUID);
break;
case PT_MV_STRING8: ulRet = 0u;
for (i = 0; i < lpProp->Value.MVszA.cValues; i++)
ulRet += (lstrlenA(lpProp->Value.MVszA.lppszA[i]) + 1u);
break;
case PT_STRING8: ulRet = lstrlenA(lpProp->Value.lpszA) + 1u;
break;
case PT_MV_UNICODE: ulRet = 0u;
for (i = 0; i < lpProp->Value.MVszW.cValues; i++)
ulRet += (strlenW(lpProp->Value.MVszW.lppszW[i]) + 1u);
ulRet *= sizeof(WCHAR);
break;
case PT_UNICODE: ulRet = (lstrlenW(lpProp->Value.lpszW) + 1u) * sizeof(WCHAR);
break;
case PT_MV_BINARY: ulRet = 0u;
for (i = 0; i < lpProp->Value.MVbin.cValues; i++)
ulRet += lpProp->Value.MVbin.lpbin[i].cb;
break;
case PT_BINARY: ulRet = lpProp->Value.bin.cb;
break;
case PT_OBJECT:
default: ulRet = 0u;
break;
}
return ulRet;
}
/*************************************************************************
* FPropContainsProp@12 (MAPI32.78)
*
* Find a property with a given property tag in a property array.
*
* PARAMS
* lpHaystack [I] Property to match to
* lpNeedle [I] Property to find in lpHaystack
* ulFuzzy [I] Flags controlling match type and strictness (FL_* flags from "mapidefs.h")
*
* RETURNS
* TRUE, if lpNeedle matches lpHaystack according to the criteria of ulFuzzy.
*
* NOTES
* Only property types of PT_STRING8 and PT_BINARY are handled by this function.
*/
BOOL WINAPI FPropContainsProp(LPSPropValue lpHaystack, LPSPropValue lpNeedle, ULONG ulFuzzy)
{
TRACE("(%p,%p,0x%08x)\n", lpHaystack, lpNeedle, ulFuzzy);
if (FBadProp(lpHaystack) || FBadProp(lpNeedle) ||
PROP_TYPE(lpHaystack->ulPropTag) != PROP_TYPE(lpNeedle->ulPropTag))
return FALSE;
/* FIXME: Do later versions support Unicode as well? */
if (PROP_TYPE(lpHaystack->ulPropTag) == PT_STRING8)
{
DWORD dwFlags = 0, dwNeedleLen, dwHaystackLen;
if (ulFuzzy & FL_IGNORECASE)
dwFlags |= NORM_IGNORECASE;
if (ulFuzzy & FL_IGNORENONSPACE)
dwFlags |= NORM_IGNORENONSPACE;
if (ulFuzzy & FL_LOOSE)
dwFlags |= (NORM_IGNORECASE|NORM_IGNORENONSPACE|NORM_IGNORESYMBOLS);
dwNeedleLen = lstrlenA(lpNeedle->Value.lpszA);
dwHaystackLen = lstrlenA(lpHaystack->Value.lpszA);
if ((ulFuzzy & (FL_SUBSTRING|FL_PREFIX)) == FL_PREFIX)
{
if (dwNeedleLen <= dwHaystackLen &&
CompareStringA(LOCALE_USER_DEFAULT, dwFlags,
lpHaystack->Value.lpszA, dwNeedleLen,
lpNeedle->Value.lpszA, dwNeedleLen) == CSTR_EQUAL)
return TRUE; /* needle is a prefix of haystack */
}
else if ((ulFuzzy & (FL_SUBSTRING|FL_PREFIX)) == FL_SUBSTRING)
{
LPSTR (WINAPI *pStrChrFn)(LPCSTR,WORD) = StrChrA;
LPSTR lpStr = lpHaystack->Value.lpszA;
if (dwFlags & NORM_IGNORECASE)
pStrChrFn = StrChrIA;
while ((lpStr = pStrChrFn(lpStr, *lpNeedle->Value.lpszA)) != NULL)
{
dwHaystackLen -= (lpStr - lpHaystack->Value.lpszA);
if (dwNeedleLen <= dwHaystackLen &&
CompareStringA(LOCALE_USER_DEFAULT, dwFlags,
lpStr, dwNeedleLen,
lpNeedle->Value.lpszA, dwNeedleLen) == CSTR_EQUAL)
return TRUE; /* needle is a substring of haystack */
lpStr++;
}
}
else if (CompareStringA(LOCALE_USER_DEFAULT, dwFlags,
lpHaystack->Value.lpszA, dwHaystackLen,
lpNeedle->Value.lpszA, dwNeedleLen) == CSTR_EQUAL)
return TRUE; /* full string match */
}
else if (PROP_TYPE(lpHaystack->ulPropTag) == PT_BINARY)
{
if ((ulFuzzy & (FL_SUBSTRING|FL_PREFIX)) == FL_PREFIX)
{
if (lpNeedle->Value.bin.cb <= lpHaystack->Value.bin.cb &&
!memcmp(lpNeedle->Value.bin.lpb, lpHaystack->Value.bin.lpb,
lpNeedle->Value.bin.cb))
return TRUE; /* needle is a prefix of haystack */
}
else if ((ulFuzzy & (FL_SUBSTRING|FL_PREFIX)) == FL_SUBSTRING)
{
ULONG ulLen = lpHaystack->Value.bin.cb;
LPBYTE lpb = lpHaystack->Value.bin.lpb;
while ((lpb = memchr(lpb, *lpNeedle->Value.bin.lpb, ulLen)) != NULL)
{
ulLen = lpHaystack->Value.bin.cb - (lpb - lpHaystack->Value.bin.lpb);
if (lpNeedle->Value.bin.cb <= ulLen &&
!memcmp(lpNeedle->Value.bin.lpb, lpb, lpNeedle->Value.bin.cb))
return TRUE; /* needle is a substring of haystack */
lpb++;
}
}
else if (!LPropCompareProp(lpHaystack, lpNeedle))
return TRUE; /* needle is an exact match with haystack */
}
return FALSE;
}
/*************************************************************************
* FPropCompareProp@12 (MAPI32.79)
*
* Compare two properties.
*
* PARAMS
* lpPropLeft [I] Left hand property to compare to lpPropRight
* ulOp [I] Comparison operator (RELOP_* enum from "mapidefs.h")
* lpPropRight [I] Right hand property to compare to lpPropLeft
*
* RETURNS
* TRUE, if the comparison is true, FALSE otherwise.
*/
BOOL WINAPI FPropCompareProp(LPSPropValue lpPropLeft, ULONG ulOp, LPSPropValue lpPropRight)
{
LONG iCmp;
TRACE("(%p,%d,%p)\n", lpPropLeft, ulOp, lpPropRight);
if (ulOp > RELOP_RE || FBadProp(lpPropLeft) || FBadProp(lpPropRight))
return FALSE;
if (ulOp == RELOP_RE)
{
FIXME("Comparison operator RELOP_RE not yet implemented!\n");
return FALSE;
}
iCmp = LPropCompareProp(lpPropLeft, lpPropRight);
switch (ulOp)
{
case RELOP_LT: return iCmp < 0;
case RELOP_LE: return iCmp <= 0;
case RELOP_GT: return iCmp > 0;
case RELOP_GE: return iCmp >= 0;
case RELOP_EQ: return iCmp == 0;
case RELOP_NE: return iCmp != 0;
}
return FALSE;
}
/*************************************************************************
* LPropCompareProp@8 (MAPI32.80)
*
* Compare two properties.
*
* PARAMS
* lpPropLeft [I] Left hand property to compare to lpPropRight
* lpPropRight [I] Right hand property to compare to lpPropLeft
*
* RETURNS
* An integer less than, equal to or greater than 0, indicating that
* lpszStr is less than, the same, or greater than lpszComp.
*/
LONG WINAPI LPropCompareProp(LPSPropValue lpPropLeft, LPSPropValue lpPropRight)
{
LONG iRet;
TRACE("(%p->0x%08x,%p->0x%08x)\n", lpPropLeft, lpPropLeft->ulPropTag,
lpPropRight, lpPropRight->ulPropTag);
/* If the properties are not the same, sort by property type */
if (PROP_TYPE(lpPropLeft->ulPropTag) != PROP_TYPE(lpPropRight->ulPropTag))
return (LONG)PROP_TYPE(lpPropLeft->ulPropTag) - (LONG)PROP_TYPE(lpPropRight->ulPropTag);
switch (PROP_TYPE(lpPropLeft->ulPropTag))
{
case PT_UNSPECIFIED:
case PT_NULL:
return 0; /* NULLs are equal */
case PT_I2:
return lpPropLeft->Value.i - lpPropRight->Value.i;
case PT_I4:
return lpPropLeft->Value.l - lpPropRight->Value.l;
case PT_I8:
if (lpPropLeft->Value.li.QuadPart > lpPropRight->Value.li.QuadPart)
return 1;
if (lpPropLeft->Value.li.QuadPart == lpPropRight->Value.li.QuadPart)
return 0;
return -1;
case PT_R4:
if (lpPropLeft->Value.flt > lpPropRight->Value.flt)
return 1;
if (lpPropLeft->Value.flt == lpPropRight->Value.flt)
return 0;
return -1;
case PT_APPTIME:
case PT_R8:
if (lpPropLeft->Value.dbl > lpPropRight->Value.dbl)
return 1;
if (lpPropLeft->Value.dbl == lpPropRight->Value.dbl)
return 0;
return -1;
case PT_CURRENCY:
if (lpPropLeft->Value.cur.int64 > lpPropRight->Value.cur.int64)
return 1;
if (lpPropLeft->Value.cur.int64 == lpPropRight->Value.cur.int64)
return 0;
return -1;
case PT_SYSTIME:
return CompareFileTime(&lpPropLeft->Value.ft, &lpPropRight->Value.ft);
case PT_BOOLEAN:
return (lpPropLeft->Value.b ? 1 : 0) - (lpPropRight->Value.b ? 1 : 0);
case PT_BINARY:
if (lpPropLeft->Value.bin.cb == lpPropRight->Value.bin.cb)
iRet = memcmp(lpPropLeft->Value.bin.lpb, lpPropRight->Value.bin.lpb,
lpPropLeft->Value.bin.cb);
else
{
iRet = memcmp(lpPropLeft->Value.bin.lpb, lpPropRight->Value.bin.lpb,
min(lpPropLeft->Value.bin.cb, lpPropRight->Value.bin.cb));
if (!iRet)
iRet = lpPropLeft->Value.bin.cb - lpPropRight->Value.bin.cb;
}
return iRet;
case PT_STRING8:
return lstrcmpA(lpPropLeft->Value.lpszA, lpPropRight->Value.lpszA);
case PT_UNICODE:
return strcmpW(lpPropLeft->Value.lpszW, lpPropRight->Value.lpszW);
case PT_ERROR:
if (lpPropLeft->Value.err > lpPropRight->Value.err)
return 1;
if (lpPropLeft->Value.err == lpPropRight->Value.err)
return 0;
return -1;
case PT_CLSID:
return memcmp(lpPropLeft->Value.lpguid, lpPropRight->Value.lpguid,
sizeof(GUID));
}
FIXME("Unhandled property type %d\n", PROP_TYPE(lpPropLeft->ulPropTag));
return 0;
}
/*************************************************************************
* HrGetOneProp@8 (MAPI32.135)
*
* Get a property value from an IMAPIProp object.
*
* PARAMS
* lpIProp [I] IMAPIProp object to get the property value in
* ulPropTag [I] Property tag of the property to get
* lppProp [O] Destination for the returned property
*
* RETURNS
* Success: S_OK. *lppProp contains the property value requested.
* Failure: MAPI_E_NOT_FOUND, if no property value has the tag given by ulPropTag.
*/
HRESULT WINAPI HrGetOneProp(LPMAPIPROP lpIProp, ULONG ulPropTag, LPSPropValue *lppProp)
{
SPropTagArray pta;
ULONG ulCount;
HRESULT hRet;
TRACE("(%p,%d,%p)\n", lpIProp, ulPropTag, lppProp);
pta.cValues = 1u;
pta.aulPropTag[0] = ulPropTag;
hRet = IMAPIProp_GetProps(lpIProp, &pta, 0u, &ulCount, lppProp);
if (hRet == MAPI_W_ERRORS_RETURNED)
{
MAPIFreeBuffer(*lppProp);
*lppProp = NULL;
hRet = MAPI_E_NOT_FOUND;
}
return hRet;
}
/*************************************************************************
* HrSetOneProp@8 (MAPI32.136)
*
* Set a property value in an IMAPIProp object.
*
* PARAMS
* lpIProp [I] IMAPIProp object to set the property value in
* lpProp [I] Property value to set
*
* RETURNS
* Success: S_OK. The value in lpProp is set in lpIProp.
* Failure: An error result from IMAPIProp_SetProps().
*/
HRESULT WINAPI HrSetOneProp(LPMAPIPROP lpIProp, LPSPropValue lpProp)
{
TRACE("(%p,%p)\n", lpIProp, lpProp);
return IMAPIProp_SetProps(lpIProp, 1u, lpProp, NULL);
}
/*************************************************************************
* FPropExists@8 (MAPI32.137)
*
* Find a property with a given property tag in an IMAPIProp object.
*
* PARAMS
* lpIProp [I] IMAPIProp object to find the property tag in
* ulPropTag [I] Property tag to find
*
* RETURNS
* TRUE, if ulPropTag matches a property held in lpIProp,
* FALSE, otherwise.
*
* NOTES
* if ulPropTag has a property type of PT_UNSPECIFIED, then only the property
* Ids need to match for a successful match to occur.
*/
BOOL WINAPI FPropExists(LPMAPIPROP lpIProp, ULONG ulPropTag)
{
BOOL bRet = FALSE;
TRACE("(%p,%d)\n", lpIProp, ulPropTag);
if (lpIProp)
{
LPSPropTagArray lpTags;
ULONG i;
if (FAILED(IMAPIProp_GetPropList(lpIProp, 0u, &lpTags)))
return FALSE;
for (i = 0; i < lpTags->cValues; i++)
{
if (!FBadPropTag(lpTags->aulPropTag[i]) &&
(lpTags->aulPropTag[i] == ulPropTag ||
(PROP_TYPE(ulPropTag) == PT_UNSPECIFIED &&
PROP_ID(lpTags->aulPropTag[i]) == lpTags->aulPropTag[i])))
{
bRet = TRUE;
break;
}
}
MAPIFreeBuffer(lpTags);
}
return bRet;
}
/*************************************************************************
* PpropFindProp@12 (MAPI32.138)
*
* Find a property with a given property tag in a property array.
*
* PARAMS
* lpProps [I] Property array to search
* cValues [I] Number of properties in lpProps
* ulPropTag [I] Property tag to find
*
* RETURNS
* A pointer to the matching property, or NULL if none was found.
*
* NOTES
* if ulPropTag has a property type of PT_UNSPECIFIED, then only the property
* Ids need to match for a successful match to occur.
*/
LPSPropValue WINAPI PpropFindProp(LPSPropValue lpProps, ULONG cValues, ULONG ulPropTag)
{
TRACE("(%p,%d,%d)\n", lpProps, cValues, ulPropTag);
if (lpProps && cValues)
{
ULONG i;
for (i = 0; i < cValues; i++)
{
if (!FBadPropTag(lpProps[i].ulPropTag) &&
(lpProps[i].ulPropTag == ulPropTag ||
(PROP_TYPE(ulPropTag) == PT_UNSPECIFIED &&
PROP_ID(lpProps[i].ulPropTag) == PROP_ID(ulPropTag))))
return &lpProps[i];
}
}
return NULL;
}
/*************************************************************************
* FreePadrlist@4 (MAPI32.139)
*
* Free the memory used by an address book list.
*
* PARAMS
* lpAddrs [I] Address book list to free
*
* RETURNS
* Nothing.
*/
VOID WINAPI FreePadrlist(LPADRLIST lpAddrs)
{
TRACE("(%p)\n", lpAddrs);
/* Structures are binary compatible; use the same implementation */
FreeProws((LPSRowSet)lpAddrs);
}
/*************************************************************************
* FreeProws@4 (MAPI32.140)
*
* Free the memory used by a row set.
*
* PARAMS
* lpRowSet [I] Row set to free
*
* RETURNS
* Nothing.
*/
VOID WINAPI FreeProws(LPSRowSet lpRowSet)
{
TRACE("(%p)\n", lpRowSet);
if (lpRowSet)
{
ULONG i;
for (i = 0; i < lpRowSet->cRows; i++)
MAPIFreeBuffer(lpRowSet->aRow[i].lpProps);
MAPIFreeBuffer(lpRowSet);
}
}
/*************************************************************************
* ScCountProps@12 (MAPI32.170)
*
* Validate and determine the length of an array of properties.
*
* PARAMS
* iCount [I] Length of the lpProps array
* lpProps [I] Array of properties to validate/size
* pcBytes [O] If non-NULL, destination for the size of the property array
*
* RETURNS
* Success: S_OK. If pcBytes is non-NULL, it contains the size of the
* properties array.
* Failure: MAPI_E_INVALID_PARAMETER, if any parameter is invalid or validation
* of the property array fails.
*/
SCODE WINAPI ScCountProps(INT iCount, LPSPropValue lpProps, ULONG *pcBytes)
{
ULONG i, ulCount = iCount, ulBytes = 0;
TRACE("(%d,%p,%p)\n", iCount, lpProps, pcBytes);
if (iCount <= 0 || !lpProps ||
IsBadReadPtr(lpProps, iCount * sizeof(SPropValue)))
return MAPI_E_INVALID_PARAMETER;
for (i = 0; i < ulCount; i++)
{
ULONG ulPropSize = 0;
if (FBadProp(&lpProps[i]) || lpProps[i].ulPropTag == PROP_ID_NULL ||
lpProps[i].ulPropTag == PROP_ID_INVALID)
return MAPI_E_INVALID_PARAMETER;
if (PROP_TYPE(lpProps[i].ulPropTag) != PT_OBJECT)
{
ulPropSize = UlPropSize(&lpProps[i]);
if (!ulPropSize)
return MAPI_E_INVALID_PARAMETER;
}
switch (PROP_TYPE(lpProps[i].ulPropTag))
{
case PT_STRING8:
case PT_UNICODE:
case PT_CLSID:
case PT_BINARY:
case PT_MV_I2:
case PT_MV_I4:
case PT_MV_I8:
case PT_MV_R4:
case PT_MV_R8:
case PT_MV_CURRENCY:
case PT_MV_SYSTIME:
case PT_MV_APPTIME:
ulPropSize += sizeof(SPropValue);
break;
case PT_MV_CLSID:
ulPropSize += lpProps[i].Value.MVszA.cValues * sizeof(char*) + sizeof(SPropValue);
break;
case PT_MV_STRING8:
case PT_MV_UNICODE:
ulPropSize += lpProps[i].Value.MVszA.cValues * sizeof(char*) + sizeof(SPropValue);
break;
case PT_MV_BINARY:
ulPropSize += lpProps[i].Value.MVbin.cValues * sizeof(SBinary) + sizeof(SPropValue);
break;
default:
ulPropSize = sizeof(SPropValue);
break;
}
ulBytes += ulPropSize;
}
if (pcBytes)
*pcBytes = ulBytes;
return S_OK;
}
/*************************************************************************
* ScCopyProps@16 (MAPI32.171)
*
* Copy an array of property values into a buffer suited for serialisation.
*
* PARAMS
* cValues [I] Number of properties in lpProps
* lpProps [I] Property array to copy
* lpDst [O] Destination for the serialised data
* lpCount [O] If non-NULL, destination for the number of bytes of data written to lpDst
*
* RETURNS
* Success: S_OK. lpDst contains the serialised data from lpProps.
* Failure: MAPI_E_INVALID_PARAMETER, if any parameter is invalid.
*
* NOTES
* The resulting property value array is stored in a contiguous block starting at lpDst.
*/
SCODE WINAPI ScCopyProps(int cValues, LPSPropValue lpProps, LPVOID lpDst, ULONG *lpCount)
{
LPSPropValue lpDest = (LPSPropValue)lpDst;
char *lpDataDest = (char *)(lpDest + cValues);
ULONG ulLen, i;
int iter;
TRACE("(%d,%p,%p,%p)\n", cValues, lpProps, lpDst, lpCount);
if (!lpProps || cValues < 0 || !lpDest)
return MAPI_E_INVALID_PARAMETER;
memcpy(lpDst, lpProps, cValues * sizeof(SPropValue));
for (iter = 0; iter < cValues; iter++)
{
switch (PROP_TYPE(lpProps->ulPropTag))
{
case PT_CLSID:
lpDest->Value.lpguid = (LPGUID)lpDataDest;
*lpDest->Value.lpguid = *lpProps->Value.lpguid;
lpDataDest += sizeof(GUID);
break;
case PT_STRING8:
ulLen = lstrlenA(lpProps->Value.lpszA) + 1u;
lpDest->Value.lpszA = lpDataDest;
memcpy(lpDest->Value.lpszA, lpProps->Value.lpszA, ulLen);
lpDataDest += ulLen;
break;
case PT_UNICODE:
ulLen = (strlenW(lpProps->Value.lpszW) + 1u) * sizeof(WCHAR);
lpDest->Value.lpszW = (LPWSTR)lpDataDest;
memcpy(lpDest->Value.lpszW, lpProps->Value.lpszW, ulLen);
lpDataDest += ulLen;
break;
case PT_BINARY:
lpDest->Value.bin.lpb = (LPBYTE)lpDataDest;
memcpy(lpDest->Value.bin.lpb, lpProps->Value.bin.lpb, lpProps->Value.bin.cb);
lpDataDest += lpProps->Value.bin.cb;
break;
default:
if (lpProps->ulPropTag & MV_FLAG)
{
lpDest->Value.MVi.cValues = lpProps->Value.MVi.cValues;
/* Note: Assignment uses lppszA but covers all cases by union aliasing */
lpDest->Value.MVszA.lppszA = (char**)lpDataDest;
switch (PROP_TYPE(lpProps->ulPropTag))
{
case PT_MV_STRING8:
{
lpDataDest += lpProps->Value.MVszA.cValues * sizeof(char *);
for (i = 0; i < lpProps->Value.MVszA.cValues; i++)
{
ULONG ulStrLen = lstrlenA(lpProps->Value.MVszA.lppszA[i]) + 1u;
lpDest->Value.MVszA.lppszA[i] = lpDataDest;
memcpy(lpDataDest, lpProps->Value.MVszA.lppszA[i], ulStrLen);
lpDataDest += ulStrLen;
}
break;
}
case PT_MV_UNICODE:
{
lpDataDest += lpProps->Value.MVszW.cValues * sizeof(WCHAR *);
for (i = 0; i < lpProps->Value.MVszW.cValues; i++)
{
ULONG ulStrLen = (strlenW(lpProps->Value.MVszW.lppszW[i]) + 1u) * sizeof(WCHAR);
lpDest->Value.MVszW.lppszW[i] = (LPWSTR)lpDataDest;
memcpy(lpDataDest, lpProps->Value.MVszW.lppszW[i], ulStrLen);
lpDataDest += ulStrLen;
}
break;
}
case PT_MV_BINARY:
{
lpDataDest += lpProps->Value.MVszW.cValues * sizeof(SBinary);
for (i = 0; i < lpProps->Value.MVszW.cValues; i++)
{
lpDest->Value.MVbin.lpbin[i].cb = lpProps->Value.MVbin.lpbin[i].cb;
lpDest->Value.MVbin.lpbin[i].lpb = (LPBYTE)lpDataDest;
memcpy(lpDataDest, lpProps->Value.MVbin.lpbin[i].lpb, lpDest->Value.MVbin.lpbin[i].cb);
lpDataDest += lpDest->Value.MVbin.lpbin[i].cb;
}
break;
}
default:
/* No embedded pointers, just copy the data over */
ulLen = UlPropSize(lpProps);
memcpy(lpDest->Value.MVi.lpi, lpProps->Value.MVi.lpi, ulLen);
lpDataDest += ulLen;
break;
}
break;
}
}
lpDest++;
lpProps++;
}
if (lpCount)
*lpCount = lpDataDest - (char *)lpDst;
return S_OK;
}
/*************************************************************************
* ScRelocProps@20 (MAPI32.172)
*
* Relocate the pointers in an array of property values after it has been copied.
*
* PARAMS
* cValues [I] Number of properties in lpProps
* lpProps [O] Property array to relocate the pointers in.
* lpOld [I] Position where the data was copied from
* lpNew [I] Position where the data was copied to
* lpCount [O] If non-NULL, destination for the number of bytes of data at lpDst
*
* RETURNS
* Success: S_OK. Any pointers in lpProps are relocated.
* Failure: MAPI_E_INVALID_PARAMETER, if any parameter is invalid.
*
* NOTES
* MSDN states that this function can be used for serialisation by passing
* NULL as either lpOld or lpNew, thus converting any pointers in lpProps
* between offsets and pointers. This does not work in native (it crashes),
* and cannot be made to work in Wine because the original interface design
* is deficient. The only use left for this function is to remap pointers
* in a contiguous property array that has been copied with memcpy() to
* another memory location.
*/
SCODE WINAPI ScRelocProps(int cValues, LPSPropValue lpProps, LPVOID lpOld,
LPVOID lpNew, ULONG *lpCount)
{
static const BOOL bBadPtr = TRUE; /* Windows bug - Assumes source is bad */
LPSPropValue lpDest = lpProps;
ULONG ulCount = cValues * sizeof(SPropValue);
ULONG ulLen, i;
int iter;
TRACE("(%d,%p,%p,%p,%p)\n", cValues, lpProps, lpOld, lpNew, lpCount);
if (!lpProps || cValues < 0 || !lpOld || !lpNew)
return MAPI_E_INVALID_PARAMETER;
/* The reason native doesn't work as MSDN states is that it assumes that
* the lpProps pointer contains valid pointers. This is obviously not
* true if the array is being read back from serialisation (the pointers
* are just offsets). Native can't actually work converting the pointers to
* offsets either, because it converts any array pointers to offsets then
* _dereferences the offset_ in order to convert the array elements!
*
* The code below would handle both cases except that the design of this
* function makes it impossible to know when the pointers in lpProps are
* valid. If both lpOld and lpNew are non-NULL, native reads the pointers
* after converting them, so we must do the same. It seems this
* functionality was never tested by MS.
*/
#define RELOC_PTR(p) (((char*)(p)) - (char*)lpOld + (char*)lpNew)
for (iter = 0; iter < cValues; iter++)
{
switch (PROP_TYPE(lpDest->ulPropTag))
{
case PT_CLSID:
lpDest->Value.lpguid = (LPGUID)RELOC_PTR(lpDest->Value.lpguid);
ulCount += sizeof(GUID);
break;
case PT_STRING8:
ulLen = bBadPtr ? 0 : lstrlenA(lpDest->Value.lpszA) + 1u;
lpDest->Value.lpszA = RELOC_PTR(lpDest->Value.lpszA);
if (bBadPtr)
ulLen = lstrlenA(lpDest->Value.lpszA) + 1u;
ulCount += ulLen;
break;
case PT_UNICODE:
ulLen = bBadPtr ? 0 : (lstrlenW(lpDest->Value.lpszW) + 1u) * sizeof(WCHAR);
lpDest->Value.lpszW = (LPWSTR)RELOC_PTR(lpDest->Value.lpszW);
if (bBadPtr)
ulLen = (strlenW(lpDest->Value.lpszW) + 1u) * sizeof(WCHAR);
ulCount += ulLen;
break;
case PT_BINARY:
lpDest->Value.bin.lpb = (LPBYTE)RELOC_PTR(lpDest->Value.bin.lpb);
ulCount += lpDest->Value.bin.cb;
break;
default:
if (lpDest->ulPropTag & MV_FLAG)
{
/* Since we have to access the array elements, don't map the
* array unless it is invalid (otherwise, map it at the end)
*/
if (bBadPtr)
lpDest->Value.MVszA.lppszA = (LPSTR*)RELOC_PTR(lpDest->Value.MVszA.lppszA);
switch (PROP_TYPE(lpProps->ulPropTag))
{
case PT_MV_STRING8:
{
ulCount += lpDest->Value.MVszA.cValues * sizeof(char *);
for (i = 0; i < lpDest->Value.MVszA.cValues; i++)
{
ULONG ulStrLen = bBadPtr ? 0 : lstrlenA(lpDest->Value.MVszA.lppszA[i]) + 1u;
lpDest->Value.MVszA.lppszA[i] = RELOC_PTR(lpDest->Value.MVszA.lppszA[i]);
if (bBadPtr)
ulStrLen = lstrlenA(lpDest->Value.MVszA.lppszA[i]) + 1u;
ulCount += ulStrLen;
}
break;
}
case PT_MV_UNICODE:
{
ulCount += lpDest->Value.MVszW.cValues * sizeof(WCHAR *);
for (i = 0; i < lpDest->Value.MVszW.cValues; i++)
{
ULONG ulStrLen = bBadPtr ? 0 : (strlenW(lpDest->Value.MVszW.lppszW[i]) + 1u) * sizeof(WCHAR);
lpDest->Value.MVszW.lppszW[i] = (LPWSTR)RELOC_PTR(lpDest->Value.MVszW.lppszW[i]);
if (bBadPtr)
ulStrLen = (strlenW(lpDest->Value.MVszW.lppszW[i]) + 1u) * sizeof(WCHAR);
ulCount += ulStrLen;
}
break;
}
case PT_MV_BINARY:
{
ulCount += lpDest->Value.MVszW.cValues * sizeof(SBinary);
for (i = 0; i < lpDest->Value.MVszW.cValues; i++)
{
lpDest->Value.MVbin.lpbin[i].lpb = (LPBYTE)RELOC_PTR(lpDest->Value.MVbin.lpbin[i].lpb);
ulCount += lpDest->Value.MVbin.lpbin[i].cb;
}
break;
}
default:
ulCount += UlPropSize(lpDest);
break;
}
if (!bBadPtr)
lpDest->Value.MVszA.lppszA = (LPSTR*)RELOC_PTR(lpDest->Value.MVszA.lppszA);
break;
}
}
lpDest++;
}
if (lpCount)
*lpCount = ulCount;
return S_OK;
}
/*************************************************************************
* LpValFindProp@12 (MAPI32.173)
*
* Find a property with a given property id in a property array.
*
* PARAMS
* ulPropTag [I] Property tag containing property id to find
* cValues [I] Number of properties in lpProps
* lpProps [I] Property array to search
*
* RETURNS
* A pointer to the matching property, or NULL if none was found.
*
* NOTES
* This function matches only on the property id and does not care if the
* property types differ.
*/
LPSPropValue WINAPI LpValFindProp(ULONG ulPropTag, ULONG cValues, LPSPropValue lpProps)
{
TRACE("(%d,%d,%p)\n", ulPropTag, cValues, lpProps);
if (lpProps && cValues)
{
ULONG i;
for (i = 0; i < cValues; i++)
{
if (PROP_ID(ulPropTag) == PROP_ID(lpProps[i].ulPropTag))
return &lpProps[i];
}
}
return NULL;
}
/*************************************************************************
* ScDupPropset@16 (MAPI32.174)
*
* Duplicate a property value array into a contiguous block of memory.
*
* PARAMS
* cValues [I] Number of properties in lpProps
* lpProps [I] Property array to duplicate
* lpAlloc [I] Memory allocation function, use MAPIAllocateBuffer()
* lpNewProp [O] Destination for the newly duplicated property value array
*
* RETURNS
* Success: S_OK. *lpNewProp contains the duplicated array.
* Failure: MAPI_E_INVALID_PARAMETER, if any parameter is invalid,
* MAPI_E_NOT_ENOUGH_MEMORY, if memory allocation fails.
*/
SCODE WINAPI ScDupPropset(int cValues, LPSPropValue lpProps,
LPALLOCATEBUFFER lpAlloc, LPSPropValue *lpNewProp)
{
ULONG ulCount;
SCODE sc;
TRACE("(%d,%p,%p,%p)\n", cValues, lpProps, lpAlloc, lpNewProp);
sc = ScCountProps(cValues, lpProps, &ulCount);
if (SUCCEEDED(sc))
{
sc = lpAlloc(ulCount, (LPVOID*)lpNewProp);
if (SUCCEEDED(sc))
sc = ScCopyProps(cValues, lpProps, *lpNewProp, &ulCount);
}
return sc;
}
/*************************************************************************
* FBadRglpszA@8 (MAPI32.175)
*
* Determine if an array of strings is invalid
*
* PARAMS
* lppszStrs [I] Array of strings to check
* ulCount [I] Number of strings in lppszStrs
*
* RETURNS
* TRUE, if lppszStrs is invalid, FALSE otherwise.
*/
BOOL WINAPI FBadRglpszA(LPSTR *lppszStrs, ULONG ulCount)
{
ULONG i;
TRACE("(%p,%d)\n", lppszStrs, ulCount);
if (!ulCount)
return FALSE;
if (!lppszStrs || IsBadReadPtr(lppszStrs, ulCount * sizeof(LPWSTR)))
return TRUE;
for (i = 0; i < ulCount; i++)
{
if (!lppszStrs[i] || IsBadStringPtrA(lppszStrs[i], -1))
return TRUE;
}
return FALSE;
}
/*************************************************************************
* FBadRglpszW@8 (MAPI32.176)
*
* See FBadRglpszA.
*/
BOOL WINAPI FBadRglpszW(LPWSTR *lppszStrs, ULONG ulCount)
{
ULONG i;
TRACE("(%p,%d)\n", lppszStrs, ulCount);
if (!ulCount)
return FALSE;
if (!lppszStrs || IsBadReadPtr(lppszStrs, ulCount * sizeof(LPWSTR)))
return TRUE;
for (i = 0; i < ulCount; i++)
{
if (!lppszStrs[i] || IsBadStringPtrW(lppszStrs[i], -1))
return TRUE;
}
return FALSE;
}
/*************************************************************************
* FBadRowSet@4 (MAPI32.177)
*
* Determine if a row is invalid
*
* PARAMS
* lpRow [I] Row to check
*
* RETURNS
* TRUE, if lpRow is invalid, FALSE otherwise.
*/
BOOL WINAPI FBadRowSet(LPSRowSet lpRowSet)
{
ULONG i;
TRACE("(%p)\n", lpRowSet);
if (!lpRowSet || IsBadReadPtr(lpRowSet, CbSRowSet(lpRowSet)))
return TRUE;
for (i = 0; i < lpRowSet->cRows; i++)
{
if (FBadRow(&lpRowSet->aRow[i]))
return TRUE;
}
return FALSE;
}
/*************************************************************************
* FBadPropTag@4 (MAPI32.179)
*
* Determine if a property tag is invalid
*
* PARAMS
* ulPropTag [I] Property tag to check
*
* RETURNS
* TRUE, if ulPropTag is invalid, FALSE otherwise.
*/
ULONG WINAPI FBadPropTag(ULONG ulPropTag)
{
TRACE("(0x%08x)\n", ulPropTag);
switch (ulPropTag & (~MV_FLAG & PROP_TYPE_MASK))
{
case PT_UNSPECIFIED:
case PT_NULL:
case PT_I2:
case PT_LONG:
case PT_R4:
case PT_DOUBLE:
case PT_CURRENCY:
case PT_APPTIME:
case PT_ERROR:
case PT_BOOLEAN:
case PT_OBJECT:
case PT_I8:
case PT_STRING8:
case PT_UNICODE:
case PT_SYSTIME:
case PT_CLSID:
case PT_BINARY:
return FALSE;
}
return TRUE;
}
/*************************************************************************
* FBadRow@4 (MAPI32.180)
*
* Determine if a row is invalid
*
* PARAMS
* lpRow [I] Row to check
*
* RETURNS
* TRUE, if lpRow is invalid, FALSE otherwise.
*/
ULONG WINAPI FBadRow(LPSRow lpRow)
{
ULONG i;
TRACE("(%p)\n", lpRow);
if (!lpRow || IsBadReadPtr(lpRow, sizeof(SRow)) || !lpRow->lpProps ||
IsBadReadPtr(lpRow->lpProps, lpRow->cValues * sizeof(SPropValue)))
return TRUE;
for (i = 0; i < lpRow->cValues; i++)
{
if (FBadProp(&lpRow->lpProps[i]))
return TRUE;
}
return FALSE;
}
/*************************************************************************
* FBadProp@4 (MAPI32.181)
*
* Determine if a property is invalid
*
* PARAMS
* lpProp [I] Property to check
*
* RETURNS
* TRUE, if lpProp is invalid, FALSE otherwise.
*/
ULONG WINAPI FBadProp(LPSPropValue lpProp)
{
if (!lpProp || IsBadReadPtr(lpProp, sizeof(SPropValue)) ||
FBadPropTag(lpProp->ulPropTag))
return TRUE;
switch (PROP_TYPE(lpProp->ulPropTag))
{
/* Single value properties containing pointers */
case PT_STRING8:
if (!lpProp->Value.lpszA || IsBadStringPtrA(lpProp->Value.lpszA, -1))
return TRUE;
break;
case PT_UNICODE:
if (!lpProp->Value.lpszW || IsBadStringPtrW(lpProp->Value.lpszW, -1))
return TRUE;
break;
case PT_BINARY:
if (IsBadReadPtr(lpProp->Value.bin.lpb, lpProp->Value.bin.cb))
return TRUE;
break;
case PT_CLSID:
if (IsBadReadPtr(lpProp->Value.lpguid, sizeof(GUID)))
return TRUE;
break;
/* Multiple value properties (arrays) containing no pointers */
case PT_MV_I2:
return PROP_BadArray(lpProp, sizeof(SHORT));
case PT_MV_LONG:
return PROP_BadArray(lpProp, sizeof(LONG));
case PT_MV_LONGLONG:
return PROP_BadArray(lpProp, sizeof(LONG64));
case PT_MV_FLOAT:
return PROP_BadArray(lpProp, sizeof(float));
case PT_MV_SYSTIME:
return PROP_BadArray(lpProp, sizeof(FILETIME));
case PT_MV_APPTIME:
case PT_MV_DOUBLE:
return PROP_BadArray(lpProp, sizeof(double));
case PT_MV_CURRENCY:
return PROP_BadArray(lpProp, sizeof(CY));
case PT_MV_CLSID:
return PROP_BadArray(lpProp, sizeof(GUID));
/* Multiple value properties containing pointers */
case PT_MV_STRING8:
return FBadRglpszA(lpProp->Value.MVszA.lppszA,
lpProp->Value.MVszA.cValues);
case PT_MV_UNICODE:
return FBadRglpszW(lpProp->Value.MVszW.lppszW,
lpProp->Value.MVszW.cValues);
case PT_MV_BINARY:
return FBadEntryList(&lpProp->Value.MVbin);
}
return FALSE;
}
/*************************************************************************
* FBadColumnSet@4 (MAPI32.182)
*
* Determine if an array of property tags is invalid
*
* PARAMS
* lpCols [I] Property tag array to check
*
* RETURNS
* TRUE, if lpCols is invalid, FALSE otherwise.
*/
ULONG WINAPI FBadColumnSet(LPSPropTagArray lpCols)
{
ULONG ulRet = FALSE, i;
TRACE("(%p)\n", lpCols);
if (!lpCols || IsBadReadPtr(lpCols, CbSPropTagArray(lpCols)))
ulRet = TRUE;
else
{
for (i = 0; i < lpCols->cValues; i++)
{
if ((lpCols->aulPropTag[i] & PROP_TYPE_MASK) == PT_ERROR ||
FBadPropTag(lpCols->aulPropTag[i]))
{
ulRet = TRUE;
break;
}
}
}
TRACE("Returning %s\n", ulRet ? "TRUE" : "FALSE");
return ulRet;
}
/**************************************************************************
* IPropData {MAPI32}
*
* A default Mapi interface to provide manipulation of object properties.
*
* DESCRIPTION
* This object provides a default interface suitable in some cases as an
* implementation of the IMAPIProp interface (which has no default
* implementation). In addition to the IMAPIProp() methods inherited, this
* interface allows read/write control over access to the object and its
* individual properties.
*
* To obtain the default implementation of this interface from Mapi, call
* CreateIProp().
*
* METHODS
*/
/* A single property in a property data collection */
typedef struct
{
struct list entry;
ULONG ulAccess; /* The property value access level */
LPSPropValue value; /* The property value */
} IPropDataItem, *LPIPropDataItem;
/* The main property data collection structure */
typedef struct
{
IPropData IPropData_iface;
LONG lRef; /* Reference count */
ALLOCATEBUFFER *lpAlloc; /* Memory allocation routine */
ALLOCATEMORE *lpMore; /* Linked memory allocation routine */
FREEBUFFER *lpFree; /* Memory free routine */
ULONG ulObjAccess; /* Object access level */
ULONG ulNumValues; /* Number of items in values list */
struct list values; /* List of property values */
CRITICAL_SECTION cs; /* Lock for thread safety */
} IPropDataImpl;
static inline IPropDataImpl *impl_from_IPropData(IPropData *iface)
{
return CONTAINING_RECORD(iface, IPropDataImpl, IPropData_iface);
}
/* Internal - Get a property value, assumes lock is held */
static IPropDataItem *IMAPIPROP_GetValue(IPropDataImpl *This, ULONG ulPropTag)
{
struct list *cursor;
LIST_FOR_EACH(cursor, &This->values)
{
LPIPropDataItem current = LIST_ENTRY(cursor, IPropDataItem, entry);
/* Note that property types don't have to match, just Id's */
if (PROP_ID(current->value->ulPropTag) == PROP_ID(ulPropTag))
return current;
}
return NULL;
}
/* Internal - Add a new property value, assumes lock is held */
static IPropDataItem *IMAPIPROP_AddValue(IPropDataImpl *This,
LPSPropValue lpProp)
{
LPVOID lpMem;
LPIPropDataItem lpNew;
HRESULT hRet;
hRet = This->lpAlloc(sizeof(IPropDataItem), &lpMem);
if (SUCCEEDED(hRet))
{
lpNew = lpMem;
lpNew->ulAccess = IPROP_READWRITE;
/* Allocate the value separately so we can update it easily */
lpMem = NULL;
hRet = This->lpAlloc(sizeof(SPropValue), &lpMem);
if (SUCCEEDED(hRet))
{
lpNew->value = lpMem;
hRet = PropCopyMore(lpNew->value, lpProp, This->lpMore, lpMem);
if (SUCCEEDED(hRet))
{
list_add_tail(&This->values, &lpNew->entry);
This->ulNumValues++;
return lpNew;
}
This->lpFree(lpNew->value);
}
This->lpFree(lpNew);
}
return NULL;
}
/* Internal - Lock an IPropData object */
static inline void IMAPIPROP_Lock(IPropDataImpl *This)
{
EnterCriticalSection(&This->cs);
}
/* Internal - Unlock an IPropData object */
static inline void IMAPIPROP_Unlock(IPropDataImpl *This)
{
LeaveCriticalSection(&This->cs);
}
/* This one seems to be missing from mapidefs.h */
#define CbNewSPropProblemArray(c) \
(offsetof(SPropProblemArray,aProblem)+(c)*sizeof(SPropProblem))
/**************************************************************************
* IPropData_QueryInterface {MAPI32}
*
* Inherited method from the IUnknown Interface.
* See IUnknown_QueryInterface.
*/
static WINAPI HRESULT IPropData_fnQueryInterface(LPPROPDATA iface, REFIID riid, LPVOID *ppvObj)
{
IPropDataImpl *This = impl_from_IPropData(iface);
TRACE("(%p,%s,%p)\n", This, debugstr_guid(riid), ppvObj);
if (!ppvObj || !riid)
return MAPI_E_INVALID_PARAMETER;
*ppvObj = NULL;
if(IsEqualIID(riid, &IID_IUnknown) ||
IsEqualIID(riid, &IID_IMAPIProp) ||
IsEqualIID(riid, &IID_IMAPIPropData))
{
*ppvObj = This;
IPropData_AddRef(iface);
TRACE("returning %p\n", *ppvObj);
return S_OK;
}
TRACE("returning E_NOINTERFACE\n");
return MAPI_E_INTERFACE_NOT_SUPPORTED;
}
/**************************************************************************
* IPropData_AddRef {MAPI32}
*
* Inherited method from the IUnknown Interface.
* See IUnknown_AddRef.
*/
static ULONG WINAPI IPropData_fnAddRef(LPPROPDATA iface)
{
IPropDataImpl *This = impl_from_IPropData(iface);
TRACE("(%p)->(count before=%u)\n", This, This->lRef);
return InterlockedIncrement(&This->lRef);
}
/**************************************************************************
* IPropData_Release {MAPI32}
*
* Inherited method from the IUnknown Interface.
* See IUnknown_Release.
*/
static ULONG WINAPI IPropData_fnRelease(LPPROPDATA iface)
{
IPropDataImpl *This = impl_from_IPropData(iface);
LONG lRef;
TRACE("(%p)->(count before=%u)\n", This, This->lRef);
lRef = InterlockedDecrement(&This->lRef);
if (!lRef)
{
TRACE("Destroying IPropData (%p)\n",This);
/* Note: No need to lock, since no other thread is referencing iface */
while (!list_empty(&This->values))
{
struct list *head = list_head(&This->values);
LPIPropDataItem current = LIST_ENTRY(head, IPropDataItem, entry);
list_remove(head);
This->lpFree(current->value);
This->lpFree(current);
}
This->cs.DebugInfo->Spare[0] = 0;
DeleteCriticalSection(&This->cs);
This->lpFree(This);
}
return (ULONG)lRef;
}
/**************************************************************************
* IPropData_GetLastError {MAPI32}
*
* Get information about the last error that occurred in an IMAPIProp object.
*
* PARAMS
* iface [I] IMAPIProp object that experienced the error
* hRes [I] Result of the call that returned an error
* ulFlags [I] 0=return Ascii strings, MAPI_UNICODE=return Unicode strings
* lppError [O] Destination for detailed error information
*
* RETURNS
* Success: S_OK. *lppError contains details about the last error.
* Failure: MAPI_E_INVALID_PARAMETER, if any parameter is invalid,
* MAPI_E_NOT_ENOUGH_MEMORY, if memory allocation fails.
*
* NOTES
* - If this function succeeds, the returned information in *lppError must be
* freed using MAPIFreeBuffer() once the caller is finished with it.
* - It is possible for this function to succeed and set *lppError to NULL,
* if there is no further information to report about hRes.
*/
static HRESULT WINAPI IPropData_fnGetLastError(LPPROPDATA iface, HRESULT hRes, ULONG ulFlags,
LPMAPIERROR *lppError)
{
TRACE("(%p,0x%08X,0x%08X,%p)\n", iface, hRes, ulFlags, lppError);
if (!lppError || SUCCEEDED(hRes) || (ulFlags & ~MAPI_UNICODE))
return MAPI_E_INVALID_PARAMETER;
*lppError = NULL;
return S_OK;
}
/**************************************************************************
* IPropData_SaveChanges {MAPI32}
*
* Update any changes made to a transactional IMAPIProp object.
*
* PARAMS
* iface [I] IMAPIProp object to update
* ulFlags [I] Flags controlling the update.
*
* RETURNS
* Success: S_OK. Any outstanding changes are committed to the object.
* Failure: An HRESULT error code describing the error.
*/
static HRESULT WINAPI IPropData_fnSaveChanges(LPPROPDATA iface, ULONG ulFlags)
{
TRACE("(%p,0x%08X)\n", iface, ulFlags);
/* Since this object is not transacted we do not need to implement this */
/* FIXME: Should we set the access levels to clean? */
return S_OK;
}
/**************************************************************************
* IPropData_GetProps {MAPI32}
*
* Get property values from an IMAPIProp object.
*
* PARAMS
* iface [I] IMAPIProp object to get the property values from
* lpTags [I] Property tage of property values to be retrieved
* ulFlags [I] Return 0=Ascii MAPI_UNICODE=Unicode strings for
* unspecified types
* lpCount [O] Destination for number of properties returned
* lppProps [O] Destination for returned property values
*
* RETURNS
* Success: S_OK. *lppProps and *lpCount are updated.
* Failure: MAPI_E_INVALID_PARAMETER, if any parameter is invalid.
* MAPI_E_NOT_ENOUGH_MEMORY, if memory allocation fails, or
* MAPI_W_ERRORS_RETURNED if not all properties were retrieved
* successfully.
* NOTES
* - If MAPI_W_ERRORS_RETURNED is returned, any properties that could not be
* retrieved from iface are present in lppProps with their type
* changed to PT_ERROR and Id unchanged.
*/
static HRESULT WINAPI IPropData_fnGetProps(LPPROPDATA iface, LPSPropTagArray lpTags, ULONG ulFlags,
ULONG *lpCount, LPSPropValue *lppProps)
{
IPropDataImpl *This = impl_from_IPropData(iface);
ULONG i;
HRESULT hRet = S_OK;
TRACE("(%p,%p,0x%08x,%p,%p) stub\n", iface, lpTags, ulFlags,
lpCount, lppProps);
if (!iface || ulFlags & ~MAPI_UNICODE || !lpTags || *lpCount || !lppProps)
return MAPI_E_INVALID_PARAMETER;
FIXME("semi-stub, flags not supported\n");
*lpCount = lpTags->cValues;
*lppProps = NULL;
if (*lpCount)
{
hRet = MAPIAllocateBuffer(*lpCount * sizeof(SPropValue), (LPVOID*)lppProps);
if (FAILED(hRet))
return hRet;
IMAPIPROP_Lock(This);
for (i = 0; i < lpTags->cValues; i++)
{
HRESULT hRetTmp = E_INVALIDARG;
LPIPropDataItem item;
item = IMAPIPROP_GetValue(This, lpTags->aulPropTag[i]);
if (item)
hRetTmp = PropCopyMore(&(*lppProps)[i], item->value,
This->lpMore, *lppProps);
if (FAILED(hRetTmp))
{
hRet = MAPI_W_ERRORS_RETURNED;
(*lppProps)[i].ulPropTag =
CHANGE_PROP_TYPE(lpTags->aulPropTag[i], PT_ERROR);
}
}
IMAPIPROP_Unlock(This);
}
return hRet;
}
/**************************************************************************
* MAPIProp_GetPropList {MAPI32}
*
* Get the list of property tags for all values in an IMAPIProp object.
*
* PARAMS
* iface [I] IMAPIProp object to get the property tag list from
* ulFlags [I] Return 0=Ascii MAPI_UNICODE=Unicode strings for
* unspecified types
* lppTags [O] Destination for the retrieved property tag list
*
* RETURNS
* Success: S_OK. *lppTags contains the tags for all available properties.
* Failure: MAPI_E_INVALID_PARAMETER, if any parameter is invalid.
* MAPI_E_BAD_CHARWIDTH, if Ascii or Unicode strings are requested
* and that type of string is not supported.
*/
static HRESULT WINAPI IPropData_fnGetPropList(LPPROPDATA iface, ULONG ulFlags,
LPSPropTagArray *lppTags)
{
IPropDataImpl *This = impl_from_IPropData(iface);
ULONG i;
HRESULT hRet;
TRACE("(%p,0x%08x,%p) stub\n", iface, ulFlags, lppTags);
if (!iface || ulFlags & ~MAPI_UNICODE || !lppTags)
return MAPI_E_INVALID_PARAMETER;
FIXME("semi-stub, flags not supported\n");
*lppTags = NULL;
IMAPIPROP_Lock(This);
hRet = MAPIAllocateBuffer(CbNewSPropTagArray(This->ulNumValues),
(LPVOID*)lppTags);
if (SUCCEEDED(hRet))
{
struct list *cursor;
i = 0;
LIST_FOR_EACH(cursor, &This->values)
{
LPIPropDataItem current = LIST_ENTRY(cursor, IPropDataItem, entry);
(*lppTags)->aulPropTag[i] = current->value->ulPropTag;
i++;
}
(*lppTags)->cValues = This->ulNumValues;
}
IMAPIPROP_Unlock(This);
return hRet;
}
/**************************************************************************
* IPropData_OpenProperty {MAPI32}
*
* Not documented at this time.
*
* RETURNS
* An HRESULT success/failure code.
*/
static HRESULT WINAPI IPropData_fnOpenProperty(LPPROPDATA iface, ULONG ulPropTag, LPCIID iid,
ULONG ulOpts, ULONG ulFlags, LPUNKNOWN *lpUnk)
{
FIXME("(%p,%u,%s,%u,0x%08x,%p) stub\n", iface, ulPropTag,
debugstr_guid(iid), ulOpts, ulFlags, lpUnk);
return MAPI_E_NO_SUPPORT;
}
/**************************************************************************
* IPropData_SetProps {MAPI32}
*
* Add or edit the property values in an IMAPIProp object.
*
* PARAMS
* iface [I] IMAPIProp object to get the property tag list from
* ulValues [I] Number of properties in lpProps
* lpProps [I] Property values to set
* lppProbs [O] Optional destination for any problems that occurred
*
* RETURNS
* Success: S_OK. The properties in lpProps are added to iface if they don't
* exist, or changed to the values in lpProps if they do
* Failure: An HRESULT error code describing the error
*/
static HRESULT WINAPI IPropData_fnSetProps(LPPROPDATA iface, ULONG ulValues, LPSPropValue lpProps,
LPSPropProblemArray *lppProbs)
{
IPropDataImpl *This = impl_from_IPropData(iface);
HRESULT hRet = S_OK;
ULONG i;
TRACE("(%p,%u,%p,%p)\n", iface, ulValues, lpProps, lppProbs);
if (!iface || !lpProps)
return MAPI_E_INVALID_PARAMETER;
for (i = 0; i < ulValues; i++)
{
if (FBadProp(&lpProps[i]) ||
PROP_TYPE(lpProps[i].ulPropTag) == PT_OBJECT ||
PROP_TYPE(lpProps[i].ulPropTag) == PT_NULL)
return MAPI_E_INVALID_PARAMETER;
}
IMAPIPROP_Lock(This);
/* FIXME: Under what circumstances is lpProbs created? */
for (i = 0; i < ulValues; i++)
{
LPIPropDataItem item = IMAPIPROP_GetValue(This, lpProps[i].ulPropTag);
if (item)
{
HRESULT hRetTmp;
LPVOID lpMem = NULL;
/* Found, so update the existing value */
if (item->value->ulPropTag != lpProps[i].ulPropTag)
FIXME("semi-stub, overwriting type (not coercing)\n");
hRetTmp = This->lpAlloc(sizeof(SPropValue), &lpMem);
if (SUCCEEDED(hRetTmp))
{
hRetTmp = PropCopyMore(lpMem, &lpProps[i], This->lpMore, lpMem);
if (SUCCEEDED(hRetTmp))
{
This->lpFree(item->value);
item->value = lpMem;
continue;
}
This->lpFree(lpMem);
}
hRet = hRetTmp;
}
else
{
/* Add new value */
if (!IMAPIPROP_AddValue(This, &lpProps[i]))
hRet = MAPI_E_NOT_ENOUGH_MEMORY;
}
}
IMAPIPROP_Unlock(This);
return hRet;
}
/**************************************************************************
* IPropData_DeleteProps {MAPI32}
*
* Delete one or more property values from an IMAPIProp object.
*
* PARAMS
* iface [I] IMAPIProp object to remove property values from.
* lpTags [I] Collection of property Id's to remove from iface.
* lppProbs [O] Destination for problems encountered, if any.
*
* RETURNS
* Success: S_OK. Any properties in iface matching property Id's in lpTags have
* been deleted. If lppProbs is non-NULL it contains details of any
* errors that occurred.
* Failure: MAPI_E_INVALID_PARAMETER, if any parameter is invalid.
* E_ACCESSDENIED, if this object was created using CreateIProp() and
* a subsequent call to IPropData_SetObjAccess() was made specifying
* IPROP_READONLY as the access type.
*
* NOTES
* - lppProbs will not be populated for cases where a property Id is present
* in lpTags but not in iface.
* - lppProbs should be deleted with MAPIFreeBuffer() if returned.
*/
static HRESULT WINAPI IPropData_fnDeleteProps(LPPROPDATA iface, LPSPropTagArray lpTags,
LPSPropProblemArray *lppProbs)
{
IPropDataImpl *This = impl_from_IPropData(iface);
ULONG i, numProbs = 0;
HRESULT hRet = S_OK;
TRACE("(%p,%p,%p)\n", iface, lpTags, lppProbs);
if (!iface || !lpTags)
return MAPI_E_INVALID_PARAMETER;
if (lppProbs)
*lppProbs = NULL;
for (i = 0; i < lpTags->cValues; i++)
{
if (FBadPropTag(lpTags->aulPropTag[i]) ||
PROP_TYPE(lpTags->aulPropTag[i]) == PT_OBJECT ||
PROP_TYPE(lpTags->aulPropTag[i]) == PT_NULL)
return MAPI_E_INVALID_PARAMETER;
}
IMAPIPROP_Lock(This);
if (This->ulObjAccess != IPROP_READWRITE)
{
IMAPIPROP_Unlock(This);
return E_ACCESSDENIED;
}
for (i = 0; i < lpTags->cValues; i++)
{
LPIPropDataItem item = IMAPIPROP_GetValue(This, lpTags->aulPropTag[i]);
if (item)
{
if (item->ulAccess & IPROP_READWRITE)
{
/* Everything hunky-dory, remove the item */
list_remove(&item->entry);
This->lpFree(item->value); /* Also frees value pointers */
This->lpFree(item);
This->ulNumValues--;
}
else if (lppProbs)
{
/* Can't write the value. Create/populate problems array */
if (!*lppProbs)
{
/* Create problems array */
ULONG ulSize = CbNewSPropProblemArray(lpTags->cValues - i);
HRESULT hRetTmp = MAPIAllocateBuffer(ulSize, (LPVOID*)lppProbs);
if (FAILED(hRetTmp))
hRet = hRetTmp;
}
if (*lppProbs)
{
LPSPropProblem lpProb = &(*lppProbs)->aProblem[numProbs];
lpProb->ulIndex = i;
lpProb->ulPropTag = lpTags->aulPropTag[i];
lpProb->scode = E_ACCESSDENIED;
numProbs++;
}
}
}
}
if (lppProbs && *lppProbs)
(*lppProbs)->cProblem = numProbs;
IMAPIPROP_Unlock(This);
return hRet;
}
/**************************************************************************
* IPropData_CopyTo {MAPI32}
*
* Not documented at this time.
*
* RETURNS
* An HRESULT success/failure code.
*/
static HRESULT WINAPI IPropData_fnCopyTo(LPPROPDATA iface, ULONG niids, LPCIID lpiidExcl,
LPSPropTagArray lpPropsExcl, ULONG ulParam,
LPMAPIPROGRESS lpIProgress, LPCIID lpIfaceIid,
LPVOID lpDstObj, ULONG ulFlags,
LPSPropProblemArray *lppProbs)
{
FIXME("(%p,%u,%p,%p,%x,%p,%s,%p,0x%08X,%p) stub\n", iface, niids,
lpiidExcl, lpPropsExcl, ulParam, lpIProgress,
debugstr_guid(lpIfaceIid), lpDstObj, ulFlags, lppProbs);
return MAPI_E_NO_SUPPORT;
}
/**************************************************************************
* IPropData_CopyProps {MAPI32}
*
* Not documented at this time.
*
* RETURNS
* An HRESULT success/failure code.
*/
static HRESULT WINAPI IPropData_fnCopyProps(LPPROPDATA iface, LPSPropTagArray lpInclProps,
ULONG ulParam, LPMAPIPROGRESS lpIProgress,
LPCIID lpIface, LPVOID lpDstObj, ULONG ulFlags,
LPSPropProblemArray *lppProbs)
{
FIXME("(%p,%p,%x,%p,%s,%p,0x%08X,%p) stub\n", iface, lpInclProps,
ulParam, lpIProgress, debugstr_guid(lpIface), lpDstObj, ulFlags,
lppProbs);
return MAPI_E_NO_SUPPORT;
}
/**************************************************************************
* IPropData_GetNamesFromIDs {MAPI32}
*
* Get the names of properties from their identifiers.
*
* PARAMS
* iface [I] IMAPIProp object to operate on
* lppPropTags [I/O] Property identifiers to get the names for, or NULL to
* get all names
* iid [I] Property set identifier, or NULL
* ulFlags [I] MAPI_NO_IDS=Don't return numeric named properties,
* or MAPI_NO_STRINGS=Don't return strings
* lpCount [O] Destination for number of properties returned
* lpppNames [O] Destination for returned names
*
* RETURNS
* Success: S_OK. *lppPropTags and lpppNames contain the returned
* name/identifiers.
* Failure: MAPI_E_NO_SUPPORT, if the object does not support named properties,
* MAPI_E_NOT_ENOUGH_MEMORY, if memory allocation fails, or
* MAPI_W_ERRORS_RETURNED if not all properties were retrieved
* successfully.
*/
static HRESULT WINAPI IPropData_fnGetNamesFromIDs(LPPROPDATA iface, LPSPropTagArray *lppPropTags,
LPGUID iid, ULONG ulFlags, ULONG *lpCount,
LPMAPINAMEID **lpppNames)
{
FIXME("(%p,%p,%s,0x%08X,%p,%p) stub\n", iface, lppPropTags,
debugstr_guid(iid), ulFlags, lpCount, lpppNames);
return MAPI_E_NO_SUPPORT;
}
/**************************************************************************
* IPropData_GetIDsFromNames {MAPI32}
*
* Get property identifiers associated with one or more named properties.
*
* PARAMS
* iface [I] IMAPIProp object to operate on
* ulNames [I] Number of names in lppNames
* lppNames [I] Names to query or create, or NULL to query all names
* ulFlags [I] Pass MAPI_CREATE to create new named properties
* lppPropTags [O] Destination for queried or created property identifiers
*
* RETURNS
* Success: S_OK. *lppPropTags contains the property tags created or requested.
* Failure: MAPI_E_NO_SUPPORT, if the object does not support named properties,
* MAPI_E_TOO_BIG, if the object cannot process the number of
* properties involved.
* MAPI_E_NOT_ENOUGH_MEMORY, if memory allocation fails, or
* MAPI_W_ERRORS_RETURNED if not all properties were retrieved
* successfully.
*/
static HRESULT WINAPI IPropData_fnGetIDsFromNames(LPPROPDATA iface, ULONG ulNames,
LPMAPINAMEID *lppNames, ULONG ulFlags,
LPSPropTagArray *lppPropTags)
{
FIXME("(%p,%d,%p,0x%08X,%p) stub\n",
iface, ulNames, lppNames, ulFlags, lppPropTags);
return MAPI_E_NO_SUPPORT;
}
/**************************************************************************
* IPropData_HrSetObjAccess {MAPI32}
*
* Set the access level of an IPropData object.
*
* PARAMS
* iface [I] IPropData object to set the access on
* ulAccess [I] Either IPROP_READONLY or IPROP_READWRITE for read or
* read/write access respectively.
*
* RETURNS
* Success: S_OK. The objects access level is changed.
* Failure: MAPI_E_INVALID_PARAMETER, if any parameter is invalid.
*/
static HRESULT WINAPI
IPropData_fnHrSetObjAccess(LPPROPDATA iface, ULONG ulAccess)
{
IPropDataImpl *This = impl_from_IPropData(iface);
TRACE("(%p,%x)\n", iface, ulAccess);
if (!iface || ulAccess < IPROP_READONLY || ulAccess > IPROP_READWRITE)
return MAPI_E_INVALID_PARAMETER;
IMAPIPROP_Lock(This);
This->ulObjAccess = ulAccess;
IMAPIPROP_Unlock(This);
return S_OK;
}
/* Internal - determine if an access value is bad */
static inline BOOL PROP_IsBadAccess(ULONG ulAccess)
{
switch (ulAccess)
{
case IPROP_READONLY|IPROP_CLEAN:
case IPROP_READONLY|IPROP_DIRTY:
case IPROP_READWRITE|IPROP_CLEAN:
case IPROP_READWRITE|IPROP_DIRTY:
return FALSE;
}
return TRUE;
}
/**************************************************************************
* IPropData_HrSetPropAccess {MAPI32}
*
* Set the access levels for a group of property values in an IPropData object.
*
* PARAMS
* iface [I] IPropData object to set access levels in.
* lpTags [I] List of property Id's to set access for.
* lpAccess [O] Access level for each property in lpTags.
*
* RETURNS
* Success: S_OK. The access level of each property value in lpTags that is
* present in iface is changed.
* Failure: MAPI_E_INVALID_PARAMETER, if any parameter is invalid.
*
* NOTES
* - Each access level in lpAccess must contain at least one of IPROP_READONLY
* or IPROP_READWRITE, but not both, and also IPROP_CLEAN or IPROP_DIRTY,
* but not both. No other bits should be set.
* - If a property Id in lpTags is not present in iface, it is ignored.
*/
static HRESULT WINAPI
IPropData_fnHrSetPropAccess(LPPROPDATA iface, LPSPropTagArray lpTags,
ULONG *lpAccess)
{
IPropDataImpl *This = impl_from_IPropData(iface);
ULONG i;
TRACE("(%p,%p,%p)\n", iface, lpTags, lpAccess);
if (!iface || !lpTags || !lpAccess)
return MAPI_E_INVALID_PARAMETER;
for (i = 0; i < lpTags->cValues; i++)
{
if (FBadPropTag(lpTags->aulPropTag[i]) || PROP_IsBadAccess(lpAccess[i]))
return MAPI_E_INVALID_PARAMETER;
}
IMAPIPROP_Lock(This);
for (i = 0; i < lpTags->cValues; i++)
{
LPIPropDataItem item = IMAPIPROP_GetValue(This, lpTags->aulPropTag[i]);
if (item)
item->ulAccess = lpAccess[i];
}
IMAPIPROP_Unlock(This);
return S_OK;
}
/**************************************************************************
* IPropData_HrGetPropAccess {MAPI32}
*
* Get the access levels for a group of property values in an IPropData object.
*
* PARAMS
* iface [I] IPropData object to get access levels from.
* lppTags [O] Destination for the list of property Id's in iface.
* lppAccess [O] Destination for access level for each property in lppTags.
*
* RETURNS
* Success: S_OK. lppTags and lppAccess contain the property Id's and the
* Access level of each property value in iface.
* Failure: MAPI_E_INVALID_PARAMETER, if any parameter is invalid, or
* MAPI_E_NOT_ENOUGH_MEMORY if memory allocation fails.
*
* NOTES
* - *lppTags and *lppAccess should be freed with MAPIFreeBuffer() by the caller.
*/
static HRESULT WINAPI
IPropData_fnHrGetPropAccess(LPPROPDATA iface, LPSPropTagArray *lppTags,
ULONG **lppAccess)
{
IPropDataImpl *This = impl_from_IPropData(iface);
LPVOID lpMem;
HRESULT hRet;
ULONG i;
TRACE("(%p,%p,%p) stub\n", iface, lppTags, lppAccess);
if (!iface || !lppTags || !lppAccess)
return MAPI_E_INVALID_PARAMETER;
*lppTags = NULL;
*lppAccess = NULL;
IMAPIPROP_Lock(This);
hRet = This->lpAlloc(CbNewSPropTagArray(This->ulNumValues), &lpMem);
if (SUCCEEDED(hRet))
{
*lppTags = lpMem;
hRet = This->lpAlloc(This->ulNumValues * sizeof(ULONG), &lpMem);
if (SUCCEEDED(hRet))
{
struct list *cursor;
*lppAccess = lpMem;
(*lppTags)->cValues = This->ulNumValues;
i = 0;
LIST_FOR_EACH(cursor, &This->values)
{
LPIPropDataItem item = LIST_ENTRY(cursor, IPropDataItem, entry);
(*lppTags)->aulPropTag[i] = item->value->ulPropTag;
(*lppAccess)[i] = item->ulAccess;
i++;
}
IMAPIPROP_Unlock(This);
return S_OK;
}
This->lpFree(*lppTags);
*lppTags = 0;
}
IMAPIPROP_Unlock(This);
return MAPI_E_NOT_ENOUGH_MEMORY;
}
/**************************************************************************
* IPropData_HrAddObjProps {MAPI32}
*
* Not documented at this time.
*
* RETURNS
* An HRESULT success/failure code.
*/
static HRESULT WINAPI
IPropData_fnHrAddObjProps(LPPROPDATA iface, LPSPropTagArray lpTags,
LPSPropProblemArray *lppProbs)
{
#if 0
ULONG i;
HRESULT hRet;
LPSPropValue lpValues;
#endif
FIXME("(%p,%p,%p) stub\n", iface, lpTags, lppProbs);
if (!iface || !lpTags)
return MAPI_E_INVALID_PARAMETER;
/* FIXME: Below is the obvious implementation, adding all the properties
* in lpTags to the object. However, it doesn't appear that this
* is what this function does.
*/
return S_OK;
#if 0
if (!lpTags->cValues)
return S_OK;
lpValues = HeapAlloc(GetProcessHeap(), HEAP_ZERO_MEMORY,
lpTags->cValues * sizeof(SPropValue));
if (!lpValues)
return MAPI_E_NOT_ENOUGH_MEMORY;
for (i = 0; i < lpTags->cValues; i++)
lpValues[i].ulPropTag = lpTags->aulPropTag[i];
hRet = IPropData_SetProps(iface, lpTags->cValues, lpValues, lppProbs);
HeapFree(GetProcessHeap(), 0, lpValues);
return hRet;
#endif
}
static const IPropDataVtbl IPropDataImpl_vtbl =
{
IPropData_fnQueryInterface,
IPropData_fnAddRef,
IPropData_fnRelease,
IPropData_fnGetLastError,
IPropData_fnSaveChanges,
IPropData_fnGetProps,
IPropData_fnGetPropList,
IPropData_fnOpenProperty,
IPropData_fnSetProps,
IPropData_fnDeleteProps,
IPropData_fnCopyTo,
IPropData_fnCopyProps,
IPropData_fnGetNamesFromIDs,
IPropData_fnGetIDsFromNames,
IPropData_fnHrSetObjAccess,
IPropData_fnHrSetPropAccess,
IPropData_fnHrGetPropAccess,
IPropData_fnHrAddObjProps
};
/*************************************************************************
* CreateIProp@24 (MAPI32.60)
*
* Create an IPropData object.
*
* PARAMS
* iid [I] GUID of the object to create. Use &IID_IMAPIPropData or NULL
* lpAlloc [I] Memory allocation function. Use MAPIAllocateBuffer()
* lpMore [I] Linked memory allocation function. Use MAPIAllocateMore()
* lpFree [I] Memory free function. Use MAPIFreeBuffer()
* lpReserved [I] Reserved, set to NULL
* lppPropData [O] Destination for created IPropData object
*
* RETURNS
* Success: S_OK. *lppPropData contains the newly created object.
* Failure: MAPI_E_INTERFACE_NOT_SUPPORTED, if iid is non-NULL and not supported,
* MAPI_E_INVALID_PARAMETER, if any parameter is invalid
*/
SCODE WINAPI CreateIProp(LPCIID iid, ALLOCATEBUFFER *lpAlloc,
ALLOCATEMORE *lpMore, FREEBUFFER *lpFree,
LPVOID lpReserved, LPPROPDATA *lppPropData)
{
IPropDataImpl *lpPropData;
SCODE scode;
TRACE("(%s,%p,%p,%p,%p,%p)\n", debugstr_guid(iid), lpAlloc, lpMore, lpFree,
lpReserved, lppPropData);
if (lppPropData)
*lppPropData = NULL;
if (iid && !IsEqualGUID(iid, &IID_IMAPIPropData))
return MAPI_E_INTERFACE_NOT_SUPPORTED;
if (!lpAlloc || !lpMore || !lpFree || lpReserved || !lppPropData)
return MAPI_E_INVALID_PARAMETER;
scode = lpAlloc(sizeof(IPropDataImpl), (LPVOID*)&lpPropData);
if (SUCCEEDED(scode))
{
lpPropData->IPropData_iface.lpVtbl = &IPropDataImpl_vtbl;
lpPropData->lRef = 1;
lpPropData->lpAlloc = lpAlloc;
lpPropData->lpMore = lpMore;
lpPropData->lpFree = lpFree;
lpPropData->ulObjAccess = IPROP_READWRITE;
lpPropData->ulNumValues = 0;
list_init(&lpPropData->values);
InitializeCriticalSection(&lpPropData->cs);
lpPropData->cs.DebugInfo->Spare[0] = (DWORD_PTR)(__FILE__ ": IPropDataImpl.cs");
*lppPropData = &lpPropData->IPropData_iface;
}
return scode;
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,492 |
HMRC repayments are to be withheld where avoidance is suspected – HMRC Brief and Rouse v CRC
Anti Avoidance HMRC VAT Income tax
The recent HMRC repayments brief 28/13 outlines a new policy on withholding repayment claims, particularly in suspected avoidance cases. In cases where it is HMRC's opinion that an avoidance scheme was used, it is their intention to withhold repayments. It is not clear on what legal basis this is to be achieved, especially bearing in mind the outcome of Cotter.
The Cotter decision was enforced in the recent case of Rouse v CRC, in which HMRC sought to enforce the payment of tax debts with immediate effect pending the resolution of enquiries into their claims for loss relief. In the Rouse case a repayment over an undisputed VAT repayment was also withheld from the taxpayer, and set against a disputed income tax liability, while waiting for the resolution of the enquiry with regards to his income tax.
Rouse v CRC
Background and facts:
Rouse had been a VAT-registered, self-employed contractor of plant and machinery since about 1993. He was also a director of a civil engineering company. For the years 2007/08 and 2008/09 he paid tax and National Insurance amounting to £1,049,061 and £998,892 respectively. In 2008/09 Mr Rouse incurred a loss of £1.5 million. He applied to 'carry back' the loss and have it offset against the tax due for 2007/08 under ITA 2007, s.132 as part of his 2007/08 tax return.
Mr Rouse had also submitted a VAT return in 2011 which stated that he was owed a repayment of over £600,000.
The Case:
HMRC rejected the claim for loss relief as they argued that the losses were made through avoidance schemes and opened an enquiry under TMA 1970 s.9A into the 2007/08 and 2008/09 tax returns, and refused to give credit for the loss in the meantime.
However, they also withheld the VAT repayment that they accepted was due to Mr Rouse to set against the income tax debt they claimed was due. The central issue in the case was whether HMRC were entitled to set-off a VAT repayment against the disputed income tax.
The Decision:
Upon Rouse's appeal, the Upper Tribunal were bound by the Court of Appeal decision in CRC v Cotter from 2012 whereby once HMRC had begun an enquiry into a return under TMA 1970, s.9A they could not also enquire under Sch 1A para 5.
Under TMA 1970, s.9A and Cotter there should be no debt on Mr Rouse's account against which the VAT credit due to him might be set off. The taxpayer's application for judicial review was therefore granted.
HMRC repayments may become increasingly hard to obtain based on their stated intentions. Fortunately, the cases of Rouse and Cotter prove that the courts and tribunals do continue to provide a mechanism to challenge HMRC decisions that exercise powers disproportionately. However, it is worrying that HMRC persist with such tactics, which they claim prevent the taxpayer from the right to appeal which should rightly be due. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,090 |
\section{Introduction}
The creation of interpersonal ties has been a fundamental question in the structural analysis of social networks. While strong ties emerge between individuals with similar social circles, forming a basis of trust and hence community structure, weak ties link two members who share few common contacts. The influential work of Granovetter reveals the vital roles of weak ties: It is weak ties that enable information transfer between communities and provide individuals positional advantage and hence influence and power \cite{Granovetter}.
\smallskip
Natural questions arise regarding the establishment of weak ties between communities: How to merge two departments in an organization into one? How does a company establish trade with an existing market? How to create a transport map from existing routes? We refer to such questions as {\em network building}. The basic setup involves two networks; the goal is to establish ties between them to achieve certain desirable properties in the combined network. A real-life example of network building is the inter-marriages between members of the Medici, the leading family of Renaissance Florence, and numerous other noble Florentine families, towards gaining power and control over the city \cite{Jackson-Medici}. Another example is by Paul Revere, a prominent Patriot during the American Revolution, who strategically created social ties to raise a militia \cite{UzziDunlap}.
\smallskip
The examples of the Medici and Paul Revere pose a more restricted scenario of network building: Here one of the two networks involved is only a single node, and the goal is to establish this node in the other network. We motivate this setup from two directions:
\begin{enumerate}
\item This setup amounts to the problem of {\em socialization}: the situation when a newcomer joins a network as an organizational member. A natural question for the newcomer is the following: How should I forge new relationships in order to take an advantageous position in the organization? As indicated in \cite{Morrison}, socialization is greatly influenced by the social relations formed by the newcomer with ``insiders'' of the network.
\item This setup also amounts to the problem of {\em network expansion}. For example, an airline expands its existing route map with a new destination, while trying to ensure a small number of legs between any cities.
\end{enumerate}
{\em Distance} refers to the length of a shortest path between two members in a network; this
is an important measure of the amount of influence one may exert to another in the network \cite{collaboration}. The {\em radius} of a network refers to the maximal distance from a central member to all others in a network. Hence when a newcomer joins an established network, it is in the interest of the newcomer to keep her distance to others bounded by the radius. The {\em diameter} of a network refers to the longest distance between any two members. It has long been argued from network science that small-world property -- the property that any two members of a network are linked by short paths -- improves network robustness and facilitates information flow \cite{robustness}.
Hence it is in the interest of the network to keep the diameter small as the network expands.
Furthermore, each relation requires time and effort to establish and maintain; thus one is interested in minimizing the number of new ties while building a network.
\paragraph*{\bf Contribution.} The novelty of this work is in proposing a formal, algorithmic study of organizational socialization. More specifically we investigate the following {\em network building problems}: Given a network $G$, add a new node $u$ to $G$ and create as few ties as possible for $u$ such that:
\begin{enumerate
\item[(1)] $u$ is in the center of the resulting network; o
\item[(2)] the diameter of the resulting network is not larger than a specific value
\end{enumerate}
Intuitively, (1) asks how a newcomer $u$ may optimally connect herself with members of $G$, so that she belongs to the center. We prove that this problem is in fact NP-complete (Theorem~\ref{thm:problemradius}). Nevertheless, we give several efficient algorithms for this problem; in particular, we demonstrate a ``simplification'' process that significantly improves performance.
Intuitively, (2) asks how a network may preserve or reduce its diameter by connecting with a new member $u$. We show that ``preserving the diameter'' is trivial for most real-life networks and give two algorithms for ``reducing the diameter''.
We experimentally test and compare the performance of all our algorithms. Quite surprisingly, the experiments demonstrate that a very small number of new edges is usually sufficient for each problem even when the graph becomes large.
\paragraph*{\bf Related works.} This work is predated by organizational behavioral studies \cite{socialization1,socialization2,Morrison}, which look at how social ties affect a newcomer's integration and assimilation to the organization. The authors in \cite{CrossThomas,UzziDunlap} argue {\em brokers} -- those who bridge and connect to diverse groups of individuals -- enable good network building; creating ties with and even becoming a broker oneself allows a person to gain private information, wide skill set and hence power. Network building theory has also been applied to various other contexts such as economics (strategic alliance of companies) \cite{Stuart1}, governance (forming inter-government contracts) \cite{federalism}, and politics (individuals' joining of political movements) \cite{Passy}. Compared to these works, the novelty here is in proposing a formal framework of network building, which employs techniques from complexity theory and algorithmics.
This work is also related to two forms of network formation: {\em dynamic models} and {\em agent-based models}, both aim to capture the natural emergence of social structures \cite{Jackson-Medici}. The former originates from random graphs, viewing the emergence of ties as a stochastic process which may or may not lead to an optimal structure \cite{entangle}. The latter comes from economics, treating a network as a multiagent system where utility-maximizing nodes establish ties in a competitive setting \cite{strategicNF,JacksonSurvey}. Our work differs from network formation as the focus here is on calculated strategies that achieve desirable goals in the combined network.
\section{Networks Building: The Problem Setup}
We view a {\em network} as an undirected unweighted connected graph $G = (V, E)$ where $V$ is a set of nodes and $E$ is a set of (undirected) edges on $V$. We denote an edge $\{u,v\}$ as $uv$. If $uv\in E$ then $v$ is said to be {\em adjacent} to $u$.
A {\em path} (of {\em length} $k$) is a sequence of nodes $u_0,u_1,\ldots,u_k$ where $u_iu_{i+1}\!\in\!E$ for any $0\!\leq\!i\!<\!k$. The {\em distance} between $u$ and $v$, denoted by $\mathsf{dist}(u,v)$, is the length of a shortest path from $u$ to $v$. The {\em eccentricity} of $u$ is the maximum distance from $u$ to any other node, i.e., $\mathsf{ecc}(u)=\max_{v\in V} \mathsf{dist}(u,v)$. The {\em diameter} of the network $G$ is $ \mathsf{diam}(G)=\max_{u\in V} \mathsf{ecc}(u)$. The {\em radius} $\mathsf{rad}(G)$ of $G$ is $\min_{u\in V} \mathsf{ecc}(u)$. The {\em center} of $G$ consists of those nodes that are closest to all other nodes; it is the set $C(G)\coloneqq \{u\in V\mid \mathsf{ecc}(u)=\mathsf{rad}(G)\}$.
\begin{definition
Let $G=(V,E)$ be a network and $u$ be a node not in $V$. For $S\subseteq V$, denote by $E_S$ the set of edges $\{uv\mid v\in S\}$. Define $G\oplus_S u$ as the graph $(V\cup \{u\}, E\cup E_S)$
\end{definition}
We require that $S\!\neq\!\varnothing$ and thus $G\oplus_S u$ is a network built by incorporating $u$ into $G$. By \cite{UzziDunlap}, for a newcomer $u$ to establish herself in $G$ it is essential to identify {\em information brokers} who connect to diverse parts of the network.
Following this intuition, we make the following definition
\begin{definition}
A set $S\subseteq V$ is a {\em broker set} of $G$ if $\mathsf{ecc}(u)=\mathsf{rad}(G\oplus_S u)$; namely, linking with $S$ enables $u$ to get in the center of the network.
\end{definition}
Formally, given a network $G=(V,E)$, the problem of {\em network building for $u$} means selecting a set $S\!\subseteq\!V$ so that the combined network $G\!\oplus_S\!u$ satisfies certain conditions. Moreover, the desired set $S$ should contain as few nodes as possible. We focus on the following two key problems:
\begin{enumerate
\item $\mathsf{BROKER}$: The set $S$ is a broker set.
\item $\mathsf{DIAM}_\Delta$: The diameter $\mathsf{diam}\!(\!G\!\oplus_S\! u\!)\!\leq\!\Delta$ for a given $\Delta\leq \mathsf{diam}(G)$.
\end{enumerate}
Note that for any network $G$, if $u$ is adjacent to all nodes in $G$, it will have eccentricity 1, i.e., in the network $G\oplus_V u$, $\mathsf{ecc}(u)\!=\!1\!=\!\mathsf{rad}(G\oplus_V u)$ and $\mathsf{diam}(G\oplus_V u)\!=\!2$. Hence a desired $S$ must exist for $\mathsf{BROKER}$ and $\mathsf{DIAM}_\Delta$ where $\Delta\geq 2$. In subsequent section we systematically investigate these two problems.
\section{How to Be in the Center? Complexity and Algorithms for $\mathsf{BROKER}$}
\subsection{Complexity}
We investigate the computational complexity of the decision problem $\mathsf{BROKER}(G,k)$, which is defined as follows:
\begin{description}
\item[INPUT] A network $G=(V,E)$, and an integer $k\geq 1$
\item[OUTPUT] Does $G$ have a broker set of size $k$?
\end{description}
The $\mathsf{BROKER}(G,k)$ problem is trivial if $G$ has radius 1, as then $V$ is the only broker set. When $\mathsf{rad}(G)>1$, we recall the following notion: A set of nodes $S\subseteq V$ is a {\em dominating set} if every node not in $S$ is adjacent to at least one member of $S$. The \emph{domination number} $\gamma(G)$ is the size of a smallest dominating set for $G$. The $\mathsf{DOM}(G,k)$ problem concerns testing whether $\gamma(G)\!\leq\!k$ for a given graph $G$ and input $k$; it is a classical NP-complete decision problem \cite{GareyJohnson}.
\begin{theorem}\label{thm:problemradius}
The $\mathsf{BROKER}(G,k)$ problem is NP-complete
\end{theorem}
\begin{proof} The $\mathsf{BROKER}(G,k)$ problem is clearly in NP. Therefore we only show NP-hardness.
We present a reduction from $\mathsf{DOM}(G,k)$ to $\mathsf{BROKER}(G,k)$. Note that when $\mathsf{rad}(G)\!=\!1$, $\gamma(G)\!=\!1$. Hence $\mathsf{DOM}(G,k)$ remains NP-complete if we assume $\mathsf{rad}(G)>1$. Given a graph $G=(V,E)$ where $\mathsf{rad}(G)>1$, we construct a graph $H$. The set of nodes in $H$ is $\{v_i\mid v\in V, 1\leq i\leq 3\}$. The edges of $H$ are as follows:
\begin{itemize
\item Add an edge $v_i v_{i+1}$ for every $v\in V$, $1\leq i<3
\item Add an edge $v_1w_1$ for every $v,w\in V
\item Add an edge $v_2 w_2$ for every edge $vw\in E
\end{itemize}
Namely, for each node $v\in V$ we create three nodes $v_1,v_2,v_3$ which form a path. We link the nodes in $\{v_1\mid v\in V\}$ to form a complete graph, and nodes in $\{v_2\mid v\in V\}$ to form a copy of $G$. Since $\mathsf{rad}(G)\geq 2$, for each node $v\in V$ there is $w\in V$ with $\mathsf{dist}(v,w)\geq 2$. Hence in $H$, $\mathsf{dist}(v_3,w_3)\geq 4$, and $\mathsf{dist}(v_2,w_3)\geq 3$. As the longest distance from any $v_1$ to any other node is $3$, we have $\mathsf{rad}(H)=3$.
Suppose $S$ is a dominating set of $G$. If we add all edges $uv$ where $v\in D=\{v_2\mid v\in S\}$, $\mathsf{ecc}(u)=3=\mathsf{rad}(H\oplus_D u)$. Hence $D$ is a broker set for $H$. Thus the size of a minimal broker set of $H$ is at most the size of a minimal dominating set of $G$.
Conversely, for any set $D$ of nodes in $H$, define the {\em projection} $p(D) = \{v\mid v_i\in D \text{ for some } 1\leq i\leq 3\}$.
Suppose $p(D)$ is not a dominating set of $G$. Then there is some $v\in V$ such that for all $w\in p(D)$, $\mathsf{dist}(v_2,w_2)\geq 2$. Thus if we add all edges in $\{ux\mid x\in D\}$, $\mathsf{dist}(u,v_3)\geq 4$. But then $\mathsf{ecc}(w_1)=3$ for any $w\in p(D)$. So $D$ is not a broker set. This shows that the size of a minimal dominating set of $G$ is at most the size of a minimal broker set.
The above argument implies that the size of a minimal broker set for $H$ coincides with the size of a minimal dominating set for $G$. This finishes the reduction and hence the proof. \qed
\end{proof}
\subsection{Efficient Algorithms}\label{subsec:radius_algorithms
Theorem~\ref{thm:problemradius} implies that computing optimal solution of $\mathsf{BROKER}$ is computationally hard. Nevertheless, we next present a number of efficient algorithms that take as input a network $G=(V,E)$ with radius $r$ and output a small broker set $S$ for $G$. A set $S\subseteq V$ is called {\em sub-radius dominating} if for all $v\in V$ not in $S$, there exists some $w\in S$ with $\mathsf{dist}(v,w)<r$. Our algorithms are based on the following fact, which is clear from definition:
\begin{fact}\label{fact:sub-radius dominating}Any sub-radius dominating set is also a broker set
\end{fact}
\subsubsection{(a) Three greedy algorithms}
We first present three greedy algorithms; each algorithm applies a heuristic that iteratively adds new nodes to the broker set $S$.
The starting configuration is $S= \varnothing$ and $U= V$. During its computation, the algorithm maintains a subgraph $F=(U,E\restriction U)$, which is induced by the set $U$ of all ``uncovered'' nodes, i.e., nodes that have distance $>(r-1)$ from any current nodes in $S$. It repeatedly performs the following operations until $U=\varnothing$, at which point it outputs $S$:
\begin{enumerate}
\item Select a node $v\in U$ based on the corresponding heuristic and add $v$ to $S$.
\item Compute all nodes at distance at most $(r-1)$ from $v$. Remove these nodes and all attached edges from $F$.
\end{enumerate}
\paragraph*{\bf Algorithm 1: $\mathsf{Max}$ (Max-Degree).} The first heuristic is based on the intuition that one should connect to the person with the highest number of social ties; at each iteration, it adds to $S$ a node with maximum degree in the graph $F$.
\paragraph*{\bf Algorithm 2: $\mathsf{Btw}$ (Betweenness).} The second heuristic is based on {\em betweenness}, an important centrality measure in networks \cite{betweeness}. More precisely, the {\em betweenness} of a node $v$ is the number of shortest paths from all nodes to all others that pass through $v$. Hence high betweenness of $v$ implies, in some sense, that $v$ is more likely to have short distance with others. This heuristic works in the same manner as $\mathsf{Max}$ but picks nodes with maximum betweenness in $F$.
\paragraph*{\bf Algorithm 3: $\mathsf{ML}$ (Min-Leaf).} The third heuristic is based on the following intuition: A node is called a {\em leaf} if it has minimum degree in the network; leaves correspond to least connected members in the network, and may become outliers once nodes with higher degrees are removed from the network. Hence this heuristic gives first priority to leaves. Namely, at each iteration, the heuristic adds to $S$ a node that has distance at most $r-1$ from $v$. More precisely, the heuristic first picks a leaf $v$ in $F$, then applies a sub-procedure to find the next node $w$ to be added to $S$. The sub-procedure determines a path $v=u_1,u_2,\ldots$ in $F$ iteratively as follows:
\begin{enumerate
\item Suppose $u_i$ is picked. If $i=r$ or $u_i$ has no adjacent node in $F$, set $u_i$ as $w$ and terminate the process
\item Otherwise select a $u_{i+1}$ (which is different from $u_{i-1}$) among adjacent nodes of $u_i$ with maximum degree.
\end{enumerate
After the process above terminates, the algorithm adds $w$ to $S$. Note that the distance between $w$ and $v$ is at most $r-1$.
We mention that Algorithms 1,3 have been applied in \cite{k-domination} to {\em regular graphs}, i.e., graphs where all nodes have the same degree. In particular, $\mathsf{ML}$ has been shown to produce small $k$-dominating sets for given $k$ in the average case for regular graphs.
\subsubsection{(b) Simplified greedy algorithms}
One significant shortcoming of Algorithms 1--3 is that, by deleting nodes from the network $G$, the network may become disconnected, and nodes that could have been connected via short paths are no longer reachable from each other. This process may produce {\em isolated} nodes in $F$, i.e., nodes having degree 0, which are subsequently all added to the output set $S$. Moreover, maintaining the graph $F$ at each iteration also makes implementations more complex. Therefore we next propose {\em simplified} versions of Algorithms 1--3.
\paragraph*{\bf Algorithms 4 $\mathsf{S}$-$\mathsf{Max}$, 5 $\mathsf{S}$-$\mathsf{Btw}$, 6 $\mathsf{S}$-$\mathsf{ML}$.} The simplified algorithms act in a similar way as their ``non-simplified'' counterparts; the difference is that here the heuristic works over the original network $G$ as opposed to the updated network $F$. Hence the graph $F$ is no longer computed. Instead we only need to maintain a set $U$ of ``uncovered'' nodes. The simplified algorithms have the following general structure: \ Start from $S= \varnothing$ and $U= V$, and repeatedly perform the following until $U=\varnothing$, at which point output $S$:
\begin{enumerate
\item Select a node $v$ from $U$ based on the corresponding heuristic and add $v$ to $S$
\item Compute all nodes with distance $<\mathsf{rad}(G)$ from $v$, and remove any of these node from $U$.
\end{enumerate
We stress that here the same heuristics as described above in Algorithms 1--3 are applied, except that we replace any mention of ``$F$'' in the description with ``$U$'', while all notions of degrees, distances, and betweenness are calculated based on the original network $G$.
As an example, in Fig.~\ref{fig:Max} we run $\mathsf{Max}$ and $\mathsf{S}$-$\mathsf{Max}$ on the same network $G$, which contains 30 nodes. The figures show the result of both algorithms, and in particular, how $\mathsf{S}$-$\mathsf{Max}$ outputs a smaller sub-radius dominating set. We further verify via experiments below that the simplified algorithms lead to much smaller output $S$ in almost all cases.
\begin{figure}[!] \centerin
\includegraphics[width=0.5\textwidth]{example.png}
\caption{ The network $G$ contains 30 nodes and has radius $\mathsf{rad}(G)=4$. The $\mathsf{Max}$ algorithm: The algorithm first puts node 3 (shown in green) into $S$. Then removes all nodes (and attached edges that are at distance three from the node 3; these nodes are considered ``covered'' by 3. In the remaining graph, there are three isolated nodes 8,14,26, as well as a line of length 2. The algorithm then puts the node 18 into $S$ which ``covers'' 27 and 13. Thus the output set is $S=\{3,18,8,14,26\}$. The $\mathsf{S}$-$\mathsf{Max}$ algorithm: The algorithm first puts 3 into the set $S$, but does not remove the covered nodes. It simply construct a set containing all ``uncovered'' nodes, namely, $\{27,18,13,14,8,26\}$. The algorithm then selects the node 13 which has max degree from these nodes, and puts into $S$. It then turns out that all nodes are covered. Therefore the output set is $S=\{3,13\}$. Thus $\mathsf{S}$-$\mathsf{Max}$ is superior in this example.}\label{fig:Max}
\end{figure}
\subsubsection{(c) Center-based algorithms}
The 6 algorithms presented above can all be applied to find $k$-dominating set for arbitrary $k\geq 1$. Since our focus is in finding sub-radius dominating set to answer the $\mathsf{BROKER}$ problem, we describe two algorithms that are specifically designed for this task. When building network for a newcomer, it is natural to consider nodes that are already in the center of the network $G$. Hence our two algorithms are based on utilizing the center of $G$.
\paragraph*{\bf Algorithm 7 $\mathsf{Center}$.} The algorithm finds a center $v$ in $G$ with minimum degree, then output all nodes that are adjacent to $v$. Since $v$ belongs to the center, for all $w\in V$, we have $\mathsf{dist}(v,w)\leq \mathsf{rad}(G)$ and thus there is $v'$ adjacent to $v$ such that $\mathsf{dist}(w,v')=\mathsf{dist}(w,v)-1<\mathsf{rad}(G)$. Hence the algorithm returns a sub-radius dominating set. Despite its apparent simplicity, $\mathsf{Center}$ returns surprisingly good results in many cases, as shown in the experiments below.
\paragraph*{\bf Algorithm 8 $\mathsf{Imp}$-$\mathsf{Center}$.} We present a modified version of $\mathsf{Center}$, which we call $\mathsf{Imp}$-$\mathsf{Center}$.
The algorithm first picks a center with minimum degree, and then orders all its neighbors in decreasing degree. It adds the first neighbor to $S$ and remove all nodes $\leq (r-1)$-steps from it. This may disconnect the graph into a few connected components. Take the largest component $C$. If $C$ has a smaller radius than $r$, we add the center of this component to $S$; otherwise we add the next neighbor to $S$. We then remove from $F$ all nodes at distance $\leq (r-1)$ from the newly added node. This procedure is repeated until $F$ is empty. See Procedure~\ref{alg:Center_improved}. Fig.~\ref{fig:Center} shows an example where $\mathsf{Imp}$-$\mathsf{Center}$ out-performs $\mathsf{Center}$.
\begin{algorithm}[!htb]
\floatname{algorithm}{Procedure}
\caption{ $\mathsf{Imp}$-$\mathsf{Center}$: Given $G=(V,E)$ (with radius $r$)}
\label{alg:Center_improved}
\begin{algorithmic}
\State Pick a center node $v$ in $G$ with minimum degree $d$
\State Sort all adjacent nodes of $v$ to a list $u_1,u_2,\ldots,u_d$ in decreasing order of degrees
\State Set $S\leftarrow \varnothing$ and $i\leftarrow 1$
\While{$U\neq \varnothing$}
\State Set $C$ as the largest connected component in $F$
\If{$\mathsf{rad}(C)<\mathsf{rad}(G)-1$}
\State Pick a center node $w$ of $C$. Set $S\leftarrow S\cup \{w\}$
\State Set $U\leftarrow U\setminus \{w'\in U\mid \mathsf{dist}(w,w')<r\}
\Else
\State Set $S\leftarrow S\cup \{u_i\}$
\State Set $U\leftarrow U\setminus \{w'\in U\mid \mathsf{dist}(u_i,w')<r\}$
\State Set $i\leftarrow i+1$
\EndIf
\State Set $F$ as the subgraph induced by the current $U$
\EndWhile
\State\Return $S$
\end{algorithmic}
\end{algorithm}
\begin{figure}[!htb
\centering
\includegraphics[width=0.7\textwidth]{sage6}
\caption{ The graph $G$ has radius $\mathsf{rad}(G)=3$. The yellow node 0 is a center with min degree 4. Thus $\mathsf{Center}$ outputs 4 nodes $\{1,4,18,29\}$. The dark green node 29 adjacent to 0 has max degree; the red nodes are ``uncovered'' by 29. Thus $\mathsf{Imp}$-$\mathsf{Center}$ outputs the 3 blue circled nodes $\{12,25,29\}$. }\label{fig:Center}
\end{figure}
Finally, we note that all of Algorithms 1--8 output a sub-radius dominating set $S$ for the network $G$. Thus the following theorem is a direct implication from Fact~\ref{fact:sub-radius dominating}.
\begin{theorem
All of Algorithms 1--8 output a brocker set for the network $G$.
\end{theorem}
\subsection{Experiments for $\mathsf{BROKER}$}
We implemented the algorithms using Sage \cite{Sage}.
We apply two models of random graphs: The first (BA) is Barabasi-Albert's preferential attachment model which generates scale-free graphs whose degree distribution of nodes follows a power law; this is an essential property of numerous real-world networks \cite{BA}. The second (NWS) is Newman-Watts-Strogatz's small-world network \cite{NewmanWattsStrogatz}, which produces graphs with small average path lengths and high clustering coefficient.
For each algorithm we are interested in two indicators of its performance: 1) {\em Output size}: The average size of the output broker set (for a specific class of random graphs). 2) {\em Optimality rate}: The probability that the algorithm gives optimal broker set for a random graph. To compute this we need to first compute the size of an optimal broker set (by brute force) and count the number of times the algorithm produces optimal solution for the generated graphs.
\paragraph*{\bf Experiment 1: Output sizes.} We generate $300$ graphs whose numbers of nodes vary between $100$ and $1000$ using each random graph model. We compute averaged output sizes of generated graphs by their number of nodes $n$ and radius $r$. The results are shown in Fig.~\ref{fig:improvedRes}. From the result we see: a) The simplified algorithms produce significantly smaller broker sets compared to their unsimplified counterparts. This shows superiority of the simplified algorithms.
b) BA graphs in general allow smaller output set than NWS graphs. This may be due to the scale-free property which results in high skewness of the degree distribution.
\begin{figure}[!htb
\centering
\includegraphics[width=\textwidth]{comaring_noMin}
\caption{ Comparing results: average performance of the $\mathsf{Max}$, $\mathsf{Btw}$, $\mathsf{ML}$, algorithms versus their simplified versions on randomly generated graphs (BA graphs on the left; NWS on the right)}\label{fig:improvedRes}
\end{figure}
\paragraph{\bf Experiment 2: Optimality rates.} For the second goal, we compute the optimality rates of algorithms when applied to random graphs, which are shown in Fig.~\ref{fig:res}. For BA graphs, the simplified algorithm $\mathsf{S}$-$\mathsf{ML}$ has significantly higher optimality rate ($\geq 85\%$) than other algorithms. On the contrary, its unsimplified counterpart $\mathsf{ML}$ has the worst optimality rate. This is somewhat contrary to Duckworth and Mans's work showing $\mathsf{ML}$ gives very small solution set for regular graphs \cite{k-domination}.
For NWS graphs, several algorithms have almost equal optimality rate. The three best algorithms are $\mathsf{S}$-$\mathsf{Max}$, $\mathsf{S}$-$\mathsf{Btw}$ and $\mathsf{S}$-$\mathsf{ML}$ which has varying performance for graphs with different sizes (See Fig.~\ref{fig:optimality_nodes}).
\begin{figure}[!htb]\centering
\includegraphics[width=\textwidth]{percentageRes_noMin}
\caption{ Optimality rates for different types of random graphs}\label{fig:res}
\end{figure}
\begin{figure}[!htb]\centering
\includegraphics[width=\textwidth]{nodes_combined_noMin}
\caption{ Optimality rates when graphs are classified by sizes}\label{fig:optimality_nodes}
\end{figure}
\paragraph*{\bf Experiment 3: Real-world datasets.}
We test the algorithms on several real-world datasets: The $\mathsf{Facebook}$ dataset, collected from survey participants of Facebook App, consists of friendship relation on Facebook \cite{facebook}. $\mathsf{Enron}$ is an email network of the company made public by the FERC \cite{enron1}. Nodes of the network are email addresses and if an address $i$ sent at least one email to address $j$, the graph contains an undirected edge from $i$ to $j$.
$\mathsf{Col} 1$ and $\mathsf{Col} 2$ are collaboration networks that represent scientific collaborations between authors papers submitted to General Relativity and Quantum Cosmology category ($\mathsf{Col} 1$), and to High Energy Physics Theory category ($\mathsf{Col} 2$) \cite{collaboration}.
\begin{table}[!htb]
\centering
\begin{tabular}{| l | l | l| l | l |}
\hline
& Facebook & Enron & Col1 & Col2\\
\hline
Number of nodes & 4,039 & 33,969 & 4,158& 8,638 \\
\hline
Number of edges & 88,234& 180,811 & 13,422& 24,806\\
\hline
Largest connected subgraph & 4,039 & 33,696 & 4,158 & 8,638 \\
\hline
Diameter & 8 & 13 & 17 & 18\\
\hline
Radius & 4 & 7 & 9 & 10\\
\hline
\end{tabular}\caption{ Network properties} \label{table:datasets}
\end{table}
Results on the datasets are shown in Fig.~\ref{fig:datasets_res}. $\mathsf{Btw}$ and $\mathsf{S}$-$\mathsf{Btw}$ algorithms become too inefficient as it requires computing shortest paths between all pairs in each iteration. Moreover, $\mathsf{S}$-$\mathsf{Max}$ also did not terminate within reasonable time for the $\mathsf{Enron}$ dataset. Even though the datasets have many nodes, the output sizes are in fact very small (within 10). For instance, the smallest output sets of the $\mathsf{Enron}$, $\mathsf{Col} 1$ and $\mathsf{Col} 2$ contain just two nodes. In some sense, it means that to become in the center even in a large social network, it is often enough to establish only very few connections.
\begin{figure}[!htb]
\centering
\includegraphics[width=.8\textwidth]{datasetRes_noMin}
\caption{ The number of new ties for the four real-world networks}\label{fig:datasets_res}
\end{figure}
Among all algorithm $\mathsf{Imp}$-$\mathsf{Center}$ has the best performance, producing the smallest output set for all networks. Moreover, for $\mathsf{Enron}$, $\mathsf{Col} 1$ and $\mathsf{Col} 2$, $\mathsf{Imp}$-$\mathsf{Center}$ returns the optimal broker set with cardinality $2$. A rather surprising fact is, despite straightforward seemingly-naive logic, $\mathsf{Center}$ also produces small outputs in three networks. This reflects the fact that in order to become central it is often a good strategy to create ties with the friends of a central person.
\section{How to Preserve or Improve the Diameter? Complexity and Algorithms for $\mathsf{DIAM}_\Delta$}\label{sec:diameter}
Let $G=(V,E)$ be a network and $u\notin V$. The $\mathsf{DIAM}_\Delta$ problem asks for a set $S\subseteq V$ such that the network $G\oplus_S u$ has diameter $\leq\Delta$; we refer to any such $S$ as {\em $\Delta$-enabling}.
\subsection{Preserving the diameter}
We first look at a special case when $\Delta=\mathsf{diam}(G)$, which has a natural motivation: How can an airline expand its existing route map with an additional destination while ensuring the maximum number of hops between any two destinations is not increased? We are interested in creating as few new connections as possible to reach this goal. Let $\delta(G)$ denote the size of the smallest $\mathsf{diam}(G)$-enabling set for $G$. We say a graph is {\em diametrically uniform} if all nodes have the same eccentricity.
\begin{theorem}\label{thm:problemdiameter}
\begin{enumerate
\item[(a)]If $G$ is not diametrically uniform,$\delta(G)\!=\!1$
\item[(b)]If $G$ is complete, then $\delta(G)=|V|$
\item[(c)]If $G$ is diametrically uniform and incomplete, then $1<\delta(G)\leq d$ where $d$ is the minimum degree of any node in $G$, and the upper bound $d$ is sharp
\end{enumerate}
\end{theorem}
\begin{proof}
For {\bf (a)}, suppose $G$ is not diametrically uniform. Take any $v$ where $\mathsf{ecc}(v)<\mathsf{diam}(G)$. Then in the expanded network $G\oplus_{\{v\}} u$, we have $\mathsf{ecc}(u)=\mathsf{ecc}(v)+1\leq \mathsf{diam}(G)$. {\bf (b)} is clear.
For {\bf (c)} Suppose $G$ is diametrically uniform and incomplete. For the lower bound, suppose $\gamma_{\mathsf{diam}(G)-1}(G)=1$. Then there is some $v\in V$ with the following property: In the network $G\oplus_{\{v\}} u$ we have $\mathsf{ecc}(u)\leq \mathsf{diam}(G)$, which means that $\mathsf{ecc}(v)<\mathsf{diam}(G)$. This contradicts the fact that $G$ is diametrically uniform. For the upper bound, take a node $v\in V$ with the minimum degree $d$. Let $N$ be the set of nodes adjacent to $v$. From any node $w\neq v$, there is a shortest path of length $\leq \mathsf{diam}(G)$ to $v$. This path contains a node in $N$. Hence $w$ is at distance $\leq \mathsf{diam}(G)-1$ from some node in $N$. Furthermore as $G$ is not complete, $\mathsf{diam}(G)\geq 2$ and $v$ is at distance $1\leq \mathsf{diam}(G)-1$ from nodes in $N$. \qed
\end{proof}
\paragraph*{Remark} We point out that in case (c) calculating the exact value of $\delta(G)$ is a hard: In \cite{hardness_diameter}, its parametrized complexity is shown to be complete for $\mathsf{W}[2]$, second level of the $\mathsf{W}$-hierarchy. Hence $\mathsf{DIAM}_\Delta$ is unlikely to be in $\mathsf{P}$. On the other hand, we argue that real-life networks are rarely diametrically uniform. Hence by Thm.~\ref{thm:problemdiameter}(a), the smallest number of new connections needed to preserve the diameter is 1.
\subsection{Reducing the diameter}
We now explore the question $\mathsf{DIAM}_\Delta$ where $2\leq \Delta<\mathsf{diam}(G)$; this refers to the goal of placing a new member in the network and creating ties to allow a closer distance between all pairs of members. We suggest two heuristics to solve this problem.
\paragraph*{\bf Algorithm 9 $\mathsf{Periphery}$.} The {\em periphery} $P(G)$ of $G$ consists of all nodes $v$ with $\mathsf{ecc}(v)=\mathsf{diam}(G)$. Suppose $\mathsf{diam}(G)>2$. Then the combined network $G\oplus_{P(G)} u$ has diameter smaller than $\mathsf{diam}(G)$. Hence we apply the following heuristic: Two nodes $v,w$ in $G$ are said to form a {\em peripheral pair} if $\mathsf{dist}(v,w)=\mathsf{diam}(G)$. The algorithm first adds the new node $u$ to $G$ and repeats the following procedure until the current graph has diameter $\leq \Delta$:\\
1) Randomly pick a peripheral pair $v,w$ in the current graph\\
2) Adds the edges $uv,uw$ if they have not been added already\\
3) Compute the diameter of the updated graph
\noindent Note that once $v,w$ are chosen as a peripheral pair and the corresponding edges $uv,uw$ added, $v$ and $w$ will have distance 2 and they will not be chosen as a peripheral pair again. Hence the algorithm eventually terminates and produces a graph with diameter at most $\Delta$.
\paragraph*{\bf Algorithm 10 $\mathsf{CP}$ (Center-Periphery).} This algorithm applies a similar heuristic as $\mathsf{Periphery}$, but instead of picking peripheral pairs at each iteration, it first picks a node $v$ in the center and adds the edge $uv$; it then repeats the following procedure until the current graph has diameter $\leq \Delta$:\\
1) Randomly pick a node $w$ in the periphery of the current graph\\
2) Add the edge $uw$ if it has not been added already\\
3) Compute the diameter of the updated graph
\noindent Suppose at one iteration the algorithm picks $w$ in the periphery. Then after this iteration the eccentricity of $w$ is at most $r+2$ where $r$ is the radius of the graph.
\subsection{Experiments for $\mathsf{DIAM}_\Delta$}
We implement and test the performance of Algorithms 9,10 for the problem $\mathsf{DIAM}_\Delta$
The performance of these algorithms are measured by the number of new ties created.
\paragraph*{\bf Experiment 4: Random graphs.}
We apply the two models of random graphs, BA and NWS, as described above. We generated $350$ graphs and considered the case when $\Delta = d(G) - 1$, i.e. the aim was to improve the diameter by one.
For both types of random graphs (fixing size and radius), the average number of new ties are shown in Fig.~\ref{fig:Barabasi_improve}.
The experiments show that $\mathsf{Periphery}$ performs better when the radius of the graph is close to the diameter (when radius is $>2/3$ of diameter), whilst $\mathsf{CP}$ is slightly better when the radius is significantly smaller than the diameter.
\begin{figure}[!htb]
\centering
\includegraphics[width=\textwidth]{improve_combined}
\caption{ Comparing two methods for improving diameter applied to BA (left) and NWS (right) graphs}\label{fig:Barabasi_improve}
\end{figure}
\paragraph*{\bf Experiment 5: Real-World Datasets.} We run both $\mathsf{Periphery}$ and $\mathsf{CP}$ on the networks $\mathsf{Col} 1$ and $\mathsf{Col} 2$ introduced above, setting $\Delta =\mathsf{diam}(G)-i$ for $1\leq i\leq 4$. The numbers of new edges obtained by $\mathsf{Periphery}$ and $\mathsf{CP}$ are shown in Figure~\ref{fig:collaboration_improve}; naturally for increasing $i$, more ties need to be created.
We point out that, despite the large total number of nodes, one needs less than $19$ new edges to improve the diameter even by four.
This reveals an interesting phenomenon: While a collaboration network may be large, a few more collaborations are sufficient to reduce the diameter of the network.
On the $\mathsf{Facebook}$ dataset, $\mathsf{Periphery}$ is significantly better than $\mathsf{CP}$: To reduce the diameter of this network from $8$ to $7$, $\mathsf{Periphery}$ requires 2 edges while $\mathsf{CP}$ requires $47$. When one wants to reach the diameter $6$, the numbers of new edges increase to 6 for $\mathsf{Periphery}$ and 208 for $\mathsf{CP}$.
\begin{figure}[!htb]
\centering
\includegraphics[width=0.8\textwidth]{collaboration_improve}
\caption{ Applying algorithms for improving diameter to Collaboration 1 and Collaboration 2 datasets}\label{fig:collaboration_improve}
\end{figure}
\section{Conclusion and Outlook}
This work studies how ties are built between a newcomer and an established network to reach certain structural properties. Despite achieving optimality is often computationally hard, there are efficient heuristics that reach the desired goals using few new edges. We also observe that the number of new links required to achieve the specified properties remain small even for large networks.
This work amounts to an effort towards an algorithmic study of network building. Along this effort, natural questions have yet to be explored include: (1) Investigating the creation of ties between two arbitrary networks, namely, how ties are created between two established networks to maintain or reduce diameter. (2) When building networks in an organizational context (such as merging two departments in a company), one normally needs not only to take into account the informal social relations, but also formal ties such as the reporting relations, which are typically directed edges \cite{LiuMoskvina}. We plan to investigate network building in an organizational management perspective by incorporating both types of ties.
\bibliographystyle{splncs03}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,209 |
Cosmopolitans Football Club is a Tanzanian football club who plays in the Tanzanian Second Division League. The club is based in Dar es Salaam.
In 1967 the team has won the Tanzanian Premier League.
Stadium
Currently the team plays at the 5000 capacity Chamazi Stadium.
Honours
Tanzanian Premier League
Champions (1): 1967
Performance in CAF competitions
African Cup of Champions Clubs: 1 appearance
1968 – First Round
References
External links
Tanzania – List of Foundation Dates – rsssf.com
Team profile – calciozz.it
Football clubs in Tanzania
Sport in Dar es Salaam | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,584 |
\section{Introduction}
\label{sec:intro}
With the development of deep learning, deep neural networks gradually demonstrate superior performance on the recognition of pre-defined classes with large amount of training data~\cite{simonyan2014very,he2016deep}.
However, the model's generalization capability on the downstream novel classes is much less explored and still needs to be improved~\cite{kornblith2021better,guobroader}.
To deal with this problem, the few-shot class-incremental learning (FSCIL) task~\cite{hou2019learning,rebuffi2017icarl,castro2018end,tao2020few,zhang2021few,zhou2022forward} comes into sight. FSCIL first (pre-)trains a model on a set of pre-defined classes (base classes), and then generalizes the model to the incremental novel classes with only few training samples, simulating human's ability of continually learning novel concepts with only few examples, and emphasizing both the performance on the pre-defined base classes and the generalization on the downstream novel classes.
However, a dilemma is recently revealed~\cite{kornblith2021better,chen2019transferability,cui2022discriminability,cui2020towards} that better loss functions, which lead to higher performance on the pre-training data, could lead to worse generalization on the downstream tasks.
As introduced by \cite{liu2020negative} and depicted in Fig.~\ref{fig:motivation}, similar phenomenon also exists in the FSCIL task that a positive classification margin~\cite{liu2020negative,wang2018cosface,deng2019arcface,sun2020circle} applied to the classification of the base-class (pre-)training could lead to higher base-class performance but lower novel-class performance, while a negative margin could result in lower base-class performance but increase the novel-class performance.
Although this dilemma widely exists in the tasks involving novel-class generalization such as few-shot learning (FSL) and FSCIL, only few works~\cite{liu2020negative} tried to explore its cause, and can hardly be used to handle it. Due to space limitation, we will provide extended related works in the appendix.
\begin{figure}[!t]
\begin{tabular*}{\textwidth}{ccccc}
& No Margin & Positive Margin & Negative Margin & Ours \\[0.05cm]
\rotatebox{90}{Base Class}
& \fbox{\includegraphics[width=0.2\linewidth,height=0.12\linewidth]{base_m=0.0_remove.jpg}}\hspace{-0.15cm}
& \fbox{\includegraphics[width=0.2\linewidth,height=0.12\linewidth]{base_m=0.3_remove.jpg}}\hspace{-0.15cm}
& \fbox{\includegraphics[width=0.2\linewidth,height=0.12\linewidth]{base_m=-0.3_remove.jpg}}\hspace{-0.15cm}
& \fbox{\includegraphics[width=0.2\linewidth,height=0.12\linewidth]{base_ours_remove.jpg}}
\\[0.1cm]
\rotatebox{90}{Novel Class}
& \fbox{\includegraphics[width=0.2\linewidth,height=0.12\linewidth]{novel_m=0.0_remove.jpg}}\hspace{-0.15cm}
& \fbox{\includegraphics[width=0.2\linewidth,height=0.12\linewidth]{novel_m=0.3_remove.jpg}}\hspace{-0.15cm}
& \fbox{\includegraphics[width=0.2\linewidth,height=0.12\linewidth]{novel_m=-0.3_remove.jpg}}\hspace{-0.15cm}
& \fbox{\includegraphics[width=0.2\linewidth,height=0.12\linewidth]{novel_ours_remove.jpg}}
\end{tabular*} \vspace{-0.2cm}
\caption{A dilemma exists between base-class performance and downstream novel-class generalization. By applying positive margins, base-class features are better separated which indicates better base-class performance, but the novel-class features are confused which indicates lower novel-class generalization. In contrast, by applying negative margins, base-class features are confused but the novel-class features are better separated. In this paper, we study the cause of such dilemma for the few-shot class-incremental learning task, and propose a method to mitigate such dilemma to better separate both base and novel classes.}\vspace{-0.4cm}
\label{fig:motivation}
\end{figure}
In this paper, we study the cause of the dilemma in the margin-based classification for the FSCIL problem from the the aspect of pattern learning. We find this dilemma can be understood as a class-level overfitting (CO) problem, which can be interpreted by the fitness of the learned patterns to each base class. The fitness determines how much the learned patterns are specific to some base classes or shared among classes, making the learned patterns either discriminative (tend to overfit base classes) or transferable (tend to underfit base classes) and causes the dilemma. Based on the interpretation, we discover the cause of the dilemma lies in the easily-satisfied constraint of learning shared or class-specific patterns. Therefore, we further design a novel FSCIL method to mitigate the dilemma of CO by providing the pattern learning process with extra constraint from margin-based patterns themselves, improving performance on both base and novel classes as shown in Fig.~\ref{fig:motivation}, and achieving state-of-the-art performance in terms of the all-class accuracy. Our contributions are:
\vspace{-0.2cm}
\begin{itemize}[$\bullet$]
\item We interpret the dilemma of the margin-based classification as a class-level overfitting problem from the aspect of pattern learning.
\item We find the cause of the class-level overfitting problem lies in the easily-satisfied constraint of learning shared or class-specific patterns.
\item We propose a novel FSCIL method to mitigate the class-level overfitting problem based on the interpretation and analysis of the cause.
\item Extensive experiments on three public datasets verify the rationale of the model design, and show that we can achieve state-of-the-art performance.
\end{itemize}
\vspace{-0.3cm}
\section{Interpreting the Dilemma of Few-Shot Class-Incremental Learning}
\label{sec:task_and_baseline}
\vspace{-0.2cm}
In this section, we first describe the Few-Shot Class-Incremental Learning (FSCIL) task and the baseline model, and then conduct experiments to analyze the dilemma.
\vspace{-0.2cm}
\subsection{Task and Baseline Description}
\vspace{-0.2cm}
The FSCIL task aims to incrementally recognize novel classes with only few training samples.
Basically, the model is first (pre-)trained on a set of base classes with sufficient training samples (a.k.a. base session), then confronted with novel classes with limited training samples (a.k.a. incremental session), and finally required to recognize test samples from all encountered classes.
Specifically, given the base session dataset $D^0=\{(x_i, y_i)\}_{i=1}^{n_0}$ with the label space $Y_0$, the model is trained to recognize all $|Y_0|$ classes from $Y_0$ by minimizing the loss
\vspace{-0.1cm}
\begin{equation}
\sum_{(x_i, y_i) \in D^0} L(\phi(x_i), y_i),
\label{eq:cls}
\end{equation}
where $L(\cdot,\cdot)$ is typically a cross-entropy loss, $\phi(\cdot)$ is the predictor which is composed of a backbone network $f(\cdot)$ for feature extraction and a linear classifier, represented as $\phi(x) = W^\top f(x)$ where $\phi(x) \in R^{N_0 \times 1}$, $W \in R^{d \times N_0}$ and $f(x) \in R^{d \times 1}$. Typically, $f(x)$ and the $W$ are $L_2$ normalized~\cite{zhang2021few}.
When the $k$th incremental session comes, the model needs to learn from its training data $D^k=\{(x_i, y_i)\}_{i=1}^{n_k}$. The weight of the classifier will be extended to represent the novel label space $Y_k$ imported by this session, represented as $W = \{w^0_1, w^0_2, ..., w^0_{|Y_0|}\} \cup ... \cup \{w^k_1, ..., w^k_{|Y_k|}\}$ where $w^k_j$ denotes the weight of the classifier corresponding to the $j$th class of the $k$th session.
A strong baseline~\cite{zhang2021few} is to freeze model's parameters to avoid the catastrophic forgetting brought by the finetuning on novel-classes. For the incremental sessions (i.e., $k$ > 0), the average of the features extracted from the training data will be used as the classifier's weight~\cite{zhang2021few} (a.k.a. prototype) as $w^k_j = \frac{1}{n_k^j} \sum_{i=1}^{n_k^j} f(x_i)$, where $n_k^j$ denotes the number of training samples in the class $j$ for the session $k$. As this baseline focuses on the base-class training, in this paper, the term \textit{training}, if not otherwise stated, refers to the base-class training.
Finally, the performance of the $k$th session will be obtained by classifying the test samples from all $\sum_{i=0}^{k} |Y_i|$ encountered classes.
\vspace{-0.3cm}
\subsection{Margin-Based Classification}
\vspace{-0.2cm}
A well known modification to base-class training loss (Eq.~\ref{eq:cls}) is to integrate a margin~\cite{liu2020negative,deng2019arcface,wang2018cosface} as
\vspace{-0.1cm}
\begin{equation}
L(x_i, y_i) = -log \frac{e^{\tau (w_{y_i} f(x_i) - m)}}{e^{\tau (w_{y_i} f(x_i) - m)} + \sum_{j \neq y_i} e^{\tau w_j f(x_i)}},
\label{eq:margin}
\end{equation}
where $w_{y_i}$ refers to the classifier weight for class $y_i$, $\tau$ is typically set to 16.0 and $m$ is the margin.
As analyzed in \cite{liu2020negative}, empirically a dilemma exists that a positive margin could improve the base-class performance but harm the novel-class generalization, and reversely, a negative margin could contribute to the novel-class performance but decrease that of the base classes as shown in Fig.~\ref{fig:motivation}. Similar phenomenon has been observed in other works such as \cite{kornblith2021better} that a better loss function for the pre-training task could harm the generalization on downstream tasks.
\vspace{-0.3cm}
\subsection{Interpretation of Class-Level Overfitting from Pattern Learning View}
\vspace{-0.2cm}
Experiments are conducted on the CIFAR100~\cite{krizhevsky2009learning} dataset and reported in Fig.~\ref{fig:analysis_of_baseline}.
CIFAR100 contains 100 classes in all. As split by \cite{tao2020few}, 60 classes are chosen as base classes, and the remaining 40 classes (with 5 training samples in each class) are chosen as novel classes\footnote{\label{foot:appendix}Please refer to the appendix for details.}. Experiments are conducted on the last incremental session, where all 100 classes are involved.
From Fig.~\ref{fig:analysis_of_baseline} (left), we can see that as the margin increases, the base-class accuracy increases while the novel-class accuracy decreases, which is consistent with \cite{liu2020negative} and validates the dilemma exists.
Compared with the well-known overfitting between the training and testing data, such dilemma, although all validated on testing data, is more like the overfitting to base classes instead of samples. Therefore, we term it as \textbf{class-level overfitting (CO)}.
Additionally, the balance is reached when no margin is added, i.e., FSCIL cannot be improved by simply applying the margin.
For such dilemma, \cite{liu2020negative} gave an explanation by the degraded mapping from novel to base classes.
However, it could hardly be used to develop methods for handling such dilemma.
In this section, we go a step further to explain this phenomenon from the aspect of pattern learning for developing methods to handle it.
A pattern denotes a part of information that the model extracts from the input, which is a finer-grained level of analyzing the model's behavior. As studied in the interpretability of deep nets~\cite{zhou2016learning,bau2017network}, each channel in the feature extracted by deep networks could correspond to a certain pattern of the input\textsuperscript{\ref{foot:appendix}}, which can be viewed as to compose the base and novel classes~\cite{zou2020compositional}. Therefore, we conduct experiments on feature channels to study the patterns learned by applying different margins.
\vspace{-0.2cm}
\subsubsection{Class-Level Overfitting Interpreted by Pattern Fitness to Base Classes}
\vspace{-0.2cm}
\paragraph{Pattern's fitness to each base class.}
We first evaluate the sparsity of the base-class patterns, which is measured by the $L_1$ norm of each feature vector. As the extracted features are $L_2$ normalized, the smaller the $L_1$ norm is, for each feature, the sparser the patterns with high activation are.
Results are plotted in Fig.~\ref{fig:analysis_of_baseline} (mid), where we can see a consistent decrease in $L_1$ when the margin increases, which means the model needs less activated patterns (channels) to represent each base class. As the number of activated patterns decreases, the effectiveness of each activated pattern must increase to account for the performance increase in the base-class pre-training in Fig.~\ref{fig:analysis_of_baseline} (left). Therefore, we hold that as the margin increases, the patterns learned by the model could fit each base class better.
\begin{figure}[t]
\centering
\includegraphics[width=0.32\linewidth]{baseline_overall_base_novel.png}
\centering
\includegraphics[width=0.32\linewidth]{baseline_L1_MTA.png}
\centering
\includegraphics[width=0.32\linewidth]{baseline_trans_relation.png}\vspace{-0.2cm}
\caption{\textbf{Left}: Class-level overfitting exists between base and novel classes, and simply applying margins to the training can not help the overall performance. \textbf{Mid}: Pattern fits base classes more as the margin increases, making it more discriminative but less transferable. \textbf{Right}: Transferability of patterns decreases as the margin increases, pushing classes away from each other.}\vspace{-0.5cm}
\label{fig:analysis_of_baseline}
\end{figure}
\vspace{-0.2cm}
\paragraph{Pattern fitness measured by the template-matching score.}
To further verify the fitness increase, we view each pattern as a semantic template and measure its matching score to each base class.
As analyzed in \cite{zhou2016learning} and \cite{bau2017network}, each pattern can be understood as a template~\cite{chen2020addernet} for the model to match the input (so that each class would has its own set of templates for recognition), and the activation can be viewed as the matching score.
Therefore, we could know how much all patterns fit (match) each class by finding the most important patterns for each class and compare their activation.
As analyzed in \cite{zhou2016learning} and \cite{zou2020compositional}, patterns (channels) with higher weights in the classification layer are more important, and the most important ones dominate the model decisions.
Therefore, given an input, we select its most important patterns by the top classification weights of its ground-truth class, and record the average activation on these patterns. The Mean value of such Top Activation across all samples is denoted as MTA in Fig.~\ref{fig:analysis_of_baseline} (mid).
As can be seen, as the margin increases, MTA increases consistently, which further verifies patterns' increase in fitting each base class.
\vspace{-0.2cm}
\paragraph{Better pattern fitness, worse pattern transferability.}
As each pattern could fit a corresponding base class better, its discriminability increases accordingly, but could it be transferred across classes?
To answer it, we test the transferability of patterns.
Since classes are related (e.g., cat and tiger), transferable patterns activated in one class could also be activated in other classes (e.g., felid patterns).
Therefore, we first find important patterns for each base class by the classification weights, then record activation of these patterns on \textbf{other} classes, and measure the transferability of patterns by the mean value of such other-class-activation. The results are plotted in Fig.~\ref{fig:analysis_of_baseline} (right). As can be seen, the transferability consistently decreases when the margin increases.
Combine this result with Fig.~\ref{fig:analysis_of_baseline} (mid), we hold that patterns tend to be less transferable when they fit each base class better.
\vspace{-0.2cm}
\paragraph{Discussion.}
The fitness also reflects the how much the given pattern is specific to a base class.
Imagine the extreme situation where each base-class only needs one pattern for representation, the fitness would reach its upper bound to make such pattern thoroughly specific to the corresponding class.
Therefore, we interpret that the higher the margin is, the more specific (overfitting) the patterns are to each base class, which makes patterns more discriminative but less transferable. Meanwhile, the lower the margin is, the more the patterns could be shared between classes (underfitting), making patterns more transferable but less discriminative.
The CO dilemma lies in that patterns can hardly be both class-specific and shared among classes by simply applying the classification margin.
\vspace{-0.2cm}
\subsubsection{Inherent Class Relations Lead to the Change in Pattern's Base-Class Fitness}
\vspace{-0.2cm}
\paragraph{Pattern's fitness negatively influences class relations.}
In Fig.~\ref{fig:analysis_of_baseline} (right), we also plot the class relations w.r.t. the margins. The class relations are measured by the average of cosine similarities between every two classes' prototypes. As can be seen, the relation drops as the margin grows, in consistent with the trend of the patterns' transferability. This is rationale because the if two prototypes share some patterns, the activation of the corresponding channels will be similar, making the cosine similarity larger.
As the transferability of patterns is negatively related to pattern's base-class fitness, we hold that the class relations are also negatively related to the base-class fitness.
\begin{figure}[t]
\centering
\begin{minipage}{0.54\textwidth}
\centering
\includegraphics[width=0.51\linewidth]{relation_sort_neg_vs_0_with_axis.jpg}\hspace{-0.3cm}
\centering
\includegraphics[width=0.51\linewidth]{relation_sort_pos_vs_0_with_axis.jpg}\vspace{-0.3cm}
\caption{The change of class relations sorted by the inherent class relations. Left: negative margin. Right: positive margin.}
\label{fig:relation_sort_vs_0}
\end{minipage}
\hspace{0.1cm}
\begin{minipage}{0.44\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{explanation3.png}\vspace{-0.2cm}
\caption{Intuitive interpretation of the pattern learning and inherent class relation.}
\label{fig:explanation}
\end{minipage}\vspace{-0.4cm}
\end{figure}
\vspace{-0.2cm}
\paragraph{Inherent class relations influence pattern's fitness.}
The margin applied to the classification directly modifies the decision boundary between every two classes, and the decision boundary is related to the relationship between every two classes.
Therefore, we study how the class relation influences the pattern's fitness to base classes.
Specifically, given 60 base classes, for the model trained without margins, we first calculate the cosine similarity between every two different classes, which gives 60 $\times$ (60 - 1) / 2 = 1,750 relations denoted as $R_0$, which represents the \textbf{inherent relations} between all classes. Similarly, we calculate 1,750 relations for the model trained with positive and negative margins respectively, denoted as $R_{pos}$ and $R_{neg}$. Then we calculate $D_{pos} = R_{pos} - R_0$ and $D_{neg} = R_{neg} - R_0$. Finally, we sort $R_0$ from small to large, and use the sorted index to arrange $D_{neg}$ and $D_{pos}$. As plotted in Fig.~\ref{fig:relation_sort_vs_0}, the blue dots denote $D_{neg}$ (left) or $D_{pos}$ (right), and the blue curve are dots fitted by multinomial functions. We can see that the change in class relations is positively related to the inherent class relations for the negative margins, while is negatively related to the inherent class relations for the positive margins, especially for index larger than 1,000.
Since class relations are negatively related to pattern's fitness to base classes, the more the class relation increases, the less the patterns could fit the corresponding base class. Therefore, the results can be understood that the more inherently similar two classes are, by applying a negative margin, the less the patterns could fit the given base class, i.e., the more the patterns are shared by given classes, making these classes' representations more similar; by applying a positive margin, the more the patterns could fit the given base classes, making these classes' representations more dissimilar.
\vspace{-0.2cm}
\paragraph{Conclusion and Discussion.}
Therefore, as shown in Fig.~\ref{fig:explanation}, we interpret the pattern learning process as follows.
Given a negative margin, the decision boundary between two classes are confused, making more samples fall into the overlapping region between two classes. This makes it possible to learn patterns shared by these two classes and hinder the pattern from fitting the given base class, and makes patterns more transferable but less discriminative. The more similar inherently two classes are (i.e., larger overlapping region in Fig.~\ref{fig:explanation}), the more shared patterns can be learned (e.g., more patterns could be shared between cats and tigers than cats and air-planes), therefore making these classes' representations more similar.
On the contrary, given a positive margin, the decision boundary between two classes should be well separated, pushing the model to learn patterns fitting (i.e., specific to) each class, which are more discriminative but less transferable. The more similar inherently two classes are, the harder the learning is and the larger the training loss will be, therefore making the patterns fit each class more and making these classes' representations more dissimilar.
\vspace{-0.3cm}
\section{Mitigating the Dilemma of Few-Shot Class-Incremental Learning}
\label{sec:method}
\vspace{-0.2cm}
In this section, we first analyze the cause of the dilemma in margin-based classification based on the above interpretation, then we propose our method (named as Class-Level Overfitting Mitigation, CLOM) to mitigate the CO dilemma based on the analysis, as shown in Fig.~\ref{fig:framework}.
\vspace{-0.2cm}
\subsection{Analysis on the Cause of Class-Level Overfitting}
\vspace{-0.1cm}
Based on the above analysis,
we can find the learning of negative-margin-based patterns loosely constrains the given pattern to be shared by both classes. However, such constraint is easily satisfied by ineffective patterns as simple as edges or corners, which could lead to the low discriminability of negative-margin-based patterns. Similarly, the learning of positive-margin-based patterns loosely constrains the given pattern to be specific to the given class. However, such constraint could be satisfied by easily finding patterns sharing no information with other classes, such as finding some complex texture only specific to the given class, which could lead to the low transferability of positive-margin-based patterns.
Such \textbf{easy-constraint} problems push patterns to be \textbf{only} class-specific or shared among classes, making pattern effective in one scenario ineffective in other ones.
To verify the above claims, we analyze how simple or complex the learned patterns are.
Inspired by \cite{kornblith2019similarity}, we quantitatively measure the simplicity/complexity of patterns by the similarity between the extracted feature and the simplest feature (e.g., corner or edge features).
We first use the baseline model to train on CIFAR100, and use the first convolutional layer as the simplest feature extractor (denoted as $f_{simple}$), since many works (e.g., \cite{yosinski2015understanding}) has shown that the first convolutional layer tends to capture corners or edges. Then, we train models with different margins, and use the backbone network for feature extraction (denoted as $f_{target}$). After that, we extract $f_{simple}$ and $f_{target}$ features from all images in base classes. Finally, we compare the CKA similarity~\cite{kornblith2019similarity} for measuring the similarity between $f_{simple}$ and the $f_{target}$.
For a sanity check, we first report the similarity between different layers within the baseline model.
\vspace{-0.4cm}
\begin{table}[h]
\caption{Sanity check for the CKA measure.}
\label{tab:CKA_sanity}
\centering
\resizebox{0.88\textwidth}{!}{\begin{tabular}{c|ccccc}
\toprule
$f_{target}$ & Conv1-output & Stage1-output & Stage2-output & second-last-Conv & backbone-output \\
\midrule
CKA & 1.0 & 0.8876 & 0.5664 & 0.2097 & 0.1306 \\
\bottomrule
\end{tabular}}\vspace{-0.2cm}
\end{table}
We can see that the shallower the layer is, the higher the CKA similarity would be, which means the more similar they are to the $f_{simple}$, i.e. the patterns are simpler, more transferable but less discriminative~\cite{yosinski2014transferable}. Then, we report the comparison of the CKA similarity between $f_{simple}$ and $f_{target}$ (backbone feature) of the baseline model trained with margins.
\vspace{-0.4cm}
\begin{table}[h]
\caption{CKA between the $f_{simple}$ and baseline backbone features trained with different margins.}
\label{tab:CKA_baseline}
\centering
\resizebox{0.95\textwidth}{!}{\begin{tabular}{c|cccccccccc}
\toprule
Margin & -0.5 & -0.4 & -0.3 & -0.2 & -0.1 & 0.0 & 0.1 & 0.2 & 0.3 & 0.4 \\
\midrule
CKA & 0.2432 & 0.2245 & 0.2010 & 0.1661 & 0.1510 & 0.1306 & 0.1149 & 0.0837 & 0.0642 & 0.0576 \\
\bottomrule
\end{tabular}}\vspace{-0.2cm}
\end{table}
We can see that by applying a negative margin, the CKA similarity clearly increases (even larger than that of the second last convolutional layer when margin < -0.3), showing that the backbone network captures patterns more similar to simplest ones such as edges or corners, which verifies that the model tends to learn simple patterns that are easily shared between classes. When applying a positive margin, the captured patterns grow to be more complex and tend to overfit base classes, making the CKA much smaller than baseline model's backbone-output, which verifies that the model tends to learn complex patterns that are easily specific to a given base class.
Therefore, the key to mitigating the CO dilemma is to extra constrain the pattern learning process.
\vspace{-0.2cm}
\subsection{Mitigating Class-Level Overfitting by Providing Extra Constraint}
\vspace{-0.1cm}
Negative-Margin-based (NM) patterns are more transferable and class-shared, while Positive-Margin-based (PM) patterns are discriminative and class-specific. These characteristics are similar to the behavior of features from the shallow and deep layers of deep networks~\cite{yosinski2014transferable,zou2021revisiting}. Generally, shallow-layer features encode low-level patterns that are easily shared by most objects therefore more transferable, such as edge or corner, while the deep-layer features encode high-level patterns that are semantically related to only few object classes therefore more discriminative. Such similarity inspires us to view PM patterns as high-level patterns while view NM patterns as (relatively) low-level patterns, and build PM patterns from the NM patterns, just like building high-level features from low-level features in deep networks. As the learning of NM patterns is influenced by PM patterns, this design could benefit the learning of NM patterns by providing extra constraint from the learning of PM patterns, and vice versa, which could therefore handle the easy-constraint problem.
\begin{figure}[t]
\centering
\includegraphics[width=0.9\linewidth]{framework.png}\vspace{-0.2cm}
\caption{Method (CLOM) framework. We construct the positive-margin-based (PM, $F(\cdot)$) feature from the negative-margin-based (NM, $f(\cdot)$) feature, and map class relations to form a set of class-specific margins, which can effectively mitigate the dilemma of class-level overfitting by providing extra constraint to the NM/PM pattern learning through the learning of PM/NM patterns respectively.}\vspace{-0.4cm}
\label{fig:framework}
\end{figure}
Specifically, given an feature extractor $f(\cdot)$, we add one more layer to construct the PM feature as
\vspace{-0.1cm}
\begin{equation}
F(x_i) = g(f(x_i)),
\label{eq:F}
\end{equation}
where $F(\cdot)$ is the PM feature extractor, $g(\cdot)$ is typically composed of a fully-connected layer, a batch-normalization layer~\cite{ioffe2015batch} and an activation layer. During training, two classification loss is applied to both $F(\cdot)$ and $f(\cdot)$ with positive and negative margins respectively, represented as
\vspace{-0.1cm}
{\small
\begin{align}
& L(x_i, y_i) = L_{neg}(x_i, y_i) + L_{pos}(x_i, y_i) \\
& = -log \frac{e^{\tau (w_{y_i} f(x_i) - m_{neg})}}{e^{\tau (w_{y_i} f(x_i) - m_{neg})} + \sum_{j \neq y_i} e^{\tau w_j f(x_i)}} - \lambda \cdot log \frac{e^{\tau (w^P_{y_i} F(x_i) - m_{pos})}}{e^{\tau (w^P_{y_i} F(x_i) - m_{pos})} + \sum_{j \neq y_i} e^{\tau w^P_j F(x_i)}},
\label{eq:ind_loss}
\end{align}
}
\hspace{-0.1cm}where $w^P_j \in R^{1 \times d^P}$ denotes the classifier weights for $F(\cdot)$ corresponding to class $j$, $m_{neg}$ denotes the negative margin for $f(\cdot)$, $m_{pos}$ is the positive margin for $F(\cdot)$, and $\lambda$ is typically set to 1.0.
Given this modification, the NM patterns $f(\cdot)$ are utilized to construct the discriminative PM patterns $F(\cdot)$. Since $g(\cdot)$ is relatively too simple to capture complex patterns, $f(\cdot)$ will be pushed to be discriminative (validated in Fig.~\ref{fig:f_neg}), instead of casually learning ineffective patterns shared between similar classes. Similarly, as $F(\cdot)$ are built from the transferable $f(\cdot)$ by the simple $g(\cdot)$, the transferability could be maintained by pushing PM patterns to fit the corresponding base class by transferable information, which improves the transferability of $F(\cdot)$ (validated in Fig.~\ref{fig:f_pos}).
During testing, given the improved $f(\cdot)$ and $F(\cdot)$, the final feature would be their concatenation.
\vspace{-0.2cm}
\subsection{Further Mitigation of Class-Level Overfitting by Integrating Class Relations}
\vspace{-0.1cm}
Based on the architecture design, we propose to further mitigate the CO problem through boosting the discriminability and tranferability of NM and PM patterns by integrating the class relations.
For easy understanding, we first introduce the modification to the classification for $f(\cdot)$, and later introduce that for $F(\cdot)$. To modify the margin-based classification from a single margin to margins related to class relations, we first move the margin from the ground-truth logit to all other logits as
\vspace{-0.15cm}
\begin{equation}
L(x_i, y_i) = -log \frac{e^{\tau w_{y_i} f(x_i)}}{e^{\tau w_{y_i} f(x_i)} + \sum_{j \neq y_i} e^{\tau (w_j f(x_i) + m(A_{ij}))}},
\label{eq:adj_m}
\end{equation}
where $A$ is the adjacency matrix between all classes, $m(\cdot)$ maps the adjacency value between two classes to a margin. Given this modification, if the margin is set to the original fixed value as Eq.~\ref{eq:margin}, the decision boundary between class $y_i$ and other classes remains the same, therefore the new loss in Eq.~\ref{eq:adj_m} could be an approximation of the original loss in Eq.~\ref{eq:margin}.
For measuring class relations, we choose to utilize the adjacency matrix of all classes. As both the feature and the classifier's weights are $L_2$ normalized, the adjacency can be measured by the cosine similarity between every two class prototypes as $A_{ij} = cos(P_i, P_j)$, where $P_i$ is the prototype for class $i$, which is typically set to $w_i$ of the classifier.
If two classes are identical, the cosine similarity would reach its upper bound to 1.0. Meanwhile, the average of class relations reflects a global margin that is effective for most classes. Therefore, we design to interpolate the margin from a global effective value to a pre-defined upper bound value as
\vspace{-0.15cm}
\begin{equation}
m(A_{ij}) = m_{ave} + \frac{m_{upper} - m_{ave}}{1.0 - A_{ave}} \cdot (A_{ij} - A_{ave}),
\label{eq:map_from_adj_to_margin}
\end{equation}
where $m_{upper}$ is a hyper-parameter controlling the margin for the upper-bound class relation (i.e., two identical classes), $m_{ave}$ is another hyper-parameter controlling the margin for the average class relations, set to the same value as that in Eq.~\ref{eq:margin}, and $A_{ave}$ is the average class relations calculated as $A_{ave} = \frac{1}{|Y_0| \cdot (|Y_0| - 1)} \sum_{j=1}^{j=|Y_0|} \sum_{k \neq j} A_{jk} $.
For $F(\cdot)$, we adopt the same modification as $f(\cdot)$, which replaces $f(\cdot)$ with $F(\cdot)$ in Eq.~\ref{eq:adj_m}, and replaces $A$ with $A^P$ in Eq.~\ref{eq:adj_m} where $A^P_{ij} = cos(w^P_i, w^P_j)$.
Therefore, the final training objective is
\vspace{-0.15cm}
\begin{flalign}\hspace{-0.35cm}
\resizebox{0.97\textwidth}{!}{
$L(x_i, y_i) = - log \frac{e^{\tau w_{y_i} f(x_i)}}{e^{\tau w_{y_i} f(x_i)} + \sum\limits_{j \neq y_i} e^{\tau (w_j f(x_i) + m_{n}(A_{ij}))}}
- \lambda \cdot log \frac{e^{\tau w^P_{y_i} F(x_i)}}{e^{\tau w^P_{y_i} F(x_i)} + \sum\limits_{j \neq y_i} e^{\tau (w^P_j F(x_i) + m_{p}(A^P_{ij}))}}$,
}
\label{eq:final_loss}
\end{flalign}
where $m_{p}(\cdot)$ and $m_{n}(\cdot)$ are two interpolate functions as Eq.~\ref{eq:map_from_adj_to_margin} with positive and negative hyper-parameters ($m^P_{upper}$, $m^P_{ave}$) and ($m_{upper}$, $m_{ave}$) respectively.
In experiments (Fig.~\ref{fig:margin_by_adj}), we find it beneficial to have $m_{upper} < m_{ave}$ and $m^P_{upper} > m^P_{ave}$, which means for classes with higher similarities, the relation mapping module enables the model to learn more shared (transferable) patterns by applying smaller negative margins, and learn more class-specific (discriminative) patterns by applying larger positive margins.
Moreover, due to the connection between NM and PM patterns, the improved transferability in NM patterns would be transmitted to PM patterns and vice versa, as validated in Fig.~\ref{fig:margin_by_adj}, which further mitigates the CO dilemma.
After the base-session training, the feature extractor $F(\cdot)$ and $f(\cdot)$ will be applied to the training data of each session to obtain the extended classifier weight $W = \{w^0_1, w^0_2, ..., w^0_{|Y_0|}\} \cup ... \cup \{w^k_1, ..., w^k_{|Y_k|}\}$ by averaging extracted features, and the final performance of each session will be obtained based on the classification of all the encountered classes' test samples.
\begin{table}[t]
\caption{Evaluation datasets.}
\label{tab:dataset}
\centering
\resizebox{1.0\textwidth}{!}{\begin{tabular}{lcccccc}
\toprule
Dataset & Total Classes & Base Classes & Novel Classes & Incremental Sessions & Novel-Class Shot & Input Size \\
\midrule
CIFAR100 & 100 & 60 & 40 & 8 & 5 & 32 $\times$ 32 \\
CUB200 & 200 & 100 & 100 & 10 & 5 & 224 $\times$ 224 \\
\textit{mini}ImageNet & 100 & 60 & 40 & 8 & 5 & 84 $\times$ 84 \\
\bottomrule
\end{tabular}}\vspace{-0.5cm}
\end{table}
\begin{table}[t]
\caption{Comparison with state-of-the-art works on the CUB200 dataset.}
\label{tab:cub_sota}
\centering
\resizebox{1.0\textwidth}{!}{\begin{tabular}{lccccccccccc}
\toprule
Method & S0 & S1 & S2 & S3 & S4 & S5 & S6 & S7 & S8 & S9 & S10 \\
\midrule
Finetune & 68.68 & 43.70 & 25.05 & 17.72 & 18.08 & 16.95 & 15.10 & 10.06 & 8.93 & 8.93 & 8.47 \\
Rebalancing~\cite{hou2019learning} & 68.68 & 57.12 & 44.21 & 28.78 & 26.71 & 25.66 & 24.62 & 21.52 & 20.12 & 20.06 & 19.87 \\
iCaRL~\cite{rebuffi2017icarl} & 68.68 & 52.65 & 48.61 & 44.16 & 36.62 & 29.52 & 27.83 & 26.26 & 24.01 & 23.89 & 21.16 \\
EEIL~\cite{castro2018end} & 68.68 & 53.63 & 47.91 & 44.20 & 36.30 & 27.46 & 25.93 & 24.70 & 23.95 & 24.13 & 22.11 \\
TOPIC~\cite{tao2020few} & 68.68 & 62.49 & 54.81 & 49.99 & 45.25 & 41.40 & 38.35 & 35.36 & 32.22 & 28.31 & 26.26 \\
Decoupled-NegCosine~\cite{liu2020negative} & 74.96 & 70.57 & 66.62 & 61.32 & 60.09 & 56.06 & 55.03 & 52.78 & 51.50 & 50.08 & 48.47 \\
CEC~\cite{zhang2021few} & 75.85 & 71.94 & 68.50 & 63.50 & 62.43 & 58.27 & 57.73 & 55.81 & 54.83 & 53.52 & 52.28 \\
FSLL+SS~\cite{mazumder2021few} & 75.63 & 71.81 & 68.16 & 64.32 & 62.61 & 60.10 & 58.82 & 58.70 & 56.45 & 56.41 & 55.82 \\
FACT~\cite{zhou2022forward} & 75.90 & 73.23 & 70.84 & 66.13 & 65.56 & 62.15 & 61.74 & 59.83 & 58.41 & 57.89 & 56.94 \\
IDLVQ-C~\cite{chen2020incremental} & 77.37 & 74.72 & 70.28 & 67.13 & 65.34 & 63.52 & 62.10 & 61.54 & 59.04 & 58.68 & 57.81 \\
\midrule
CLOM (Ours) & \textbf{79.57} & \textbf{76.07} & \textbf{72.94} & \textbf{69.82} & \textbf{67.80} & \textbf{65.56} & \textbf{63.94} & \textbf{62.59} & \textbf{60.62} & \textbf{60.34} & \textbf{59.58} \\
\bottomrule
\end{tabular}}\vspace{-0.4cm}
\end{table}
\vspace{-0.3cm}
\section{Experiments}
\label{sec:exp}
\vspace{-0.2cm}
\begin{figure}[t]
\centering
\includegraphics[width=0.45\linewidth]{results_cifar.png}
\centering
\includegraphics[width=0.45\linewidth]{results_mini.png}\vspace{-0.2cm}
\caption{Comparison with state-of-the-art works on CIFAR100 and \textit{mini}ImageNet.}\vspace{-0.5cm}
\label{fig:cifar_and_mini}
\end{figure}
In this section, we first introduce the experiment settings, then compare the proposed method with the state-of-the-art methods, and finally report the ablation study for the effectiveness of each design.
\vspace{-0.3cm}
\subsection{Datasets}
\vspace{-0.1cm}
Datasets include CIFAR100~\cite{krizhevsky2009learning}, Caltech-UCSD Birds-200-2011 (CUB200)~\cite{wah2011caltech} and \textit{mini}-ImageNet~\cite{Vinyals2016Matching} as listed in Tab.~\ref{tab:dataset} following the split in \cite{tao2020few}. For details, please refer to the appendix.
\vspace{-0.3cm}
\subsection{Implementation Details}
\vspace{-0.1cm}
The implementation is based on CEC's code~\cite{zhang2021few}, and our code will be released\footnote{https://github.com/Zoilsen/CLOM}. For CIFAR100, we set $d^P$=256, set $m_{ave}$=-0.2, set $m_{upper}$=-0.5, and we have $m^P_{ave}$=0.1 and $m^P_{upper}$=0.2. For CUB200, we scale the learning rate of the backbone network to 10\% of the global learning rate since the pre-training of the backbone is adopted~\cite{zhou2022forward,zhang2021few}, and set $d^P$ to 8192. Then we have $m_{ave}$=-0.2 and $m_{upper}$=-0.25 and $m^P_{ave}$=0.3 and $m^P_{upper}$=0.6. For \textit{mini}ImageNet, we set $d^P$ to 4096, and have $m_{ave}$=-0.2 and $m_{upper}$=-0.5 and $m^P_{ave}$=0.1 and $m^P_{upper}$=0.2.
Please refer to appendix for details.
\begin{table}[t]
\begin{center}
\caption{Ablation study of modules on the last incremental session of three datasets. }
\label{tab:ablation}
\resizebox{0.8\textwidth}{!}{
\begin{tabular}{lccccccccc}
\toprule
\multirow{2}{*}{\tabincell{c}{Method}} & \multicolumn{3}{c}{CUB200} & \multicolumn{3}{c}{CIFAR100} & \multicolumn{3}{c}{\textit{mini}ImageNet} \\
\cmidrule{2-10}
& Overall & Novel & Base & Overall & Novel & Base & Overall & Novel & Base \\
\midrule
Baseline & 57.78 & 45.97 & 79.48 & 47.02 & 37.40 & 72.32 & 46.58 & 31.02 & 72.33 \\
+ $g(\cdot)$& 57.21 & 47.13 & 79.39 & 48.37 & 39.55 & 72.70 & 46.79 & 30.97 & 72.60 \\
+ Margin & 58.73 & 49.93 & 79.47 & 49.21 & 40.22 & 73.72 & 47.30 & 32.07 & 72.93 \\
+ Relation & \textbf{59.58} & \textbf{50.89} & \textbf{79.57} & \textbf{50.25} & \textbf{41.17} & \textbf{74.20} & \textbf{48.00} & \textbf{33.60} & \textbf{73.08} \\
\bottomrule
\end{tabular}}\vspace{-0.6cm}
\end{center}
\end{table}
\vspace{-0.3cm}
\subsection{Comparison with the State-of-the-Art}
\vspace{-0.1cm}
Comparisons with the state-of-the-art works are listed in Tab.~\ref{tab:cub_sota} and Fig.~\ref{fig:cifar_and_mini}, where we can achieve state-of-the-art performance on all three datasets.
Specifically, we can first see that our method, as a prototype-based method (e.g., CEC~\cite{zhang2021few}), could significantly outperform finetune-based methods (e.g., iCaRL~\cite{rebuffi2017icarl}). This is because the few-shot training data could not provide sufficient information for novel-class learning, therefore directly freezing parameters on novel classes could reduce the catastrophic-forgetting problem brought by the finetuning. Our method also outperforms other prototype-based methods, this is because the core of the prototype-based methods lies in the metric learning~\cite{snell2017prototypical}. As empirically proved by current works~\cite{sun2020circle,wang2018cosface,deng2019arcface,liu2020negative}, applying the margin-based classification could effectively improve the embedding space learned by metric-based methods. Since our method handles the difficulty of applying margin-based classification (i.e., class-level overfitting) to the FSCIL task, our method could beat these prototype-based ones. For detailed numbers of CIFAR100 and \textit{mini}ImageNet, please refer to the appendix.
\vspace{-0.3cm}
\subsection{Ablation Study of Modules}
\vspace{-0.1cm}
The ablation study of each module is reported in Tab.~\ref{tab:ablation}, where $g(\cdot)$ denotes adding another layer, \textit{Margin} denotes the applicant of margin-based classification, and \textit{Relation} refers to the relation mapping of the margin. We study from three aspects: \textit{Base}-class, \textit{Novel}-class, and \textit{Overall} accuracy of the last incremental session.
From Tab.~\ref{tab:ablation}, we can see that
$\bullet$ \textit{Simply applying another layer cannot consistently improve the performance}.
$\bullet$ \textit{The designed architecture could mitigate class-level overfitting}. Compared with experiments in Fig.~\ref{fig:analysis_of_baseline} where no \textit{Overall} performance improvements can be obtained by simply adding margins, the performance here is clearly improved by adding margins on the designed architecture. Moreover, the overall improvements originate from not only the improved \textit{Base} performance, but also the boosted \textit{Novel} performance, demonstrating the mitigation of class-level overfitting (CO) problem.
$\bullet$ \textit{Relation mapping could further improve performance by mitigating class-level overfitting}.
\vspace{-0.2cm}
\subsection{Verification of Class-Level Overfitting Mitigation}
\vspace{-0.1cm}
\begin{figure}[t]
\centering
\includegraphics[width=0.26\linewidth]{f_neg_overall.png}
\centering
\includegraphics[width=0.26\linewidth]{f_neg_novel.png}
\centering
\includegraphics[width=0.26\linewidth]{f_neg_base.png}\vspace{-0.2cm}
\caption{NM feature's accuracy compared with the baseline with negative margins (CIFAR100).}
\label{fig:f_neg}
\centering
\includegraphics[width=0.26\linewidth]{f_pos_overall.png}
\centering
\includegraphics[width=0.26\linewidth]{f_pos_novel.png}
\centering
\includegraphics[width=0.26\linewidth]{f_pos_base.png}\vspace{-0.2cm}
\caption{PM feature's accuracy compared with the baseline with positive margins (CIFAR100).}\vspace{-0.5cm}
\label{fig:f_pos}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.24\linewidth]{margin_by_adj_neg_cifar.png}
\centering
\includegraphics[width=0.24\linewidth]{margin_by_adj_pos_cub.png}
\centering
\includegraphics[width=0.24\linewidth]{F_relation_novel.png}
\centering
\includegraphics[width=0.24\linewidth]{f_relation_base.png}\vspace{-0.1cm}
\caption{Upper margins in the relation mapping module ($m_{upper}$: CIFAR100, $m^P_{upper}:$ CUB200).}\vspace{-0.5cm}
\label{fig:margin_by_adj}
\end{figure}
\vspace{-0.2cm}
\paragraph{Mitigating the class-level overfitting by the architecture design.} We first compare our method with the baseline method which applies the margin directly to the backbone feature in Fig.~\ref{fig:f_neg} and \ref{fig:f_pos}, so as to validate the mitigation of CO. In Fig.~\ref{fig:f_neg}, we apply the negative margin to the backbone (NM) feature while applying no margin to the PM feature, and see the accuracy of the NM feature against the baseline. As can be seen, while keeping the improvements on novel classes (mid), our method (orange) maintains the base-class accuracy (right), in contrast to a sharp decrease of the baseline (blue) when the margin decreases, which makes our NM feature significantly outperforms the baseline in terms of the overall accuracy (left). Similarly, in Fig.~\ref{fig:f_pos}, our PM feature also maintains a higher performance on novel classes when the margin increases. These experiments validate the mitigation of the CO dilemma: by implicitly guiding the learning of the PM and NM features by each other, the transferability or discriminability drop on them can be mitigated respectively.
\vspace{-0.3cm}
\paragraph{Further mitigating class-level overfitting by relation mapping.} To validate the relation mapping module, in Fig.~\ref{fig:margin_by_adj} we fixed the average margin (i.e., $m_{ave}$=-0.2 in Fig.~\ref{fig:margin_by_adj}.1st and $m^P_{ave}$=0.3 in Fig.~\ref{fig:margin_by_adj}.2nd) and experiment with different upper margins (i.e., $m_{upper}$ in Fig.~\ref{fig:margin_by_adj}.1st and $m^P_{upper}$ in Fig.~\ref{fig:margin_by_adj}.2nd). Similar to Fig.~\ref{fig:f_neg}, when conducting experiments on one branch, the hyper-parameters of the other branch are fixed. We can see the relation mapping module can indeed help the learning, as the upper margins are significantly different from the average margin.
We also plot the NM and PM performance on base classes with negative margins in Fig.~\ref{fig:margin_by_adj}(3rd), and those on novel classes with positive margins in Fig.~\ref{fig:margin_by_adj}(4th). We can see the NM and PM performance are improved simultaneously, which validates the improvements brought by the extra guidance of each features. Therefore, the relation mapping module can also help to mitigate the class-level overfitting problem by building PM features with more transferable patterns, and vice versa.
Moreover, it is interesting to find the performance is improved when $m_{upper} < m_{ave}$ and $m^P_{upper} > m^P_{ave}$, which means for classes with higher similarities, the relation mapping module enables the model to learn more shared patterns by applying negative margins with larger absolute value, and learn more class-specific patterns by applying larger positive margins. Also, this is consistent with the results in Fig.~\ref{fig:relation_sort_vs_0}.
\vspace{-0.2cm}
\subsection{Mitigating the Easy-Constraint Problem}
\vspace{-0.1cm}
Following experiments in Tab.~\ref{tab:CKA_sanity}, we also utilize the CKA similarity between the simplest feature $f_{simple}$ and the NM/PM feature to validate the mitigation of the easy-constraint problem.
By applying our method, for the NM (or PM) patterns, we set $f_{target}$ to $f(\cdot)$ (or $F(\cdot)$) in Fig.5 and set the positive/negative margin to 0.3/-0.4, and report the CKA below.
\begin{minipage}{\textwidth}
\tabcaption{CKA between $f_{simple}$ and NM (left) or PM (right) features trained with different margins.}\vspace{-0.2cm}
\label{tab:CKA_NM+PM}
\begin{minipage}[t]{0.53\textwidth}
\resizebox{1.0\textwidth}{!}{
\begin{tabular}{c|cccccc}
\toprule
Margin & -0.5 & -0.4 & -0.3 & -0.2 & -0.1 & 0.0 \\
\midrule
CKA & 0.1867 & 0.1779 & 0.1724 & 0.1638 & 0.1552 & 0.1427 \\
\bottomrule
\end{tabular}
}
\end{minipage}
\begin{minipage}[t]{0.45\textwidth}
\resizebox{1.0\textwidth}{!}{
\begin{tabular}{c|ccccc}
\toprule
Margin & 0.0 & 0.1 & 0.2 & 0.3 & 0.4 \\
\midrule
CKA & 0.1476 & 0.1439 & 0.1430 & 0.1214 & 0.1100 \\
\bottomrule
\end{tabular}
}
\end{minipage}
\end{minipage}
In Tab.~\ref{tab:CKA_NM+PM} (left), compared with Tab.~\ref{tab:CKA_baseline}, CKA decreases clearly compared with the baseline method, which validates that NM patterns learned by our method are more complex and less similar to edges and corners than the baseline method, verifying the mitigation of the easily-satisfied constraint problem by extra supervision from the learning of PM patterns.
Also, in Tab.~\ref{tab:CKA_NM+PM} (right), compared with Tab.~\ref{tab:CKA_baseline}, CKA is much larger than that of the baseline method, which validates that the PM patterns are more similar to the simplest patterns than the baseline method, which makes it less overfitting the base classes, verifying the mitigation by extra supervision from the learning of NM patterns.
\vspace{-0.3cm}
\section{Conclusion}
\vspace{-0.2cm}
In this paper, we focus on the dilemma in the margin-based classification for FSCIL. We first interpret the dilemma as a class-level overfitting problem from the aspect of pattern learning, then find the cause of this problem lies in the easily-satisfied constraint of learning shared or class-specific patterns. Based on the analysis, we design a method (CLOM) to mitigate the dilemma by constructing PM patterns from NM patterns, and mapping class relations into class-specific patterns. Extensive experiments on three public datasets validate the effectiveness and outstanding performance of the proposed method.
\vspace{-0.3cm}
\section*{Acknowledgements}
\vspace{-0.2cm}
This work is supported by National Natural Science Foundation of China under grants U1836204, U1936108, 62206102, Science and Technology Support Program of Hubei Province under grant 2022BAA046, and 2022 CCF-DiDi Gaiya Young Scholar Research Fund.
\bibliographystyle{plain}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,891 |
Contour half height shower enclosure. Made to measure, 750 or 900mm high. Bi-folding door hinging from the wall with single door hinging from the fixed panel. White finish. PET glazing. Includes curtain and rail. British made. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,533 |
{"url":"https:\/\/gamedev.stackexchange.com\/questions\/176916\/how-to-have-a-point-light-with-no-falloff-in-unity-3d?noredirect=1","text":"# How to have a point light with no falloff in Unity 3D?\n\nDoes anyone know of a good strategy to make point lights with no falloff, eg. anything in the radius of the light will be 100% illumination while anything outside the radius will be 0% illumination.\n\nI want to use this as a sort of 3D line of sight drawing. The line of sight is a big sphere originating from the character's head that I thought would be a clever idea to have that represented as a point light, however with the falloff it doesn't correctly show where the edge of the line of site is.\n\nI've tried a few solutions, such as a custom falloff package (doesn't seem to have the options I want) and a few shaders, but I am terrible at shaders so it's difficult for me to debug. I'll post the shader here in case somebody can tell me if I'm being dumb, otherwise if someone can point me to a different solution that would be great. Thanks!\n\nHere's a shader I tried, but it just turns everything invisible and it's not immediately obvious to me why:\n\nShader \"Custom\/No Falloff v2\"\n{\nProperties\n{\n_Color(\"Color\", Color) = (1,1,1,1)\n_MainTex(\"Texture\", 2D) = \"white\" {}\n}\n{\nTags{ \"Queue\" = \"Transparent\" }\n\nPass\n{\nBlend SrcAlpha OneMinusSrcAlpha\n\nCGPROGRAM\n#pragma vertex vert\n#pragma fragment frag\n#include \"UnityCG.cginc\"\n#include \"Lighting.cginc\"\n\n\/\/ shadow helper functions and macros\n#include \"AutoLight.cginc\"\n\nsampler2D _MainTex;\nfloat4 _MainTex_ST;\n\nfixed4 _Color;\n\nstruct v2f\n{\nfloat2 uv : TEXCOORD0;\nfloat4 pos : SV_POSITION;\n};\n\nv2f vert(appdata_base v)\n{\nv2f o;\no.pos = UnityObjectToClipPos(v.vertex);\no.uv = TRANSFORM_TEX(v.texcoord, _MainTex);\nreturn o;\n}\n\nfixed4 frag(v2f i) : SV_Target\n{\nfixed4 col = tex2D(_MainTex, i.uv) * _Color;\n\n\/\/ compute shadow attenuation (1.0 = fully lit, 0.0 = fully shadowed)\n\n{\nreturn fixed4(0.0156862745, 0, 0.23529411764,1.0); \/\/or any other color for shadow\n}\n\nreturn col;\n}\nENDCG\n}\n\n}\n}\n\n\u2022 If it's just a big sphere, you could also use a sphere collider and use a semi-transparent mesh as your \"line of sight\" to go along with it. It should take fewer resources than a point light too. Is there a specific reason you're picking a point-light? This old answer of mine might be of use to you as a shader with a few modifications. \u2013\u00a0John Hamilton Nov 13 at 11:24\n\u2022 @JohnHamilton Well I guess it's a bit deceptive to say it's a sphere. It's a sphere under ideal circumstances with no obstructions but the main point of having it be a point light rather than a sphere collider is that line of sight will be blocked by other objects in the way. Does that make sense? Also there will only ever be a small number (under 10 generally) of these in a scene at any time so resources aren't the biggest priority. \u2013\u00a0William Miles Nov 14 at 23:57","date":"2019-12-13 20:36:55","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.32707563042640686, \"perplexity\": 1517.8975865064551}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-51\/segments\/1575540569146.17\/warc\/CC-MAIN-20191213202639-20191213230639-00096.warc.gz\"}"} | null | null |
Q: Trying to map inputs to the questions of radio type Trying to get the radio type inputs but it doesn't work. BUt Iam getting this response I need legit iputs there. Tried multiple things like adding it into array and in that case, I got [data data] something like this.
function showRadioOptions(questionId, data, mKey, qLevel) {
data = data.sort((a, b) => a.orderNo - b.orderNo);
let radioOptions: any = [];
for (let i = 0; i < data.length; i++) {
radioOptions = radioOptions.concat(
`<input type='radio' className='showChildQuestion qid_${questionId}' name='${questionId}' id='${questionId}' value=${data[i].id} data-key='${mKey}' data-ishierarchy='${data[i].isHierarchy}' data-hierarchy-question='${data[i].hierarchyQuestionId}' data-current-question='${questionId}' data-question-level='${data[i].questionLevel}'> ${data[i].optionText} <br/>`
);
}
return radioOptions;
}
function Radio(questionId, options, mKey, qLevel) {
return "<br/>" + showRadioOptions(questionId, options, mKey, qLevel);
}
A: function showRadioOptions(questionId, data, mKey, qLevel) {
data = data.sort((a, b) => a.orderNo - b.orderNo);
let radioOptions: any = "";
radioOptions = data.map((item, i) => {
return (
<label>
<input
type="radio"
className={"showChildQuestion qid_" + questionId}
name={questionId}
id={questionId}
value={item.id}
data-key={mKey}
data-ishierarchy={item.isHierarchy}
data-hierarchy-question={item.hierarchyQuestionId}
data-current-question={questionId}
data-question-level={item.questionLevel}
/>
{item.optionText} <br />
</label>
);
});
return radioOptions;
}
function Radio(questionId, options, mKey, qLevel) {
return showRadioOptions(questionId, options, mKey, qLevel);
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,752 |
{"url":"https:\/\/googology.wikia.org\/wiki\/Kudi-Chan%27s_Number","text":"10,974 Pages\n\nKudi-Chan's Number is an example of a naive extension to Graham's number, and of a salad number. It is defined as follows:\n\n\u2022 $$k_0=4$$\n\u2022 $$k_1=G\\uparrow\\uparrow\\uparrow\\uparrow G$$ where $$G$$ is Graham's number\n\u2022 $$k_2=G \\underbrace{\\uparrow\\uparrow\\cdots\\uparrow\\uparrow}_{k_1 \\text{ arrows}} G$$\n\u2022 $$k_n=G \\underbrace{\\uparrow\\uparrow\\cdots\\uparrow\\uparrow}_{k_{n-1} \\text{ arrows}} G$$\n\u2022 $$k_G=\\text{Kudi-Chan's number}$$\n\nIt can be shown that this number is upper-bounded by GG64+64 in Graham's function.\n\nIt was named as such by\u00a0Cookiefonster, who found this number in an unpublished article in Sbiis Saibian's Large Numbers website.[1]\n\n## Sources\n\nCommunity content is available under CC-BY-SA unless otherwise noted.","date":"2021-08-05 11:08:01","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6621124148368835, \"perplexity\": 1511.2062131487605}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-31\/segments\/1627046155529.97\/warc\/CC-MAIN-20210805095314-20210805125314-00028.warc.gz\"}"} | null | null |
Books purchased from sastasahityamandal.org, that are requested for return within 30 days of dispatch are eligible for full refund, deducting the freight charges as applicable. The customer will receive the refund for the shipping charges also, if the return is a result of our error. We do not replace or exchange the books have been purchased. Our customer can only receive the refund. Moreover, can only return items that have been received. Customers are requested to accept the conditions placed by us before proceeding with the return.
Books that are purchased online from sastasahityamandal.org can be refunded subject to certain conditions sastasahityamandal.org. penalizes the customer for the delivery and processing charges incurred upon for the fulfillment of the order. The books that have been purchased from our showroom, Sasta Sahitya Mandal, N-77, First Floor New Delhi-110001, India, cannot be returned online. Such returns and requests for refunds have to be made manually at the Showroom. The customer will be informed via e-mail about the refund once we have received and processed the returned item. This is applicable to the items that are returned to the store without receiving them. Items can be returned if they are in the original condition. A request for return has to be submitted within 30 days of dispatch of the items. The customer will obtain the refund in the same form of payment used to purchase the books by the customer, within Seven days of our receiving the returns. Customer can request for returns using the My Account option. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,523 |
{"url":"http:\/\/finding-palindromes.blogspot.com\/2012\/07\/","text":"## Sunday, July 22, 2012\n\n### Palindromes in DNA\n\nDNA strings are often millions of letters long. For example, the copy I have of the DNA of the Amycolatopsis marina, a bacteria discovered in 2009, consists of $6,503,724$ characters.\n\nBacteria are of course some of the smallest living beings. The DNA in the copy I have of the human chromosome 18 is an astounding $74,657,229$ characters long. Chromosome 18 is one of the 23 chromosomes of human beings, and is estimated to contain in between $300$ and $500$ genes. A gene is responsible for part of our functioning as human beings. An example of a gene contained within chromosome 18 is the NPC1 gene, named after the Niemann-Pick disease, type C1. This gene is named after the disease you get when you have an error on the gene. Experiments with mice show that this gene probably controls the appetite, and people with a mutation on the NPC1 gene often suffer from obesitas. Interestingly, the same mutation on NPC1 might make you immune for the ebola virus, one of the deadliest known viruses. So the mutation is partly a blessing in disguise.\n\nIf I search for the keyword palindrome in the electronic publications available at the library of my university, I get more than $500$ hits. The first ten of these hits are all about palindromes in DNA. My guess is that at least $90$% of the list of publications are about palindromes in DNA. So what is the interest in palindromes in DNA? Surely, if your strings only contain A', T', C', and G' you are likely to get many palindromes?\n\nLet us look at how many palindromes we expect to find in the DNA of the Amycolatopsis marina. Suppose I want to find palindromes of length twelve. I calculate the chance that an arbitrary DNA string of length twelve is a palindrome as follows. The first six characters of the string don't matter. The seventh character needs to match with the sixth character, for which we have a chance of one in four. Remember that in DNA, A' matches with T' and vice versa, but both A' and T' do not match with themselves, or with C' and G'. The eighth character needs to match with the fifth character, for which we also have a chance of one in four. This goes on until the twelfth character, so I get a chance of $\\frac{1}{4} \\times \\frac{1}{4} \\times \\frac{1}{4} \\times \\frac{1}{4} \\times \\frac{1}{4} \\times \\frac{1}{4} = (\\frac{1}{4})^6 = \\frac{1}{4^6} = \\frac{1}{4096}$ Since the Amycolatopsis marina is $6,503,724$ characters long, there are $6,503,713$ substrings of length twelve. Multiplying this number with the chance that it is a palindrome, I expect to get $1,589$ palindromes. Using my algorithm for finding palindromes I get $1,784$ palindromes. This is slightly above $10$% more than expected, but that might be an accident. If I look at the palindromes of length fourteen in the $18$th human chromosome, using a similar calculation I expect to find $4,556$ palindromes. My algorithm finds $25,323$. More than five times as many palindromes as expected! This is not an accident anymore. Palindromes play a role in DNA. But what role?\n\nPalindromes perform several tasks in human DNA. I will discuss one particularly intriguing task in this blog post.\n\nWhy do we humans have sex? I can answer this question from several perspectives. The most common answer will probably mention love, pleasure, or children. Looking at the question from a biological perspective, the answer discusses the biological advantages of having sex. Our children get their genes from two parents. They get two complete copies of human DNA, and merge these copies in some way to make them the way they are. In this merging process, damaged DNA from one parent can be replaced by functioning DNA from the other parent. There is a lot of DNA to repair: every day, each cell may be damaged at one million different places. Most of this damage is harmless, since a lot of DNA does not seem to have any function at all, but some kinds of damage may cause a lot of problems. Being able to repair non-functioning DNA when combining DNA from parents is essential for keeping the human race in a good state. The American geneticist Hermann Joseph Muller used this argument to explain why sexual reproduction is favored over asexual reproduction in organisms. When an organism reproduces asexually it passes all its DNA errors on to its offspring, without a possibility to repair them, eventually leading to the extinction of the population. The process has been dubbed Muller's ratchet, after the ratchet device, which can only turn in one direction. This is the theory, practice is slightly more complicated.\n\nMuller's ratchet should already be at work in humans, since there is one human chromosome in which no combination of parental genes takes place: chromosome 23. Chromosome 23 determines whether we become a man or a woman. A man gets a copy of the chromosome called X from his mother and a copy called Y from his father. A woman gets an X from her mother and an X from her father. A woman can merge both X's and repair possible errors, but a man has two different copies of the chromosome, and has no possibility to combine, let alone repair, them. The Y is passed on from father to son, with no involvement of women. Muller's ratchet theory says the genes on the chromosome that make a man should deteriorate quickly, and men should soon become extinct.\n\nThere is no sign of men becoming extinct. Apparently there are some other mechanisms at work to repair DNA on the Y chromosome. If I cannot obtain a copy of some piece of DNA from my mother, maybe I can store copies of the DNA in the Y chromosome itself? If I maintain two copies, I can always check if a piece of DNA is correct by comparing it against the copy of this piece of DNA. This is the mechanism used by the Y chromosome, where the copies are stored as palindromes, with some noise in the middle of these palindromes. Such palindromes with gaps in the middle are often called inverted repeats in biology. The Y chromosome contains eight huge palindromes, the longest of which consists of almost three million characters. Around 25% of the Y chromosome consists of palindromes. The DNA in the palindromes carries genes for describing the male testes. So the mechanism by means of which men survive is called palindrome...\n\n## Sunday, July 1, 2012\n\n### Other implementations and solutions\n\nNext year it is 25 years ago since I constructed an algorithm for finding palindromes efficiently. I was 22 years old then, and had just started as a PhD student. I was quite excited about having solved the problem of finding palindromes efficiently, and wrote a scientific paper about it. My excitement wasn't shared by the scientific community though. In the last twenty-five years this paper has been cited less than ten times, and appears at the bottom end of my most cited papers list.\n\nIn 2007 we developed the ICFP programming contest. The ICFP programing contest is a very challenging contest, in which thousands of programmers try to show off their programming talents in their programming language of choice. Our contest asked participants to transform the left picture below to the right picture using as few commands as possible. We included a problem related to palindromes in our contest, since this was my pet-problem ever since 1988. After the contest I explained how to solve the palindrome problem efficiently in a blog post.\n\nWhen some years later I looked at the number of hits the contest pages received, I found that each month, thousands of people are reading the blog message about finding palindromes. Why does this page attract so many visitors?\n\nThe palindrome problem is a common question in interviews for jobs for software developers, so the blog post attracts software developers looking for a new job, and preparing themselves for an interview. Another reason the blog post attracts visitors is that I think quite a few people are genuinely interested in the problem and its solution. The concept of palindromes appears in almost all languages, which means that the question of finding palindromes is asked all over the world. The blog post indeed attracts visitors from all over the world. The last 100 (July 1, 2012) visitors come from all continents except Australia.\n\nMany people ask the question of how to find palindromes, but also many people try to answer the question. You can find hundreds of solutions for finding palindromes on the internet. Some of these are variants of my linear-time solution, others are more naive quadratic or sometimes even cubic-time solutions. Below I give the list I found, ordered on the programming language used for the implementation. If there exists a linear-time implementation, I don't list less efficient solutions.","date":"2017-09-25 18:33:02","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 2, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.3211069405078888, \"perplexity\": 951.5886501191045}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-39\/segments\/1505818693240.90\/warc\/CC-MAIN-20170925182814-20170925202814-00375.warc.gz\"}"} | null | null |
Acanthinus rohweri is een keversoort uit de familie snoerhalskevers (Anthicidae). De wetenschappelijke naam van de soort is voor het eerst geldig gepubliceerd in 1967 door Werner.
Snoerhalskevers | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,101 |
Tributes to five 'men of courage'
The soldiers had been living in a compound at a police checkpoint
Tributes have been paid to five British soldiers killed in an attack by a "rogue" Afghan policeman on Tuesday.
Defence Secretary Bob Ainsworth said they had been "men of courage" whose memories would live on.
They were Guardsman Jimmy Major, Warrant Officer Darren Chant, Sgt Matthew Telford, Cpl Steven Boote and Cpl Nicholas Webster-Smith.
Meanwhile a soldier from 3rd Battalion, The Rifles, has been killed in an explosion in Helmand, the MoD has said.
'Incomparable courage'
Mr Ainsworth said of the five men killed earlier in the week: "My deepest sympathies and condolences lie with their grieving families, friends, and all those who served alongside them who will feel the pain of loss most intensely. They are in all our thoughts."
WO1 Darren 'Daz' Chant was due to become a father for the fourth time.
The most senior British general in Afghanistan, Lt Gen Jim Dutton, has warned that this kind of atrocity will not be the last.
Gen Dutton told BBC1's Politics Show, to be broadcast on Sunday: "It's not the first time that an Afghan policeman or an Afghan soldier or indeed soldiers of other nations in other theatres have carried out this sort of atrocity.
"And regrettably I think we have to say it probably won't be the last. But it is a very rare event."
WO1 Chant's "devastated" pregnant wife, Nausheen, said: "Our unborn son will never meet his father, but he will know him through his legacy.
'Gentle giant'
"For whether in uniform or out, his incomparable courage and selflessness humbled all those who knew and loved him."...
There is more at the BBC here.
For more, go to Helmand Blog here
The courage of these men WILL live on. Always remembered and honoured.
Labels: Guardsman Jimmy Major Warrant Officer Darren Chant Sgt Matthew Telford Cpl Steven Boote Cpl Nicholas Webster-Smith Men of Courage | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 119 |
package com.github.alebabai.tg2vk.service.tg.update.command.handler.impl;
import com.github.alebabai.tg2vk.domain.Role;
import com.github.alebabai.tg2vk.domain.User;
import com.github.alebabai.tg2vk.repository.UserRepository;
import com.github.alebabai.tg2vk.security.service.JwtTokenFactoryService;
import com.github.alebabai.tg2vk.service.core.PathResolver;
import com.github.alebabai.tg2vk.service.tg.common.TelegramService;
import com.github.alebabai.tg2vk.service.tg.update.command.TelegramCommand;
import com.github.alebabai.tg2vk.service.tg.update.command.handler.TelegramCommandHandler;
import com.pengrad.telegrambot.model.Chat;
import com.pengrad.telegrambot.model.Message;
import com.pengrad.telegrambot.model.request.InlineKeyboardButton;
import com.pengrad.telegrambot.model.request.InlineKeyboardMarkup;
import com.pengrad.telegrambot.model.request.ParseMode;
import com.pengrad.telegrambot.request.SendMessage;
import org.springframework.beans.factory.annotation.Autowired;
import org.springframework.context.support.MessageSourceAccessor;
import org.springframework.stereotype.Service;
import java.util.Optional;
import static com.github.alebabai.tg2vk.util.CommandUtils.getClientRedirectUrl;
import static java.util.Optional.of;
@Service("loginCommandHandler")
public class TelegramLoginCommandHandler implements TelegramCommandHandler {
private final UserRepository userRepository;
private final PathResolver pathResolver;
private final TelegramService tgService;
private final JwtTokenFactoryService tokenFactory;
private final MessageSourceAccessor messages;
@Autowired
public TelegramLoginCommandHandler(UserRepository userRepository,
PathResolver pathResolver,
TelegramService tgService,
JwtTokenFactoryService tokenFactory,
MessageSourceAccessor messages) {
this.userRepository = userRepository;
this.pathResolver = pathResolver;
this.tgService = tgService;
this.tokenFactory = tokenFactory;
this.messages = messages;
}
@Override
public void handle(TelegramCommand command) {
final Message context = command.context();
final SendMessage loginMessage = of(context.chat().type())
.filter(Chat.Type.Private::equals)
.map(type -> createLoginSendMessage(context))
.orElseGet(() -> createAccessDeniedSendMessage(context.chat().id(), messages.getMessage("tg.command.login.msg.denied")));
tgService.send(loginMessage);
}
private SendMessage createAccessDeniedSendMessage(Long id, String message2) {
return new SendMessage(id, message2);
}
private SendMessage createLoginSendMessage(Message message) {
final Optional<User> userOptional = userRepository.findOneByTgId(message.from().id());
final String loginText = userOptional
.map(user -> String.join(
"\n\n",
messages.getMessage("tg.command.login.msg.warning"),
messages.getMessage("tg.command.login.msg.instructions")
))
.orElse(messages.getMessage("tg.command.login.msg.instructions"));
final Role[] roles = userOptional
.map(user -> user.getRoles().toArray(new Role[0]))
.orElse(new Role[]{Role.USER});
return createAccessDeniedSendMessage(message.chat().id(), loginText)
.parseMode(ParseMode.Markdown)
.replyMarkup(new InlineKeyboardMarkup(new InlineKeyboardButton[]{
new InlineKeyboardButton(messages.getMessage("tg.command.login.label.button.get_token"))
.url(pathResolver.resolveServerUrl("/api/redirect/vk-login")),
new InlineKeyboardButton(messages.getMessage("tg.command.login.label.button.send_token"))
.url(getClientRedirectUrl(tokenFactory.generate(message.from().id(), roles), pathResolver.resolveServerUrl("/api/redirect/client"), "revoke-token")),
}));
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,712 |
function [run_result] = OHT_run_distribKSs(params,domain,bdrys,experiment,run_type)
%OHT_run_distribKSs: Function takes ln(K) and ln(Ss)
%values in a column vector and calculates either simulated observations,
%simulated phasor fields (representing real / imaginary components of phasor at all points
%within the model), or the sensitivity matrix.
%
% Syntax:
%[run_result] = OHT_run_distribKSs(params,domain, bdrys, experiment,run_type)
%
%where:
% OUTPUTS:
% -run_result is the result of the model run, consisting of either a set
% of simulated observations, a full picture of the forward model phasor
% field (representing real and imaginary components of the phasor at all
% points within the model), or a sensitivity matrix for the given
% observations. The output type depends on the "run_type" input.
% -Run_type 1: run_result is a vector, representing all of the real
% coefficients of the phasor (for each observation) followed by the
% imaginary coefficients of the phasor. If the number of observations
% is numobs, then the output is a (2*numobs x 1) vector.
% -Run_type 2: run_result is a vector, representing the real part of
% the phasor values at all points in space, followed by the imaginary
% parts of the phasor at all points in space. If there are numcells
% cells in the numerical model, the output is a (2*numcells x 1)
% vector.
% -Run_type 3: run_result is a matrix, representing the sensitivity
% of all observations to K and Ss values in all grid cells. Rows
% consist of all real phasor part sensitivities, followed by all
% imaginary phasor part sensitivities. Columns consist of sensitivity
% to ln(K) in all model cells, followed by sensitivity to ln(Ss) in
% all model cells. Matrix size is thus (2*numobs x 2*numcells)
% INPUTS:
% -params is the set of all K and Ss values for the model, as a
% vector with 2*numcells ln(K) and ln(Ss) values, ordered in the
% standard format according to meshgrid and reshape conventions.
% -domain is a structure containing fields x,y, and z that describe
% the boundaries of domain grid cells
% -bdrys is a structure containing fields types and vals, that
% describe the type and values at the 6 boundaries.
% -experiment is a structure containing all information about
% testing, as generated by OHT_create_inputs.m. See documentation of
% that function for further information.
%
% Code by Michael Cardiff, 2015-2016
%Error checking, extract all needed information from structures
if ~isfield(domain,'x')
error('X boundary information must be supplied in domain_disc.x');
else
xb = domain.x;
end
if ~isfield(domain,'y')
error('Y boundary information must be supplied in domain_disc.y');
else
yb = domain.y;
end
if isfield(domain,'z')
zb = domain.z;
else
zb = [0 1];
end
bdry_types = bdrys.types;
bdry_vals = bdrys.vals;
num_omegas = size(experiment,1);
num_x = numel(xb)-1;
num_y = numel(yb)-1;
num_z = numel(zb)-1;
num_cells = num_x*num_y*num_z;
lnK = params(1:num_cells);
lnSs = params((num_cells+1):(2*num_cells));
lnK_mat = reshape(lnK,num_y,num_x,num_z);
lnSs_mat = reshape(lnSs,num_y,num_x,num_z);
num_obs = zeros(num_omegas,1);
for i = 1:1:num_omegas
num_obs(i) = size(experiment(i).tests,1);
end
num_totalobs = sum(num_obs);
cum_totalobs = [0; cumsum(num_obs)];
num_stims = zeros(num_omegas,1);
for i = 1:1:num_omegas
num_stims(i) = size(experiment(i).stims,2);
end
num_totalstims = sum(num_stims);
cum_totalstims = [0; cumsum(num_stims)];
sim_obs = zeros(2*num_totalobs,1);
Phis = zeros(num_cells,num_totalstims);
if nargout == 2
H = zeros(2*num_totalobs,2*num_cells);
end
for i = 1:1:num_omegas
firstobs = cum_totalobs(i) + 1;
lastobs = cum_totalobs(i+1);
firststim = cum_totalstims(i) + 1;
laststim = cum_totalstims(i+1);
switch run_type
case 1
[sim_obs_omega] = phasor_model_obssens...
(experiment(i).omega,experiment(i).tests,experiment(i).obs,experiment(i).stims,bdry_vals,exp(lnK_mat),exp(lnSs_mat),...
xb,yb,zb,bdry_types);
sim_obs(firstobs:lastobs) = real(sim_obs_omega);
sim_obs((num_totalobs+firstobs):(num_totalobs+lastobs)) = imag(sim_obs_omega);
run_result = sim_obs;
case 2
[~,Phi_omega] = phasor_model_obssens...
(experiment(i).omega,experiment(i).tests,experiment(i).obs,experiment(i).stims,bdry_vals,exp(lnK_mat),exp(lnSs_mat),...
xb,yb,zb,bdry_types);
Phis(1:num_cells,firststim:laststim) = real(Phi_omega);
Phis((num_cells+1):(2*num_cells),firststim:laststim) = imag(Phi_omega);
run_result = Phis;
case 3
[~,~,H_omega] = phasor_model_obssens...
(experiment(i).omega,experiment(i).tests,experiment(i).obs,experiment(i).stims,bdry_vals,exp(lnK_mat),exp(lnSs_mat),...
xb,yb,zb,bdry_types);
%Convert sensitivity from wrt K and Ss to wrt ln(K) and ln(Ss)
for j = 1:1:num_obs
H_omega{1}(j,:) = H_omega{1}(j,:).*exp(lnK)';
H_omega{2}(j,:) = H_omega{2}(j,:).*exp(lnSs)';
end
H(firstobs:lastobs,:) = real([H_omega{1} H_omega{2}]);
H((num_totalobs+firstobs):(num_totalobs+lastobs),:) = imag([H_omega{1} H_omega{2}]);
run_result = H;
otherwise
error('Only run type of 1, 2, or 3 is supported');
end
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,318 |
Letters and feedback: Oct. 22, 2016
Reader: Do not be fooled by the so-called "solar energy" amendment. The summary is deliberately misleading.
Letters and feedback: Oct. 22, 2016 Reader: Do not be fooled by the so-called "solar energy" amendment. The summary is deliberately misleading. Check out this story on floridatoday.com: http://on.flatoday.com/2eBdCxx
BrevardCounty Published 1:29 p.m. ET Oct. 21, 2016
Liberty University President Jerry Falwell, Jr., left, chats with Gary Johnson, Libertarian Party presidential candidate, Oct. 17 at Liberty University.(Photo: AP)
'Vote no on Amendment 1'
Do not be fooled by the so-called "solar energy" amendment. The summary is deliberately misleading. If you read the whole text, you will see that it is a clever conspiracy originated and funded by the electric power industry to discourage people from solar energy generation. Vote NO on Amendment 1.
John C. Webber, Melbourne Beach
Thank you from FPL's CEO
It's been a little more than a decade since a major hurricane impacted Florida, but that all changed with Hurricane Matthew. The Category 4 storm left a trail of destruction, and unfortunately, claimed the lives of several Floridians.
We understand how frustrating it is to be without power, and want to thank you for your understanding and patience, especially those of you who received multiple estimates of when your service would be restored. As the extent of the damage became more apparent, our efforts in certain communities for our 15,000-strong workforce shifted from restoration to rebuilding of electric infrastructure. Instead of replacing wires and fuses, we found ourselves clearing massive trees, replacing and installing new poles and conducting labor-intensive work that extended original estimates.
In the aftermath of the storm, I made it a priority to be in the field to gain a full appreciation of what we together were up against. The same debris and trees that damaged homes damaged our power lines and electric equipment. I also met with a number of state and local leaders, and I was impressed by their strong leadership and how quickly communities rallied together during this challenging time. Here at FPL, we know that returning life to normal after a major hurricane is important and getting the lights back on is a big part of that.
We're grateful to be part of the larger first responder team upon which your community depends. On behalf of the entire FPL family, we thank everyone for their patience and support.
Eric Silagy, Juno Beach
Editor's note: Silagy is president and chief executive officer of Florida Power & Light.
Debris complaint doesn't wash
That West Melbourne resident who is whining about a pile of yard waste on her curb has good reason to complain. Why, my goodness, her grass is dying and those selfish workers won't pick up the junk by hand. How unspeakable is that? What is this world coming to? Lady, get out there and take care of your own yard.
Jack Sayles, Melbourne
Issues should guide our votes
The direction this presidential campaign has taken is nothing but diversion.
There should be only three things influencing our choice:
Who can best address the long-term national debt and the economy?
Who will appoint Supreme Court justices that will rule based on the Constitution, its purpose as outlined in that document, rather than their political philosophies?
Who can best protect the USA from its enemies? There have been less-than-perfect presidents in the past. Richard Nixon and Bill Clinton come to mind. Yet, these men also did good things. Nixon opened up China to us and Clinton balanced the budget.
The media, especially the TV news channels, has determined that it will be the conscience of our nation that will protect us from our obviously ignorant and misguided choices. Don't be taken in by the biased and selective reporting of NBC, CNN, Fox and all. The actual issues should be the only deciding factors in your voting choice.
Chuck Deming, Melbourne
Other-party votes not wasted
I am disturbed by Carl Hiaasen's opinion printed Oct. 20 in FLORIDA TODAY, that casting a vote for any party other than the Democratic or Republican party is a wasted vote.
Hiaasen advocates the two-party system is what's best for America. He could not be more wrong. Republicans and Democrats are too polarized to work for the common good and are paid by special interests to protect those interests. People should vote for the candidate or party they feel provides the best solutions. Gary Johnson was a respected two-time governor of New Mexico, who will more effectively address the issues than either Trump or Clinton.
Hugh Kroehling, Merritt Island
Hiaasen's comments 'spew contempt'
No. The Democrats in 2000 did not lose because of Ralph Nader. They lost because they insisted on trying to shove an unpopular (but decent, policy wonk sort of guy) politician down our throats. After the prosperous Bill Clinton years, 2000 should have been a slam dunk for the Democrats. But not with Al Gore.
If the Democrats lose this election, it will not be because of people who voted their convictions. It will be because the DNC and media have tried to force-feed us another candidate. A candidate well liked and well funded by the inner circles for sure, but not one popular with the people, not one who can provide the inspirational leadership we so yearn for.
This is exactly why we have a Trump this year, and a Sanders and a Johnson. I'm sick of hearing Aleppo. Has anyone listened to interviews with Johnson? Read his position papers? Taken a look at what he accomplished as a governor? I was living in Massachusetts when Weld was governor. He was fantastic. Johnson/Weld is the only ticket in this race that could fix our dysfunctional government. They've both worked with Republicans and Democrats. They've both worked to make the lives of people better in the states they governed.
Carl Hiaasen's words are just like all the others this year, spewing contempt for people who don't see the election as he does. Contempt for people who will vote their convictions, who refuse to accept the lesser of two evils, again.
Dennis Merritt, Rockledge
Read or Share this story: http://on.flatoday.com/2eBdCxx
Proposed Starbucks drive-thru splits residents
Do not rename Airport Blvd for MLK | Opinion
An artificial inlet for the Indian River Lagoon is irresponsible
Letters and feedback: July 11, 2019
Letters and feedback: July 7, 2019
Where is the outrage over migrant children being caged? | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,538 |
Rikis Poppy 3 is a photograph by G Wigler which was uploaded on August 18th, 2011.
Close up photo of golden poppies in Colorado.
There are no comments for Rikis Poppy 3. Click here to post the first comment. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,840 |
\section{Introduction}
Let $X$ be a compact (closed and connected) Riemann surface $X$ of genus $g \geq 2$. Higgs bundles over $X$ were introduced by Hitchin in \cite{H87a}. A \textit{Higgs bundle} over $X$ is a pair $(E, \phi)$ consisting of a a holomorphic vector bundle $E$ and a Higgs field $\phi : E \to E \otimes K$, where $K$ is the canonical bundle over $X$. The coefficients of the characteristic polynomial of $\phi$ defines a morphism
$h : \mathcal{M}_{\mathrm{Higgs}}(r,d) \longrightarrow \mathcal{A}\coloneqq \bigoplus_{i=1}^{r} H^0(X, K^i)$,
from the moduli space of stable Higgs bundles over $X$ of fixed rank $r$ and degree $d$ to a vector space $\mathcal{A}$, called the \textit{Hitchin map} (see \cite{H87}). Hitchin in \cite{H87}, showed that the generic fibers of $h$ are abelian varieties and this map gives the Higgs bundles moduli space a structure of an algebraically completely integrable system. Later in \cite{M94}, Markman generalized this result for the moduli space of $L$-twisted Higgs bundles $(E,\phi_L)$, where $L$ is a line bundle over $X$ and $\phi_L : E \to E \otimes L$.
Let $D \subset X$ be a fixed finite subset. The notion of parabolic bundles over a curve and their moduli spaces were constructed by Mehta and Seshadri in \cite{MS80}. Their motivation was to extend the Narasimhan-Seshadri correspondence in the case of irreducible unitary representations of $\pi_1(X-D)$. A \textit{parabolic bundle} is a holomorphic vector bundle together with a weighted flag over each parabolic point $p\in D$. A \textit{parabolic Higgs bundle} on $X$ is a parabolic bundle $E$ on $X$ together with a parabolic Higgs field $\phi : E \to E \otimes K(D)$. The moduli space of parabolic Higgs bundles was constructed by Yokogawa \cite{Y93}.
Symplectic (resp. orthogonal) parabolic bundles are parabolic bundles with a suitably defined nondegenerate anti-symmetric (resp. symmetric) form taking values in a line bundle $L$ (see \cite{BMW11} for more details). In \cite{BR89}, Bhosle and Ramanathan described the notion of parabolic principal $G$-bundles, where $G$ is a connected reductive group, and also constructed its moduli space. When all weights are rational, the notion of symplectic (resp. orthogonal) parabolic bundles coincides with the notion of parabolic principal $G$-bundles where $G$ is a symplectic (resp. orthogonal) complex group (see \cite{BMW11}). A symplectic (resp. orthogonal) parabolic Higgs bundle is a symplectic (resp. orthogonal) parabolic bundle together with a parabolic Higgs field which is compatible (in a suitable sense) with the symplectic (resp. orthogonal) structures.
In \cite{H87}, Hitchin also showed that the moduli space of stable symplectic/orthogonal Higgs bundles also forms an algebraically completely integrable system, fibered over a vector space, either by a Jacobian or a Prym variety of so-called spectral curves. In \cite{R20}, Roy generalized this result for the moduli space of stable parabolic symplectic/orthogonal Higgs bundles.
In this paper, we consider the moduli space of parabolic projective symplectic/orthogonal Higgs bundles with fixed rank and degre and fixed parabolic structure. We know that for odd rank the projective orthogonal group is same as the odd orthogonal group, i.e. $\mathrm{PSO}(2m+1,\mathbb{C}) =\mathrm{SO}(2m+1,\mathbb{C})$. Therefore, we only consider the projective symplectic group $\mathrm{PSp}(2m,\mathbb{C})$ and projective even orthogonal group $\mathrm{PSO}(2m,\mathbb{C})$. A parabolic $\mathrm{PSp}(2m,\mathbb{C})$-Higgs (resp. $\mathrm{PSO}(2m,\mathbb{C})$-Higgs) bundle lifts to a parabolic $\mathrm{GSp}(2m,\mathbb{C})$-Higgs (resp. $\mathrm{GSO}(2m,\mathbb{C})$-Higgs) bundle. We gave an alternative description of the Prym varieties and we consider an action of the Jacobian group $\mathrm{Jac}(X)$ on the Prym varieties. We showed that the generic Hitchin fibers are isomorpic to the quotient variety (see Theorem \ref{PSp} and Theorem \ref{PSO_theorem}).
Finally we consider the case when the symplectic/orthogonal form takes values in the trivial line bundle $\mathcal{O}_X$ and we fix this line bundle for the respective moduli spaces. In this case, the $2$-torsion subgroup $\mathrm{Jac}_2(X)\subset \mathrm{Jac}(X)$ acts on the Prym varieties and the generic fibers of the Hitchin map are isomorphic to the quotient variety (see Theorem \ref{trivial_PSp} and Theorem \ref{trivial_PSO}).
\section{Preliminaries}
\subsection{Parabolic bundles}\label{parabolic}
Let $X$ be a compact Riemann surface of genus of genus $g\geq 2$. Fix a subset $D = \{p_1,\dots , p_n\} \subset X$ of $n$ distinct marked points.
\begin{definition}
A \textit{parabolic bundle} $E_*$ of rank $r$ over $X$ is a vector bundle $E$ of rank $r$ over $X$ with a parabolic structure over the subset $D$, i.e. for each point $p \in D$
\begin{enumerate}
\item every fiber $E_p$ has a filtration of subspaces, i.e.
\[
E_p \eqqcolon E_{p,1}\supsetneq E_{p,2} \supsetneq \dots \supsetneq E_{p,r_p} \supsetneq E_{p,r_p+1} =\{0\},
\]
\item an increasing sequence of real numbers (parabolic weights) satisfying
\[
0\leq \alpha_1(p) < \alpha_2(p) < \dots < \alpha_{r_p}(p) < 1,
\]
\end{enumerate}
where $1\leq r_p \leq r$ is an integer.
\end{definition}
We denote the collection of all parabolic weights by $\alpha =\{(\alpha_1(p),\alpha_2(p),\dots ,\alpha_{r_p}(p))\}_{p\in D}$ corresponding to a fixed parabolic structure. The parabolic structure $\alpha$ is said to have \textit{full flags} if
\[\mathrm{dim}(E_{p,i}/E_{p,i+1}) = 1
\]
for all $i \in \{1,\dots, r_p\}$ and for each $p\in D$, or equivalently $r_p=r$ for each $p\in D$. In this paper, we will assume that the parabolic structure have full flag at every parabolic point $p \in D$.
The \textit{parabolic degree} of $E_*$ is defined by
\[
\operatorname{pardeg}(E_*) \coloneqq \deg(E)+ \sum\limits_{p\in D}\sum\limits_{i=1}^{r_p} \alpha_i(p) \cdot \dim(E_{p,i}/E_{p,i+1})
\]
and the \textit{parabolic slope} of $E_*$ is defined by
\[
\mu_{\mathrm{par}}(E_*) \coloneqq \frac{\text{pardeg}(E_*)}{r}.
\]
\begin{definition}
A \textit{parabolic homomorphism} $\phi : E_* \to E^\prime_*$ between two parabolic bundles is a homomorphism between underlying vector bundles such that at each parabolic point $p \in D$ we have
\[
\alpha_i(p) > \alpha_j^\prime(p) \implies \phi(E_{p,i}) \subseteq E_{p,j+1}^\prime.
\]
Furthermore, we call such a homomorphism \textit{strongly parabolic} if
\[ \alpha_i(p) \geq \alpha_j^\prime(p) \implies \phi(E_{p,i}) \subseteq E_{p,j+1}^\prime
\]
for every $p \in D$.
\end{definition}
We denote by $\mathrm{PEnd}(E_*)$ and $\mathrm{SPEnd}(E_*)$ the parabolic and strongly parabolic endomorphisms of $E_*$ respectively.
The \textit{dual} and \textit{tensor product} of parabolic bundles can be defined in a natural way (see \cite{Y95}).
\begin{definition}
A \textit{parabolic subbundle} $F_*$ of a parabolic bundle $E_*$ is a subbundle $F\subset E$ of the underlying vector bundle endowed with an induced parabolic structure. An induced parabolic structure on $F$ is defined as follows. For every parabolic point $p\in D$, the quasi-parabolic structure on $F$, i.e. the flag in $F_p$ is given by
\[
F_p \eqqcolon F_{p,1}\supsetneq F_{p,2} \supsetneq \dots \supsetneq F_{p,r'_p} \supsetneq \{0\},
\]
where $F_{p,i}= F_p \cap E_{p,i}$, i.e. we are considering the intersection with the already given flag in $E_p$, and also scrapping all the repetitions of subspaces in the filtration. The weights $0\leq \alpha'_1(p) < \alpha'_2(p) < \dots < \alpha'_{r'_p}(p) < 1$ are taken to be the largest possible among the given weights which are allowed after the intersections, i.e.
\[
\alpha'_i(p) = \mathrm{max}_j\{\alpha_j(p)| F_p \cap E_{p,j}=F_{p,i} \}=\mathrm{max}_j\{\alpha_j(p)| F_{p,i}\subseteq E_{p,j} \}
\]
That is to say, the weight associated to $F_{p,i}$ is the weight $\alpha_j(p)$ such that $F_{p,i}\subseteq E_{p,j}$ but $F_{p,i}\nsubseteq E_{p,j+1}$.
\end{definition}
\begin{definition}
A parabolic bundle $E_*$ is called \textit{semistable} (resp. \textit{stable}) if every nonzero proper subbundle $F_* \subset E_*$ satisfies
\[
\mu_{\mathrm{par}}(F_*) \leq \mu_{\mathrm{par}}(E_*) \hspace{0.2cm} (\mathrm{resp. } \hspace{0.2cm} <).
\]
\end{definition}
The moduli space $\mathcal{M}(\alpha,r,d)$ of stable parabolic bundles over $X$ of fixed rank $r$ and degree $d$ and parabolic structure $\alpha$ was constructed by Mehta and Seshadri in \cite{MS80}. They also showed that $\mathcal{M}(\alpha,r,d)$ is a normal projective variety of dimension
\[
\dim \mathcal{M}(\alpha,r,d) =r^2(g-1) + 1 + \dfrac{n(r^2-r)}{2},
\]
where $n$ is the number of marked points and the last summand comes from the fact that the parabolic structure have full flags over each parabolic point.
\subsection{Parabolic Higgs bundles}
Let $K$ be the canonical bundle over $X$. We write $K(D) \coloneqq K \otimes \mathcal{O}(D)$.
\begin{definition}
A \textit{(strongly) parabolic Higgs bundle} over $X$ is a parabolic bundle $E_*$ over $X$ together with Higgs field $\Phi : E_* \to E_* \otimes K(D)$, such that $\Phi$ is strongly parabolic i.e. $\Phi(E_{p,i}) \subset E_{p,i+1} \otimes \left.K(D)\right|_p$ for all $p \in D$.
\end{definition}
There is also a notion of parabolic Higgs bundle where the Higgs field $\Phi$ is only assumed to be parabolic, i.e. $\Phi(E_{p,i}) \subset E_{p,i} \otimes \left.K(D)\right|_p$ for all $p \in D$. However in this paper we will always assume that the Higgs field is strongly parabolic.
\begin{definition}
A subbundle $F_* \subset E_*$ is called $\Phi$\textit{-invariant} if $\Phi(F_*)\subset F_*\otimes K(D)$.
\end{definition}
\begin{definition}
A parabolic Higgs bundle $(E_*,\Phi)$ is called \textit{semistable} (resp. \textit{stable}) if every nonzero proper $\Phi$-invariant subbundle $F_* \subset E_*$ satisfies
\[
\mu_{\mathrm{par}}(F_*) \leq \mu_{\mathrm{par}}(E_*) \hspace{0.2cm} (\mathrm{resp. } \hspace{0.2cm} <).
\]
\end{definition}
The moduli space $\mathcal{M}_{\mathrm{Higgs}}(\alpha,r,d)$ of stable parabolic Higgs bundles of fixed rank $r$, degree $d$ and parabolic structure $\alpha$ was constructed by Yokogawa in \cite{Y93} (see \cite{BY96} for more details). It is a normal quasi-projective complex variety of dimension
\[
\dim\mathcal{M}_{\mathrm{Higgs}}(\alpha,r,d) = 2\dim \mathcal{M}(\alpha,r,d).
\]
The parabolic version of the Serre duality (see \cite{BY96}, \cite{Y95}) says that
\[
H^1(\mathrm{PEnd}(E_*)) \cong H^0(\mathrm{SPEnd}(E_*) \otimes K(D))^*.
\]
Therefore, there is an open embedding $T^*\mathcal{M}(\alpha,r,d) \xhookrightarrow{} \mathcal{M}_{\mathrm{Higgs}}(\alpha,r,d)$ and that is why the dimension of $\mathcal{M}_{\mathrm{Higgs}}(\alpha,r,d)$ is twice the dimension of $\mathcal{M}(\alpha,r,d)$. Thus the moduli space of parabolic Higgs bundles $\mathcal{M}_{\mathrm{Higgs}}(\alpha,r,d)$ has a symplectic structure induced from the natural symplectic structure of the cotangent space.
Let $\mathrm{Jac}^d(X)$ denote the space of degree $d$ line bundles over $X$. Consider the determinant map
\begin{align*}
\mathrm{det} : \mathcal{M}_{\mathrm{Higgs}}(\alpha,r,d) &\longrightarrow \mathrm{Jac}^d(X) \times H^0(X,K)\\
(E_*,\Phi) &\longmapsto (\wedge^rE_*,\mathrm{trace}(\Phi)).
\end{align*}
Since $\Phi$ is strongly parabolic, $\mathrm{trace}(\Phi) \in H^0(X,K)\subset H^0(X,K(D))$. The moduli space $\mathcal{M}^{\xi}_{\mathrm{Higgs}}(\alpha,r,d)$ of stable parabolic Higgs bundles with fixed determinant $\xi$ is defined by the fiber $\mathrm{det}^{-1}(\xi,0)$, i.e.
\[
\mathcal{M}^{\xi}_{\mathrm{Higgs}}(\alpha,r,d) \coloneqq \mathrm{det}^{-1}(\xi,0).
\]
If the Higgs field is zero, then the dimension of the moduli space $\mathcal{M}^{\xi}_{\mathrm{Higgs}}(\alpha,r,d)$ is given by
\[
\dim\mathcal{M}^{\xi}_{\mathrm{Higgs}}(\alpha,r,d) = 2(g-1)(r^2-1)+nr(r-1).
\]
\subsection{Spectral correspondence}
We will give a description of the spectral correspondence for the moduli space $\mathcal{M}^{\xi}_{\mathrm{Higgs}}(\alpha,r,d)$ (although a similar description can be given for the moduli space $\mathcal{M}_{\mathrm{Higgs}}(\alpha,r,d)$).
Let $p : \mathrm{Tot}(K(D)) \to X$ be the natural projection from the total space of $K(D)$ to $X$ and let $x \in H^0(\mathrm{Tot}(K(D)),p^*K)$ denote the tautological section. Since the Higgs field $\Phi$ is strongly parabolic, the residue at every point of $D$ is nilpotent. Therefore the trace of the map $$\wedge^i\Phi : \wedge^iE_* \to \wedge^iE_* \otimes K(D)^i$$ lies in $K^i(D^{i-1})$ for each $2\leq i \leq r$, where $K^i(D^{j})$ denote the tensor product of the $i$-th power of $K$ and the $j$-th power of $\mathcal{O}(D)$. The coefficients of the characteristic polynomial of $\Phi$ are precisely given by $s_i=\mathrm{trace}(\wedge^i\Phi)$. Therefore, we have the $\textit{Hitchin map}$
\begin{align*}
h_{\mathrm{par}} : \mathcal{M}^{\xi}_{\mathrm{Higgs}}(\alpha,r,d) &\longrightarrow \mathcal{H}_{\mathrm{par}} \coloneqq \bigoplus_{i=2}^{r} H^0(X, K^{i}(D^{i-1}))\\
(E_*,\Phi) &\longmapsto (s_2,\dots , s_{r}).
\end{align*}
By Riemann-Roch theorem, the dimension of the base $\mathcal{H}_{\mathrm{par}}$ is
\[
r^2(g-1) +\dfrac{nr(r-1)}{2},
\]
which is same as the half the dimension of the moduli space $\mathcal{M}^{\xi}_{\mathrm{Higgs}}(\alpha,r,d)$.
Given $s=(s_2,s_3\dots, s_{r}) \in \mathcal{H}_{\mathrm{par}}$ with
\[
s_i \in H^0(X, K^{i}(D^{i-1})) \subset H^0(X, K(D)^i),
\]
the \textit{spectral curve} $X_s$ in $\mathrm{Tot}(K(D))$ is defined by
\[
x^{r} + \tilde{s}_2 x^{r-2} + \tilde{s}_3 x^{r-3} \cdots + \tilde{s}_{r}=0
\]
where $\tilde{s}_i = p^*(s_i)$ and $x \in H^0(\mathrm{Tot}(K(D)),p^*K)$ is the tautological section. Let $$\pi : X_s \to X$$ be the restriction of the projection $p$. For a generic point $s \in \mathcal{H}_{\mathrm{par}}$, the spectral curve $X_s$ is smooth and by \cite{GL11} the fiber $h_{\mathrm{par}}^{-1}(s)$ of the Hitchin map is isomorphic to
\[
\mathrm{Prym}(X_s/X) = \{L \in \mathrm{Pic}(X) : \det \pi_*L \cong \xi\}.
\]
\subsection{Parabolic $\mathrm{GSp}(2m,\mathbb{C})$-Higgs bundles}
Let us consider the standard symplectic form
\[
J = \begin{bmatrix} 0 & I_{m} \\ -I_{m} & 0 \end{bmatrix}
\]
on $\mathbb{C}^{2m}$. Then the general symplectic group defined by
\[
\mathrm{GSp}(2m,\mathbb{C}) = \{ A \in \mathrm{GL}(2m,\mathbb{C}) : AJA^t=\lambda_A J \text{ for some } \lambda_A \in \mathbb{C}^*\}
\]
is an extension of $\mathbb{C}^*$ by the symplectic group $\mathrm{Sp}(2m,\mathbb{C})$, i.e. there exist an exact sequence
\[
1 \to \mathrm{Sp}(2m,\mathbb{C}) \xrightarrow{} \mathrm{GSp}(2m,\mathbb{C}) \xrightarrow[]{p} \mathbb{C}^* \xrightarrow[]{} 1,
\]
where $p(A)=\lambda_A$. Therefore, $\det(A)=p(A)^m=\lambda_A^m$ for every $A \in \mathrm{GSp}(2m,\mathbb{C})$.\\
The Lie algebra of $\mathrm{GSp}(2m,\mathbb{C})$ is given by
\[
\mathfrak{gsp}(2m,\mathbb{C}) = \{M \in \mathfrak{gl}(2m,\mathbb{C}) : MJ + JA^t = \frac{\mathrm{tr}(M)}{m}J\} \cong \mathfrak{sp}(2m,\mathbb{C}) \oplus \mathbb{C}.
\]
The decomposition $M = N + \frac{\mathrm{tr}(M)}{m}I_{2m}$, with $N \in \mathfrak{sp}(2m,\mathbb{C})$ produces the above isomorphism.
Let $L$ be a line bundle over $X$ of degree $l$. Let $E_*$ be a parabolic bundle and let
\begin{equation}\label{bilinear}
\varphi : E_* \otimes E_* \to L
\end{equation}
be a homomorphism of parabolic bundles. The trivial line bundle $\mathcal{O}_X$ equipped with the trivial parabolic structure is realized as a parabolic subbundle of $E_* \otimes E^\vee_*$ by sending a locally defined function $f$ to the locally defined endomorphism of $E$ given by pointwise multiplication with $f$. Let
\begin{equation}\label{nondegenerate}
\tilde{\varphi} : E_* \to L \otimes E^\vee_*
\end{equation}
be the homomorphism defined by the composition
\[
E_* = E_* \otimes \mathcal{O}_X \xhookrightarrow{} E_* \otimes (E_* \otimes E^\vee_*) = (E_* \otimes E_*) \otimes E^\vee_* \xrightarrow{\varphi \otimes Id} L \otimes E^\vee_*.
\]
\begin{definition}\label{maindefn}
A \textit{symplectic parabolic bundle} is a pair $(E_*,\varphi)$ of the above form such that $\varphi$ is anti-symmetric and the homomorphism $\tilde{\varphi}$ is an isomorphism.
\end{definition}
Suppose $E$ is the underlying vector bundle of a symplectic parabolic bundle $(E_*,\varphi)$. The tensor product $E \otimes E$ is a coherent subsheaf of the vector bundle underlying the parabolic bundle $E_*\otimes E_*$. Therefore, $\varphi$ induces a homomorphism
\begin{equation}\label{underlying}
\hat{\varphi} : E \otimes E \to L
\end{equation}
of vector bundles.
\begin{definition}
A subbundle $F \subset E$ of the underlying bundle of $(E_*,\varphi)$ is called \textit{isotropic} if $\hat{\varphi}(F \otimes F)=0$.
\end{definition}
A parabolic Higgs field $\Phi$ on a symplectic parabolic bundle $(E_*,\varphi)$ is said to be \textit{compatible} with $\varphi$ if $\tilde{\varphi}$ takes $\Phi$ to the induced parabolic Higgs field on $L \otimes E^\vee_*$ (we are considering the zero section as the Higgs field on $L$). We can describe this compatibility condition locally. A strongly parabolic Higgs field $\Phi$ on $E_*$ can be viewed as a holomorphic section of $\mathrm{SPEnd}(E_*) \otimes K(D)$. Let $s$ and $t$ be any holomorphic sections of $E_*$ defined over an open subset $U \subset X$. Consider
\[
\hat{\varphi}_{\Phi}(s,t) \coloneqq \hat{\varphi}(\Phi(s)\otimes t)+ \hat{\varphi}(s \otimes \Phi(t)) \in \Gamma(U,L\otimes K(D)),
\]
where $\hat{\varphi}$ is the pairing defined in \ref{underlying}. The Higgs field $\Phi$ is said to be compatible with $\phi$ if and only if $\hat{\varphi}_{\Phi}(s,t)=0$ for all sections $s$ and $t$.
\begin{definition}
A \textit{symplectic parabolic Higgs bundle} $(E_*,\varphi,\Phi)$ is a symplectic parabolic bundle $(E_*,\varphi)$ equipped with a parabolic Higgs field $\Phi$ on $E_*$ which is compatible with $\varphi$.
\end{definition}
When the parabolic weights are all rational, the notion of symplectic parabolic Higgs bundle is equivalent to the notion of parabolic $\mathrm{GSp}(2m,\mathbb{C})$-Higgs bundles (see \cite{BMW11}).
\subsection{Parabolic $\mathrm{GSO}(2m,\mathbb{C})$-Higgs bundles}
Let $B$ be a non-degenerate symmetric bilinear form on $\mathbb{C}^{2m}$. Also for notational convenience, we denote the corresponding symmetric matrix by $B$. Then the general (even) orthogonal group
\[
\mathrm{GO}(2m,\mathbb{C}) = \{ A \in \mathrm{GL}(2m,\mathbb{C}) : ABA^t=\lambda_A J \text{ for some } \lambda_A \in \mathbb{C}^*\}.
\]
therefore,, we have $(\det(A))^2=\lambda_A^{2m}$ for every $A \in \mathrm{GO}(2m,\mathbb{C})$.
In this case, there is a \textit{sgn} morphism
\[
\textit{sgn} : \mathrm{GO}(2m,\mathbb{C}) \longrightarrow \{\pm 1\}
\]
sending $A$ to $\det(A)/\lambda_A^m$. The general special orthogonal group is the kernel of this \textit{sgn} morphism and it is denoted by $\mathrm{GSO}(2m,\mathbb{C})= \mathrm{ker}(\textit{sgn})$. So we have a short exact sequence
\[
1 \to \mathrm{GSO}(2m,\mathbb{C}) \xrightarrow{} \mathrm{GO}(2m,\mathbb{C}) \xrightarrow[]{\textit{sgn}} \{\pm 1\} \xrightarrow[]{} 1.
\]
\begin{definition}\label{maindefn}
An \textit{orthogonal parabolic bundle} is a pair $(E_*,\varphi)$, where $\varphi$ (as in \ref{bilinear}) is symmetric and the homomorphism $\tilde{\varphi}$ (as in \ref{nondegenerate}) is an isomorphism.
\end{definition}
\begin{definition}
An \textit{orthogonal parabolic Higgs bundle} $(E_*,\varphi,\Phi)$ is an orthogonal parabolic bundle $(E_*,\varphi)$ equipped with a parabolic Higgs field $\Phi$ on $E_*$ which is compatible with $\varphi$.
\end{definition}
As in the $\mathrm{GSp}$-case, for rational weights the notions of (even) orthogonal parabolic Higgs bundles and parabolic $\mathrm{GSO}(2m,\mathbb{C})$-Higgs bundles coincide.
\subsection{Moduli space}
\begin{definition}
A symplectic/orthogonal parabolic Higgs bundle $(E_*,\varphi,\Phi)$ is said to be \textit{semistable} (resp. \textit{stable}) if every nonzero isotropic subbundle $F \subset E$ such that $\Phi(F_*) \subset F_* \otimes K(D)$ satisfies
\[
\mu_{par}(F_*) \leq \mu_{par}(E_*) \hspace{0.4cm}(\text{resp.} \hspace{0.15cm} < )
\]
holds, where $F_*\subset E_*$ has the induced parabolic structure.
\end{definition}
The moduli space $\mathcal{M}_G(\alpha)$ of stable parabolic $G$-bundles of a fixed topological type and with a fixed parabolic structure $\alpha$ is a normal quasi-projective variety (see \cite{BBN01}, \cite{BR89}) of dimension
\[
\dim\mathcal{M}_G(\alpha)= \dim Z(G)+ (g-1)\dim(G) + n\dim(G/B),
\]
where $Z(G)$ denotes the the center of $G$ and $n$ is the number of parabolic points. The last summand comes from the fact that the flags we are considering over each point of $D$ are full flags and $B$ is the Borel subgroup of $G$ determined by $\alpha$. The moduli space $\mathcal{M}_{G\mathrm{-Higgs}}(\alpha)$ of stable parabolic $G$-Higgs bundles (see \cite{R16}) is a normal quasi-projective variety of dimension
\[
\dim \mathcal{M}_{G\mathrm{-Higgs}}(\alpha) = 2\dim \mathcal{M}_G(\alpha).
\]
In particular when $G=\mathrm{GSp}(2m,\mathbb{C})$, the moduli space $\mathcal{M}_{\mathrm{GSp-Higgs}}(\alpha,2m,d)$ of stable parabolic $\mathrm{GSp}(2m,\mathbb{C})$-Higgs bundle of fixed degree $d$ has dimension
\[
\dim \mathcal{M}_{\mathrm{GSp-Higgs}}(\alpha,2m,d) = 2m(2m+1)(g-1) + 2m^2n.
\]
Similarly, the moduli space $\mathcal{M}_{\mathrm{GSO-Higgs}}(\alpha,2m,d)$ of stable parabolic $\mathrm{GSO}(2m,\mathbb{C})$-Higgs bundle of fixed degree $d$ has dimension
\[
\dim \mathcal{M}_{\mathrm{GSO-Higgs}}(\alpha,2m,d) = 2m(2m-1)(g-1) + 2mn(m-1).
\]
\subsection*{Notation:} From now on, for notational convenience we shall denote a parabolic bundle $E_*$ by $E$.
\section{Parabolic $\mathrm{PSp}(2m,\mathbb{C})$-Higgs bundles}
In this section we will discuss the Hitchin fibration for the moduli space of parabolic $\mathrm{PSp}(2m,\mathbb{C})$-Higgs bundles. The projective symplectic group $\mathrm{PSp}(2m,\mathbb{C})$ is given by the following exact sequence:
\begin{equation*}
1 \xrightarrow{} \mathbb{C}^* \xrightarrow{c \to cI_{2m}} \mathrm{GSp}(2m,\mathbb{C}) \xrightarrow{} \mathrm{PSp}(2m,\mathbb{C}) \xrightarrow{} 1.
\end{equation*}
The sheaf version of this sequence induces the following exact sequence in homology :
\begin{equation}
H^1(X,\mathcal{O}_X^*) \xrightarrow{} H^1(X,\mathrm{GSp}(2m,\mathcal{O}_X)) \xrightarrow{q} H^1(X,\mathrm{PSp}(2m,\mathcal{O}_X)) \to 0
\end{equation}
The surjectivity of the map $q$ implies that there is a bijective correspondence between the parabolic $\mathrm{PSp}(2m,\mathbb{C})$-bundles and the equivalence classes of parabolic $\mathrm{GSp}(2m,\mathbb{C})$-bundles with respect to the action given by the tensor product of line bundles on the associated bundles. If $V$ is a parabolic $\mathrm{PSp}(2m,\mathbb{C})$-bundle and $\Tilde{V}$ is a parabolic $\mathrm{GSp}(2m,\mathbb{C})$-bundle such that $q(\Tilde{V}) = V$, then we call that $\Tilde{V}$ is a lifting of $V$ to a parabolic $\mathrm{GSp}(2m,\mathbb{C})$-bundle.
The group $\mathrm{PSp}(2m,\mathbb{C})$ can also be defined by the quotient of the symplectic group $\mathrm{Sp}(2m,\mathbb{C})$ by the action of a finite group by the following exact sequence :
\begin{equation*}
1 \xrightarrow{} \{\pm 1\} \xrightarrow{1 \to I_{2m}} \mathrm{Sp}(2m,\mathbb{C}) \xrightarrow{} \mathrm{PSp}(2m,\mathbb{C}) \xrightarrow{} 1.
\end{equation*}
Since it is a quotient by a finite group, the Lie algebras are equal, i.e.
\begin{equation}\label{equalLie}
\mathfrak{sp}(2m,\mathbb{C}) = \mathfrak{psp}(2m,\mathbb{C}).
\end{equation}
Consider a parabolic $\mathrm{PSp}(2m,\mathbb{C})$-Higgs bundle $(V,\eta)$ which lifts to a parabolic $\mathrm{GSp}(2m,\mathbb{C})$-Higgs bundle $(\Tilde{V},\Tilde{\eta})$, i.e. $q(\Tilde{V},\Tilde{\eta}) = (V,\eta)$. Let $(E,\Phi,\varphi,L)$ be the bundle corresponding to $(\Tilde{V},\Tilde{\eta})$. Then $(V,\eta)$ corresponds to the equivalence class $[(E,\Phi,\varphi,L)]$ where the equivalence relation $\sim_{\mathrm{Jac}(X)}$ is given by :
\[
(E,\Phi,\varphi,L) \sim_{\mathrm{Jac}(X)} (E\otimes M,\Phi \otimes \mathrm{Id}_M,\varphi_M,L\otimes M^2) \hspace{1cm} \mathrm{for} \hspace{0.1cm} \mathrm{any} \hspace{0.2cm} M\in {\mathrm{Jac}(X)}
\]
where
\[
\varphi_M : (E\otimes M) \otimes (E\otimes M) \to L \otimes M^2
\]
is the induced symplectic form on $E\otimes M$ taking values in $L\otimes M^2$.
Let $M_{\mathrm{GSp-Higgs}}(\alpha,2m) = \coprod_{d\in \mathbb{Z}} \mathcal{M}_{\mathrm{GSp-Higgs}}(\alpha,2m,d)$ denote the moduli stack of parabolic $\mathrm{GSp}$-Higgs bundles of fixed rank $2m$ with any degree. Then the $\mathrm{Jac}(X)$-orbit $[(E,\Phi,\varphi,L)]$ of $(E,\Phi,\varphi,L)$ under the action of $\mathrm{Jac}(X)$ on the stack $M_{\mathrm{GSp-Higgs}}(\alpha,2m)$ is defined by:
\begin{align*}
M_{\mathrm{GSp-Higgs}}(\alpha,2m) \times \mathrm{Jac}(X) &\longrightarrow M_{\mathrm{GSp-Higgs}}(\alpha,2m) \\
((E,\Phi,\varphi,L),M) &\mapsto (E\otimes M,\Phi \otimes \mathrm{Id}_M,\varphi_M,L\otimes M^2).
\end{align*}
Note that $\deg(L\otimes M^2) = \deg(L)+2\deg(M)$, i.e. it changes the degree of $L$ by a multiple of $2$. For a parabolic $\mathrm{PSp}(2m,\mathbb{C})$- Higgs bundle $[(E,\Phi,\varphi,L)]$, the isomorphism $E \cong E^\vee \otimes L$ implies that $\mathrm{pardeg}(E) = m\deg(L)$. So the parabolic degree of $E$ is determined by the degree of $L$. Therefore,
\begin{align*}
\mathrm{pardeg}(E\otimes M) &= m\deg(L\otimes M^2) \\ &=m\deg(L) + 2m\deg(M)\\ &= \mathrm{pardeg}(E) + 2m\deg(M).
\end{align*}
Therefore, the parabolic degree of a parabolic $\mathrm{PSp}(2m,\mathbb{C})$-Higgs bundle can be defined as follows :
\begin{definition}
Let $(V,\eta)$ be a parabolic $\mathrm{PSp}(2m,\mathbb{C})$-Higgs bundle and let $(\Tilde{V},\Tilde{\eta})$ be a lifting to a parabolic $\mathrm{GSp}(2m,\mathbb{C})$-Higgs bundle. Let $(E,\Phi,\varphi,L)$ be the datum corresponding to $(\Tilde{V},\Tilde{\eta})$ with parabolic degree $ml$, where $l=\deg(L)$. Then the $\textit{parabolic degree}$ of $(V,\eta)$ is given by the class $\overline{ml} \in \mathbb{Z}/m\mathbb{Z}$.
\end{definition}
Threfore, we will consider the moduli space $\mathcal{M}_{\mathrm{PSp-Higgs}}(\alpha,2m,\overline{ml})$ of stable parabolic $\mathrm{PSp}(2m,\mathbb{C})$-Higgs bundles of fixed rank $2m$ and parabolic degree $\overline{ml}$.
Consider a basis $\{s_{2i}\}_{i =1,\dots , m}$ of invariant polynomials of the lie algebra $\mathfrak{sp}(2m,\mathbb{C}) = \mathfrak{psp}(2m,\mathbb{C})$, where $s_{2i}=\mathrm{tr}(\wedge^{2i}\eta)$. Therefore the Hitchin morphism for the moduli space of parabolic $\mathrm{PSp}(2m,\mathbb{C})$-Higgs bundles is of the form
\begin{align*}
h_{\mathrm{PSp-par}} : \mathcal{M}_{\mathrm{PSp-Higgs}}(\alpha,2m,\overline{ml}) &\longrightarrow \mathcal{H}_{\mathrm{PSp-par}} \coloneqq \bigoplus_{i=1}^{m} H^0(X, K^{2i}(D^{2i-1}))\\
(V,\eta) &\longmapsto (s_2,\dots , s_{2m}).
\end{align*}
Observe that if $(\Tilde{V},\Tilde{\eta})$ is a lifting of $(V,\eta)$ to a parabolic $\mathrm{GSp}(2m,\mathbb{C})$-Higgs bundle then
\[
h_{\mathrm{GSp-par}}(\Tilde{V},\Tilde{\eta}) = h_{\mathrm{PSp-par}}(V,\eta).
\]
Let $s=(s_2,s_4,\dots , s_{2m}) \in \mathcal{H}_{\mathrm{PSp-par}}$ be a generic point of the Hitchin base. The spectral curve
\[
\pi : X_s \to X
\]
is defined by the equation
\[
x^{2m} + s_2x^{2m-2} + s_4x^{2m-4}+ \cdots + s_{2m} = 0.
\]
For a generic $s\in \mathcal{H}_{\mathrm{PSp-par}}$, the corresponding spectral curve $X_s$ is smooth (see \cite{BNR89}).
Since all odd coefficients of the above equation are zero, the spectral curve $X_s$ possesses an involution $\sigma : X_s \to X_s$ defined by $\sigma(\lambda)=-\lambda$. Therefore, we can define a $2$-fold covering map
\[
q : X_s \to X_s/\sigma.
\]
Since $\sigma$ sends a degree zero line bundle on $X_s$ to a degree zero line bundle, it acts on the Jacobian $\text{Jac}(X_s)$. The \textit{Prym variety} $P_{s,\sigma}\coloneqq \mathrm{Prym}(X_s, X_s/\sigma)$ is given by
\[
P_{s,\sigma} \coloneqq \mathrm{Prym}(X_s, X_s/\sigma) = \{N \in \text{Jac}(X_s): \sigma^*N \cong N^\vee\}
\]
and it is of dimension
\[
\dim P_{s,\sigma}=g(X_s) - g(X_s/\sigma).
\]
Following \cite[Theorem 4.1]{R20}, we can give a different description of the Prym variety. Let $J \in P_{s,\sigma}$ be an element in the Prym variety. Consider the line bundle
\[
U=J \otimes R^\vee,
\]
where $R = (K_{X_s} \otimes \pi^*K^\vee \otimes \pi^*L^\vee)^{1/2}$ is a holomorphic square root. Then it follows that $U$ satisfies the isomorphism
\begin{equation}\label{alternative}
\sigma^*U \cong U^\vee \otimes (K_{X_s} \otimes \pi^*K^\vee)^{-1} \otimes \pi^*L.
\end{equation}
Similarly, let $U \in \text{Jac}(X_s)$ be a line bundle satisfying the equation (\ref{alternative}). Then the line bundle $J = U \otimes R \in P_{s,\sigma}$ is an element in the Prym variety. Therefore, there is a bijective correspondence between the Prym variety $P_{s,\sigma}$ and
\begin{equation}\label{prym}
\Omega_{s,\sigma} \coloneqq \{(U,L,\tau)\hspace{0.1cm}|\hspace{0.1cm} U \in \text{Jac}(X_s), \tau : \sigma^*U \cong U^\vee \otimes (K_{X_s} \otimes \pi^*K^\vee)^{-1} \otimes \pi^*L \}.
\end{equation}
From now on, we will refer an element in the Prym variety as an element of $\Omega_{s,\sigma}$.
The Jacobian $\mathrm{Jac}(X)$ acts on $\Omega_{s,\sigma}$ as follows:\
\begin{align*}
\Omega_{s,\sigma} \times \mathrm{Jac}(X) &\longrightarrow \Omega_{s,\sigma}\\
((U,L,\tau),M) &\longmapsto (U \otimes \pi^*M, L \otimes M^2, \tau_M)
\end{align*}
where $\tau_M= \tau \otimes \text{Id}_{\pi^*M}$ is the following isomorphism
\begin{align*}
\sigma^*(U \otimes \pi^*M) &\cong \sigma^*U \otimes \pi^*M\\
&\cong U^\vee \otimes (K_{X_s} \otimes \pi^*K^\vee)^{-1} \otimes \pi^*L \otimes \pi^*M\\
&\cong U^\vee \otimes \pi^*M^\vee \otimes \pi^*M \otimes (K_{X_s} \otimes \pi^*K^\vee)^{-1} \otimes \pi^*L \otimes \pi^*M\\
&\cong (U\otimes \pi^*M)^\vee \otimes (K_{X_s} \otimes \pi^*K^\vee)^{-1} \otimes \pi^*(L\otimes M^2).
\end{align*}
\begin{theorem}\label{PSp}
For a generic point $s\in \mathcal{H}_{\mathrm{PSp-par}}$, the fiber $h_{\mathrm{PSp-par}}^{-1}(s)$ is isomorphic to the quotient $\Omega_{s,\sigma}/\mathrm{Jac}(X)$.
\end{theorem}
\begin{proof}
Let $(V,\eta) \in h_{\mathrm{PSp-par}}^{-1}(s)$ be a parabolic $\mathrm{PSp}(2m,\mathbb{C})$-Higgs bundle in the fiber of the Hitchin morphism and let $(\Tilde{V},\Tilde{\eta})$ be a lifting to a parabolic $\mathrm{GSp}(2m,\mathbb{C})$-Higgs bundle which corresponds to the datum of $(E,\Phi,\varphi,L)$. Then $(V,\eta)$ corresponds to the datum of the $\mathrm{Jac}(X)$-orbit $[(E,\Phi,\varphi,L)]$ uniquely. By \cite[Theorem 4.1]{R20}, the datum of $(E,\Phi,\varphi,L)$ corresponds to an element of the Pyrm variety $\Omega_{s,\sigma}$ via the spectral correspondence. Consider an element $M \in \mathrm{Jac}(X)$, i.e. a degree zero line bundle over $X$. Then by the projection formula the datum of $(E \otimes M,\Phi \otimes \text{Id}_M,\varphi_M,L\otimes M^2)$ corresponds uniquely to an element of $\Omega_{s,\sigma}/\mathrm{Jac}(X)$. Therefore, we conclude that the datum of the $\mathrm{Jac}(X)$-orbit $[(E,\Phi,\varphi,L)]$ corresponds uniquely to the datum of the $\mathrm{Jac}(X)$-orbit of an element of $\Omega_{s,\sigma}$ via the spectral correspondence.
\end{proof}
\section{Parabolic $\mathrm{PSO}(2m,\mathbb{C})$-Higgs bundles}\label{PSO}
As in the previous section, there is a bijective correspondence between the parabolic $\mathrm{PSO}(2m,\mathbb{C})$-bundles and the equivalence classes of parabolic $\mathrm{GSO}(2m,\mathbb{C})$-bundles. So, for every parabolic $\mathrm{PSO}(2m,\mathbb{C})$-bundle $V$ there is a lifting $\Tilde{V}$ of $V$ to a parabolic $\mathrm{GSO}(2m,\mathbb{C})$-bundle.
The group $\mathrm{PSO}(2m,\mathbb{C})$ can be defined by the quotient of $\mathrm{SO}(2m,\mathbb{C})$ by the action of a finite group by the following exact sequence :
\begin{equation*}
1 \xrightarrow{} \{\pm 1\} \xrightarrow{1 \to I_{2m}} \mathrm{SO}(2m,\mathbb{C}) \xrightarrow{} \mathrm{PSO}(2m,\mathbb{C}) \xrightarrow{} 1.
\end{equation*}
Therefore, we have
\begin{equation}\label{equalLie}
\mathfrak{so}(2m,\mathbb{C}) = \mathfrak{pso}(2m,\mathbb{C}).
\end{equation}
Let $(V,\eta)$ be a parabolic $\mathrm{PSO}(2m,\mathbb{C})$-Higgs bundle lifting to the parabolic $\mathrm{GSO}(2m,\mathbb{C})$-Higgs bundle $(\Tilde{V},\Tilde{\eta})$ and let $(E,\Phi,\varphi,L)$ be the bundle corresponding to $(\Tilde{V},\Tilde{\eta})$. Then $(V,\eta)$ corresponds to the equivalence class $[(E,\Phi,\varphi,L)]$ where the equivalence relation $\sim_{\mathrm{Jac}(X)}$ is given by :
\[
(E,\Phi,\varphi,L) \sim_{\mathrm{Jac}(X)} (E\otimes M,\Phi \otimes \mathrm{Id}_M,\varphi_M,L\otimes M^2) \hspace{1cm} \mathrm{for} \hspace{0.1cm} \mathrm{any} \hspace{0.2cm} M\in {\mathrm{Jac}(X)}
\]
where
\[
\varphi_M : (E\otimes M) \otimes (E\otimes M) \to L \otimes M^2
\]
is the induced symmetric bilinear nondegenerate form on $E\otimes M$ taking values in $L\otimes M^2$.
As in the previous case, the parabolic degree of a parabolic $\mathrm{PSO}(2m,\mathbb{C})$-Higgs bundle can be defined as follows:
\begin{definition}
Let $(V,\eta)$ be a parabolic $\mathrm{PSO}(2m,\mathbb{C})$-Higgs bundle and let $(\Tilde{V},\Tilde{\eta})$ be a lifting to a parabolic $\mathrm{GSO}(2m,\mathbb{C})$-Higgs bundle. Let $(E,\Phi,\varphi,L)$ be the datum corresponding to $(\Tilde{V},\Tilde{\eta})$ with parabolic degree $ml$, where $l=\deg(L)$. Then the $\textit{parabolic degree}$ of $(V,\eta)$ is given by the class $\overline{ml} \in \mathbb{Z}/m\mathbb{Z}$.
\end{definition}
Let $\{s_{2i}\}_{i =1,\dots , m}$ be a basis of invariant polynomials of the lie algebra $\mathfrak{so}(2m,\mathbb{C}) = \mathfrak{pso}(2m,\mathbb{C})$
In this case, the coefficient $s_{2m}$ is a square of a polynomial $p_m\in H^0(X,K(D)^m)$, the Pfaffian, of degree $m$. A basis for the invariant polynomials on the Lie algebra $\mathfrak{so}(2m,\mathbb{C}) = \mathfrak{pso}(2m,\mathbb{C})$ is given by the coefficients $\{s_2, ..., s_{2m-2},p_m\}$. Therefore, the Hitchin map for the moduli space of parabolic $\mathrm{PSO}(2m,\mathbb{C})$-Higgs bundles is given by
\begin{align*}
h_{\mathrm{PSO-par}} : \mathcal{M}_{\mathrm{PSO-Higgs}}(\alpha,2m,\overline{ml}) &\longrightarrow \mathcal{H}_{\mathrm{PSO-par}} \coloneqq \bigoplus_{i=1}^{m-1} H^0(X, K^{2i}(D^{2i-1})) \oplus H^0(X,K(D)^{m})\\
(V,\eta) &\longmapsto (s_2,\dots , s_{2m-2},p_m).
\end{align*}
For $s=(s_2,\dots, s_{2m-2},p_m)\in \mathcal{H}_{\mathrm{PSO-par}}$, the corresponding spectral curve $X_s$ is given by the equation
\[
x^{2m} + s_2x^{2m-2} + \cdots + s_{2m-2}x^2 + p_m^2 = 0.
\]
The zeroes of $p_m$ are singularities of $X_s$ and these are the only singularities. Since $p_m \in H^0(X,K(D)^{m})$, there are $K(D)^m = m(2g-2+n)$ many singularities. As in the previous case, $X_s$ possesses an involution $\sigma(\eta)=-\eta$. Then the fixed points of this involution $\sigma$ are exactly the singularities of $X_s$. Let $\hat{X_s}$ denote the desingularisation of $X_s$ with genus
\[
g(\hat{X_s})= g(X_s) - \mathrm{number \hspace{0.1cm }of\hspace{0.1cm} singularities}.
\]
Since the singularities of $X_s$ are double points, the involution $\sigma$ on $X_s$ extends to an involution $\hat{\sigma}$ on $\hat{X_s}$. Therefore the \textit{Prym variety} $P_{s,\hat{\sigma}}\coloneqq \mathrm{Prym}(\hat{X_s}, \hat{X_s}/\hat{\sigma})$ is given by
\begin{equation}\label{desingular}
P_{s,\hat{\sigma}} \coloneqq \mathrm{Prym}(\hat{X_s}, \hat{X_s}/\hat{\sigma}) = \{N \in \text{Jac}(\hat{X_s}): \hat{\sigma}^*N \cong N^\vee\}
\end{equation}
As in the symplectic case, there is a bijective correspondence between the Prym variety $P_{s,\hat{\sigma}}$ and
\[
\Omega_{s,\hat{\sigma}} \coloneqq \{(U,L,\tau)\hspace{0.1cm}|\hspace{0.1cm} U \in \text{Jac}(\hat{X_s}), \tau : \hat{\sigma}^*U \cong U^\vee \otimes (K_{\hat{X_s}} \otimes \pi^*K^\vee)^{-1} \otimes \pi^*L \}.
\]
Again the action of the group $\mathrm{Jac}(X)$ on $\Omega_{s,\hat{\sigma}}$ is given by\
\begin{align*}
\Omega_{s,\hat{\sigma}} \times \mathrm{Jac}(X) &\longrightarrow \Omega_{s,\hat{\sigma}}\\
((U,L,\tau),M) &\longmapsto (U \otimes \pi^*M, L \otimes M^2, \tau_M)
\end{align*}
where $\tau_M= \tau \otimes \text{Id}_{\pi^*M}$ is the isomorphism
\[
\hat{\sigma}^*(U \otimes \pi^*M) \cong (U\otimes \pi^*M)^\vee \otimes (K_{\hat{X_s}} \otimes \pi^*K^\vee)^{-1} \otimes \pi^*(L\otimes M^2).
\]
\begin{theorem}\label{PSO_theorem}
For a generic point $s\in \mathcal{H}_{\mathrm{PSO-par}}$, the fiber $h_{\mathrm{PSO-par}}^{-1}(s)$ is isomorphic to $\Omega_{s,\hat{\sigma}}/\mathrm{Jac}(X)$.
\end{theorem}
\begin{proof}
Let $(V,\eta) \in h_{\mathrm{PSp-par}}^{-1}(s)$ lifts to a parabolic $\mathrm{GSO(2m,\mathbb{C})}$-Higgs bundle $(\Tilde{V},\Tilde{\eta})$ whose corresponding datum is $(E,\Phi,\varphi,L)$. Then $(V,\eta)$ corresponds to the datum of the $\mathrm{Jac}(X)$-orbit $[(E,\Phi,\varphi,L)]$ uniquely. By \cite[Theorem 4.2]{R20}, the datum of $(E,\Phi,\varphi,L)$ corresponds to an element of the Pyrm variety $\Omega_{s,\hat{\sigma}}$. Let $M \in \mathrm{Jac}(X)$. Then the datum of $(E \otimes M,\Phi \otimes \text{Id}_M,\varphi_M,L\otimes M^2)$ corresponds uniquely to an element of $\Omega_{s,\hat{\sigma}}/\mathrm{Jac}(X)$. Therefore, we conclude that the datum of the $\mathrm{Jac}(X)$-orbit $[(E,\Phi,\varphi,L)]$ corresponds uniquely to the datum of the $\mathrm{Jac}(X)$-orbit of an element of $\Omega_{s,\hat{\sigma}}$.
\end{proof}
\section{Fixed line bundle : $\mathcal{O}_X$}
In this section, we will assume that the symplectic/orthogonal form in \ref{bilinear} takes values in the trivial line bundle $\mathcal{O}_X$. In particular, we consider the moduli space of parabolic symplectic/orthogonal Higgs bundles with fixed rank, degree and fixed line bundle $\mathcal{O}_X$. In other words, we are considering the moduli spaces with trivial determinant.
\subsection{Parabolic $\mathrm{PSp}(2m,\mathbb{C})$-Higgs bundles with fixed line bundle $\mathcal{O}_X$}
Let $\mathrm{Jac}_2(X)$ be the subgorup of the Jacobian $\mathrm{Jac}(X)$ which contains the $2$-torsion elements of $\mathrm{Jac}(X)$, i.e.
\[
\mathrm{Jac}_2(X) \coloneqq \{M \in \mathrm{Jac}(X) \hspace{0.1cm} | \hspace{0.1cm} M^2 \cong \mathcal{O}_X \}.
\]
Let $(E,\Phi,\varphi,\mathcal{O}_X)$ be a parabolic symplectic Higgs bundle with the symplectic form $\varphi : E \otimes E \to \mathcal{O}_X$ taking values in $\mathcal{O}_X$ and let $M \in \mathrm{Jac}_2(X)$. Then
\[
\varphi_M : (E \otimes M) \otimes (E\otimes M) \longrightarrow \mathcal{O}_X \otimes M^2 \cong \mathcal{O}_X \otimes \mathcal{O}_X \cong \mathcal{O}_X
\]
defines a symplectic form on $E \otimes M$ with values in $\mathcal{O}_X$.
Since the symplectic form takes values in $\mathcal{O}_X$, a parabolic $\mathrm{PSp}(2m,\mathbb{C})$-Higgs bundle $(V,\eta)$ lifts to a parabolic $\mathrm{Sp}(2m,\mathbb{C})$-Higgs bundle $(\Tilde{V},\Tilde{\eta})$. Let $(E,\Phi,\varphi,\mathcal{O}_X)$ be the parabolic symplectic Higgs bundle corresponding to $(\Tilde{V},\Tilde{\eta})$. Then $(V,\eta)$ corresponds to the equivalence class $[(E,\Phi,\varphi,\mathcal{O}_X)]$ where the equivalence relation $\sim_{\mathrm{Jac}_2(X)}$ is given by :
\[
(E,\Phi,\varphi,\mathcal{O}_X) \sim_{\mathrm{Jac}_2(X)} (E\otimes M,\Phi \otimes \mathrm{Id}_M,\varphi_M,\mathcal{O}_X) \hspace{1cm} \mathrm{for} \hspace{0.1cm} \mathrm{any} \hspace{0.2cm} M\in {\mathrm{Jac}_2(X)}.
\]
As in \ref{prym}, the Prym variety is given by
\[
\Omega_{s,\sigma} \coloneqq \{(U,\mathcal{O}_X,\tau)\hspace{0.1cm}|\hspace{0.1cm} U \in \text{Jac}(X_s), \tau : \sigma^*U \cong U^\vee \otimes (K_{X_s} \otimes \pi^*K^\vee)^{-1} \otimes \pi^*\mathcal{O}_X\}.
\]
Also, the group $\mathrm{Jac}_2(X)$ acts on $\Omega_{s,\sigma}$ by
\begin{align*}
\Omega_{s,\sigma} \times \mathrm{Jac}_2(X) &\longrightarrow \Omega_{s,\sigma}\\
((U,\mathcal{O}_X,\tau),M) &\longmapsto (U \otimes \pi^*M, \mathcal{O}_X\otimes M^2, \tau_M) \cong (U \otimes \pi^*M, \mathcal{O}_X, \tau_M)
\end{align*}
where $\tau_M= \tau \otimes \text{Id}_{\pi^*M}$ is the isomorphism
\[
\sigma^*(U \otimes \pi^*M) \cong (U\otimes \pi^*M)^\vee \otimes (K_{X_s} \otimes \pi^*K^\vee)^{-1} \otimes \pi^*\mathcal{O}_X.
\]
\begin{theorem}\label{trivial_PSp}
For a generic point $s\in \mathcal{H}_{\mathrm{PSp-par},\mathcal{O}_X}$ of the parabolic $\mathrm{PSp}$-Hitchin map with the fixed line bundle $\mathcal{O}_X$, the fiber $h_{\mathrm{PSp-par},\mathcal{O}_X}^{-1}(s)$ is isomorphic to the quotient $\Omega_{s,\sigma}/\mathrm{Jac}_2(X)$.
\end{theorem}
\begin{proof}
The proof is similar to the proof of the Theorem \ref{PSp}.
\end{proof}
\subsection{Parabolic $\mathrm{PSO}(2m,\mathbb{C})$-Higgs bundles with fixed line bundle $\mathcal{O}_X$}
As in the symplectic case, a parabolic $\mathrm{PSO}(2m,\mathbb{C})$-Higgs bundle $(V,\eta)$ with the orthogonal form taking values in $\mathcal{O}_X$ lifts to a parabolic $\mathrm{SO}(2m,\mathbb{C})$-Higgs bundle $(\Tilde{V},\Tilde{\eta})$. Let $(E,\Phi,\varphi,\mathcal{O}_X)$ be the parabolic (even) orthogonal Higgs bundle corresponding to $(\Tilde{V},\Tilde{\eta})$. Then $(V,\eta)$ corresponds to the equivalence class $[(E,\Phi,\varphi,\mathcal{O}_X)]$ where the equivalence relation $\sim_{\mathrm{Jac}_2(X)}$ is given by:
\[
(E,\Phi,\varphi,\mathcal{O}_X) \sim_{\mathrm{Jac}_2(X)} (E\otimes M,\Phi \otimes \mathrm{Id}_M,\varphi_M,\mathcal{O}_X) \hspace{1cm} \mathrm{for} \hspace{0.1cm} \mathrm{any} \hspace{0.2cm} M\in {\mathrm{Jac}_2(X)}.
\]
Following the above section \ref{PSO}, the alternative description of the Prym variety (\ref{desingular}) is given by
\[
\Omega_{s,\hat{\sigma}} \coloneqq \{(U,\mathcal{O}_X,\tau)\hspace{0.1cm}|\hspace{0.1cm} U \in \text{Jac}(\hat{X_s}), \tau : \hat{\sigma}^*U \cong U^\vee \otimes (K_{\hat{X_s}} \otimes \pi^*K^\vee)^{-1} \otimes \pi^*\mathcal{O}_X \}.
\]
Also, the action of the subgroup $\mathrm{Jac}_2(X)$ on $\Omega_{s,\hat{\sigma}}$ is given by
\begin{align*}
\Omega_{s,\hat{\sigma}} \times \mathrm{Jac}_2(X) &\longrightarrow \Omega_{s,\hat{\sigma}}\\
((U,\mathcal{O}_X,\tau),M) &\longmapsto (U \otimes \pi^*M, \mathcal{O}_X, \tau_M)
\end{align*}
where $\tau_M= \tau \otimes \text{Id}_{\pi^*M}$ is the isomorphism
\[
\hat{\sigma}^*(U \otimes \pi^*M) \cong (U\otimes \pi^*M)^\vee \otimes (K_{\hat{X_s}} \otimes \pi^*K^\vee)^{-1} \otimes \pi^*\mathcal{O}_X.
\]
\begin{theorem}\label{trivial_PSO}
For a generic point $s\in \mathcal{H}_{\mathrm{PSO-par},\mathcal{O}_X}$ of the parabolic $\mathrm{PSO}$-Hitchin map with the fixed line bundle $\mathcal{O}_X$, the fiber $h_{\mathrm{PSO-par},\mathcal{O}_X}^{-1}(s)$ is isomorphic to the quotient $\Omega_{s,\hat{\sigma}}/\mathrm{Jac}_2(X)$.
\end{theorem}
\begin{proof}
The proof is similar to the proof of the Theorem \ref{PSO_theorem}.
\end{proof}
\begin{remark}
We can actually consider any degree zero line bundle in place of $\mathcal{O}_X$.
\end{remark}
\section*{Acknowledgement}
This work was supported by the Institute for Basic Science (IBS-R003-D1).
\section*{Data Availability}
Data sharing is not applicable to this article as no new data were created or analyzed in this study.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,783 |
When you think of a good workout, no doubt you picture yourself covered in sweat, muscles aching, lungs burning, and pushing yourself to your limits. There's no doubt about it: vigorous exercise is one of the best ways to burn fat, improve your cardiovascular conditioning, raise testosterone levels, and enhance your overall health.
But as one study proved, it's not the only way to get healthy. In fact, as researchers from Boston University Medical Center found, moderate exercise can yield beautifully visible results as well. The researchers gathered participants and gave them accelerometers to wear during the day. The accelerometers measured physical activity, as well as the amount of time spent in active and inactive activities. The researchers also measured insulin resistance, metabolic rate, and inflammation of each participant throughout the study.
The data showed that even minor increases in physical activity led to visible improvement in both insulin sensitivity and inflammation. Even if the participants didn't spend sufficient time exercising or engage in vigorous exercise, they still improved their metabolic health. Their blood contained higher levels of leptin (a satiety hormone that reduces appetite) as well as FABP4, a protein needed for the transportation of fat molecules.
The NIH estimates insulin resistance is a common problem affecting 86 million adults at least in the U.S. alone. With a resistance to insulin (the hormone that regulates blood glucose) comes excessively high blood sugar levels, which can lead to pre-diabetes and diabetes. Pre-diabetics have a significant risk of developing both type 2 diabetes and cardiovascular disease, which may lead to a stroke or heart attack.
As this study proved, even a small amount of exercise can help to reduce your risk of insulin resistance. It will also decrease inflammation, which plays a role in cancer, heart disease, metabolic syndrome, and many other health problems.
Can't make it to the gym for a proper 60-minute sweat-fest? Don't worry about it. Even if you only spend 15 to 30 minutes walking around a nearby park, your office building, or your neighborhood, it's good for your body. You'll decrease inflammation and improve your glucose control, leading to a decreased risk of both heart disease and diabetes. Given how prominent these two health conditions are in our country, it's in your best interest to spend more time moving around, even if it's just mild to moderate exercise.
1. N. L. Spartano, M. D. Stevenson, V. Xanthakis, M. G. Larson, C. Andersson, J. M. Murabito, R. S. Vasan. Associations of objective physical activity with insulin sensitivity and circulating adipokine profile: the Framingham Heart Study. Clinical Obesity, 2017; DOI: 10.1111/cob.12177. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,727 |
Q: Go structures defined from common package in test This is a packaging question in Go. I have a package schema that looks like this:
project
|
--- common_test.go
|
--- sub1
| |
| --- file.go
| --- file_test.go
--- sub2
| |
| |
| --- file.go
| --- file_test.go
common_test.go would look like that:
package project
type Test struct {
...
}
And sub1 and sub2 would look like:
package sub1
import "project"
func SomeTest(t *testing.T) {
test := project.Test{}
...
}
The project package only contains the common_test.go file, which defines structures that are used in both sub1/file_test.go and sub2/file_test.go. But if I try to trigger the test, I get the following error:no buildable Go source files in project.
I understand that it is because common_test.go includes the _test suffix and because it is the only file in the project package. But I use the suffix for the reason I don't want this file to be compiled except for test execution.
Is there a way to define test-specific common structures in a package that does not contain non-test files? What is the good practice in this case?
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,459 |
Edelweiss Canyon Rope is designed specifically for canyoneering applications. With an Everdry coating and a sheath made from nylon and polyester, this rope sheds water like a husky sheds hair. All Edelweiss static lines are designed to take up to a factor 1 fall. This extra margin of safety is key to your success in wild canyons across the globe. See the static rope section for more info on Edelweiss low-stretch ropes. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,912 |
Q: Show That a Power Of 17 Exists Between $2^{16},2^{17}$
With minimal computation (i.e., without computing any of the actual quantities), show that $$\exists \; k \in \mathbb Z \; \text{such that} \; 2^{16} \lt 17^{k} \lt 2^{17}$$
My 17th birthday is coming up soon, and one of my friends at school sent me this question- she said that she had made it up for my birthday. I tried solving it by expanding $(2^{16}-1)+1=(2^{15}+2^{14}+2^{13}+...+2^{2}+2^{1}+2^{0})+1$ and $(2^{17}-1)+1=(2^{16}+2^{15}+2^{14}+...+2^{2}+2^{1}+2^{0})+1$, and seeing if "polynomializing" the powers of $2$ in this way helped find the power of $17$ somehow (I know from computation that the power is $17^{4}$).
I also noted that $17=2^{4}+1\gt2^{4}$, so $17^{4}\gt (2^{4})^{4}=2^{16}$, but I don't think I can perform the same sort of trick for the upper bound of $2^{17}$ without exactly the type of computation my friend wants me to avoid.
So my question is: can I actually solve my friend's problem without any sort of heavy calculation, or is she just messing with me? (Note: I would not be shocked if it were to be the latter, because that is exactly the sort of thing I have known her to do in the past.)
A: You suspect (correctly) that $2^{16} < 17^4 < 2^{17}$ since $17=2^4+1$
So try $$17^4=(16+1)^4 $$ $$= 16^4 + 4\times 16^3 +6\times 16^2+4\times 16^1 +1\times 16^0$$ $$< 16^4+4\times 16^3 +6\times 16^3+4\times 16^3 +1\times 16^3$$ $$=16^4+15 \times 16^3 $$ $$< 2 \times 16^4 = 2^{17}$$
and clearly $2^{16} = 16^4 < 17^4 $
A: Consider the fairly general case of showing, for some positive real numbers $a$, $b$, $c$ and $d$, with $a \neq 1$ and $d \gt c$, that
$$\exists \; k \in \mathbb Z \; \text{such that} \; a^{c} \lt b^{k} \lt a^{d} \tag{1}\label{eq1A}$$
Taking logs, say natural ones, of all of the parts gives
$$c\ln a \lt k \ln b \lt d \ln a \implies \frac{c\ln a}{\ln b} \lt k \lt \frac{d\ln a}{\ln b} \tag{2}\label{eq2A}$$
Thus, you just need to see if there's an integer value $k$ between the lower & upper limit values. In your case, you have $a = 2$, $b = 17$, $c = 16$ and $d = 17$. Plugging these values into \eqref{eq2A} gives
$$\begin{equation}\begin{aligned}
& \frac{16\ln 2}{\ln 17} \lt k \lt \frac{17\ln 2}{\ln 17} \\
& 3.914\ldots \lt k \lt 4.159\ldots
\end{aligned}\end{equation}\tag{3}\label{eq3A}$$
This shows that just $k = 4$ works, as you've found and the other answers have shown as well. However, this method shown here will work in any more general, difficult cases to fairly easily determine if there's any $k$ and, if so, which value(s) of $k$ will work.
A: Note $17^4=(2^4+1)^4=2^{16}+4\cdot2^{12}+6\cdot2^8+4\cdot2^4+1$
$<2^{16}+2^{14}+2^{11}+2^6+1<2^{16}+2^{15}+2^{14}+\dots+2^2+2^1+2^0<2^{17}$.
A: Similar Isaac YIU Math Studio's answer, we could instead use hexadecimal.
$2^{16}=16^4=10000_{16}$
$2^{17}=2\cdot2^{16}=20000_{16}$
As done in Henry's answer, we can binomially expand (or really just multiply out $11^4$) to see we then have:
$17^4=11_{16}^4=14641_{16}$
which is clearly between $10000_{16}$ and $20000_{16}$.
A: Just another way, it is enough to compare $17^4 $ and $2\cdot16^4$. By Bernoulli's inequality,
$$\frac{17^4}{16^4} = \frac1{\left(1-\frac{1}{17}\right)^4} < \frac1{1-\frac4{17}}=\frac{17}{13}<2$$
A: $$\begin{align}&\left[\frac{17}{16}\right]^{\large 4\cdot\color{#c00}4}\!\! = \left[1+\frac{1}{\color{#0a0}{16}}\right]^{\large\color{#0a0}{16}}\!\!<\, \color{#0a0}e\, < \color{c00}2^{\large\color{#c00} 4}\\[.2em]
\overset{\large (\ {\phantom{|_|}} )^{\LARGE 1/\color{#c00}4}\!\!\!}\Longrightarrow\ \ \ &\left[\frac{17}{16}\right]^{\large 4}\! <\, \color{c00}2,\ \ {\rm so}\,\ \ \bbox[6px,border:1px solid #c00]{17^{\large 4} <\ 2\cdot 16^{\large 4} =\, 2^{\large17}}\end{align}\qquad$$
A: You can use binary to find that:
$2^{16}=1000000000000000_{(2)} \\ 2^{17}=10000000000000000_{(2)} \\ 17^4=10001\times10001\times10001\times10001_{(2)}=10100011001000001_{(2)}$
Obvious that $2^{16}<17^4<2^{17}$.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,914 |
//
using Alachisoft.NosDB.Common.Configuration.DOM;
using Alachisoft.NosDB.Common.Recovery;
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
namespace Alachisoft.NosDB.Common.Recovery
{
// individual backup/restore job
public interface IClusteredRecoveryJob : IRecoveryCommunicationInitiater, IJobProgressHandler, IDisposable
{
RecoveryOperationStatus Initialize(RecoveryConfiguration config, object additionalParams);
RecoveryOperationStatus Start(RecoveryConfiguration config);
RecoveryOperationStatus End(RecoveryConfiguration config);
RecoveryOperationStatus Cancel(RecoveryConfiguration config, string shard = "", RecoveryFileState folderState = RecoveryFileState.Failed, bool explicitCancel = false);
object CurrentState(RecoveryConfiguration config);
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,410 |
Obertilliach – gmina w Austrii, w kraju związkowym Tyrol, w powiecie Lienz. Według danych z Austriackiego Urzędu Statystycznego liczyła 687 mieszkańców (1 stycznia 2015).
Współpraca
Miejscowość partnerska:
Althütte, Niemcy
Przypisy
Gminy w powiecie Lienz | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,664 |
Q: regex fails to match simple string GNU regex fails to match
string st = "ABCDAZEA";
cout<<"S:\t"<<st<<endl;
std::regex expr("[A-Z]+", std::regex::extended );
std::smatch matches;
std::regex_search(st, matches, expr);
cout<<"found "<<matches.size()<<endl;
fflush (stdout);
what's wrong in my code ?
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,922 |
{"url":"https:\/\/www.physicsforums.com\/threads\/center-of-mass-of-a-hemisphere.527548\/","text":"# Center of mass of a hemisphere\n\nAs part of a problem I need to find the center of mass of a hollow hemisphere.\n\nI did it in the following way.\n\nI considered the lower hemisphere ( around negative z axis )\n\nFirst by definition, the position vector of the center of mass is obtained as\n\n$\\vec{R}$= $\\frac{\\int\\vec{r}dm}{\\int dm}$\n\nNow I set up the problem in spherical coordinates\n\nhttp:\/\/img855.imageshack.us\/img855\/4741\/spherev.png [Broken]\n\nNow, $\\vec{r}$ = R$\\hat{r}$\n\nwhere $\\hat{r}$ = sin$\\theta$cos$\\phi$$\\hat{i}$+sin$\\theta$sin$\\phi$$\\hat{j}$+cos$\\theta$$\\hat{k}$\n\nAnd dm = $\\sigma$ds, where ds is the area element, and $\\sigma$ is the mass per unit area.\n\nds = R2sin$\\theta$d$\\theta$d$\\phi$\n\nSo then $\\vec{R}$= $\\frac{\\int\\vec{r}ds}{\\int ds}$\n\ni.e (2$\\pi$R2) $\\vec{R}$=$\\int$$\\int$R3 (sin2$\\theta$cos$\\phi$$\\hat{i}$+sin2$\\theta$sin$\\phi$$\\hat{j}$+$\\frac{sin2\\theta}{2}$$\\hat{k}$)d$\\theta$d$\\phi$\n\nintegrating $\\theta$ from $\\pi$\/2 to $\\pi$ and $\\phi$ from 0 to 2$\\pi$\n\nI get $\\vec{R}$ = -$\\frac{\\hat{k}}{2}$\n\nnow I know this is not the right answer, the correct answer is -$\\frac{2\\pi}{R}$$\\hat{k}$\n\nI know that the problem can be simplified with symmetry and done with much more ease than what I have attempted, but I'd like to know what is wrong with the way I formulated the problem.\n\nLast edited by a moderator:\n\nRelated Calculus and Beyond Homework Help News on Phys.org\nvela\nStaff Emeritus\nHomework Helper\nI get $\\vec{R}$ = -$\\frac{\\hat{k}}{2}$\nnow I know this is not the right answer, the correct answer is -$\\frac{2\\pi}{R}$$\\hat{k}$\nThis can't possible be the correct answer. The units aren't correct.\n\nThis can't possible be the correct answer. The units aren't correct.\n\nsorry I made a mistake, the correct answer is- $\\frac{2R}{\\pi}$$\\hat{K}$\n\nYou can arrive at that answer with this argument, the sphere can be thought of to be made of a number of semicircular line segments of radius R ( or taking one and rotating it about the negative Z axis )\n\nSo the C.M of the sphere will be the same as one such segment, since the rotation occurs about an axis through the center of mass.\n\nhttp:\/\/img97.imageshack.us\/img97\/3853\/semit.png [Broken]\n\nnow, if I consider a horizontal strip, the CM of the strip is on the Z axis at a distance Rcos$\\theta$ and the mass dm = 2$\\lambda$dl , where $\\lambda$ is the linear mass density, $\\lambda$ gets canceled out and the cm is at\n\n$\\vec{R}$ =$\\frac{1}{\\pi R}$$\\int$2R2cos$\\theta$d$\\theta$\n\ndl = Rd$\\theta$\n\n$\\theta$ varies from 0 to $\\frac{\\pi}{2}$\n\nthen $\\vec{R}$ = $\\frac{2R}{\\pi}$\n\nLast edited by a moderator:\nvela\nStaff Emeritus\nHomework Helper\nThe radius of the circular strip isn't R. It's only equal to R when theta is pi\/2.\n\nThe radius of the circular strip isn't R. It's only equal to R when theta is pi\/2.\nyou misunderstand what i mean, I am taking a semicircle of radius R and rotating it , think of it like an electric drill with a leaf as a semicircle,\n\nwhen it spins it will form a hemisphere right?\n\nP.S\n\nThis is not what is happening in the integral, i just used the notion of rotation to infer that the CM of both the semicircle and hemisphere have to be the same\n\nvela\nStaff Emeritus","date":"2020-11-28 14:32:29","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7917582988739014, \"perplexity\": 435.947517870302}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-50\/segments\/1606141195656.78\/warc\/CC-MAIN-20201128125557-20201128155557-00335.warc.gz\"}"} | null | null |
{"url":"https:\/\/worldbuilding.stackexchange.com\/questions\/137585\/what-if-earth-became-a-rogue-planet\/137619","text":"# What if Earth became a rogue planet? [duplicate]\n\nI'm writing a science fiction novel in which Earth essentially gets the boot out of our Solar System. No sun, no moon, just Earth flying through the depths of deep space. I need to nail down rogue Earth's climate before I can really dig into it.\n\n\u2022 What temperature would the surface be?\n\u2022 How quickly would it drop year to year?\n\u2022 Would the atmosphere freeze, and if so, what would it look like?\n\u2022 What would happen to cities, trees, the ocean, etc?\n\nAny tips\/ideas you guys have would be a huge help!\n\n## marked as duplicate by Mo\u0142ot, Renan, KerrAvon2055, Separatrix, user535733Jan 24 at 16:25\n\n\u2022 From the answers received so far I feel that you might want to add the science-based tag. Otherwise you'll just get random \"I think it would be X because space is cold\". \u2013\u00a0pipe Jan 24 at 12:18\n\u2022 another possible duplicate worldbuilding.stackexchange.com\/questions\/15094\/\u2026 \u2013\u00a0Pelinore Jan 24 at 15:58\n\n\u2022 What temperature would the surface be?\n\nCold, really really cold, imagine the coldest winter you can remember, it's going to be colder than that, a lot colder, honestly it's going to be really cold, so cold you can't imagine how cold it's going to be.\n\nLets just say extra socks & a bobble hat aren't going to be much help.\n\n\u2022 How quickly would it drop year to year?\n\nYear by year? it wouldn't even take one year, assuming you start by moving away from the sun you'll be in a new ice age colder than any before it long before you get half way to the orbit of Mars.\n\n\u2022 Would the atmosphere freeze, and if so, what would it look like?\n\nYes, it'll look like ice, what else would it look like.\n\n\u2022 What would happen to cities, trees, the ocean, etc?\n\nThey'll all freeze.\n\nAny other questions?\n\nThe only place you've any real possibility of life persisting for a bit is going to be in close proximity to an active geothermal of some sort. Yellowstone for instance might provide a haven for a few humans for a bit longer than elsewhere on the planet.\n\n\u2022 Unfortunately, once the earth is beyond the orbit of Mars, certain gases will start to condense and fall out of the atmosphere. Atmospheric pressure would drop causing all surface life to die. Best bet, once the oceans freeze over is geothermal vents at the bottom of the ocean. the ice sheets should prevent the loss of at least some of the ocean to space. \u2013\u00a0Sonvar Jan 24 at 5:09\n\u2022 @Sonvar : Yup, the Yellowstone locals would have to have got some sort of domes built before then if they wanted to survive past that point, a thriving industry of oxygen mining would spring up (driving out in trucks & returning with trucks piled with frozen oxygen-snow for the thermals to melt so they've something to breath), they'll basically need spacesuits as well of course by then. \u2013\u00a0Pelinore Jan 24 at 5:17\n\u2022 Note that engineering a super-tree that could split CO2 for you is not going to be possible. Reprogramming the DNA to crank up chlorophyll concentration is one thing, finding enough energy to power your sun lamps is another. You can't just put a solar farm out there in the desert because there won't be anything shining on it. No atmosphere = no wind farms, hydroelectricity is a no-go as well, and we barely have enough crude oil to fuel the oxygen trucks, can't afford to just shove it into a furnace and get electricity from that. \u2013\u00a0John Dvorak Jan 24 at 8:21\n\u2022 @John Dvorak Nuclear energy is the obvious choice. \u2013\u00a0Emilio M Bumachar Jan 24 at 13:38\n\u2022 @EmilioMBumachar In addition to that, Iceland already uses about 30% geothermal power, so if they got their act together quickly enough, they might be able to maintain a small colony indefinitely. Nuclear stockpiles would run out eventually, so I think geothermal is probably a better long-term choice. \u2013\u00a0Gryphon Jan 24 at 13:39\n\nEarth pretty much looked like this not so very long ago:\n\nWelcome to Snowball Earth, some 650 million or more years ago. And this happened within the Goldilocks Zone!\n\nRogue Earth probably won't end up looking like a cue ball, simply because absent the Sun's influence, there won't be much weather. Whatever's in the atmosphere will rain or snow until Earth is far enough away that incoming solar energy no longer affects ocean currents and winds. Eventually, the surface will just be nut (and bolt!) freezing temperatures and rapidly diminishing amounts of incoming heat and light. Bad news for us.\n\nLiquid water would likely persist in the oceans, meaning those buggers that live deep down won't even notice that us surface dwellers have turned into ice cubes.\n\nHow quickly depends on several factors:\n\n\u2022 Where Earth is, at the time of its ejection, with respect to the direction of the Sun's travel around the Galaxy;\n\u2022 Which direction Earth gets ejected (this is very important, because if Earth is ejected in the wrong direction, it will just plummet into the Sun and your whole project will be moot)\n\u2022 How fast Earth is traveling\n\nRogue planets can zip right along, and if Earth is positioned \"behind\" the Sun's direction of travel and gets ejected back the way it came and at speed, we could be waving bye-bye to the Sun pretty quickly! If we end up heading in the Sun's direction, perhaps we won't notice much difference?\n\nNothing would happen for a while, assuming we're talking about Sun just disappearing overnight. You can even demonstrably witness what happens with 12 hours of no sunlight in the Equator (or during polar night in the Arctic, for that matter). The primary reason for this is the vast amount of water.\n\nWe can, however, give some ballpark estimate[*] for the speed of process.\n\nIf the Sun would just disappear Earth would begin to cool at a rate of roughly 300 W\/m\u00b2. Now, Earth is not an ideal blackbody and temperature is not uniform, but in terms of estimates, well, close enough. This is equivalent to having a 1 mm layer of water drop 1 Kelvin in temperature in ca. 14 seconds (1 mm of water per square meter = 1 kg of water).\n\nTropics have an ocean mixing layer of roughly 1000 meters and average water temperature (in that layer) of something like 20 C, so if we exclude currents and atmospheric convection it would take $$\\frac{1000 \\text{m}}{0.001 \\text{m}} \\cdot 14 \\text{ s\/K} \\cdot 20 \\text{K} \\sim 10$$ years before tropical oceans would begin to freeze over.\n\nNow, this is not what would happen, but it gives some insight on the speed of the process.\n\nIn reality things are much more complicated: Solar energy received in Equator is partly transferred in ocean and atmospheric convection to polar latitudes. If we assume hurricane-ish type energy transfer, we could be looking at something like extra 10-100 W per square meter of ocean, add ocean currents for 200-300 watts extra, and we'd still end up with a time window that is closer to few years rather than few weeks.\n\nNow, on land, on the other hand. We might be talking of an equivalent of several meters of water. So it would take perhaps a month, considering energy transfer by the atmosphere, to turn land into inhabitable snowfield and another month or two to make it uncomfortable for the Nordics or Canadians. This is naturally talking in averages, so locally it could be better...or much worse.\n\nIn terms of freezing, enthalpy of fusion of water is roughly 333 kJ per kilogram, much higher compared to heat capacity of 4.2 kJ\/kg$$\\cdot$$K. Therefore, assuming no geothermal energy, it would still take in order of centuries to have an ice-sheet in tropics that would measure in kilometers.\n\n[*] Correct to few orders of magnitude...If lucky, then an order of magnitude.\n\nI would say:\n\n1. No moon - means static oceans. The total amount of water grows only if Earth changes orbit. 1.1. Slower motion - more hours in a day, e.g. 28.\n2. More contrast in the climate: colder in the North and warmer in the South.\n3. More green forests, jungles. Higher trees.\n\nGood luck!\n\n\u2022 Welcome to Worldbuilding, PirrenCode! If you have a moment, please take the tour and visit the help center to learn more about the site. You may also find Worldbuilding Meta and The Sandbox useful. Here is a meta post on the culture and style of Worldbuilding.SE, just to help you understand our scope and methods, and how we do things here. Have fun! \u2013\u00a0Gryphon Jan 24 at 14:21\n\u2022 Welcome to the site, PirrenCode. You might want to reconsider this answer, as all three points are refuted by science. \u2013\u00a0Frostfyre Jan 24 at 14:55\n\u2022 more hours in a day without a sun? \u2013\u00a0Andrey Jan 24 at 15:13\n\u2022 Do you mind explaining how can you get 2 and 3? \u2013\u00a0L.Dutch Jan 24 at 15:17","date":"2019-08-23 04:54:19","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 2, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.3617251515388489, \"perplexity\": 1597.1019382566387}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-35\/segments\/1566027317847.79\/warc\/CC-MAIN-20190823041746-20190823063746-00046.warc.gz\"}"} | null | null |
ACCEPTED
#### According to
Index Fungorum
#### Published in
null
#### Original name
Didymosphaeria nitidula Sacc.
### Remarks
null | {
"redpajama_set_name": "RedPajamaGithub"
} | 4,484 |
/********************************************************************************
** Form generated from reading UI file 'openuridialog.ui'
**
** Created by: Qt User Interface Compiler version 5.3.2
**
** WARNING! All changes made in this file will be lost when recompiling UI file!
********************************************************************************/
#ifndef UI_OPENURIDIALOG_H
#define UI_OPENURIDIALOG_H
#include <QtCore/QVariant>
#include <QtWidgets/QAction>
#include <QtWidgets/QApplication>
#include <QtWidgets/QButtonGroup>
#include <QtWidgets/QDialog>
#include <QtWidgets/QDialogButtonBox>
#include <QtWidgets/QHBoxLayout>
#include <QtWidgets/QHeaderView>
#include <QtWidgets/QLabel>
#include <QtWidgets/QPushButton>
#include <QtWidgets/QSpacerItem>
#include <QtWidgets/QVBoxLayout>
#include "qvalidatedlineedit.h"
QT_BEGIN_NAMESPACE
class Ui_OpenURIDialog
{
public:
QVBoxLayout *verticalLayout;
QLabel *label_2;
QHBoxLayout *horizontalLayout;
QLabel *label;
QValidatedLineEdit *uriEdit;
QPushButton *selectFileButton;
QSpacerItem *verticalSpacer;
QDialogButtonBox *buttonBox;
void setupUi(QDialog *OpenURIDialog)
{
if (OpenURIDialog->objectName().isEmpty())
OpenURIDialog->setObjectName(QStringLiteral("OpenURIDialog"));
OpenURIDialog->resize(564, 109);
verticalLayout = new QVBoxLayout(OpenURIDialog);
verticalLayout->setObjectName(QStringLiteral("verticalLayout"));
label_2 = new QLabel(OpenURIDialog);
label_2->setObjectName(QStringLiteral("label_2"));
verticalLayout->addWidget(label_2);
horizontalLayout = new QHBoxLayout();
horizontalLayout->setObjectName(QStringLiteral("horizontalLayout"));
label = new QLabel(OpenURIDialog);
label->setObjectName(QStringLiteral("label"));
horizontalLayout->addWidget(label);
uriEdit = new QValidatedLineEdit(OpenURIDialog);
uriEdit->setObjectName(QStringLiteral("uriEdit"));
horizontalLayout->addWidget(uriEdit);
selectFileButton = new QPushButton(OpenURIDialog);
selectFileButton->setObjectName(QStringLiteral("selectFileButton"));
selectFileButton->setText(QString::fromUtf8("\342\200\246"));
selectFileButton->setAutoDefault(false);
horizontalLayout->addWidget(selectFileButton);
verticalLayout->addLayout(horizontalLayout);
verticalSpacer = new QSpacerItem(20, 40, QSizePolicy::Minimum, QSizePolicy::Expanding);
verticalLayout->addItem(verticalSpacer);
buttonBox = new QDialogButtonBox(OpenURIDialog);
buttonBox->setObjectName(QStringLiteral("buttonBox"));
buttonBox->setOrientation(Qt::Horizontal);
buttonBox->setStandardButtons(QDialogButtonBox::Cancel|QDialogButtonBox::Ok);
verticalLayout->addWidget(buttonBox);
retranslateUi(OpenURIDialog);
QObject::connect(buttonBox, SIGNAL(accepted()), OpenURIDialog, SLOT(accept()));
QObject::connect(buttonBox, SIGNAL(rejected()), OpenURIDialog, SLOT(reject()));
QMetaObject::connectSlotsByName(OpenURIDialog);
} // setupUi
void retranslateUi(QDialog *OpenURIDialog)
{
OpenURIDialog->setWindowTitle(QApplication::translate("OpenURIDialog", "Open URI", 0));
label_2->setText(QApplication::translate("OpenURIDialog", "Open payment request from URI or file", 0));
label->setText(QApplication::translate("OpenURIDialog", "URI:", 0));
#ifndef QT_NO_TOOLTIP
selectFileButton->setToolTip(QApplication::translate("OpenURIDialog", "Select payment request file", 0));
#endif // QT_NO_TOOLTIP
} // retranslateUi
};
namespace Ui {
class OpenURIDialog: public Ui_OpenURIDialog {};
} // namespace Ui
QT_END_NAMESPACE
#endif // UI_OPENURIDIALOG_H
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,193 |
719 Salem Avenue, HAGERSTOWN, MD 21740 (#1009976310) :: Arlington Realty, Inc.
Newly remolded 3 BR house with large 2 story garage & off street parking Fenced in back yard with bonus storage shed. Enclosed back porch/sun room. New Kitchen w/ ceramic tile floors. New wood floors in LR & DR. Finished attic bedroom. Fresh paint inside and out. New AC unit w/ gas heat. Close to Interstate and shopping. House Qualifies for $7,500 City Down payment/Closing Cost Grant. 100% Financing with Grant and Seller Contribution. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,923 |
If you'd like to acquire one of my paintings, there are a number of them available Pablo Center. There are no prices. You set the price and 100% of the proceeds go to Beacon House, a homeless shelter here in Eau Claire. Please spread the word and help do some good.
I'm very excited to announce that I have been invited to participate in PERSONAL STRUCTURES organized by the GAA Foundation and hosted by the European Cultural Centre in Venice Italy during the Venice Biennale, 2019.
In Broad Daylight – Foundations Near Hughitt Slip – Łódź. 12″x12″ each.
I want to express my thanks to all involved, especially to artists Jo Ellen Burke and Terry Meyer who took it upon themselves to organize this Go Fund Me page on my behalf. Thank you all who have contributed!
Thank you to Volume One for this writeup.
And a special thank you to Al Ross and Wisconsin Public Radio for this interview.
Very excited to be at this point in the painting. I prepped & "sunk" the boat last night. A beautiful day at 46°F to spend under the sky. I have my work cut out for me now.
Video can bee seen at my Instagram.
Today I'm getting a start on weathering the boat keel for the current painting. Next I'll prep/mount the cladding. Eventually I'll submerge the whole thing in water, freeze it, put it in the oven & start it on fire (among other things) before placing it in the painting. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,358 |
\section{Introduction}
\label{intro}
The problem of the transformation of the galaxy population from star-forming to quiescent is still an open one in modern astrophysics. General agreement has been reached on the fact that galaxy mass, galaxy environment and AGN feedback play a major role in star formation quenching. It has been suggested (see \cite{hickox09}) that the central AGN co-evolves with the host-galaxy: while the host-galaxy transforms from a star-forming to a quiescent one, the AGN passes from a quasar, X-ray emitter phase to a radio-galaxy one. These transformations happen at earlier epochs for haloes of higher mass, that were found to reside primarily in high-density environments, where early-type galaxies dominate at low redshifts (\cite{quadri12,chuter11}). Moreover, it was already known that many radio AGNs reside in early-type galaxies (\cite{ledlow96}), that the probability that a galaxy hosts a radio AGN is increasing with stellar mass (\cite{bardelli09}), and that the fraction of radio active early-type galaxies is an increasing function of local density (\cite{bardelli10}). In this work (see \cite{malavasi15}) the environment of radio sources of the VLA-COSMOS survey (\cite{schinnerer07}), cross-identified with the COSMOS photometric redshift sample (\cite{ilbert09}), is explored.
\section{Data And Method}
\label{data}
The analysis was performed on the environment of a selection of 272 radio AGNs from the VLA-COSMOS survey (\cite{schinnerer07,schinnerer10}). This survey (which is composed of 1.4 GHz data, with a sensitivity of about 11 $\mu$Jy r.m.s.) was cross-correlated with the COSMOS photometric survey (\cite{scoville07}) with photometric redshifts measured by \cite{ilbert09}. The COSMOS survey catalogue is composed of optical galaxies down to $i^+ < 26.5$, which were used both as tracers for the environment around AGN sources and as extraction pool for the control samples. The accuracy of the photometric redshifts ($z_p$) is estimated to be $\sigma_{\Delta z/(1+z)} = 0.06$.
\begin{figure}
\begin{minipage}{0.5\textwidth}
\resizebox{\hsize}{!}{\includegraphics{ssfrmstarO.pdf}}
\end{minipage}
\begin{minipage}{0.5\textwidth}
\resizebox{\hsize}{!}{\includegraphics{Bin3AGNQOShade.pdf}}
\end{minipage}
\caption{Left: \textit{SSFR - $M^{\ast}$ plane}. The black crosses refer to the full sample of VLA-COSMOS radio sources, the red triangles to the radio AGN sample, and the green lines to the cuts in stellar mass and SSFR described in text. In yellow, the full COSMOS optical sample is reported for comparison. Right: \textit{Galaxy overdensity richness distribution, samples AGN and QO}. This plot refers to $1 \le z \le 2$. The solid red line represents the AGN sample, the grey lines the 100 extractions of the QO control sample. The dashed black line is the richness distribution of the control sample extraction corresponding to the median value of the KS probability value distribution.}
\label{figure}
\end{figure}
AGNs were extracted among radio sources as those hosted by massive and quiescent galaxies through a cut to $\log(M^{\ast}/M_{\odot}) \ge 10$ and $\log(SSFR/yr^{-1}) \le -11$, as shown in the left panel of Figure \ref{figure}. The control sample of normal galaxies QO has been extracted from the same lower-right region of the SSFR - $M^{\ast}$ plane. In order to have a fully representative control sample, galaxies were randomly extracted with the same mass distribution of the radio AGNs.
The environment has been estimated around every AGN source and control galaxy by counting optical galaxies in a parallelepiped with a base side of 1 Mpc (comoving) and height $2 \cdot \varDelta z = 2 \times 3 \times \sigma_{\varDelta z/(1+z)} \times (1+z_p)$, in three different redshift bins: $z \in [0.0-0.7[$, $[0.7-1.0[$ and $[1.0-2.0]$. The number of radio AGNs is respectively 119 sources, 100 sources, and 53 sources.
\section{Results And Conclusions}
\label{results}
It was found that the environment around radio AGNs is significantly denser than the environment around sources from the control sample (\emph{i.e.} that show no sign of radio emission). This is visible in the right panel of Figure \ref{figure}, which shows the overdensity richness distribution for the AGN sample and for the 100 independent extractions of the control sample in the farthest redshift bin. A Kolmogorov-Smirnov test between the distributions results in median values of the KS test probability value distribution of $8.6 \times 10^{-5}$, $6.0 \times 10^{-6}$, and 0.006 in each redshift bin respectively.
The AGN sample has been further divided according to its radio power: a high-power sub-sample ($\log(L_{1.4 GHz}) \ge 24.5$) and a low-power one ($24 \le \log(L_{1.4 GHz}) < 24.5$) were created, with the distinction between the two that roughly corresponds to the canonical division between FRI and FRII objects.
It was found that the significance in the environmental segregation signal is maintained only for low-power radio AGNs in the lowest and intermediate redshift bins, while for the high-power radio AGNs no significant signal is present. Therefore, higher overdensity richness enhance the probability that a galaxy hosts a low-power radio AGN. In conclusion, we found a clear correlation between radio AGN presence and environment up to $z \sim 2$, consistent with the scenario sketched in \cite{hickox09}.
\subsection*{Acknowledgements}
The authors acknowledge the financial contributions by grants ASI/INAF I/023/12/0 and PRIN MIUR 2010-2011 \textquotedblleft The dark Universe and the cosmic evolution of baryons: from current surveys to Euclid\textquotedblright.\\
\textcopyright$\:$ Springer International Publishing Switzerland 2016 \\
N.R. Napolitano et al. (eds.), \textit{The Universe of Digital Sky Surveys}, Astrophysics
and Space Science Proceedings 42, DOI 10.1007/978-3-319-19330-4\_17 \\
\input{malavasibib}
\end{document}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,383 |
"Ждрело" је југословенска ТВ драма из 1982. године. Режирао ју је Миљенко Дерета а сценарио је написао Новица Тадић.
Улоге
|-
|Данило Лазовић ||
|}
Спољашње везе
Југословенски филмови
Српски филмови
Филмови 1982.
Телевизијски филмови
Српски телевизијски филмови
Српске телевизијске драме
Телевизијске драме
Филмске драме | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 461 |
Gregorio María Kardinal Aguirre y García OFM (* 12. März 1835 in La Pola de Gordón, Spanien; † 10. Oktober 1913 in Toledo) war Erzbischof von Burgos und Toledo.
Leben
Gregorio María Aguirre García trat nach seiner Schulzeit in den Franziskanerorden ein und studierte die Fächer Katholische Theologie und Philosophie in León. 1860 empfing er das Sakrament der Priesterweihe und unterrichtete anschließend an verschiedenen Seminaren und Hochschulen in Spanien und auf den Philippinen.
1885 ernannte ihn Papst Leo XIII. zum Bischof von Lugo. Die Bischofsweihe spendete ihm Mariano Kardinal Rampolla del Tindaro. 1894 wurde Aguirre y García zum Erzbischof von Burgos, 1899 zusätzlich zum Apostolischen Administrator von Calahorra und La Calzada ernannt. Papst Pius X. nahm Gregorio María Aguirre García 1907 als Kardinalpriester mit der Titelkirche San Giovanni a Porta Latina in das Kardinalskollegium auf und ernannte ihn zwei Jahre später zum Erzbischof von Toledo sowie zum Patriarchen der Westindischen Inseln. 1911 entsandte er ihn als päpstlichen Legaten zum Eucharistischen Kongress in Madrid.
Gregorio María Aguirre García starb am 10. Oktober 1913 in Toledo und wurde in der dortigen Kathedrale bestattet.
Ehrungen
1910 wurde er mit dem Großkreuz mit Collane des Ordens Karls III. ausgezeichnet.
Weblinks
Kardinal (20. Jahrhundert)
Patriarch von Westindien
Erzbischof von Toledo
Erzbischof von Burgos
Römisch-katholischer Bischof (19. Jahrhundert)
Römisch-katholischer Bischof (20. Jahrhundert)
Römisch-katholischer Theologe (19. Jahrhundert)
Römisch-katholischer Theologe (20. Jahrhundert)
Franziskaner (OFM)
Person (Calahorra)
Spanier
Geboren 1835
Gestorben 1913
Mann | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,443 |
{"url":"https:\/\/socratic.org\/questions\/an-object-is-thrown-vertically-from-a-height-of-9-m-at-5-m-s-how-long-will-it-ta","text":"# An object is thrown vertically from a height of 9 m at 5 m\/s. How long will it take for the object to hit the ground?\n\nMar 11, 2018\n\nLet's construct an equation for the motion of the object.\n\nWell here we will be using, $S = u t + \\frac{1}{2} a {t}^{2}$\n\nConsidering upward direction to be positive, we get, $S = - 9$ as on reaching the ground it will have a net displacement of $9 m$ downwards.\n\n$u = 5 m {s}^{-} 1$ and $g = - 9.8 m {s}^{-} 2$\n\nSo,if it takes time $t$ to reach the ground we can write,\n\n-9 = 5t -1\/2 \u00d79.8\u00d7t^2\n\nSolving this we get, $t = 1.96 s$","date":"2019-09-22 14:52:19","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 8, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5324309468269348, \"perplexity\": 318.1976347695931}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-39\/segments\/1568514575515.93\/warc\/CC-MAIN-20190922135356-20190922161356-00456.warc.gz\"}"} | null | null |
(function () {
var app = angular.module("tepEx1", ["ngMaterial"]);
app.controller("MainController", ["$scope", "$mdSidenav", function ($scope, $mdSidenav) {
$scope.toggleSidenav = function (menuId) {
$mdSidenav(menuId).toggle();
};
}]);
})(angular, window); | {
"redpajama_set_name": "RedPajamaGithub"
} | 1,862 |
module V1
class ItemJSONDecorator < Draper::Decorator
delegate :id, :name, :children, :parent, :user_defined_id, :collection, :unique_id, :updated_at
def self.display(item, json)
new(item).display(json)
end
def at_id
h.v1_item_url(object.unique_id)
end
def parent_url
if object.parent
h.v1_item_url(object.parent.unique_id)
end
end
def collection_url
h.v1_collection_url(collection_id)
end
def children_url
if object.children && !object.children.empty?
h.v1_item_children_url(object.unique_id)
end
end
def showcases_url
h.v1_item_showcases_url(object.unique_id)
end
def pages_url
h.v1_item_pages_url(object.unique_id)
end
def collection_id
object.collection.unique_id
end
def description
object.description.to_s
end
def additional_type
"https://github.com/ndlib/honeycomb/wiki/Item"
end
def slug
CreateURLSlug.call(object.name)
end
def media
if object.media
SerializeMedia.to_hash(media: object.media)
end
end
def metadata
V1::MetadataJSON.metadata(object)
end
def display(json)
if object.present?
set_json_keys(json)
end
end
def to_builder
Jbuilder.new do |json|
display(json)
end
end
def to_json
to_builder.target!
end
def to_hash
JSON.parse(to_json)
end
private
def set_json_keys(json) # rubocop:disable Metrics/AbcSize
json.set! "@context", "http://schema.org"
json.set! "@type", "CreativeWork"
json.set! "@id", at_id
json.set! "isPartOf/item", parent_url
json.set! "isPartOf/collection", collection_url
json.set! "hasPart/children", children_url
json.set! "hasPart/showcases", showcases_url
json.set! "hasPart/pages", pages_url
json.set! "additionalType", additional_type
json.id unique_id
json.user_defined_id user_defined_id
json.collection_id collection_id
json.slug slug
json.name name
json.description description.to_s
json.media media
json.metadata metadata
json.last_updated updated_at
end
end
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,664 |
5 minutes to explain the Dean phenomenon
I have been invited to present to a high-powered group of Illinois Democrats on Monday morning. The group will include Governor Blagojevich — and the subject is how to energize the grassroots people who've felt out of touch with the Democratic Party. My role is, as someone closely involved with the Dean technology crew, to explain what we did and why it worked.
Here's what I think were the keys to the Dean success (the barometer for purposes of this discussion is fundraising, not election results):
Give people a voice. The minute people think their voice is heard is the minute they feel empowered. Empowered individuals want to see their group succeed, and money was universally acknowledged as an important yardstick of the group's success.
Talk back. Communication is about dialogue, not press releases. When campaigns engage their supporters, let their supporters (as well as their opponents) speak up, and most importantly, when they respond, the campaign appears more authentic. Authenticity breeds respect. And respect earns the people's commitment.
Make supporters visible. By making individuals' contributions visible (and the results of those contributions apparent), Dean was sharing the spotlight. When he said "this race is about you, not me", he meant it. People who were either disenchanted with traditional politics, or cynical about their ability to make a difference to begin with, suddenly felt like they really could do something.
Be transparent. Transparency affected all levels of the campaign — from being open about fundraising goals (and their progress to those goals) to sharing works-in-progress on policy issues — and it made supporters feel like they were contributing to that progress instead of taking orders from above. Furthermore, as goals got within reach, supporters dug in and gave more (money, time, energy) to reach the goal.
Give up control. This is perhaps the scariest for those who've been around a while. Dean's campaign didn't try to control the message or the medium, instead choosing to let the grassroots run with it. This is where the barometer mentioned above is important: from a fundraising perspective, this lack of control gave each contributor a sense of ownership. It's arguable that this lack of control is what contributed to the transition from a campaign to a movement, but that's fodder for a different analysis.
So… I've got five minutes on Monday to boil this down and try to ensure that those in the crowd understand that it's not about bits and bytes, not about keyboards and mousepads, but about energy. And enthusiasm. And passion. And I believe the bullets above highlight how the Dean campaign tapped each of those. I'd love to hear your thoughts. What have I missed?
6 responses to "5 minutes to explain the Dean phenomenon"
Besides for expecting them to wonder how their buddies can profit. After all it is Illinois politics… It would be interesting to see if Chicago ward machines can or even would have any interest in actaully trying this stuff.
Don't forget the need for a strong candidate. I don't think any of your ideas would have done much good for Joe Lieberman, for example.
I think you might want to read Howard Kurtz's piece in the Washington Post this morning…….it gives an interestingperspective.http://www.washingtonpost.com/ac2/wp-dyn/A15741-2004Feb28?language=printer
Read this. Joe Trippi gave a talk about the conflicts between netroots and the campaign. This is a summary.http://www.baselinemag.com/article2/0,3959,1530690,00.asp
It is astonishing that you are spending tons of energy talking about what went right with the Dean Campaign and absolutly nothing about what went wrong with it and how to go about changing what went wrong.Was it Dean's message no one liked? Dean was far too liberal? Dean was just a baffoon who made too many mistakes? Dean's attack ads went over board? Dean Youth Organization spent tons of their time HATING BUSH and absolutely none of their time giving any voters an explanation or re-assurance of how a Radical like Howard Dean could get the Moderate/Centrist vote and Swing Voters, Split Ticket Voters. How Dean could reach out to Hispanics? Youth People organize well with heated anger but they don't vote? The real question is why in the hell bother organizing the Youth if they don't vote?There's plenty of questions that have so far gone unanswered. And that is astonishing, albeit not surprising. I don't carry much weight with the Left that they would ever blame themselves for any mistakes they make.
visit my dean for america website @ http://www3.deanforamerica.com/site/TR?pg=personal&fr_id=1090&px=1880890&s_tafId=1100&s_oo=cU0UT-O2TNCPLhYs3lnipg and jeff the only thing that ruined dean was the fact that too many americans are corporate mainstream media fed idiots. dean screamed. and so what? so the fact he cut child abuse 50% in vermont doesn't matter now? the fact that he raised school test scores with his genius reworking of the distribution of the funding there? made it so over 33% of the energy vermont uses is renewable, that vermont ranks 1st in the percentage of children with health insurance? that he set aside 470,000 acres of land for conservation balanced the budget 11 years cut taxes set aside a rainy day fund i could go on.. all the amazing things he's done and wants to do for us don't matter? they do matter just dem. voters didn't research these candidates and all they know about dean is that he screamed because the mainstream media played that soundbite 700 times a day, more times than they showed twin towers falling, more times than they've ever played anything it broke every record. it's not deans fault americans are shallow and watch too much tv and it's not his fault kerry and bush have spent a combined 6 million on negative adds about dean, some even showing deans picture then osamas. this is the same country that was 90% behind the war in iraq the day after it started and now we know there were no weapons we never saw any proof in the first place. americans don't like smart honest people like dean. they like idiots who lie to them and remind them of themselves and that's not deans problem. another thing you getting mad at dean for involving youth in his campaign because they aren't old enough to vote is absurd, also dean is not a radical he's fiscal concervative social liberal and nothing radical about him i can see unless, as you seem to think, caring for childrens rights i.e. americas future, makes a person radical because children can't vote. mayb all you know about him yourself is that he screamed, you can't know anything about him and still think he's a radical. you're a weirdo, i'm glad you don't like dean. people like you, make people who don't like dean, look like creeps. i'm 28 years old btw so now you know there are people supporting dean who aren't children who can and will vote. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,467 |
/// @defgroup cpGrooveJoint cpGrooveJoint
/// @{
/// Check if a constraint is a slide joint.
cpBool cpConstraintIsGrooveJoint(const cpConstraint *constraint);
/// Allocate a groove joint.
cpGrooveJoint* cpGrooveJointAlloc(void);
/// Initialize a groove joint.
cpGrooveJoint* cpGrooveJointInit(cpGrooveJoint *joint, cpBody *a, cpBody *b, cpVect groove_a, cpVect groove_b, cpVect anchorB);
/// Allocate and initialize a groove joint.
cpConstraint* cpGrooveJointNew(cpBody *a, cpBody *b, cpVect groove_a, cpVect groove_b, cpVect anchorB);
/// Get the first endpoint of the groove relative to the first body.
cpVect cpGrooveJointGetGrooveA(const cpConstraint *constraint);
/// Set the first endpoint of the groove relative to the first body.
void cpGrooveJointSetGrooveA(cpConstraint *constraint, cpVect grooveA);
/// Get the first endpoint of the groove relative to the first body.
cpVect cpGrooveJointGetGrooveB(const cpConstraint *constraint);
/// Set the first endpoint of the groove relative to the first body.
void cpGrooveJointSetGrooveB(cpConstraint *constraint, cpVect grooveB);
/// Get the location of the second anchor relative to the second body.
cpVect cpGrooveJointGetAnchorB(const cpConstraint *constraint);
/// Set the location of the second anchor relative to the second body.
void cpGrooveJointSetAnchorB(cpConstraint *constraint, cpVect anchorB);
/// @}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,452 |
package com.github.marcobjorge.anytime.interchange;
import com.github.marcobjorge.anytime.dispatcher.Dispatcher;
import com.github.marcobjorge.anytime.event.AnytimeEvent;
import com.github.marcobjorge.anytime.util.ActiveChecker;
import org.slf4j.Logger;
import org.slf4j.LoggerFactory;
import static com.google.common.base.Preconditions.checkNotNull;
public abstract class BaseInterchange<T extends AnytimeEvent, V extends AnytimeEvent> implements Interchange<T, V> {
private static final Logger logger = LoggerFactory.getLogger(BaseInterchange.class);
private final String id;
private boolean active;
private ActiveChecker activeChecker;
protected BaseInterchange(final String id) {
checkNotNull(id, "id cannot be null");
this.active = true;
this.id = id;
this.activeChecker = () -> false;
}
@Override
public String id() {
return id;
}
@Override
public void start(final ActiveChecker activeChecker, final Dispatcher<T, V> dispatcher) {
checkNotNull(activeChecker, "activeChecker cannot be null");
checkNotNull(dispatcher, "dispatcher cannot be null");
this.activeChecker = activeChecker;
logger.trace("Interchange {} started", id());
while (isActive()) {
run(dispatcher);
}
logger.trace("Interchange {} stopped", id());
}
@Override
public void stop() {
this.active = false;
}
@Override
public boolean isActive() {
return this.active && this.activeChecker.isActive();
}
protected abstract void run(final Dispatcher<T, V> dispatcher);
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,902 |
<?xml version="1.0" encoding="UTF-8"?>
<html xmlns="http://www.w3.org/1999/xhtml" xmlns:wicket="http://wicket.apache.org">
<head>
<title>Wicket Examples - component reference</title>
<link rel="stylesheet" type="text/css" href="style.css"/>
</head>
<body>
<span wicket:id="mainNavigation"/>
<h1>wicket.markup.html.form.RadioChoice</h1>
<wicket:link><a href="Index.html">[back to the reference]</a></wicket:link>
<p>
A RadioChoice component lets users select a single value from a group of radio buttons.
</p>
<p>
<form wicket:id="form">
<table style="border: 2px dotted #fc0; width: 400px; padding: 5px;">
<tr>
<td valign="top">Select your favorite site</td>
<td>
<span valign="top" wicket:id="site">
<input type="radio">site 1</input>
<input type="radio">site 2</input>
</span>
</td>
</tr>
<tr>
<td colspan="2" align="center">
<input type="submit" value="submit" />
</td>
</tr>
</table>
</form>
<span wicket:id="feedback">feedbackmessages will be put here</span>
</p>
<span wicket:id="explainPanel">panel contents come here</span>
</body>
</html>
| {
"redpajama_set_name": "RedPajamaGithub"
} | 360 |
{"url":"http:\/\/openstudy.com\/updates\/55c2b507e4b0c7f4a9786311","text":"## Loser66 one year ago An open box with the length is twice its width. Find the maximum value that the volume of the box can have. Please, help.\n\n1. Loser66\n\n|dw:1438823690584:dw|\n\n2. anonymous\n\nis the length 2 or width 2?\n\n3. Loser66\n\nno numbers there, just that\n\n4. anonymous\n\n2w = length of box?\n\n5. Loser66\n\nyes\n\n6. anonymous\n\nis the box a rectangle as stated or is it a square\n\n7. Loser66\n\nMy attempt: Volume of the box is V = x*W*L = x*W (2W) = 2x W^2\n\n8. anonymous\n\nThey said the length is 2x bigger, so w = 1 in that case\n\n9. Loser66\n\nSurface are of the open box is $$S = (L+2x)(W+2x) - 4x^2= (2W+2x)(W+2x)-4x^2$$ $$S = 2W^2+4xW+2xW+4x^2-4x^2= 2W^2+6xW$$\n\n10. anonymous\n\nI gotta run, continue with someone else\n\n11. Mertsj\n\nHow can you find the volume if you know nothing about the height of the box?\n\n12. Loser66\n\nThanks\n\n13. anonymous\n\n@Mer I thought the same, anyways np loser.\n\n14. Loser66\n\n@Mertsj maximum of volume at the point of derivative of the Volume =0\n\n15. Loser66\n\nWe consider Volume of the box is the function with respect to x and W\n\n16. Loser66\n\n$$V = 2xW^2\\\\V'= 2W^2 + 2xW*W'$$\n\n17. anonymous\n\ni think we're presuming maximum volume for a fixed perimeter, $$P=2(2x+w+2x+2w)$$\n\n18. anonymous\n\nso $$P=8x+6w\\implies x=\\frac18(P-6w)$$\n\n19. Loser66\n\nNo, not that , the perimeter of what? if it is of the box, then you don't have x there, right?\n\n20. Loser66\n\n|dw:1438825047689:dw| that is perimeter. One more thing, the prof asked us to work with Surface area. I am sorry for this information\n\n21. anonymous\n\nfixed perimeter of the original rectangle that we cut the box out of. so now we have $$V(x,w)=w\\cdot2w\\cdot x=2xw^2$$now using $$x=\\frac18(P-6w)$$ we have: $$V=\\frac14(P-6w)w^2=\\frac14(Pw^2-6w^3)$$now take: $$\\frac{dV}{dw}=\\frac14(2Pw-18w^2)=\\frac12(Pw-9w^2)$$so $$\\frac{dV}{dw}=0\\\\Pw-9w^2=0\\\\w(P-9w)=0\\implies w=P\/9$$\n\n22. Loser66\n\noh, sorry, you are correct with perimeter. :)\n\n23. anonymous\n\nwhich gives $$V^*=\\frac14(P-6\\cdot P\/9)(P\/9)^2=\\frac14(P\/3)(P^2\/81)=\\frac{P^3}{4\\cdot 243}$$\n\n24. Loser66\n\nI got what you meant. How about Surface area? how to work on it?\n\n25. misty1212\n\ni get $v(x) = 2 w^2 x-6 w x^2+4 x^3$does that help?\n\n26. anonymous\n\nand also $$x=\\frac18(P-6w)=\\frac18(P\/3)=\\frac{P}{24}$$\n\n27. Loser66\n\n@misty we have to find x such that the Volume is maximum @oldrin.bataku Again, I have to work with Surface area, not perimeter. Please\n\n28. misty1212\n\nyeah take the derivative wrt x get$2 (w^2-6 w x+6 x^2)$\n\n29. misty1212\n\nset equal to zero, solve for $$x$$ get something like $(\\frac{1}{2}+\\frac{\\sqrt3}{6})w$\n\n30. misty1212\n\nor maybe that is the wrong one\n\n31. Loser66\n\n@misty1212 because V = 2xW^2 , not as your equation.\n\n32. misty1212\n\n$(\\frac{1}{2}-\\frac{\\sqrt3}{6})w$ is probably better\n\n33. misty1212\n\nhmm i get $V=(w-2x)(2w-2x)x$ as a start\n\n34. misty1212\n\nmultiply out get $V(x)=2 w^2 x-6 w x^2+4 x^3$\n\n35. Loser66\n\n@misty1212 not that, the w is the width of the box, 2w is the length of the box and x is its height\n\n36. misty1212\n\nnot the way you drew it for sure\n\n37. misty1212\n\noh i see then the way the question is asked, there is no maximum make w bigger and bigger the volume increases without bound\n\n38. Loser66\n\n@oldrin.bataku I saw you wrote something, where is it?? why don't you post it?\n\n39. anonymous\n\nso for a fixed surface area? $$A=2w^2+2wx+2(2w)x\\\\\\quad =2w^2+6wx$$so we have $$x=\\frac{A-2w^2}{6w}=\\frac{A}{6w}-\\frac{w}3$$ so $$V=2xw^2=2\\left(\\frac{A}{6w}-\\frac{w}3\\right)w^2=\\frac{Aw}{3}-\\frac{2w^3}{3}\\\\\\frac{dV}{dw}=\\frac{A}3-2w^2$$so $$\\frac{A}3-2w^2=0\\\\2w^2=A\/3\\\\w^2=A\/6\\\\w=\\sqrt{A\/6}$$\n\n40. Loser66\n\nWhere is your x? is it not that we have to find the ratio of x and W such that V is max? I meant like x = 1\/5 W, then V is max.\n\n41. anonymous\n\nplug it in: $$x=\\frac{A-2\\cdot A\/6}{6\\cdot\\sqrt{A\/6}}=\\frac{A-A\/3}{\\sqrt{6A}}=\\frac{2A\/3}{\\sqrt{6A}}=\\frac{\\sqrt{2A}}{3\\sqrt3}$$\n\n42. anonymous\n\nso the ratio of w, x is just $$x\/w=\\frac{\\sqrt{2A}\\cdot\\sqrt6}{3\\sqrt3\\cdot\\sqrt{A}}=\\frac{\\sqrt2\\cdot\\sqrt2}{3}=\\frac23$$\n\n43. Loser66\n\nI got exactly the same, but it is impossible when $$0<x<W\/2$$\n\n44. Loser66\n\n|dw:1438827425494:dw|","date":"2017-01-23 23:24:36","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 2, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8321963548660278, \"perplexity\": 2454.465047032384}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-04\/segments\/1484560283301.73\/warc\/CC-MAIN-20170116095123-00397-ip-10-171-10-70.ec2.internal.warc.gz\"}"} | null | null |
Cipriano de Rore, född 1516 i Antwerpen, död 1565 i Parma, var en nederländsk kontrapunktist, verksam under renässansen i den nederländska skolan.
Biografi
Cipriano de Rore var lärjunge till Adrian Willaert i Venedig, blev kapellsångare vid Markuskyrkan där, var hovkapellmästare hos Alfonso II d'Este i Ferrara 1553-58, hovkapellmästare i Parma 1561-63 och 1564-65, dessemellan ett år kapellmästare i Markuskyrkan.
Han räknas bland de mest betydande av sin tids tonsättare och var en bland de första som i praxis tillgodogjorde sig de experiment med kromatiska toner, som Willaert och Zarlino anställde. Hans madrigaler trycktes (1577) i partiturform, på den tiden en stor sällsynthet. Rore skrev även flera böcker motetter, mässor, psalmer o. s. v.
Verkförteckning
Sekulär musik
I madrigali (Venedig, 1542, fem röster)
Il primo libro de madregali cromatici (Venedig, 1544, fem röster; utvidgning av publikationen 1542)
Il secondo libro de madregali (Venedig, 1544, fem röster)
Il terzo libro di madrigali, (Venedig, 1548, fem röster)
Musica ... sopra le stanze del Petrarcha ... libro terzo (Venedig, 1548, fem röster)
Il primo libro de madrigali (Ferrara, 1550, fem röster) (innehåller också chansoner på franska)
Il quarto libro d'imadregali (Venedig, fem röster)
Il secondo libro de madregali, (Venedig, 1557, fyra röster)
Li madrigali libro quarto, (Venedig, 1562, fem röster)
Le vive fiamme de' vaghi e dilettevoli madrigali, (Venedig, 1565, fyra och fem röster) (innehåller också sekulrära stycken på latin)
Il quinto libro de madrigali (1566, fem röster) (innehåller också sekulära stycken på latin)
Flera andra verk i antologier, mellan 1547 och 1570
Sakral musik
Motectorum liber primus (Venedig, 1544, fem röster)
Motetta (Venedig, 1545, fem röster)
Il terzo libro di motetti (Venedig, 1549, fem röster)
Passio ... secundum Joannem (Paris, 1557; två till sex röster)
Motetta (Venedig, 1563, fyra röster)
Sacrae cantiones (Venedig, 1595; fem till sju röster)
Källor
Nederländska klassiska kompositörer
Renässanskompositörer
Nederländska musiker under 1500-talet
Födda 1516
Avlidna 1565
Män
Ugglan | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,144 |
\section{Introduction}
\label{intro}
Topological Field Theories have stirred a lot of interest, both in
two and in four dimensions \cite{suit1}. Their general feature is that of
recasting intersection theory in the moduli-space of some suitable
geometrical structure into the language of standard quantum
field-theory, specifically into the framework of the path-integral.
Indeed the point-independent correlation functions of these peculiar
field-theories represent intersection integrals of cohomology classes in
the given moduli-space.
Hystorically, the first topological field-theory
that has been introduced, is the topological version of 4D Yang-Mills
theory \cite{witten}, sometimes named Donaldson theory. It deals with the
moduli-spaces of Yang-Mills instantons and its correlation functions
describe Donaldson invariants \cite{donaldson}.
A lot of attention has also been
devoted to topological sigma models in two dimensions
\cite{witten2}. In this case
one probes the moduli-space of holomorphic mappings from the world-sheet
to a complex target space. Theories that have a close relation
with topological sigma-models are the topological versions of N=2
Landau-Ginzburg models \cite{suitA}.
They have provided an interesting arena for
the study of the moduli-spaces associated with Calabi-Yau manifolds
\cite{suitB},
a topic of primary interest in connection with the effective
Lagrangians of superstring models.
In a different, but closely related set up, the coupling of topological
matter multiplets to topological 2D gravity \cite{suit3} has been used to
investigate
non critical string theories
and relations have been established
with the integrable hierarchies discovered in matrix models \cite{suit4}.
{}From a formal field-theoretic point of view the general framework
of topological field-theories is that of geometrical
BRST-quantization \cite{suit5}. One
deals with a classical Lagrangian that has a very large symmetry,
such as the group of continuous deformations of a gauge-connection
or of a metric and which, therefore, is a topological-invariant-density
(i.e. some characteristic class of some fibre-bundle). To this symmetry
one applies the standard BRST quantization scheme and, in this way, one
obtains a topological BRST-cohomology, namely a double elliptic
complex involving both the standard exterior derivative $d^2=0$
and a second nilpotent operator (the Slavnov operator $s^2=0$) that
anticommutes with the first: $sd+ds=0$. The true geometrical and physical
content of the theory emerges when one fixes the gauge: indeed the gauge fixing
condition is, normally, some kind of self-duality condition that reduces the
space of physical states to the space of suitable {\sl instantons}.
In this perspective the relevance of the topological twist is appreciated.
This is a procedure, discovered by Witten \cite{witten}, that extracts a
topological
field-theory with its gauge already fixed to a suitable instanton condition
from an
N=2 supersymmetric ordinary field-theory. Actually the very first example of
topological field-theory, namely
Donaldson theory, was constructed in this way
starting from N=2 super Yang-Mills theory. The basic ingredients of the twist
procedure are:
i) the possibility of changing the spins of the fields, by redefining
a new Lorentz group as the diagonal of the old one (or a factor thereof)
with an internal symmetry group, in such a way that, after the twist,
the top spin boson of each supersymmetric multiplet and one of its
fermionic partners acquire the same spin in the new theory;
ii) the existence of an additional U(1)-symmetry of the old theory, such that,
redefining also the ghost number as the old one plus this particular
U(1)-charge, the anticommuting partners of the bosons, that have acquired the
same spin in the twist procedure, have, in the new theory, ghost number one,
while their bosonic partners remain with ghost number zero. In this way the
old fermions become the ghost associated with the topological symmetry.
The twist not only provides a constructive procedure for topological
field-theories but also illuminates the topological character of a sector of
the
parent theory. This way of thinking has been most successfully implemented
in two-dimensions. There the (Euclidean) Lorentz group is SO(2) and it can
be easily redefined by taking its diagonal with the U(1) automorphism group
of N=2 supersymmetry. In this simple case, the same U(1) provides also the
charge to shift the ghost numbers. The result, as already mentioned, is given
by either the topological sigma-models, or the topological Landau-Ginzburg
models, or their coupling to topological 2D gravity. The topological sector
of the original N=2 theory that is unveiled by this twist procedure is that of
the chiral correlation functions.
\par
In four-dimensions the twist procedure relies once more on the properties of
N=2 supersymmetry, but involves many more subtleties, so that the programme of
topologically twisting all N=2, D=4 theories needs deeper thinking. This
programme has been started in \cite{anselmifre} by twisting pure N=2
supergravity: in the present
paper we push this programme one step further by twisting
N=2 supergravity coupled to vector multiplets and by discussing the effect
of the twist on N=2 hypermultiplets. The accomplished result of the present
paper is given by a D=4 topological Yang-Mills theory coupled to topological
D=4 gravity, the space of physical states being the moduli-space
of gauge-instantons living in the background of gravitational instantons. One
of the properties of this theory is that it does not seem to reduce to
Donaldson
theory in the limit where the gravitational coupling is switched off. Hence
it seems to define a different topological Yang-Mills theory. Whether this
difference is substantial or not is still to be clarified; anyhow it is not
accidental rather it is deeply rooted in the properties of N=2 supersymmetry.
\par
Indeed the subtleties one encounters in twisting N=2,D=4 theories
relate mostly to the second item of the twisting programme, namely to the
identification of the U(1) symmetry needed to shift the ghost-number. This
identification is involved with the non-linear sigma model structure
of the original N=2 theory, in particular with the special
K\"ahler geometry of the vector multiplet coupling. In this paper we find out
that the required U(1)-symmetry, named by us R-duality, exists, in the
supergravity coupled case, if the Special K\"ahler manifold is chosen to be
$SU(1,n)/SU(n) \times U(1)$, the so named minimal coupling case. In the
flat case the needed U(1) also exists, as Witten construction shows, if the
minimal coupling is selected. The point is that the minimal coupling in
flat space and in curved space correspond to different unequivalent
sigma model geometries: the flat $C^n$-manifold versus the special
K\"ahler manifold $SU(1,n)/SU(n) \times U(1)$. This shows how the
flat space limit of the gravity coupled topological Yang-Mills theory is
in principle different from Donaldson theory as constructed by Witten.
Other subtleties of the D=4 topological twist were already encountered
and resolved in our previous paper on pure N=2 supergravity. Indeed
the greater complexity of N=2 supergravity with respect to N=2 super Yang-Mills
forced us \cite{anselmifre} to generalize the procedure of topological twist
as introduced by Witten \cite{witten} in N=2 super Yang-Mills and
at the same time lead us to reach a deeper understanding of its structure.
In particular, we stress that the twist acts only on
the Lorentz indices and not on the space-time indices \cite{anselmifre}
and this is quite natural in the formalism of differential forms.
This feature of the twist
avoids the problem encountered by Witten in Ref.\ \cite{witten},
namely that the twisting procedure is meaningful only when space-time is
$R^4$. We shall come back on this aspect extensively in this paper.
When one studies the topological sector of N=2 matter coupled supergravity,
one soon realizes that other aspects of the twist
still need a better understanding.
In particular, as we already pointed out, the fundamental question is the
following: what is, in general terms, the $U(1)$ symmetry that leads to the
ghost number of the topological version of a given theory? In N=2 super
Yang-Mills,
as well as in N=2 pure supergravity there is only one $U(1)$ internal
symmetry (apart from global dimensional
rescalings, that are not relevant to our
discussion)
and so either it works or not. Fortunately it works. However,
in N=2 supergravity coupled to vector multiplets, there can be more that
one internal $U(1)$; think for example of
the $U(1)$ K\"ahler transformation or some $U(1)$
subgroup of the group of duality transformations \cite{gaillardzumino}
(at least when the vectors are not gauged). Anyway neither of these
two known possibilities has the correct properties to become a ghost
number and further on
we show that indeed they cannot do the job.
On the other hand one expects that a twist is possible, since the theory
of topological gravity coupled to topological Yang-Mills should
exist. In Ref.\ \cite{anselmifre} we have shown how to produce a
gauge-free algebra and generic observables
for {\sl any} topological theory and it would be very surprising to find
that it is impossible to choose any kind of instantons to fix the topological
symmetry and a gauge fermion to give a lagrangian to the theory. So, we
start our work with the belief that if a suitable $U(1)$ internal charge
is missing, this is because it is not known and not because it does not exist.
As anticipated,
it will be named R-duality, for reasons that we shall explain.
First we define it
and this lead us to single out the basic properties
an internal $U(1)$ symmetry should have
in order to give ghost number. Then
we shall explicitly
prove invariance of the minimally coupled theory
under this symmetry.
Our paper is organized as follows.
In section II we make some general
remarks on the possibility that minimal N=2 matter coupled
supergravity possesses the desired internal $U(1)$ symmetry (R-duality).
In section III we recall the structure of N=2 matter coupled supergravity
in the rheonomy framework. In section IV we fully determine R-duality
and prove that it is indeed an on shell symmetry of the theory. In section V
we present the topologically twisted-topologically shifted theory
(the gauge-free algebra, the complete BRST algebra,
the topological gauge-fixings, the observables, the gauge-fermion).
Finally, in section VI we discuss the twist of quaternionic matter multiplets
coupled to N=2 supergravity and along with this discussion, we
summarize all the steps of the twisting procedure in four dimensions,
improved by the experience of the present paper.
\section{General remarks on R-duality}
\label{general}
In this section we discuss the possibility that
minimal N=2 matter coupled supergravity is
R-duality invariant.
This internal $U(1)$ charge will add to the ghost number to define
the ghost number of the topologically twisted theory.
Thus we shall be able to extend the procedure of topological twist
and topological shift of Ref.\ \cite{anselmifre} in a rather direct way.
Let us first make some
simple remarks about the properties of the
chiral-dual invariance displayed by N=2 simple supergravity.
These properties
will guide us in finding the desired generalization to the
matter coupled case. We use the same notation
of Ref.\ \cite{anselmifre}. Consider the Bianchi identity of
the graviphoton $A$, that is
\begin{equation}
{\cal D}\!R^{\otimes}+2\epsilon_{AB}\bar\psi_A\wedge\rho_B=0,
\end{equation}
its equation of motion,
\begin{equation}
4i\epsilon_{AB}\bar\rho_A\wedge \gamma_5\psi_B-{\cal D}(F^{ab}V^c\wedge V^d)
\epsilon_{abcd}=0,
\label{eqmo}
\end{equation}
the rheonomic parametrization of the graviphoton
curvature $R^\otimes$,
\begin{equation}
R^\otimes=F_{ab}V^a\wedge V^b,
\end{equation}
and the on shell chiral-dual transformation, i.\ e.\
\begin{eqnarray}
\hat\delta \psi_A=i\gamma_5\psi_A\nonumber\\
\hat\delta F_{ab}=-2i\tilde F_{ab}=\epsilon_{abcd}F^{cd}.
\label{chidua}
\end{eqnarray}
In Ref.\ \cite{anselmifre} it was noted that the chiral-dual variation
of the Bianchi identity is the equation of motion and {\sl viceversa}. This is
evident if we re-write the Bianchi identity of the graviphoton and its equation
of
motion in the following form
\begin{eqnarray}
d[R^\otimes-\epsilon_{AB}\bar\psi_A\wedge\psi_B]&=&0,\nonumber\\
d[\epsilon_{abcd}F^{ab}V^c\wedge V^d-
2i\epsilon_{AB}\bar\psi^A\wedge\gamma_5\psi^B]&=&0.
\end{eqnarray}
Moreover, let us see what is the condition for the transformation
(\ref{chidua}) to be well defined,
i.\ e.\ what is required for the existence of
a $\hat\delta A$ compatible with (\ref{chidua}).
One immediately finds
\begin{eqnarray}
\epsilon_{abcd}F^{cd}V^a\wedge V^b&=&\hat\delta[F_{ab}V^a\wedge
V^b]=\nonumber\\
=\hat\delta R^\otimes&=&d\hat\delta
A+2i\epsilon_{AB}\bar\psi_A\wedge\gamma_5\psi_B.
\end{eqnarray}
So, $\epsilon_{abcd}F^{cd}V^a\wedge V^b-2i\epsilon_{AB}\bar\psi_A\wedge
\gamma_5\psi_B$
must be an exact form and we focus on the case
in which a necessary and sufficient condition
for this to be true
is that the form is closed, i.\ e.\
$d[\epsilon_{abcd}F^{cd}V^a\wedge V^b-2i\epsilon_{AB}\bar\psi_A\wedge\gamma_5
\psi_B]=0$. This is precisely the equation of
motion for the graviphoton (\ref{eqmo}). Consequently, the
$U(1)$ transformation is defined on shell and only on shell.
This way of reasoning is a natural generalization of the
well known case of electromagnetism and it will directly extend
to N=2 matter coupled supergravity.
What do we expect R-duality to be like? Obviously, it should
reduce to the known results both on the gravitational multiplet when matter is
suppressed and on the vector multiplets when gravity is switched off.
In other words, it should be a dual transformation on the graviphoton
(that is why we call it {\sl duality}),
a chiral transformation on the fermions and should leave the graviton and
the matter vectors inert. The scalars of the vector multiplets should
have charges $+2$ and $-2$. Consequently, on the fields of the
vector multiplets the symmetry we are
seeking should act as the usual internal $U(1)$ symmetry of
N=2 super Yang-Mills, which is an R-symmetry \cite{fayetferrara}.
Finally, it should be possible to gauge
the matter vectors (but not the graviphoton) while preserving the symmetry.
We expect R-duality not to be present in the most general case,
i.\ e.\ with any special K\"ahler manifold, but only in the simplest case,
namely for
minimal coupling \cite{dewit}. This is suggested by the fact
that something similar seems to
happen even in the case of flat N=2 super Yang-Mills theory.
As a matter of fact, the theory involves
the choice of an arbitrary flat special geometry prepotential $F(X)$,
which is a holomorphic homogeneous function of degree two
of the simplectic sections $X_\Lambda$ \cite{cremmer}.
As a result, the lagrangian involves a coupling matrix
$f^{ij}(z)$, which, in flat coordinates $z_i={X_i\over X_0}$,
depends holomorphically on the scalars $z_i$
and is given by the second derivative of $F$, $f^{ij}(z)=
{\partial\over \partial z_i}
{\partial\over \partial z_j}F(X(z))$ \cite{cremmer}.
The kinetic lagrangian of the vectors has the following form
\begin{equation}
F^i_{\mu\nu}F^{j\mu\nu}{\rm Re} \hskip .05truecm f^{ij}-
{1\over 2}\varepsilon^{\mu\nu\rho\sigma}
F^i_{\mu\nu}F^j_{\rho\sigma}{\rm Im} \hskip .05truecm f^{ij}.
\end{equation}
Only when $f^{ij}(z)=\delta^{ij}$,
namely when $F$ is quadratic, there is an evident R-invariance, since if $z$
has a nonvanishing charge, then
the only neutral holomorphic function of $z$ is the constant.
In other words, the topological twist appears to be
possible only in one case,
although the negative result that R-symmetry is barred in nonminimal
coupling has not been established in a conclusive way.
Indeed, we shall prove that R-duality exists in minimal matter
coupled N=2 supergravity, but we shall not prove
that this is the only possible case. There could
be some unexpected field redefinitions that make it work in more general
cases, even if they presumably cannot
make it suitable for a topological twist.
Uniqueness remains, for the time being, just our conjecture.
We recall that in topological Yang-Mills theory the chiral anomaly
becomes ghost number anomaly after the twist and can be described by
saying that the functional measure has a definite nonvanishing
ghost number. Consequently, only the amplitudes of observables that
have a total ghost number opposite to this value
can be nontrivial. These features of
ghost number are present also in topological gravity with or
without matter. In Ref.\ \cite{dolgov} it is shown that
the dual invariance of Maxwell theory in external gravity is anomalous.
In topological gravity
we thus expect a ghost number anomaly which is due not only to the
anomalous chiral behaviour of the fermions, but also to the
anomalous dual behaviour
of the graviphoton. In other words one has to take care of
the zero modes of the graviphoton, besides those of the fermions.
Let us now derive some {\sl a priori} information about R-duality.
As in Ref.\ \cite{anselmifre} to each field of the theory we assign
a set of labels $^c(L,R,I)^g_f$,
where $L$ is the representation of $SU(2)_L$,
$R$ is the representation of $SU(2)_R$,
$I$ is the representation of $SU(2)_I$, $c$ is the $U(1)_I$ charge,
$g$ the ghost number and $f$ the form number.
If the twist acts on $SU(2)_R$,
then after the twist we have objects described by
$(L,R\otimes I)^{g+c}_f$. In this case the left handed
components of gravitinos and gauginos must necessarily have $U(1)_I$ charge
$+1$, since they are the only fermions that have the correct spin content
to give the topological ghosts after the twist.
For example, the left handed components of the gravitinos are characterized by
$({1\over 2},0,{1\over 2})^0_1$ and give $({1\over 2},{1\over 2})_1$
after the twist,
and the vierbein $V^{a}$ is also a $({1\over 2},{1\over 2})_1$ object.
Similarly, the left handed components of the gauginos become
$({1\over 2},{1\over 2})_0$ after the twist: let us call them $\lambda_a$.
The vector bosons,
however, are Lorentz scalars, so they give $(0,0)^0_1$. Consequently, the
correct topological ghosts can only be $\lambda_a V^a$.
The charge of the right handed components of gravitinos and gauginos
is fixed to be
$-1$ by the fact that they are the natural candidates
to become the topological antighosts,
as far as their Lorentz
transformation properties
are concerned. As a check, we can also see that
the charge of the right handed gravitinos is independently fixed by the
following argument to the value $c=-1$.
The supersymmetry charges must also transform. In fact,
the right handed components of the supersymmetry ghosts,
which are the ghost partners of the right handed gravitinos
and so must have the same charge,
are characterized by $(0,{1\over 2},{1\over 2})^1_0$ and give
$(0,1)_0\oplus (0,0)_0$ after the twist. This is the only possibility
to obtain a scalar zero form from the supersymmetry ghosts and we recall
\cite{anselmifre} that the $(0,0)_0$ component must be topologically shifted
by a constant in order to define the BRST symmetry of the topological theory.
This implies $g+c=0$ for the right handed components of the supersymmetry
ghosts, and so $c=-1$.
We conclude that on any of the so far considered
fermions, collectively denoted by $\lambda$ (supersymmetry
ghosts included),
R-duality acts as follows
\begin{eqnarray}
\hat\delta \lambda_L=\lambda_L\nonumber\\
\hat\delta \lambda_R=-\lambda_R,
\end{eqnarray}
where $\hat\delta$ denotes R-duality and $\lambda_L$, $\lambda_R$ are the left
and right handed components, respectively. This automatically rules out the
$U(1)$ K\"ahler transformation as a candidate for R-duality,
since the $U(1)$ K\"ahler charges of the gaugino
and gravitino left handed components are
opposite to each other \cite{dauriaferrarafre}. Note that the
previous reasonings are not applicable to the case of
hypermultiplets. Indeed, we shall find that the left handed
components of the spinors contained in these multiplets
have charge $-1$, while the
right handed ones have
charge $+1$ (Section \ref{sectquater}).
Once we have fixed the charges of the fermions,
the R-duality transformations
of the bosons are uniquely
fixed by requiring on shell consistency with supersymmetry,
$\delta_\varepsilon$, i.\ e.\
\begin{equation}
[\hat\delta,\delta_\varepsilon]=0.
\end{equation}
Before giving the complete result obtained from this requirement, we recall
the structure of N=2 matter coupled supergravity.
\section{N=2 supergravity plus vector multiplets in the minimal coupling case}
\label{sugra}
By definition, N=2 supergravity minimally coupled to $n$ vector multiplets
corresponds to the case where the special K\"ahler
manifold spanned by the vector multiplet scalars
is the homogeneous manifold
${\cal M}={SU(1,n)\over SU(n)\otimes U(1)}$. In the language
of holomorphic prepotentials this corresponds to the choice
$F(X)={1\over 4}({X_0}^2-\sum_{i=1}^n {X_i}^2)$.
An easy way to obtain the explicit
form of this theory, in the rheonomy framework
that we use throughout the paper,
is by truncation of N=3
matter coupled supergravity \cite{castdauriafre,ceresole}.
If we are interested in
the case of just one vector multiplet, it is more convenient to
truncate pure N=4 $SO(4)$ supergravity \cite{cremmerscherk}. As a matter of
fact, we first tested our conjectures using this trick
(which we do not discuss here) and, after
having found that their were correct, we extended them to
$n$ vector multiplets in the way we now present.
The gravitational multiplet is $(V^a,\psi_A,\psi^A,A_0)$
(the index $A$ taking the values $1,2$),
where $V^a$ is the vierbein, $\psi_A$ are the
gravitino left handed components
($\gamma_5 \psi_A=\psi_A$),
$\psi^A$ are the right handed ones ($\gamma_5 \psi^A=-\psi^A$)
and $A_0$ is the graviphoton. The $n$ vector multiplets are labelled by an
index $i=1, \ldots n$ and are denoted by
($A_i,\lambda^A_i,\lambda^i_A,z_i,\bar z^i)$, $A_i$ being the vector bosons,
$\lambda^A_i$ the gaugino left handed components,
$\lambda_A^i$ the right handed ones, $z_i$ and $\bar z^i$ the scalars.
Vierbein, gravitinos, graviphoton and vector bosons are 1-forms,
all the other fields being 0-forms.
A special K\"ahler manifold $SK(n)$
is a Hodge K\"ahler manifold providing the base manifold for
a flat $Sp(2n+2)$ simplectic vector bundle
${\cal S}\stackrel{\pi}{\rightarrow} SK(n)$, whose holomorphic sections
$(X_\Lambda,{\partial F\over \partial X_\Lambda})$, $\Lambda=
0,1\ldots n,$ are given in terms of a prepotential $F(X)$,
homogeneous of degree two in the $n+1$ variables $X_\Lambda(z)$
($z$ belonging to $SK(n)$).
It is common to introduce the following expressions
\begin{eqnarray}
F^{\Lambda \Sigma}&=&\partial^\Lambda\partial^\Sigma F(X),\nonumber\\
N^{\Lambda \Sigma}&=&F^{\Lambda \Sigma}+\bar F^{\Lambda \Sigma},\nonumber\\
G&=&-{\rm ln}(N^{\Lambda\Sigma}X_\Lambda\bar X_\Sigma),\nonumber\\
L_\Lambda&=&e^{G\over 2}X_\Lambda,\nonumber\\
f_\Lambda^i&=&\partial^iL_\Lambda
+{1\over 2} G^i L_\Lambda,\nonumber\\
{\cal N}^{\Lambda \Sigma}&=&-\bar F^{\Lambda \Sigma}+{1\over
N^{\Delta\Gamma}L_\Delta L_\Gamma}N^{\Lambda\Pi}L_\Pi N^{\Sigma \Xi}L_\Xi,
\label{definitions}
\end{eqnarray}
where $G$ is the K\"ahler potential,
$\partial^\Lambda={\partial\over \partial X_\Lambda}$,
$\partial^i={\partial\over \partial z_i}$,
$G^i=\partial^i G$.
In the minimal case, if we use the special
coordinates $z_\Lambda={X_\Lambda\over X_0}$ ($z_0=1$) and furthermore
we impose $X_0\equiv 1$, then $F(z)={1\over 4}(1-\sum_{i=1}^n
z_i z_i)$ and
\begin{eqnarray}
F^{\Lambda \Sigma}&=&={1\over 2}\eta^{\Lambda\Sigma}=
{1\over 2}{\rm diag}(1,-1,\ldots -1),
\nonumber\\
N^{\Lambda \Sigma}&=&\eta^{\Lambda\Sigma},\nonumber\\
G&=&-{\rm ln}a,\nonumber\\
L_\Lambda&=&{z_\Lambda\over \sqrt{a}},\nonumber\\
f_\Lambda^i&=&\left(\matrix{f^i_0\cr
f^i_j}\right)={1\over a\sqrt{a}}
\left(\matrix{\bar z^i\cr
a\delta^i_j+z_j\bar z^i}\right),\nonumber\\
{\cal N}^{\Lambda \Sigma}&=&\left(\matrix{{\cal N}^{00}&{\cal N}^{0j}\cr
{\cal N}^{i0}&{\cal N}^{ij}}\right)=
{1\over 2(1-z_i z_i)}\left(\matrix{1+z_l z_l & -2 z_j\cr
-2z_i & \delta_{ij}(1-z_lz_l)+2z_iz_j}\right),
\label{minimaldefinitions}
\end{eqnarray}
where $a=1-z_i\bar z^i$.
In the notation of N=3 matter coupled supergravity
\cite{castdauriafre,ceresole},
the manifold ${{\cal G}\over{\cal H}}={SU(3,n)\over SU(3)\otimes SU(n)\otimes
U(1)}$ (which becomes ${\cal M}={SU(1,n)\over SU(n)\otimes
U(1)}$ when truncating to N=2), is described by
a matrix ${L_\Lambda}^\Sigma(z,\bar z)$ that depends on the coordinates
$z_i^A,\bar z_i^A\equiv z^i_A$,
where $A=1,2,3$, $i=1,\ldots n$, $\Lambda=(A,i)$. The N=2 truncation
is realized by setting to zero the fermions that have index $A=3$,
the bosons with $A=1,2$, the spin 1/2 of the N=3 graviton multiplet
and the $SU(3)$-singlet spin 1/2 fields of the vector multiplets.
The $L$ matrix is \cite{castdauriafre,ceresole}
\begin{equation}
{L_\Lambda}^\Sigma(z,\bar z)=\left(\matrix{
{L_1}^1 & {L_1}^2&{L_1}^3&{L_1}^j\cr
{L_2}^1 & {L_2}^2&{L_2}^3&{L_2}^j\cr
{L_3}^1 & {L_3}^2&{L_3}^3&{L_3}^j\cr
{L_i}^1 & {L_i}^2&{L_i}^3&{L_i}^j}\right)={1\over
\sqrt{a}}\left(\matrix{
1&0&0&0\cr
0&1&0&0\cr
0&0&1 & \bar z^j\cr
0&0&z_i & {M_i}^j}\right),
\end{equation}
where ${M_i}^j=\sqrt{a}\delta_i^j+{z_i\bar z^j\over |z|^2}(1-\sqrt{a})$.
The correspondence with the N=2 notation is the following
\begin{equation}
{L_\Lambda}^\Sigma=\left(\matrix{
1&0&0&0\cr
0&1&0&0\cr
0&0&L_0 & f_0^k {(g^{-{1\over 2}})_k}^j\cr
0&0&L_i & f_i^k {(g^{-{1\over 2}})_k}^j}\right),
\end{equation}
where ${(g^{-{1\over 2}})_i}^j=\sqrt{a}\delta_i^j+
{z_i\bar z^j\over |z|^2}(a-\sqrt{a})$. Note that
${{1\over a}M_i}^k {{1\over a}M_k}^j={g_i}^j
\equiv \partial_i \partial^j G$, where $\partial_i={\partial^i}^*$;
${g_i}^j$ is the metric tensor
of the K\"ahler manifold $\cal M$. We thus define ${{1\over a}M_i}^j=
{(g^{1\over 2})_i}^j$, and $a{{M^{-1}}_i}^j=
{(g^{-{1\over 2}})_i}^j$.
The N=2 truncation of the ${\cal G}\over {\cal H}$ connection
${\Omega_\Lambda}^\Sigma$ is
\begin{equation}
{\Omega_\Lambda}^\Sigma={(L^{-1})_\Lambda}^\Pi (d{L_\Pi}^\Sigma+g {f_\Pi}^
{\Delta \Gamma} A_\Delta {L_\Gamma}^\Sigma)\equiv
\left(\matrix{0&0&0&0\cr
0&0&0&0\cr
0&0&-i Q & P^j\cr
0&0&P_i & Q_i^j+{i\over n}\delta_i^j Q}\right).
\label{piequation}
\end{equation}
In particular, $Q$ is the gauged K\"ahler connection and $P^i$ is the gauged
vierbein on $\cal M$,
\begin{eqnarray}
Q&=&-{i\over 2}(G^i\nabla z_i-G_i\nabla\bar z^i),\nonumber\\
P_i&=&{(g^{1\over 2})_i}^j\nabla z_j
\label{extra}
\end{eqnarray}
and $P^i=P_i^*$.
{}From now on, let $\Lambda$ take only the values $(A=3,i=1,\ldots n)$. For
convenience, the index $3$ will be eventually replaced by a $0$
or simply omitted, when there can be no misunderstanding.
At this point, truncating the N=3 curvature definitions (see
Eq.s (IV.7.46) and (IV.7.48) of Ref.\ \cite{castdauriafre}),
we obtain the N=2 curvature definitions already adapted to the minimal
coupling.
\begin{eqnarray}
R^a&=&dV^a-\omega^{ab}\wedge V_b-i\bar\psi_A\gamma^a\wedge\psi^A\equiv
{\cal D}V^a-i\bar\psi_A\wedge\gamma^a\psi^A,\nonumber\\
R^{ab}&=&d\omega^{ab}-\omega^{ac}\wedge {\omega_c}^b,\nonumber\\
\rho_A&=&d\psi_A-{1\over 4}\omega^{ab}\gamma_{ab}\wedge\psi_A+
{i\over 2}Q\wedge\psi_A=
{\cal D}\psi_A+{i\over 2}Q\wedge\psi_A\equiv\nabla\psi_A,\nonumber\\
\rho^A&=&d\psi^A-{1\over 4}\omega^{ab}\gamma_{ab}\wedge\psi^A-
{i\over 2}Q\wedge\psi^A=
{\cal D}\psi^A-{i\over 2}Q\wedge\psi^A\equiv\nabla\psi^A,\nonumber\\
F_\Lambda&=&dA_\Lambda+{f_\Lambda}^{\Omega\Delta}A_\Omega\wedge A_\Delta+
\epsilon_{AB}L_\Lambda \bar\psi^A\wedge\psi^B+\epsilon^{AB}\bar L_\Lambda
\bar\psi_A\wedge\psi_B,\nonumber\\
\nabla\lambda_{iA}&=&d\lambda_{iA}-{1\over 4}\omega^{ab}\wedge\gamma_{ab}
\lambda_{iA}+{i\over 2}\left(1+{2\over n}\right)Q
\lambda_{iA}+{Q_i}^j\lambda_{jA},\nonumber\\
\nabla\lambda^{iA}&=&d\lambda^{iA}-{1\over 4}\omega^{ab}\wedge\gamma_{ab}
\lambda^{iA}-{i\over 2}\left(1+{2\over n}\right)Q \lambda^{iA}
+{Q^i}_j\lambda^{jA},\nonumber\\
\nabla z_i&=&dz_i+gA_\Lambda k^\Lambda_i(z),\nonumber\\
\nabla \bar z^i&=&d\bar z^i+gA_\Lambda k^{i\Lambda}(\bar z),
\label{curvatures}
\end{eqnarray}
where $\gamma_{ab}={1\over 2}[\gamma_a,\gamma_b]$ and ${Q^i}_j=({Q_i}^j)^*$.
$k_{\Lambda i}(z)$ and $k^i_{\Lambda}(\bar z)$ are respectively
the holomorphic
and antiholomorphic Killing vectors generating the special
K\"ahler manifold isometries.
The explicit expression of these Killing vectors can be read from Eq.s
(\ref{piequation}) and (\ref{extra}), isolating the
term
proportional to $A_\Lambda$ in the definition of
$P_i={(g^{1\over 2})_i}^j(dz_j+gA_\Lambda k_j^\Lambda(z))$.
One finds $k_i^\Lambda(z)={f_i}^{\Lambda k}z_k$ in the
case in which only the matter vectors are gauged (this
point will be justified in the following section).
In the N=2 notation it is useful to introduce the new definitions
\begin{eqnarray}
\lambda_i^A&=&-\epsilon^{AB}{(g^{-{1\over 2}})_i}^j\lambda_{jB},\nonumber\\
\lambda^i_A&=&-\epsilon_{AB}{(g^{-{1\over 2}})_j}^i\lambda^{jB}.
\label{redef}
\end{eqnarray}
Since $z$ and $\bar z$ will be shown to have
opposite R-duality charges, the matrix $g^{1\over 2}$ is R-duality invariant
and so
the above definitions do not change the R-duality transformation
properties of the fermions. Formulae (\ref{redef}) are determined in such
a way as to match the following rheonomic parametrizations
\begin{eqnarray}
P_i&=&P_{i|a}V^a+\epsilon^{AB}\bar\lambda_{iA}\psi_B,\nonumber\\
\nabla z_i&=&Z_{i|a}V^a+\bar\lambda_i^A\psi_A,
\label{rheo1}
\end{eqnarray}
that appear in the N=3 and N=2 formulations, respectively.
In the N=2 notation the gaugino curvatures are
\begin{eqnarray}
\nabla \lambda_i^A&=&{\cal D}\lambda_i^A-{i\over 2}Q\lambda^A_i
-{\Gamma_i}^j\lambda_j^A,\nonumber\\
\nabla \lambda^i_A&=&{\cal D}\lambda^i_A+{i\over 2}Q\lambda_A^i
-{\Gamma^i}_j\lambda^j_A,
\end{eqnarray}
where ${\Gamma_i}^j=-{(g^{-1})_i}^l(\partial^j {g_l}^k)\nabla z_k
-gA_\Lambda\partial^j k_i^\Lambda$
is the gauged Levi-Civita holomorphic connection on $\cal M$
and ${\Gamma^i}_j=({\Gamma_i}^j)^*$.
In the variables $\lambda_{iA}$, $P_i$ inherited from the N=3 truncation,
the standard N=2 Bianchi identities
(see Eq.s (3.35) of Ref.\ \cite{dauriaferrarafre})
take the following form
\begin{eqnarray}
{\cal D}R^a&+&R^{ab}\wedge V_b+i\bar \rho^A\wedge \gamma^a\psi_A-i
\bar\psi^A\wedge \gamma^a \rho_A=0,\nonumber\\
{\cal D}R^{ab}&=&0,\nonumber\\
\nabla \rho_A&+&{1\over 4}R^{ab}\wedge\gamma_{ab}\psi_A-
{i\over 2}K\wedge\psi_A=0,\nonumber\\
\nabla \rho^A&+&{1\over 4}R^{ab}\wedge\gamma_{ab}\psi^A+
{i\over 2}K\wedge\psi^A=0,\nonumber\\
\nabla F_\Lambda&-&f_\Lambda^i \nabla z_i \epsilon_{AB}\bar\psi^A\wedge\psi^B
-\bar f_{\Lambda i} \nabla \bar z^i \epsilon^{AB}\bar\psi_A\wedge\psi_B+
\nonumber\\
&+&
2L_\Lambda\epsilon_{AB}\bar\psi^A\wedge\rho^B+
2\bar L_\Lambda\epsilon^{AB}\bar\psi_A\wedge\rho_B=0,\nonumber\\
\nabla^2 \lambda_{iA}&+&{1\over 4}R^{ab}\wedge\gamma_{ab}\lambda_{iA}-
{R_i}^j\lambda_{jA}-{i\over 2}\left(1+{2\over n}\right)K\lambda_{iA}=0,
\nonumber\\
\nabla^2 \lambda^{iA}&+&{1\over 4}R^{ab}\wedge\gamma_{ab}\lambda^{iA}-
{R^i}_j\lambda^{jA}+{i\over 2}\left(1+{2\over n}\right)K\lambda^{iA}=0,
\nonumber\\
\nabla P_i&=&dP_i+{Q_i}^j\wedge P_j+i\left(1+{1\over n}\right)
Q\wedge P_i=0,\nonumber\\
\nabla P_i&=&dP^i+{Q^i}_j\wedge P^j-i\left(1+{1\over n}\right)
Q\wedge P^i=0,
\label{bianchi}
\end{eqnarray}
where $K=dQ$, ${R_i}^j=d{Q_i}^j+{Q_i}^k\wedge{Q_k}^j$ and
${R^i}_j=({R_i}^j)^*$.
The rheonomic parametrizations are
\begin{eqnarray}
R^a&=&0,\nonumber\\
R^{ab}&=&{R^{ab}}_{cd}V^c\wedge V^d-i\bar\psi_A(2\gamma^{[a}\rho^{A|b]c}
-\gamma^c\rho^{A|ab})\wedge V_c+\nonumber\\
&-&i\bar\psi^A(2\gamma^{[a}{\rho_A}^{|b]c}
-\gamma^c{\rho_A}^{|ab})\wedge V_c+2G^{-ab}\epsilon^{AB}\bar\psi_A\wedge\psi_B+
\nonumber\\
&+&2G^{+ab}\epsilon_{AB}\bar\psi^A\wedge\psi^B
+{i\over 4}\varepsilon^{abcd}\bar\psi_A\wedge\gamma_c\psi^B
(2\bar\lambda_{iB}\gamma_d\lambda^{iA}-
\delta^A_B\bar\lambda_{iC}\gamma_d\lambda^{iC}),\nonumber\\
\rho_A&=&\rho_{A|ab}V^a\wedge V^b-2i \epsilon_{AB}G^+_{ab}\gamma^a\psi^B
\wedge V^b+{i\over 4}\psi_B\bar
\lambda^{iB}\gamma^a\lambda_{iA}\wedge V_a+\nonumber\\
&+&{i\over 8}\gamma_{ab}\psi_B\left(2\bar
\lambda^{iB}\gamma^a\lambda_{iA}-
\delta^B_A\bar
\lambda^{iC}\gamma^a\lambda_{iC}\right)\wedge V^b,\nonumber\\
\rho^A&=&\rho^A_{ab}V^a\wedge V^b-2i \epsilon^{AB}G^-_{ab}\gamma^a\psi_B
\wedge V^b+{i\over 4}\psi^B\bar
\lambda_{iB}\gamma^a\lambda^{iA}\wedge V_a+\nonumber\\
&+&{i\over 8}\gamma_{ab}\psi^B\left(2\bar
\lambda_{iB}\gamma^a\lambda^{iA}-
\delta_B^A\bar
\lambda_{iC}\gamma^a\lambda^{iC}\right)\wedge V^b,\nonumber\\
F_\Lambda&=&F_\Lambda^{ab}V_a\wedge V_b+i(f_\Lambda^i\bar\lambda^A_i
\gamma^a\psi^B\epsilon_{AB}+\bar f_{\Lambda i}\bar\lambda_A^i
\gamma^a\psi_B\epsilon^{AB})\wedge V_a,\nonumber\\
\nabla \lambda_{iA}&=&\nabla_a\lambda_{iA}V^a+iP_{i|a}\gamma^a\psi^B
\epsilon_{AB}+G^{+ab}_i\gamma_{ab}\psi_A
+gC_i\psi_A,\nonumber\\
\nabla \lambda^{iA}&=&\nabla_a\lambda^{iA} V^a+iP^i_{|a}\gamma^a\psi_B
\epsilon^{AB}+G_{ab}^{-i}\gamma^{ab}\psi^A
+gC^i\psi^A,\nonumber\\
\nabla z_i&=&Z_{i|a}V^a+\bar\lambda_i^A\psi_A,\nonumber\\
\nabla \bar z^i&=&\bar Z^i_{|a} V^a+ \bar\lambda^i_A\psi^A.
\label{rheo2}
\end{eqnarray}
where $C^i={(L^{-1})_3}^k{L_j}^i{L_l}^3{f_k}^{jl}$ are obtained from the
N=2 truncation as particular instances of the N=3 boosted structure constants,
$C_i=({C^i})^*$, $\bar f_{\Lambda i}=
(f_\Lambda^i)^*$. $G^{+ab}$, $G^{+ab}_i$, $G_{ab}^-$ and $G_{ab}^{-i}$
are determined by the equation
\begin{equation}
{1\over 2}F^{ab}_\Lambda+{L_\Lambda}^3 G^{-ab}+{L_\Lambda}^i G_i^{+ab}+
{(L^{-1})_3}^\Pi J_{\Pi\Lambda}G^{+ab}-
{(L^{-1})_i}^\Pi J_{\Pi\Lambda}G^{i-ab}=0,
\end{equation}
where $J_{\Lambda\Pi}$ is the $SU(1,n)$-invariant metric
\begin{equation}
J_{\Lambda\Pi}=\left(\matrix{1&0\cr 0&-\delta_{ij}}\right).
\end{equation}
One finds
\begin{eqnarray}
G^{+ab}&=&-{1\over 4}\sqrt{a}(1+2\bar {\cal N})^{0\Lambda}
F^{+ab}_\Lambda,\nonumber\\
G^{+ab}_i&=&-{1\over 4}\sqrt{a}{(g^{1\over 2})_i}^j
{(1+2\bar{\cal N})_j}^\Lambda F^{+ab}_\Lambda
+{1\over a}\bar z^2 z_i G^{+ab}+{1\over 2\sqrt{a}}z_i\bar z^k F^{+ab}_k.
\label{eigenstates}
\end{eqnarray}
and $G_{ab}^-=(G_{ab}^+)^*$ and $G_{ab}^{-i}=({{G_i}_{ab}}^+)^*$.
The rheonomic parametrizations are on-shell consistent
with the Bianchi identities (\ref{bianchi}).
We can now write down the lagrangian of N=2 supergravity minimally coupled to
$n$ vector multiplets.
\begin{equation}
{\cal L}={\cal L}_{kin}+{\cal L}_{Pauli}+{\cal L}_{torsion}+{\cal L}_{4Fermi}
+\Delta {\cal L}_{gauging}+\Delta {\cal L}_{potential},
\label{lagra}
\end{equation}
where
\begin{eqnarray}
{\cal L}_{kin}&=&\varepsilon_{abcd}R^{ab}\wedge V^c\wedge V^d
-4(\bar\psi^A\wedge\gamma_a\rho_A+\bar\rho^A\wedge\gamma_a\psi_A)\wedge V^a+
\nonumber\\
&-&{i\over 3}{g_i}^j(\bar\lambda^A_j\gamma_a\nabla\lambda_A^i+
\bar\lambda^i_A\gamma_a\nabla\lambda_j^A)\wedge V_b \wedge V_c\wedge V_d
\varepsilon^{abcd}+\nonumber\\
&+&{2\over 3}{g_i}^j
[\bar Z^i_{|a}(\nabla z_j-\bar\lambda_j^A\psi_A)
+Z_{j|a}(\nabla\bar z^i-\bar\lambda^i_A\psi^A)]\wedge V_b\wedge V_c
\wedge V_d \varepsilon^{abcd}+\nonumber\\
&+&{1\over 6}(\bar {\cal N}^{\Lambda\Sigma}F^{+ab}_\Lambda
F^{+}_{\Sigma ab}+{\cal N}^{\Lambda\Sigma}F^{-ab}_\Lambda
F^{-}_{\Sigma ab}-{g_i}^j \bar Z^i_{|a}{Z_{j|}}^a)\varepsilon_{cdef}
V^c\wedge V^d\wedge V^e\wedge V^f+\nonumber\\
&-&4i(\bar {\cal N}^{\Lambda\Sigma}F^{+ab}_\Lambda-
{\cal N}^{\Lambda\Sigma}F^{-ab}_\Lambda)\wedge (F_\Sigma+\nonumber\\
&-&i(f_\Sigma^i\bar\lambda^A_i
\gamma^c\psi^B\epsilon_{AB}+\bar f_{\Sigma i}\bar\lambda_A^i
\gamma^c\psi_B\epsilon^{AB})\wedge V_c)\wedge V_a\wedge V_b,\nonumber\\
{\cal L}_{Pauli}&=&-4iF_\Lambda\wedge({\cal N}^{\Lambda\Sigma} L_\Sigma
\epsilon_{AB}\bar\psi^A\wedge\psi^B-\bar{\cal N}^{\Lambda\Sigma}\bar L_\Sigma
\epsilon^{AB}\bar\psi_A\wedge\psi_B)+\nonumber\\
&+&4F_\Lambda\wedge(\bar{\cal N}^{\Lambda\Sigma}f_\Sigma^i\bar \lambda_i^A
\gamma_a\psi^B\epsilon_{AB}-{\cal N}^{\Lambda\Sigma}
\bar f_{\Sigma i}\bar \lambda^i_A
\gamma_a\psi_B\epsilon^{AB})\wedge V^a+\nonumber\\
&-&2i{g_i}^j(\nabla z_j\wedge \bar \lambda^i_A\gamma_{ab}\psi^A-
\nabla \bar z^i\wedge \bar \lambda_j^A\gamma_{ab}\psi_A)
\wedge V^a\wedge V^b,\nonumber\\
{\cal L}_{torsion}&=&R^a\wedge V_a\wedge {g_i}^j
\bar\lambda_A^i\gamma_b\lambda_j^A
\wedge V^b,\nonumber\\
{\cal L}_{4Fermi}&=&i(W\epsilon_{AB}\bar\psi^A\wedge\psi^B\wedge
\epsilon_{CD}\bar\psi^C\wedge\psi^D-\bar W
\epsilon^{AB}\bar\psi_A\wedge\psi_B\wedge
\epsilon^{CD}\bar\psi_C\wedge\psi_D)+\nonumber\\
&-&2i{g_i}^j\bar\lambda^i_A\gamma_a
\lambda_j^B\bar\psi_B\wedge\gamma_b\psi^A\wedge V^a\wedge V^b+\nonumber\\
&+i&(W_{ij}\epsilon^{AB}\bar\lambda^i_A\gamma_a\psi_B\wedge V^a\wedge
\epsilon^{CD}\bar\lambda^j_C\gamma_b\psi_D\wedge V^b+\nonumber\\
&-&W^{ij}\epsilon_{AB}\bar\lambda_i^A\gamma_a\psi^B\wedge V^a\wedge
\epsilon_{CD}\bar\lambda_j^C\gamma_b\psi^D\wedge V^b)+\nonumber\\
&+&{1\over 18}\epsilon_{abcd}V^a\wedge V^b\wedge V^c\wedge V^d
{g_i}^j\bar\lambda^i_A\gamma^m\lambda_j^A
{g_k}^l\bar\lambda^k_B\gamma_m\lambda_l^B,
\nonumber\\
\Delta {\cal L}_{gauging}&=&{2i\over 3}g(
\bar \lambda_i^A\gamma^a\psi^B W^i_{AB}
+\bar \lambda^i_A\gamma^a\psi_B W_i^{AB})
\wedge V^b\wedge V^c\wedge V^d\varepsilon_{abcd}+\nonumber\\
&+&{1\over 6}g(M^{ij}\bar\lambda_i^A\lambda_j^B\epsilon_{AB}+
M_{ij}\bar\lambda^i_A\lambda^j_B\epsilon^{AB})\varepsilon_{abcd}
V^a\wedge V^b\wedge V^c\wedge V^d,\nonumber\\
\Delta {\cal L}_{potential}&=&-{1\over 12}g^2 {g_i}^j W^i_{AB} W_j^{AB}
\varepsilon_{abcd} V^a\wedge V^b \wedge V^c \wedge V^d,
\label{lagra2}
\end{eqnarray}
where $W=2L_\Lambda L_\Sigma {\cal N}^{\Lambda \Sigma}$,
$W^{ij}=2\bar {\cal N}^{\Lambda\Sigma}f_\Lambda^i f_\Sigma^j$ and
$W_{ij}=(W^{ij})^*$, while
$M^{ij}=k^{l\Lambda}f_\Lambda^{[i}{g_l}^{j]}$
and $M_{ij}=(M^{ij})^*$,
$W^i_{AB}=\epsilon_{AB}k^{i\Lambda}L_\Lambda$,
$W_i^{AB}=(W^i_{AB})^*$.
The lagrangian in Eq.s (\ref{lagra}) and (\ref{lagra2})
agrees with the lagrangian
(4.13) of Ref.\ \cite{dauriaferrarafre} upon suppression of
the hypermultiplets and up to ${\cal L}_{4Fermi}$ and
the second term of
$\Delta{\cal L}_{gauging}$, that were not calculated in
\cite{dauriaferrarafre}.
Indeed, the very reason why we have performed the above described N=2
truncation of
the N=3 theory was that of obtaining these terms without calculating
them explicitly. Our purpose is that of checking R-duality in the
minimal coupling, however, as a byproduct, we have also obtained the complete
form of the lagrangian of N=2 supergravity coupled to vector multiplets
for an arbitrary choice of the special K\"ahler manifold.
All the objects entering (\ref{lagra2}) have already been
interpreted in a general N=2 setup
(in which the graviphoton can be gauged).
As a matter of fact,
the N=3 theory does not admit the most general gauging
of the vectors \cite{castdauriafre,ceresole}, but it surely admits
any gauging of the matter vectors. Even if the minimal N=2 theory exists
in any case, the truncation from N=3 can only give the minimal N=2
theory in which the graviphoton is not gauged.
As promised, in the following section we define R-duality and prove
that it is indeed an on-shell symmetry of the above theory.
\section{R-duality for N=2 matter coupled supergravity}
\label{Rduality}
Now, starting form the R-duality transformation properties of the fermions,
as derived in Section \ref{general}, we determine the transformations
of the bosons by simply requiring $[\hat \delta,\delta_\varepsilon]=0$
on-shell, if $\delta_\varepsilon$ is the supersymmetry transformation with
parameters $\varepsilon$ (let $\varepsilon_A$
and $\varepsilon^A$ be the left and right handed components,
respectively). The supersymmetry transformations can be read in the
usual way
from the rheonomic parametrizations (\ref{rheo1}) and (\ref{rheo2}). In any
case, their explicit expression will be written down later on in the context
of the BRST-quantization of the theory (see formula (\ref{brstalgebra})).
So, we start from
\begin{equation}
\matrix{\hat\delta \psi_A=\psi_A, & \hat\delta \varepsilon_A=\varepsilon_A, &
\hat\delta \lambda^A_i=\lambda_i^A, \cr
\hat\delta \psi^A=-\psi^A, & \hat\delta \varepsilon^A=-\varepsilon^A, &
\hat\delta \lambda_A^i=-\lambda^i_A.}
\end{equation}
First of all, consistency of R-duality
with supersymmetry states that, if a field $\phi$ has an R-duality
charge equal to $q$, then $\delta_\varepsilon \phi$ has the same charge $q$
and viceversa. It is immediate to see that $\hat \delta
\delta_\varepsilon V^a=0$ and so we deduce $\hat \delta V^a=0$.
This is good, because in our mind, R-duality is to become ghost number
and the vierbein should remain of zero ghost number together with
all the matter vectors.
Similarly, $\hat \delta \delta_\varepsilon z_i= 2 \delta_\varepsilon z_i$,
requiring $\hat\delta z_i=2 z_i$. An analogous reasoning gives,
when applied to $\bar z^i$, $\hat \delta \bar z^i=-2 \bar z^i$, thus
confirming that $z_i$ and $\bar z^i$ have opposite charges. This immediately
rules out the possibility that the $U(1)$ symmetry we are looking for might be
a subgroup of the group of duality transformations
\cite{gaillardzumino,castdauriafre}.
Indeed, in that case $z_i$ and $\bar z^i$
would have the same charge. This is welcome, because, if $U(1)_I$
were a subgroup of the duality group, we could not maintain the symmetry
in the presence of gauging, as, on the contrary, we expect to be able to do.
We immediately
see that the K\"ahler potential $G$ is invariant, as well as
the metric ${g_i}^j$
(note that this fact would not hold true in the nonminimal case).
It remains to find the transformation properties of the vector bosons.
Let us concentrate on the ungauged case ($g=0$) for the moment.
One can verify that $[\hat\delta,\delta_\varepsilon]\psi_A=0$ and
$[\hat\delta,\delta_\varepsilon]\psi^A=0$ imply
\begin{equation}
\hat\delta G^{\pm ab}=\pm 2 G^{\pm ab},
\label{dual1}
\end{equation}
while
$[\hat\delta,\delta_\varepsilon]\lambda_i^A=0$ and
$[\hat\delta,\delta_\varepsilon]\lambda^i_A=0$ imply
\begin{equation}
\matrix{\hat\delta G^{+ab}_i=0, & \hat \delta G^{i-ab}=0}.
\label{dual2}
\end{equation}
respectively.
Equations (\ref{dual1}) and (\ref{dual2}) form a linear system of equations in
$\hat\delta F^{\pm ab}_\Lambda$, in which the number of unknowns equals the
number of equations. The unique solution is
\begin{equation}
\matrix{
\hat\delta F^{0+ab}=4\bar{\cal N}^{0\Lambda}F^{+ab}_\Lambda,
& \hat \delta F^{+ab}_i=0,\cr
\hat\delta F^{0-ab}=-4{\cal N}^{0\Lambda}F^{-ab}_\Lambda,
& \hat \delta F^{i-ab}=0.}
\label{utility1}
\end{equation}
The graviphoton is thus transformed in a way that resembles the duality
transformations and this forbids its gauging it if we want R-duality.
Consequently, when considering the gauged case,
we must assume that only the matter vectors are
gauged, i.\ e.\ $f_\Lambda^{\Sigma\Omega}=0$ whenever one of the indices
$\Lambda,$ $\Sigma$, $\Omega$ takes the value zero. There is
no restriction, on the contrary,
on the gauge group of the matter vectors.
Let us
rewrite the rheonomic parametrization of the vectors
and the definition of their curvatures
\begin{eqnarray}
F_\Lambda&=&F_\Lambda^{ab}V_a\wedge V_b+i(f_\Lambda^i\bar\lambda^A_i
\gamma^a\psi^B\epsilon_{AB}+\bar f_{\Lambda i}\bar\lambda_A^i
\gamma^a\psi_B\epsilon^{AB})\wedge V_a,\nonumber\\
F_\Lambda&=&dA_\Lambda+{f_\Lambda}^{\Omega\Delta}A_\Omega\wedge A_\Delta+
\epsilon_{AB}L_\Lambda \bar\psi^A\wedge\psi^B+\epsilon^{AB}\bar L_\Lambda
\bar\psi_A\wedge\psi_B.
\label{utility2}
\end{eqnarray}
These expressions show that, under the above conditions on the structure
constants
$f_\Lambda^{\Sigma\Omega}$, the transformations $\hat \delta F^{+ab}_i=0$
and $\hat \delta F^{i-ab}=0$ imply $\hat \delta A_i=0$,
i.\ e.\ all the matter vectors are inert under R-duality (they will
have ghost number zero after the twist and this is good in order
to recover topological Yang-Mills theory).
Summarizing, R-duality acts on-shell as follows
\begin{eqnarray}
\matrix{\hat\delta V^a=0,&\cr
\hat\delta \psi_A=\psi_A,&
\hat\delta\psi^A=-\psi^A,\cr
\hat\delta F^{0+ab}=4\bar{\cal N}^{0\Lambda}F^{+ab}_\Lambda,&
\hat\delta F^{0-ab}=-4{\cal N}^{0\Lambda}F^{-ab}_\Lambda,\cr}\nonumber
\end{eqnarray}
\begin{eqnarray}
\matrix{
\hat\delta A_i=0&\cr
\hat\delta \lambda^A_i=\lambda^A_i,&
\hat\delta \lambda_A^i=-\lambda_A^i,\cr
\hat\delta z_i=2 z_i,&
\hat\delta \bar z^i=-2 \bar z^i.\cr}
\label{utility3}
\end{eqnarray}
One easily checks that formulas
(\ref{dual1}), (\ref{dual2}), (\ref{utility1}) and (\ref{utility3})
are still valid when all
the vectors but the graviphoton are gauged.
What about $\hat \delta A_0$? As in all duality-type transformations,
$\hat \delta A_0$ should be meaningful only on-shell (see Section
\ref{general}). In fact, (\ref{utility1}) and (\ref{utility2})
imply (using the explicit expressions (\ref{minimaldefinitions}))
\begin{eqnarray}
\hat \delta F_0^{ab} V_a\wedge V_b&=&4({{\bar{\cal N}}_0}^{\Lambda}
F^{+ab}_\Lambda-{{\cal N}_0}^{\Lambda}
F^{-ab}_\Lambda)V_a\wedge V_b=\nonumber\\
&=&\hat \delta [dA_0+\epsilon_{AB}L_0\bar\psi^A\wedge\psi^B+
\epsilon^{AB}\bar L_0\bar\psi_A\wedge\psi_B+\nonumber\\
&-&i(f^i_0\bar\lambda_i^A\gamma^a\psi^B
\epsilon_{AB}+\bar f_{0i}\bar\lambda^i_A\gamma^a\psi_B\epsilon^{AB})
\wedge V_a]=
\nonumber\\
&=&d\hat\delta A_0-4{{\cal N}_0}^\Lambda L_\Lambda
\epsilon_{AB}\bar\psi^A\wedge\psi^B +4 {{\bar{\cal N}}_0}^\Lambda\bar L_\Lambda
\epsilon^{AB}\bar\psi_A\wedge\psi_B+\nonumber\\
&-&4i({{\bar{\cal N}}_0}^\Lambda
f^i_\Lambda\bar\lambda_i^A\gamma^a\psi^B\epsilon_{AB}-
{{\cal N}_0}^\Lambda \bar f_{\Lambda i}\bar\lambda^i_A
\gamma^a\psi_B\epsilon^{AB})\wedge V_a.
\end{eqnarray}
Imposing $d^2 \hat \delta A_0=0$, we get
\begin{eqnarray}
d[({\bar{\cal N}_0}^{\Lambda}
F^{+ab}_\Lambda&-&{{\cal N}_0}^{\Lambda}
F^{-ab}_\Lambda)V_a\wedge V_b+{{\cal N}_0}^\Lambda L_\Lambda
\epsilon_{AB}\bar\psi^A\wedge\psi^B -{{\bar{\cal N}}_0}^\Lambda\bar L_\Lambda
\epsilon^{AB}\bar\psi_A\wedge\psi_B+\nonumber\\
&+&i({{\bar{\cal N}}_0}^\Lambda
f^i_\Lambda\bar\lambda_i^A\gamma^a\psi^B\epsilon_{AB}-
{{\cal N}_0}^\Lambda \bar f_{\Lambda i}\bar\lambda^i_A
\gamma^a\psi_B\epsilon^{AB})\wedge V_a]=0.
\end{eqnarray}
One can easily verify that this is the equation of motion of the graviphoton
as derived from the lagrangian (\ref{lagra}).
Furthermore, the R-duality variation
of the $A_0$ equation of motion is proportional to the $A_0$-Bianchi identity
and viceversa. It is easily checked that the other curvatures of
(\ref{curvatures}) and the remaining Bianchi identities of (\ref{bianchi})
transform correctly, so the last step in order to establish R-duality of
the theory is the proof of invariance for the remaining field equations.
The equations of motion of the vector bosons can be written in the
following form
\begin{equation}
dS^\Lambda+2f_\Delta^{\Lambda \Sigma}A_\Sigma\wedge
S^\Delta+R^\Lambda=0,
\end{equation}
where $S^\Lambda$ is, by definition, the coefficient in the lagrangian
of the field strength $F_\Lambda$, namely
\begin{eqnarray}
S^\Lambda&=&(\bar{\cal N}^{\Lambda\Sigma}
F^{+ab}_\Sigma-{\cal N}^{\Lambda\Sigma}
F^{-ab}_\Sigma)V_a\wedge V_b+{\cal N}^{\Lambda\Sigma} L_\Sigma
\epsilon_{AB}\bar\psi^A\wedge\psi^B -\bar{\cal N}^{\Lambda\Sigma}
\bar L_\Sigma
\epsilon^{AB}\bar\psi_A\wedge\psi_B+\nonumber\\
&+&i(\bar{\cal N}^{\Lambda\Sigma}
f^i_\Sigma\bar\lambda_i^A\gamma^a\psi^B\epsilon_{AB}-
{\cal N}^{\Lambda\Sigma} \bar f_{\Sigma i}\bar\lambda^i_A
\gamma^a\psi_B\epsilon^{AB})\wedge V_a,
\end{eqnarray}
and $R^\Lambda$ is the remainder that comes from
the ${\delta\over \delta A_\Lambda}$-variation of those terms that are
manifestly R-duality invariant and do not depend on the graviphoton $A_0$.
Since one can easily verify that
$\hat \delta S^\Lambda$ vanishes whenever $\Lambda\neq 0$
(to this purpose,
note that $\hat \delta(\bar{\cal N}^{i\Sigma}F_\Sigma^{+ab})=
\hat \delta({\cal N}^{i\Sigma}F_\Sigma^{-ab})=0$
and use the explicit expressions
(\ref{minimaldefinitions})), then the field equations
of the matter vectors are all R-duality invariant.
In order to prove R-duality
invariance of the remaining field equations, we note that it is not necessary
to study the entire lagrangian $\cal L$ (\ref{lagra}), because various terms
can give only contributions with the correct $\hat \delta$-transformation
properties. These are precisely the R-duality invariant terms of $\cal L$
that do not depend on $A_0$.
On the other hand, since
$\hat \delta F_0^{ab}$ depends on all the fields, we cannot neglect
a term $\Delta \cal L$
only because it is $\hat\delta$-invariant ($\hat\delta \Delta {\cal L}=0$)
if it contains $A_0$. Indeed,
if $\phi$ is a field of charge $q$ ($\hat\delta \phi=q\phi$;
we can take $\phi\neq A_0$ since the $A_0$-equation has already been
studied), then the
contributions to its field equation (i.\ e.\ ${\partial \over
\partial \phi}\Delta {\cal L}$) must have charge $-q$ in order to transform
correctly
($\hat\delta{\partial \over
\partial \phi}\Delta {\cal L}=-q{\partial \over
\partial \phi}\Delta {\cal L}$) and it must happen that
\begin{equation}
\left[\hat\delta,{\partial \over
\partial \phi}\right]\Delta {\cal L}=-q {\partial \over
\partial \phi}\Delta {\cal L}.
\end{equation}
For this to be true it is sufficient (and necessary, if
$\Delta {\cal L}$ has not a special form) to have
\begin{equation}
\left[\hat\delta,{\partial \over
\partial \phi}\right]\phi^\prime=-q {\partial \over
\partial \phi}\phi^\prime,
\end{equation}
for all fields $\phi^\prime$.
However, this is not true for $\phi^\prime=F^{ab}_0$ and so, if
$\Delta {\cal L}$ depends on $A_0$ one should analyze it explicitly.
Summarizing, it is sufficient to test R-duality invariance of the
contributions to the
field equations that come from the terms of the lagrangian
either containing $A_0$ or not $\hat\delta$-invariant. This
part of the lagrangian is given by
\begin{eqnarray}
\Delta {\cal L}&\equiv&
{1\over 6}(\bar {\cal N}^{\Lambda\Sigma}F^{+ab}_\Lambda
F^{+}_{\Sigma ab}+{\cal N}^{\Lambda\Sigma}F^{-ab}_\Lambda
F^{-}_{\Sigma ab})\varepsilon_{cdef}V^c\wedge V^d\wedge V^e \wedge V^f+
\nonumber\\
&-&4i(\bar {\cal N}^{\Lambda\Sigma}F^{+ab}_\Lambda-
{\cal N}^{\Lambda\Sigma}F^{-ab}_\Lambda)\wedge V_a\wedge V_b\wedge
(F_\Sigma+\nonumber\\
&-&i(f_\Sigma^i\bar\lambda^A_i
\gamma^c\psi^B\epsilon_{AB}+\bar f_{\Sigma i}\bar\lambda_A^i
\gamma^c\psi_B\epsilon^{AB})\wedge V_c),\nonumber\\
&-&2i{1\over \sqrt{a}}F^0\wedge(
\epsilon_{AB}\bar\psi^A\wedge\psi^B-
\epsilon^{AB}\bar\psi_A\wedge\psi_B)+\nonumber\\
&-&{2\over a\sqrt{a}}F^0\wedge(\bar z^i\bar \lambda_i^A
\gamma_a\psi^B\epsilon_{AB}-z_i\bar \lambda^i_A
\gamma_a\psi_B\epsilon^{AB})\wedge V^a+\nonumber\\
&+&{i\over a}(\epsilon_{AB}\bar\psi^A\wedge\psi^B\wedge
\epsilon_{CD}\bar\psi^C\wedge\psi^D-
\epsilon^{AB}\bar\psi_A\wedge\psi_B\wedge
\epsilon^{CD}\bar\psi_C\wedge\psi_D)+\nonumber\\
&-&{i\over a^3}(z_i z_j\epsilon^{AB}\bar\lambda^i_A\gamma_a
\psi_B\wedge V^a\wedge
\epsilon^{CD}\bar\lambda^j_C\gamma_b\psi_D\wedge V^b+\nonumber\\
&-&\bar z^i \bar z^j\epsilon_{AB}\bar\lambda_i^A\gamma_a\psi^B\wedge V^a\wedge
\epsilon_{CD}\bar\lambda_j^C\gamma_b\psi^D\wedge V^b),
\end{eqnarray}
where $W$ and $W^{ij}$ have been replaced
by their explicit expressions in terms of
$z_i$, and $\bar z^i$ and, after replacement, the manifestly
$\hat\delta$-invariant terms not containing $A_0$ have been deleted.
At this point, the check that the contributions to the field equations of the
fermions, the vierbein and the scalars transform correctly
is rather direct and we leave it to the reader. We thus conclude that
{\bf Proposition.}
{\it N=2 supergravity minimally coupled to $n$ vector multiplets
gauging an arbitrary $n$ dimensional group
(in which the graviphoton is not gauged),
is on-shell R-duality invariant\footnotemark
\addtocounter{footnote}{0}\footnotetext{\rm
Note that for an N=2 theory
without hypermultiplets, the statement that a certain
vector is not gauged is equivalent to the statement that it corresponds
to a $U(1)$ subgroup of the full gauge group.}.}
The possibility that R-duality exists also in the N=3 theory or in
more extended supergravity theories as well as the possibility to have
it in N=2 matter coupled supergravity in nonminimal cases (even
if, we presume, it might not be suitable for a topological twist)
remain open problems. Here we have restricted our attention to
that internal $U(1)$ symmetry
that was relevant to our purposes, that is the topological twist.
We have so far neglected the coupling of matter hypermultiplets to
N=2 supergravity, since it is immediately verified that the generalization
of R-duality due to the presence of
them is trivial. The scalars have $0$ charge, however
the left handed components of
fermions must have $-1$ charge and the right handed components
must have $+1$ charge, differently from the case of the other fermions
so far encountered.
The twist is by no means
trivial. As a matter of fact, it turns out that it is interesting as we
shall see at the end of this paper.
\section{Topological twist of the minimal theory}
\label{sectiontwist}
In this section, we discuss the twisted topological theory. First
of all, let us note that the gauge-free algebra (i.\ e.\ the minimal BRST
algebra, with neither antighosts nor gauge-fixings, nor Lagrange multipliers)
is simply the tensor product of the gauge-free algebras for topological gravity
and topological Yang-Mills \cite{anselmifre}, that is to say
\begin{eqnarray}
sA&=&-\nabla c-\psi,\nonumber\\
sc&=&\phi-{{1}\over{2}}\left[c,c\right],\nonumber\\
s\psi&=&\nabla\phi-\left[c,\psi\right],\nonumber\\
s\phi&=&-\left[c,\phi\right],\nonumber\\
sV^a&=&\psi^a-{\cal D}_0\varepsilon^a+\varepsilon^{ab}\wedge V_b,\nonumber\\
s\omega_0^{ab}&=&\chi^{ab}-{\cal D}_0\varepsilon^{ab},\nonumber\\
s\varepsilon^a&=&\phi^a+\varepsilon^{ab}\wedge\varepsilon_b,\nonumber\\
s\varepsilon^{ab}&=&\eta^{ab}+{\varepsilon^a}_c\wedge\varepsilon^{cb},
\nonumber\\
s\psi^a&=&-{\cal D}_0\phi^a+\varepsilon^{ab}\wedge\psi_b-\chi^{ab}\wedge
\varepsilon_b-\eta^{ab}\wedge V_b,\nonumber\\
s\phi^a&=&\varepsilon^{ab}\wedge\phi_b-\eta^{ab}\wedge\varepsilon_b,\nonumber\\
s\chi^{ab}&=&-{\cal D}_0\eta^{ab}+\varepsilon^{ac}\wedge{\chi_c}^b-\chi^{ac}
\wedge{\varepsilon_c}^b,\nonumber\\
s\eta^{ab}&=&\varepsilon^{ac}\wedge{\eta_c}^b-\eta^{ac}\wedge{\varepsilon_c}^b.
\label{gaugefree}
\end{eqnarray}
We have grouped the $n$ matter vectors $A_i$ into the column $A=(A_i)$.
Similarly, $\psi=(\psi_i)$, $\phi=(\phi_i)$ and $c=(c_i)$.
For the definitions of the other symbols, refer to Ref.\ \cite{anselmifre}.
The observables and the corresponding descent
equations can be derived from
the hatted extensions of the identities
$d\hskip .1truecm{\rm tr}[F\wedge F]=0$, $d\hskip .1truecm{\rm tr}[R\wedge
R]=0$ and $d\hskip .1truecm{\rm tr}[R\wedge \tilde R]=0$, in the usual way
\cite{anselmifre}.
The BRST algebra of N=2 matter coupled supergravity can be found as explained
in Ref.\ \cite{anselmifre}, that is to say by extending all
differential forms to ghost forms.
We report only the final result, that, together with the translation
ghosts $\varepsilon^a$, the Lorentz ghosts $\varepsilon^{ab}$,
the supersymmetry ghosts $c_A,c^A$, involves also the gauge ghosts $c^\Lambda$.
\begin{eqnarray}
sV^a&=&-{\cal D}\varepsilon^a+\varepsilon^{ab}\wedge V_b
+i(\bar\psi_A\wedge\gamma^a c^A
+\bar c_A \wedge \gamma \psi^A),\nonumber\\
s\varepsilon^a&=&\varepsilon^{ab}\wedge \varepsilon_b
+i\bar c_A\wedge\gamma^a c^A,\nonumber\\
s\omega^{ab}&=&-{\cal D}\varepsilon^{ab}
+2{R^{ab}}_{cd}V^c\varepsilon^d+i(\varepsilon_c\bar\psi_A+V_c \bar c_A)
(2\gamma^{[a}\rho^{A|b]c}
-\gamma^c\rho^{A|ab})+\nonumber\\
&+&i(\varepsilon_c \bar\psi^A+ V_c \bar c^A)
(2\gamma^{[a}{\rho_A}^{|b]c}
-\gamma^c{\rho_A}^{|ab})+4G^{-ab}\epsilon^{AB}\bar\psi_A\wedge c_B+
\nonumber\\
&+&4G^{+ab}\epsilon_{AB}\bar\psi^A\wedge c^B
+{i\over 4}\varepsilon^{abcd}(\bar\psi_A\wedge\gamma_c c^B
+\bar c_A\wedge\gamma_c\psi^B)
(2\bar\lambda_{iB}\gamma_d\lambda^{iA}-
\delta^A_B\bar\lambda_{iC}\gamma_d\lambda^{iC}),\nonumber\\
s\varepsilon^{ab}&=&{\varepsilon^a}_c\wedge\varepsilon^{cb}
+{R^{ab}}_{cd}\varepsilon^c \varepsilon^d-i\bar c_A(2\gamma^{[a}\rho^{A|b]c}
-\gamma^c\rho^{A|ab})\varepsilon_c+\nonumber\\
&-&i\bar c^A(2\gamma^{[a}{\rho_A}^{|b]c}
-\gamma^c{\rho_A}^{|ab})\varepsilon_c+
2G^{-ab}\epsilon^{AB}\bar c_A\wedge c_B+
\nonumber\\
&+&2G^{+ab}\epsilon_{AB}\bar c^A\wedge c^B
+{i\over 4}\varepsilon^{abcd}\bar c_A\wedge\gamma_c c^B
(2\bar\lambda_{iB}\gamma_d\lambda^{iA}-
\delta^A_B\bar\lambda_{iC}\gamma_d\lambda^{iC}),\nonumber\\
s\psi_A&=&-{\cal D}c_A+{1\over 4}\varepsilon^{ab}\gamma_{ab}\wedge\psi_A
-{i\over 2}Q\wedge c_A-{i\over 2}Q_{(0,1)}\wedge\psi_A+
2\rho_{A|ab}V^a\wedge \varepsilon^b+\nonumber\\
&-&2i \epsilon_{AB}G^+_{ab}\gamma^a (c^B\wedge V^b
+\psi^B\wedge \varepsilon^b)
+{i\over 4}(c_B V_a+\psi^B \varepsilon_a)
\bar\lambda^{iB}\gamma^a\lambda_{iA}+\nonumber\\
&+&{i\over 8}\gamma_{ab}(c_B V^b+\psi_B \varepsilon^b)
\left(2\bar\lambda^{iB}\gamma^a\lambda_{iA}-
\delta^B_A\bar
\lambda^{iC}\gamma^a\lambda_{iC}\right),\nonumber\\
sc_A&=&{1\over 4}\varepsilon^{ab}\gamma_{ab}\wedge c_A
-{i\over 2}Q_{(0,1)}\wedge c_A+\rho_{A|ab}\varepsilon^a\wedge
\varepsilon^b-2i \epsilon_{AB}G^+_{ab}\gamma^a c^B
\wedge \varepsilon^b+\nonumber\\
&+&{i\over 4}c_B\bar
\lambda^{iB}\gamma^a\lambda_{iA}\wedge \varepsilon_a+
{i\over 8}\gamma_{ab}c_B\left(2\bar
\lambda^{iB}\gamma^a\lambda_{iA}-\delta^B_A\bar
\lambda^{iC}\gamma^a\lambda_{iC}\right)\wedge \varepsilon^b,\nonumber\\
s\psi^A&=&-{\cal D}c^A+{1\over 4}\varepsilon^{ab}\gamma_{ab}\wedge\psi^A
+{i\over 2}Q\wedge c^A+{i\over 2}Q_{(0,1)}\wedge\psi^A+
2\rho^A_{|ab}V^a\wedge \varepsilon^b+
\nonumber\\
&-&2i \epsilon^{AB}G^-_{ab}\gamma^a (c_B\wedge V^b
+\psi_B\wedge \varepsilon^b)
+{i\over 4}(c^B V_a+\psi^B \varepsilon_a)
\bar\lambda_{iB}\gamma^a\lambda^{iA}+\nonumber\\
&+&{i\over 8}\gamma_{ab}(c^B V^b+\psi^B \varepsilon^b)
\left(2\bar\lambda_{iB}\gamma^a\lambda^{iA}-
\delta_B^A\bar
\lambda_{iC}\gamma^a\lambda^{iC}\right),\nonumber\\
sc^A&=&{1\over 4}\varepsilon^{ab}\gamma_{ab}\wedge c^A
+{i\over 2}Q_{(0,1)}\wedge c^A+\rho^A_{|ab}\varepsilon^a\wedge
\varepsilon^b-2i \epsilon^{AB}G^-_{ab}\gamma^a c_B
\wedge \varepsilon^b+\nonumber\\
&+&{i\over 4}c^B\bar
\lambda_{iB}\gamma^a\lambda^{iA}\wedge \varepsilon_a+
{i\over 8}\gamma_{ab}c^B\left(2\bar
\lambda_{iB}\gamma^a\lambda^{iA}-\delta_B^A\bar
\lambda_{iC}\gamma^a\lambda^{iC}\right)\wedge \varepsilon^b,\nonumber\\
sA_\Lambda&=&-dc_\Lambda-2{f_\Lambda}^{\Omega\Delta}A_\Omega\wedge c_\Delta
-2\epsilon_{AB}L_\Lambda \bar\psi^A\wedge c^B
-2\epsilon^{AB}\bar L_\Lambda
\bar\psi_A\wedge c_B+\nonumber\\
&+&2F_\Lambda^{ab}V_a\wedge \varepsilon_b
+i(f_\Lambda^i\bar\lambda^A_i
\gamma^a c^B\epsilon_{AB}+\bar f_{\Lambda i}\bar\lambda_A^i
\gamma^a c_B\epsilon^{AB})\wedge V_a+\nonumber\\
&+&i(f_\Lambda^i\bar\lambda^A_i
\gamma^a\psi^B\epsilon_{AB}+\bar f_{\Lambda i}\bar\lambda_A^i
\gamma^a\psi_B\epsilon^{AB})\wedge \varepsilon_a,\nonumber\\
sc_\Lambda&=&-{f_\Lambda}^{\Omega\Delta}c_\Omega\wedge c_\Delta-
\epsilon_{AB}L_\Lambda \bar c^A\wedge c^B-\epsilon^{AB}\bar L_\Lambda
\bar c_A\wedge c_B+F_\Lambda^{ab}\varepsilon_a\wedge \varepsilon_b+\nonumber\\
&+&i(f_\Lambda^i\bar\lambda^A_i
\gamma^a c^B\epsilon_{AB}+\bar f_{\Lambda i}\bar\lambda_A^i
\gamma^a c_B\epsilon^{AB})\wedge \varepsilon_a,\nonumber\\
s\lambda_{iA}&=&{1\over 4}\varepsilon^{ab}\wedge\gamma_{ab}
\lambda_{iA}-{i\over 2}\left(1+{2\over n}\right)Q_{(0,1)}\wedge
\lambda_{iA}-{{Q_{(0,1)}}_i}^j\lambda_{jA}+
\nabla_a\lambda_{iA}\varepsilon^a+\nonumber\\
&+&iP_{i|a}\gamma^a c^B
\epsilon_{AB}+G^{+ab}_i\gamma_{ab}c_A
+gC_i c_A,\nonumber\\
s\lambda^{iA}&=&{1\over 4}\varepsilon^{ab}\wedge\gamma_{ab}
\lambda^{iA}+{i\over 2}\left(1+{2\over n}\right)Q_{(0,1)}\wedge \lambda^{iA}
-{{Q_{(0,1)}}^i}_j\lambda^{jA}+\nabla_a\lambda^{iA}\varepsilon^a+\nonumber\\
&+&iP^i_{|a}\gamma^a c_B
\epsilon^{AB}+G_{ab}^{-i}\gamma^{ab}c^A
+gC^i c^A,\nonumber\\
s z_i&=&-gc_\Lambda k^\Lambda_i(z)+Z_{i|a}\varepsilon^a+\bar\lambda_i^A
c_A,\nonumber\\
s \bar z^i&=&-gc_\Lambda k^{i\Lambda}(\bar z)+\bar Z^i_{|a} \varepsilon^a
+\bar\lambda^i_A c^A.
\label{brstalgebra}
\end{eqnarray}
In Eq.\ (\ref{brstalgebra})
$Q_{(0,1)}$ and ${{Q_{(0,1)}}_i}^j$ are obtained by the one-forms
$Q$ and ${Q_i}^j$
upon substitution of $\nabla z_i$ with
$Z_{i|a}\varepsilon^a+\bar\lambda_i^A c_A$,
$\nabla z_i$ with $\bar Z^i_{|a} \varepsilon^a
+\bar\lambda^i_A c^A$ and of $A_\Lambda$ with $c_\Lambda$.
In particular, $Q_{(0,1)}=-{i\over 2}(G^i Z_{i|a}-G_i\bar
Z^i_{|a})\varepsilon^a
-{i\over 2}(G^i\bar\lambda_i^A c_A-G_i\bar \lambda^i_A c^A)$.
The BRST algebra of the twisted theory is the above algebra when one
implements the topological twist and the topological shift,
as explained in Ref.\ \cite{anselmifre}. From now on, when we shall refer to
the above algebra, this implementation will be understood.
The explicit twist is realized as follows
\begin{equation}
\matrix{
\psi_{\alpha A}\rightarrow \psi_{\alpha \dot A}, &
\psi^{\dot \alpha A}\rightarrow \psi^{\dot\alpha \dot A},\cr
\lambda_{i\alpha A}\rightarrow \lambda_{i\alpha\dot A},&
\lambda^{i\dot\alpha A}\rightarrow \lambda^{i\dot\alpha\dot A},\cr
\epsilon_{AB}\rightarrow \epsilon_{\dot A \dot B},&
\epsilon^{AB}\rightarrow -\epsilon^{\dot A \dot B},}
\label{explicittwist}
\end{equation}
while the topological shift is obtained by
\begin{equation}
c^{\dot\alpha\dot A}\rightarrow -{i\over 2}e\varepsilon^{\dot \alpha\dot A}
+c^{\dot\alpha\dot A}.
\end{equation}
Here, $e$ is an object that rearranges the form-number, ghost-number
and statistics in the correct way and that appears only in
the intermediate steps of the twist. It will be called
the {\sl broker}.
The broker
is a zero-form with fermionic statistics and ghost number one.
$e^2$ has even ghost number and Bose statistics, hance it
can be set equal to a number and in our
notation we normalize it as $e^2=1$.
We now rewrite the most relevant twisted-shifted BRST transformations
up to nonlinear terms. To this purpose, note that, when $z_i$ and $\bar z^i$
tend to zero, then
$a\rightarrow 1$; $L_0,\bar L_0\rightarrow 1$;
${f_i}^j\rightarrow \delta_i^j$;
${(g^{1\over 2})_i}^j\rightarrow \delta_i^j$; $G^+_{ab}\rightarrow
-{1\over 2}F^{+0}_{ab}$;
$G^{i-}_{ab}\rightarrow -{1\over 2}F^{i-}_{ab}$. Let us define
(note that the gauginos are expressed in the N=3 notation, namely
$\lambda_{iA}$, $\lambda^{iA}$)
\begin{equation}
\matrix{
\tilde\psi^a={e\over 2}\psi_{\alpha \dot A}(\bar\sigma^a)^{\dot A
\alpha}, & \tilde\psi^{ab}=-e{(\bar \sigma^{ab})^{\dot A}}_{\dot \alpha}
{\psi^{\dot \alpha}}_{\dot A},&\tilde \psi=-e{\psi_{\dot \alpha}}^{\dot A}
\delta_{\dot A}^{\dot \alpha},\cr
C^a={e\over 2}c_{\alpha \dot A}(\bar\sigma^a)^{\dot A
\alpha}, & C^{ab}=-e{(\bar \sigma^{ab})^{\dot A}}_{\dot \alpha}
{c^{\dot \alpha}}_{\dot A},& C=-e{c_{\dot \alpha}}^{\dot A}
\delta_{\dot A}^{\dot \alpha},\cr
\lambda_i={e\over 2}{\lambda_i}_{\alpha \dot A}(\bar\sigma^a)^{\dot A
\alpha}V_a, & {\lambda^i}^{ab}=-e{(\bar \sigma^{ab})^{\dot A}}_{\dot \alpha}
{{\lambda^i}^{\dot \alpha}}_{\dot A},& \tilde\lambda^i=-e
{{\lambda^i}_{\dot \alpha}}^{\dot A}\delta_{\dot A}^{\dot \alpha}.}
\label{definit}
\end{equation}
As an example of the action of the broker $e$, note that,
while ${1\over 2}\psi_{\alpha \dot A}(\bar\sigma^a)^{\dot A
\alpha}$ is a one-form, is a fermion
and has ghost number zero,
the true topological ghost
$\tilde\psi^a$ must be a one-form, with ghost number one and it is a boson.
In Ref.\ \cite{anselmifre} the broker was not explicitly introduced,
although it was implicitly assumed.
Up to nonlinear terms, we obtain
\begin{equation}
\matrix{
s V^a=\tilde\psi^a-d\varepsilon^a+\varepsilon^{ab}\wedge V_b,&
s \varepsilon^{a}=C^a,\cr
s \varepsilon^{ab}=-{1\over 2}{F_0^+}^{ab},&
s\tilde\psi^a=-d C^a+{1\over 2}{F^+}^{ab}_0\wedge V_b,\cr
s\tilde\psi^{ab}=-dC^{ab}+{i\over 2}{\omega^-}^{ab},&
s \tilde\psi =-dC ,\cr
s C^{a}=0,&
s C^{ab}={i\over 2}{\varepsilon^-}^{ab},\cr
s C=0,&
s\lambda_i={1\over 2}dz_i,\cr
s{\lambda^i}^{ab}=i F_i^{-ab},&
s\tilde\lambda^i=0,\cr
sA_i=-dc_i+\lambda_i,&
sc_i=-{1\over 2}z_i,\cr
sz_i=0,&
s\bar z^i={i\over 2}\tilde \lambda^i,\cr
s A_0= i \tilde\psi -dc_0,&
s c_0 =-{1\over 2}+iC.}
\label{simply}
\end{equation}
Here $F^{-ab}={1\over 2}(F^{ab}+{i\over 2}\varepsilon^{abcd}F_{cd})$
(with respect to Ref.\ \cite{anselmifre} there is, in particular,
a sign difference
in the conventions for $\gamma_5$ and $\varepsilon_{abcd}$).
{}From Eq.s (\ref{definit}) and (\ref{simply})
we can directly identify what are the topological ghosts,
the topological antighosts (up to
interaction terms) and the topological gauge-fixings.
More generally, one retrieves the topological meaning of the twisted versions
of all the fields of the original theory.
$\tilde\psi^a$ are the topological ghosts associated to the graviton,
$\lambda_i$ those associated to the matter vectors;
the corresponding topological antighosts are $\tilde\psi^{ab}$
and $\lambda^{iab}$, respectively.
The ghosts for ghosts are
$C^a$, $F^{+ab}_0$ and $z_i$, respectively
for diffeomorphisms, Lorentz rotations and
gauge transformations.
$\bar z^i$ are antighosts for ghosts, while
$C^{ab}$ and $C$ are extraghosts.
Let us discuss the gauge-fixings.
They involve complicated expressions depending on the various fields
(even in the
topological $\sigma$-model in two dimensions \cite{witten2}
one finds convenient to impose a topological gauge-fixing depending
on the ghosts),
but they can be equivalently read when all the ghosts are set to zero,
because in the
minimum of the BRST action all the ghosts are zero by definition.
To this purpose, the interaction terms are negligible (they
always contain ghosts). Our expectations are confirmed: the theory
does indeed describe
Yang-Mills instantons $F_i^{-ab}=0$ in a background gravitational
instanton $\omega^{-ab}=0$ (the Wick rotation to the Euclidean is
of course understood, as in Ref.\ \cite{anselmifre})\footnotemark
\footnotetext{Note that the BRST variation of the topological gravitational
antighost $\tilde\psi^{ab}$ contains, in addition to the gauge-fixing
$\omega^{-ab}$, also the derivative of the extraghost $C^{ab}$.
As explained in Ref.\ \cite{anselmifre}, this is due to the redundancy of
the gauge conditions $\omega^{-ab}=0$.}.
We note that there are more observables than those we have constructed
by means of the minimal BRST algebra (\ref{gaugefree}). They involve
also antighosts. In fact there is
another noticeable
differential form which is closed but not exact and which could be a source
of nontrivial observables, namely the K\"ahler form $K$. In fact the K\"ahler
potential $G$ exists only locally and $K=dQ$ is only a local statement.
The associated descent equations still give observables, however so far
we have not revealed their deep meaning (if any). The K\"ahler form
and its extended version are constructed with both
ghosts and antighosts, while one usually uses only ghosts.
We must remark that the topological Yang-Mills theory we have found
is {\sl not} exactly Witten's topological Yang-Mills theory coupled to gravity.
In fact, Witten's theory is described by a flat K\"ahler manifold
(and $Q$ exists globally, so $K$ is not interesting), while our theory
corresponds to ${SU(1,n)\over SU(n)\otimes U(1)}$ and $K$ cannot be
globally exact \cite{dauriaferrarafre}, so it cannot be
{\sl a priori} discarded. One has
\begin{equation}
K=i{g_i}^j\nabla z_j\wedge \nabla \bar z^i+
{i\over 2}g(G_i k^{i\Lambda}-G^i k_i^\Lambda)
(dA_\Lambda+{f_\Lambda}^{\Sigma\Pi}A_\Sigma\wedge A_\Pi).
\end{equation}
The descent equations derived from $\hat d \hat K=0$ give
the following observables
\begin{eqnarray}
{\cal O}^{(0)}&=&K_{(0,2)},\nonumber\\
{\cal O}^{(1)}_\gamma&=&\int_\gamma K_{(1,1)},\nonumber\\
{\cal O}^{(2)}_S&=&\int_S K,
\end{eqnarray}
where $\gamma$ and $S$ are one- and two-dimensional cycles, while
\begin{eqnarray}
K_{(0,2)}&=&i{g_i}^j(Z_{j|a}\varepsilon^a+\bar\lambda_j^A c_A)
\wedge (\bar Z^i_{|a} \varepsilon^a
+\bar\lambda^i_B c^B)+\nonumber\\
&-&{i\over 2}g(G_j k^{j\Lambda}-G^j k_j^\Lambda)
(\epsilon_{AB}L_\Lambda \bar c^A\wedge c^B+\epsilon^{AB}\bar L_\Lambda
\bar c_A\wedge c_B-F_\Lambda^{ab}\varepsilon_a\wedge \varepsilon_b+\nonumber\\
&-&i(f_\Lambda^i\bar\lambda^A_i
\gamma^a c^B\epsilon_{AB}+\bar f_{\Lambda i}\bar\lambda_A^i
\gamma^a c_B\epsilon^{AB})\wedge \varepsilon_a),\nonumber\\
K_{(1,1)}&=&i{g_i}^j(Z_{j|a}\varepsilon^a+\bar\lambda_j^A c_A)
\wedge \nabla \bar z^i+i{g_i}^j\nabla z_j
(\bar Z^i_{|a} \varepsilon^a
+\bar\lambda^i_A c^A)+
\nonumber\\
&-&{i\over 2}g(G_j k^{j\Lambda}-G^j k_j^\Lambda)
(2\epsilon_{AB}L_\Lambda \bar\psi^A\wedge c^B
+2\epsilon^{AB}\bar L_\Lambda
\bar\psi_A\wedge c_B+\nonumber\\
&-&2F_\Lambda^{ab}V_a\wedge \varepsilon_b
-i(f_\Lambda^i\bar\lambda^A_i
\gamma^a c^B\epsilon_{AB}+\bar f_{\Lambda i}\bar\lambda_A^i
\gamma^a c_B\epsilon^{AB})\wedge V_a+\nonumber\\
&-&i(f_\Lambda^i\bar\lambda^A_i
\gamma^a\psi^B\epsilon_{AB}+\bar f_{\Lambda i}\bar\lambda_A^i
\gamma^a\psi_B\epsilon^{AB})\wedge \varepsilon_a).
\end{eqnarray}
The correspondence between the gauge-free algebra (\ref{gaugefree})
and the complete BRST algebra (\ref{brstalgebra}) is realized by the
following identifications
\begin{eqnarray}
\psi^a&=&i(\bar c_A\wedge \gamma^a\psi^A+
\bar \psi_A\wedge \gamma^a c^A)-A^{ab}\wedge \varepsilon^b
=\tilde\psi^a+\cdots,\nonumber\\
\phi^a&=&i\bar c_A\wedge\gamma^a c^A=C^a+\cdots,\nonumber\\
\chi^{ab}&=&sA^{ab}-A^{ac}{\varepsilon_c}^b+\varepsilon^{ac}
{A_c}^b+2{R^{ab}}_{cd}V^c\varepsilon^d+i(\varepsilon_c\bar\psi_A+V_c \bar c_A)
(2\gamma^{[a}\rho^{A|b]c}
-\gamma^c\rho^{A|ab})+\nonumber\\
&+&i(\varepsilon_c \bar\psi^A+ V_c \bar c^A)
(2\gamma^{[a}{\rho_A}^{|b]c}
-\gamma^c{\rho_A}^{|ab})+4G^{-ab}\epsilon^{AB}\bar\psi_A\wedge c_B+
\nonumber\\
&+&4G^{+ab}\epsilon_{AB}\bar\psi^A\wedge c^B
+{i\over 4}\varepsilon^{abcd}(\bar\psi_A\wedge\gamma_c c^B
+\bar c_A\wedge\gamma_c\psi^B)
(2\bar\lambda_{iB}\gamma_d\lambda^{iA}-
\delta^A_B\bar\lambda_{iC}\gamma_d\lambda^{iC}),\nonumber\\
\psi_i&=&2\epsilon_{AB}L_i \bar\psi^A\wedge c^B
+2\epsilon^{AB}\bar L_i
\bar\psi_A\wedge c_B+\nonumber\\
&-&2F_i^{ab}V_a\wedge \varepsilon_b
-i(f_i^j\bar\lambda^A_j
\gamma^a c^B\epsilon_{AB}+\bar f_{i j}\bar\lambda_A^j
\gamma^a c_B\epsilon^{AB})\wedge V_a+\nonumber\\
&-&i(f_i^j\bar\lambda^A_j
\gamma^a\psi^B\epsilon_{AB}+\bar f_{i j}\bar\lambda_A^j
\gamma^a\psi_B\epsilon^{AB})\wedge \varepsilon_a
=-\lambda_i+\cdots,\nonumber\\
\phi_i&=&-\epsilon_{AB}L_i \bar c^A\wedge c^B
-\epsilon^{AB}\bar L_i
\bar c_A\wedge c_B+F_i^{ab}\varepsilon_a\wedge \varepsilon_b+\nonumber\\
&+&i(f_i^j\bar\lambda^A_j
\gamma^a c^B\epsilon_{AB}+\bar f_{i j}\bar\lambda_A^j
\gamma^a c_B\epsilon^{AB})\wedge \varepsilon_a
=-{1\over 2}z_i+\cdots,\nonumber\\
\eta^{ab}&=&{R^{ab}}_{cd}\varepsilon^c \varepsilon^d-i\bar
c_A(2\gamma^{[a}\rho^{A|b]c}
-\gamma^c\rho^{A|ab})\varepsilon_c
-i\bar c^A(2\gamma^{[a}{\rho_A}^{|b]c}
-\gamma^c{\rho_A}^{|ab})\varepsilon_c+\nonumber\\&+&
2G^{-ab}\epsilon^{AB}\bar c_A\wedge c_B+
2G^{+ab}\epsilon_{AB}\bar c^A\wedge c^B+\nonumber\\
&+&{i\over 4}\varepsilon^{abcd}\bar c_A\wedge\gamma_c c^B
(2\bar\lambda_{iB}\gamma_d\lambda^{iA}-
\delta^A_B\bar\lambda_{iC}\gamma_d\lambda^{iC})=-{1\over 2}F^{+ab}_0+\cdots,
\label{matching}
\end{eqnarray}
where $A^{ab}\wedge V_b=i\bar \psi_A\wedge\gamma^a\psi^A$
and the dots stand for nonlinear corrections.
Now we write the gauge fermion $\Psi$, the BRST variation of which is the
quadratic part of the N=2 lagrangian, after topological
twist and topological shift.
\begin{eqnarray}
\Psi&=&-16i(B^{ab}-i\omega^{-ab}+2dC^{ab})\wedge \tilde\psi_{ac}
\wedge V_b\wedge V^c+8iF_0\wedge \psi^a\wedge V_a+\nonumber\\
&+&\left({2\over 3}\eta^{ab}\varepsilon_{ab}
-{1\over 6}(M_{iab}-2iF^{-}_{iab})\lambda^{iab}\right)
\varepsilon_{cdef}V^c\wedge V^d\wedge V^e\wedge V^f+\nonumber\\
&+&{4\over 3}\lambda_i^a d\bar z^i\wedge\varepsilon_{abcd}V^b\wedge V^c\wedge
V^d.
\end{eqnarray}
Here,
$B^{ab}$ and $M^{iab}$ are Lagrange multipliers ($s\tilde\psi^{ab}=B^{ab}$,
$s\lambda^{iab}=M^{iab}$, $sB^{ab}=0$, $sM^{iab}=0$), while
$\lambda_i^a$ is such that $\lambda_i=\lambda_i^a V_a$.
\section{The general structure of
the twisting procedure and quaternionic topological $\sigma$-model}
\label{sectquater}
In this section we discuss the topological twist of quaternionic matter
multiplets \cite{dauriaferrarafre} coupled to N=2 supergravity.
We shall not develop the entire formalism in full detail, living it for a
future publication, but we shall concentrate on some of its relevant aspects.
We already anticipated in the introduction that the twisting procedure
as described by Witten \cite{witten} needs some modifications in order to
work correctly. First of all, as shown in Ref.\ \cite{anselmifre},
the twist acts on the Lorentz group and does not touch the space-time indices.
This was straightforward in the case of pure supergravity, since
all the fields are one-forms, i.\ e.\ they are all on the same footing
as far as space-time indices are concerned. Consequently the twist on
the Lorentz group works in exactly the same way as the twist
described by Witten. However, when studying the
case of the Yang-Mills theory, one has to face the problem that the vector
bosons $A_i$ are one forms and Lorentz scalars, while the gauginos
$\lambda_i^A$ and $\lambda^i_A$ are zero-forms and Lorentz spinors. If you
are in flat space, you can mix Lorentz and Einstein indices and so the
twist can work in the way described by Witten. However, Witten
himself notes \cite{witten} that his method works only in flat
space, even if the result is valid in any curved space. If we
follow our method, this problem is simply absent. We remain
in the most general curved space and act only on the Lorentz indices.
At this point, the twisted vector boson is still a one-form and a Lorentz
scalar, while the twisted left handed gaugino $\lambda_i^a\equiv
{e\over 2}{\lambda_i}_{\alpha \dot A}(\bar\sigma^a)^{\dot A
\alpha}$ is a zero-form and a Lorentz vector.
{}From (\ref{simply}) you immediately read that the true topological antighost
is not simply $\lambda_i^a$, but $\lambda_i=
\lambda_i^aV_a$, i.\ e.\ the object that you obtain from the simple
twist ($\lambda_i^a$) must be contracted with the vierbein $V^a$.
$\lambda_i$ is a one-form and a Lorentz scalar, as desired.
In order to show that the contraction with a vielbein
plays a substantial role in the twisting procedure,
one would like to exhibit a case in which
this step is so important that no result can be
obtained without it (even in flat space). This
is precisely the case of the quaternionic $\sigma$-model. The multiplet
consists of $(q^i,\zeta_I,\zeta^I)$, where $\zeta_I$ and
$\zeta^I$ are the left
handed and right handed components of the spinors ($I=1,\ldots 2m$),
while $q^i$ are the
coordinates of a 4$m$-dimensional manifold ${\cal Q}(m)$
($i=1,\ldots 4m$),
with a quaternionic structure, namely
possessing three complex structures
$J^x$, $x=1,2,3$, fulfilling the quaternionic
algebra. Specifically ${\cal Q}(m)$ is a Hyperk\"ahler manifold when gravity
is not dynamical (i.e.\
it is external), while it is a quaternionic manifold
when gravity is dynamical.
As you see, no field has indices of $SU(2)_I$, i.\ e.\
all the fields are singlets under the internal $SU(2)$.
Consequently, the usual twisting procedure acts trivially on hypermultiplets:
the Lorentz scalars remain Lorentz scalars
and the spinors remain spinors. In a moment we shall show
how this problem can be solved by means of a contraction with
a suitable vielbein.
The general feature of ${\cal Q}(m)$ is that
its holonomy group $Hol({\cal Q}(m))$ is contained in $SU(2)\otimes Sp(2m)$.
This $SU(2)$ is nothing but $SU(2)_I$ \cite{dauriaferrarafre}.
In the Hyperk\"ahler case, the $SU(2)$ part of the spin connection
of ${\cal Q}(m)$ is flat, while in the quaternionic case
its curvature is proportional to $\Omega^x=h_{ik}(J^x)^k_jdq^i\wedge dq^j$,
where $h_{ij}$ is the metric of ${\cal Q}(m)$. In both
cases one can exploit another $SU(2)$, which will be
denoted by $SU(2)_Q$, namely the $SU(2)$ factor in
the $SU(2)\otimes SO(m)$ maximal subgroup of $Sp(2m)$. We shall see that
the twisting procedure requires also a redefinition of
$SU(2)_L$, namely
\begin{equation}
SU(2)_L\rightarrow SU(2)_L^\prime={\rm diag}[SU(2)_L\otimes SU(2)_Q].
\end{equation}
Summarizing, the complete twisting procedure can be divided in the
following three steps. Step A corresponds to the redefinitions of $SU(2)_L$,
$SU(2)_R$
and ghost number $U(1)_g$
\begin{eqnarray}
SU(2)_L&\rightarrow & SU(2)_L^\prime={\rm diag}
[SU(2)_L\otimes SU(2)_Q],\nonumber\\
SU(2)_R&\rightarrow & SU(2)_R^\prime={\rm diag}
[SU(2)_R\otimes SU(2)_I],\nonumber\\
U(1)_g&\rightarrow & U(1)_g^\prime={\rm diag}
[U(1)_g\otimes U(1)_I],\nonumber\\
^c(L,R,I,Q)^g_f &\rightarrow & (L\otimes Q,R\otimes I)^{g+c}_f,
\end{eqnarray}
where $Q$ denotes the representation of $SU(2)_Q$.
Step B is the correct identification
of the topological ghosts
(fields with $g+c=1$ from $g=0$, $c=1$) by contraction
with a suitable vielbein (if it exists). Step 3 is the topological shift,
namely the shift by a constant
of the $(0,0)^0_0$ field coming from the right handed components
of the supersymmetry ghosts.
Let us see how the contraction with a suitable vielbein can help
when the usual twisting procedure does not give directly
the true topological ghosts (i.e.\ it gives objects with the
wrong spin assignment).
Since the hypermultiplets are made of zero-forms,
the vierbein $V^a$ cannot help us. Fortunately, however, there
{\sl is } a vielbein that does the job, namely the quaternionic vielbein
${\cal U}^{A I}_i$ ($A=1,2$ is an index of $SU(2)_I$)
\cite{dauriaferrarafre}.
We can for example take the contraction ${\cal U}^i_{A I}\bar c^A
\zeta^I$, where ${\cal U}^i_{A I}$ is the inverse vielbein.
After the topological shift, this expression becomes
$-{i\over 2}e{\cal U}^i_{\dot A I}\zeta^{\dot A I}$, up
to interaction terms, and is the natural candidate to become the topological
ghost (it is also the {\sl only} candidate).
Here is another novelty: the topological ghost is constructed with
the {\sl right} handed components of the fermions, {\sl not}
the left handed ones. This means that the R-duality charge of
$\zeta^I$ is $+1$ and that of $\zeta_I$ is $-1$, the opposite
of what happens in the other cases that we have studied. This is not
completely surprising, because the reasoning of Section \ref{general}
that established the R-duality charges of gravitinos and gauginos
was essentially based on the effects of the usual redefinition
of $SU(2)_R$ on the
representations of the Lorentz group, effects that are absent
in the present case. From Ref.\ \cite{dauriaferrarafre} one can convince
oneself
that this is in fact the correct charge assignment.
We report here only those terms in the BRST variations of the fields that
correspond to supersymmetries, in order to spot the nature of
the instantons described by the theory,
reading the topological gauge-fixings.
\begin{eqnarray}
\delta q^i&=&{\cal U}^i_{A I}(\epsilon^{AB}C^{IJ}
\bar c_B\zeta_J+\bar c^A\zeta^I),\nonumber\\
\delta\zeta_I&=&i{\cal U}_a^{BJ}\gamma^a c^A\epsilon_{AB}C_{IJ},\nonumber\\
\delta\zeta^I&=&i{\cal U}_a^{AI}\gamma^a c_A,
\end{eqnarray}
where $C_{IJ}$ is the flat $Sp(2m)$ invariant metric
while ${\cal U}_a^{AI}$ is the supercovariantized
derivative of the quaternionic field $q^i$ with indices flattened both with
respect to spacetime and with respect to the quaternionic manifold
via the corresponding vielbeins.
\begin{equation}
{\cal U}_a^{A I}=V^\mu_a({\cal U}_i^{A I}\partial_\mu q^i-
\epsilon^{AB}C^{IJ}
\bar{\psi_\mu}_B\zeta_J-\bar {\psi_\mu}^A\zeta^I).
\end{equation}
The topological shift gives, up to nonlinear terms,
\begin{eqnarray}
\delta q^i&=&-{i\over 2}e{\cal U}^i_{\dot A I}(\zeta^{\dot A})^I
\equiv\xi^i,\nonumber\\
\delta(\zeta_\alpha)_I&=&{e\over 2}{\cal U}_a^{\dot B J}
(\sigma^a)_{\alpha \dot B}C_{IJ},\nonumber\\
\delta(\zeta^{\dot A})^I&=&0,
\end{eqnarray}
{}From this equation we realize that the topological symmetry is indeed
the expected one for a $\sigma$-model, namely
the map $q^i:M_{spacetime}\rightarrow {\cal Q}(m)$ can be continuously
deformed.
The topological ghosts $\xi^i$ are exactly what we anticipated.
In order to correctly identify the topological antighosts, we have to write
the index $I$ as the pair $(\alpha,k)$, where $\alpha=1,2$ is the doublet
index of $SU(2)_Q$ and $k=1,\ldots m$ is the
vector index of $SO(m)$, such that
$C_{IJ}=C_{(\alpha,k)(\beta,l)}$
takes the form $\epsilon_{\alpha\beta}\delta_{kl}$.
Now we write
\begin{equation}
\delta(\zeta_\alpha)_{\beta k}={e\over 2}{\cal U}_a^{\dot B \gamma l}
(\sigma^a)_{\alpha \dot B}\epsilon_{\beta\gamma}\delta_{kl}=
{e\over 2}{\cal U}_a^{\dot B \gamma k}
(\sigma^a)_{\alpha \dot B}\epsilon_{\beta\gamma}.
\end{equation}
At this point we can introduce the vielbein $E_i^{ak}\equiv {1\over 2}
{\cal U}^{\dot A
\alpha k}_i(\sigma^a)_{\alpha \dot A}$ and the true topological antighosts
$\zeta^{+ab}_k=-e{(\sigma^{ab})_\alpha}^\beta\epsilon^{\alpha\gamma}(\zeta_\beta)
_{\gamma k}$ and $\zeta_k=-e\epsilon^{\alpha\beta}(\zeta_\alpha)_{\beta k}$,
which, under the Lorentz group transform
as $(1,0)$ and $(0,0)$ respectively. One finds
\begin{eqnarray}
\delta \zeta^{+ab k}&=&2V^{\mu[a}E_i^{b]^+k}\partial_\mu q^i,
\nonumber\\
\delta\zeta^k&=&V^\mu_a E^{ak}_i\partial_\mu q^i,
\end{eqnarray}
where $[ab]^+$ means selfdualization in the indices $a,b$. Thus we see
that {\sl both} $\zeta^{+ab}_k$ and $\zeta_k$ are topological antighosts
(otherwise we would have not enough equations to fix the gauge completely).
In the previously studied cases, instead, the $(0,1)$ components
were the only topological antighosts, while
the $(0,0)$ component permitted to
fix the gauge freedom of the topological ghosts (directly related to
the gauge freedom of the gauge freedom, which now is missing).
Thus, the instantons described by this
theory (which we name {\sl hyperinstantons}) are
given by the following equations
\begin{eqnarray}
V^{\mu[a}E_i^{b]^+k}\partial_\mu q^i &=&0,
\nonumber\\
V^\mu_a E^{ak}_i \partial_\mu q^i &=&0.
\label{inst1}
\end{eqnarray}
In a certain sense, Eq.\ (\ref{inst1}) define a condition of holomorphicity
of the maps $M_{spacetime}\rightarrow {\cal Q}(m)$ with respect to
the three complex (or almost complex) structures $J^x$ of
${\cal Q}(m)$. For this reason we find it proper to name
triholomorphic a map $q$ satisfying Eq.\ (\ref{inst1}). In conclusion,
in the same way as the instantons of topological $\sigma$-models in D=2
are given by holomorphic maps, those of topological $\sigma$-models
in D=4 are given by triholomorphic maps.
If gravity is external (${\cal Q}(m)$ is Hyperk\"ahler) then the gravitational
background should be restricted by the need to have N=2 global supersymmetry,
however, the proof that the solutions to the above equations are indeed
instantons works for any background and is based on the following identity
\begin{eqnarray}
\int_{\cal M} &d^4x\sqrt{-g}& g^{\mu\nu}\partial_\mu q^i\partial_\nu q^j
h_{ij}=
2\int_{\cal M} d^4x
\sqrt{-g}[(V^\mu_a E^{ak}_i \partial_\mu q^i)^2+4(V^{\mu[a}E_i^{b]^+k}
\partial_\mu q^i)^2]+\nonumber\\
&-4i\int_{\cal M}&E^{[a k}\wedge E^{b]^- k}\wedge V_a \wedge V_b,
\label{dim1}
\end{eqnarray}
where $h_{ij}=2E_i^{ak}E_j^{bk}\eta_{ab}$ is the metric of ${\cal Q}(m)$,
while $E^{ak}\equiv E_i^{ak} dq^i$. The form
$E^{[a k}\wedge E^{b]^- k}\wedge V_a \wedge V_b$ is proportional to
$\Omega^x\wedge V_a \wedge V_b$ (the coefficient, which is a
numerical matrix $M_x^{ab}$ antiselfdual in $ab$ is not important),
where $\Omega^x$ are the 2-forms introduced above.
$\Omega_x$ are closed forms of ${\cal Q}(m)$ if ${\cal Q}(m)$ is
Hyperk\"ahler. Consequently, in such a case the last term of
(\ref{dim1}) is a topological invariant and this completes our proof.
In the case gravity is dynamical (${\cal Q}(m)$ is quaternionic)
there exist three forms $\omega_x$ such that
\begin{eqnarray}
d&\Omega_x&+\varepsilon_{xyz}\omega^y\wedge\Omega^z=0,\nonumber\\
d&\omega_x&+{1\over 2}\varepsilon_{xyz}\omega^y\wedge\omega^z=\Omega_x.
\end{eqnarray}
The definition of the curvatures changes drastically with respect
to the case of pure N=2 supergravity
\cite{dauriaferrarafre}, in the sense that the curvature
$\rho^A$ of the right handed components of the gravitinos contains
a term that modifies the gravitational topological gauge-fixing,
after performing the topological twist and the topological shift.
This means that the gravitational instantons
are no longer described by an antiselfdual spin connection.
As a matter of fact $\rho^A={\cal D}\psi^A+{i\over 2}\epsilon^{AB}\epsilon_{CD}
{(\sigma_x)_B}^C \psi^D$, where ${(\sigma_x)_A}^B$ are the Pauli matrices
and the resulting instantons are given by
\begin{equation}
\omega^{-ab}-{i\over 2}M_x^{ab}\omega^x=0.
\label{inst2}
\end{equation}
There exist only one matrix with the properties of $M^{ab}_x$, up
to a multiplicative constant, and this constant can be fixed by the fact
that $M_x^{ac}M_y^{db}\eta_{cd}{\varepsilon^{xy}}_z=2i M_x^{ab}$
(see for example section 5 of \cite{billo}).
The proof that the hyperinstantons that solve Eq.s (\ref{inst1})
and (\ref{inst2}) are effectively instantons follows from the fact that the
total kinetic
lagrangian (Einstein lagrangian plus $\sigma$-model kinetic lagrangian)
can be written as a sum of squares of the left hand sides of the above
equations up to a total derivative
\begin{eqnarray}
{\cal L}_{kin}&=&\varepsilon_{abcd}R^{ab}\wedge V^c\wedge V^d
-{1\over 6}\varepsilon_{abcd}V^a\wedge V^b\wedge V^c\wedge V^d
g^{\mu\nu}h_{ij}\partial_\mu q^i\partial_\nu q^j=\nonumber\\
&=&4i(\omega^{-ab}-{i\over 2}M^{ab}_x\omega^x)\wedge
(\omega_{-ac}-{i\over 2}{M_{ac}}_y\omega^y)\wedge V_b\wedge V^c+\nonumber\\
&-&{1\over 3}\varepsilon_{cdef}V^c\wedge V^d\wedge V^e\wedge V^f
[4(V^{\mu[a}E_i^{b]^+k}\partial_\mu q^i)^2+
(V^\mu_a E^{ak}_i \partial_\mu q^i)^2]+\nonumber\\
&+&{\rm total \hskip .1truecm derivative}.
\end{eqnarray}
As an example, let us consider the simplest case, namely the case
$m=1$, ${\cal Q}(1)=H^1$, with the standard flat metric. We have
${\cal U}^i_{\dot A\alpha}=(\sigma^i)_{\alpha \dot A}$
and $E^a_i=\delta^a_i$.
The hyperinstantons satisfy
\begin{eqnarray}
V^{\mu [a}\partial_\mu q^{b]^+} &=&0,
\nonumber\\
V^\mu_a \partial_\mu q^a &=&0.
\end{eqnarray}
If we further specialize the example, namely we choose flat spacetime metric,
we have
\begin{eqnarray}
\partial_{[\mu} q_{\nu]^+}&=&0,
\nonumber\\
\partial_\mu q^\mu &=&0.
\end{eqnarray}
If you imagine that $q_\mu$ is an abelian four vector,
the hyperinstantons are
the selfdual solutions in the Lorentz gauge. But now $\partial_\mu q^\mu=0$
is a true equation and not a choice of gauge. In particular, all harmonic
forms $q=q_\mu dx^\mu$ are solutions (they would be the residual gauge freedom
in the interpretation of $q_\mu$ as a four potential and so they would
not be true solutions).
\section{Conclusions}
\label{concl}
We have seen that, with appropriate procedure and relying on an appropriate
symmetry (R-duality), all N=2, D=4 theories can be topologically twisted,
just as it happens of N=2 theories in two dimensions. This possibility
introduces a set of new topological field theories, each of which describes
intersection theory in the moduli-space of certain interesting geometrical
structures. Some of these structures are, as far as we know, new or at
least not well estabilished in the mathematical literature.
To be specific, let us enumerate these theories.
i) The twist of N=2 $\sigma$-models in flat background, whose target space is a
Hyperk\"ahler manifold, introduces the notion of a topological
hyperk\"ahlerian $\sigma$-model, where the appropriate instantons are the
triholomorphic maps (hyperinstantons). Correlation functions in this theory
will be intersection integrals in the moduli-space of triholomorphic maps:
a subject that to our knowledge has not been so far developed and certainly
deserves careful investigation.
ii) The twist of N=2 supergravity minimally coupled to vector multiplets
yields a topological theory where the instantons are gauge instantons living
in the background of gravitational instantons. The moduli-space of these
structures is the arena
where correlation functions of our theory have to be calculated. Making
an analogy with the 2-dimensional world, our theory stands to
topological Yang-Mills theory as the
topological matter models coupled to topological gravity stand to
pure topological minimal models in D=2.
iii) Similarly, twisting N=2 $\sigma$-models coupled to N=2
supergravity, one obtains a topological $\sigma$-model where the target space
is
quaternionic and which interacts with topological gravity.
The instantons of this theory are interesting objects. They
correspond to the quaternionic analogue of triholomorphic maps
living in the background of generalized
gravitational instantons.
The space-time spin connection is no longer selfdual
but its antiselfdual part is identified with the $SU(2)$ part of the spin
connection
on the quaternionic manifold. This is a phenomenon similar to the embedding of
the
spin connection into the gauge connection occurring in string
compactifications.
iv) Twisting the complete N=2 matter coupled supergravity, one obtains
a topological theory where all the above instantons are fused together:
gravitational, gauge and hyperinstantons. To our knowledge, no study
of the moduli-space of such structures has been attempted.
v) Alternatively, one can also study the twist on N=2
hyperk\"ahlerian $\sigma$-models coupled to N=2 super Yang-Mills. In this
case we have the fusion of gauge and hyperinstantons.
\acknowledgments
Its a pleasure to thank
R.\ D'Auria, S.\ Ferrara and P.\ Soriani
for illuminating and inspiring discussions.
\references
\bibitem{suit1} For a review see D. \ Birmingham, M.\ Blau and M.\ Rakowski,
Phys.\ Rep. \ 209 (1991) 129.
\bibitem{witten} E.\ Witten, Commun.\ Math.\ Phys.\ 117 (1988) 353.
\bibitem{donaldson} S.\ K.\ Donaldson, J.\ Diff.\ Geom.\ 18 (1983) 279.
\bibitem{witten2} E.\ Witten, Commun.\ Math.\ Phys.\ 118 (1988) 411.
\bibitem{suitA} C.\ Vafa, Mod.\ Phys.\ Lett.\ A6 (1990) 337;
R.\ Dijkgraaf, E. \ Verlinde and H. \ Verlinde, Nucl.\ Phys.\ B352 (1991) 59,
B.\ Block and A. \ Varchenko, Prepr. IASSNS-HEP-91/5;
E. \ Verlinde and N.\ P.\ Warner, Phys. \ Lett.\ 269B (1991) 96;
A.\ Klemm, S.\ Theisen and M.\ Schmidt, Prepr. TUM-TP-129/91,
KA-THEP-91-00, HD-THEP-91-32;
Z.\ Maassarani, Prepr. USC-91/023;
P.\ Fre' , L.\ Girardello, A.\ Lerda, P.\ Soriani, Preprint SISSA/92/EP,
Nucl.\ Phys.\ in press; B.\ Dubrovin, Napoli preprint, INFN-NA-4-91/26.
\bibitem{suitB}P.\ Candelas, C.\ T.\ Horowitz, A.\ Strominger and E.\ Witten,
\ Nucl.\ Phys. B258 (1985) 46;
P.\ Candelas, A.\ M.\ Dale, C.\ A.\ Lutken and R.\ Schimmrick,
Nucl.\ Phys.\ B298 (1988) 493;
M.\ Linker and R. \ Schimmrick, Phys.\ Lett.\ 208B (1988) 216;
C.\ A.\ Lutken and G.\ C.\ Ross, Phys. Lett. 213B (1988) 152;
P.\ Zoglin, Phys.\ Lett.\ 218B (1989) 444;
P.\ Candelas, Nucl.\ Phys.\ B298 (1988) 458;
P.\ Candelas and X.\ de la Ossa, Nucl.\ Phys.\ B342 (1990) 246.
\bibitem{suit3} E.\ Witten, Nucl.\ Phys.\ B340 (1990) 281;
E.\ Verlinde, H.\ Verlinde, Surveys in Diff.\ Geom.\ 1 (1991) 243;
M.\ Kontsevich, Comm.\ Math.\ Phys.\ 147 (1992) 1;
R.\ Dijkgraaf, E.\ Witten, Nucl.\ Phys.\ B342 (1990) 486;
L.\ Bonora, C.S.\ Xiong, SISSA preprint 161/92/EP.
\bibitem{suit4} A.M.\ Polyakov, Mod.\ Phys.\ Lett.\ A2 (1987) 893;
V.G.\ Knizhnik, A.M.\ Polyakov, A.B. Zamolochikov, Mod.\ Phys.\ Lett.\
A3 (1988) 819; J.\ Distler, H.\ Kawai, Nucl.\ Phys.\ B231 (1989) 509;
F.\ David, Mod.\ Phys.\ Lett.\ A3 (1988) 207;
V.\ Kazakov, Phys.\ Lett.\ 150B (1985) 282;
D.J.\ Gross, A.A.\ Migdal, Phys.\ Rev.\ Lett.\ 64 (1990) 717;
M.\ Douglas, S.\ Schenker, Nucl.\ Phys.\ B335 (1990) 635;
E.\ Brezin, V.\ Kazakov, Phys.\ Lett.\ 236B (1990) 144.
\bibitem{suit5} L.\ Bonora, P.\ Pasti, M.\ Tonin, Ann.\ Phys.\ 144
(1982) 15; L.\ Beaulieu, M.\ Bellon, Nucl.\ Phys.\ B294 (1987) 279.
\bibitem{anselmifre} D.\ Anselmi, P.\ Fr\`e, ``Twisted N=2 Supergravity as
Topological Gravity in Four Dimensions'', preprint, SISSA 125/92/EP,
July, 1992; to appear in Nucl.\ Phys.\ B.
\bibitem{gaillardzumino} M.\ K.\ Gaillard, B.\ Zumino, Nucl.\ Phys.\ B193
(1981) 221.
\bibitem{fayetferrara} See, for example,
P.\ Fayet, S.\ Ferrara, Phys.\ Rep.\ 35C (1977) 249;
R.\ Barbieri, S.\ Ferrara, D.V.\ Nanopoulos, K.S.\ Stelle, Phys.\ Lett.\
113B (1982) 219; A.\ Salam, J.\ Strathdee, Nucl.\ Phys.\ B87 (1985) 85.
\bibitem{dewit} B.\ de Wit, A.\ van Proeyen, Nucl.\ Phys.\ B245 (1984) 89.
\bibitem{cremmer} E.\ Cremmer in ``Supersymmetry and its Applications'',
edited by G.W.\ Gibbons, S.W.\ Hawking, P.K.\ Townsend, Cambridge
University Press, 1985.
\bibitem{dolgov} A.\ D.\ Dolgov, I.\ B.\ Khriplovich, A.\ I.\
Vainshtein, V.\ I.\ Zakharov, Nucl.\ Phys.\ B315 (1989) 138.
\bibitem{dauriaferrarafre} R.\ D'Auria, S.\ Ferrara, P.\ Fr\`e,
Nucl.\ Phys.\ B359 (1991) 705.
\bibitem{castdauriafre} L.\ Castellani, R.\ D'Auria, P.\ Fr\`e,
``Supergravity and Superstrings'', World Scientific, 1991.
\bibitem{ceresole}
L.\ Castellani, A.\ Ceresole, R.\ D'Auria, S.\ Ferrara, P.\ Fr\`e,
E.\ Maina, Nucl.\ Phys.\ B286 (1986) 317.
\bibitem{cremmerscherk} E.\ Cremmer, J.\ Scherk, Nucl.\ Phys.\ B127 (1977) 259.
\bibitem{billo} M.\ Bill\`o, P.\ Fr\`e, L.\ Girardello, A.\ Zaffaroni,
preprint SISSA 159/92/EP, IFUM/431/FT.
\end{document}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,150 |
\section{Introduction}
Modulation recognition has been widely investigated over the past decades using likelihood-based (LB) or feature-based (FB) approaches\cite{nandi1998algorithms, wei2000maximum, yang1997suboptimal, swami2000hierarchical}. Inspired by the remarkable success of deep neural networks in computer vision and natural language processing\cite{krizhevsky2012imagenet}\cite{socher2014recursive}, modulation recognition using deep neural networks, also called deep modulation recognition, has shown promising performance improvements over conventional methods and attracted increasing research interests.
O'Shea first introduced convolutional neural networks (CNNs) for modulation recognition and showed performance advantages by learning features with deep neural networks over conventional expert features \cite{o2016convolutional}. In \cite{hong2017automatic}, recurrent neural network (RNN) was used to exploit temporal sequence characteristic of modulated signals.
Several deep architectures based on both CNN and RNN were examined and analyzed in \cite{west2017deep}. A CNN based unit classifier was proposed in \cite{meng2018automatic} to accommodate varying input lengths. In \cite{o2018over}, over-the-air radio machine learning dataset was generated using GNU Radio and USRP, showing the performance of deep modulation recognition under different channel conditions.
In \cite{peng2018modulation}, the author provided a new way for CNN-based modulation recognition by representing received signal samples as images.
A two-stage hybrid method based on short-time Fourier transform (STFT) and CNN was proposed in \cite{daldal2020Automatic} which demonstrates the effectiveness of time-frequency features.
In \cite{shisheng2020deep}, the author introduced a deep neural network (DNN)-based modulation classifiers and shown its robustness to uncertain noise conditions.
However, most existing research focuses on single receive antenna. In radio propagation environments, modulation recognition signals may suffer from multipath fading,
which would cause performance degradations, especially when the signal experiences deep fade at the receiver\cite{o2018over}. One of the most powerful techniques to mitigate effects of fading is to use diversity-combining. Therefore, it is important to investigate deep neural networks for modulation recognition with multiple receive antennas.
In \cite{wang2020deep}, cooperative automatic modulation classification (Co-AMC) by multiple receive antennas is proposed for modulation recognition in MIMO systems, where the CNN is trained with each receive antenna independently, and the final decision on modulation type is reached by cooperative decision. Results show that Co-AMC by multiple receive antennas performs better than single receive antenna. However, the performance of this method is limited by feature learning with single-antenna signals.
In this paper, two end-to-end feature learning deep architectures, which can be trained directly using multi-antenna signals, are introduced for modulation recognition with multiple receive antennas, including:
\begin{itemize}
\item Multi-View CNN (MVCNN). This was initially proposed for 3D shape recognition using view-based descriptors of 2D images, which combines information from multiple views of a 3D shape into a single and compact descriptor for end-to-end feature learning\cite{su2015multi}. By treating signals from different receive antennas as different views of a 3D object and designing a suitable view-pooling location
and operation for feature fusion of multi-antenna signals, MVCNN can then be applied for modulation recognition with multiple receive antennas.
\item Weight-Learning CNN (WLCNN). Considering that in wireless communications, the instantaneous signal-to-noise ratio (SNR) of signals could be different among receive antennas, the feature combing weight of each antenna need to be adaptive according to input signal characteristics. We propose WLCNN based on a weight-learning module (WLM) which is designed to automatically learn the weights for combining features of different receive antennas, and incorporated with CNNs to enable end-to-end feature learning for modulation recognition with multiple receive antennas.
\end{itemize}
Results show that comparing with existing Co-AMC method, both end-to-end feature learning deep architectures, MVCNN and WLCNN, gain better modulation classification accuracy. Further, the proposed WLCNN, which automatically learns the weights for feature combing, results in the best performance.
The rest of this paper is organized as follows. System model is given in Section \uppercase\expandafter{\romannumeral2}, and deep architectures for multi-antenna modulation recognition are described in Section \uppercase\expandafter{\romannumeral3}. In Section \uppercase\expandafter{\romannumeral4}, simulation results are given and analyzed. Finally, conclusions are discussed in Section V.
\section{System Model}
\subsection{Signal Model and Representations}
Suppose that signals are sent from a transmitter and experience flat-fading in radio channels, our goal is to determine the modulation type of the signals using a receiver with $N_r$ antennas. Given the transmitted signal $\mathbf{s}$, the equivalent baseband received signal $\mathbf{r}$ can be expressed as:
\begin{equation}
\mathbf{r}=\mathbf{H} \mathbf{s}+\mathbf{n}
\end{equation}
\noindent where $\mathbf{H}$ is a complex-valued $N_r \times 1$ vector where its $n_r$-th ($1\leq n_r\leq N_r$) element denotes the complex channel coefficient between the transmitter and the $n_r$-th receiving antenna. $\mathbf{n}$ denotes the additive Gaussian noise.
Within one observation interval, $N$ signal samples are collected from each antenna to form a $1 \times N$ complex-valued vector, which is further decomposed into a $2 \times N$ matrix, where the first and second row correspond to the in-phase and quadrature components, respectively. Signal samples from $N_r$ receive antennas are then collected to form a $N_r \times 2 \times N$ tensor. For more efficient training, we normalize the power of these vectors.
Let $\bold{x}^{(i)}$ denote the three-dimensional tensor collected in the $i$-th observation interval, and $\bold{y}^{(i)}$ be its corresponding label denoting the transmitted signal's modulation type. Then $\big(\bold{x}^{(i)}, \bold{y}^{(i)} \big)$ forms one training example, and training examples from randomly different time instants with identical observation durations are collected as datasets for deep modulation recognition.
\subsection{Deep Learning Based Modulation Recognition}
The optimization of supervised deep learning process can be formulated as finding the parameters $\boldsymbol{\theta}$ of a neural network that significantly reduce a cost function $J(\boldsymbol{\theta})$\cite{goodfellow2016deep}, which can be written as,
\begin{equation}
J(\boldsymbol{\theta})=\mathbb{E}_{(\bold{x}, \bold{y}) \sim \hat{p}_{\text {data }}} L(f(\bold{x} ; \boldsymbol{\theta}), \bold{y})
\end{equation}
\noindent where $L(\cdot)$ denotes the loss function, $f(\bold{x} ; \boldsymbol{\theta})$ corresponds to the neural network predicted output when the input signal is $\bold{x}$, and $y$ denotes the true modulation class label, $\hat{p}_{\text {data }}$ is the empirical distribution of modulated signals.
In our experiments, the cross-entropy is chosen as the loss function, given by
\begin{equation}
L(\hat{y}, y)= - \sum_{m=1}^{M}y_m\log(\hat{y}_m)
\end{equation}
\noindent where $M$ is the total of modulation types.
For the training of neural network parameters, Adam optimizer\cite{kingma2014adam} with initial learning rate 0.001 is used for mini-batch optimization.
\section{Deep Architectures for Multi-Antenna Modulation Recognition}
In this section, two end-to-end feature learning deep architectures including MVCNN and WLCNN, which can be trained directly using multiple antenna signals, are introduced for modulation recognition with multiple receive antennas, Co-AMC \cite{wang2020deep} is also introduced for comparison.
\begin{figure}[!t]
\centering
\includegraphics[width=1.3in]{model_resBlock.eps}
\caption{The architecture of a residual block.}
\label{fig:resBlock}
\end{figure}
\begin{table}[!t]
\renewcommand{\arraystretch}{1.3}
\caption{Base CNN LAYOUT}
\label{tab:ResNet34}
\centering
\begin{tabular}{llccc}
\toprule
& Layer &Kernel Size &Stride &Output Size \\
\midrule
& Input & & &$1 \times 2 \times 512$ \\
\multirow{3}{*}{CNN$_1$} & Conv, BN, Tanh &$7$ &2 &$64 \times 1 \times 256$ \\
& Max Pooling &$3$ &2 &$64 \times 1 \times 128$ \\
& Residual Block * 3 &$3$ &1 &$64 \times 1 \times 128$ \\
\midrule
\multirow{5}{*}{CNN$_2$} & Residual Block * 4 &$3$ &2 &$128 \times 1 \times 64$ \\
& Residual Block * 6 &$3$ &2 &$256 \times 1 \times 32$ \\
& Residual Block * 3 &$3$ &2 &$512 \times 1 \times 16$ \\
& Average Pooling &$16$ &1 &$512 \times 1 \times 1$ \\
& FC, Softmax & & &$ 20$ \\
\bottomrule
\end{tabular}
\end{table}
ResNet\cite{he2016deep} is chosen as a base network to construct the multi-antenna modulation recognition architectures. This is because: (1) ResNet can reduce the effect of degradation of deeper networks; (2) Existing research on modulation recognition with single receive antenna has shown that ResNet is a suitable candidate\cite{o2018over}. The ResNet is tuned such that when we further increase the network size or change network parameters, the modulation classification performance does not further improve, given the single-antenna dataset. In this way, a 34-layer ResNet is obtained and shown in Table \ref{tab:ResNet34}, which consists of 1 convolutional (Conv) layer, 16 residual blocks, and a fully-connected (FC) layer, the total parameters of the base CNN is about 7.23M. Each residual block consists of two convolutional layers and batch normalization (BN)\cite{ioffe2015batch} operations, as illustrated in Fig. \ref{fig:resBlock}. Considering that the modulated signals are composed of both positive and negative values, Tanh is used as the activation function on first convolutional layer and ReLU on the others, and Softmax function is used to normalize the output distributions.
\subsection{Multi-View Convolutional Neural Network}
MVCNN was initially proposed for 3D view-based shape recognition, which combines information from multiple views of a 3D shape into a single and compact shape descriptor for end-to-end feature learning and has shown to be quite effective in 3D shape recognition.
To realize the MVCNN in our considered scenario, the received signals from one antenna is taken as one view of 3D object, and the base network, i.e., the ResNet in Table \ref{tab:ResNet34}, is split by a view-pooling layer into two parts: CNN$_1$ and CNN$_2$. As illustrated in Fig. \ref{fig:mvcnn}, features from individual antennas are first extracted by CNN$_1$, and then fused by view-pooling layer. Note that all the $N_r$ branches of MVCNN, i.e., CNN$_1$, share the same parameters. Operations of the view-pooling layer are similar to conventional pooling layers in CNN, e.g., max pooling or average pooling, the difference lies in that the view-pooling operations are carried out across the dimension of receive antennas. Features fused by view-pooling layer are then passed through CNN$_2$, where the information obtained across multiple antennas is further processed and then the output of MVCNN is obtained, which corresponds to a vector consisting of the empirical conditional probabilities of different modulation types.
\begin{figure}[!t]
\centering
\includegraphics[height=1.3in]{model_mvcnn.eps}
\caption{The architecture of MVCNN for multiple-antenna modulation recognition.}
\label{fig:mvcnn}
\end{figure}
Different locations of the view-pooling layer determine different network architectures and could lead to different modulation recognition performances. So the location of view-pooling layer and pooling operations need to be carefully designed. In this work, we test the performance of MVCNN with different locations and operations, including maximum and mean operations of the view-pooling layer. Results show that the best performance is obtained by locating the view-pooling layer after the first 3 residual blocks, and by using max-view-pooling operation for feature fusion across the dimension of receive antennas, where the max-view-pooling is an element-wise maximum operation across different receive antennas as:
\begin{equation}
F_{o}=\max\{F_1, F_2, ..., F_{N_r}\}
\label{eq:wb}
\end{equation}
\noindent where $F_{nr} (nr=1, 2, ..., N_r)$ denotes signal features from the $nr$-th antenna, and $F_o$ is the output feature of max-view-pooling.
\subsection{Weight-Learning Convolutional Neural Network}
In wireless communications, radio signals received from different receive antennas could experience different fading, and the weight of feature combing for each antenna need be adaptive to the input signal characteristics. We propose WLCNN for automatically learning the weights for feature combing from different receive antennas.
The architecture of WLCNN is shown in Fig. \ref{fig:wlcnn}, the base CNN, as Table \ref{tab:ResNet34}, is split into two parts: CNN$_1$ and CNN$_2$. Features from individual antennas are first extracted by CNN$_1$, then the signal features from different receive antennas are combined with learned weights from the WLM. Combined features are then passed through CNN$_2$ to reach the output of WLCNN. Similar as that for MVCNN, the location of feature combing is located after the first 3 residual blocks, and CNN$_1$ share the same parameters.
The process of learning the weights of signal feature combing from different receive antennas is described as follows.
Given the signal feature from the $nr$-th receive antenna $F_{nr}$, the combined feature $F_o$ can be written as an element-wise weighted sum of each receive antenna, given by
\begin{equation}
F_{o}=\sum_{nr=1}^{N_r}w_{nr}*F_{nr}
\label{eq:wb}
\end{equation}
\noindent where ${w}_{nr}$ is the normalized learned weight of the $nr$-th receive antenna.
\begin{figure}[!t]
\centering
\includegraphics[height=1.3in]{model_wlcnn.eps}
\caption{The architecture of WLCNN for multiple-antenna modulation recognition.}
\label{fig:wlcnn}
\end{figure}
Due to the unknown characteristics of received signals, a neural network based WLM is adopted to learn the weight for each receive antenna.
Specifically, for the input signal from the $nr$-th receive antenna $x_{nr}$, the WLM with parameters $\boldsymbol{\theta_{w}}$ outputs a single value $\hat{w}_{nr}$ representing the estimated weight of the $nr$-th receive antenna, which can be expressed as,
\begin{equation}
\hat{w}_{nr}=g(x_{nr}; \boldsymbol{\theta_{w}})
\label{eq:wl}
\end{equation}
The estimated weights of different receive antennas are normalized by softmax function as,
\begin{equation}
w_{nr}=\frac{e^{\hat{w}_{nr}}}{\sum_{nr=1}^{N_r}e^{\hat{w}_{nr}}}
\label{eq:softmaxnorm}
\end{equation}
The WLM is designed as a CNN based neural network with only one neuron in the last layer representing the weight of feature combing, and this weight automatically varies with different input signals. The WLM for each receive antenna shares the same structure and is composed of 1 convolutional layer and 3 residual blocks with 16 convolutional filters. Similar to MVCNN, the parameters of WLCNN are trained in an end-to-end manner with multi-antenna signals. The total parameters of the WLM is around 6.34k and is roughly 0.1\% parameters of base CNN. The specific architecture of the WLM is as Table \ref{tab:wlm}.
\begin{table}[!t]
\renewcommand{\arraystretch}{1.3}
\caption{Architecture of WLM}
\label{tab:wlm}
\centering
\begin{tabular}{@{}llcccl@{}}
\toprule
& Layer &Kernel Size &Stride &Output Size \\
\midrule
& Input & & &$1 \times 2 \times 512$ \\
& Conv, BN, Tanh &$7$ &2 &$16\times 1 \times 256$ \\
& Residual Block &$3$ &2 &$16 \times 1 \times 128$ \\
& Residual Block &$3$ &2 &$16 \times 1 \times 64$ \\
& Residual Block &$3$ &2 &$16 \times 1 \times 32$ \\
& FC & & &$1$ \\
\bottomrule
\end{tabular}
\end{table}
\subsection{Cooperative automatic modulation classification}
In Co-AMC, a base CNN is first trained by signals from each receive antenna, and then the decision on the modulation type is cooperatively made based on outputs of trained CNNs from all branches. Direct averaging (DA) method is used due to the identical signal distribution of each receive antenna in our experiment, where each branch of predicted distribution is averaged to make a cooperative prediction as Fig. \ref{fig:egcnn}. Also, these CNNs share the same parameters and have the same structure.
The output of the $k$-th CNN is a $M \times 1$ vector denoted as $\bold{\hat{p}_k}$, where $M$ corresponds to the total number of modulation types, and its $m$-th element, $\hat{p}_{km}$, can be seen as the predicted possibility of the $m$-th modulation type from the $k$-th receive antenna.
The predicted distributions from different antennas are averaged to obtain the global estimate of modulation type. Let $\hat{p}^{(m)}$ denote the estimated probability of the $m$-th modulation format, given by
\begin{equation}
\hat{p}^{(m)}=\sum_{k=1}^{N_{r}} \hat{p}_{k m} / N_{r}
\label{eq:egc}
\end{equation}
The decision on the modulation format can be reached by choosing the index $m^*$ that maximizes $\hat{p}^{(m)}$, given by
\begin{equation}
m^* = \mathop{\arg\max }_{m}\hat{p}^{(m)}
\label{eq:egcdecision}
\end{equation}
\begin{figure}[!t]
\centering
\includegraphics[width=2.8in]{model_egcnn.eps}
\caption{The decision process of cooperative automatic modulation classification (Co-AMC) for multiple-antenna modulation recognition.}
\label{fig:egcnn}
\end{figure}
\section{Results and Analysis}
In this section, we present the performances of end-to-end feature learning deep architectures and compare with the Co-AMC for modulation recognition with multiple receive antennas.
\begin{figure}[!t]
\centering
\subfigure[MVCNN]{\includegraphics[width=2.95in]{acc_MVCNN-MP.eps}}
\vfil
\subfigure[WLCNN]{\includegraphics[width=2.95in]{acc_WLCNN.eps}}
\vfil
\subfigure[Co-AMC]{\includegraphics[width=2.95in]{acc_CPCNN.eps}}
\caption{Modulation recognition performances versus SNR with different numbers of receive antennas with different deep architectures.}
\label{fig:acc_snr_model}
\end{figure}
Datasets in our experiments are generated using GNU Radio\cite{o2016radio}. A square root raised cosine filter with a roll-off factor of 0.35 is used for pulse shaping. Transmitted signals experience independent and identically distributed Rayleigh fading, where the second moment of the Rayleigh fading coefficient is normalized to unit. Received signals are filtered and down converted to baseband, and are up sampled by a factor of 8. $N_r \times 2 \times N$ real samples are collected from $N_r $ receive antennas to form one example, where $N_r$ ranges from 1 to 8 and $N$ is set to 512 in our experiments. 2000 examples are generated for each SNR and each modulation type, and the size of training set and test set is 1 : 1. There are 20 different modulation formats including both analog and digital modulation types, including: BPSK, QPSK, 8PSK, 16PSK, 16QAM, 32QAM, 64QAM, 128QAM, 256QAM, 16APSK, 32APSK, 64APSK, 128APSK, OOK, 4ASK, GMSK, FM, AM, DSB, SSB.
\begin{figure}[!t]
\centering
\includegraphics[width=3in]{acc_nr_model.eps}
\caption{The classification accuracy comparison of different multi-antenna modulation recognition methods versus number of receive antennas with SNR = 0dB and 10dB.}
\label{fig:acc_nr_model}
\end{figure}
The classification accuracy versus SNR for MVCNN, WLCNN and Co-AMC with different numbers of receive antennas are presented in Fig. \ref{fig:acc_snr_model}.
It is shown that, when the number of receive antennas increases, the classification accuracies increase accordingly.
When $Nr$ increases from 1 to 8, the noise tolerance of the MVCNN and WLCNN is improved by around 10 dB, and the classification accuracy is improved by up to 30\% when the SNR is about 0dB.
This coincides with the analysis in conventional modulation recognition for multiple receive antennas that recognition performance can be improved by utilizing spatial diversity. Note that performances of the multi-antenna modulation recognition methods are identical for the case $N_r=1$. This is because when the number of receive antenna is equal to 1, these networks reduce to the same computational architecture.
The classification accuracy versus $N_r$ for different architectures with given SNRs are plotted in Fig. \ref{fig:acc_nr_model}. It is shown that comparing with Co-AMC, the two end-to-end feature learning deep architectures have better performance. When the $Nr$ increases to 8, MVCNN improves the modulation classification accuracy by about 7.4\% while WLCNN improves by about 9.8\% in 0dB, this is due to the advantages of end-to-end feature learning with radio signals from multiple antennas simultaneously. The proposed WLCNN results in the best performance among them, this is because, although the channel coefficients are independent and identically distributed, at any particular instant of time, the fading coefficient for one channel is different from another. That is, the instantaneous SNR from different receive antennas are different. The proposed WLCNN automatically learns the weights for feature combining of different antennas with a WLM. In this way, the weights for multiple receive antennas are adaptively varied according to different input signals. In other words, the proposed WLCNN, which learns the weights for feature combing through end-to-end training, can better fuse features from multiple receive antennas than MVCNN which uses a pooling operation.
\section{Conclusion and Discussion}
In this paper, two end-to-end learning deep architectures are introduced for modulation recognition with multiple receive antennas: MVCNN and WLCNN. Compared with existing Co-AMC algorithm, the two end-to-end deep architectures for modulation recognition with multiple receive antennas achieve better performance. Further, the proposed WLCNN obtains the best performance among them by automatically learning the feature combining weights of different antennas with adapting to different input signals.
\section{Acknowledgment}
We would like to thank Professor Nuno Vasconcelos for his valuable comments and discussions on deep architectures for modulation recognition.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,453 |
Apple quietly released an all-new iPad and iPhone edition on Monday, with the smartphone going on sale from today. Whilst Apple's flagship smartphones see a new colour join the line-up, the iPad which debuted on the Apple store this week is a completely new device. Strange, then, that this device was added to their site without even a whisper of a launch beforehand. Where the iPad franchise has previously been showcased via headline events, this press release announcement went by almost unnoticed… Almost. What does this say about the importance of the iPad to Apple, and of the tablet market in general?
Apple quietly unveiled their new iPad, now without the "Air" branding.
Apple note that there's over one million apps designed specifically for iPad.
The new iPad starts at just £339 for the 32GB wifi-only model, whilst the 128GB cellular-model comes in at £559; this demonstrates Apple's intentions to split their tablet line-up into three – the high-end iPad Pro, mid-range iPad and cheaper iPad Mini. The device also sees it's processor upgraded to an A9 chip, the chip used in iPhone 6S and iPhone SE. The powerful A9X chip seems to be reserved exclusively for the iPad Pro. Other than the device's notable speed boost, the iPad displays most of the same features as its predecessor.
As CNET explains, this is the first time that Apple has unveiled a new iPad in such a low-key manner, highlighting the overall downturn in the tablet market. We've now seen global tablet sales drop for a ninth consecutive quarter, with Apple's tablet sales decreasing across 12 straight quarters.
Recently, Samsung also announced new tablets to its range, with the Samsung Galaxy Tab S3 – a 9.7" device running Google's Android software – and two larger iterations of the Samsung Galaxy Book, which runs Microsoft's Windows 10 software. Samsung's announcement still made the main stage at Mobile World Congress in Barcelona, but the purpose of the event seemed more about keeping momentum going in advance of their Samsung Galaxy S8 launch – scheduled for next Wednesday, March 29th. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,418 |
Duckett Construction is located in Albertville, Al. We specialize in building your dream home, cabinets, remodeling, additions, and dirt work. Whit Duckett, owner of Duckett Construction, has been building for 25 years, and has owned his own company for 16 years. Whit and his employees are hard working and motivated to start your project and finish it in a timely manner. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,562 |
Q: Browser sync for Angular SPA I'm using browser sync with an Angular SPA. Serving the site looks like this:
gulp.task('serve', function() {
browserSync.init(null, {
server: {
baseDir: './',
middleware: [historyApiFallback()]
}
});
});
This works fine. The use of historyApiFallback (npm module) means browser sync doesn't freak out when going to a new URL path when all it needs to do is continue serving the index.html.
The problem I have is with watching files. I've tried this:
gulp.task('watch', function() {
watch('./src/scss/**/*.scss', function() {
runSequence('build-css', browserSync.reload);
});
});
The watch task does work because the build-css task triggers fine. Then the console logs Reloading Browsers... and just hangs. The browser never gets the CSS injection or reload. What am I doing wrong here?
Note that I'm using gulp-watch not the native gulp watch purposely.
A: I recommend you to use the lite-server. It is a simple customized wrapper around BrowserSync to make it easy to serve SPAs(you don't even have to configure the history api).
You can use it by simple adding an entry in the scripts object in your package.json and running the following command: npm run dev.
"scripts": {
"dev": "lite-server"
},
The module will automatically watch for changes of your files and keep then sync. So it will work with your gulp, because it will update your browser after the build-css task is executed(because the output files will change).
I am currently using it with angular 1, angular 2 and vue.js and worked fine with all.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,033 |
@interface JavaNetUnknownHostException : JavaIoIOException {
}
- (instancetype)init;
- (instancetype)initWithNSString:(NSString *)detailMessage;
@end
J2OBJC_EMPTY_STATIC_INIT(JavaNetUnknownHostException)
CF_EXTERN_C_BEGIN
J2OBJC_STATIC_FIELD_GETTER(JavaNetUnknownHostException, serialVersionUID, jlong)
CF_EXTERN_C_END
J2OBJC_TYPE_LITERAL_HEADER(JavaNetUnknownHostException)
#endif // _JavaNetUnknownHostException_H_
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,343 |
Home » Criminal Procedure » Rights Of The Accused At The Trial
Rights Of The Accused At The Trial
The rule enumerates the rights of a person accused of an offense, which are both constitutional as well as statutory, save the right to appeal which is purely statutory in character.
1. Substantive – considers the intrinsic validity of the law
2. Procedural – based on the principle that a court hears before it condemns. Requirement of notice and hearing.
Section 1. Rights of accused at trial
A. To Be Presumed Innocent
- In all criminal prosecutions, the accused is presumed innocent until the contrary is proved beyond a reasonable doubt.
- The conviction should be based on the strength of the prosecution and not on the weakness of the defense, an accusation is not synonymous with guilt.
- It is the doubt engendered by an investigation of the whole proof and inability, after such investigation, to let the mind rest easy upon the certainty of guilt. Absolute certainty of guilt is not demanded by the law to convict of any criminal charge but moral certainty is required as to every proposition of proof requisite to constitute the offense.
REASON: the slightest possibility of an innocent man being convicted for an offense he has not committed would be far more dreaded than letting a guilty person go unpunished or for a crime, he may have perpetrated.
EQUIPOSE RULE
- where the evidence of the parties in a criminal case are evenly balanced, the constitutional presumption of innocence should tilt in favor of the accused who must be acquitted.
Exceptions To The Presumption Of Innocence
People v. Mingoa, 92 Phil 856 (1953)
- The legislature may enact that when certain facts have been proved, they shall be prima facie evidence of the existence of guilt of the accused and shift the burden of proof provided there be a rational connection between the facts proved and the ultimate fact presumed so that the inference of the one from proof of the other is not unreasonable and arbitrary experience.
In cases of self-defense, the person who invokes self-defense is presumed guilty. In this case, a REVERSE TRIAL will be held.
B. To Be Informed Of The Nature And The Cause Of The Accusation Against Him
The right requires that the information should state the facts and the circumstances constituting the crime charged in such a way that
a person of common understanding may easily comprehend and be
informed of what it is about.
People v. Ortega, 276 SCRA 166 (2003)
- An accused may not be convicted of an offense unless it is
clearly charged in the complaint or information. To convict
him of an offense other than that charged in the complaint or
information would be a violation of this constitutional right.
When a person is charged in a complaint with a crime and the evidence
does not show that he is guilty thereof, but does show that he is
guilty of some other crime or a lesser offense, the court may sentence
him for the lesser offense, PROVIDED that the lesser offense is a
cognate offense and is included in the complaint filed in court.
The qualifying or aggravating circumstances must be ALLEGED and
PROVED in order to be considered by the court.
C. To Be Present And Defend In Person And By Counsel At Every Stage Of The Proceeding
Presence of the Accused is required:
1. During arraignment (Sec. 1b, Rule 116)
2. Promulgation of judgment EXCEPT when the conviction is for a light offense, in which case, it may be pronounced in the presence of his counsel or representative.
3. when ordered by the court for purposes of identification
Not applicable in the SC and CA
- The law securing to an accused person the right to be present at every stage at the proceedings has no application to the proceedings before the CA and the SC nor to the entry and promulgation of the judgments. The defendant need not be present during the hearing of the appeal. (Sec. 9, Rule 124)
Aquino, Jr. v. Military Commission, 63 SCRA 546 (1975)
- The accused may waive his right to be present during the trial.
However, his presence may be compelled when he is to be identified.
Effects Of Waiver Of The Right To Appear By The Accused
1. waiver of the right to present evidence
2. prosecution can present evidence if the accused fails to appear
3. the court can decide without the evidence of the accused
Trial in Absencia
- It is important to state that the provision of the Constitution
authorizing the trial in absentia of the accused in case of his
non-appearance AFTER ARRAIGNMENT despite due notice simply means
that he thereby waives his right to meet the witnesses
face to face, among others.
Such waiver of a right of the accused does not mean a release
of the accused from his obligation under bond to appear in court
when so required. The accused may waive his right but not his
duty or obligation to the court.
REQUIREMENTS FOR TRIAL IN ABSENTIA
1. accused has been arraigned
2. he has been duly notified of the trial
3. his failure to appear is unjustified
Gimenez v. Nazareno, 160 SCRA 1 (1988)
- an escapee who has been duly tried in absentia waives his
right to present evidence on his own behalf and to confront
and cross-examine witnesses that have testified against him.
D. Right To Counsel
Importance: Without the aid of counsel, a person may be convicted,
not because he is guilty but because he does not know how to
establish his innocence.
The right covers the period beginning from the custodial investigation,
well into the rendition of the judgment and even on appeal.
(People v. Serzo, Jr. 274 SCRA 553) the right to counsel can be invoked
at any stage of the proceedings, even on appeal.
CUSTODIAL INVESTIGATION
- It is the questioning by law enforcement officers of a SUSPECT
taken into custody or otherwise deprived of his freedom of
action in a significant way. it includes the practice of issuing
an "invitation" to a person who is investigated in connection
with an offense he is suspected to have committed. (RA 7437)
People v. Morial, 363 SCRA 96 (2001)
- If during the investigation the assisting lawyer leaves, comes
and goes, the statement signed by the accused is still
inadmissible because the lawyer should assist his client from
the time the confessant answers the first question asked by the
investigating officer until the signing of the extrajudicial
The right to counsel and the right to remain silent do not cease even
after a criminal complaint/information has already been filed
against the accused AS LONG AS he is still in custody.
The duty of the court to appoint a counsel de oficio when the
accused has no legal counsel of choice and a desire to employ one is
MANDATORY only at the time of ARRAIGNMENT (sec. 6, Rule 116)
Difference Between The Right To Counsel During Custodial Investigation
and During The Trial
A. During trial – the right to counsel means EFFECTIVE counsel.
Counsel is here not to prevent the accused from a confessing but to
defend the accused.
B. Custodial Investigation – stricter requirement, it requires the
presence of competent and independent counsel who is preferably
the choice of the accused. Since a custodial investigation is
not done in public there is a danger that confessions can be
exacted against the will of the accused.
The right to counsel is NOT ABSOLUTE, it subject to being exercised
within a reasonable time and manner (Laranaga v. CA, 281 SCRA 254)
he cannot insist on one that he cannot afford, one who is not a
member of the bar and one who declines for a valid reason such as
conflict of interest. (People v. Servo, 274 SCRA 553)
- This is when the accused voluntarily submits himself to the
jurisdiction of the court and proceeds with his defense.
Jurisprudence provides that the defendant cannot raise the question
of his right to have an attorney the first time on appeal.
The accused may defend himself in person only if the court is
convinced that he can properly protect his rights even without the
assistance of counsel.
US v. Escalante, 36 Phil. 743 (1917)
- If the question is not raised in the trial court, the
prosecution may go to trial.
People v. Nang Kay, 88 Phil. 515 (1951)
- the question will not be considered in the appellate court for
the first time when the accused fails to raise it in the
lower court.
Delgado v. CA, 145 SCRA 357 (1986)
- The mistake of counsel will bind his client. The only exception
is when the counsel represents himself as a lawyer and is not
one because in that case the accused is denied of his right to
counsel and due process.
E. To Testify As A Witness In His Own Behalf
People v. Santiago, 46 Phil 734 (1922)
- A denial of the defendant's right to testify on his own behalf
would constitute an unjustifiable violation of his
constitutional right.
If the accused testifies, he may be cross-examined ONLY on matters
covered by his direct examination, unlike an ordinary witness who
can be cross-examined as to any matter stated in the direct
examination or connected therewith (Section 6, Rule 132). His failure
to testify will not be taken against him but his failure to present
evidence in his behalf shall be taken against him (US v. Bay, 97 SCRA 495).
The testimony of an accused who testifies on his own behalf but
refuses to be cross examined will not be given weight and will
have no probative value because the prosecution will not be able
to test its credibility.
F. Right Against Self-Incrimination
The scope of this right covers only testimonial compulsion only and
not the compulsion to produce real and physical evidence using the
body of the accused.
DNA TESTING is not covered in the right against self-incrimination
Rationale For Protecting The Right Against Self Incrimination:
1. humanitarian reasons, to prevent the state from using its
coercive powers.
2. practical reasons - the accused is more likely to commit perjury.
The accused in protected under this rule from questions that tend
to incriminate him, which means those that may subject him to
penal liability.
The right may be waived by the failure of the accused to invoke the
privilege at the proper time, that is AFTER the incriminating
question is asked and BEFORE his answer.
The privilege of the accused to be exempt from testifying as a
witness, involves a prohibition against testimonial compulsion
only and the production by the accused of incriminating documents
and articles demanded off him. (US v. Tan Teng, 23 Phil, 145)
EXCEPTIONS: immunity statutes such as:
1. RA 1379 (Forfeiture of illegally obtained wealth)
2. RA 749 – Bribery and Graft cases
Right Of The Accused Vs. Right Of An Ordinary Witness
- The ordinary witness may be compelled to take the witness
stand and claim the privilege as each and every incriminating
question is thrown at him while an accused may refuse to take
the witness stand and refuse to answer any and all questions.
The accused may also refuse to answer on his past criminality
only if he can still be prosecuted for it.
However, if the accused testifies in his own behalf, then he may
be cross-examined as any other witness. He may NOT on cross
examination refuse to answer any question on the ground that the
answer he will give or the evidence that he will produce would have
the tendency to incriminate him for the crime that he was charged.
But he MAY refuse to answer any question incriminating him for an
offense distinct from that for which he is charged.
Rights Of The Accused In The Matter Of Testifying Of Producing Evidence
Before the case:
2. Right to remain silent and to counsel
3. Right not to be subjected to force or violence or any other means
which vitiate free will
4. Right to have the evidence obtained in violation of these rights
After the case is filed in court:
1. Right to refuse to be a witness
2. Right to not have any prejudice whatsoever result to him by
such refusal
3. The right to testify on his own behalf subject to cross-examination
by the prosecution
4. While testifying the right to refuse a specific question which
ends to incriminate him for some other crime.
Use Immunity
- Witness' compelled testimony and the fruits thereof cannot be
used in subsequent prosecution of a crime against him
- Witness can still be prosecuted but the compelled testimony
cannot be used against him.
Transactional Immunity
- Witness immune from prosecution of a crime to which his compelled
testimony relates.
- Witness cannot be prosecuted at all
Effect of Refusal of Accused to Testify
General Rule:
- Silence should not prejudice the accused.
EXCEPTION: Unfavorable inference is drawn when:
1. the prosecution has already established a prima facie case,
the accused must present proof to overturn the evidence
2. the defense of the accused is an alibi and he does not testify,
the interference is that the alibi is not believable.
G. Right To Confront And Cross Examine Witnesses Against Him At Trial (Right Of Confrontation)
- It is the act of setting a witness face to face with the
accused so that the latter may make any objection he has to
the witness, and the witness may identify the accused, and
this must take place in the presence of the court having
jurisdiction to permit the privilege of cross examination.
The main purpose of this right to confrontation is to secure the
opportunity of cross examination and the second purpose is to enable
the judge to observe the demeanor of the witness.
By way of exception to this rule, it is provided that the court may
utilize as part of its evidence the testimony of a witness who is
deceased, out of or with due diligence cannot be found in the
Philippines, unavailable or otherwise unable to testify, given in
another proceeding, judicial or administrative, involving the same
parties and subject matter, the adverse party having had the
opportunity to cross-examine him. (Rule 130, Sec 47)
In any criminal proceeding, the defendant enjoys the right to have
compulsory process to secure the attendance of witnesses and the
production of evidence on his behalf.
WAIVER OF RIGHT TO CONFRONTATION
a. May be done expressly or impliedly.
b. It is implied when the accused waives his right to be present at
trial or when he was given the opportunity but fails to take
advantage of it.
H. Right To Compulsory Process
This is the right of the accused to have a subpoena and/or a
subpoena duces tecum issued in his behalf in order to compel the
attendance of witnesses and the production of other evidence.
If a witness refuses to testify when required is in contempt of court.
The court may order a witness to give bail or to be arrested.
I. Right To A Speedy, Impartial Public Trial
The right to a speedy trial is intended to avoid oppression and to
prevent delay by imposing on the courts and on the prosecution
an obligation to proceed with reasonable dispatch.
Facts Considered To Determine If Right To Speedy Trial Has Been Violated
1. length of the delay
2. reason for the delay
3. the accused's assertion or non assertion of the right
4. prejudice to the accused resulting from the delay.
Rules on Speedy Trial
- The limitation of this right is that the State must not
be deprived of its day in court and the right of the State and
the prosecution of due process must be respected.
There is NO violation of the right where the delay is immutable to the accused. (Solis v. Agloro, 64 SCRA 370)
The right to a speedy trial is violated when there are UNJUSTIFIED
postponements (People v. Declaro, 170 SCRA 143)
REMEDIES AVAILABLE TO THE ACCUSED WHEN HIS RIGHT
TO A SPEEDY TRIAL IS VIOLATED
1. He should ask for the trial of the case, not the dismissal.
2. Unreasonable delay of the trial of a criminal case as to make
the detention of the defendant illegal gives ground for habeas
corpus as a remedy for obtaining release as to avoid detention
for a reasonable period of time.
3. Accused would be entitled to relief in a mandamus proceeding to
compel the dismissal of the information.
4. ask for the trial of the case and then move to dismiss
(Gandicela v. Lutero, 88 Phil. 790)
Impartial Trial
- Due process requires a hearing before an impartial and
disinterested tribunal and that every litigant is entitled to
nothing less that the cold neutrality of an impartial judge.
(Mateo, Jr. v. VIllaluz, 50 SCRA 180)
"Like Caesar's wife, a judge must not be only pure but beyond
suspicion." (Palang v. Zosa, 58 SCRA 776)
Public Trial
- One held open or publicly; anyone interested in observing the
way the judge conducts his proceedings in a courtroom may do
so (Garcia v. Domingo, 52 SCRA 143) it is sufficient that
relatives and friends who want to watch the proceedings are
given the opportunity to witness the proceedings. It is done
in public to prevent abuses that may be committed by the court
and the accused is entitled to moral support from his friends
and relatives. If it is done in the judges chambers, it is
still valid because the public is not excluded.
(Garcia v. Domingo, 52 SCRA 143)
EXCLUSION OF THE PUBLIC IS VALID WHEN:
1. evidence to be produced is offensive to decency or public morals
2. upon motion of the accused (Section 21, Rule 119)
Rule on Trial by Publicity
- The right of the accused to a fair trial is NOT incompatible
to free press. Pervasive publicity is no per se as prejudicial
to the right to a fair trial. To warrant the finding of
prejudicial publicity, there must be allegations and proof
that judges have been unduly influenced, not simply that they
might be due to the barrage of publicity. (People v. Teehankee,
249 SCRA 54)
J. Right To Appeal On All Cases Allowed By Law And In The Manner Prescribed By Law
The right to appeal from a judgment of the conviction is
fundamentally of statutory origin. It is not a matter of absolute
right that is independent of constitutional or statutory provisions
allowing such appeal.
Waiver of Right to Appeal
- The right to appeal is personal to the accused and it may be
waved either expressly or by implication. HOWEVER, where the
death penalty is imposed, such right cannot be waived as the
review of the judgment by the SUPREME COURT is automatic and
mandatory (A.M. No. 00-5-03 SC)
Ozaeta v. CA, 179 SCRA 800 (1989)
- Anyone who seeks to exercise the right to appeal must comply
with the requirements of the rules. Otherwise the right to
appeal is lost.
People v. Ang Gioc, 74 Phil. 366 (1941)
- When the accused flees, after the case has bee submitted to
court for decision, he will be deemed to have waived his right
to appeal from the judgment rendered against him.
NOTE: such may no be reviewed by the CA.
THE SPEEDY TRIAL ACT OF 1998 (RA 8493)
DUTY OF THE COURT AFTER THE ARRAIGNMENT OF THE ACCUSED
- The court SHALL order a pre-trial conference to
1. plea bargaining
2. stipulation of facts
3. marking and identification of evidence
4. waiver of objections to admissibility of evidence
5. such other matters as will promote a fair and
expeditious trial
Time Limit for Trial in Criminal Cases Shall not exceed 180 days
from the first day of trial, however the rule is not absolute.
The EXCEPTIONS:
1. those governed by the Rules on Summary Procedure
2. where the penalty prescribed by law does NOT exceed 6 months
imprisonment or a fine of P1,000 or both
3. those authorized by the Chief Justice of the SC
Period of Arraignment of Accused
- Within 30 days from the filing of the information, or from the
date the accused appealed before the justice/judge/court in
which the charge is pending, whichever date last occurs.
When Shall Trial Commence After Arraignment
- Within 30 days from arraignment, HOWEVER, it may be extended
BUT only:
1. for the 180 days for the first 12 calendar month period
from the effectivity of the law
2. 120 days for the second 12 month period
3. 80 days for the third 12 month period | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,384 |
\section{Introduction}
Interesting phenomena, such as the Aharonov-Bohm effect and Berry
phases, can occur in physical systems with nontrivial topology in
real or parameter space. Topological quantum systems are now
attracting considerable interest because of their fundamental
importance in diverse areas ranging from quantum field theory to
semiconductor physics,\cite{RMP} with the most recent example being
the exploration of topological insulators.\cite{Kane_PRL05,Bernevig}
These topological physical systems may also have potential
applications because they are robust against local perturbations.
Specifically, a topologically protected quantum state degeneracy
cannot be lifted by any local interactions.\cite{RMP,Wen} It is
therefore natural to consider using topological phases for
applications requiring a high degree of quantum coherence.\cite{RMP}
For example, it has recently been pointed out that non-Abelian
anyons\cite{Leinaas,Wilczek,Wilczek_book} in a fractional quantum
Hall system can lead to topological quantum computing.\cite{Sarma05}
Anyons are neither bosons nor fermions, but obey anyonic braiding
statistics.\cite{Leinaas,Wilczek,Wilczek_book} Unfortunately, they
have not yet been convincingly observed experimentally in any
physical system.
Instead of only looking for naturally existing topological phases, one could also design artificial lattice structures that possess desired
topological phases. One example is the Kitaev honeycomb model,\cite{Kitaev} which requires that the spin (natural or artificial) at each node
of a honeycomb lattice interacts with its three nearest neighbors through three different interactions: $\sigma_{x}\sigma_{x}$, $\sigma_{y}
\sigma_{y}$, and $\sigma_{z}\sigma_{z}$. Depending on the bond parameters, this anisotropic spin model supports both Abelian and non-Abelian
anyons.~\cite{Kitaev} Its realization could potentially lead to experimental demonstration of anyons and implementation of topological quantum
computing. However, the requirement for anisotropic interactions is tremendously demanding and generally cannot be satisfied by natural spin
lattices.
Various artificial lattices may possess interesting topological phases. For instance, it has been proposed that a triangular Josephson junction
array may have a two-fold degenerate ground state that is topologically protected.\cite{Ioffe_Nat02,Albuquerque_PRB08} A recent proposal
suggests the use of capacitively coupled Josephson junction arrays to simulate a two-component fermion model that has topological
excitations.\cite{Xue_PRA09} There is also a suggestion that a Josephson junction array with properly designed interactions and topology can be
local-noise resistant.\cite{Gladchenko} With respect to the physical realization of the Kitaev model, there are proposals using neutral atoms in
optical lattices.\cite{Duan,Zoller,Sarma07} One similarity among all of these proposals, whether based on Josephson junction arrays or on
optical lattices, is that they all require extremely low temperatures and precise single-atom manipulations. The reason is that topologically
interesting properties are not generally contained in the symmetry of the system Hamiltonian. Instead they are only emergent properties at very
low temperatures.
Here we propose a quantum emulation of the Kitaev lattice using
superconducting quantum circuits (Ref.~\onlinecite{YSN_preprint}
gives a brief summary of this work). As for the topic of quantum
analog simulations, see Ref.~\onlinecite{Buluta} for a brief review.
In our superconducting network, a Josephson charge qubit is placed
at each node of a honeycomb lattice. These charge qubits behave
like artificial spins and are tunable via external
fields.\cite{YN05,Mak,Wendin} Each charge qubit interacts with its
three nearest neighbors through three different types of circuit
elements. One advantage of our proposal is that some circuit
elements involved and their functionalities at low energies have
already been demonstrated experimentally---for example, the
$\sigma_{z}\sigma_{z}$ and $\sigma_{x} \sigma_{x}$ couplings between
charge qubits have been studied in experiments.\cite{NEC,Yamamoto}
Here we show theoretically that they can indeed provide the needed
anisotropic interactions when included in a honeycomb lattice. We
then identify the ground states of this network in two different
parameter regimes and show that it can have both vortex and
bond-state excitations. We also describe how they can be generated
using spin-pair operations.
\begin{figure}
\includegraphics[width=3in,
bbllx=100,bblly=213,bburx=467,bbury=725] {cfig1.eps} \caption{(Color
online)(a)~Schematic diagram of the basic building block of a Kitaev
lattice, consisting of four superconducting charge qubits (labelled
1 to 4): (i)~Qubits 1 and 2 are inductively coupled via a mutual
inductance $M$; (ii)~qubits 1 and 3 are coupled via an $LC$
oscillator; and (iii)~qubits 1 and 4 are capacitively coupled via a
mutual capacitance $C_m$. Inset: The three types of inter-qubit
couplings are denoted as $x$-, and $y$- or $z$-type bonds. Here each
charge qubit consists of a Cooper-pair box (green dot) that is
linked to a superconducting ring via two identical Josephson
junctions (each with coupling energy $E_J$ and capacitance $C_J$),
to form a SQUID loop. Also, each qubit is controlled by both a
voltage $V_i$ (applied to the qubit via the gate capacitance $C_g$)
and a magnetic flux $\Phi_i$ (piercing the SQUID loop). (b) A
partial Kitaev lattice (honeycomb lattice) constructed by repeating
the building block in (a), where a charge qubit is placed at each
site. A plaquette is defined as a hexagon in the lattice. The
plaquette operator is defined as $W_p = \sigma_1^x \sigma_2^y
\sigma_3^z \sigma_4^x \sigma_5^y \sigma_6^z$ and is shown for a
given plaquette $p$.} \label{fig1}
\end{figure}
\section{Kitaev lattice based on superconducting quantum circuits}
At low energies, superconducting (SC) qubits can behave as
artificial spins. Among the varieties of SC qubits (charge, flux,
phase,\cite{YN05,Mak,Wendin} and other
hybrids~\cite{YTN06,YHAN07,transmon}), only charge qubits are known
to interact with each other in all the individual forms of
$\sigma_{x}\sigma_{x}$, $\sigma_{y}\sigma_{y}$, and
$\sigma_{z}\sigma_{z}$ (via a mutual inductance, an $LC$ oscillator,
and a capacitance, respectively).\cite{YTN,Schon,NEC,Bruder}
Therefore, to emulate a Kitaev honeycomb lattice, we propose to
build a two-dimensional SC circuit network based on SC charge
qubits. More specifically, on a honeycomb lattice a charge qubit is
placed on each node [Fig.~\ref{fig1}(b)], and one of the three
circuit elements is inserted along each bond of the lattice (denoted
as the $x$-, $y$-, or $z$-type bond). A building block of this
lattice is shown in Fig.~\ref{fig1}(a), which consists of four
charge qubits that are connected via an $x$-, a $y$-, and a $z$-type
bond. Each charge qubit is a Cooper-pair box connected to a
superconducting ring by two identical Josephson junctions to give it
tunability: Each qubit is controlled by both the magnetic flux
$\Phi_i$ piercing the SQUID loop and the voltage $V_i$ applied via
the gate capacitance $C_g$.
Naively, a circuit element should maintain its basic characteristics
when inserted in a larger network, at least in the linear regime.
However, as it has been shown in previous studies of hybrid
qubits,\cite{YTN06,YHAN07,transmon} a superconducting qubit based on
one particular variable (for example, charge) can acquire characters
of another (for example, flux) when additional circuit elements are
added to it. Therefore, here we first clarify whether the different
circuit elements in our honeycomb network maintain their basic
individual characteristics (particularly the forms and strengths of
the interactions) at low energies when lumped together.
We first write down the Lagrangian of the quantum circuits, choosing
the average phase drop $\varphi_i$ across the two Josephson
junctions of each charge qubit as the canonical coordinates. After
identifying the corresponding canonical momenta, we then derive
(this derivation is shown in the appendix) the total Hamiltonian of
the quantum circuits as
\begin{eqnarray}
H\!&\!=\!&\!\sum_i H_i + \sum_{x-{\rm link}}K_x(j,k) + \sum_{y-{\rm
link}}K_y(j,k) \nonumber \\
&&\!+\sum_{z-{\rm link}}K_z(j,k).
\label{hamiltonian}
\end{eqnarray}
Here the free Hamiltonian of the $i$th charge qubit is
\begin{equation}
H_i=E_c(n_i-n_{gi})^2 - E_{Ji}(\Phi_i)\cos\varphi_i,
\end{equation}
where $E_c = 2e^2 C_\Sigma/\Lambda$ is the charging energy of the
Cooper pair box, with the total capacitance $C_{\Sigma} =
2C_J+C_g+C_m$, and $\Lambda = C_{\Sigma}^2 -C_m^2$; $n_i =
-i\partial/\partial\varphi_i$ the number operator of the Cooper
pairs in the $i$th box (which is conjugate to $\varphi_i$); $n_{gi}
= C_g V_i/2e$ the reduced offset charge induced by the gate voltage
$V_i$; and $E_{Ji}(\Phi_i) = 2E_J \cos(\pi\Phi_i/\Phi_0)$ the
effective Josephson coupling energy of the $i$th charge qubit, with
$\Phi_0=h/2e$ the flux quantum.
The three nearest-neighbor couplings, shown as the $x$, $y$, and $z$ bonds in Fig.~1(a), are given by
\begin{eqnarray}
K_x(1,2)\!&\!=\!&\!M I_1 I_2, \nonumber\\
K_y(1,3)\!&\!=\!&\! -4 \xi E_{J1}(\Phi_1) E_{J3}(\Phi_3) \sin\varphi_1
\sin\varphi_3,\\
K_z(1,4)\!&\!=\!&\!E_m (n_1 - n_{g1}) (n_4 - n_{g4}),\nonumber
\end{eqnarray}
where
\begin{eqnarray}
&&\xi=L \left[\frac{\pi C_{\Sigma} (C_g + C_m)}{\Lambda\Phi_0}\right]^2, \nonumber\\
&&E_m =\frac{4e^2 C_m}{\Lambda}, \nonumber \\
&&I_i = -I_c \sin\left(\frac{\pi\Phi_i}{\Phi_0}\right) \cos\varphi_i \,.
\end{eqnarray}
Here $I_c = 2\pi E_J/\Phi_0$ is the critical current through the Josephson junctions of the charge qubits (we assume identical junctions for
simplicity), while $I_i$ is the circulating supercurrent in the SQUID loop of the $i$th charge qubit. Note that the coupling strength between
nodes 1 and 3 (along a $y$-link), $\xi \propto (C_g + C_m)^2$, is affected by the mutual capacitance $C_m$ that connects qubit 1 (3) with its
nearest-neighbor along a $z$-link. Compared to the case of two qubits coupled by an $LC$ oscillator,\cite{Schon} where $\xi \propto C_g^2$, the
capacitive inter-node coupling along the $z$-link in the present circuit greatly increases the inter-node coupling along the $y$-link because
usually $C_m \gg C_g$. This is an important and positive consequence when multiple circuit elements are introduced to create different
inter-node interactions.
At low temperatures, only the lowest-energy states of a superconducting circuit element are involved in the system dynamics, which is quantum
mechanical. For the particular case of a charge qubit, where $E_c \gg E_J$, the lowest-energy eigenstates are mixtures of having zero and one
Cooper pair in the box, when the gate voltage $V_i$ is near the optimal point $e/C_g$ (i.e., $n_{gi}\sim \frac{1}{2}$). Defining $|0\rangle_i$
and $|1\rangle_i$ as the two charge states having zero and one extra Cooper pair in the box, we now have a two-level system as a quantum bit, or
qubit. In the spin-$\frac{1}{2}$ representation based on these charge states $|0\rangle_i\equiv|\!\!\uparrow\rangle_i$ and $|1\rangle_i \equiv
|\!\!\downarrow\rangle_i$ ($i$ is the index of the nodes), the system variables can be expressed as
\begin{eqnarray}
&&n_i = \frac{1}{2}(1 -\sigma_i^z),\nonumber\\
&&\cos\varphi_i=\frac{1}{2}\sigma_i^x,\\
&&\sin\varphi_i =-\frac{1}{2}\sigma_i^y.\nonumber
\end{eqnarray}
Here we consider the simple case with $n_{gi} = n_g$ (i.e., all gate voltages on the different nodes are identical: $V_i=V_g$) and $\Phi_i =
\Phi_e$ for all qubits. The low-energy Hamiltonian of the system is then reduced to
\begin{eqnarray}
H\!&\!=\!&\!J_x\sum_{x-{\rm link}}\sigma_j^x\sigma_k^x + J_y\sum_{y-{\rm link}}\sigma_j^y\sigma_k^y +J_z\sum_{z-{\rm link}}
\sigma_j^z\sigma_k^z \nonumber \\
&&\!+\sum_i(h_z\sigma_i^z+h_x\sigma_i^x),
\label{model}
\end{eqnarray}
where
\begin{eqnarray}
J_x\!&\!=\!&\!\frac{1}{4}MI_c^2\sin^2\left(\frac{\pi
\Phi_e}{\Phi_0}\right)\geq 0,\nonumber\\
J_y\!&\!=\!&\!-\xi[E_{J}(\Phi_e)]^2\leq 0, \nonumber \\
J_z\!&\!=\!&\!\frac{1}{4}E_m>0,\\
h_z\!&\!=\!&\!\left(E_c+\frac{1}{2}E_m\right)\left(n_{g}-\frac{1}{2}\right), \nonumber \\
h_x\!&\!=\!&\!-\frac{1}{2}E_{J}(\Phi_e),\nonumber
\end{eqnarray}
with $E_{J}(\Phi_e) = 2E_J\cos(\pi\Phi_e/\Phi_0)$. The reduced
Hamiltonian (\ref{model}) is the Kitaev model with an effective
magnetic field with $z$- and $x$-components. Here $h_x$ and $h_z$
play the role of a ``magnetic'' field. Since $J_y\propto h_x^2$, to
maintain finite inter-qubit couplings, $h_x$ cannot vanish.
Therefore our Hamiltonian represents a Kitaev model in an
always-finite magnetic field, although the field direction can be
adjusted. This Hamiltonian has an extremely complex quantum phase
diagram because of all the (experimentally) adjustable parameters.
Here we are particularly interested in whether it has
topologically-interesting phases and when such topological
properties might emerge.
\section{Vortex and bond-state excitations}
Below we focus on two particular parameter regimes of the
finite-field Kitaev model of (\ref{model}), under the general
condition that the $z$-bond interaction dominates over the other
interactions. In particular, when $J_z \gg J_x, |J_y| \gg |h_z|,
|h_x|$, we identify a vortex state excitation. This case is
described in Section A below. When $h_z = 0$ but $h_x$ is of the
same order as $J_x$ and $J_y$, we identify a new excitation that we
call the bond state. We describe this case in Sec.~III.B. The
vortex state is a known topological excitation in the zero-field
Kitaev model, while the bond state is specific to the finite-field
Kitaev model.
\subsection{Kitaev lattice with dominant $z$-bonds in a weak ``magnetic'' field}
We first consider the case when
\begin{equation}
J_z \gg J_x, |J_y| \gg |h_z|, |h_x|,
\end{equation}
and treat $V=\sum_i(h_z\sigma_i^z+h_x\sigma_i^x)$ as the perturbation. Using perturbation theory in the Green function formalism,~\cite{Kitaev}
one can construct an effective Hamiltonian for the lattice:
\begin{equation}
H' = -\frac{2h_z^2}{\Delta\varepsilon_z} \sum_{z-{\rm link}} \sigma_j^z \sigma_k^z - \frac{2h_x^2}{\Delta\varepsilon_x} \sum_{x-{\rm link}}
\sigma_j^x \sigma_k^x,
\end{equation}
where $\Delta\varepsilon_{z(x)}$ is the excitation energy of the
state $\sigma_i^{z(x)}|g_0\rangle$, i.e., the energy difference
between states $\sigma_i^{z(x)}|g_0\rangle$ and $|g_0\rangle$. Here
$|g_0\rangle$ is the ground state of the unperturbed Hamiltonian,
i.e., Hamiltonian (\ref{model}) with the perturbation term $V$
excluded. Note that the effective Hamiltonian $H'$ only contains
contributions from the second-order terms because both the first-
and third-order terms vanish. Including the zeroth-order term
(unperturbed Hamiltonian), the total Hamiltonian of the system can
be written as
\begin{equation}
H = J'_x \sum_{x-{\rm link}} \sigma_j^x \sigma_k^x + J_y
\sum_{y-{\rm link}} \sigma_j^y \sigma_k^y + J'_z \sum_{z-{\rm link}}
\sigma_j^z \sigma_k^z \,,
\end{equation}
where the effective $z$- and $x$-couplings are
\begin{eqnarray}
J'_z \!&\!=\!&\! J_z - \frac{2h_z^2}{\Delta \varepsilon_z}, \nonumber\\
J'_x \!&\!=\!&\! J_x - \frac{2h_x^2}{\Delta \varepsilon_x}\,.
\end{eqnarray}
\begin{figure}
\includegraphics[width=2.6in,
bbllx=145,bblly=445,bburx=418,bbury=745] {cfig2.eps} \caption{(Color online) (a)~Two types of vortex excitations $w_1$ and $w_2$. A pair of
vortices are generated along the horizontal direction for $w_1$ (vertical direction for $w_2$) by the spin-pair operator
$\tilde{\sigma}^z\equiv\sigma^zI$ ($\tilde{\sigma}^y\equiv\sigma^y\sigma^x$) acting on a $z$-link. (b)~Two bond-state excitations $b_1$ and
$b_2$, which are also generated by the spin-pair operators $\tilde{\sigma}^z$ and $\tilde{\sigma}^y$ on a $z$-link. } \label{fig2}
\end{figure}
Below we focus on the Abelian excitations. When $J'_z \gg J'_x,
|J_y|$, the dominant part of the Hamiltonian $H$ is that along the
vertical links,
\begin{equation}
H_0 = J'_z\sum_{z-{\rm link}} \sigma_j^z\sigma_k^z.
\end{equation}
Under $H_0$, the two spins along each $z$-link tend to be aligned opposite to each other ($|\!\!\uparrow \downarrow \rangle$ or $|\!\!\downarrow
\uparrow \rangle$) in order to lower their energies. Indeed, the highly degenerate ground state $|g\rangle$ of $H_0$ is an arbitrary vector in
the Hilbert subspace spanned by $\bigotimes_{i=1}^N |\sigma \bar{\sigma} \rangle_i$, where $N$ denotes the total number of $z$-links and $\sigma
= \;\uparrow, \downarrow$. Within the ground-state subspace of $H_0$ and up to fourth order,\cite{Kitaev} the effective Hamiltonian of the
Kitaev lattice takes the form
\begin{equation}
H_{\rm eff}=-J_{\rm eff}\sum_p W_p, \label{vortex}
\end{equation}
where
\begin{eqnarray}
J_{\rm eff} & = & \frac{{J'}_x^2 J_y^2}{16{J'}_z^3}\approx\frac{J_x^2 J_y^2}{16J_z^3},
\nonumber \\
W_p & = & \sigma_1^x \sigma_2^y \sigma_3^z \sigma_4^x \sigma_5^y
\sigma_6^z \,.
\end{eqnarray}
Here $W_p$ is the plaquette operator for a given plaquette $p$ [see Fig.~\ref{fig1}(b)]. The operator $W_p$ for any plaquette $p$ commutes with
the unperturbed Hamiltonian $H_0$:
\begin{equation}
[H_0, W_p] = 0;
\end{equation}
so that $[H_0, H_{\rm eff}] = 0$ as well, and the ground states $|g\rangle_w$ of $H_{\rm eff}$ form a {\it subset} of the degenerate ground
states ${|g\rangle}$ of $H_0$. It is straightforward to show that $W_p^2 |g\rangle = |g\rangle$, or $W_p |g\rangle = \pm |g\rangle$. Since
$J_{\rm eff} > 0$, to minimize the energy of $|g\rangle_w$, we need
\begin{equation}
W_p|g\rangle_w=|g\rangle_w.
\end{equation}
In other words, the eigenvalues of the $W_p$ operators in the ground state $|g\rangle_w$ are $w_p = 1$ for all plaquettes $p$.
When some plaquettes undergo transformations that lead to $w_p=1\rightarrow-1$, the system gets into an excited state. The lowest-energy
excitation corresponds to the generation of a pair of vortices when $w_p=1\rightarrow-1$ for two neighboring plaquettes. In this excitation
process, each of the two neighboring plaquettes acquires a phase $\pi$, which is equivalent to the addition of a flux quantum $\Phi_0$ through
each plaquette. As shown in Fig.~2(a), such an excitation can be generated by applying either of the following two spin-pair operators on the
ground state $|g\rangle_w$:
\begin{equation}
|\widetilde{Z}_i\rangle = \tilde{\sigma}_i^z|g\rangle_w,~~~
|\widetilde{Y}_i\rangle = \tilde{\sigma}_i^y|g\rangle_w,
\end{equation}
with
\begin{equation}
\tilde{\sigma}_i^z\equiv\sigma_{i}^zI_{i},~~~
\tilde{\sigma}_i^y\equiv\sigma_{i}^y\sigma_{i}^x.
\end{equation}
Here the two operators $\sigma^z_i$ ($\sigma^y_i$) and $I_i$ ($\sigma^x_i$) act on the ground state $|g\rangle_w$ at the bottom and top sites of
the $i$th $z$-link, respectively. This pair of vortices, generated by either $\tilde{\sigma}_i^z$ or $\tilde{\sigma}_i^y$, are topological
states with an excitation energy of
\begin{equation}
\Delta\varepsilon=4J_{\rm eff}
\end{equation}
above the ground state. As shown in Refs.~\onlinecite{Kitaev} and \onlinecite{Sarma07}, these excitations exhibit the braiding statistics of
Abelian anyons. The ratio between this excitation gap for the anyons and $J_z$ is
\begin{equation}
\frac{\Delta\varepsilon}{J_z} \sim \left(\frac{J_x J_y}{J_z^2}\right)^2 \ll 1.
\end{equation}
For example, if $J_z \sim 10$GHz and both $J_x$ and $J_y$ are one tenth of $J_z$, this gap would be about 1 MHz, corresponding to a temperature
of 0.1 mK. This small gap requires an extremely low experimental temperature for suppressing the thermal activation of the ground state to the
vortex states. Note that a different perturbative approach~\cite{Vidal} shows that in the parameter region $J'_z \geq J'_x, |J_y|$, the
spin-pair operators $\tilde{\sigma}^z_i$ and $\tilde{\sigma}^y_i$ generally create both vortex and fermionic excitations. However, in the limit
of $J'_z \gg J'_x, |J_y|$, the dominant excitations are vortex states,~\cite{Vidal} which is consistent with the conclusion drawn above.
\subsection{Kitaev lattice with dominant $z$-bonds in a uniform ``magnetic'' field along the $x$-direction}
If we stay in the regime where the $z$-bond couplings are dominant
($J_z \gg J_x, |J_y|$), but place each charge qubit at the optimal
point where $n_g=\frac{1}{2}$, so that $h_z=0$, a different quantum
phase arises when $|h_x|$ is comparable to $J_x, |J_y|$. In other
words, we now consider the regime
\begin{equation}
J_z \gg J_x, |J_y|, |h_x|, \ \ {\rm and} \ h_z = 0 \,.
\end{equation}
Here the zeroth-order Hamiltonian is again the coupling along the $z$-bonds: $H_0 = J_z \sum_{z-{\rm link}} \sigma_j^z \sigma_k^z$ (notice that
here the coupling strength is $J_z$, not $J'_z$), with the same highly degenerate ground state $|g\rangle$ as discussed in the previous
subsection. To clarify the low-energy excitation spectrum in this regime, we again use perturbation theory in the Green's function formalism to
remove the linear terms and derive an effective Hamiltonian in the ground state sub-Hilbert space of $H_0$. Up to second-order, the effective
Hamiltonian takes the form
\begin{equation}
H_{\rm eff}^{(z)} = -K_{\rm eff} \sum_{z-{\rm link}} \sigma_j^x \sigma_k^x,
\label{bond}
\end{equation}
where
\begin{equation}
K_{\rm eff} = \frac{h_x^2}{J_z}.
\end{equation}
The spin pair operator $P_z=\sigma_j^x\sigma_k^x$ at a $z$-bond
(again the two Pauli operators act on the bottom and top nodes of
the particular $z$-bond) commutes with the zeroth-order Hamiltonian
$[P_z, H_0] = 0$ (although it anti-commutes with the four plaquette
operators $W_p$ connected to this $z$-bond). Similar to $W_p$, the
pair operator $P_z$ also has two eigenvalues $p_z = \pm 1$. Thus the
ground state $|g\rangle_b$ of $H_{\rm eff}^{(z)}$ should satisfy
$p_z=1$ for all the $z$-bonds in the system. In other words,
\begin{equation}
P_z|g\rangle_b=|g\rangle_b,
\end{equation}
for all $z$-bonds. Since no two $z$-bonds share a node in the honeycomb lattice, and the lattice is completely covered by all the $z$-bonds, we
can solve the eigenstates of $P_z$ of each $z$-bond and obtain the ground state of $H_{\rm eff}^{(z)}$ as
\begin{equation}
|g\rangle_b = \frac{1}{2^{N/2}} \bigotimes_{i=1}^N(|\!\!\uparrow\downarrow\rangle_i+ |\!\!\downarrow\uparrow\rangle_i).
\end{equation}
This is a nondegenerate ground state, which forms a simple subset of the highly degenerate ground states ${|g\rangle}$ of $H_0$. It is
maximally entangled within each $z$-bond, but not entangled at all between different $z$-bonds. In other words, the two-spin correlation
function decays to identically zero beyond a $z$-bond. The lattice is now an ensemble of maximally entangled ``spin'' pairs that are completely
independent from each other. This ground state is reminiscent of (and simpler than) the dimerized valence bond solid state discussed in the
context of spin Hamiltonians.\cite{Affleck_87,Majumdar_69} There valence bond states refer to a singlet $|\uparrow\downarrow
-\downarrow\uparrow\rangle$ for the electron spins, which is dictated by the Coulomb interaction and Pauli principle between electrons.
When the pair operators $\tilde{\sigma}_i^z$ and $\tilde{\sigma}_i^y$ are separately applied to the ground state at the $i$th $z$-bond [see
Fig.~2(b)], the excited states
\begin{equation}
|\widetilde{Z}_i\rangle=\tilde{\sigma}_i^z|g\rangle_b,~~~
|\widetilde{Y}_i\rangle=\tilde{\sigma}_i^y|g\rangle_b
\end{equation}
are called a bond state---while the pair operators are different,
the states they generate are only different by an overall phase
because $|g\rangle_b$ is a factored state for all $z$-bonds. A bond
state at the $i$th $z$-bond corresponds to the change of
$p_z=1\rightarrow -1$ at that particular bond. It is $2 K_{\rm
eff}$ above the ground state in energy. Notice that a bond state is
an excitation that is completely localized to a particular $z$-bond.
Furthermore, bond states are generated by the same pair operators
that generate the vortex excitations, although the ground states of
the system are different in these two cases. In contrast to
$|g\rangle_w$, the ground state $|g\rangle_b$ is {\it
nondegenerate}, and the bond state excitations are very different
from the vortex states. This transition from vortex excitations to
bond states occurs when we vary the parameters of the system (i.e.,
tuning $n_g$ to $1/2$ and reduce $\Phi_e$ from close to $\Phi_0/2$
so that $h_x$ increases to the same magnitude as $J_x$ and/or
$J_y$), during which the topological property of the system changes.
\begin{figure}
\includegraphics[width=3in,
bbllx=136,bblly=217,bburx=494,bbury=742]{cfig3.eps} \caption{(Color
online) Schematic diagram of the procedures for braiding
excitations. (a)~The operations $U_h$ and $U_v$ for creating
excitations, which are achieved by successively applying spin-pair
operators at $z$-bonds along the horizontal ($P_h$) and vertical
($P_v$) paths. Here the paths $P_v$ and $P_h$ intersect at a
$z$-bond. (b)~A combined operation $U_h^{-1}U_v^{-1}U_hU_v$ for
both, braiding the excitations created in (a), and fusing them to
the vacuum. (c)~The operations $U_h$ and $U_v$ for creating
excitations, which are also achieved by successively applying
spin-pair operators at $z$-bonds along $P_h$ and $P_v$, but the
paths $P_v$ and $P_h$ do not intersect at a $z$-bond.} \label{fig3}
\end{figure}
\section{The braiding of excitations}
A vortex looping around another vortex can produce either a sign
change or {\it no} sign change to the wave function. The first case
is denoted as an $e$-type vortex looping around an $m$-type vortex,
and the second case corresponds to an $e$-type vortex looping around
an $e$-type vortex (see, e.g., Ref.~\onlinecite{Sarma07} for a more
detailed discussion). This indicates anyonic statistics between the
$e$ and $m$ vortex states. Therefore, braiding, which refers to
moving one quasi-particle around another, is an important tool to
determine the statistics of the quasi-particles (in the present case
the vortices). Here we show an alternative procedure for braiding an
excitation with another, which can be applied to both vortex and
bond states.
Let us consider two particular evolutions for the system. The first evolution $U_v$ contains spin-pair operations $\tilde{\sigma}_i^y =
\sigma_{i}^y \sigma_{i}^x$ applied to the ground state $|\tilde{g}\rangle$ at three successive $z$-bonds along the vertical path $P_v$, as shown
in Fig.~\ref{fig3}(a). Here $|\tilde{g}\rangle \equiv |g\rangle_w$ for the vortex case and $|\tilde{g}\rangle \equiv |g\rangle_b$ for the
bond-state case. The second evolution $U_h$ contains spin-pair operations $\tilde{\sigma}_i^z=\sigma_{i}^zI_{i}$ applied at four successive
$z$-bonds along the horizontal path $P_h$ as shown in Fig.~\ref{fig3}(a). After these two operations in series, the state of the system is $U_h
U_v |\tilde{g}\rangle$, where
\begin{equation}
U_h = \tilde{\sigma}_4^z\tilde{\sigma}_3^z\tilde{\sigma}_2^z\tilde{\sigma}_1^z, ~~~U_v=\tilde{\sigma}_3^y\tilde{\sigma}_2^y\tilde{\sigma}_1^y
\,.
\label{operation}
\end{equation}
Now we turn the evolutions backward by applying $U_v^{-1}$ and $U_h^{-1}$ to the system successively, so as to fuse~\cite{RMP,Kitaev} the
excitations to the vacuum (i.e., the ground state) [see Fig.~\ref{fig3}(b)]. The final state of the system is now
\begin{equation}
|\Psi_f\rangle=U_h^{-1}U_v^{-1}U_hU_v|\tilde{g}\rangle.
\label{braid}
\end{equation}
When the paths $P_v$ and $P_h$ intersect at a $z$-bond, such as in the example given in Fig.~\ref{fig3}(a), where $\tilde{\sigma}^y_2$ and
$\tilde{\sigma}^z_2$ anti-commute, $U_h$ and $U_v$ anti-commute as well: $U_hU_v=-U_vU_h$. The final state thus becomes
\begin{equation}
|\Psi_f\rangle=-|\tilde{g}\rangle \,.
\end{equation}
In other words, a phase flip $e^{i\pi}$ resulted from the
evolutions. For vortex excitations, this is equivalent to the case
of an $e$-type vortex looping around an $m$-type vortex, as shown in
Ref.~\onlinecite{Sarma07}. In contrast, when similar operations are
applied but the paths $P_v$ and $P_h$ do not intersect at a $z$-bond
[see Fig.~\ref{fig3}(c), for example], $U_h U_v = U_v U_h$, so that
\begin{equation}
|\Psi_f\rangle=|\tilde{g}\rangle,
\end{equation}
yielding no phase flip in the final state as compared to the initial state. For vortex excitations, this is equivalent to the case of an
$e$-type vortex looping around another $e$-type vortex.
The braiding of excitations, i.e. whether there is or there is no
phase flip, can be revealed by means of Ramsey-type
interference.~\cite{Sarma07,Pachos} To achieve this, one can keep
the same $U_v$ as above, but use
\begin{equation}
U_h=(\tilde{\sigma}^z_4)^{\frac{1}{2}}(\tilde{\sigma}^z_3)^{\frac{1}{2}}
(\tilde{\sigma}^z_2)^{\frac{1}{2}}(\tilde{\sigma}^z_1)^{\frac{1}{2}},
\end{equation}
where
\begin{equation}
(\tilde{\sigma}^z_i)^{\frac{1}{2}}\equiv(\sigma^z_i)^{\frac{1}{2}}I_i \,,
\end{equation}
i.e., each $\sigma^z_i$ is replaced by half of the rotation. In the braiding case shown in Fig.~\ref{fig3}(a),
\begin{equation}
\tilde{\sigma}^y_2(\tilde{\sigma}^z_2)^{\frac{1}{2}}
=i(\tilde{\sigma}^z_2)^{-\frac{1}{2}}\tilde{\sigma}^y_2
\end{equation}
at the crossing point of paths $P_h$ and $P_v$. Thus,
\begin{eqnarray}
|\Psi_f\rangle\!&\!=\!&\!U_h^{-1}U_v^{-1}U_hU_v|\tilde{g}\rangle
=(\tilde{\sigma}^z_2)^{-\frac{1}{2}}[i(\tilde{\sigma}^z_2)^{-\frac{1}{2}}]|\tilde{g}\rangle \nonumber\\
\!&\!=\!&\!i(\tilde{\sigma}^z_2)^{-1}|\tilde{g}\rangle=i\tilde{\sigma}^z_2|\tilde{g}\rangle =i|\widetilde{Z}_2\rangle,
\end{eqnarray}
similar to the case with an $e$ vortex braiding with a superposition state of an $m$ vortex and the vacuum.~\cite{Sarma07} However, in the case
without braiding [see Fig.~\ref{fig3}(c)],
\begin{equation}
|\Psi_f\rangle=U_h^{-1}U_v^{-1}U_hU_v|\tilde{g}\rangle=|\tilde{g}\rangle.
\end{equation}
Therefore, the braiding of excitations can be distinguished by verifying if an excited state $|\widetilde{Z}_2\rangle$ occurs at the crossing
point of paths $P_h$ and $P_v$.
While the vortex state described by Eq.~(\ref{vortex}) and the bond state described by Eq.~(\ref{bond}) are very different excitations, they
have similar braiding properties. In the braiding procedure shown above, the system is initially in the vacuum (either $|g\rangle_w$ or
$|g\rangle_b$); after the braiding operations in Eq.~(\ref{braid}), the system is fused to the vacuum again, but with a sign change to the
ground-state wave function no matter which ground state the system starts with. In order to distinguish the difference between the vortex and
bond-state excitations, one needs to focus on the intermediate steps of the braiding operations. Take $U_v$ in Eq. (\ref{operation}) as an
example. When it is applied to $|g\rangle_w$, the spin-pair operator $\tilde{\sigma}_1^y$ in it first creates a pair of $e$ vortices, and then
the other spin-pair operations $\tilde{\sigma}_2^y$ and $\tilde{\sigma}_3^y$ successively move one vortex downward along the vertical path
$P_v$. The final state $U_v |g\rangle_w$ is also a pair of vortices, but the two vortices are separated by three $z$-bonds in the vertical
direction [see Fig.~\ref{fig3}(a)]. Importantly, this pair of vortices $U_v|g\rangle_w$ is {\it degenerate} with the pair of vortices
$\tilde{\sigma}_1^y|g\rangle_w$. However, in sharp contrast to the vortex case, when $U_v$ in Eq. (\ref{operation}) is applied to
$|g\rangle_b$, each of the spin-pair operations $\tilde{\sigma}_i^y$, $i=1,2,3$ creates a bond state and the final state $U_v |g\rangle_b$ is
{\it nondegenerate} with the bond state $\tilde{\sigma}_1^y|g\rangle_b$.
\section{Implementation of quantum rotations}
As indicated in previous sections, single-qubit rotations are needed to create vortex and bond-state excitations, and to perform braiding
operations. Below we show that these quantum rotations of individual qubits in the honeycomb lattice can be achieved via electrical and
magnetic controls. The key is to reduce the coupling between a specific qubit and its neighboring qubits to such a degree that its single-qubit
dynamics dominates for a period of time.
To generate a $\sigma_z$ rotation at a particular charge qubit, we consider the following approach by controlling both the magnetic flux through
SQUID loops and the local electric field. Specifically, when the magnetic flux in the SQUID loop of each charge qubit is set to
$\Phi_e=\Phi_0/2$, $h_x = 0$ and $J_y = 0$, so that the honeycomb lattice is now decoupled into a series of one-dimensional chains. For a
charge qubit $E_c \gg E_J(\Phi_e)$, thus $E_c \gg J_x$. We further assume that $E_c \gg E_m$, so that $E_c\gg J_z$ as well. One can now shift
the gate voltage for a period of time $\tau$ at the $i$th lattice point far away from the usual working point $n_g \sim \frac{1}{2}$ of the
Kitaev lattice, so that the corresponding single-qubit energy $\delta h_z \sim E_c$ (instead of $\sim E_J$) is much larger than both $J_z$ and
$J_x$. Such a parameter regime should be reasonably easy to achieve for a charge qubit. This operation of shifting $n_g$ should yield a local
$z$-type rotation on the $i$th qubit:
\begin{equation}
R_i^z(\theta)=\exp[-i(\delta h_z\tau/\hbar)\sigma_i^z]\equiv\exp(-i\theta\sigma_i^z/2),
\end{equation}
where
\begin{equation}
\delta h_z=h_z(n_{gi})-h_z(n_g).
\end{equation}
When $\theta \equiv 2\delta h_z\tau/\hbar=\pi$ (where $n_{gi}>n_g$), $R_i^z(\pi)=-i\sigma_i^z$, so the $\sigma_i^z$ operation on the $i$th qubit
is given by
\begin{equation}
\sigma_i^z=e^{i\pi/2}R_i^z(\pi),
\end{equation}
while half of the rotation is
\begin{equation}
(\sigma_i^z)^{\frac{1}{2}}=e^{i\pi/4}R_i^z(\pi/2).
\end{equation}
The corresponding inverse rotations can be achieved by shifting the gate voltage to $n_{gi}<n_g$.
A $\sigma_x$ rotation of a particular charge qubit can be generated by a similar approach. Specifically, when $n_g = \frac{1}{2}$ and
$\Phi_e=0$, one has $h_z=0$, $h_x=-E_J$, and $J_x=0$. Again the honeycomb lattice is separated into a series of one-dimensional chains. Here
we assume that $E_J \gg |J_y|,J_z$, achievable in this charge-qubit system, which allows us to perform a single-qubit rotation driven by $E_J$.
When $n_g = \frac{1}{2}$, for a time $\tau$ we switch off the flux in the SQUID loop of the $i$th qubit (the working point of this Kitaev
lattice is usually at $0<\Phi_e<\Phi_0/2$), producing a local $x$-type rotation on the $i$th qubit:
\begin{equation}
R_i^x(\theta)=\exp[i(\delta E_J\tau/\hbar)\sigma_i^x] \equiv \exp(i\theta\sigma_i^x/2),
\end{equation}
where
\begin{equation}
\delta E_J=E_J-\frac{1}{2}E_J(\Phi_e).
\end{equation}
The $\sigma_i^x$ rotation on the $i$th qubit is
\begin{equation}
\sigma_i^x = e^{-i\pi/2}R_i^x(\pi),
\end{equation}
where $2\delta E_J\tau/\hbar=\pi$. Note that when the flux in the SQUID loop of the $i$th qubit is switched off to produce a local
$x$-rotation, the flux in the SQUID loop of the nearest-neighbor qubit that is connected to the $i$th qubit via an $LC$ oscillator should be
simultaneously shifted to a value around $\Phi_0/2$, so as to keep $|J_y|$ between these two qubits much smaller than $E_J$.
With both $\sigma_i^z$ and $\sigma_i^x$ rotations available for the $i$th qubit, the $\sigma_i^y$ rotation is given by
\begin{equation}
\sigma_i^y=e^{-i\pi/2}\sigma_i^z\sigma_i^x.
\end{equation}
Therefore, one can construct all the wanted operations $\tilde{\sigma}_i^z$ and $\tilde{\sigma}_i^z$ for generating both vortex and bond-state
excitations by using the single-qubit rotations $\sigma_i^z$ and $\sigma_i^x$.
In order to obtain accurate $z$- and $x$-type single-qubit rotations, we assume that $E_c$ and $E_J$ are much larger than the inter-qubit
coupling. Actually this somewhat stringent condition can be loosened for realistic systems. As shown in Ref.~\onlinecite{Wei}, accurate
effective single-qubit rotations can still be achieved using techniques from nuclear magnetic resonance when the inter-qubit coupling is small
compared to single-qubit parameters (instead of much smaller than $E_c$ and $E_J$).
\section{Discussion and conclusion}
In this paper, our main objective is to construct an experimentally
feasible proposal to emulate the Kitaev model on a network made of
superconducting nanocircuits. To focus on the topological
properties of the system, we choose the limit of identical qubits
and identical coupling strength. Furthermore, we fix the mutual
inductances and the capacitances of the various circuit elements
involved. There are basically two tunable parameters: the gate
voltage on the Cooper pair boxes ($n_g$) for each charge qubit, and
the magnetic flux $\Phi_e$ through the SQUID loops connected to the
Cooper pair boxes. Within the regime where $z$-bonds dominate in
interaction energy scale ($J_z$ much larger than all other
couplings, including $J_x$, $J_y$, $h_x$, and $h_z$), we have
explored two limiting cases: one with weak effective magnetic fields
($|h_x|$, $|h_z|$ $\ll$ $J_x$, $|J_y|$), the other with the
effective field only along the $x$-direction. We have identified
some properties of the relevant ground states, and the low-energy
excitations, the vortex and bond states. However, much more study
is needed to completely clarify the energy spectrum, the phase
diagram, and the dynamics of this superconducting network.
One observation we have made is that the vortex excitations and bond-state excitations can be generated using the same spin-pair operations,
starting from different ground states ($|g\rangle_w$ and $|g\rangle_b$) that depend on the system parameters. We have also shown that while
$|g\rangle_w$ is highly entangled, $|g\rangle_b$ is only entangled locally but not globally. This quantum phase transition requires more
extensive studies to identify the critical point and related critical phenomena, such as how system entanglement changes near the transition
point, and most importantly how its topological properties change. It would also be worthwhile to investigate the system spectrum (from vortex
excitation to bond state excitation) and dynamics during this transition, similar to our study of quantum phase transitions between Abelian and
non-Abelian phases of the Kitaev model.\cite{Shi_PRB09} While such studies are generally numerically intensive, it would help reveal the exotic
topological properties of this many-body model.
With the elementary building blocks given in Fig.~\ref{fig1}(a), one can construct Kitaev spin models on {\it other} types of lattices as well
(see, e.g., Refs.~\onlinecite{Yao} and \onlinecite{Sun}). In particular, it has been shown that in the absence of a magnetic field, the Kitaev
model on a decorated honeycomb lattice~\cite{Yao} can support gapped non-Abelian anyons. The quantum analog simulation of Kitaev models on
different lattices using superconducting circuits should shed light on the novel properties of these topological systems.
There are two important open issues in the study of building a
superconducting qubit network to emulate a spin lattice. One is the
role played by the decoherence of individual qubits, and the other
is the measurement of correlated states on a qubit network. It is
well known that charge qubits suffer from fast decoherence. However,
it is not clear how decoherence would affect the topological
excitations. Indeed, the faster decoherence of charge qubits may
allow the system to reach its ground state faster. Furthermore,
topological excitations are supposed to be robust against local
fluctuations, so that decoherence in individual nodes may not easily
destroy excitations such as the vortex state. Quantum measurement
is another open issue in the study of collective states, whether
ground states or low-energy excitations, of a qubit lattice. While
single-qubit measurement of superconducting qubits can now be done
with quite high fidelity,\cite{Holfheinz_Nature,DiCarlo_Nature} and
two-qubit correlation measurements have been done,\cite{Bell_UCSB}
measuring multi-qubit correlations requires further theoretical and
experimental studies. We hope that our proposal acts as another
incentive for researchers in the field of superconducting qubits to
look for ways to perform measurements that can reveal quantum
correlations.
In conclusion, we have proposed an approach to emulate the Kitaev model on a honeycomb lattice using superconducting quantum circuits, and shown
that the low-energy dynamics of the superconducting network should follow a finite-field Kitaev model Hamiltonian. We analytically study two
particular limits for system parameters, explore their ground state characteristics, and identify their low-energy excitations as vortex states
and bond states. We further show that both vortex- and bond-state excitations can be generated using the same spin-pair operations, starting
from different ground states. Our proposal points to an experimentally realizable many-body system for the quantum emulation of the Kitaev
honeycomb spin model.
\begin{acknowledgments}
We thank J. Vidal and Yong-Shi Wu for useful discussions. J.Q.Y. and
X.F.S. were supported by the National Basic Research Program of
China Grant Nos. 2009CB929300 and 2006CB921205, and the National
Natural Science Foundation of China Grant Nos. 10625416 and
10534060. X.H. and F.N. acknowledge support by the National
Security Agency and the Laboratory for Physical Sciences through the
US Army Research Office, X.H. acknowledges support and hospitality
by the Kavli Institute of Theoretical Physics at the University of
California at Santa Barbara, and F.N. thanks support by the National
Science Foundation Grant No.~0726909.
\end{acknowledgments}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,360 |
namespace metrics {
// Define parameters for a collection trigger.
struct TriggerParams {
// Trigger profile collection with 1/|sampling_factor| probability.
int64_t sampling_factor = 1;
// Upper bound of a random delay added before collection on a trigger event.
// The delay is uniformly chosen between 0 and this value.
base::TimeDelta max_collection_delay;
};
// Defines collection parameters for metric collectors. Each collection trigger
// has its own set of parameters.
struct CollectionParams {
CollectionParams();
// Time a profile is collected for, where it makes sense.
base::TimeDelta collection_duration;
// For PERIODIC_COLLECTION, partition time since login into successive
// intervals of this duration. In each interval, a random time is picked to
// collect a profile.
base::TimeDelta periodic_interval;
// Parameters for other collection triggers.
TriggerParams resume_from_suspend;
TriggerParams restore_session;
};
} // namespace metrics
#endif // CHROME_BROWSER_METRICS_PERF_COLLECTION_PARAMS_H_
| {
"redpajama_set_name": "RedPajamaGithub"
} | 107 |
A Dr. Jeuss Christmas - I Hope You Don't Miss This
By Brandon Monahan
Mulberry-Pew (right) introducing Sammy (left) to the Airzooka
Wedged in a rather obscure area of Los Angeles, The Sacred Fools Theater has gone over and beyond the duty of compensating for local. With revolutionary plays and some of the best talent in the business, they certainly are worth their while.
A Dr. Jeuss Christmas was written by the mysterious nom-de-plume named Dr. Jeuss (pronounced 'juice'). An obvious parody of Dr. Seuss, Dr. Jeuss mixed the innocent pleasure of Seuss' rhyme scheme, with more adult dialogue. Whether belting out an emphatic rhyme to 'fit' or referring crassly and comically to heavy petting, the actors continued to exude the feel of Cat in the Hat gone bad. And it was oh, so good!
Sammy (left) and Tammy (right) with the bizarre creature, The Whizzit
There were several scene stealers, most notably Russ Jones as The Whizzit. If a large green monstrosity isn't enough to elicit a laugh or two, audiences will be tickled by his Scottish accent and demeanor. One can scarcely hold back a chuckle when an 'arse' appears in dialogue.
The brother and sister, Tammy and Sammy, played by Christiane Cannon and Ryan Schwartzman, respectively, were also delightful to watch. Without a bit of awkwardness and with perfect precision, they played, hugged, flew about the stage, bumped into objects and in general acted like siblings.
Jason Frazier played the darker side of the play, Simon Thaddeus Mulberry-Pew, the rich snob with a dark heart. With a foul mouth and fouler intentions, he encompassed all that villains must.
There were plenty of other excellent cast members, such as Samuel Rhymes in drag as Mom, Mikal Hanna as the lovable Mumpus, Christopher Carrington as the lewd Santa Claus-like Mr. Christmas, and of course, C.M. Gonzalez, playing a giant robot.
Intimacy between enemies
This piece, brought together by the directing styling of John Mitchell, benefited from well versed dialogue, perfectly timed comedy and talented actors, willing to be roughed up for the sake of art.
A Dr. Jeuss Christmas is playing at The Sacred Fools Theater, located at 660 N. Heliotrope Dr. (off the 101 Freeway at Melrose) in Los Angeles from December 8th until the 30th. Tickets are $20 and can be purchased by calling (310) 281-8337 or going online at www.SacredFools.org. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,493 |
\section{Introduction}
One of the original (and still most compelling) motivations for
building the SKA was realizing the ability to study the evolving
neutral gas content of galaxies throughout cosmic time. The
distribution and kinematics of \ion{H}{i} as traced by the red-shifted
$\lambda$21~cm emission line provide unique insights into galaxy
formation and evolution. In addition to permitting direct assessment
of the atomic and dynamical mass, \ion{H}{i} imagery retains the
signature of galaxy interactions from the previous Gyr, rather than
only providing a snap-shot of gas content. However, current
instrumentation permits \ion{H}{i} detection and imaging in only the
very local universe. It will require some two orders of magnitude
greater instantaneous sensitivity to push back the \ion{H}{i} frontier
to the early universe.
Realization of the complete SKA is envisioned to take place by about
2020. However, a series of path-finding instruments will become
available as early as 2009, and these should pave the way for a 10\% SKA by
about 2015. What exactly might these instruments make possible and
what types of surveys might be optimally pursued with each? In this
short contribution we will consider this question from the perspective
of \ion{H}{i} surveys.
\section{Surveys}
The complete SKA is currently specified as providing 20000~m$^2$/K of
effective sensitivity (between at least 0.5 and 5 GHz) with an
instantaneous FOV significantly exceeding 1~deg$^2$ at low
frequencies. About 25\% of the collecting area is envisaged to be
within a region of 1~km diamter, 50\% within 5~km, and 75\% within
150~km. For simplicity, we will assume that the effective sensitivity
of the complete SKA for \ion{H}{i} galaxy surveys will be about
10000~m$^2$/K at all frequencies, while applications requiring the
highest brightness sensitivity will have 5000~m$^2$/K. This should be
accurate out to at least z~=~3, since at lower frequencies it will be
possible to use longer baselines without over-resolving targets, but
will certainly no longer apply at z~=~6, without a higher sensitivity
than currently envisaged between 200 and 300~MHz.
The currently funded path-finder instruments: APERTIF (APERture Tile
In Focus) in the Netherlands, xNTD (the eXtended New Technology
Demonstrator) in Australia and KAT (Karoo Array Telescope) in South
Africa each amount to about a 1\% SKA in terms of their surveying
power. APERTIF, by placing Focal Plane Array (FPA) receivers in each
of the 14 dishes of 25~m diameter that make up the WSRT array will
provide 100~m$^2$/K sensitivity over a 8~deg$^2$ FOV; while each of
xNTD and KAT is planned to provide about 50~m$^2$/K sensitivity over a
22~deg$^2$ FOV. Since survey speed scales as
BW$\times$FOV$\times$Sens$^2$, all three of these systems will provide
similar survey performance, given a similar instantaneous bandwidth of
about 300~MHz and frequency tuning range of at least 850 to 1700 MHz.
The distinguishing features of these SKA path-finders are: (1)
simultaneous, wide-field, wide-band data acquisition, followed by (2)
parallel multi-topical astrophysical analysis. Specialized science
teams (continuum, polarimetry, spectral line, transients, etc.) will
capitalize on the full survey potential and maximize the scientific
return on these instrument investments. This should permit world-class
science to be carried out in the decade preceding full SKA deployment.
Some of the major scientific themes that drive \ion{H}{i} science to higher
sensitivities are:
\begin{enumerate}
\item Quantifying the evolving gas content as well as the baryonic power
spectrum of galaxies by detecting statistical samples spanning the
widest possible range of red-shifts.
\item Witnessing galaxy formation and evolution via resolved imaging
studies at the highest possible look-back times.
\item Imaging the local cosmic web by pushing back the N$_{HI}$
frontier into the optically thin (to ionizing photons) regime below
10$^{18}$cm$^{-2}$.
\end{enumerate}
We have assessed the types of \ion{H}{i} surveys which might be
carried out to address these three science themes by instruments
having 1, 10 and 100\% of the full SKA sensitivity (taken to be
10000~m$^2$/K for galaxy surveys and 5000~m$^2$/K for the low N$_{HI}$
application) in order to determine their relative and absolute
utility. We have assumed an unevolved HIPASS HIMF (Zwaan et
al. \cite{zwaa03}) at all red-shifts to allow a conservative
prediction of galaxy detection rates. We further assume that the
effective line-width of the \ion{H}{i} signal is given by $\Delta
V(M_{HI})~=~0.105 M_{HI}^{1/3}$ in statistical agreement with local
galaxy populations. The assumed comological parameters are H$_0$~=~73
km/s/Mpc, $\Omega_m$~=~0.24 and $\Omega_\Lambda$~=~0.76. We also
assume that the facility has a instantaneous FOV of 8~deg$^2$
independent of frequency (as will be the case for APERTIF) and then
consider surveys which cover a total area on the sky of 8000, 800, 80
and 8~deg$^2$. The total observing time for each survey was 1000~days
for the 1\% SKA cases and 100~days for 10 and 100\% SKA cases. As will
be seen below, these appear to be realistic survey durations to reach
interesting depths.
The galaxy results are summarized in Tables~\ref{tab:dete} and
\ref{tab:imag}, where we have assumed a 7$\sigma$ threshold (at a
velocity resolution matched to the line-width) for simple detection
and a 100$\sigma$ threshold for imaging. This second criterion stems
from allowing for some 10's of high significance resolution elements
across each source. We tabulate the (base 10) logarithm of the number
of detections (rounded to the nearest integer) of each survey in a
sequence of red-shift bins.
Several of the detection surveys are illustrated in more detail in
Fig.~\ref{fig:surv}. The number of detections per half dex \ion{H}{i}
mass bin are plotted in the figure. Separate curves are drawn for each
of the red-shift intervals to permit assessment of the achieved mass
depth in each interval. The dotted vertical line near
M$_{HI}$~=~10$^{10}$~M$_{\odot}$ is M$_{HI}^*$ of the HIMF. Good
sampling of a red-shift interval demands a significant detection rate
down to below M$_{HI}^*$.
The results for low column density surveys are summarized in
Table~\ref{tab:cweb} where the 1$\sigma$ column density sensitivity is
indicated over a 20 km~s$^{-1}$ line-width for a beam size of
60~arcsec. This beam size correponds roughly to that of the central km
of an array configuration. Such a beam size might reasonably be
expected to be filled with diffuse \ion{H}{i} emission out to
distances of about 30~Mpc where it subtends less than 10~kpc. At
larger distances, the effective column density sensitivity will likely
be diminished due to beam dilution.
\begin{table*}
\caption{Survey Size and Detection Results}
\label{tab:dete}
\centering
\begin{tabular} {c c c c c c c c c c c c c }
\hline\hline
Sensitivity & Time & Area &\multicolumn{10}{c} {log(Number Detections)
in z-range $>7\sigma$}\\
(\% SKA)& (Days) & (deg$^2$) & 0--0.01&0.01--0.02&0.02-0.05&0.05--0.1&0.1--0.2&0.2--0.5&0.5--1&1--2&2--5&$>$5\\
\hline
1 & 1000 & 8000 & 3&4&5&5&5&4&-&-&-&-\\
1 & 1000 & 800 & 3&3&4&5&5&5&1&-&-&-\\
1 & 1000 & 80 & 2&2&3&4&4&5&4&-&-&-\\
1 & 1000 & 8 & 1&1&2&3&4&5&5&4&-&-\\
\hline
10 & 100 & 8000 & 4&4&5&6&6&6&2&-&-&-\\
10 & 100 & 800 & 3&3&4&5&5&6&5&1&-&-\\
10 & 100 & 80 & 2&2&3&4&5&6&6&5&1&-\\
10 & 100 & 8 & 1&2&3&3&4&5&5&6&5&-\\
\hline
100 & 100 & 8000 & 4&4&5&6&7&8&8&7&3&-\\
100 & 100 & 800 & 3&4&5&5&6&7&7&8&7&1\\
100 & 100 & 80 & 2&3&4&5&5&6&7&8&8&6\\
100 & 100 & 8 & 1&2&3&4&5&6&6&7&8&8\\
\hline
\end{tabular}
\end{table*}
\begin{table*}
\caption{Survey Size and Imaging Results }
\label{tab:imag}
\centering
\begin{tabular} {c c c c c c c c c c c c c }
\hline\hline
Sensitivity & Time & Area &\multicolumn{10}{c} {log(Number Detections)
in z-range $>100\sigma$}\\
(\% SKA)& (Days) & (deg$^2$) & 0--0.01&0.01--0.02&0.02-0.05&0.05--0.1&0.1--0.2&0.2--0.5&0.5--1&1--2&2--5&$>$5\\
\hline
1 & 1000 & 8000 & 3&3&3&2&-&-&-&-&-&-\\
1 & 1000 & 800 & 2&2&3&3&2&-&-&-&-&-\\
1 & 1000 & 80 & 1&2&3&3&3&1&-&-&-&-\\
1 & 1000 & 8 & 1&1&2&2&3&2&-&-&-&-\\
\hline
10 & 100 & 8000 & 3&3&4&4&3&-&-&-&-&-\\
10 & 100 & 800 & 2&3&4&4&4&2&-&-&-&-\\
10 & 100 & 80 & 2&2&3&3&4&3&-&-&-&-\\
10 & 100 & 8 & 1&1&2&3&3&4&2&-&-&-\\
\hline
100 & 100 & 8000 & 4&4&5&5&6&5&-&-&-&-\\
100 & 100 & 800 & 3&3&4&5&5&6&4&-&-&-\\
100 & 100 & 80 & 2&2&3&4&5&5&5&4&-&-\\
100 & 100 & 8 & 1&2&3&3&4&5&5&5&3&-\\
\hline
\end{tabular}
\end{table*}
\begin{table}
\caption{Survey Size and Cosmic Web Depth }
\label{tab:cweb}
\centering
\begin{tabular} {c c c c c }
\hline\hline
Sensitivity & Time & Area & log($\Delta N_{HI}$)\\
(\% SKA)& (Days) & (deg$^2$) & ($\theta$=60'', $\Delta$V=20 km/s)\\
\hline
1 & 1000 & 8000 & 18.5\\
1 & 1000 & 800 & 18\\
1 & 1000 & 80 & 17.5\\
1 & 1000 & 8 & 17\\
\hline
10 & 100 & 8000 & 18\\
10 & 100 & 800 & 17.5\\
10 & 100 & 80 & 17\\
10 & 100 & 8 & 16.5\\
\hline
100 & 100 & 8000 & 17\\
100 & 100 & 800 & 16.5\\
100 & 100 & 80 & 16\\
100 & 100 & 8 & 15.5\\
\hline
\end{tabular}
\end{table}
\begin{figure*}
\centering
\includegraphics[width=14.8cm]{RBraun_1_fig1.eps}
\caption{Detected numbers of galaxies as a function of their
\ion{H}{i} mass for representative surveys. The different curves
correspond to the red-shift intervals indicated at the top of the
plots. The logarithm of the integrated number of detections in
a red-shift interval is indicated by the integer above the
peak of each curve. A detection threshold of 7$\sigma$ is
assumed and an instantaneous FOV of 8~deg$^2$. The dotted
vertical line is M$_{HI}^*$. The label above each panel
gives the survey sensitivity, duration and total area. }
\label{fig:surv}
\end{figure*}
\section{Conclusions}
Achieving interesting \ion{H}{i} galaxy sample sizes with 1\% SKA
surveys requires very substantial survey durations, of about 1000
days. Good sampling (log(N)$\sim$5 down to below M$_{HI}^*$) can then
be achieved out to z~=~0.2 over 8000 deg$^2$ or even to z~=~0.5 over
800 deg$^2$ as shown in Table~\ref{tab:dete}. The same surveys would
permit the resolved imaging of order 1000 galaxies in each of several
red-shift bins as can be seen from Table~\ref{tab:imag} as well as
detection of faint neutral filaments in the vicinity of galaxies with
a logarithmic column density {\sc RMS} of 18.5 or 18 (from
Table~\ref{tab:cweb}). This should be compared with the current
detection threshold in all but the deepest \ion{H}{i} imaging surveys
of about 10$^{19}$cm$^{-2}$ over $\Delta$V~=~20~km~s$^{-1}$. Since the
surface area of diffuse \ion{H}{i} is known to increase by a factor of
three for each decade of column density in this regime (from the
statistics of QSO absorption lines, cf. Braun \& Thilker
\cite{brau04}) it is clear that the first step towards systematic
mapping of the diffuse \ion{H}{i} filament distribution will be
achieved in such surveys. Simultaneous with the various \ion{H}{i}, OH
and other spectral line applications that would be served by these
surveys are the continuum, polarimetric and variability applications
that other members of a survey science team would exploit.
The very long survey durations, underline the great utility of having
several of such 1\% SKA path-finders operational in the same
timeframe; APERTIF in the North and xNTD and KAT in the
South. Complimentary surveys could then be carried out by the different
facilities to maximize the total scientific return.
Once 10\% SKA sensitivities are achieved, then ground-breaking surveys
are possible with only 100 day duration. Sample sizes of log(N)$\sim$6
extending below M$_{HI}^*$ are possible over 800 deg$^2$ out to
z~=~0.5 and over 80 deg$^2$ out to z~=~1. Such surveys will permit
very competitive measurement of acoustic oscillations in the galaxy
power spectrum (eg. Blake \& Glazebrook \cite{blak03}). Given the more
modest survey duration (relative to the 1\% SKA case), one can
envision a series of surveys probing different depths. The diffuse
\ion{H}{i} sensitivity is such ($\sim$10$^{17}$cm$^{-2}$) that the next
factor of three in sky area will become accessible to imaged detection
and kinematic study of the cosmic web.
With the 100\% SKA sensitivity the capabilities are truly
phenomenal. Survey sample sizes in the range log(N)~=~7--8 are
feasible over the red-shift range of 0.2 to about 5. The SKA will
easily be the most productive red-shift engine in astronomy. This
finally brings \ion{H}{i} imaging into completely new terrain. Precise
tracking of potential time evolution of dark energy (via the baryonic
acoustic oscillation signature) should be possible out to
z~$\sim$~3. The local cosmic web will be imaged down to
N$_{HI}$~=~10$^{16}$cm$^{-2}$. What exactly will be seen at z~$>$3?
This will depend crucially on the SKA sensitivity in the critical
frequency window of 350 to 200~MHz. If this can be maintained at the
level of 10000~m$^2$/K, then the prospects are extremely good for
detecting large populations of early universe objects. Given the very
low cost of collecting area in this frequency range, this seems to be
a very worthwhile area for investment.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 921 |
<?php
namespace PHPExiftool\Driver\Tag\Leica;
use JMS\Serializer\Annotation\ExclusionPolicy;
use PHPExiftool\Driver\AbstractTag;
/**
* @ExclusionPolicy("all")
*/
class Contrast extends AbstractTag
{
protected $Id = 12298;
protected $Name = 'Contrast';
protected $FullName = 'Panasonic::Subdir';
protected $GroupName = 'Leica';
protected $g0 = 'MakerNotes';
protected $g1 = 'Leica';
protected $g2 = 'Camera';
protected $Type = 'int32u';
protected $Writable = true;
protected $Description = 'Contrast';
protected $flag_Permanent = true;
protected $Values = array(
0 => array(
'Id' => 0,
'Label' => 'Low',
),
1 => array(
'Id' => 1,
'Label' => 'Medium Low',
),
2 => array(
'Id' => 2,
'Label' => 'Normal',
),
3 => array(
'Id' => 3,
'Label' => 'Medium High',
),
4 => array(
'Id' => 4,
'Label' => 'High',
),
);
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,587 |
\section*{Appendix:
Decay rate from a condensate in presence of a thermal component}
\vskip 1cm
The decay rate of the condensate via a $p$-body
collision may involve either
a collision between $p$ condensate atoms, or a
collision between $q$ ($<p$) condensate
atoms and $p-q$ atoms from the thermal fraction. Taking into account
symmetrization
\cite{Kagan85}, this decay rate can be written:
\begin{equation}
{\dot N}=-G\left( \int n^2({\bf r})\;
d^3 r+2\int n({\bf r})\;n_{\rm th}({\bf r})\; d^3r\right)
\label{twobodyapp}
\end{equation}
for a two-body process and
\begin{equation}
{\dot N}=-L\left( \int n^3({\bf r})\; d^3 r+6
\int n^2({\bf r})\;n_{\rm th}({\bf r})\; d^3r + 6\int n({\bf r})
\;n^2_{\rm th}({\bf r})\; d^3r \right)
\label{threebodyapp}
\end{equation}
for a three-body process.
As indicated in the text we calculate the first integral of
each of these two expressions
in the Thomas-Fermi limit, assuming a parabolic condensate density
$$
n({\bf r})= (\mu-V({\bf r}))/g
$$
inside the condensate, and $n({\bf r})=0$ outside. Here $\mu$ is the chemical potential, $V({\bf r})$ denotes
the harmonic trapping potential and $g=4\pi \hbar^2 a/m$.
To evaluate $n_{\rm th}({\bf r})$ in
the remaining terms, we use the Hartee-Fock
approximation \cite{Dalfovo98}.
Since the density of the thermal fraction is
always smaller than the central density
of the condensate by an order of magnitude, we
neglect the effect of the thermal component on the condensate distribution.
In this approximation the density of the thermal
fraction is given by:
$$
n_{\rm th}({\bf r})=\Lambda_T^{-3}\; g_{3/2}
\left(e^{-|\mu-V({\bf r})|/k_BT} \right)
$$
where $\Lambda_T=h/\sqrt{2\pi m k_B T}$ and
$g_{3/2}(z)=\sum z^\ell \ell^{-3/2}$. This
expression takes into account the repulsion
of the uncondensed atoms from
the condensate by the interaction potential
$2gn({\bf r})$. The overlap integrals entering
in (\ref{twobodyapp}) and
(\ref{threebodyapp}) are then calculated numerically.
The final results can be cast in the form:
$$
\frac{\dot N}{N}= -L\left(\langle n^2\rangle +
6 \,\langle n\rangle \;\tilde n_{\rm th}({\bf 0}) \; \alpha(\tilde \mu)
+ 6\, \tilde n^2_{\rm th}({\bf 0}) \; \beta(\tilde \mu)\right)
\ -G \left(\langle n\rangle + 2 \,\tilde n_{\rm th}({\bf 0})
\gamma(\tilde \mu)
\right)
$$
with $\tilde \mu=\mu/(k_B T)$.
The quantity $\tilde n_{\rm th}({\bf 0})=
\Lambda_T^{-3}g_{3/2}(1)$ represents the
uncondensed density of an ideal Bose gas at
temperature $T$ (below $T_c$)
at the center of the trap. The functions
$\alpha, \beta,\gamma$ are equal to 1 in the
limit $\mu \ll k_B T$ and smaller than 1
otherwise, since the overlap between
$n({\bf r})$ and $n_{\rm th}({\bf r})$ is then reduced.
For instance, for
$\mu=k_B T$ (experimental situation at $t=0$), we
find $\alpha(1)=0.26$,
$\beta(1)=0.11$ and $\gamma(1)=0.31$. The effect
of mixed collisions (condensate +
thermal fraction) is therefore notably reduced with
respect to the ideal gas case.
\newpage
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,332 |
Kim Hollis: So that just happened. Jurassic World just became the biggest opener of all-time, earning $208.8 million domestically in its first three days. Its global debut was a massive $524.1 million. Please explain how this was possible.
Edwin Davies: I've been thinking about this a lot since the Friday grosses started coming in and suggested that basically everyone - including Universal, though they may have just been playing coy - underestimated Jurassic World by a frankly historic amount. The best explanation I can come up with is that the original Jurassic Park is to Millennials what Star Wars was to Gen X-ers. It was this huge adventure that instilled a love of movies in a whole generation of kids, many of whom passed their affection for the film on to their own children, as evidenced by the fact that 55% of tickets were sold to people under the age of 25. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,001 |
{"url":"https:\/\/forums.novell.com\/showthread.php\/136498-Can-t-delete-directories","text":"## Can't delete directories\n\nHello,\n\nI'm having a problem deleting a directory structure. Looks like this:\n\nvol1\\data\\cm\\yvonnet\\candicel\\.. on for 19 directories deep. They are all empty. When I try to delete the last directory by right-clicking, I don't even get the delete option. One level up when I try to delete I get \"cannot delete file\". Also tried it via DOS.\n\nSuggestions? I've discovered it is what is crashing the Netware source server when using the Consolidation utility (but somehow it was copied to the target and I have the same problem on that server now too.\n\nThanks,\nTom","date":"2018-10-16 16:46:55","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8052313327789307, \"perplexity\": 2442.8388895552735}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-43\/segments\/1539583510853.25\/warc\/CC-MAIN-20181016155643-20181016181143-00335.warc.gz\"}"} | null | null |
{"url":"https:\/\/www.semanticscholar.org\/paper\/An-%24%5Cwidetilde%5CmathcalO(m%2F%5Cvarepsilon%5E3.5)%24-Cost-Lee-Padmanabhan\/c99223d0f78dddc03d1682a37ff27c4d3b8b6f0d","text":"Corpus ID: 67877060\n\n# An $\\widetilde\\mathcalO(m\/\\varepsilon^3.5)$-Cost Algorithm for Semidefinite Programs with Diagonal Constraints\n\n@inproceedings{Lee2020AnA,\ntitle={An \\$\\widetilde\\mathcalO(m\/\\varepsilon^3.5)\\$-Cost Algorithm for Semidefinite Programs with Diagonal Constraints},\nbooktitle={COLT},\nyear={2020}\n}\n\u2022 Published in COLT 2020\n\u2022 Mathematics, Computer Science\n\u2022 We study semidefinite programs with diagonal constraints. This problem class appears in combinatorial optimization and has a wide range of engineering applications such as in circuit design, channel assignment in wireless networks, phase recovery, covariance matrix estimation, and low-order controller design. In this paper, we give an algorithm to solve this problem to $\\varepsilon$-accuracy, with a run time of $\\widetilde{\\mathcal{O}}(m\/\\varepsilon^{3.5})$, where $m$ is the number of non-zero\u2026\u00a0CONTINUE READING\n2 Citations\n\n#### Tables and Topics from this paper\n\nA Faster Interior Point Method for Semidefinite Programming\n\u2022 Mathematics, Computer Science\n\u2022 ArXiv\n\u2022 2020\n\u2022 8\n\u2022 PDF\nRiemannian Langevin Algorithm for Solving Semidefinite Programs\n\u2022 Mathematics, Computer Science\n\u2022 ArXiv\n\u2022 2020\n\u2022 1\n\u2022 PDF\n\n#### References\n\nSHOWING 1-10 OF 48 REFERENCES\nPhase recovery, MaxCut and complex semidefinite programming\n\u2022 Mathematics, Computer Science\n\u2022 Math. Program.\n\u2022 2015\n\u2022 442\n\u2022 PDF\nA Direct Formulation for Sparse PCA Using Semidefinite Programming\n\u2022 Computer Science, Mathematics\n\u2022 SIAM Rev.\n\u2022 2007\n\u2022 561\n\u2022 PDF\nMax-Cut based overlapping channel assignment for 802.11 multi-radio wireless mesh networks\n\u2022 Weiqi Wang, Bo Liu\n\u2022 Computer Science\n\u2022 Proceedings of the 2013 IEEE 17th International Conference on Computer Supported Cooperative Work in Design (CSCWD)\n\u2022 2013\n\u2022 2\nRank minimization and applications in system theory\n\u2022 Mathematics\n\u2022 Proceedings of the 2004 American Control Conference\n\u2022 2004\n\u2022 220\n\u2022 PDF\nMinimizing finite sums with the stochastic average gradient\n\u2022 Mathematics, Computer Science\n\u2022 Math. Program.\n\u2022 2017\n\u2022 786\n\u2022 PDF\nA Faster Cutting Plane Method and its Implications for Combinatorial and Convex Optimization\n\u2022 Mathematics, Computer Science\n\u2022 2015 IEEE 56th Annual Symposium on Foundations of Computer Science\n\u2022 2015\n\u2022 166\n\u2022 PDF\nPrimal-dual subgradient methods for convex problems\n\u2022 Y. Nesterov\n\u2022 Mathematics, Computer Science\n\u2022 Math. Program.\n\u2022 2009\n\u2022 694\n\u2022 PDF","date":"2021-01-20 14:26:18","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7516468167304993, \"perplexity\": 10963.068800414581}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-04\/segments\/1610703520883.15\/warc\/CC-MAIN-20210120120242-20210120150242-00279.warc.gz\"}"} | null | null |
Q: Why Did the Empire Use Storm/Clone Troopers and not Droids? The Separatists were under control of Count Dooku, who was under Sidious' control, who was essentially running the entire Clone War and using it as an excuse to take over the Republic and turn it into an empire. So when the Clone War is over and Sidious is in control of everything, he could easily obtain the battle droids the Separatists used without much trouble and a minimum of expense.
While I can understand why one would not want a 100% droid army, it would make sense to use droids when possible, since they don't need food (just plug 'em in), parts are interchangeable, and they're even more disposable than clonetroopers are.
So why doesn't the Empire use them whenever possible instead of stormtroopers? Is there an in-universe reason why the Deathstar isn't carrying clone troops or why a backwater world like Tatooine isn't left to mostly droid troopers instead of human stormtroopers?
Note that I'm talking in-universe, but since Lucas had supposedly mapped out the prequel trilogy before finishing writing Episode IV, he still could have used humans in droid costumes if he felt he could do so realistically (like with C-3P0) instead of using stormtrooper uniforms. In other words, if Lucas was already thinking of such widespread droid usage in the early stages, he could have used that in Episode IV.
A: Droids lack certain qualities that most humans (even clones) possess: grasping the subtleties of situations, thinking on their feet and adapting. I believe the Kaminoans touted this to Obi-Wan when they gave him a tour of the cloning facility.
There are droids with advanced faculties, but they are likely to be expensive models -- not ones that you can mass produce in the millions and deploy into battle fields. The battle on Naboo was a good example of how poorly droid soldiers fare against human ones.
Although why nobody bothered to built a battle droid with an R2 unit's brain is beyond me. That's one droid I'd not want to meddle with. There is, however, an extended universe example of a protocol droid being converted to an assassin.
From Star Wars Wikia on Droids:
Despite these advantages, however, combat droids suffered from several drawbacks. Most importantly, in order to create total obedience and foil any chance of rebellion, droid units were often crippled with extremely sub-par artificial intelligence.
A: Well, the Empire wasn't really in any major wars. The Rebellion was mostly backwater, and small scale, until after the first Death Star. When you already have an army of mixed Clone and Storm Troopers why start a whole new infrastructure for droids? Most Stormtroopers functioned day to day as the international police force. You don't want a robot with the limited ability to reason or to consider the situation in that role (Also one that would require an expensive robo-brain)
A: The expanded universe has established that the Empire adhered to a policy called "Human High Culture" which believed that humans were superior, and the Empire actively discriminated on that policy. On assumes that as a reasonable extension of this speciesist discrimination that the Empire believed that human (by the time of A New Hope) troopers would be superior to droids.
A: I seem to recall (and admittedly, I read this in the 90's, so that makes it a dubious source) that the emperor was able to provide some cohesion to the storm-troupers by using the force — and this was so noticeable that there was commentary about sudden disorganization and chaos about the time of the Emperor's death. This would not have been possible with droids, but using regular humans would have been cost prohibitive. Clones seem like the obvious solution.
A: The Senate banned them.
According to the Rogue One Ultimate Guide,
the Senate rolled out a mandate prohibiting the creation of battle droids
This was done due to negative feeling towards battle droids following the Clone Wars. It goes on to say that a loophole was used to create the KX-series "Security" Droids, as seen in Rogue One. The same is probably true of the Imperial "sentry" droids seen in Rebels.
According to Star Wars: Absolutely Everything You Need to Know, this didn't stop the Empire from occasionally utilizing battle droids.
[Emphasis theirs']
Thrawn uses outlawed droid technology... a terrifying murder bot.
Various other Imperial battle droids also show up in Legends, such as the dark troopers, Carbonite and Incinerator war droids and Orbot Droids.
A: In episode three, all the separatists were killed, so that means all the factories probably shut down, since no more separatists were alive.
Technically, stormtroopers are not clones. The clone factory stopped producing since they ran out of DNA from Jango. So storm troopers are just volunteers who serve the Empire. And that is why they have horrible accuracy.
A: Darth Vader and Lord Sidious were afraid of Count Dooku using the droids to continue his war against the Empire, therefore he had Anakin kill him off and then go on to kill off all of the Separatist leaders.
Also don't forget that the Empire still is operated by the Senate of Coruscant, which is made up of survivors of the clone wars. Therefore they probably would not want to use the same things they fought against as their allies.
As well as the people of the galaxy... Just because Emperor Palpatine became Lord Sidious doesn't mean he doesn't have to work for votes to stay in power.
A: I can't provide sources, but this is more of an opinion.
Firstly, if the Emperor were to maintain complete control after the crisis was averted, he would need a common enemy for all of the (Old) republic to get behind. This would be the Droids; even with his special powers and knowledge of the dark side the emperor couldn't force an entire galaxy to follow him despite the fact he's using the droids they elected him to destroy to control them.
Secondly is a complete theory, but given the humano-centric nature of the Empire, with a few very rare exceptions, notice that most of the droid-manufacturers are not human, the only exception (correct me if I'm wrong) being Dooku, who was the emperor's apprentice anyway; perhaps he favoured the clones as being a way of establishing human dominance unquestionably (despite the clones being made by non-humans, no one knew this), when the droids would've represented an alien power. You know, common enemy and suchlike. Having a vague knowledge of the species Sith, and the tension between the Sith and humans who learn the Dark Side, this would hold a personal goal for the emperor: proving that the humans are dominant, that he is not inferior to the "true Sith" in any way, for he created an empire of humans, made by humans.
Warning: this part of my argument is a stray... I'm just babbling now.
In a way, doesn't it seem peculiar that in the original three aliens are in the rebellion, whereas the empire is all human, yet in the second trilogy of prequels, both the senate has a high rate of alien life, and the separatists. It seems to me that I get a mixture of signals from this, firstly that the Emperor has tried to focus all the aliens of considerable power to the Separatist movement for extermination, so that when he takes utter control he can get rid of those less powerful but still with influence via the dissolution of the senate, and have an all-human regime without anybody catching on.
Secondly, it makes me think that the separatists were meant to be all-alien, and the desperate shots of aliens in the senate were trying to emphasize equality of an entire galaxy before the Empire was formed, while hoping not to take away from the idea that the separatists were "THE ALIENS!" XD, okay my brain hurts.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,863 |
Tag / manga
March 26, 2015 by GirlGoneGeekBlog
The Melancholy of 'Oyasumi Punpun'
depression, Inio Asano, manga, Oyasumi Punpun
Wikipedia describes Oyasumi Punpun (Goodnight Punpun) as:
"…a Japanese manga written and illustrated by Inio Asano about Punpun Punyama (later Onodera), a normal child depicted in the form of a bird. The story follows him as he copes with his dysfunctional family and friends, his love interest, his oncoming adolescence and his hyperactive mind… The manga follows Punpun as he grows up, splitting the book into around 4 stages of his life: Elementary school, Middle school, High school, and his early 20s."
But nowhere in that description is the word "depression", which is what I think Punpun is essentially about.
Inio Asano (Solanin, Nijigahara Holograph) tells the stories no one else wants to tell because they are too embarrassing. Because it's how a lot of us secretly feel and act. No one wants to really admit how truly dark their dark side is. Not only does Asano admit it, but he depicts heavy and depressing themes with genius level beauty and authenticity.
I went through the 'motions while reading Goodnight Punpun. One page would make me incredibly sad and another would have me laughing out loud at its dark sense of humor. But the one thing that was consistent throughout was Asano's creative and compelling storytelling.
Despite how childishly drawn the bird-like Punpun is, he is one of the most realistic characters I've ever read. Sometimes a long-legged bird is the best character to depict the utter despair we sometimes feel as we come to age. The two-dimensional bird-boy and the realistically illustrated world and characters combine to create one of the most interesting and beautiful aesthetics I've found in a manga. The depiction of Punpun changes during different stages in his life. Mostly he's a bird-like character, when he's in deep depression he's a floating pyramid and during other, darker times, he is an four-eyed monster with horns. The reader is able to see how Punpun feels because of these various representations, even though the characters in the manga see Punpun as a normal human being.
It wouldn't be a far cry to describe Punpun as a psychological horror. However, the horrors in this manga aren't monsters hiding under the bed, instead they hide in the dark corners of our minds. Asano removes the filters that we usually use to cover up our sad and unwanted thoughts. It's the brutal honesty of Punpun that makes it seem so unreal because it's the reality we tend to ignore.
As someone who was (and still kind of is) suffering from depression, there was an undeniable comfort I found in the melancholy of Punpun Onodera. Punpun helped me not only accept my depression, but justify it.
March 8, 2015 by GirlGoneGeekBlog
My Favorite Shoujo Manga
Anime, Ao Haru Ride, bitou lollipop, Blue Spring Ride, Daytime Shooting Star, From Me To You, Hirunaka no Ryuusei, Kimi Ni Todoke, manga, shoujo manga, six half
I recently posted My Favorite Josei Manga and now it's time for shoujo to shine! I'm super picky when it comes to anime and manga and extremely picky when it comes to shoujo. I don't even know why I'm so particular with shoujo because so much of them are the same story. But even so, I want the shy main characters (MCs) not too cold, the love triangles not too hot and the art to be juuussst riiiigghht.
Because I prefer josei manga to shoujo this list is shorter. But I hope with my extreme pickiness that the manga on this list are some of the best of the best… Or at least in my eyes. You may be wondering why you don't see certain (super awesome) shoujo on this list. The likely reason is that I've only watched the anime and not read the manga. This is the case for Sailor Moon, Ouran High School Host Club, Fruits Basket and Kids on the Slope to name a few.
Although I end up reading more josei than shoujo, I still have a deep love for the shoujo genre. It's a nice change from the content I usually consume. Instead of stressing about characters dying, I'm stressing about whether the MC will get the courage to give her crush Valentine's Day chocolates. With so much sexualization in the media, I truly enjoy freaking out when the MC holds her boyfriend's hand for the first time. I'm all about innocence, love triangles, amazing friendships and sensei crushes!
Have any shoujo titles you'd recommend? Comment below! I'm always looking for new manga to read.
Kimi Ni Todoke (From Me To You)
Genre: comedy, drama, romance, school life, shoujo, slice of life
Mangaka: Shiina, Karuho
Status: Ongoing (95 Chapters)
Synopsis: Kuronuma Sawako is completely misunderstood by her classmates. Her timid and sweet demeanor is often mistaken for malicious behavior. This is due to her resemblance to the ghost girl from "The Ring", which has led her peers to give her the nickname Sadako. Longing to make friends and live a normal life, she is naturally drawn to Kazehaya Shouta, the most popular guy in class, whose "100% refreshing" personality earns him great admiration from Sawako. So when Kazehaya starts talking to her, maybe there is hope for the friendships Sawako has always longed for. Maybe…there is even a little hope for some romance in her future. [Source]
Why I Love It: OMG you guys this manga is just the best. It's like the Saga of shoujo it's practically perfect. You can't help but fall in love with Sawako. She's such an honest and sweet character. I think what sets KnT apart from other shoujo is the focus on friendship. The friendship between Sawako, Chizuru and Ayane is so beautiful and healthy. I love that it's either more important, or on the same level of importance, as the romance in this manga. I think it's what young girls (and women!) need to read. The anime is great, too!
Read This If You Like: Ao Haru Ride, Sukitte li Na Yo (Say I Love You), great friendships, shy MC's
Ao Haru Ride (Blue Spring Ride)
Genre: shoujo, romance, school, comedy, slice of life, drama
Mangaka: Sakisaka, Io
Status: Complete (49 Chapters)
Synopsis: Yoshioka Futaba has a few reasons why she wants to "reset" her image & life as a new high school student. Because she's cute, she was ostracized by her female friends in junior high, and because of a misunderstanding, she couldn't get her feeling across to the one boy she has ever liked, Tanaka-kun.
Now in high school, she is determined to be as unladylike as possible so that her friends won't be jealous of her. While living her life this way contentedly, she meets Tanaka-kun again, but he now goes under the name of Mabuchi Kou. He tells her that he felt the same way as she did when they were younger, but now things can never be the same again. Will Futaba be able to continue her love that never even started from three years ago? [Source]
Why I love it: Definitely one of my top three on this list. The art is so lovely! The premise sounds typical: unrequited childhood love, bad boy with black hair, nice boy with blonde hair and a love triangle. But this manga does the typical shoujo manga plot well very well. There's an anime, too!
Read This If You Like: Kimi ni Todoke, love triangles, Say "I Love You"
Hirunaka no Ryuusei (Daytime Shooting Star)
Genre: shoujo, romance, school life, slice of life, comedy
Mangaka: Yamamori, Mika
Status: Complete (87 Chapters, 12 Volumes)
Synopsis: 15-year-old country girl, Suzume Yosano, has to move to Tokyo to live with her uncle due to her father's transfer. She bumps into a mysterious man who ends up taking her to her uncle's place after she gets lost. Turns out, Suzume will be seeing him a lot more often once she starts school because… he's her homeroom teacher!? [Source]
Why I love it: This was one of my favorite mangas of 2014! There's a love triangle (of course) but this one includes a sensei! OH THE SCANDAL! I found myself conflicted because I liked both of the guys in this story so much I didn't know who I wanted Suzume to pick. I was so stressed out! The art is really fantastic, I plan to read more from Mika Yamamori.
Read This If You Like: Kimi ni Todoke, love triangles, sensei-student love affairs
Kanjuku Summer End
Genre: shoujo/josei, romance, slice of life
Mangaka: Fuji, Momo
Status: Complete (One Shot, 58 Pages)
Synopsis: Tae helps with her family's tomato field. However, when Gen, a ruffian-looking boy from Tokyo, is sent to help with the field, it seems that a sweet love will also be ripening. [Source]
Why I love it: Summer love, shoujo style. What's not to love?!
Read This If You Like: summer love
Shoujo/Josei Manga Border Lands
Basically this section is manga by Rikako Iketani. I fell in love with her after reading Six Half. I find that most of her manga overlap between shoujo and josei themes. The content and romance are a bit more mature than most shoujo, but not completely josei/adult either, which I think is a great mix.
Six Half
Genre: drama, romance, school, shoujo/josei
Mangaka: Iketani, Rikako
Status: Ongoing (I've been able to find up to 21 chapters in English but there are up to 44 raw chapters not translated online)
Synopsis: Kikukawa Shiori woke up from a bad motorcycle accident with severe loss of memory. Although she is allowed to go home, she has trouble adjusting to a seemingly new life with a brother who is always worried, a sister who apparently hates her, and a boyfriend who won't give her space. Awful rumors about her previous lifestyle abound in school, too, and even her friends tell her she had it coming. Alone, scared, and angry at the seemingly awful person she used to be, Shiori tries desperately to find a place of comfort she can call her own… [Source]
Why I Love It: I really really love this story. I love that the MC, Shiori, was a terrible person before her accident. Because of this, she's given a reset on her life and decides to be a better person. She's depressed, for obvious reasons, and I think it's represented in a genuine way.
Read This If You Like: amnesia stories, second chances
Bitou Lollipop (Bitter Sweet Lollipop)
Genre: romance, comedy, slice of life, shoujo/josei
Status: Complete/Ongoing? (34 Chapters is the complete manga apparently, but I've only found 19 chapters in English online)
Synopsis: When Madoka learns that her parents won hundred million yens at the lottery, she does not believe her ears. But while she begins to dream about a princess's new life, she meets to live only in a small appendix of the house of the family Asagi… For Madoka, a new life begins. [Source]
Why I Love It: This is one of my favorites. I really like the MC, I think Madoka is strong and independent in her own way. There are layers to the story, it's not just about romance, but there's some family drama involved as well.
Read This If You Like: MC's living on their own, older girl/younger guy romance, family drama
February 28, 2015 by GirlGoneGeekBlog
My Favorite Josei Manga
josei, manga, romance, shoujo
Juggling comics, books, TV shows, games, anime and manga is difficult. So I came up with a "hack". I try to put more focus on one thing at a time. A few years ago it was comics. I read a lot of old stuff and got into ongoing series. I'm now at a comfortable place, and pace, with comics.
At the end of 2013 manga took over. It wasn't even a decision like comics was, it just happened. When it came to manga, I stayed in the shallow end of the pool. I read a few things here and there, but never consistently. There's so much out there I felt lost and didn't know where to start. Which is the same feeling I had with comics. I think part of me knew that it would consume my life, so I wanted to make sure I had time to swim in the deep end.
My newfound manga obsession is all 'Nana's' fault really. I started watching the anime, and after a few episodes I decided to stop and switch to the manga instead. Now it wasn't that I didn't like the anime, I did, and that was the point. Since I knew I liked 'Nana' a lot already, I figured it was a great place to start as far as manga go. After 'Nana' left me hangin' I needed to fill that hole she left in my heart. In order to figure out what to read next I went on My Anime List to look up recommendations based on 'Nana', then recommendations on those recommendations.
Now dozens of josei manga later, I have this lovely list of some of my favorite of the genre manga. Have any manga suggestions you'd like to share? Comment below! I'm always looking for new manga to read.
Genre: josei, romance, drama, music, slice of life
Mangaka: Yazawa, Ai
Status: Forever Ongoing (84 Chapters)
Synopsis: Nana Komatsu is a young woman who's endured an unending string of boyfriend problems. Moving to Tokyo, she's hoping to take control of her life and put all those messy misadventures behind her. She's looking for love and she's hoping to find it in the big city.
Nana Osaki, on the other hand, is cool, confident and focused. She swaggers into town and proceeds to kick down the doors to Tokyo's underground punk scene. She's got a dream and won't give up until she becomes Japan's No. 1 rock'n'roll superstar.
This is the story of two 20-year-old women who share the same name. Even though they come from completely different backgrounds, they somehow meet and become best friends. The world of Nana is a world exploding with sex, music, fashion, gossip and all-night parties. [Source]
Why I Love It: Why don't I love it is the real question. This is my all time favorite josei and probably in my top 3 favorite manga period. The friendship between Nana and Hachi is so strong and dynamic. All of the characters have depth are and really interesting. I stayed up 'till the early hours of the morning reading Nana. I was so obsessed. The only bad thing is that sadly Ai Yazawa got sick and wasn't able to complete Nana. Now she's all better (yay!), but she hasn't announced any plans on finishing it. For more about Nana, check out my blog post 'Doki Doki: Anime & Manga Helped Me Get Over A Heartbreak'.
Read This If You Like: mangas about music (punk specifically), fashion, friendships, opposites attract, soap opera romances
Genre: josei, romance, drama, fashion
Synopsis: Yukari wants nothing more than to make her parents happy by studying hard and getting into a good college. One afternoon, however, she is kidnapped by a group of self-styled fashionistas calling themselves "Paradise Kiss." Yukari suddenly finds herself flung into the roller-coaster life of the fashion world, guided by George, art-snob extraordinaire. In a glamorous makeover of body, mind and soul, she is turned from a hapless bookworm into her friends 22 own exclusive clothing model. [Source]
Why I Love It: 'Nana' and 'ParaKiss' are must reads for josei manga. Ai Yazawa is the shit. This was another one of my obsessions. Paradise Kiss is like a soap opera with a fashion focus. You'll go through the 'motions reading this but it's totally worth all the stress and frustration.
Read This If You Like: fashion, modeling, soap opera romance, glam, makeovers
The One (manhua)
Genre: josei, romance, drama
Mangaka: Lee, Nicky
Status: Complete (110 Chapters)
Synopsis: Cane Lele was born into the fashion industry. Her mother, Ye Feii, was Taiwan's top model, and her father was also a model, until a tragic airplane crash left Lele without parents.
Therefore it is no surprise that Lele hates the industry, thinking it superficial and unnecessary. Raised by her paternal grandmother, Lele reaches seventeen before she is sucked in by her mother's sister, Ye Feihung, a fashion agent (and former model) who "convinces" the reluctant Lele to pose. Lele is adverse to the whole idea, until she sees a photo shoot of popular American model Angus Lanson, and she begins to see modeling as an art form.
When Aunt Feihung's magazine editor invites Angus Lanson to a meeting of all the top fashion ambassadors, Lele is invited along and is curious to meet Angus. She does not know that Angus' twin elder brother, Eros Lanson, is secretly accompanying his brother. This obviously leads to confusion and is the start of a charming story of Lele's goals: Fashion in New York city, becoming a top model, and love? [Source]
Why I Love It: If you love 'Paradise Kiss' and 'Nana' then you might like 'The One'. The art is similar (long skinny legs and arms and big old eyes) and it's about the modeling industry. Oh and there are sexy twins! But beware; there are some twincesty things that turned me off for a while. :/
Read This If You Like: 'Paradise Kiss', 'Nana', modeling, fashion, twins
& (And Okazaki Mari)
Genre: josei, romance, slice of life
Mangaka: Okazaki, Mari
Synopsis: Aoki Kaoru is a 26-year-old woman who has never had a boyfriend. Normally unattached and unmotivated in her own life, Kaoru decides to start up a nail salon as a side business in addition to her regular job as a medical clerk. Although having a severe adversity to physical touch, she is forced to confront that fear when she meets a man who can't help but touch others. [Source]
Why I Love It: I'm in the middle of this manga but I really dig it. This is probably the most shoujo-like of this list. I like how Aoki started her own business and is independent, but at the same time very naïve about love and relationships since she's never been in one.
Read This If You Like: older man/younger woman relationships, first love, 'Cousin'
Futago
Status: Complete (25 Chapters but only 20 chapters easily available in English)
Synopsis: Yukari was enjoying her single life when her elder twin sister, Ayaka, suddenly pays her a visit to ask her for some strange advice, and brings along a guy she met in Hong Kong with her! This is a story about a set of mismatched twins. [Source]
Why I Love It: I love Futago so much; it's such a great story about sisterhood and friendship. They are both strong in their own ways and as their bond grows, they both change a little for the better. Futago is by the same mangaka as 'Bitou Lollipop' and 'Six Half' which I talk about in My Favorite Shoujo Manga post.
Read This If You Like: 'Nana', stories about sisters, opposites attract, 'Kabocha to Mayonnaise' (Pumpkin and Mayonnaise)
Genre: josei, romance, comedy, slice of life
Mangaka: Ikuemi, Ryou
Synopsis: Tsubomi Shirakawa, 18 years old, has graduated from high school, but while her circle of friends have already found jobs or started college, she has decided to become a "Freeter," and begins part-timing at a local rental video store. Meanwhile, she finds out that her cousin Noni, whom she has not met from as long as she can remember, has debuted in show business.
Tsubomi may have a chubby figure but was uninterested in beauty of figure or form. However that is about to change when, a coworker, Shiro, brings her to visit a bar where she falls in love with the owner, Nasukawa. [Source]
Why I Love It: I was hooked from the start, which is good since I usually only give manga one chapter to woo me. I love how the story deals with serious issues like self-esteem and weight and lighter stuff like crushes.
Read This If You Like: older man/younger woman romance, body and self-esteem, '&' (the manga)
Love Vibes
Genre: romance, drama, josei
Mangaka: Sakurazawa, Erica
Status: Complete (7 Chapters, 1 Volume)
Synopsis: Shouji has a girlfriend but is seeing Mako on the side. Unable to continue pretending she feels less for him than she does, she breaks it off with him. Shortly before that happens, Mako is hit on by Mika, who tells Mako that she's interested in both men and women. Mika wants to pursue a relationship with Mako, but is Mako's relationship with Shouji really over? [Source]
Why I Love It: I really loved the art, realism and complexity of the relationships.
Read This If You Like: LGBT love stories, love triangles, simple art, 'Nana'
Genre: josei, romance
Mangaka: Nananan, Kiriko
Synopsis: The life of high school student Kayako Kirishima is altered forever when she meets the class outsider Masami Endo. The two quickly form a close friendship, and as time goes by that friendship turns into deep love. But as personal traumas of Endo's past conflict with Kirishima's future, their love is put in jeopardy. [Source]
Why I Love It: Kiriko Nananan is becoming one of my favorite mangaka. She manages to make complex subjects feel light because her art style is so delicate. 'Blue' was no different. This was the story about the challenges that come with young lesbian relationships in a society that isn't accepting of them.
Read This If You Like: LGBT love stories, young love, simple art
Kabocha to Mayonnaise (Pumpkin and Mayonnaise)
Genre: josei, slice of life
Synopsis: Ten stories about the everyday life of Sei, an idealistic songwriter, and Miho, the young woman who struggles to support the both of them. [Source]
Why I Love It: The beauty of the art is in its simplicity. I really liked how realistic and "slice of life" this manga actually felt.
Read This If You Like: complicated MC's, josei slice of life
Anata no Koto wa Sorehodo
Genre: josei, drama, romance
Managka: Ikuemi, Ryou
Status: Ongoing (2 Chapters)
Synopsis: A fortuneteller once told her, "Marry the second man you fall in love with." Medical clerk Miyoshi did just that. However, upon leaving a drinking party, she bumps into her first love, Arishima. Feelings reignite between the two, yet Arishima is married and Miyoshi already has an amicable husband… [Source]
Why I Love It: I like how problematic the main character is. She's not inherently good, but you don't really dislike her for it, but you still don't like her either. This is the same mangaka that did 'Cousin'.
Read This If You Like: Futago, complicated MC's
Genre: josei, drama, romance, slice of life,
Mangaka: Yoshizumi, Wataru
Status: Complete (7 Chapters)
Synopsis: A bittersweet love story about the life of Kojima Ari and Fujitani Sousuke as they live together in a de facto relationship (without marriage) in modern-day Japan. [Source]
Why I Love It: It's short and simple but its realism about the challenges that come with relationships is what hooked me. It manages to tell a five year story in seven chapters without feeling rushed.
Read This If You Like: realistic relationships
Helter Skelter: Fashion Unfriendly
Genre: josei, horror, psychological
Managka: Okazaki, Kyoko
Synopsis: If you are aware of fashion in Japan, you must have seen Liliko's face. For the last few years, she has been at the top of the modeling world, with her face and body promoting the biggest brands. But as everyone who is in this world admits, staying on top is a constant and never ending battle. There are always new faces introduced to the public. Younger models and new looks are brought into the fold every season. And keeping that position means learning to adapt and learning to cope with change.
To maintain her position Liliko has decided to under the knife. This is not her first go with this service. It is yet another round of plastic surgery, all done to keep herself looking young and vibrant. However, in this case just a little nip and tuck was not enough. Liliko is bent on undergoing a full body makeover. From head-to-toe, every inch of her will undergo cosmetic surgery, and thus begins her madness. [Source]
Why I Love It: The premise reminds me so much of a Chuck Palahniuk story. Liliko is such a sick and terrible character. But the story doesn't rely on Liliko's cruel actions alone, the entire story is interesting and the ending is one of my favorites.
Read This If You Like: fashion, modeling dark stories, twisted MC's, simple art, 'Heart O Uchinomese'
Heart O Uchinomese (Knock Your Heart Out)
Mangaka: Asakura, George
Synopsis: No matter how many times she sleeps with Arai, Negishi realizes that neither her nor Arai's feelings were being conveyed. Their sex felt "like" love, but only feeling "like" something wasn't the same thing as having it. What is Negishi supposed to do in order to get Arai to understand her true feelings?
Why I Love It: It's dark with an interesting art style. It's the opposite of the squeaky clean and cute shoujo romances, which is what attracted me to it. It's actually pretty fucked up.
Read This If You Like: dark romances, Inio Asano's manga, 'River's Edge', 'Helter Skelter'
Girl Gone Geek's Best of 2014
Art, Comics, Manga, Videogames
Anime, bayonetta, bitch planet, comics, east of west, image comics, interstellar, manga, Nintendo, Saga, sex criminals, shoujo, space dandy, the wicket and the divine, true detective, vertigo comics
Best Use of Boobies and Philosophy in an Anime- Space Dandy
When I first heard about Space Dandy it sounded a lot like Cowboy Bebop, which is my favorite anime series, so I was all for it. To be honest, Shinichiro Watanabe could make an anime that was just a black screen for twelve episodes I would praise the animation and character development. However, when I saw the first episode of Dandy I thought it was ridiculous. Like really ridiculous. And so was the next one and the one after that… "Well, this isn't very "Bebopy." I thought to myself. I wasn't sure if I liked it. Which made me start to lose faith in Watanabe. Which made me question what was even real. Which made me fall into an existential crisis. But then that crisis was averted when I realized I had to stop expecting Space Dandy to be something it wasn't. Once that happened, everything changed. By the time the series ended, I loved Dandy just as much as he loves Boobies. (Which is a lot.) When watching Space Dandy, you just have to lose all expectations, relax and enjoy the ride, baby.
Best Reason to Call Out of Work- Bayonetta 2
Dat ass, though.
Best Creepy Manga- Nijigahara Holograph by Inio Asano
My love for Inio Asano grows every time I read one of his messed up coming-of-age stories and Nijigahara Holograph was no different. I dedicated blog posts to Solanin and Nijigahara Holograph because they are that good and I'll probably do the same for Oyasumi Punpun, which I just finished reading. Asano-sensei is a brilliant storyteller, artist and one of my favorite mangakas. Despite the strangeness and disturbing quality to his stories, there is a unique beauty to them as well. His brutal honesty about humanity forces us to confront things we're too scared to address. His manga tell us more about ourselves than we may be comfortable with, and with that, he creates a story so beautiful it's unnerving.
Best Love Triangle to Obsess About- Hirunaka no Ryuusei (Daytime Shooting Star) by Mika Yamamori
I'm a sucker for a shoujo romance love triangle, but throw a sensei up in that triangle and I. CAN'T. EVEN.
Smartest Hour On Sunday- True Detective
Since time is a flat circle, I can't wait until my consciousness inevitability experiences True Detective again for the first time.
Best Cowboy in Space Movie- Interstellar
I was personally offended when I heard some people didn't like this movie.
The Comic That Was So Good It Was Perfect- Saga story by Brian K. Vaughan and art by Fiona Staples
Coolest World You Really Like But Don't Want to Live in Because It's Pretty Messed Up– East of West story by Jonathan Hickman and drawn by Nick Dragotta
This was a way for me to dedicate a category to East of West because it's too good not to mention. Seriously, you should be reading this.
Best Eyegasm- J.H. Williams III for Sandman Overture story by Neil Gaiman
So the real reason Sandman Overture comes out every leap year isn't because of the creators busy schedules, it's because if JHW3 releases Sandman Overture art once a month it will literally rip a hole in time and space and the universe will collapse in on itself.
Best Reason to Judge a Comic Book by Its Cover- The Wicked + The Divine written by Kieron Gillen and drawn by Jamie McKelvie
When you first see the cover for WicDiv you're all like, "Oooh shiny!" So you buy it because you like pretty things and figure if the story is terrible at least it's pretty to look at. And then you realize you're a really shallow person and get depressed. To distract yourself from this distressing realization you read WicDiv and everything about it is super cool and yet surprisingly smart. Then you stop feeling sad and shallow and pretend that you didn't just buy the comic for the pretty cover, you bought it for the smart story. I'll keep your secret if you keep mine.
Best Comic to Read in Public Places- Sex Criminals written by Matt Fraction and drawn by Chip Zdarsky
Fraction and Zdarsky prove as long as your comic is great, you can make as many sex jokes as you want.
The Comic That Will Take Over 2015- Bitch Planet written by Kelly Sue DeConnick and art by Valentine De Landro
Feminism means equal rights for all women, but more often than not, the fight for equality is separated between white women and women of color *cough cough* "separate but equal". Bitch Planet is a punch in the face to patriarchy and I'm so damn happy that all women are included. Thank you Kelly Sue and Valentine!
October 28, 2014 by GirlGoneGeekBlog
NYCC 2014: Cosplay Roundup
Anime, Comics, Cosplay, Events, Fantasy, Manga, Movies & Television, Random Geekery, Sci-fi/ Cyberpunk, Videogames
Anime, comic books, comic-con, comics, cosplay, Fantasy, film, manga, New York Comic Con, NYCC, sci-fi, tv, Video Games
Every year I attend New York Comic Con more and more fans cosplay. It's tough picking favorites but the ones that made me squee the most were Death from East of West and Steven Universe and his mom as Garnet. Oh and here's my post about my Sailor Moon (aka Sailor Goon), Space Dandy and Spike Spiegel cosplay.
Movies & TV & Video Games & Podcasts
NYCC 2014 Cosplay & Coords: Sailor Moon, Space Dandy, Cowboy Bebop & More
Anime, Apparel, Cosplay, Events, Manga, Wear
Anime, comic-con, cosplay, cowboy bebop, manga, New York Comic Con, NYCC, omocat, Rule 63, sailor moon, space dandy, spike spiegel
It was only a year ago when I posted "Confessions of a First Time Cosplayer" and look how far I've come! Now I know I'm not building Tali armor from scratch like my friend and cosplay sensei @DarthRachel, but I'm proud of myself nonetheless.
Sailor Goon
I introduce to you, Sailor Goon; my gangsta take on Sailor Moon. I was Sailor Moon and my friend Roshi was Black Lady. People at the con said we were cute and creepy so mission accomplished basically.
I was inspired by Asiey Barbie's Sailor Gang art on tumblr. I tagged her on Instagram and Twitter and her reaction was, "O.M.G. yasss, this is EXCELLENT!" and "I am SLAYED. absolute perfection." She reblogged my post on Tumblr which is probably why it's getting so many notes and said "YOU GUYS LOOK AT THIS FLAWLESSNESS". So of course that made my day!
Artist Babs Tarr was also a big influence on this cosplay design. Her Bōsōzoku Sailor Scouts helped us come up with accessories like the denim jacket, patches and ripped nylon stockings. Babs was at NYCC this weekend so I was lucky enough to meet her and get a gorgeous Bōsōzoku Sailor Moon print and my Batgirl #1 signed. She is super sweet and LOVED our cosplay. She took a photo of us and gave us some of her stickers I actually planned on buying anyway!
As you can probably tell, Sailor Goon was my favorite cosplay of the weekend. The fun part was coming up with ways I could change Usagi's regular costume from Moon to Goon. And my friend Roshi was giving some Black Moon realness with her Black Lady cosplay. Surprisingly, Sailor Goon was the first woman I've cosplayed. Up until now I've Rule 63'ed my favorite male characters like the Eleventh Doctor, Spike Spiegel and Space Dandy.
Space Dandy is amazing… and weird. But that WTFness of Dandy is why I love it so much. I remember how excited I was hearing that the GAWD Shinichirō Watanabe of (Cowboy Bebop and Samurai Champloo) was going to be the chief director of a new anime called Space Dandy. I admit, I had to adjust to the randomness of the show. Although Dandy is a bounty hunter in space, it's nothing like Cowboy Bebop. Once I got that through my head, I fully enjoyed each episode. The finale was mind-blowing. Maybe I'll blog about it at some point.
I'm not a crafty person at all, but I managed to make some Dandy accessories and I'm really proud myself, even though these are nothing compared to other stuff people make. It's my first time "crafting" so give me a break! I made the necklace out of poster paper and paint marker, but it doesn't look that bad as it sounds. It's kind of like a Monet and looks better the further away you are. I hope my kanji isn't too embarrassing; it was my first time writing that as well. The jacket is from Csddlink and I made Dandy's belt buckle out of foam which I painted gold. Since I decided to go with high-waisted leggings, I turned the buckle into a pin. The items that are missing are his hair, bracelet and shoes. So this is kinda a half-assed cosplay. But now that I have most of his costume, I'll give the rest my best effort next year.
I made a Space Dandy playlist because the music is so good. Shake your booty while reading this blog post, baby. It's the Dandy way!
Spike Spiegel from Cowboy Bebop
If this looks familiar it's because I did this female Spike last year. I loved it so much I had to do it again! Nothing new added to this except I'm channelling Faye Valentine with my purple hair. It's been a year and I still haven't gotten a good Spike wig. I'm the worst you guys I'm so sorry.
Kawaii Street Style
I wasn't always dressing up as someone else at NYCC. These last two additions aren't cosplay but some pretty cute coords if I do say so myself.
On Friday I wore my new favorite sweater from Omo-Cat called 'Toast Girl'. I was going for kawaii street style and did cult party kei makeup which is hard to notice because… you know… I'm black.
Subtle Princess Serenity
*Queue magical girl transformation* I went from Sailor Goon by day, to Subtle Princess Serenity by night (with an emphasis on "subtle"). I wore this to the Fan Girls' Night Out Party hosted by Geek Girl Brunch (that org I co-founded). There are tons of galaxy prints out there but the Shadowplay NYC shop on Etsy has some of the best I've seen. Better than Black Milk (yeah I said it!). I ordered two dresses from them and wore the Mystic Mountain Jersey Dress to FGNO. It features a Hubble Print of the Mystic Mountain, a Pillar in the Carina Nebula. I kept the space theme going and paired my Nebula dress with Princess Serenity and Sailor Moon accessories. I got so many compliments!
Well, that's a wrap for my New York Comic Con cosplay and coords! Stay tuned to see if I get better at cosplay crafting next year. Spoiler: I probably won't. I already have my eyes set on Leatherface Joker, Spider Jerusalem from Transmetropolian and a a young Major Motoko Kusanagi from Ghost in the Shell: Arise (all featured in my Cosplay Bucket List post).
April 28, 2014 by GirlGoneGeekBlog
My Cosplay Bucket List
Anime, Apparel, Comics, Manga, Movies & Television, Other, Other, Random Geekery, Sci-fi/ Cyberpunk
adventure time, Anime, comics, cosplay, gatchaman crowds, ghost in the shell, manga, Saga, sandman, space dandy, the joker, Transmetropolitan
As you may remember from my post "Confessions of a First Time Cosplayer" I really like cosplaying. Who doesn't?! But I have to admit, even after my numerous Doctor Who cosplays and my pretty killer femme Spike Spiegel, I still feel like a newb since I didn't actually make anything from scratch myself. The designs and concepts were my own, but the crafting was not. But after some drinks with Cosplay Queen Darth Rachel and a little pep talk from her, I feel less newby.
I let my lack of craftiness limit me, but I will not let it hold me back any longer! I will try my damndest to create most of these clothes and items by hand. I will probably fail many times, but that's part of the fun… right?
Either way, here's my cosplay queue, the characters I want to cosplay badly because they are my faves and they look super freaking cool. Shout out to the homie Leslie IRL for the blog post idea!
Space Dandy is ridiculous anime and I love it so much. I'll genderbend this up with a '50s style dress or skirt and top to match Dandy's Greaser look. But this cosplay is actually happening irl at NYCC this year! I have the jacket already, I just need to make shoes. >_<
Delirium from Sandman
Sandman is one of my favorite comic book series of all time. Delirium was definitely one of my favorite of the Endless. I was always excited when she would pop up in the comic. Making the wig for her cosplay will be fun!
Gwendolyn from Saga
I freaked out when I first saw Gwendolyn. Being a black girl who reads comics, I don't see lots of characters that look like me, so when BVK created bomb-ass Gwendolyn I was beside myself. Just look at her looking all magnificent!
Motoko Kusanagi from Ghost in the Shell
The Major is one of my favorite characters ever! She's a shining example of a strong female character. The Ghost in the Shell series is one of my all time faves as well. I even wrote a 20 page paper in grad school about GITS and post-humanism (see: Cyborgs Exist and Humanity is Doomed for a shorter version). There are so many looks I could go with; but I already know I'm not going to do the one where she doesn't wear pants.
Spider Jerusalem from Transmetropolitan
Spider is one of my favorite comic book characters. He's such a lovable bastard! I'm doing a genderbent Spider. I really like genderbending.
Leatherface Joker
I want to cosplay Leatherface Joker so badly. He's so gorgeously gruesome! I'll probably genderbend this one too because I just like to do that and it's fun. I have no idea how I will even do the mask but if there's a will, there' s a way!
Lumpy Space Princess from Adventure Time
Oh my glob! How can you not love LSP?! With this cosplay I'll probably go the non-lumpy route and wear a over-the-top dress instead, with a big purple wig and of course, the star.
Berg Katze from Gatchaman Crowds
So I actually haven't finished this anime yet… whoops. BUT I immediately fell in love with Berg Katze and his character design. I MEAN LOOK AT THAT HAIR! He's fun villain and reminds me of the Joker because you don't really hate him as much as you probably should. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,101 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.